VandV_pdf.pdf

8/10/2019 VandV_pdf.pdf

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We will learn:

The difference between validation and vericationWhat is the MMSHow to implement a new solver in openFOAM (can be useful!)How to run a set of simulations efciently using Linux tools:

sedforgrep

How to sample our openFOAM results and transfer them to MatlabWhat can we do in real life to (try to) ensure that our CFD results are

corect


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Lets begin !


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By default, everything is wrong


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until you can prove it is correct

(or x it)


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NUMERICALMODEL

REALITY PHYSICALMODEL

Qualification

VerificationValidation

1: Write a set of

equations (usuallyODE/PDE) thatimplement physicallaws

2: Two steps-Write a computer code to solve theequations-Choose a mesh, time step, residuals..and solve numerically the equations

0: The reality(inntely subtle)


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NUMERICALMODEL


Qualification


Qualify the physicaland numericalmodels for their use

in PREDICTIONS

Check that thenumerical solutionof the equations iscorrect

Check that the

physical model isaccurate enough


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NUMERICALMODEL


Qualification


Holy Grial ofcomputational

enginering !!

SOLVE THEEQUATIONS RIGHT

SOLVE THERIGHT EQUATIONS


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NUMERICALMODEL


Qualification


MATHEMATICAL

process

EXPERIMENTALprocess


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Reality

Ideal world

Imperfect,cheap copy

Mathematical reasoning

Validation and Verication in Platos mind

Discovering (actually,remembering) the ideal world

Never ! Are you joking ??

If you want to understand the stars, dont observe them,think about them


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MMS: Method of Manufactured Solutions

DP DE (u ) = b dD:

Problem to be solved with OpenFoam:

d: = bc u

n=

u

n bc

Only a few trivial cases have known analytic solutionThen, how can we check our solutions ?

.. what could be wrong?-the code itself (not to be expected in OpenFoam)-the mesh and/or the time step (most likely)-the model used (e.g, turbulence model)

Given b and the b.c., nd u


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MMS inverts the process to generate arbitrarily complexsolutions of our problem:

Generate u and then obtain (by derivation) b

Exemple: Conduction heat transfer equation

fvm::ddt(T) - fvm::laplacian( D, T) == B

T

t D

2 T

x2

+ 2 T

y2

+ 2 T

z2

= B

T

t D2

T = B D: thermal diffusivity

OpenFoam representation:


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We choose our domain to be [0:1]^3

To keep things simple, we choose T to beonly a function of x, and thermal diffusivityto be 1. Example:

T=-4*x*(x-1)*(x-3/4)*(x-1/4) but in general, it can be (has to be) function of x,y,z

Then, by analitic derivation:

B=8*(x - 1)*(x - 1/4) + 8*(x - 1)*(x - 3/4) +8*(x - 1/4)*(x - 3/4) + 8*x*(x - 1) + 8*x*(x - 1/4) +8*x*(x - 3/4)


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For lazy people.. use Matlab : (generaB)clear allclose allsyms x y z t T BDT=1 % difusividad termica (demasiado elevada, sin sentido sico)

T=-4*x*(x-1)*(x-3/4)*(x-1/4)

% ecuacion que vamos a resolver en OpenFOAM:% fvm::ddt(T) - fvm::laplacian(DT, T) == B

% por tanto, este debe ser el termino fuente ..B = diff(T,t) - DT*(diff(T,x,2)+diff(T,y,2)+diff(T,z,2) )

% Si queremos examinar los valores numericos de T o B, podemos hacerlo

% creando funciones a partir de los resultados simbolicosfB=@(x,y,z,t) eval(char(B));fT=@(x,y,z,t) eval(char(T));

fB(0,0.5,0.5,0.5)

fT(0,0.5,0.5,0.5)


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Another example, again with thermal diffusivity equal to 1

T=-4*x*(x-1)*(x-3/4)*(x-1/4)*(x-7/10)*(x-8/10)*x^2

B = diff(T,t) - DT*(diff(T,x,2)+diff(T,y,2)+diff(T,z,2) )

B=simplify(B)

this yields:

B=(x*(11200*x^5 - 29400*x^4 + 28485*x^3 - 12355*x^2 + 2271*x - 126))/50


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B=(x*(11200*x^5 - 29400*x^4 + 28485*x^3 - 12355*x^2 + 2271*x - 126))/50


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T=-4*x*(x-1)*(x-3/4)*(x-1/4)*(x-7/10)*(x-8/10)*x^2


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Now, we will use openFOAM to solve the problem:

T t D

2 T x 2 +

2 T y 2 +

2 T z 2 = B

whereB= (x*(11200*x^5 - 29400*x^4 + 28485*x^3 - 12355*x^2 + 2271*x - 126))/5 the domain is a 1x1x1 cube and:T=0 in the x boundariesdT/dn=0 in the y,z boundaries (to make it 1d)

Laplace equation,can be solved with the available laplacianFoam, but tointroduce a RHS, we need to write a new solver

2 T = 0


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A new solver in this case is surprisingly easy, it will only beabout 50 lines of C++ code !

very easy (with the help of a friend) :)Copy laplacianFoam source code in a separate folder

In createFields.h, add the following:

Info


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In laplacianFoam.C change the following:

1-Assign a value for B to each domain cell, as a functionof x,yz, and t2-Change the PDE equation to be solved

in the original laplacianFoam, the equation is:

solve(

fvm::ddt(T) - fvm::laplacian(DT, T)

);

This is C++ !


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while (simple.loop()){

Info


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W l d t t d iti f B ld i 0 f ld


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We also need to create a denition for B eld in 0 folderThis is a bit nonsense as B will be assigned later, but it is the easiersolution

FoamFile{version 2.0;format ascii;class volScalarField;object T;

}

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //dimensions [0 0 -1 1 0 0 0];internalField uniform 0;boundaryField{

adiabatic{

type zeroGradient;}cold{

type xedValue;value uniform 0;

}

}

K/s

T

t D

2 T

x2

+ 2 T

y2

+ 2 T

z2

= B

K/s K/m^2 K/sm^2/s

I t t/t tP ti if th l diff i it 1


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In constant/transportProperties, we specify thermal diffusivity=1

DT DT [ 0 2 -1 0 0 0 0 ] 1.0;

m^2/s

I constant/pol Mesh/blockMeshDict if


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In constant/polyMesh/blockMeshDict, we specify:

blocks(

hex (0 1 2 3 4 5 6 7) ( 10 10 10 ) simpleGrading (1 1 1));

In system/controlDict we specify:


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In system/controlDict , we specify:

application laplacianFoam;

startFrom startTime;

startTime 0;

stopAt endTime;

endTime 1e6;

deltaT 1e6;


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Steps to run a simulation (once the solver is ready):

1-With paper&pencil (or Matlab if you are as lazy as me)

-Generate T-Obtain B

2-Insert B in OpenFoam and solve for T-Edit solver to change B (take care with integers!!)

-Compile (wmake) and copy (cp) the executable-Change constant/polyMesh/blockMeshDict (to testdifferent meshes)

-Change system/controlDict (if needed)

-Change system/sampleDict (if needed)-Generate the new mesh: blockMesh-Run the solver: ./ newlaplacianFoam


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We can see the results in paraview, but it will be better to usethe openFoam sample utility


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specications for sample are in system/sampleDict

// Fields to sample:elds

( T B);

sets(

lineX1{

type uniform;axis x;

start (0.0 0.5 0.5);end (1.0 0.5 0.5);nPoints 100;

});

100 points in line parallel to x axis,from (0, 1/2, 1/2) to (1,1/2,1/2)


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in /postProcessing/sets/1e+06, the le with our sample is created:

manel@ubuntu:~/Dropbox/CFD/cubeBpar/postProcessing/sets/1e

+06$ head lineX1_B_T.xy 0 0 00.010101 0.000825318 1.97071e-050.020202 0.00165064 3.94142e-050.030303 0.00247595 5.91213e-050.040404 0.00330127 7.88284e-050.0505051 0.00412659 9.85355e-050.0606061 0.00495191 0.0001182430.0707071 0.00577723 0.00013795

0.0808081 0.00660255 0.0001576570.0909091 0.00742786 0.000177364

x B T

We use Matlab to plot it and compare with the exact solution:


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We use Matlab to plot it and compare with the exact solution:

clear

leID = fopen('../cubeBpar/postProcessing/sets/1e+06/lineX1_B_T.xy','r');formatSpec = '%f %f %f';sizeA = [3 Inf];DAT = fscanf(leID,formatSpec,sizeA); % DAT contains: row 1: x positions; row 2: B; row 3: T% Calculamos solucion analitica

y=0.5; % irrelevante, es unidimensionalz=0.5; % idemt=0.5; % No se usafor i=1:size(DAT,2) % numero de puntos

xv(i)=DAT(1,i);

x=xv(i);Ta(i)=-4*x*(x-1)*(x-3/4)*(x-1/4)*(x-7/10)*(x-8/10)*x^2;endplot(xv,Ta,'b');hold onplot(DAT(1,:),DAT(3,:),'r');

T


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x

T

openFoam (10)

exact


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Steps to compare with the imposed analytical solution(once the numerical solution has been computed):

1-Obtain a 1d cut of the T eld-sample

we choose a 1d cut just because our problem is 1d, ingeneral, all the domain has to be sampled

2-Read it, and compute the exact solution in the samepositions, you can do it with Matlab or octave, gnuplot,

T


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T

Not too bad, but is it correct ?How many control volumes do we need for an accurate solution?

We need to repeat the simulation with increasingly ne meshes ! Can we do it automatically ?Yes, we can !

x


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FoamFile{

version 2.0;format ascii;class dictionary;object blockMeshDict;

} // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //convertToMeters 1.0;

vertices(

(0 0 0)(1 0 0)(1 1 0)(0 1 0)(0 0 1)

(1 0 1)(1 1 1)(0 1 1)

);

blocks(

hex (0 1 2 3 4 5 6 7) ( #NX# 10 10) simpleGrading (1 1 1));

constant/polyMesh/blockMeshDict.NX (arbitrary name)


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We will use the command sed to replace the string #NX#with the number of control volumes that we want, for instance 123:

cat blockMeshDict.NX | sed "s/#NX#/123/" > blockMeshDict

Type the leblockMeshDict.NX, butdirect the output to thenext command (symbol|)

Substitute #NX# by123

Direct the outputto the leblockMeshDict


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We create a le called (for instance) doit_once , with :

cd constant/polyMesh/cat blockMeshDict.NX | sed "s/#NX#/ $1 /" > blockMeshDictcd ../..blockMesh./newlaplacianFoam

samplecp -r postProcessing postProcessing_ $1

Then, we type:chmod +x doit_once./doit_once 20

First parameter

Make it executable

Run with 20

Copy theresults to anotherfolder


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for x in 5 10 20 40 80 160 320 640 1280do

echo $x./doit_once $x

done

We create doit_all, with :

T


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x

T

error


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! x

error

! x2

Max (T! x-Ta)


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The meaning of it all

u i +1 u i 1

2 x

u

x

x i

= x 2

3! 3u

x 3

x i

+ . . .

xi 1

xi

xi+1

u (x i + x i ) u (x i ) x i

u

x x i=

x i

2! 2u

x 2x i+

x 2i

3! 3u

x 3x i+ . . .

| {z } Truncation ErrorFirst orderapproximation

Second orderapproximation

http://www.rsmas.miami.edu/personal/miskandarani/Courses/MSC321/lectniteDifference.pdf
http://www.rsmas.miami.edu/personal/miskandarani/Courses/MSC321/lectfiniteDifference.pdf


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T l d


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To conclude

DO Begin with a similar problem that has a known solution, look in

literature for experimental results; reproduce it and understand theresuts

Use AT LEAST two or (better) three different meshes of differentdensities to evaluate how close to the mesh independent solution areyou

Be specially careful with turbulence models !!

NEVER DO Never launch a simulation with a model (ie, multiphase, compressible,

LES..) if you dont understand the equations and the physics behind .

Never be satised with a single solution. If you can rene the mesh,generate a coarser mesh.Be impressed by the beauty of possibly wrong solutions

REMEMBER: One day CFD will be reliable and wind tunnels will be closed; until then,

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