Solution - Institute for Computational Physics

Solution - Institute for Computational Physics
Simulation Methods in Physics II
SS 2015
Worksheet 5:
Hydrodynamics and the
Lattice-Boltzmann-Method
Solutions
Georg Rempfer
Johannes Zeman
July 23, 2015
Institute for Computational Physics, University of Stuttgart
Contents
1 Introduction
2
2 Short Questions - Short Answers
2
3 Stokes’ Law, Computer Game Superheros and Low Reynolds Numbers
4
4 Simulation of Hydrodynamic Flow Around a Rigid Object
6
5 Visualization
7
5.1 Laminar Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.2 Turbulent Video . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 Unattended Simulations
12
6.1 Batched Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.2 Automatic Analyzation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Investigating the Force-Flowrate Relation
16
7.1 Analytical Theory for low Re . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.2 Results from the Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 18
8 Bonus Questions
20
1
Remarks
The Solutions provided here show possible approaches to solve the tasks from the corresponding worksheet and may not be exhaustive.
1 Introduction
In this worksheet, you will first have to answer some theoretical questions about hydrodynamics. In the remaining practical part, you will investigate the transition from
laminar to turbulent flow using one of the Lattice-Boltzmann implementations included
in ESPResSo. There are actually two implementations, one running on the CPU and
one running on the GPU. The GPU version is significantly faster (up to a factor of 50
depending on the graphics card), so you are better off using the GPU implementation.
There are two other topics this tutorial will cover, which are relevant to simulations in
general: visualization of your results with with high quality pictures and videos and how
to run simulations unattendedly over long periods of time.
2 Short Questions - Short Answers
Task
Answer the following questions:
(4 points)
• What could be considered the main advantages of the Lattice-BoltzmannMethod (LBM) over “classical” computational fluid dynamics (CFD)
methods (Finite Volume or Finite Elements Navier-Stokes solvers)?
• In which cases does the LBM not work?
• What’s the meaning of a Reynolds number and how is it defined?
• How is Stokes’ law defined and what does it describe?
Solution
• One major advantage of the LBM is its linear scalability in parallel computing,
since collisions are computed locally. This makes the LBM an ideal candidate for
GPU computing.
Also, multiphase flows are no big deal with LB. In fact, inter-phase interactions
are automatically included in the collisions.
Furthermore, the LBM facilitates the realization of boundaries for arbitrarily complex geometries.
2
• The LBM can only be used at small Mach numbers and low compressibilities.
Also, the method introduces an implicit compressibility, so that simulations of
completely incompressible fluids are not possible.
• The Reynolds number Re describes the ratio of inertial (momentum) to viscous
forces in a fluid under given flow conditions. It is defined as Re = vL
ν , where v is the
velocity of the considered object/boundary relative to the fluid, L its characteristic
length scale (“linear dimension”), and ν is the fluid’s kinematic viscosity. At low
Reynolds numbers (Re 1), where viscous forces dominate, the flow is laminar,
while at high Reynolds numbers (Re 1), the flow is turbulent.
• Stokes’ law describes the drag force (frictional force) exerted on a spherical object
in a fluid at low Reynolds numbers. It is defined as
F d = 6πµRv
,
where F d is the drag force acting on a sphere with radius R moving at velocity v
through a fluid with dynamic viscosity µ.
3
3 Stokes’ Law, Computer Game Superheros and Low Reynolds Numbers
Imagine you are in your living room after a long day of studying. As a refreshment,
you just got yourself a nice glass of delicious dihydrogen monoxide (H2 O), as suddenly
Duke Nukem kicks in the door, shouts “Hail to the king, baby!”, and fires at you with
his supercharged shrink ray. Ouch! Within the blink of an eye, you are shrunk to
micrometer size and, to complete your misery, fall into your own glass of water.
Fortunately, supercharged shrink rays cause the majority of the target’s mass to be
temporarily beamed to a parallel universe, so that the mass density of the target remains
unchanged. However, you just happen to wear that brand new heavy-duty spherical body
armor you got from www.gimmemoreofthatfreakygeekstuffilikesomuch.com, so your mass
density is about twice the density of H2 O.
Task
(5 points)
• Now being a sphere of 2R = 1 µm in diameter, how do you expect to
fall into the water? Will you plunge in smoothly or will you be stopped
rather abruptly? Give reasons for your answer!
• Use Stokes’ law in combination with Archimedes’ principle to calculate
kg
your terminal velocity under water. Assume ρs = 2000 m
3 (your density),
kg
m
(density
of
water),
g
=
9.81
(gravitational
acceleraρH2 O = 1000 m
3
s2
tion), and µ = 0.001 P a s.
• Why is Stokes’ law applicable in this case?
• As you just figured out, without doing anything, you will sink deeper and
deeper into the water. Since giving up and drowning is not an option, you
start swimming to get back to the surface. Is it better to use butterfly
strokes, breast strokes, or freestyle strokes to propel yourself upwards? If
you had the choice, would you prefer a paddle or a ship’s propeller to
facilitate moving?
• Could Stokes’ law still be applied in air? (µair = 1.8 × 10−5 P a s, ρair =
kg
µm
1.225 m
3 and v ≈ 1
s )
Hint
• In order to get an idea what this task is about, read Purcell’s (very entertaining!)
paper Life at low Reynolds number.
You can find it under /group/sm/2015/purcell77a.pdf.
4
Solution
• Since you are a micron-sized shpere, your Reynolds number is very low and inertial forces are negligible. This means that your velocity depends entirely on the
viscosity of the fluid within which you are moving. Thus, an abrupt change of
viscosity also means an abrupt change in velocity, so that you will be slowed down
rather abruptly when crossing the air/water interface.
• The terminal velocity is reached as soon as the force Fg caused by gravitational
acceleration and the viscous drag force Fd in the fluid cancel out:
!
|Fg | = |Fd |
According to Archimedes’ principle, the gravitational force is reduced by the buoyant force in the fluid, so that
4
|Fg | = (ρs − ρH2 O ) V g = (ρs − ρH2 O ) πR3 g
3
.
Plugging this into Stokes’ law and solving for v yields
v=
2g
µm
(ρs − ρH2 O ) R2 ≈ 0.545
9µ
s
• Since you’re in your spherical body armor, you are... right, you’re spherical. Your
2vRρH2 O
Reynolds number is Re =
= 0.545 × 10−7 (which can be considered as
µ
low), so it is justified to apply Stokes’ law.
• Using the butterfly stroke, only the whip-like motion of your body will be of
use, since the motion of your arms is pretty symmetrical. For breast strokes, the
situation is different, since the motion of your arms is not symmetrical. Moving
your arms forward, their effective cross section is smaller than in the backward
motion, so this should be less exhausting and would still propel you forward.
Mounting paddles to your armor like in a row boat would only allow completely
symmetric motion, so the ship propeller is the better option.
• In air, the Reynolds number for the given values is 6.806 × 10−8 , so Stokes’ law is
still applicable.
5
4 Simulation of Hydrodynamic Flow Around a Rigid Object
Now we will go back to more serious problems and perform hydrodynamic simulations
using the LBM. Our system consists of a pattern of 3d periodic stripes as obstacles,
surrounded by a newtonian fluid and a homogeneous force density acting on the fluid as
shown in fig. 1.
Figure 1: System to investigate. Light gray is the simulation volume with periodic boundaries and the LB fluid. Dark
grey represents an obstacle with no-slip boundary conditions and red stands for the applied constant force
density
Turbulence is a phenomenon comprising processes on very different time- and length
scales, which makes it hard to simulate with moderate computational effort. You are
welcome to experiment with the parameters to produce turbulence and any working set
of parameters will be appreciated, but be warned that this is a time-consuming task.
The following set of parameters will work for this task:
• fluid viscosity ν = 1.1
• fluid density ρ = 1
• fluid force density f = 0 ... 0.2
• fluid grid spacing a = 1
• time step τ = 0.01
• box size = 200 × 60 × 20
• obstacle dimensions = 10 × 30 × 20
You will need to compile ESPResSo with the following features defined in myconfig.h:
EXTERNAL_FORCES, LB, and LB_BOUNDARIES.
Should you want to take advantage of the GPU Lattice-Boltzmann implementation, you
need to compile ESPResSo with CUDA support:
$ > . / configure --with-cuda= / usr / local / cuda
6
To set up the simulation, you will need the following ESPResSo commands, apart from
the commands you already used in previous tutorials:
• thermostat off
• lbfluid <cpu|gpu> dens ρ agrid a tau τ visc ν ext_force f 0 0
• lbboundary rhomboid corner px py pz a ax ay az
b bx by bz c cx cy cz direction outside
Please consult the ESPResSo user’s guide if you need further information on those commands. The commands to produce output are given in section 5.1. Also note that
although we are not going to use particles, you will have to set a Verlet skin with
setmd skin 0.1
For the following simulations, you can use the Tcl scripts provided on the course website.
5 Visualization
For this part of the tutorial we are going to use paraview, an easy to use, open source
visualization program for scientific data. You should find a preinstalled version on the
CIP pool computers.
If you want to use it on other computers, it is part of most linux distributions’ repositories
or you can get a copy at http://www.paraview.org/. Versions up to 3.13 may contain a
bug that causes a memory leak while rendering movies. So if your repositories contain
a version up to that, better get the current release from the website.
5.1 Laminar Picture
Task
(2 points)
• Create a nice plot of the stationary state of a laminar flow field around
the obstacle and include it in your report. Using a body force density of
0.00001 should produce laminar flow.
Hints
• You can output the LB velocity field and the boundary geometry from ESPResSo
in a paraview compatible format with the following commands:
– lbfluid print vtk boundary filename
– lbfluid print vtk velocity filename
7
• For this tutorial, it will be sufficient to know about the 7 paraview controls shown
in fig. 2.
1
2
7
3
4
6
5
Figure 2: Paraview with a sample visualization and the most important controls.
1. Load data files into the pipeline for visualization
2. Adjust color settings of the filter selected in the pipeline browser
3. Add one of the most used filters to the element selected in the pipeline browser
4. The pipeline browser – shows loaded data files and filter applied to them for
visualization
5. Configuration panel for the filter chosen in the pipeline browser
6. The preview panel – you can rotate, zoom and move this with the mouse
7. Controls for videos
With a little exploration, these controls should be self explanatory. If you need
help, consult your tutor, the paraview help (F1) or the paraview online documentation at http://www.paraview.org.
8
Solution
Running the following code with ESPResSo will generate the required files for the visualization in paraview:
# s e t parameters
set force 0 .00001
set box_x 200
set box_y 60
set box_z 20
set visc 1 .10
set n_steps 40000
setmd box_l $box_x $box_y $box_z
setmd time_step 0 .01
setmd skin 0 .1
thermostat off
# s e t up LB
lbfluid gpu dens 1 agrid 1 tau $dt visc $visc ext_force $force 0 0
# s e t up o b s t a c l e
lbboundary rhomboid corner 5 [ expr $box_y /4] 0 a 10 0 0 \
b 0 [ expr $box_y /2] 0 c 0 0 $box_z direction outside
# s t o r e geometry o f box with o b s t a c l e f o r v i s u a l i z a t i o n
lbfluid print vtk boundary b.vtk
# run s i m u l a t i o n
integrate $n_steps
# store flow f i e l d
lbfluid print vtk velocity v.vtk
After loading the files in paraview, adding streamlines and tweaking the visualization
parameters, the resulting picture should look similar to figure 3.
Figure 3: Laminar flow visualization with streamlines.
9
5.2 Turbulent Video
Task
(2 points)
• Create a video of the time evolution of the system with a force density
from the upper end. Choose the simulation time between frames such
that the video looks smooth and the occuring turbulences are interesting
to watch. Using a body force density of 0.2 should produce turbulent flow.
Hints
• Outputting so much data significantly slows down the simulation and requires
roughly 6.2 MB per frame. If you don’t want to spend that much time on the
simulation and the rendering or do not have that much memory available, you can
just output a few hundred frames from time step 9000 on. On the ICP CIP pool
machines, consider running the simulations under /work/<username>.
• Creating videos with paraview works the same way as creating images. The only
difference is that you have to output one vtk file for every frame in ESPResSo
(about one frame every 10 integration steps will do) and include a counter in the
file names. You can then open all those files at once with paraview and use the
video controls to watch a preview of your animation.
You can save an animation through the "Files" menu in ogv format.
• If the resulting video is too big to hand in, store it on your CIP account and just
hand in the path. Also, there are example videos available on the website or under
https://youtu.be/A2zCOfrjH-Q and https://youtu.be/GYRRGuYoNF4.
10
Solution
The following ESPResSo script will do the job:
#s e t up s i m u l a t i o n
set box_x 200
set box_y 60
set box_z 20
set force 0 .2
set visc 1 .10
set n_cycles
4000 ;# measurement c y c l e s
set steps_per_cycle
10 ;# s t e p s p e r c y c l e
setmd box_l $box_x $box_y $box_z
setmd time_step 0 .01
setmd skin 0 .1
thermostat off
# s e t up LB
lbfluid gpu dens 1 agrid 1 tau $dt visc $visc ext_force $force 0 0
# s e t up o b s t a c l e
lbboundary rhomboid corner 5 [ expr $box_y /4] 0 a 10 0 0 \
b 0 [ expr $box_y /2] 0 c 0 0 $box_z direction outside
# s t o r e geometry o f box with o b s t a c l e f o r v i s u a l i z a t i o n
lbfluid print vtk boundary b.vtk
# run s i m u l a t i o n
for { set cycle 0} { $cycle < $n_cycles } { incr cycle } {
integrate $steps_per_cycle
lbfluid print vtk velocity v_$cycle.vtk ;# v i d e o f r a m e
}
11
6 Unattended Simulations
After having verified that the chosen system actually produces laminar and turbulent
flow, we want to quantitatively investigate the flow rate. In section 7.1 you will show
that for low forces, the net flow rate scales linearily with the force density. For flow
velocities producing a non-neglible convective contribution, however, there is no theory that predicts the scaling for such a complex geometry. We will therefore run the
simulation for many force densities ranging from f = 0.00001 to f = 0.2 . Since it is
a computation time intensive task to obtain data of sufficient resolution, you should
automize this process so that no user interaction is required.
Task
(1 point)
• Change the simulation script such that the force density can be specified as
a command line argument. Determine the time needed for one simulation
and adjust the given bash script such that it takes a couple of hours to
sample the specified range of force densities.
Hint
• Command line arguments can be accessed in an ESPResSo Tcl script through the
list $argv, and the number of passed arguments can be accessed through $argc.
Solution
The following code snippet reads in a command line argument and stores it in a variable
force. If no argument is given, a usage message is printed to the screen.
set force 0 .0
if { $argc > 0 } then {
set force [ lindex $argv 0]
} else {
puts " USAGE: ␣ Espresso ␣ template.tcl ␣ FORCE "
exit 1
}
12
6.1 Batched Execution
Running the same simulation several times while varying a parameter can easily be
achieved by writing a second program that executes the simulation in a loop and passes
the parameter as a command line argument. A simple bash script will suffice in this
case.
One problem is that the bash itself does not support calculations involving floating point
numbers. This can however be fixed by using awk. The following example demonstrates
how to do this:
f=0
f=$ ( awk " BEGIN { printf ␣ \ " % e \ " , ␣ $f ␣ + ␣ 0 .00001 } " )
Due to the lack of support for floating point numbers, you will only be able to compare
the equality of strings in loops and conditions. Therefore, if more complicated calculations are needed for the batched execution, a full featured language such as Python
might be better suited for this task.
The given bash script batch_execute.sh works in the way described above. Adjust it
so that it will run on one of the CIP pool machines for a couple of hours, sampling the
force density range f ∈ [0.00001 ... 0.2]. Of course you can also prepare several of those
scripts to split the work amongst several computers.
6.2 Automatic Analyzation
Task
(1 point)
• Change the existing simulation script to calculate the time dependent
net flow rate of each run and store it into a file. Complete the existing
analyzation so that it reliably calculates the average asymptotic net flow
rate for each run and briefly explain how it works.
Hints
• The ESPResSo command
lbnode x y z print <velocity|boundary|...>
allows you to access LB information during the simulation.
• Change the simulation script so that the simulation is interrupted every 100 steps
and the time and the current flow rate is written to a file called flowrate_$f.dat
with gnuplot readable format in a folder results.
The flow rate can be calculated by summing up the flow in force direction of all
elements on a plane perpendicular to the force direction.
13
• Computing averages of time series can be done directly in the simulation script.
Use the tcl command lappend to append measured values to a list and use the
function uwerr to compute the corresponding average and statistical error.
• It is important that the determination of the averaged asymptotic flow rate from
this time resolved flow rate profile works in a completely automatic fashion for all
force densities ranging from f = 0.00001 to f = 0.2 . So first run some simulations
in this range (specifically at the upper and lower limit) by hand to see how long
it takes to reach a stationary or oscillatory state and how much time you need
for averaging to get a reliable mean value in the oscillatory case. Implement an
equilibration routine before the measurement run during which no flow rate data
is recorded. If needed, adjust the routine in batch_execute.sh gathering the
computed averages.
Solution
The following script will work:
# r e a d f o r c e from command l i n e
set force 0 .0
if { $argc == 1 } then {
set force [ lindex $argv 0] ;# 0 . 2 f o r t u r b u l e n t v i d e o
} else {
puts " USAGE: ␣ Espresso ␣ template.tcl ␣ FORCE "
exit 1
}
#s e t up s i m u l a t i o n
set box_x 200
set box_y 60
set box_z 20
set visc 1 .10
set dt
0 .01
set eq_cycles
800 ;# e q u i l i b r a t i o n s t e p s
set n_cycles
800 ;# measurement c y c l e s
set steps_per_cycle 100 ;# s t e p s p e r c y c l e
setmd box_l $box_x $box_y $box_z
setmd time_step $dt
setmd skin 0 .1
thermostat off
# s e t up LB
lbfluid gpu dens 1 agrid 1 tau $dt visc $visc ext_force $force 0 0
# s e t up o b s t a c l e
lbboundary rhomboid corner 5 [ expr $box_y /4] 0 a 10 0 0 \
b 0 [ expr $box_y /2] 0 c 0 0 $box_z direction outside
# equilibration routine
14
for { set cycle 0} { $cycle < $eq_cycles } { incr cycle } {
integrate $steps_per_cycle
}
# open f l o w r a t e o u t p u t f i l e
set flowrate_file [ format " flowrate_f %10 .8f.dat " $force ]
set fp [ open $flowrate_file " w " ]
puts $fp " # time ␣ flow_rate "
# run measurement c y c l e s
set flowrate_list " "
for { set cycle 0} { $cycle < $n_cycles } { incr cycle } {
integrate $steps_per_cycle
# calculeat flow rate
set flowrate 0 .0
for { set y 0} { $y < $box_y } { incr y } {
for { set z 0} { $z < $box_z } { incr z } {
set flowrate [ expr $flowrate + \
[ lindex [ lbnode 100 $y $z print velocity ] 0]]
}
}
# store flow rate
set t [ expr $cycle * $steps_per_cycle * $dt ]
puts $fp " $t ␣ $flowrate "
lappend flowrate_list " $flowrate ␣ "
}
# f l u s h b u f f e r and c l o s e f l o w r a t e f i l e
flush $fp
close $fp
# run a n a l y s i s
set uwerr_frate [ uwerr $flowrate_list [ llength $flowrate_list ] 1]
set mean_frate [ lindex $uwerr_frate 0]
set error_frate [ lindex $uwerr_frate 1]
# print the values to screen
puts [ format " flow_rate ␣ = ␣ %17 .10e " $mean_frate ]
puts [ format " flow_rate_error ␣ = ␣ %17 .10e " $error_frate ]
15
7 Investigating the Force-Flowrate Relation
7.1 Analytical Theory for low Re
Task
(2 points)
• Prove that in the stationary state for low force densities, there is a linear
relation between the applied force density and the net volume flux in our
system.
Hint
• Flow of Newtonian fluids is conventionally described by the equation of continuity (1) and the Navier-Stokes equations (2)
∂ρ ~
+ ∇ · (ρ~v ) = 0 ,
∂t
ρ
∂~v
~ ~v = −∇p
~ +
+ ρ ~v · ∇
∂t |
{z
}
convective
momentum
transport
(1)
η∆~v
+f~ ,
(2)
| {z }
viscous
momentum
transport
~ and ∆ stand for the (in
with density ρ, velocity ~v and viscosity of the fluid η. ∇
cartesian coordinates) component-wise gradient and laplacian, respectively. By
rescaling all the occuring variables, one can eliminate one of the two occuring
parameters and formulate (2) in a completely unitless way, depending only on the
dimensionless Reynolds number:
∂~v 0 0 ~ 0
~ 0 + 1 ∆~v 0 + f~0
+ ~v · ∇ ~v = −∇p
0
∂t
Re
(3)
Check http://en.wikipedia.org/wiki/Reynolds_number#Derivation for more
information on the nondimensionalization.
From equation (3), one deduces that for very low Re, e.g. small densities, small systems, low velocities or big viscosities, the viscous momentum transport dominates
over the convective one. Assuming this is the case, one can neglect the convective
contribution and find the stationary state of an incompressible Newtonian fluid by
solving the resulting Stokes’ equations
∆~v =
1~
1
∇p − f~ ,
η
η
~ v = 0.
∇~
(4)
(5)
Pay attention to the fact that the pressure in this equation is not a given quantity
but also a variable. In fact, it is what couples the two equations and makes the
16
solutions interesting. As boundary conditions we impose ~v = 0, so called no-slip
boundary conditions.
From this you should be able to deduce that the resulting ~v scales linearily with
the applied force density, as it would be the case for a simple linear, uncoupled
differential equation.
Solution
For low Reynold’s numbers Stokes’ equations (4),(5) are a valid approximation for the
Navier-Stokes-Equations (1),(2).
Suppose ~v is a solution of Stokes’ equations with body force density f~. Then, by multi~ and~, we obtain
plying with a constant factor α and using the linearity of ∇
∆ (α~v ) =
1~
1 ~
∇ (αp) −
αf ,
η
η
~ (α~v ) = 0 ,
∇
which means that α~v is a solution to Stokes’ equations with a body force αf~. The
resulting pressure is just scaled by α as well. Obviously, where ~v = 0, it is also true that
α~v = 0, and therefore α~v fulfills the same no-slip boundary conditions that ~v originally
fulfilled.
This means that if one obtains a solution for a certain body force density, in the limit
of low Reynolds numbers, all solutions for other body force densities can be obtained
by just scaling the original solution appropriately. Furthermore, this also means that in
the limit of low force, there is a linear relation between force density and flow rate with
a geometry-dependent proportionality constant.
17
7.2 Results from the Simulations
Task
(3 points)
• Plot the relation between applied force density and asymptotic average
net flow rate and discuss the result.
• For which range of f does the law deduced in section 7.1 hold?
• Where does the flow field start to really look turbulent?
• Can you propose any law that describes the scaling of the asymptotic
average net flow rate on the whole investigated interval of f ?
• Is there information in the literature for this behaviour, and, if yes, is it
theoretically deduced and from what or is it experimentally or heuristically obtained?
Hint
• In order to make the different regimes clearly visible, you might want to show your
results in a log-log plot.
Solution
• The results of the simulations are plotted below:
asymptotic average net flow [a. u.]
104
103
102
101
Simulations
100
Fit 1: 249924 × f 1.0
Fit 2: 10518 × f 0.471
10−1
10−5
10−4
10−3
10−2
body force density f [a. u.]
10−1
Figure 4: Average asymptotic flow rate as a function of applied body force density. Green dots ( ): simulations, red
line (
): fit to lower 20% of data, purple line (
): fit to upper 20% of data.
18
• The proprotionality deduced in the previous task is clearly reproduced in the first
part of the data up to about f = 3 × 10−4 (compare fig. 4 fit 1 (
)).
• From figure 4 it is not possible to draw conclusions at which body force density
the fluid really shows turbulent flow. In order to answer the question, one has to
investigate the time series of flow rates for different body force densities, including
equilibration:
6,000
f = 1 × 10−4
f = 5 × 10−4
5,000
flow rate [a. u.]
f = 1 × 10−3
f = 5 × 10−3
4,000
f = 1 × 10−2
f = 5 × 10−2
3,000
f = 1 × 10−1
2,000
1,000
0
0
200
400
600
800
1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400 2,600 2,800
time t [a. u.]
Figure 5: Flow rate over time for different body force densities f . At high force densities (
and
), the
transition from laminar to turbulent flow occurs early is clearly visible. At low force densities (
,
),
the flow stays laminar as expected. At intermediate force densities (
and
), however, the flow stays
laminar up to f = 0.005, even though this is already far in the transitional regime. Note the small but visible
change from laminar to turbulent flow for f = 0.01 (
) at t ≈ 1500.
Figure 5 shows the flow rate over time for different body force densities f . While
for high values of f , the transition from laminar to turbulaent flow occurs early and
is clearly visible, the flow stays laminar for f ≤ 0.005, even though, according to
figure 4, this value already lies far in the transitional regime. For f = 0.01 (
),
the transition to turbulent flow occurs late (t ≈ 1500) and is not as pronounced as
for higher values of f .
Thus, the flow field starts to look turbulent at 0.005 < f < 0.01.
• For the laminar regime, the asymptotic average flow rate is proportional to the
applied body force density and in the turbulent regime it shows a square root
behavior, with a smooth transition in between. However, a unified formulation for
the whole investigated range of force densities cannot be given.
19
• The square root dependence is experimentally well tested and there are a lot of
older, heuristic works on it:
http://en.wikipedia.org/wiki/Darcy%E2%80%93Weisbach_equation
However, turbulence is one of the phenomena for which physics does not provide
a complete description and so there is no derivation from first principles.
8 Bonus Questions
Task
(3 points)
• ESPResSo does not specify a unit system and depending on how one sets
certain units, other ones are rescaled. Assuming our system consists of
water with density ρ = 1000 kg/m3 and kinematic viscosity ν = 10−6 m2 /s
and the video you created shows the system in real time using 2000 frames
with a spacing of 10 time steps (of 0.01 t each) in between and a frame
rate of 60 fps – how big is the system?
• Assuming one naively set kB T = 1, what temperature would that correspond to in the real world?
• In the produced video you saw that the flow lost its mirror symmetry in
the y-direction after a while although the geometry of the problem and the
applied force density is symmetric. What caused this symmetry break in
the simulation? What causes it in reality? How could one cause it earlier
to maybe save some computational time?
Hint
• You can find information about how to figure out the unit system in section 4.1 of
the bachelor thesis delivered with this homework.
Solution
• First, the time scale of the simulation has to be determined. The duration of the
video in seconds is
2000 frames
100
tvideo =
=
s,
(6)
60 fps
3
which corresponds to a simulation time of
tsim = 2000 frames × 10
20
steps
t
× 0.01
= 200 t .
frame
step
(7)
!
Requiring tvideo = tsim yields the time scale
[t] =
100
1
1
s×
= s.
3
200
6
(8)
Next, we know that the kinematic viscosity of water is ν = 10−6 m2 /s, so that
2
ν = [x]
[t] yields the length scale
[x] =
q
ν × [t] ≈ 0.408 mm .
(9)
In the simulation, the dimensions of the box were set to Lx = 200 , Ly = 60 , Lz =
20, thus corresponding to Lx = 8.165 cm , Ly = 2.449 cm , Lz = 0.816 cm. The box
volume thus corresponds to
V = 1.632 × 10−5 m3 .
(10)
• Now that we know the length scale [x] of the system, we can get its mass scale
from the density of water:
[m] =
ρ
≈ 0.068 mg
[x]3
(11)
This allows us to answer the question about setting the temperature to 1: ESPResSo
implicitly sets kB = 1, so that with [E] = [m][x]2 /[t]2 = kB T one obtains
T ≈ 2.95 × 1010 K .
(12)
This seams reasonable, considering that simulations at this temperature show
macroscopic density fluctuations which do definitely not occur at room temperature.
• The symmetry break comes from the fact that the algorithm iterates over the grid
in a certain order, which does not preserve the symmetry. The actual difference
is then caused by the fact that in floating point arithmetic, adding numbers of
different order of magnitude causes a different error, depending on the order of
summation. (Small + small) + big gives a smaller error than (big + small) +
small, since for the latter, the accuracy is already lower after the first addition.
In reality, this is a flow of microscopic particles whose dynamics are to some degree chaotic, leading to local fluctuations causing the symmetry break. This is a
phenomenon essentially caused by a non-vanishing temperature. One could incorporate random fluctuations into the Lattice-Boltzmann algorithm to account for
this. In fact, ESPResSo’s implementations of LB provide this thermalization.
21
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