Dwyer_MScThesis.

Dwyer_MScThesis.
Defining Ventilation Boundary
Conditions for a Greenhouse
Climate Model
Master Thesis
D.E. Dwyer
Department of Process and Energy Technology
Defining Ventilation Boundary
Conditions for a Greenhouse Climate
Model
Master Thesis
D.E. Dwyer
August 29, 2014
Faculty of Mechanical, Maritime and Materials Engineering (3mE) · Delft University of Technology
The work in this thesis was supported by TNO. Their cooperation is hereby gratefully acknowledged.
c
Copyright All rights reserved.
Abstract
Presently 10% of natural gas supplied to the Netherlands is used to maintain a stable climate and
continuous electricity within horticultural greenhouses. As a result, technologies that reduce this
energy consumption are in high demand. Theoretical models of heat flows in greenhouses can be
used as a tool to increase the efficiency of these developments. This work seeks to improve such a
tool by modeling the effects of various window opening angles and wind directions on the ventilation
boundary condition for a greenhouse climate model.
Using OpenFOAM, Computational Fluid Dynamics (CFD) simulations of various window opening
angles and wind directions were run for both external and internal flow of a Venlo-type greenhouse. A
thorough verification of the simulation results examined spatial convergence, temporal convergence,
model implementation, iterative convergence, and consistency. The results of these simulations were
deemed insufficient for a boundary analysis due to initial flow field errors. While the variable results
could not be applied to a boundary condition analysis, the constant case (with 30 degree window
openings and a zero degree azimuth angle) was used to reproduce the internal flow field of the
greenhouse.
The resulting model contained concentrated error at windows where fixed velocity values are not
prescribed, correcting for continuity. This boundary condition is a first step towards more accurate
internal greenhouse flow simulations of ventilation. Further, methods for generating a ventilation
boundary condition using verified variable simulation results are discussed for future use.
Master Thesis
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D.E. Dwyer
Master Thesis
Table of Contents
Acknowledgements
ix
1 Introduction
1-1 Project Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-1-1
1
1
Current state of the Kasklimaat Model (KKM) . . . . . . . . . . . . . . . .
2
1-1-2 Future Plans for the Greenhouse Climate Model . . . . . . . . . . . . . . . .
1-2 Project Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
1-3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
2 Theoretical Background
7
2-1 Ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1-1 Wind-Driven Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1-2 Buoyancy-Driven Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1-3
7
7
8
Combined Effect of Buoyancy and Wind . . . . . . . . . . . . . . . . . . . .
8
2-2 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2-2-1
Finite Volume Approach
2-2-2
Numerical Model: Discretization for the Navier-Stokes System . . . . . . . . 10
2-2-3
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Methodology
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
17
3-1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3-2 Wind Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3-3 Window Opening Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 CFD Simulation
21
4-1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4-2 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4-2-1
Solution and Algorithm Control . . . . . . . . . . . . . . . . . . . . . . . . . 24
4-2-2 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4-3 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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Table of Contents
5 Verification and Validation of Results
27
5-1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5-1-1 Examine Spatial Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5-1-2
Examining Temporal Convergence . . . . . . . . . . . . . . . . . . . . . . . 32
5-1-3
Examine the Implementation of the Model . . . . . . . . . . . . . . . . . . . 32
5-1-4
Examine Iterative Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 33
5-1-5
Examine Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5-2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5-3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 Boundary Analysis
47
6-1 Selecting parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6-2 Parameter value specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6-2-1
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6-2-2
Multiple regression analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Recommendations
53
8 Conclusion
55
A Appendix: Boundary Conditions
57
B Appendix: Calculation Details
59
Bibliography
63
Glossary
67
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
D.E. Dwyer
Master Thesis
List of Figures
1-1 Terminology diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2-1 Staggered grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3-1 Wind azimuth angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3-2 15 and 30 degree window openings . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3-3 53.3 Degree window opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4-1 3-Dimensional greenhouse models . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4-2 Overview of computational domain from the top and side views . . . . . . . . . . . 23
5-1
5-2
5-3
5-4
5-5
5-6
5-7
5-8
Top view of larger computational domain . . . . . . . . . . . . . . . . . .
Larger Computational Domain Mesh . . . . . . . . . . . . . . . . . . . . .
Velocity magnitude flow field in large computational domain . . . . . . . .
Velocity magnitude flow field in computational domain . . . . . . . . . . .
Top view of finely meshed grid . . . . . . . . . . . . . . . . . . . . . . . .
Side view of finely meshed grid with twice as many cells in each dimension
Spatial Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . .
Monitoring of Courant value for each time step . . . . . . . . . . . . . . .
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5-9 Initial residual error for each time step for (a) a zero degree wind approach angle and
(b) a 75 degree wind approach angle . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5-10 Temporal fluctuations of the drag coefficient . . . . . . . . . . . . . . . . . . . . .
5-11 Temporal fluctuations of the drag coefficient . . . . . . . . . . . . . . . . . . . . .
5-12 Temporal fluctuations of calculated fields for a 0 degree wind approach angle and 30
degree window openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-13 Temporal fluctuations of calculated fields for a 75 degree wind approach angle and
30 degree window openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-14 Comparison between initial and final velocity magnitude at the window openings .
5-15 Comparison between initial and final pressure at the window openings . . . . . . .
5-16 Volumetric flow rate at windows for varied window opening angle calculations . . .
5-17 Vector plot of velocity for a slice located along the leeward facing windows in a
simulation containing closed windward facing windows . . . . . . . . . . . . . . .
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D.E. Dwyer
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List of Figures
5-18 Velocity boundary condition [m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5-20 x component of velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5-21 Results for y component of velocity field . . . . . . . . . . . . . . . . . . . . . . . . 44
5-22 Results for z component of velocity field . . . . . . . . . . . . . . . . . . . . . . . . 44
5-23 Velocity vector field at a plane parallel to the ground located (a) 1 meter from the
ground and (b) 3.7 meters from the ground, colored by velocity magnitude . . . . . 44
5-24 Volumetric flow rate at windows of the ’constant’ case . . . . . . . . . . . . . . . . 45
5-25 Pressure field of the ’constant’ simulation . . . . . . . . . . . . . . . . . . . . . . . 46
5-26 Results for turbulent kinetic energy field . . . . . . . . . . . . . . . . . . . . . . . . 46
5-27 Results for turbulent kinetic energy dissipation field . . . . . . . . . . . . . . . . . . 46
6-1 Selecting boundary condition parameters . . . . . . . . . . . . . . . . . . . . . . . . 49
6-2 Selecting boundary condition parameters: effect on pressure distribution . . . . . . . 49
6-3 Selecting boundary condition parameters: Effect on velocity distribution . . . . . . . 50
D.E. Dwyer
Master Thesis
List of Tables
4-1 Atmospheric Boundary Layer (ABL) Conditions . . . . . . . . . . . . . . . . . . . . 24
4-2 Residual error tolerances used in the Pressure Implicit with Splitting of Operators
(PISO) algorithm for each parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4-3 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5-1 Spatial convergence details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5-2 Percentage of stationary window probes in varied wind azimuth angle calculations
(with fixed 30 degree windward and leeward window opening angles . . . . . . . . . 38
5-3 Percentage of stationary window probes in simulations of varied window opening
angle combinations (with fixed 0 degree wind approach angle) . . . . . . . . . . . . 38
A-1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
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List of Tables
D.E. Dwyer
Master Thesis
Acknowledgements
These past two years in the Netherlands have brought many memories and learning experiences that
I am grateful for. In particular, the past nine months of thesis work would not have been possible
without the help and support of many people. I would like to thank my industry supervisors Ivo
Kalkman and Leonard Baart de la Faille for their time and patience working with me over the past
9 months. Additionally, thanks to my TU Delft supervisors Rene Delfos and Mathieu Pourquie.
Besides my academic support, my time in the Netherlands would not have been the same without my
friends and roommates, who showed me the bright side of Delft. Finally, without the encouragement
and support of my family I never would have considered moving to the Netherlands.
Delft, University of Technology
August 29, 2014
Master Thesis
D.E. Dwyer
D.E. Dwyer
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D.E. Dwyer
Acknowledgements
Master Thesis
Chapter 1
Introduction
The highest concentration of greenhouses in the world is located in the Netherlands. Greenhouses
can provide controlled crop growth, shelter from external climate conditions, and protection from
pest invasions. Therefore, the Dutch greenhouses enable the country to take the lead as the largest
exporter of tomatoes in the world [1]. With their expertise in glass greenhouse technology, the Dutch
are also the largest exporter of greenhouse equipment [2]. Hence, Netherlands Organization for
Applied Scientific Research (TNO)’s greenhouse research and development initiatives have followed
suit, spanning 40 years of greenhouse knowledge creation. The focus of their work has targeted
increased access to sunlight and reduced energy consumption.
Within the industry, it is commonly known that greenhouses are a major consumer of energy, maintaining their warm temperatures and powering their electricity needs with 10% of the natural gas
supplied to the Netherlands. The latest technologies that have been developed to address this issue
include screens that retain heat within different layers of height in the greenhouse, external screens
to increase exterior insulation of the greenhouse, and dehumidification systems that minimize heat
loss. Furthermore, an evenly distributed climate can increase crop productivity. In order to test the
impact of new technologies such as these, theoretical models of heat flows can increase the efficiency
of these developments.
Computational fluid dynamics is a powerful tool for simulating internal greenhouse climate. It is
used because of its wide range of applicability in comparison to simpler models, which come with
limitations. Extensive research on greenhouse Computational Fluid Dynamics (CFD) modeling has
already been done: Between the years 2000 and 2012, 214 papers and 88 articles were published
about CFD modeling of greenhouses [3]. Developing an accurate CFD solver, however, can be very
complicated, and is not always a field of knowledge that is familiar to the engineers developing
energy efficient greenhouse technologies.
1-1
Project Motivation
As a result, TNO has set out to develop a Kasklimaat Model (KKM), which translates in English
to Greenhouse Climate Model. While a multitude of scientists have developed greenhouse models
that predict thermal flows, humidity control, and carbon dioxide exchanges, what makes this model
unique is the ultimate end-use goals. The model will be distributed as an easy-to-use greenhouse
climate modeling software. Possible uses for this technology can include development of greenhouse
regulations, testing new technological developments for greenhouses, and testing new greenhouse
Master Thesis
D.E. Dwyer
2
Introduction
designs. The second use for this model would be for model-based control. The conditions inside the
greenhouse (i.e. temperature, humidity, and carbon dioxide levels) must be very precise for optimal
plant growth. However, in such a large space, physical effects like buoyancy and wind can cause
gradients in greenhouse parameters to occur, requiring control systems for adjustment.
In order to meet these two goals, the ultimate product must use numerical methods and simplifications to accurately calculate the physical conditions inside the greenhouse at a fast rate. The
parameters modeled in the KKM are density, pressure, temperature, gas composition, flow velocity,
radiation intensity (both solar and heat), and turbulence. Components included in the model are
heaters, fans, screen fabric, and windows. A major challenge in developing this model is predicting
the internal flows without modeling the entire external environment.
1-1-1
Current state of the KKM
Although TNO has been modeling greenhouse climate for nearly two years, the development of
the latest KKM has been underway since June, 2013. The project began with an initial review of
the physical parameters that must be included in the model in order to accurately and realistically
model a greenhouse. A review of existing greenhouse climate models was also completed, to avoid
redundancy. Next, a simplified 3D CFD model was implemented in MATLAB. The model was then
transitioned to OpenFOAM in order speed up the calculation time and further develop the model
while making optimal use of the functionality that was already available.
Both transient and steady-state preliminary greenhouse solvers have been developed in OpenFOAM.
The steady-state model is necessary in order to initialize the simulation prior to the transient solver.
It can also be used when the user has no interest in dynamical processes but merely in the situation
which will develop after sufficient time has passed. The model runtime has been optimized using
local time stepping methods for model based control applications. There is still progress to be made
within each of the following physical parameters.
• The temperature parameter currently accounts for buoyancy, wall heat transfer (by indicating
an outside temperature and thermal conductivity), heat radiation (considered as a constant
or direction dependent emissivity), and heater sources (specified on boundaries or within the
volume). Planned improvements on this parameter include incorporating specific heater power
per volume, implementing the effect of obstacles on radiation view factors, and adding the
effect of solar radiation heat transfer.
• The gas composition parameter accounts for component gases (with variable compositions),
water condensation on cold surfaces (using the Colburn-Hougen equation), and evaporation (as
a constant). The calculation of this parameter will be improved by changing the evaporation
rate to depend on other variables, validating the condensation model, and incorporating sources
and sinks for non-water components.
• The flow velocity parameter accounts for buoyancy-driven flow, the effect of porous media,
and the effect of fans as either a pressure difference or outflow velocity. This parameter will
be improved by coupling the flow and temperature of porous media, defining fans and screens
independently of the mesh, and including turbulence effects of flow through porous media and
past fans. At the beginning of this project, the effect of flow around windows, and the effect
of external wind flow on internal velocities was not yet included.
• The radiation intensity parameter currently accounts for heat radiation, and will eventually
incorporate solar radiation into the calculation.
D.E. Dwyer
Master Thesis
1-2 Project Objectives
3
Beyond the physical parameters, the model will be tested and validated using existing greenhouse
measurements and tests in an experimental greenhouse facility.
Lattice Boltzmann computing techniques are also under consideration as a possible alternative to
conventional CFD methods. Rather than modeling fluxes over control volumes, as simulated using
CFD, Lattice Boltzmann simulations model the fluid as a series of propogations and collisions of
particles. This method is known for its computational efficiency, eliminating the need to explicitly
solve pressure dynamics. It is considered a robust method for large scale simulations.
1-1-2
Future Plans for the Greenhouse Climate Model
The plan for the KKM in 2014 is to improve the model by incorporating the missing physical elements, improving the speed of the calculation, and testing Lattice Boltzmann computing techniques.
Moving in the direction of the ultimate product goal, the model will also be expanded to include
parameters that are greenhouse specific, installation specific, crop specific and location specific.
Finally, the model will be validated using greenhouse measurements and controlled experiments in
a local experimental greenhouse to prove that it is consistent with reality.
Given the current state and future plans for the KKM project, the flow velocity parameter was
deemed a critical component for the model, and was selected as the focus for this work. Analyzing
the effect of external climatic conditions on the internal greenhouse climate can ultimately lead to
the development of more efficient designs and enhanced ventilation management. At the start of
this project, there were two calculation methods missing from the model that rely on external flow
conditions: external air exchanges resulting from flow around windows and from pressure differences
caused by external flows.
1-2
Project Objectives
Ventilation in greenhouses is driven by both buoyancy and wind. While stack-driven ventilation
can be modeled relatively accurately without the need for CFD, using zonal modeling, it is the air
exchanges that occur near the window that are less predictable. Eddies near windows can cause an
outflow containing the same air carried into the greenhouse, rather than allowing the greenhouse to
exchange air with the external environment. While numerous studies have investigated the effects
of external flows on the overall ventilation rate of greenhouses, there is still a lack of information
detailing how external flow conditions may be used to characterize the flow field at the ventilation
boundary. This aspect of the KKM can most greatly benefit from CFD modeling.
The research question defining this graduation project is
Can CFD be used to define a realistic boundary condition at the window openings of an
interior greenhouse climate model to represent the effect of external wind?
The conditions this study accounts for are wind direction, wind speed, window opening angle, and
greenhouse geometry. The following procedure is used to address this research question and arrive
at the end results:
• Complete an initial literature review of relevant existing research
• Model a greenhouse with varying external environments to be used as a basis for determining
boundary condition trends
Master Thesis
D.E. Dwyer
4
Introduction
• Validate models using available measurements
• Analyze results of various external (and resulting internal) environments and deduct a methodology for prescribing these trends on an enclosed greenhouse model
• Prescribe unique boundary conditions at the windows in the interior KKM, which only models
the greenhouse interior, and compare the results with the validated external models
1-3
Thesis Overview
This report is organized with the following structure:
• The thesis begins with an overview of the theoretical background in Chapter 2, covering both
the driving forces behind greenhouse ventilation and a discussion of the CFD algorithm used
in this work.
• Chapter 3 goes into detail about the methods used to select and test variables that can be
used to derive a greenhouse ventilation boundary condition.
• The computational methods for calculating the effects of different variables are discussed in
detail in Chapter 4. This section discusses both the mesh generation, boundary conditions,
and numerical schemes used to stabilize the calculation.
• Chapter 5 provides a discussion of the convergence of the resulting simulations, an overview
of the flow behavior within the greenhouse, and a validation of the results using available
experimental data.
• In Chapter 6, methods for formulating and implementing a ventilation boundary conditions
using the external flow simulations are discussed.
• Chapter 7 provides recommendations for further work and improvements,
• followed by concluding remarks in Chapter 8.
1-4
Terminology
Beyond the abbreviations and symbols provided in the appendix of this report, there is some basic
horticultural greenhouse terminology that is used throughout this work that must be identified for
the reader.
Horticultural greenhouses with glass coverings are often made up of repeated long narrow segments
with either an A-shaped or arched roof. The term span refers to one of these segments. Ridges are
the peak of the span roof and gutters are the valleys between two spans. These values are identified
in Figure 1-1.
Windows are sometimes referred to as ventilators. Along an A-shaped roof, windows are differentiated by the direction they face. Windows located upwind of the roof ridge are windward-facing,
while widows downwind of the roof ridge are leeward-facing. The opening angle of the window commonly discussed in this work is an angle (in degrees) measured from the plane of the roof surface to
the window flap, with the ridge as the vertex. Examples of three different window opening angles
are depicted in Figures 3-2 and 3-3.
The angle at which the wind approaches the greenhouse is called the wind azimuth angle. A wind
azimuth angle of zero is perpendicular to the ridges of the greenhouse. The angle is measured
D.E. Dwyer
Master Thesis
1-4 Terminology
5
Figure 1-1: Terminology
counterclockwise from the zero degree wind azimuth angle. Figure 3-1 provides a visualization for
this concept.
Sometimes, references are made to the position of a window within the greenhouse. Span number
identifies the span where the window is located, with the first span located upwind, and the last span
located downwind. Span number increases in the stream-wise direction. On an x, y, z coordinate
system, the origin is located at the most upwind point of the greenhouse, along the ground on
the far right-hand corner (as observed from upwind). The x-axis is perpendicular to the ridges of
the greenhouse, the y-axis is oriented parallel to the ridge, and the z-axis is representative of the
greenhouse height.
Master Thesis
D.E. Dwyer
6
D.E. Dwyer
Introduction
Master Thesis
Chapter 2
Theoretical Background
This chapter presents the relevant theory used to calculate the greenhouse ventilation boundary
condition. It begins with an overview of the driving forces that cause ventilation, and continues with
the numerical methods used to compute this behavior.
2-1
Ventilation
Ventilation air exchanges result from temperature and pressure differences between the inside and
surroundings of a greenhouse. Crop production in greenhouses is most greatly influenced by ventilation efficiency [4]. This is because ventilation enhances parameters that effect crop development and
production: it removes excessive heat within the greenhouse; enhances carbon dioxide and oxygen
exchanges; maintains acceptable humidity levels [5]; and manages biotic transfers such as fungal
spores [6]. Ventilation has a seasonal function, reducing excess humidity during the winter and
cooling the greenhouse interior during the summer.
Greenhouse ventilation can be naturally or mechanically driven. While mechanical ventilation is
driven by electric fans, natural ventilation through windows is driven by both wind and buoyancy
forces. Natural ventilation is generally more cost effective, and is therefore the more widely used
ventilation method. However, sensitivity analyses show that external factors like wind speed, wind
direction, and temperature can greatly influence natural ventilation rates [7].
Ventilation is most commonly described in terms of pressure differences. The wind force causes
pressure differences near the windows, while a gradient in air density between the inside and outside
result in a vertical pressure gradient which drives the buoyancy force (also known as the stack effect).
Natural ventilation is dominated by the wind force at velocities greater than 2 m/s [8] [9], and by
the buoyancy force in the winter, when temperature differences are greater between the interior and
exterior of the greenhouse.
2-1-1
Wind-Driven Flow
Wind-driven ventilation occurs when air interacts with an obstacle, generating a pressure distribution
around the greenhouse. Pressure drops at the windows can be related to velocity through a window
opening, as described by Bernoulli’s equation (assuming that air is incompressible):
1
∆Pw = ρCp U 2
2
Master Thesis
(2-1)
D.E. Dwyer
8
Theoretical Background
where ∆Pw is the pressure drop due to wind, U is wind speed, ρ is the air density, and Cp is the
non-dimensional pressure coefficient. The pressure coefficient is defined as
Cp =
P − P0
1
2
1 ρU0
(2-2)
where P is the static pressure at some point in the flow, P0 is the static pressure at the undisturbed
stream, U0 is the velocity of the undisturbed stream.
2-1-2
Buoyancy-Driven Flow
Temperature differences between the building interior and exterior, or within spaces in a building
cause buoyancy forces, thereby driving the airflow. For the case of buoyancy-driven flow, the Boussinesq approximation is used to determine the pressure difference, ∆Ps which drives airflow through
a window opening. This method assumes incompressible flow, and homogeneous temperature distributions in- and outside the window.
∆Ps (y) = ∆P0 − ρg
∆T
z
T
(2-3)
where ∆P0 is the pressure at ground level, T is the outside temperature, ∆T is the temperature
difference between the inside and outside, rho is density, and z is the vertical coordinate.
2-1-3
Combined Effect of Buoyancy and Wind
The weight of buoyancy force compared to wind force is expressed with the dimensionless Richardson,
0
Ri, (or Archimedes) number Ri = gu2h , where g 0 represents reduced gravitational acceleration,
g 0 = ∆ρ
ρ0 , and u represents the wind speed. Gravitational acceleration is applied for Boussinesq
approximations, when density differences in the fluid are low. A Richardson number less than unity
indicates that the flow is dominated by the wind force, while a Richardson number much greater
than unity indicates dominance of buoyancy force. However, the less dominant force cannot be
ignored. Mistriotis et al. observed significant effects of buoyancy forces despite the dominance of
wind forces [10].
The combined effect of buoyancy and wind for greenhouse ventilation calculations has been studied in
detail by Boulard and Baille [11], concluding that the model that best fits experimental measurements
calculates ventilation from the sum of the two pressure differences.
∆P = ∆Ps + ∆Pw
(2-4)
The resulting velocity through the windows is related to this pressure drop by
1
∆P = ρζU 2
2
(2-5)
where ζ is a pressure drop coefficient.
D.E. Dwyer
Master Thesis
2-2 Computational Fluid Dynamics
2-2
9
Computational Fluid Dynamics
Several methods are used to study the behavior of greenhouse climate due to ventilation. Until
recently, field experiments have been the predominant method, using techniques such as impulse
peak and continuous injection tracer gas tests to visualize the behavior; and sonic anemometry and
thermocouples to quantify air velocity and temperature. However, these techniques require many
sensors in order to provide an accurate description of velocity, temperature, and humidity patterns.
Wind tunnel tests have also been a popular experimental method, providing more control over
the external wind conditions [12]. While theoretical ventilation can be simply calculated assuming
perfect mixing, Computational Fluid Dynamics (CFD) makes it possible to model the turbulence,
complex flow patterns, and temperature and humidity distributions that occur within the greenhouse,
resulting in varied crop quality.
This method eliminates the need for costly facilities and equipment. Furthermore, several studies
have compared experimental methods to CFD, finding results qualitatively in agreement with wind
tunnel tests and field measurements. However, evidence from literature shows that CFD studies of
greenhouses tend to differ quantitativly, generating results that deviate between 16.3 and 44% from
experimental measurements [13], [4], [14], [15].
CFD, a method in which flow equations are solved numerically, has become a commonly used
method for calculating ventilation patterns for the purpose of evaluating ventilation efficiency [16].
An iterative process is used to apply the conservation equations to each control volume and calculate
model variables. Both wind- and stack-driven ventilation can be modeled using CFD.
2-2-1
Finite Volume Approach
The discretization method used for this work divides the computational domain into finite volumes,
where the center of each domain defines the solution. In contrast to the finite element and finite
difference methods, the finite volume method utilizes the integral form of the conservation equations
to calculate the fluxes at the surfaces of each finite control volume. An advantage of this method in
comparison to the finite difference method is that conservation can be enforced as long as the surface
integral is consistent along adjacent boarders. The sum of all control volume surface fluxes results in
the global conservation, because internal surface fluxes cancel out. While second order calculations
are generally accepted to be an accurate calculation method, higher order methods (above second
order) are more difficult to achieve with the finite volume approach [17].
Staggered Grid
The finite volume method begins with discretizing the computational domain. Given the complex
pressure-velocity coupling discussed in the following section, it is common to use a staggered grid,
in which scalar variables (pressure, in this case) are defined at ordinary nodal points, while velocity
components are calculated on staggered grids centered at cell faces [18]. If velocities and pressures
are both defined at the same node, then a oscillating non-uniform pressure field can act like a uniform
field (in instances where oscillations are present). The staggered grid method eliminates the need to
interpolate velocity at the cell faces for the transport calculations. It also allows pressure differences
at adjacent nodes to act as the driving force for velocity. A depiction of the 2-dimensional staggered
grid is seen in Figure 2-1. The u-velocities are stored at e and w cell faces, while the v-velocities
are stored at n and s cell faces.
In this staggered arrangement, the pressure gradient term,
∂p
pP − pW
=
∂x
δxu
Master Thesis
∂p
∂x ,
for example, can be calculated with
(2-6)
D.E. Dwyer
10
Theoretical Background
Figure 2-1: 2-dimensional staggered grid used for coupled pressure-velocity systems. Pressure is
stored at nodes marked (•) and velocity is defined at the cell faces, marked with arrows (horizontal
arrows (→) indicate u-velocities and vertical arrows (↑) indicate v-velocities) [18]
Rhie-Chow Interpolation
While the staggered grid is very beneficial, it does not translate well to unstructured meshes, and
could result in unphysical pressure oscillations. The OpenFOAM CFD software makes use of an
interpolation method closely related to the Rhie-Chow interpolation method [19], which satisfies
the need for discretized calculations on unstructured grids. Rhie-Chow interpolation is used for
the discretization of the transport equation, and is elaborated on in the derivation of the pressure
equation in the following section.
The method involves interpolating discretized variables by introducing a coefficient for pressurevelocity coupling. The interpolation can be seen as a correction proportional to the difference
between the pressure gradient at the face and the interpolated pressure gradient at the face [20].
2-2-2
Numerical Model: Discretization for the Navier-Stokes System
The calculation method used to evaluate the effect of wind assumes air to be an incompressible
fluid, due to the fact that pressure does not vary greatly within the model. With a nearly zero mach
u
number, M = usound
≈ 0, density is independent of dynamic pressure. Incompressible fluids can be
described using the following general conservation equation:
∂ϕ ∂(U ϕ) ∂(V ϕ) ∂(W ϕ)
+
+
+
= Γϕ ∇2 ϕ + Sϕ
∂t
∂x
∂y
∂z
D.E. Dwyer
(2-7)
Master Thesis
2-2 Computational Fluid Dynamics
11
where ϕ represents the non-dimensional form of the transported quantity; U , V , and W are the
three velocity vector components; Γϕ is the diffusion coefficient; and Sϕ is the source term. In
words, the transport equation can be stated as:
rate of increase of ϕ + net flow rate of ϕ out of the control volume = rate of increase of ϕ due to
diffusion + rate of increase of ϕ due to sources
CFD flow solvers use a set of coupled Partial Differential Equations (PDEs) to iteratively calculate
flow variables for each control volume. The unsteady nature of flow within greenhouses requires the
use of an incompressible, transient flow solver to model the various external situations imposed on
the greenhouse model. The momentum equation can be derived from the general transport equation
for incompressible flow (Eq 2-7) by replacing ϕ with the velocity vector, u.
Momentum equation
∂u
+ ∇ · (uu) − ∇ · (ν∇u) = −∇p
∂t
(2-8)
∇·u=0
(2-9)
Continuity equation
For the greenhouse case considered, three momentum equations exist (one for each velocity component), and all three velocity components are present in the continuity equation. The purely
incompressible system eliminates coupling between density and pressure, in addition to coupling
between energy and the rest of the system.
The solver used in this work makes use of the Pressure Implicit with Splitting of Operators (PISO)
algorithm, proposed by Issa [21], to solve the incompressible continuity and momentum equations
with pressure-velocity coupling. The idea behind the PISO algorithm is to use multiple pressure
corrections to solve the two complex coupling terms present in the pressure-velocity systems:
• a non-linear convection term with u-u coupling
• intricate linear pressure-velocity coupling (due to all three velocity components appearing in
each momentum equation and the continuity equation), making it challenging to solve for
pressure.
Solving Non-Linearity
The non-linear convection term represents the transport of velocity. An iterative cycle is used to
solve for the non-linear convection term,
∇ · (uu) ≈ ∇ · (uo un )
(2-10)
where uo is the current solution and un is the new solution for each iteration, repeating the calculation
until uo = un [22] [18]. While iterative calculations can increase computational cost for large time
steps, a small time step can produce a more accurate temporal solution.
Deriving the pressure equation
Coupling between pressure and velocity introduces a constraint on the solution. With no pressure
equation for incompressible flow, an equation can be derived from the continuity and momentum
equations. The momentum equation is discretized for the velocity at location (P) as
auP uP +
X
auN uN = r − ∇p
(2-11)
N
Master Thesis
D.E. Dwyer
12
Theoretical Background
o
u
is the momentum source term. The momentum equation has been divided through
where r = ∆t
by volume to allow for interpolation of the coefficients at faces. The summation of the north south,
P
east, and west neighbors (N ) is defined as auN uN . A H(u) operator containing both the transport
N
and source terms,
H(u) = r −
X
auN uN
(2-12)
N
simplifies the equation, to isolate uP .
auP = H(u) − ∇p
(2-13)
uP = (auP )−1 (H(u) − ∇p)
(2-14)
Substitution of the incompressible continuity equation (∇ · u = 0) results in a pressure equation for
incompressible flow [22]
∇ · [(auP )−1 ∇p] = ∇ · [(auP )−1 H(u)]
(2-15)
The pressure gradient is not discretized to conform with the Rhie-Chow interpolation procedure.
Interpolation of the H(u) operator and ∇p based on auP is the essence of Rhie-Chow interpolation.
Deriving the Face Flux Equation
The conservative face fluxes are corrected before each pressure-corrector step. This section derives
the calculation for the face flux, F . Beginning with the discretized continuity equation,
∇·u=
X
f
sf · u =
X
F
(2-16)
f
where the face flux is the surface normal of the velocity, F = sf · u. Substituting velocity from
Equation 2-14, the face flux becomes
F = sf
1
(H(u) − ∇p)
auP
(2-17)
Satisfying this equation guarantees face fluxes to be conservative.
Sequence of Operations
The PISO algorithm uses a segregated guess-and-correct method to solve a sequence of equations.
The first pressure corrector establishes a velocity field, while the following corrector(s) determine the
pressure distribution.
The PISO algorithm uses the following method:
1. Momentum predictor: Solve the discretized momentum equation (Eq 2-11) to compute an
intermediate velocity field, using the conservative fluxes in Equation 2-17, and pressure, p,
from the previous time step
D.E. Dwyer
Master Thesis
2-2 Computational Fluid Dynamics
13
fvVectorMatrix UEqn
(
fvm::ddt(U)
+ fvm::div(phi, U)
- fvm::laplacian(nu, U)
);
solve(UEqn == -fvc::grad(p));
2. Compute the mass fluxes at the cells faces by interpolation, using an approximate velocity
corresponding to (auP )−1 H(u)
3. Pressure solution: Solve the pressure equation using the H(u) operator from the predicted
velocities.
fvScalarMatrix pEqn
(
fvm::laplacian(rAU, p) == fvc::div(phi)
);
where rAU corresponds to (auP )−1 .
4. Correct the mass fluxes at the cell faces with 2-17
5. Explicit velocity correction: Correct the velocities using the new pressure field with Equation
2-14.
6. Update the boundary conditions
7. Repeat from 3 for the prescribed number of non-orthogonal corrector steps
8. Increase the time step and repeat from 1[22][23]
2-2-3
Turbulence
Instabilities from non-linear inertial and viscous terms in the Navier-Stokes equations result in turbulent flow. Given the rotational nature, vortex stretching is required to maintain the constantly
fluctuating vorticity. With vortex stretching absent from 2-dimensional flows, turbulence must be
modeled in the 3-dimensional domain. Turbulence is also time-dependent, requiring statistical averaging techniques to estimate fluctuations.
Turbulent eddies are unsteady vortices. The largest eddies are a result of energy from the mean
velocity, called turbulent kinetic energy. The energy is passed on to smaller eddies, and eventually
dissipated into heat. The scale of the smallest eddies are called the Kolmogorov scales, consisting
of length, velocity, and time. Resolving these very small quantities would require a decent amount
of computational power. For complex problems with high Reynolds numbers, new variables that
approximate these small turbulence quantities are introduced into the equations of motion. The
addition of these variables results in an under-defined system of equations. Closure models are used
to describe assumed correlations within the equations of motion.
Master Thesis
D.E. Dwyer
14
Theoretical Background
RANS Methodology
The three commonly used methods for turbulence modeling are Direct Numerical Simulation (DNS),
Large Eddy Simulation (LES), and Reynolds Averaged Navier-Stokes (RANS) modeling. RANS
turbulence modeling determines time-averaged flow parameters with turbulence modeling. DNS
numerically solves the Navier-Stokes equations along with the entire spatial and temporal scales of
turbulence without the use of a turbulence model. Memory storage for this method can be very
high, in addition to the slow integration over time, requiring a small enough time step to satisfy a
Courant number below 1. LES separates fluid movement into large and small eddies, calculating the
large eddies in three dimensions, with time dependence, and calculating small eddies on a subgrid
scale.
This study is limited to the the most widely used, RANS methodology, due to computational memory
and speed limitations. This method involves Reynolds decomposition of pressure and velocity coupled
with time-averaging of these equations. The Reynolds decomposition breaks down pressure and
velocity into a mean and a time-averaged component.
u = U + u0
(2-18)
p = P + p0
(2-19)
This process results in the additional −ρu0i¯u0j term, known as the Reynolds stress tensor, when
applied to the Navier Stokes equation.
The shortened simulation time and reduced memory requirements come with several inaccuracies.
Jiang et al. have identified two major problems with the use of RANS for the purpose of calculating
both external and internal ventilation flows [24]. The first issue is the inability of RANS to predict
airflow around buildings. One study indicates that RANS modeling has difficulty generating the
separation region on the roof, and has tendencies to overpredict the recirculation region beyond the
bluff body. The second issue observed by Jiang et al. is the modeling of mean flow parameters,
with the example of the mean velocity for a single window with uniform flow to be zero, canceling
out the air exchanges present in reality. Jiang et al. conclude that RANS is not ideal for modeling
ventilation with both the internal and external environment in steady state calculations. However, for
the case of large, complex geometries, the method was found to be appropriate, given the necessary
computational time and memory for alternatives. For unsteady RANS simulations (used in this
work), the problem of overpredicting the recirculation region is less extreme, and the cancelation of
air exchanges is not present.
Of the various RANS methodologies, the standard k − model has been commonly used among
previous greenhouse ventilation studies [12]. This method solves two additional transport equations:
turbulent kinetic energy, k, and turbulent kinetic energy dissipation, [25]. This model determines
eddy viscosity using a single turbulence length scale of motion, assuming turbulent diffusion occurs
at a set length scale. There are more accurate alternatives for the situation being modeled that
require the same computational capacity. The Re-Normalization Group (RNG) model better predicts
detachment and reattachment of flow by introducing modified transport coefficients, accounting for
small spatial scales. A more recent model, the realizable k − model, addresses eddy viscosity
differently and uses an additional transport equation for the viscous dissipation, which is derived
from the vorticity fluctuation transport equation. As a result, this model better represents rotation,
flux separations and boundary layers where large pressure gradients occur. The calculations in this
work make use of the realizable k − model.
D.E. Dwyer
Master Thesis
2-2 Computational Fluid Dynamics
15
Realizable k − Model
The realizable k − model solves for turbulent kinetic energy and turbulent kinetic energy dissipation using the following simplified transport equations, derived from the mean-square vorticity
fluctuations.
∂
∂
∂
(ρkuj ) =
(ρk) +
∂t
∂xj
∂xj
"
µt
µ+
σk
#
∂k
+ Pk + Pb − ρ − YM + Sk
∂xj
(2-20)
and
∂
∂
∂
(ρuj ) =
(ρ) +
∂t
∂xj
∂xj
"
µt
µ+
σ
#
∂
2
√ + C1 C3 Pb + S (2-21)
+ ρC1 S − ρC2
∂xj
k + ν
k
i
where Pk is the generation of turbulence kinetic energy due to mean velocity gradients (Pk = τij δU
δxj ),
Pb is the generation of turbulence kinetic energy due to buoyancy, and YM represents the fluctuating
dilation in compressible turbulence that contributes to the total dissipation rate. σk and σ are the
turbulent Prandtl numbers for the turbulent kinetic energy and its dissipation rate. Sk and S are
the user-defined source terms.
The turbulent viscosity, µ, is dependent on a variable, Cµ .
k2
1
Cµ =
∗
A0 + AS kU
µt = ρCµ
where
∗
U =
q
(2-22)
(2-23)
Sij Sij + Ω̃ij Ω̃ij
(2-24)
and
Ω̃ij = Ω̄ij − 3ijk ωk
(2-25)
Ω̄ij is the mean rate-of-rotation tensor viewed in a√rotating reference frame with angular velocity
ωk . Constants are defined as A0 = 4.04 and AS = 6 cos φ, where
√
1
cos−1 ( 6W )
3
Sij Sjk Ski
W =
S̃ 3
φ=
S̃ =
Sij =
1
2
q
Sij Sij
∂uj
∂ui
+
∂xi ∂xj
(2-26)
(2-27)
(2-28)
(2-29)
Finally, the C constants are experimentally determined by Shih et al. as C1 = 1.44, C2 = 1.92,
Cµ = 0.09 [26].
Master Thesis
D.E. Dwyer
16
D.E. Dwyer
Theoretical Background
Master Thesis
Chapter 3
Methodology
With the goal of modeling the internal conditions of a greenhouse independent of modeling external
flows, trends can be derived from simulations of greenhouses with various external environment
conditions and geometry configurations. The most important variables relating to external conditions
are selected and systematically varied. The calculations of the external greenhouse environment
are validated for accuracy using available measurements, before a ventilation boundary condition
is deduced from the observed trends. In order to implement the observed ventilation boundary
condition trends on an internal greenhouse model, the necessary parameters for running a stable
internal greenhouse calculation are determined. The results of the internal models are compared
to the calculations containing the the full external environment to show that the results reasonably
portray the expected flow patterns within the greenhouse.
3-1
Model Parameters
Several parameters are selected on the basis of making the model scientifically accurate and providing
the appropriate variables for the intended applications. An extensive review of past Computational
Fluid Dynamics (CFD) studies on greenhouse ventilation by Bournet and Boulard, 2010 [12], found
the following parameters to have varying degrees of impact (organized by driving force).
• buoyancy-driven ventilation
– temperature difference between inside and outside air
– differences in relative humidity inside and outside the greenhouse
• wind-driven ventilation
–
–
–
–
–
–
–
–
–
Master Thesis
wind velocity
wind direction
window opening angle
window direction (leeward vs. windward)
span number and greenhouse length
window location (distance from windward side)
greenhouse height
combination of roof and side wall openings
presence of nearby buildings
D.E. Dwyer
18
Methodology
– specific devices such as insect-proof or shading screens, etc.
– greenhouse design
– window size
Due to the time limitations of this work, buoyancy calculations are not included in the analysis.
Among the parameters that influence wind-driven ventilation, wind opening angle and wind direction
are the primary variables that are investigated. However, the analysis also considers wind velocity.
An empty Venlo-style greenhouse was modeled for this study. The Venlo greenhouse is a pitched roof
multi-span greenhouse. The greenhouse is ventilated with 1.2-by-2.2 meter staggered windows along
the roof that are open in both the leeward and windward directions. It is made up of 15 greenhouse
modules (otherwise known as spans), each 4 meters wide (in the x direction), 37.5 meters long (in
the y direction), with 5 meter high ridges and 4 meter high gutters (in the z direction). Details
about greenhouse dimensions and terminology, can be found in Figure 1-1. The roof is pitched
at 26.7 degrees from the horizontal plane. Furthermore, to prevent water from collecting on the
roof, the land the greenhouse sits on is elevated at an inclination of 1.5 mm/m. This is accounted
for in the model by placing the greenhouse on a narrow wedge base. While it is ideal to simplify
such models for the purpose of CFD calculations, an appropriate number of spans was necessary
to observe the reconnecting flow over the roof of the greenhouse. Given the focus on the effect of
external conditions on the greenhouse, crops and greenhouse equipment such as screens and heaters
were not modeled for this analysis.
The average wind speed was modeled for the Westland region, an area well known for its horticulture
and greenhouse industry. Local weather stations belonging to Koninklijk Nederlands Meteorologisch
Instituut (KNMI) (in English, the Royal Netherlands Meteorological Institute), indicate an average
wind speed of 3.7 m/s at the height of the greenhouse windows [27]. This value is extrapolated
from 10-meter high weather station measurements assuming atmospheric boundary layer conditions
discussed in Section 4.2 with the relation
uz =
u
10
ln
10+z0
z0
z + z0
ln
z0
(3-1)
where uz is the velocity at a specified height, z, u10 is the weather station velocity measured at a
height of 10 meters, and surface roughness length, z0 is set to 0.1m.
3-2
Wind Direction
Wind azimuth angle was varied in 15 degree increments from 0 to 90 degrees, where a 0-degree
azimuth angle represents wind perpendicular to the greenhouse ridge, and a 90-degree azimuth angle
indicates wind flowing parallel to the ridge. The symmetry of the greenhouse structure can be used
to deduce ventilation patterns resulting from approach angles outside the 0 to 90 degree range.
3-3
Window Opening Angle
Venlo greenhouses are characterized by alternating (windward- and leeward-facing) discontinuous
window openings, which result in complex 3-dimensional flows. It is common for the leeward- and
windward-facing windows to be open at different angles. Hence, for this study, combinations of
opening angles between 0 and 100% are investigated (0, 15, 30, and 53.3 degree window opening
angles). The maximum window angle occurs when the window panel is parallel to the adjacent roof
slope. For the roof slope of 26.7 degrees, the maximum ventilation opening occurs at an angle 53.3
degrees from the roof surface.
D.E. Dwyer
Master Thesis
3-3 Window Opening Angle
19
Figure 3-1: Wind azimuth angles
(a)
(b)
Figure 3-2: 15 and 30 degree window openings
Figure 3-3: 53.3 degree window opening angle
Master Thesis
D.E. Dwyer
20
D.E. Dwyer
Methodology
Master Thesis
Chapter 4
CFD Simulation
This section discusses the approach used to develop the meshes, calculate flow patterns, and simulate
the various external conditions.
OpenFOAM is a powerful Computational Fluid Dynamics (CFD) tool that was selected on the conditions that it is efficient, extensive, and as an open source software package, it is easily modifiable,
flexible, and has no licensing costs. Working with OpenFOAM also presents several drawbacks:
integrating the software with optimization methods and experiments can be a challenge, and as an
open source software licensed under the GNU General Public License (GPL), the commercialized
Kasklimaat Model (KKM) model source code must be made freely available to customers and may
be freely distributed thereafter [28].
Mesh Generation
The mesh is modeled three-dimensionally to account for the three-dimensional behavior of turbulent flows. While many greenhouse climate models are developed in 2-dimensions, when wind is
not perpendicular to the opening surface, 3-dimensional simulations more realistically represent the
circulation patterns inside a greenhouse [12], such as the exchanges between staggered leeward and
windward facing windows.
OpenFOAM’s SnappyHexMesh utility was used to snap the environment grid around the imported
greenhouse Stereolithography (STL) file developed in Sketchup. Meshes generated by the SnappyHexMesh utility contain hexahedra and split-hexahedra derived from triangulated surface geometries.
Simulating various window opening angles and wind approach angles was achieved in this work by
generating a different mesh for each situation. The different window opening geometries and greenhouse orientations were modeled in Sketchup and meshed in the same computational domain.
The mesh grids were developed with finer cells near the greenhouse structure and along the ground
(where strong gradients are expected to occur from cell to cell), while cells are coarser further away
from the greenhouse, where relatively fewer changes in the flow are expected. This optimal mesh
arrangement minimizes the necessary computational cost while maintaining solution accuracy. In
order to minimize truncation errors, gradual variations in cell size were used within the grid. Cell
volumes generally ranged between 5e-5 and 12.5 m3 .
The following methods in OpenFOAM were used to develop the mesh in Figure ??b.
• External computational domain dimensions, grid density, and external patches (i.e. earth,
atmosphere, inlet, outlet, and surrounding) were defined
Master Thesis
D.E. Dwyer
22
CFD Simulation
• New mesh generated using the snappyHexMesh utility to snap the external environment grid
to the greenhouse STL geometry modeled in SketchUp (Figure ??a).
• TopoSet utility used to select and group greenhouse faces (walls, ceiling and floor) within the
mesh
• CreatePatch used to specify greenhouse boundary patches according to groupings from topoSet.
These patches are later used to define boundary conditions when running simulations.
(a)
(b)
Figure 4-1: 3-Dimensional greenhouse model developed in SketchUp (a)
and integrated into the mesh (b)
Computational Domain
The boundaries of the mesh containing the environment surrounding the greenhouse can influence
the simulation results. The computational domain must be large enough to encompass the largest
relevant flow structures. Furthermore, if the cross-section is too small, artificial acceleration occurs
around the object of interest. As a rule of thumb: the area of concern should be less than 10%
of computational domain[29]. The selected blockage (ratio of projected greenhouse area in the
flow direction to the computational domain cross section) is 5%. Following the recommendations
of Blocken et al., the computational domain ratios are equivalent to the greenhouse cross-section
[30]. The extension of the domain in the flow direction is 4 times the height of the greenhouse, as
suggested by The Association of German Engineers (VDI) [29]. The outflow boundary should be
far enough away from the object of interest that flow does not enter through this boundary, as this
would hinder or even prevent convergence [31]. To allow the flow to redevelop after the greenhouse
and prevent backflow, the outflow boundary is located at a distance of 15 times the height of the
greenhouse. Figure 4-2 presents an over
4-1
Boundary Conditions
The constraints set for the various external environment calculations are inlet velocity profile, a
prescribed pressure outlet, no-slip conditions along surfaces, and k and wall functions. The nonuniform inlet wind profile is directed perpendicular to the flow. Details of the boundary conditions
can be found in Table A-1 of the Appendix.
D.E. Dwyer
Master Thesis
4-2 Numerical Approach
23
(a)
(b)
Figure 4-2: Overview of computational domain from the top and side views
Fully developed Atmospheric Boundary Layer (ABL) profile resulting from the k-epsilon
model
The wind velocity profile at the inlet of the model exhibits higher velocities farther away from the
ground as a result of frictional drag. The widely adopted profile is based on Richards and Hoxey’s
work [32], which assumes the vertical velocity to be zero, the pressure to be constant in the vertical
and streamwise directions, and the shear stress to be constant throughout the boundary layer. While
wind is driven by large pressure differences, constant pressure can be assumed for the derivation of
the atmospheric boundary layer profile because the pressure differences are very small over small
segments of the vertical and horizontal dimensions:
u∗
z + z0
U (z) =
ln
K
z0
(4-1)
The turbulent kinetic energy k is defined by
u2
k = p∗
Cµ
(4-2)
and turbulent kinetic energy dissipation is
=
u3∗
K(z + z0 )
where the friction velocity u∗ is calculated from a velocity Uh at a reference height h.
KUh
u∗ =
0
ln( h+z
z0 )
(4-3)
(4-4)
z is the vertical height, K is von Karman’s constant ( 0.4), and z0 is surface roughness length. The
input values for the atmospheric boundary layer profile are identified in Table 4-1.
4-2
Numerical Approach
The fvSchemes and fvSolutions files used by OpenFOAM to set numerical schemes and equation
solvers can be found in Appendix B. The details of these two documents are discussed in the following
two subsections.
Master Thesis
D.E. Dwyer
24
CFD Simulation
Table 4-1: ABL Conditions
4-2-1
Variable
Variable
Value
Reference Velocity
Uref
5.5
Reference Height
Href
30
Surface Roughness Length
z0
uniform 0.1
Turbulent Kinetic Energy
turbulentKE
0.50497
Turbulent Kinetic Energy Dissipation
turbulentEpsion
0.1734
z-axis Orientation
zDirection
(0 0 1)
Minimum coordinate value in z direction
zGround
uniform 0
Solution and Algorithm Control
The simpleFoam solver was initially used to simulate fluid flow in the external environment. This
is a steady-state solver for incompressible, turbulent flow. However, it became apparent that the
transient characteristics of the situation prevent the model from converging, and a transient solver
can better achieve a solution. Therefore, the Pressure Implicit with Splitting of Operators (PISO)
algorithm described in the theoretical background was used for the wind flow simulations presented.
The pisoFoam solver models fluid characterized as incompressible, transient, and turbulent. Three
pressure correction iterations and one non-orthogonal corrector were used during these calculations.
The realizable k − turbulence model (also presented in the theoretical background) was used for
turbulence modeling.
Linear Solvers
The pisoFoam solver using the realizable k− turbulence model solves equations for velocity, pressure,
turbulent kinetic energy, and turbulent kinetic energy dissipation. Each discretization equation
uses linear solvers to solve a set of linear equations (not to be confused with application solvers,
which define the set of equations and algorithms). ThePreconditioned Conjugate Gradient (PCG)
linear solver is used to solve the symmetric pressure matrix equation using a Diagonal IncompleteCholesky (DIC) preconditioner. However, the Generalized Geometric-Algebraic Multi-Grid (GAMG)
linear solver would have been more ideal. This method generates a fast solution for a selection of
cells, which is then mapped onto a finer mesh as the initial guess for the solution. The velocity,
turbulent kinetic energy, and turbulent kinetic energy dissipation all use the Gauss Seidel smooth
solver. Smooth solvers reduce oscillatory error and transfer "smooth" error to coarse grids. While
the Gauss Seidel method is one of the more reliable smooth solvers, it is relatively slow due to its
sequential nature. A preconditioner, such as Preconditioned bi-Conjugate Gradient (PBiCG), could
have considerably reduced the number of iterations for solving these sets of equations.
Convergence Criteria
With each iteration of the solvers, the residual error is evaluated. The solver stops when one of the
following is satisfied.
• the residual falls below the solver tolerance,
D.E. Dwyer
Master Thesis
4-2 Numerical Approach
25
• the ratio of current to initial residuals drops below the solver relative tolerance,
• or the number of iterations exceeds the specified maximum number of iterations.
The tolerances used are listed in Table 4-2.
Table 4-2: Residual error tolerances used in the PISO algorithm for each parameter
Parameter
Relative Tolerance
Absolute Tolerance
Velocity, U
0.1
1e-8
Pressure, p
0
1e-6
Turbulent kinetic energy, k
0.1
1e-8
Turbulent kinetic energy dissipation, 0.1
1e-8
A pressure relative tolerance of zero indicates that the tolerance is calculated solely on the basis
of the relative tolerance. A higher pressure tolerance was used in order to speed up the pressure
correction steps.
4-2-2
Numerical Schemes
Table 4-3 provides details about the discretization schemes used for the calculation.
Table 4-3: Numerical Schemes
Calculation
Scheme
Time
Backward (second order, implicit)
Gradient
Gauss linear (second order, Gaussian)
Divergence
Gauss linear (second order)
Laplacian
Gauss linear limited corrected 0.333
Interpolation
Linear
Surface normal gradient
Explicit non-orthogonal correction
The backward second order time scheme is used to maintain an accurate transient calculation.
Gradient terms, ∇, are discretized using the standard finite volume discretization of Gaussian integration, requiring interpolation of values from cell centers to face centers. In this case, linear
interpolation, more commonly known as central differencing, is used.
The divergence scheme, ∇·, which determines the convection term of a fluid, ∇ · (ρuu), can only
be discretized using the Gauss scheme, along with an interpolation scheme. Generally, simulations
were initialized using upwind (first order) interpolation to calculate a more stable flow field before
upgrading to the more accurate linear (second order) interpolation scheme. Furthermore, the positive
turbulent kinetic energy and turbulent kinetic energy dissipation scalars are bounded. Limited linear
differencing is used for the velocity vector to account for direction.
Laplacian schemes discretize the Laplacian term, ∇ · (ν∇u). The interpolation scheme specified
for the diffusion coefficient is Gauss linear. The surface normal gradient uses the coefficient 0.333,
which corresponds to limited non-orthogonal correction.
Master Thesis
D.E. Dwyer
26
CFD Simulation
Surface normal gradient scheme, ∇u, specifies the gradient of two adjacent cell centers normal to
the face between the two cells. This scheme uses explicit non-orthogonal correction.
4-3
Simulation Methods
With a total of 15 window opening angle combinations and 6 wind approach angles, 21 calculations
of various external environment conditions were simulated. The calculations were initialized by
mapping the flow patterns from a "constant" simulation used for testing spatial convergence and
computational domain accuracy. The "constant" simulation contains a greenhouse with both leeward
and windward windows open at a 30 degree angle from the roof, and an azimuth angle of 0 degrees
(approaching the greenhouse perpendicular to the roof).
Methods used to maintain stability in the calculations (particularly the cases with varied wind azimuth
angle) were
• mapping a flow field from another simulation as the initial condition,
• initializing the flow with a first order calculation (in space), before calculating the flow using
a more accurate second order calculation scheme
• altering the numerical schemes (i.e. increasing the number of pressure correctors, increasing
the number of non-orthogonal correctors, and using more effective preconditioners), and
• creating a new mesh with restricted skewness and finer refinement ratios.
• maintaining a Courant (dimensionless transport of each time step) number below 1.
The following section uses multiple methods to evaluate the convergence of the 21 cases. More
details about Courant number are discussed in the section to follow.
D.E. Dwyer
Master Thesis
Chapter 5
Verification and Validation of Results
This section presents observations which verify that the simulations accurately implement the conceptual theory modeled, and further validate the model’s representation of physical reality.
5-1
Verification
Verification assessments are critical for determining correct implementation of conceptual models
and whether the results can be used in an analysis of the boundary condition. The following methods
are used to verify the simulations in this work [33]:
• Examine spatial (grid) convergence
• Examine temporal convergence
• Examine the implementation of the model
• Examine iterative convergence
• Examine consistency
5-1-1
Examine Spatial Convergence
Two tests were done to ensure mesh accuracy. Resolution independence was tested by running the
solver with the same boundary conditions in a coarser and finer mesh, containing twice as many
cells in each dimension. Similarly, the effect of the computational domain was tested by increasing
the dimensions in each direction by a factor of 2.
Computational Domain
To ensure that the computational domain is large enough to avoid artificial acceleration, tests were
conducted to optimize mesh accuracy. Figures 5-1 and 5-2 depict the top and side views of the
larger computational domain examined.
Two points were selected to investigate error between the two calculations. At a point located at the
center of a windward facing window, the velocity magnitude of the smaller computational domain
Master Thesis
D.E. Dwyer
28
Verification and Validation of Results
Figure 5-1: Top view of larger computational domain, twice as long in each dimension (370m x
400m)
Figure 5-2: Larger Computational Domain Mesh, twice as long in each dimension (370m x 50m)
is 38.8% below the velocity of the larger computational domain. However, at a point located along
the floor of the greenhouse, the velocity calculated in the smaller computational domain is 62.1%
above the velocity calculated for the larger computational domain. The zoomed in velocity profiles
for a slice in both the large and small computational domains can be seen in Figures 5-3 and 5-4,
respectively.
Figure 5-3: Velocity magnitude flow field in the larger computational domain
The two flow fields are presented with matching scales. It is apparent that the larger computational
domain models lower velocities within the greenhouse. Furthermore, there appears to be some
artificial acceleration present in the smaller computational domain. When animated over time, the
flow within the greenhouse is not steady. Therefore, the smaller computational domain was used to
for the parametric tests ventilation tests due to time limitations. Qualitatively, the various greenhouse
simulations can be expected to show reasonable flow trends, with low velocities near the windward
side of the greenhouse and higher velocities at the floor near the leeward side of the greenhouse.
However, given the transient nature of the flow, the results will most accurately represent mean flow
D.E. Dwyer
Master Thesis
5-1 Verification
29
Figure 5-4: Velocity magnitude flow field in the computational domain
fields, rather than flow fields at a given time.
Mesh Fineness and Spatial Convergence
The performance of high Reynolds turbulent flow calculations is dependent on cell size near the wall,
in order to accurately represent the approximate logarithmic behavior of velocity near walls due to
friction. However, wall functions can also be used to account for this behavior using experimental
correlations based on wall roughness. Wall functions, however, remain a boundary condition in this
work to optimize the model accuracy (particularly of turbulence quantities), a common practice
among greenhouse climate studies [12].
To ensure that the mesh cells are fine enough to minimize discretization error, tests were conducted
to optimize the mesh accuracy (within the computational limits of TNO’s computing equipment). A
flow field was computed for three grids, each with twice the number of grid points in each dimension
of the previous grid. With each level of grid refinement, the spatial discretization errors should
asymptotically approach zero.
Figure 5-5: Top view of finely meshed grid with twice as many cells in each dimension (185m x
200m)
Master Thesis
D.E. Dwyer
30
Verification and Validation of Results
Figure 5-6: Side view of finely meshed grid with twice as many cells in each dimension (185m x
25m)
Table 5-1 below indicates the grid details and the resulting pressure coefficient computed at a single
point located in the opening of a windward facing window in the center of the first span of the
greenhouse.
Table 5-1: Spatial convergence details
Normalized Grid Spacing
Pressure Coefficient, Cp
1
0.2694
2
0.2515
4
-0.0201
From these values, the observed order of convergence is determined. The error can then be defined
as the difference between the discrete and exact solutions.
E = f (h) − fexact = Chp + HigherOrderT erms
(5-1)
where f is a calculated quantity (velocity, pressure, etc.),C is a constant, h is a grid spacing measure,
and p is the order of convergence. For a second order discretization scheme, the order of convergence,
p, would be 2. In this case, the numerical CFD schemes provide a theoretical second order of
convergence. However, an observed order of convergence, p, is usually lower as a result of boundary
conditions, numerical models, and the grid. Neglecting higher order terms, Equation 5-1 can be
rewritten as
log(E) = log(C) + p log(h)
(5-2)
by taking the logarithm on both sides of the equation. The format of this equation indicates that
the order of convergence is the slope of the curve of log(E) and log(C). While a least squares fit
could be used to determine this value, a more accurate method for such few data points would be
to assume a constant grid refinement ratio, r with Equation 5-3.
p=
2
ln ff23 −f
−f1
ln r
(5-3)
where r is the grid refinement ratio, r = h1 /h2 . The pressure coefficients provided in Table 5-1
result in an order of convergence of 3.92, while the expected order is 2. This indicates that the three
calculations are not in the asymptotic range of convergence.
Richardson extrapolation, a method based on Taylor series representation, uses the observed order
of convergence, p, to estimate a value at zero grid spacing using lower-order discrete values [33].
The continuum value can be obtained by
D.E. Dwyer
Master Thesis
5-1 Verification
31
(f1 − f2 )rp
fh=0 ∼
= f2 +
rp − 1
(5-4)
For the quantities given in Table 5-1, the Richardson extrapolation method estimates a zero grid
spacing pressure coefficient of 0.2707. However, given that the three calculations are not in the
asymptotic range of convergence, a more reasonable order of convergence of 1.7 (rather than 3.92),
would result in a zero grid spacing pressure coefficient of 0.2774. The values are plotted in Figure
5-7
Figure 5-7: Spatial convergence study. The value in red indicates the Richardson extrapolation for
an assumed order of convergence of 1.7
The method can also be used to estimate the discretization error of values obtained from CFD. The
estimated fractional error of the fine mesh is defined as
−1
(5-5)
f2 − f1
f1
(5-6)
∗ rp
rp − 1
(5-7)
E1 =
rp
where the relative error is given as
=
intermediate mesh is defined as
E2 =
Based on values reported in Table 5-1, the estimated fractional error for calculations using the fine
mesh was determined to be 0.0047, while the fractional error for the intermediate grid refinement
is 0.071. Assuming an order of convergence of 1.7 (rather than 3.92), the error becomes 0.030 and
0.096 for the fine and intermediate grids, respectively. Despite the increased error, for the purpose of
this work, the intermediate mesh refinement will be used, given the available computational power
and limited time.
Master Thesis
D.E. Dwyer
32
5-1-2
Verification and Validation of Results
Examining Temporal Convergence
Temporal accuracy and numerical stability of transient flows are dependent on mesh fineness. The
Courant number, Co, represents the dimensionless transport for each time step, and must be kept
below 1.
U ∆t
Co =
(5-8)
∆x
Monitoring the Courant number (discussed in Section ??) and maintaining a value below 1 was the
main method used to stabilize the calculations and monitor convergence. To achieve this, time steps
ranged from 0.001 to 0.01 seconds. The Courant value was constant throughout the calculation
with negligible fluctuations when window opening angle was varied. In some cases, a very low time
step was necessary for the first few time iterations. The time step was later increased as the solution
stabilized, in order to speed up the calculation. Sudden jumps were observed in the Courant value
throughout the calculations for various wind azimuth angle, although the maximum value remained
below 1 for nearly all iterations. These calculations often required smaller time steps to maintain a
Courant value close to 0.5, avoiding spikes in excess of 1. These spikes are likely caused by errors
within the mesh, such as skewed cells and adjacent cells of difference fineness ratios. An example of
the monitored Courant value for an azimuth angle of 75 degrees, compared with a 0 degree azimuth
angle can be found in Figure 5-8.
(a)
(b)
Figure 5-8: Monitoring Courant value for (a) 0 and (b) 75 degree wind azimuth angles at each time step. This image shows the degree
of Courant fluctuations when calculating varied wind approach
angles, in comparison to the constant Courant number throughout calculations of varied window opening angles.
5-1-3
Examine the Implementation of the Model
This verification method requires an investigation of the simulation calculation visually and experimentally. This evaluation revealed that one of the simulations was redundant. The simulation
containing closed windward-facing windows and 15 degree leeward-facing windows contained an error in the greenhouse geometry, modeling instead 15 degree windward facing windows and closed
leeward facing windows. Therefore, the effect of closed windward-facing windows and 15 degree
leeward-facing windows cannot be included in the analysis.
D.E. Dwyer
Master Thesis
5-1 Verification
5-1-4
33
Examine Iterative Convergence
Given the iterative nature of Computational Fluid Dynamics (CFD) calculations using the Pressure
Implicit with Splitting of Operators (PISO) algorithm, monitoring the convergence is necessary in
order to obtain an accurate solution. Within the PISO algorithm, residual error tolerances were set
either as absolute or relative tolerance for each model parameter (Table 4-2).
Time Step Residuals
An example of the monitored initial residual for the calculated parameters and continuity is presented
in Figure 5-9.
(a)
(b)
Figure 5-9: Initial residual error for each time step for (a) a zero degree
wind approach angle and (b) a 75 degree wind approach angle
The time step was increased for several cases following initial stabilization, to speed up the calculation. Increasing the time step resulted in an initial increase in the residual error, followed by a
steady decrease. In other situations, it was necessary to decrease the time step, in order to lower
the Courant number and the residual error (as seen in Figure 5-9b). While residual error decreased
below 1e-5 for the cases where window opening angle was varied, the wind azimuth angle cases
produced larger residual error (sometimes exceeding 1e-4). This residual could have been reduced
by lowering the time step, however, time constraints prevented the time step from being further
decreased. The residual error could also have been altered by using a different initialization field.
The higher residual error for the wind azimuth angle cases indicates that convergence has not been
reached, and close attention should be paid in the verification and validation of these calculations.
Integral Values
Stationarity of external flow is determined both visually and theoretically. Animating time steps
of planes parallel to the wind direction can be valuable for observing flow stabilize and vortex
shedding. Further observations of convergence can be deduced by evaluating integral quantities
over the greenhouse (lift, drag, and yield). Sometimes, it is possible that these external forces
converge before residual decreases and stabilizes. Therefore, this test is can be useful for evaluating
the convergence of the azimuth angle calculations.
The temporal fluctuations and stabilizing of the drag coefficient are depicted in Figure 5-10. The
drag coefficient is a ratio of the drag force to the product of dynamic pressure and area.
Master Thesis
D.E. Dwyer
34
Verification and Validation of Results
Cd =
Fd
1
2
2 ρU Aref
(5-9)
where Fd is the drag force, ρ is density, U is the velocity far away from the greenhouse at the height
of the window, and Aref is the reference cross sectional area, taken as the projected greenhouse on
a plane whose normal points in the direction of the wind. Drag force, Fd , is the sum of the viscous
drag and pressure drag forces (Fd = Fv + Fp ). These two forces are integral quantities calculated in
OpenFOAM over the entire greenhouse surface. The viscous force is proportional to the air speed,
u, greenhouse surface area, A, and inversely proportional to separation, z:
Fv = µA
u
z
(5-10)
where µ represents dynamic viscosity. The pressure force is given as
Fp = pA
(5-11)
where p represents the pressure on a surface with area, A.
Wind Direction
Window Opening Angle
0.7
0.68
0.66
Drag Coefficient
0.64
0.62
0.6
0.58
15 degrees
30 degrees
45 degrees
60 degrees
75 degrees
90 degrees
0.5
0.4
0.3
Drag Coefficient
0|15 degrees
0|30 degrees
0|54 degrees
15|0 degrees
15|15 degrees
15|30 degrees
15|54 degrees
30|0 degrees
30|15 degrees
30|30 degrees
30|54 degrees
54|0 degrees
54|15 degrees
54|30 degrees
54|54 degrees
0.2
0.1
0.56
0.54
0
0.52
−0.1
0.5
0
50
100
150
Time [s]
200
250
300
0
50
100
150
Time [s]
200
250
300
Figure 5-10: Temporal fluctuations of drag coefficient for variations in window opening angle (left)
and variations in wind direction (right)
The highest drag forces occur when either a windward or leeward facing window is completely open
(54 degrees). When compared with the varied window opening simulations, the varied wind direction
simulations have a very unsteady drag coefficient, although the drag fluctuates over a constant mean
value. Figure 5-11 breaks down the drag coefficient into pressure and viscous components. While
pressure drag coefficients appear stable, (despite fluctuations for window directions), the viscous
coefficients are not completely stable for wind azimuth angle simulations. However, it is important
to recognize that the viscous forces have a much smaller magnitude, and therefore have little influence
on drag force. There is an unexpected negative pressure drag coefficient for the 45 degree wind
approach angle calculation, which results in a negative drag coefficient. This indicates that the
calculation has likely not yet reached convergence, an argument that is further supported in the
following sections.
Flow Parameters Near Windows
Given that the focus of this work is to evaluate the flow fields near the windows of the greenhouse,
it is necessary to perform a more detailed analysis of stationarity at the window openings. While
monitoring the residual for each time step can be valuable, it is also important to monitor the
D.E. Dwyer
Master Thesis
5-1 Verification
35
−4
−3
Window Opening Angle
x 10
2.5
0|15 degrees
0|30 degrees
0|54 degrees
15|0 degrees
15|15 degrees
15|30 degrees
15|54 degrees
30|0 degrees
30|15 degrees
30|30 degrees
30|54 degrees
54|0 degrees
54|15 degrees
54|30 degrees
54|54 degrees
4
3
2
15 degrees
30 degrees
45 degrees
60 degrees
75 degrees
90 degrees
2
Viscous Drag Coefficient
Viscous Drag Coefficient
5
Wind Direction
x 10
1.5
1
0.5
1
0
0
0
50
100
150
Time [s]
200
250
300
0
50
100
Window Opening Angle
150
Time [s]
200
250
300
200
250
300
Wind Direction
0.06
0.07
0.058
0.06
Pressure Drag Coefficient
Pressure Drag Coefficient
0.056
0.054
0.052
0.05
0.048
0.046
0.05
0.04
0.03
0.02
0.01
0
0.044
−0.01
0.042
−0.02
0.04
0
50
100
150
Time [s]
200
250
300
0
50
100
150
Time [s]
Figure 5-11: Temporal fluctuations of drag coefficient for variations in window opening angle (left)
and variations in wind direction (right), broken down by pressure and viscous forces
parameter values over time to ensure that a sufficient number of time steps have allowed the
calculation to overcome the high variance due to unphysical initial conditions. Figures 5-12 and
5-13 give an example of several values measured over time probed at various window openings of
the "constant" run and a 75 degree wind azimuth angle respectively. From the constant run, it is
evident that fluctuations due to initialization take place during the first 80 seconds. However, there
remain some minor fluctuations throughout the remaining duration of the calculation due to the
transient nature of the flow. A similar trend was observed for the simulations of varying window
opening angles.
Alternatively, there is higher variance in the varied wind azimuth angle calculations. The reduced
time step in the first 50 seconds (consistent with the residuals plot in Figure 5-9b) indicates slightly
more stable calculations, although large fluctuations are still abundant. The large jumps in turbulence
quantities (at time = 40 seconds, for example) are are an indication that smaller flow features are
resolved due to a reduction in the time step. When the time step is reduced, the small scale turbulent
motion with fast time scales can be resolved. A time scale for the small eddies, teta can be estimated
with
tη =
νl
u3
1
2
(5-12)
with velocity, u, length scale, l, and kinematic viscosity, ν. Therefore, halving the time step, would
reduce the length scale of resolved turbulence features by one quarter (provided they can be spatially
resolved with a Courant number below 1). The change in turbulence kinetic energy and turbulent
kinetic energy dissipation values in Figure 5-13 is in accordance with the change in length scale.
Resolving k and for different sized flow features results in a significant change for some values but
smaller change for others, because the size of turbulent eddies varies throughout the flow region.
Similar observations were made for the five other varied wind approach angle simulations.
In both Figures 5-12 and 5-13, the turbulent kinetic energy, k, is constant, while the expected
Master Thesis
D.E. Dwyer
36
Verification and Validation of Results
Figure 5-12: Temporal fluctuations of calculated fields colored by window direction for a 0 degree
wind approach angle and 30 degree window openings
behavior would follow the fluctuations of the velocity, given the dependence of turbulent kinetic
energy on velocity in Equation 2-20. This error is due to the the high tolerance set in Table 4-2
for turbulent kinetic energy. As a result, the solver does not solve for this field at each time step.
This is further confirmed by the residual plot in Figure 5-9, which shows an initial turbulent kinetic
energy residual below 1e-8 for each time step. A lower tolerance for k would have produced more
physical results, but would have also increased the computation time.
Further investigation into the stationarity of the calculations over the latest 30 seconds of computation time can be determined with regression modeling. To test that the simulations are statistically
stationary, a regression model
Umag = C + βt,
(5-13)
was fit to the velocity time series for all points sampled at the greenhouse windows (8495 probes for
varied wind azimuth angle and 8665 for varied window opening angles). C represents the stationary
velocity magnitude in m
s , with a rate of change, β, over time, t. A stationary velocity probe would
have β = 0, indicating that fluctuations occur over a constant trend (rather than an increasing or
decreasing trend).
The significance of the coefficient β was evaluated based on the p-value and the estimated β
coefficient value. The p-value significance was judged according to the F-statistic of the hypothesis
test that the corresponding coefficient is equal to zero. A p-value below 0.05 rejects this hypothesis.
β coefficients within the range of ±0.0033 sm2 (indicating a an increase or decrease in velocity by
0.1 m/s over a 30 second period) were also considered insignificant. The resulting percentage of
stationary window probes are listed in Tables 5-2 and 5-3.
Nearly all probed points in the the window opening angle calculations were found to be stationary
according to the specified criteria. However, the wind azimuth angle calculations are not stationary.
This is due to the initialized flow fields. It appears that the degree to which the initialized greenhouse
D.E. Dwyer
Master Thesis
5-1 Verification
37
Figure 5-13: Temporal fluctuations of calculated fields colored by window direction for a 75 degree
wind approach angle and 30 degree window openings
structure overlapped with the rotated greenhouse influences the number of stationary probes. A 15
degree azimuth angle has the largest amount of overlap with the 0 degree azimuth angle initialization
field, and therefore has the fewest stationary calculations at the windows. Alternatively, the 75 and
90 degree wind azimuth angle calculations have the highest percentage of stationary probes, due to
the smaller overlap of the rotated and ’mapped’ greenhouses.
The analysis of CFD calculations in this work is based on the mean values and their standard
deviations over the latest 30 seconds worth of time steps (found in Appendix ??).
5-1-5
Examine Consistency
Verification of simulation consistency involves an investigation of whether the simulation initial
conditions change over time and an evaluation of the net flux through the windows.
Comparison with Initial Conditions
In addition to low residual tolerances, inaccurate initial conditions require the calculation to run for
a sufficient number of time steps before reasonable results are produced. Calculations for all 21
variable cases (varying window angles and wind directions) of the external environment were run for
a minimum of 3.5 minutes of computational time, allowing enough time for the calculated fields to
stabilize and for initial residual error to be sufficiently low.
Master Thesis
D.E. Dwyer
38
Verification and Validation of Results
Table 5-2: Percentage of stationary window probes in varied wind azimuth angle calculations (with
fixed 30 degree windward and leeward window opening angles
Azimuth Angle
Stationarity
15
55.6%
30
57.1%
45
63.5%
60
64.5%
75
72.5%
90
72.2%
Table 5-3: Percentage of stationary window probes in simulations of varied window opening angle
combinations (with fixed 0 degree wind approach angle)
Leeward Angle
Windward Angle
0
0
15
30
54
100%
100%
100%
15
100%
100%
100%
100%
30
100%
100%
99.4%
100%
45
100%
100%
100%
100%
54
100%
100%
100%
100%
In order to stabilize the varied wind azimuth angle simulations and minimize calculation time, the
flow field of these calculations were initialized with the results from the ’constant’ run, in which
windows are open at 30 degree angles and wind approaches at a zero degree azimuth angle. In
this section, the influence of the initial conditions is evaluated. Given that a ventilation boundary
condition will be developed based on the flow at the windows, Figures 5-14 and 5-15 compare the
initial and final velocity and pressure values at the windows of the greenhouse after 5 minutes of
computation time.
There is a clear indication that after 5 minutes, the flow field still resembles the initial flow field
(with the greenhouse rotated 75 degrees counterclockwise of the simulated greenhouse). This could
be due to a longer time scale for internal flow compared to the external environment, requiring more
time for the internal flow to converge. This hypothesis is further validated with Equation 5-12, in
3
which the relationship between time scale and velocity (tη ∝ u− 2 ) indicates lower velocities within
the greenhouse are expected to have longer time scales compared to the higher velocities in the
external environment. The low velocities near the greenhouse, coupled with the unsteadiness in the
flow field, indicates that the solution time could take significantly longer to produce a flow field at
the windows representative of a 75 degree wind azimuth angle. Furthermore, it is possible that the
solver is simply unable to achieve a proper solution when starting from the applied initialization field.
The results of the azimuth angle calculations are deemed insignificant based on the high residual
error, insufficient runtime (and remaining influence of initialization), and evidence that the turbulent
kinetic energy is not solved correctly (due to a high residual tolerance setting). However, alternative
D.E. Dwyer
Master Thesis
5-1 Verification
39
Initialized Values
Final Values
6
5
4
3
2
1
0
Figure 5-14: Comparison between initial and final velocity magnitude [m/s] at the window openings
for the 75 degree wind azimuth angle simulation. The "final value" indicates a sample taken after
5 minutes of calculation.
Initialized Values
Final Values
100004
100002
100000
99998
99996
99994
2
Figure 5-15: Comparison between initial and final pressure [ m
s2 ] at the window openings for the
75 degree azimuth angle simulation. The "final value" indicates a sample taken after 5 minutes of
calculation.
initialization methods, such as a calculation from a coarser grid, or the use of the PotentialFoam
solver may have been better methods for initialization for the purpose of this study. These calculations can also be run for a longer duration to determine a more realistic flow pattern along the
greenhouse roof once the calculation finally converges.
Window Fluxes
A closer look at the flux through the windows can provide further detail about the greenhouse
ventilation. The volumetric flow rate is determined using a mean velocity for each window.
~
Q = ~umean · dA
(5-14)
~ is a vector normal to the window surface, whose magnitude amounts to the
where vector area, A
~
window area, A = An̂.
Figure 5-16 visualizes the fluxes at each window for all 14 calculations (recalling that an error in
the greenhouse geometry required discarding the 15 degree leeward and closed windward facing
window calculation). There are some unexpected trends observed for various window angles. Most
Master Thesis
D.E. Dwyer
40
Verification and Validation of Results
3
Volumetric Flow Rate [m /s]
2.5
30 degrees LW v
53 degrees LW v
2
1.5
1
15 degrees WW >
15 degrees LW v
0.5
0
30 degrees WW >
−0.5
−1
−1.5
53 degrees WW >
−2
−2.5
Figure 5-16: Volumetric flow rate at the plane of the windows for the 14 varied window opening
angle simulations in m3 /s. The location of the greenhouse profile (row and column) determine
the combination of windward (WW) and leeward (LW) opening angles, respectively. The first
row represents simulations where the windward-facing windows are closed, while the first column
represents leeward-facing windows that are closed. Due to errors in simulated greenhouse geometry,
15 degree leeward and close windward window opening angle simulation is not shown.
Figure 5-17: Vector plot of velocity for a slice located along the leeward facing windows in a
simulation containing closed windward facing windows
obvious is in the first row and first column of greenhouse simulations, representing closed leeward,
D.E. Dwyer
Master Thesis
5-1 Verification
41
and windward-facing windows, respectively.
When the windward-facing windows are closed, there is no inflow into the greenhouse. These results
show that the continuity equation is not satisfied. A closer look at the vector plot for a slice along
the leeward facing window in a simulation where the windward facing windows are closed (Figure
5-17) further validates this observation that purely outflow occurs in leeward facing windows.
(a)
(b)
Figure 5-18: f
or the greenhouse geometry of (a) a simulation with 53 degree windward and leeward facing windows and (b) a simulation with closed windward and 30 degree leeward facing
windows]Velocity boundary condition for the greenhouse geometry of
(a) a simulation with 53 degree windward and leeward facing
windows and (b) a simulation with closed windward and 30
degree leeward facing windows
This is caused by erroneous initialization of the flow field. Rather than just initializing the flow
field for each window configuration, it is evident that the boundary condition of the greenhouse is
changed from no slip conditions to the velocity of the initialized field. Therefore, a closed window
acts as an imposed velocity directed into the greenhouse. A window open 100% imposes a velocity
on the external flow of the greenhouse, rather than acting as a barrier to flow. The velocity boundary
condition of the greenhouse structure for these two situations are presented in Figure 5-18. The
imposed velocity where a closed window is expected results in outflow through the window openings
in order to satisfy continuity.
The ’constant’ case is still expected to have a mean flow rate of zero, given that no initialization error
occurs at the greenhouse geometry for this case. However, when the volumetric flow rate at each
window is summed, the result is a 14.2 m3 /s net inflow. This could be due to sampling limitations.
The values sampled at the greenhouse windows are not evenly distributed over the window, while the
volumetric flow rate in each is determined using an average velocity of the sampled values in each
window. No leaks were found in the greenhouse geometry (aside from intentional window openings),
and the boundary condition of the greenhouse was set to no slip conditions.
The inability to verify mass conservation consistency indicates that none of the data from the variable
cases can be evaluated further due to erroneous initialization. Following the examination of spacial
convergence, temporal convergence, model implementation, iterative convergence, and consistency,
the only data that can be verified is the ’constant’ case.
Master Thesis
D.E. Dwyer
42
Verification and Validation of Results
5-2
Validation
While verification showed that the constant simulation has converged, numerically, validation is
necessary in order to show that the results are physically realistic.
Several experimental greenhouse ventilation studies have found consistent results with respect to
inflow and outflow windows. They have found that airflow generally enters through downwind
ventilators and exits through upwind ventilators, irrespective of wind speed and window opening
angle [35] [36]. This pattern is observed in the results presented in the previous section.
Quantitative validations must compare the results to wind tunnel tests, or field tests where the
velocity is large enough (> 2m/s) that the buoyancy effect is negligible. While experimental field
tests lack control over the external wind conditions (i.e. constant wind direction and wind velocity)
to match with the present study, wind tunnel tests provide more control in this regard.
Given that the wind-induced pressure distribution over the greenhouse roof is the driving force for
ventilation (neglecting buoyancy), Mistriotis et al. suggest that correct pressure differences predicted
by CFD are an indication that ventilation pressures are accurately modeled [37]. This is a common
practice within the greenhouse climate modeling sector, and is used to validate the ventilation models
used to develop a ventilation boundary condition. Experimental measurements were collected by
Wells and Hoxey [38] for a Venlo greenhouse with the same roof pitch as the greenhouse used
for the calculations of this study. The experimental measurements correct for fluctuations in wind
direction by assuming a quadratic relationship between pressure coefficient and wind direction. The
weighted function was derived by minimizing residuals between wind load and (Cp ∆q). Figure 519(a) presents a comparison of the pressure coefficients for a closed roof calculation of the geometry
used in this study.
While this validation method shows good agreement, the previous discussion of CFD turbulence
calculation methodologies (RANS versus LES) indicates that the actual velocities calculated can
still be inaccurate.
0
0
Pressure Coefficient
−0.5
−1
−1.5
CFD Calculation (15 degree window openings)
Wells and Hoxey Experimental Measurements (5.4 degree window openings)
Ventilator Pressure Coefficient
CFD Calculation
Wells and Hoxey Experimental Measurements
5
10
15
20
25
30
35
Distance from windward edge [m]
(a)
40
45
50
55
−0.5
−1
−1.5
5
10
15
20
25
30
35
Distance from windward edge [m]
40
45
50
55
(b)
Figure 5-19: Pressure coefficient validation comparing the pressure distribution over a closed greenhouse roof (a) and over the window
openings along a greenhouse roof (b)[38]
Wells and Hoxey [38] also determined a constant pressure coefficient for open windows (at a 5.4
degree opening angle). Despite a 24.6 degree difference in window opening angle from the measured
data, Figure 5-19(b) compares the pressure coefficient to the CFD calculation. The values are
plotted to the same scale as (a) to show that the pressure varies much less over window ventilators.
When compared to the mean value for this window configuration determined by Wells and Hoxey[38],
the measured pressure coefficient, -0.25, falls within one standard deviation of the mean calculated
pressure coefficient at the windows, −0.27 ± 0.10. Furthermore, in order for flow to move across
D.E. Dwyer
Master Thesis
5-3 Results
43
the length of the greenhouse from the windows downwind to windows upwind, a gradient in the
pressure distribution over the windows is necessary. A constant pressure distribution determined by
Wells and Hoxey is not an accurate estimation.
While these measurements have been widely used for CFD ventilation calculation validation, it is
also important to note that the values are intended for predicting structural stability of greenhouses
due to wind loading.
5-3
Results
Following the verification and validation, this section presents a discussion of the resulting flow
patterns of the ’constant’ case. None of the variable cases will be evaluated due to the inability to
verify the results.
Flow Field
The following figures present the latest time step calculated for a plane perpendicular to the greenhouse ridges located along the windward window openings. Figure 5-20 shows flow separation
occurring, with a clear boundary layer over the greenhouse. Introducing the greenhouse with its
pitched roofs and window flaps into the flow alters the structure of the boundary layer near the
ground, increasing wall shear stress and skin friction.
This plot also indicates the flow within the greenhouse is opposite to the flow in the external
environment, flowing from right to left within the greenhouse, and from left to right outside the
greenhouse. Figures 5-20 and 5-22 both show air entering through the vents downwind and near the
center of the greenhouse, while wind exits through upwind vents. There still exists both inflow and
outflow in each window. The z velocity plot shows the very first window having much higher inflow
and outflow magnitudes when compared with the other windward facing windows in the contour
plot.
The y component of velocity presented in Figure 5-21 shows low velocities across the width of the
greenhouse. As expected, y velocities are very low, yet present due to the influence of the staggered
leeward-facing windows which act as an outlet to the windward-facing velocity field presented in the
figure.
Figure 5-20: x component of velocity flow field [m/s] for the ’constant’ simulation
The vector plot colored according to velocity magnitude in Figure 5-23 provides more insight into
the velocities in the y direction. The figure shows the velocity at a plane (a) 1 meter and (b) 3.7
meters from the floor of the greenhouse (1 meter below the midpoint of the windows). Vectors are
oriented based on velocity direction. In both figures, the highest velocities occur near the walls.
Master Thesis
D.E. Dwyer
44
Verification and Validation of Results
Figure 5-21: y component of velocity flow field [m/s] for the ’constant’ simulation
Figure 5-22: z component of velocity flow field [m/s] for the ’constant’ simulation
In addition to internal flow moving opposite to the external wind direction, vortices can be seen in
the corners of the greenhouse near the floor. There is a slightly diagonal flow pattern across the
greenhouse.
Closer to the windows, in Figure 5-23(b), the influence of the window direction on the flow patterns
within the greenhouse can be seen. Velocities just below windward facing windows generally point in
the in the +x direction (particularly at the downwind end of the greenhouse), while near the leeward
facing windows the flow generally is directed upwind inside the greenhouse (particularly near the
center span of the greenhouse). Flow near the leeward-facing windows, however, is less consistent
with regard to direction, indicating turbulent flow activity in these regions. The large vortices on
the windward side of the greenhouse extend over the height of the greenhouse, spanning from the
floor to the region just below the windows. These vortices occur as a result of flow directed towards
the edges as it enters the first row of windward facing windows, and flow directed to the center of
(a)
(b)
Figure 5-23: Velocity vector field at a plane parallel to the ground located (a)
1 meter from the ground and (b) 3.7 meters from the ground,
colored by velocity magnitude [m/s]
D.E. Dwyer
Master Thesis
5-3 Results
45
the greenhouse at windward facing windows along the edges.
Window Fluxes
A closer look at the flux through the windows of the ’constant’ case can provide further detail about
the greenhouse ventilation. Figure 5-24 presents the volumetric flow rate at the windows of the
’constant’ case.
Volumetric Flow Rate [m3/s]
2
1
0
−1
−2
−3
Figure 5-24: Volumetric flow rate at windows of the ’constant’ case in m3 /s
With wind moving in the +x direction, the windward facing windows of the first span where the wind
first interacts with the building shows a very high volumetric inlet flow rate and the corresponding
leeward facing windows on the span show high outlet rates. The windward facing windows have a
pattern of initial high volumetric inlet flow rate at the first span followed by a net air outlet from
the windward facing windows upwind. From the center to the downwind spans, the inlet flow rate
gradually increases through the windward facing windows. The air inlet rate is also unusually high
on the edges of the span, while the leeward facing windows show a initial high outflow rate, followed
by a decrease in outflow towards the center and increase in outflow rates at the downwind leeward
facing windows.
Over the latest 30 seconds of stabilized computation time, the time-averaged velocity magnitude
through the windows, normalized by the free stream velocity at the window height ranged from
2.76E-3 to 1.12, averaging 0.60. The standard deviations of the velocity vector components gives
an indication of turbulence effects. Standard deviations for the x, y, and z components averaged
4.82E-3, 4.25E-3, and 1.69E-3, respectively.
(
Pressure Field)
The pressure plot in Figure 5-25 shows the stagnation point, where the free flowing wind first
meets the greenhouse at the left side. The very small pressure difference across the length of the
greenhouse induces the flows from high to low pressure (in the −x direction) observed in the velocity
plots. Furthermore, despite the generally higher pressure within the greenhouse, the pressures along
the outside of the roof adjacent to windows where air enters the greenhouse are higher than the
pressures inside the greenhouse, inducing flow into the greenhouse.
Master Thesis
D.E. Dwyer
46
Verification and Validation of Results
Figure 5-25: Pressure field of the ’constant’ simulation (p/ρ [m2 /s2 ]). White circles identify
a pressure difference between the inside and outside of the greenhouse, driving airflow into the
greenhouse
Turbulence Properties
The turbulent kinetic energy depicted in Figure 5-26 indicates there are large velocity fluctuations
at the first two spans of the greenhouse, where the wind first interacts with the structure. The low
turbulent kinetic energy within the greenhouse indicates fairly stable flow within the greenhouse,
with the largest effect occurring as a result of the roof spans and window flaps.
Figure 5-26: Results for turbulent kinetic energy field
The turbulent kinetic energy dissipation, in Figure 5-27, is also fairly high at the front edges of
the greenhouse, preventing the turbulent fluctuations from spreading far beyond this region. A high
turbulent kinetic energy dissipation is expected in regions with higher velocities (and resulting kinetic
energy) and smaller turbulent eddy length scales.
Figure 5-27: Turbulent kinetic energy dissipation field for the ’constant’ case
D.E. Dwyer
Master Thesis
Chapter 6
Boundary Analysis
While errors in the initialization of variable simulations prevent further analysis in this work, this
section presents a discussion of how valid and verified results could be used to formulate a greenhouse
ventilation boundary condition. The methodology for developing a boundary condition can be broken
down into two steps: First, an analysis of which parameters must be assigned to the ventilation
boundary of an internal greenhouse CFD calculation is needed in order to maintain a stable simulation
and produce a physically meaningful result. Next, the assigned values for the selected parameters
are determined based on the results of the test calculations.
6-1
Selecting parameters
When calculating flows in a greenhouse as incompressible and neglecting the effects of buoyancy, the
modeled parameters are pressure (normalized by density, m2 /s2 ), velocity, turbulent kinetic energy,
and turbulent kinetic energy dissipation. It is not necessary to prescribe values for all parameters,
as this can over-constrain the calculation. For example, pressure and velocity are coupled in the
momentum equation. Therefore it is necessary to prescribe one of the two values at a boundary.
Given that velocity is a vector of three components, and pressure is a scalar, pressure is the ideal
parameter of the two to prescribe at the boundary. Furthermore, the continuity equation requires
the net flux in and out of the greenhouse to be zero. The pressure and velocity boundary conditions
must not over constrain this requirement. Turbulent eddies are present at the windows, requiring
turbulence properties to be specified at the boundary as well. Turbulent kinetic energy, turbulent
kinetic energy dissipation, and velocity are also interrelated, and it is therefore not possible to
prescribe all three at a single boundary.
Several tests investigated the parameters to be implemented at the boundary of an internal greenhouse model using the mapped values from the CFD simulation results. With pressure driven flow
being the most ideal, two dimensional tests were initially simulated to determine whether a boundary condition was feasible before adding the complexity of 3-dimensionally staggered greenhouse
windows. Three pressure boundary options were tested in a simplified two-dimensional calculation
containing two windows:
• A "pressure directed inlet velocity" boundary condition determines velocity based on static
pressure and a specified velocity direction vector
• Static pressure
Master Thesis
D.E. Dwyer
48
Boundary Analysis
• Total pressure
The "pressure directed inlet velocity" is less ideal for a boundary condition, because it requires
the specification of a velocity direction vector in addition to the static pressure (four components
altogether). The most stable calculation resulted in a combination of total pressure at the inlet
boundary face and static pressure prescribed at the outlet boundary face. However, testing this
combination of boundary conditions for internal flow of a greenhouse using mapped pressure values
(from an external environment calculation) resulted in an unstable calculation. This test was unstable
because of the complexity of introducing 134 windows distributed over a greenhouse roof in three
dimensions.
Prescribing velocity over the window boundaries was also not possible, because continuity (Equation
2-9) must be satisfied. The prescribed velocity must be very accurate. Even a small roundoff error can
easily prevent continuity from being achieved. Therefore, as is common practice in Computational
Fluid Dynamics (CFD) calculations, a combination of velocity and pressure were prescribed at the
boundaries of the internal greenhouse model.
In general, pressure is prescribed at an outlet, and velocity at the inlet. However, most of the
windows act as both an inlet and outlet. Several combinations of pressure and velocity boundaries
were simulated to determine the optimal combination. It was determined that the calculation is less
stable when pressure is prescribed on the majority of windows because flow direction is only specified
for a single window. With this in mind, the following two configurations were tested:
• Pressure prescribed at a single window and velocity prescribed at all other windows;
• Pressure prescribed at four windows which act as a net outlet and velocity prescribed at all
other windows;
The internal flows that result from these mapped calculations give similar results to the initial
calculations which simulate the entire external environment. However, large errors exist in both
cases. Recalling back to the flux analysis of the constant case, there was a very large unphysical net
inflow into the greenhouse of 14.2 m3 /s. This large inflow is also captured when the velocities at
the windows were interpolated to the closed greenhouse model. In order to satisfy continuity, the
window(s) in which velocity is not fixed must correct for the large inflow. In the first case, when the
pressure is prescribed at a single window, an extreme outlet velocity is observed. The second case,
in which pressure is prescribed at four net outlet windows resulted in larger errors at the downwind
side of the greenhouse, where the pressure boundaries were specified. A conclusion of the two tests
is that the pressure inlet windows should be selected according to the desired error distribution.
The figures below show the two tests that determine where the ideal pressure prescribed window
should be located. The middle row of Figure 6-1 identifies a window which behaves mostly as an
inlet, while the bottom row identifies four leeward-facing windows that act mostly as outlets. The
results show that the inlet window becomes an outlet with a very high velocity, while the outlet
window selections remain outlets, but simulate a higher velocity than expected. Furthermore, the
pressure within the greenhouse increases for a pressure prescribed on an inlet window, while the
pressure decreases within the greenhouse when the pressure is prescribed on the outlet windows.
With the goal of replicating the internal flows, the internal flows are shown in Figures 6-2 and 63. The velocity flows and pressure distribution for the slide single inlet pressure specification are as
expected, while large errors appear to be present when pressure was specified on the selection of four
outlet (leeward-facing) windows. It is important to note that despite different pressures within the
greenhouse, the distribution remains the same for the single window pressure prescription, resulting
in the same velocity field as expected.
D.E. Dwyer
Master Thesis
6-1 Selecting parameters
49
Figure 6-1: Selecting boundary condition parameters: two tests show the results of prescribing a
mapped (top row) pressure on a single window (middle row) or selection of windows (bottom row),
while velocity is prescribed on all others
Figure 6-2: Selecting boundary condition parameters: Results of prescribing a mapped (top) pressure on a single window (middle row) or selection of windows (bottom row), while velocity is
prescribed on all others
Despite the similarities in the flows shown for prescribing the pressure at the single window, there
remains high errors near the window. However, the other test produced errors in the vicinity of
all four pressure-prescribed windows. To minimize this error, different numerical schemes were
Master Thesis
D.E. Dwyer
50
Boundary Analysis
Figure 6-3: Selecting boundary condition parameters: results of prescribing a mapped (top) pressure
on a single window (middle row) or selection of windows (bottom row), while velocity is prescribed
on all others
tested (additional non-orthogonal correctors and higher order gradient schemes), as well as different
boundary condition types (i.e. ’pressure directed inlet velocity’, static pressure, and total pressure).
Specifying the direction of the velocity at the pressure-prescribed boundary, for example, appears
not to satisfy continuity.
Tests were also done to determine that optimal results occur when both turbulent kinetic energy
and turbulent kinetic energy dissipation are prescribed at all boundaries.
6-2
Parameter value specification
The complex air exchanges that occur between the staggered windward and leeward windows requires
a simplified method for predicting and prescribing boundary conditions at the windows.
There are two options for specifying parameters at the ventilation boundary condition. The first
option interpolates measurements of the test calculations to generate an estimation of the boundary
conditions for a specified window opening angle and wind azimuth angle. This option does not
provide flexibility with respect to the greenhouse geometry and window geometries. The second
option develops trends using regression analysis to determine the estimated ventilation boundary
condition. This option has the potential of applying for a wider range of greenhouse geometries.
Both options are investigated in this study.
6-2-1
Interpolation
Interpolation of the boundary condition is done by sampling the boundary fields of external flow
simulations with slightly larger and slightly smaller window opening angles, and using these interpolated values to prescribe the boundary conditions of an internal greenhouse flow calculation. This
D.E. Dwyer
Master Thesis
6-2 Parameter value specification
51
method makes the assumption that a linear trend exists between the boundary conditions in two
simulations of window opening angle, which was not observed to realistically portray the window
opening trends, according the multiple regression analysis in the following section.
6-2-2
Multiple regression analysis
Multiple regression analysis is used to fit data to a trend and determine the relationship between
dependent (response) variables and independent (predictor) variables. In this case, a multiple regression analysis is performed for each of the boundary variables, as well as for the pressure coefficient
difference when comparing the value at an open window to a closed window. The pressure and
velocity response variables are analyzed as non-dimensional variables, in order to scale boundary
values according to different wind velocities and outdoor pressure conditions. A stepwise method
can be used to determine statistically significant relations and interactions of predictor variables.
This method, available in the Matlab statistics toolbox, begins with an assumed trend (i.e. constant, quadratic) between the predictor and response variables, along with interaction terms among
the predictor variables. The method iteratively removes (or adds in the case of a constant base
trend) coefficients that are insignificant according to the p-value discussed in the stationarity test
in the previous chapter, and recalculates the coefficients until all remaining terms are statistically
significant.
Five predictor variables to be assessed:
• Span number (a categorical version of x-coordinate)
• Y-coordinate
• Height (z-coordinate)
• window direction (leeward-facing or windward-facing)
• windward window area
• leeward window area
Span number was selected because it is typical for pressure distribution over a greenhouse to be
determined according to the span number. Furthermore, for greenhouses with many spans, central
spans often exhibit the same behavior, making it easier to match this model to other greenhouse
geometries. With the window plane at a tilt, there is a strong correlation between the fixed x and
z coordinates of points in the window boundary. There is furthermore a correlation between the x
coordinate and window direction, which alternates along the x axis. Therefore, the x-coordinate is
represented as the categorical "Span number" predictor variable, which eliminates these dependencies. The z-coordinate is expected to indicate trends over the height of each individual window. The
observed inlet and outlet tendencies of the window flux in Figure 5-24 suggest that window direction
is a significant nominal variable to test.
The reported results from literature has shown strong trends in window direction as well as significance of window angle. Window direction is to be represented as a nominal predictor, generating
a separate trend for leeward and windward facing windows. Stronger regression models can be produced using the effective window area in comparison to window opening angle, as expected, given
that ventilation rates are directly proportional to vent area. The effective window area is calculated
as
A=W
Master Thesis
q
2L2 (1 − cos(α)) + L2 sin(α)
(6-1)
D.E. Dwyer
52
Boundary Analysis
where L is the length of the window, W is the width of the window (along the width of the
greenhouse), and α is the window opening angle from the plane of the greenhouse roof. A similar
technique was used by Bailey et. al. [39] to determine the effect of window opening angles on
ventilation rate.
Assuming Reynolds independence, a common assumption used in wind engineering, the effect of
different wind speeds can be determined by scaling the velocity. Therefore, the non-dimensional
velocities should be modeled in the regression analysis.
Given the concurrent initiative to develop a "Cp Generator" at TNO, which estimates the pressure
distribution over the roof and walls of various building geometries (without windows), a regression
model can also be developed for the pressure coefficient difference that results from opening windows
along the surface of the greenhouse geometry.
D.E. Dwyer
Master Thesis
Chapter 7
Recommendations
While it is unfortunate that the variable simulations of window opening angles and wind directions
were not verifiable, many lessons were learned that can further improve the implementation of these
calculations in future work. These simulations can be improved by ensuring the wall boundary
conditions are unchanged during initialization. Furthermore, the linear solvers can be updated using
the suggestions in Section 4-2-1 to speed up the calculations. Other forms of initialization are also
possible, such as beginning the calculation on a coarser mesh before mapping the results to the finer
mesh, or using the PotentialFoam solver to generate the starting fields for the full Navier-Stokes
PisoFoam solver.
Simulations of other external parameters that influence ventilation (identified in Section 3-1) can
greatly improve this work. Looking back on the work, it would be be beneficial to find a correlation
between greenhouse geometry and ventilation as well. Or at the very least, implement these boundary
conditions on a greenhouse with different geometry and validate the results.
A possibility for improving the calculations further would be the much more computationally expensive Large Eddy Simulation (LES) methodology, using a simplified greenhouse geometry. This
method may more accurately model the external and internal greenhouse flow given the range of
turbulence length scales that are limited when using the Reynolds Averaged Navier-Stokes (RANS)
methodology.
Validation of the simulations using either experimental velocity values for various window opening
angles or in a controlled wind tunnel simulation would improve the confidence in the various window
opening angle simulations.
Implementation of the boundary condition and the accuracy of such simulations are highly dependent
on the configuration of pressure and velocity boundaries. A solution to this problem could be to alter
the solver so that it distributes the concentration of error (correcting for continuity) that presently
only exists at the pressure boundary.
Master Thesis
D.E. Dwyer
54
D.E. Dwyer
Recommendations
Master Thesis
Chapter 8
Conclusion
This graduation project focussed on developing a ventilation boundary condition for a horticultural
greenhouse climate model with the goal of replicating the internal flows of the greenhouse. The
selected boundary condition variables were window opening angle and wind direction. The method
used to develop this boundary condition was through simulation of these variables in an external
flow environment using Reynolds Averaged Navier-Stokes (RANS) turbulence modeling. A thorough
verification of the resulting simulations examined spatial convergence, temporal convergence, model
implementation, iterative convergence, and consistency.
The results of the wind azimuth angle simulations were found to not yet be completely converged.
Evidence of this included negative pressure force for one of the simulations, high residual error, failure
of a statistical stationarity test, and strong correlations between the initialized conditions and final
values. However, while the results of varied window opening angles reached reasonable convergence,
they failed to satisfy continuity, leading to the further conclusion that initialization errors were also
present in these simulations.
The resulting flows for the constant case were consistent with qualitative flow trends reported in
other greenhouse studies. Flow typically enters through the downwind windows and exits through
upwind windows. This flow behavior also induced large eddies in the corners of the greenhouse
near the windward side. The pressure distributions of this case were validated using Wells and
Hoxey’s experimental pressure distributions [38]. A valid pressure distribution indicates that the
pressure-induced ventilation calculations should also be accurate.
The complexity of an internal greenhouse Computational Fluid Dynamics (CFD) model due to the
extensive window layout required boundary specifications of both pressure and velocity at different
greenhouse windows. Various configurations were tested for pressure and velocity specification
along the greenhouse ventilation boundary faces, resulting in concentrated error near the pressureprescribed windows. With more pressure-prescribed windows, the erroneous flow was less extreme,
distributed among the windows.
Two methods, interpolation and regression analysis, were discussed as methods for implementing
a boundary condition at the greenhouse windows. Regression modeling can be used to predicted
pressure and velocity distributions using the span number (an ordinal variable), y position, height
(z position), window direction, and effective windward and leeward window opening areas. This
method has the potential to eventually provide more flexibility in greenhouse geometry.
While errors were present when implementing a boundary condition for the ’constant’ case, there remains the benefit that reasonable internal flow may be simulated without the computational cost and
Master Thesis
D.E. Dwyer
56
Conclusion
time of simulating the entire external environment. Approximate solutions obtained with numerical
methods inherently contain discretization and modeling errors. When compared to ventilation calculations in an operational greenhouse, uncertainties and errors are unavoidable in such uncontrolled
environments. Therefore, for the purpose of estimating the general flows within a greenhouse, this
model is sufficient. When monitoring the climate at a precise location (rather than a general region
in the greenhouse) and when using model values for climate control applications, a more accurate
measurement is necessary. Further improvements are needed for applications requiring more accurate
estimates of flow within the greenhouse.
D.E. Dwyer
Master Thesis
Appendix A
Appendix: Boundary Conditions
Master Thesis
D.E. Dwyer
D.E. Dwyer
m/s
fixedValue (0 0 0)
fixedValue (0 0 0)
fixedValue (0 0 0)
zeroGradient
zeroGradient
fixedValue (0 0 0)
atmBoundaryLayerInletVelocity
inletOutlet
Dimensions
Greenhouse
Floor
Greenhouse
Ceiling
Greenhouse
Walls
Atmosphere
Surroundings
Earth
Inlet
Outlet
Velocity, U
zeroGradient
fixedValue 0.50497
kqRWallFunction
zeroGradient
zeroGradient
kqRWallFunction
kqRWallFunction
Table A-1: Boundary conditions
fixedValue 100000
zeroGradient
zeroGradient
zeroGradient
zeroGradient
zeroGradient
zeroGradient
kqRWallFunction
m2 /s2
m2 /s2
zeroGradient
Turbulent Kinetic Energy,
k
Pressure, p
zeroGradient
atmBoundaryLayerInletEpsilon
epsilonWallFunction
zeroGradient
zeroGradient
epsilonWallFunction
epsilonWallFunction
epsilonWallFunction
m2 /s3
Turbulent Kinetic Energy Dissipation, 58
Appendix: Boundary Conditions
Master Thesis
Appendix B
Appendix: Calculation Details
Finite Volume Calculation Schemes
2
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fvSchemes ;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* //
16
17
18
19
20
ddtSchemes
{
default
}
backward ;
21
22
23
24
25
gradSchemes
{
default
}
Gauss linear ;
26
27
28
29
divSchemes
{
default
Master Thesis
none ;
D.E. Dwyer
60
Appendix: Calculation Details
div ( phi , U )
Gauss limitedLinearV 1 . 0 phi ;
div ( ( nuEff ∗ dev ( T ( grad ( U ) ) ) ) )
Gauss linear ;
div ( phi , epsilon )
bounded Gauss linear ;
div ( phi , k )
bounded Gauss linear ;
30
31
32
33
34
}
35
36
37
38
39
laplacianSchemes
{
default
}
Gauss linear limited corrected 0 . 3 3 3 ;
40
41
42
43
44
interpolationSchemes
{
default
}
linear ;
45
46
47
48
49
snGradSchemes
{
default
}
limited corrected 0 . 3 3 3 ;
50
51
52
53
54
55
fluxRequired
{
default
p;
}
no ;
56
57
58
//
*************************************************************************
//
Finite Volume Solution Methods
2
/* --------------------------------*- C++
-*----------------------------------*\
| =========
|
3
| \\
4
|
5
|
1
6
7
8
9
10
11
12
13
14
15
|
/
\\
/
\\
\\/
/
F ield
|
O peration
|
| OpenFOAM : The Open Source CFD Toolbox
A nd
| Version :
|
| Web:
|
M anipulation
|
2.2.0
www. OpenFOAM .org
|
\*--------------------------------------------------------------------------*/
FoamFile
{
version
2.0;
format
ascii ;
class
dictionary ;
object
fvSolution ;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* //
16
D.E. Dwyer
Master Thesis
61
17
18
19
20
solvers
{
p
{
solver
preconditioner
tolerance
relTol
21
22
23
24
PCG ;
DIC ;
1e −06;
0;
}
25
26
pFinal
{
solver
preconditioner
tolerance
relTol
}
27
28
29
30
31
32
33
PCG ;
DIC ;
1e −06;
0;
34
U
{
35
36
37
38
39
40
41
solver
smoother
tolerance
relTol
nSweeps
smoothSolver ;
GaussSeidel ;
1e −8;
0.1;
1;
solver
smoother
tolerance
relTol
nSweeps
smoothSolver ;
GaussSeidel ;
1e −8;
0.1;
1;
}
42
43
k
{
44
45
46
47
48
49
50
}
51
52
epsilon
{
solver
smoother
tolerance
relTol
nSweeps
}
53
54
55
56
57
58
59
60
61
smoothSolver ;
GaussSeidel ;
1e −8;
0.1;
1;
}
62
63
64
PISO
{
nCorrectors
3;
nNonOrthogonalCorrectors 1 ;
pRefCell
0;
pRefValue
0;
65
66
67
68
69
}
70
71
72
//
*************************************************************************
//
Master Thesis
D.E. Dwyer
62
D.E. Dwyer
Appendix: Calculation Details
Master Thesis
Bibliography
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vol. 13 of Crop Production Science in Horticulture, pp. 257–304, Wallingford, Oxfordshire
OX10 8DE, UK: CABI Publishing, 2005.
[2] W. Baltussen, “Greenhouse technology suppliers conquer the world,” Survey results of AVAG
Plus members 13-008, LEI Wageningen UR, Wageningen, March 2013.
[3] P.-E. Bournet, “Assessing greenhouse climate using CFD: a focus on air humidity issues,” tech.
rep., Agrocampus Ouest, Angers, France, 2013.
[4] P.-E. Bournet, S. Ould Khaoua, T. Boulard, C. Migeon, and G. Chassériaux, “Effect of roof
and side opening combinations on the ventilation of a greenhouse using computer simulation,”
Transactions of the ASABE, vol. 50, no. 1, pp. 201–212, 2007.
[5] T. Bartzanas, T. Boulard, and C. Kittas, “Effect of vent arrangement on windward ventilation
of a tunnel greenhouse,” Biosystems Engineering, vol. 88, no. 4, pp. 479–490, 2004.
[6] T. Boulard, H. Fatnassi, A. Kichah, F. Girardot, J. C. Roy, and I. B. Lee, Development and validation of a global CFD model of heat, water vapour and fungal spores transfers in a greenhouse,
vol. 801 PART 2 of Acta Horticulturae. 2008. Cited By (since 1996):1.
[7] E. Romano, F. Nucci, and C. Bisaglia, “A theoretical approach for a sensitive analysis of
the influences of external factors on greenhouse microclimate,” Acta Horticulturae, vol. 919,
pp. 81–88, 2011. cited By (since 1996)0.
[8] G. Bot, Greenhouse climate: from physical processes to a dynamic model. PhD thesis, Wageningen University, 1983. cited By (since 1996)1.
[9] J. de Jong, Natural ventilation of large multi-span greenhouses. PhD thesis, Wageningen
University, 1990. cited By (since 1996)73.
[10] A. Mistriotis, G. Bot, P. Picuno, and G. Scarascia-Mugnozza, “Analysis of the efficiency of
greenhouse ventilation using computational fluid dynamics,” Journal of Agricultural Engineering
Research, vol. 85, pp. 217–228, 1997.
[11] T. Boulard and A. Baille, “Modelling of air exchange rate in a greenhouse equipped with
continuous roof vents,” Journal of Agricultural Engineering Research, vol. 61, no. 1, pp. 37–48,
1995.
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Bibliography
[12] P.-E. Bournet and T. Boulard, “Effect of ventilator configuration on the distributed climate
of greenhouses: A review of experimental and CFD studies,” Computers and Electronics in
Agriculture, vol. 74, no. 2, pp. 195 – 217, 2010.
[13] M. Teitel, G. Ziskind, O. Liran, V. Dubovsky, and R. Letan, “Effect of wind direction on greenhouse ventilation rate, airflow patterns and temperature distributions,” Biosystems Engineering,
vol. 101, pp. 351–369, 2008.
[14] H. Fatnassi, T. Boulard, C. Poncet, and M. Chave, “Optimization of greenhouse insect screening
with computational fluid dynamics,” Biosystems Engineering, vol. 93, pp. 301–312, 2006.
[15] S. Ould Khaoua, P. Bournet, and G. Chassériaux, “Mathematical modelling of the climate
inside a glasshouse during daytime including radiative and convective heat transfers,” Acta
Horticulturae, vol. 718, pp. 255–262, 2006. cited By (since 1996)1.
[16] S. Reichrath and T. Davies, “Using CFD to model the internal climate of greenhouses: Past,
present and future,” Agronomie, vol. 22, no. 1, pp. 3–19, 2002. cited By (since 1996)65.
[17] J. Ferziger and M. Peric, Computational Methods for Fluid Dynamics. Springer-Verlag, third ed.,
2002.
[18] H. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics: The finite
volume method. Longman Group, 1995.
[19] C. Rhie and W. Chow, “Numerical study of the turbulent flow past an airfoil with trailing edge
separation,” AIAA Journal, vol. 21, no. 11, pp. 1525–1532, 1983.
[20] F. P. Kärrholm, “Rhie-Chow interpolation in OpenFOAM,” tech. rep., Chalmers University of
Technology, Göteborg, Sweden, 2006.
[21] R. Issa, “Solution of the implicitly discretized fluid flow equations by operator-splitting,” Journal
of Computational Physics, vol. 62, pp. 40–65, 1986.
[22] H. kan Nilsson, “A look inside icofoam (and pisofoam),” Presented at the MSc/PhD course in
CFD with OpenSource software, 2013.
[23] H. Jasak, Error analysis and estimation for the finite volume method with applications to fluid
flows. PhD thesis, Imperial College Londen, 1996.
[24] Y. Jiang and Q. Chen, “Study of natural ventilation in buildings by large eddy simulation,”
Journal of Wind Engineering and Industrial Aerodynamics, vol. 89, no. 13, pp. 1155–1178,
2001.
[25] B. Launder and D. Spalding, “The numerical computation of turbulent flows,” Computer Methods in Applied Mechanics and Engineering, vol. 3, no. 2, pp. 269–289, 1974.
[26] T.-H. Shih, W. Liou, A. Shabbir, Z. Yang, and J. Zhu, “A new k- eddy-viscosity model for
high rehnolds number turbulent flows - model development and validation,” Computers Fluids,
vol. 24, no. 3, pp. 227–238, 1995.
[27] KNMI, “Climatology: Daily weather data of the netherlands,” May 2014.
[28] “Gnu general public license, version 3.” http://www.gnu.org/licenses/gpl.html, June
2007. Last retrieved 2012-05-10.
[29] “Blatt 9: 2005-11, umweltmeteorologie - prognostische mikroskalige windfeldmodelle evaluierung für gebäude- und hindemisumströmung, beuth,” 2005. cited By (since 1996)1.
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[30] B. Blocken, S. Roels, and J. Carmeliet, “Modification of pedestrian wind comfort in the silvertop
tower passages by an automatic control system,” Journal of Wind Engineering and Industrial
Aerodynamics, vol. 92, no. 10, pp. 849–873, 2004.
[31] T. Wintergerste, M. Casey, and A. Hutton, “The best practice guidelines for CFD - a european
initiative on quality and trust,” in American Society of Mechanical Engineers 2002 Pressure
Vessels and Piping Conference, vol. 448, pp. 1–9, 2002. cited By (since 1996)1.
[32] P. Richards and R. Hoxey, “Appropriate boundary conditions for computational wind engineering
models using the k-epsilon model,” Journal of Wind Engineering and Industrial Aerodynamics,
1993.
[33] J. W. Slater, “Nparc alliance cfd verification and validation.” http://www.grc.nasa.gov/
WWW/wind/valid/, March 2012. Last retrieved 2014-08-10.
[34] T. Norton, D.-W. Sun, J. Grant, R. Fallon, and V. Dodd, “Applications of computational fluid
dynamics (CFD) in modelling and design of ventilation systems in the agricultural industry: A
review,” Bioresource Technology, vol. 98, no. 12, pp. 2386–2414, 2007.
[35] S. Wang and J. Deltour, “Lee-side ventilation-induced air movement in a large-scale multi-span
greenhouse,” Journal of Agricultural Engineering Research, vol. 719, pp. 103–110, 1999.
[36] T. Boulard, G. Papadakis, C. Kittas, and M. Mermier, “Air flow and associated sensible heat
exchanges in a naturally ventilated greenhouse,” Agricultural and Forest Meteorology, vol. 88,
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[37] A. Mistriotis, T. de Jong, M. Wagemans, and G. Bot, “Computational fluid dynamics (cfd)
as a tool for the analysis of ventilation and indoor microclimate in agricultural buildings,”
Netherlands Journal of Agricultural Science, vol. 45, pp. 81 – 96, 1997.
[38] D. Wells and H. Hoxey, “Measures of wind loads on full-scale glasshouses,” Journal of Wind
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[39] B. Bailey, A. Robertson, and A. Lockwood, “The influence of wind direction on greenhouse
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2003. cited By (since 1996)73.
Master Thesis
D.E. Dwyer
66
D.E. Dwyer
Bibliography
Master Thesis
Glossary
List of Acronyms
ABL
Atmospheric Boundary Layer
CFD
Computational Fluid Dynamics
DIC
Diagonal Incomplete-Cholesky
DNS
Direct Numerical Simulation
GAMG
Generalized Geometric-Algebraic Multi-Grid
GPL
General Public License
KKM
Kasklimaat Model
KNMI
Koninklijk Nederlands Meteorologisch Instituut
LES
Large Eddy Simulation
PBiCG
Preconditioned bi-Conjugate Gradient
PCG
Preconditioned Conjugate Gradient
PDEs
Partial Differential Equations
PISO
Pressure Implicit with Splitting of Operators
RANS
Reynolds Averaged Navier-Stokes
RNG
Re-Normalization Group
STL
Stereolithography
TNO
Netherlands Organization for Applied Scientific Research
VDI
The Association of German Engineers
Master Thesis
D.E. Dwyer
68
Glossary
List of Symbols
Γϕ
ϕ
Diffusion coefficient
Transported quantity in non-dimensional form
∆P0
∆Ps
∆Pw
∆T
ρ
ζ
Cp
g0
h
K
k
P
P0
Sϕ
U
U
U
u
u∗
U0
Uh
V
W
z
z0
C
h
p
r
Ri
T
Pressure distribution at ground level
Buoyancy pressure drop
Wind pressure drop
Temperature difference between inside and outside
Density
Pressure drop coefficient
Pressure coefficient
Reduced gravity
Reference height
Von Karman constant
Turbulent kinetic energy
Static pressure at some point in the flow
Static pressure at the undisturbed stream
Source term
Velocity
Velocity
Velocity vector in the x direction
Wind speed
Friction velocity
Velocity of the undisturbed stream
Velocity at reference height
Velocity vector in the y direction
Velocity vector in the z direction
Vertical height
Surface roughness length
Constant
Measure of grid spacing
Order of convergence
Grid refinement ratio
Richardson number
Outside temperature
D.E. Dwyer
Master Thesis
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