Computational fluid dynamics (CFD) 9 th SIMLAB Course
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1 Computational fluid dnamics (CFD) 9 th SIMLAB Course Janos Benk October 3-9, Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
2 Overview Introduction Potential flow Stokes equation and discretization Boundar Conditions Navier-Stokes equation and its dicretization (Parallelization) Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
3 Introduction What defines a flow? What are the quantities in such a incompressible flow field? Velocit vector pressure scalar Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
4 Introduction We are looking for a relation between the velocit vector and the pressure We note vel (u,v) the velocit vector in D The pressure is noted b p For the case of simplicit we consider onl stationar scenarios though the whole tutorial We use a regular structured grid (as the simplest grid) The cells form a mesh One cell Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
5 Introduction Grid based method Finite difference (reuse some knowledge from the previous lecture) Other discretization techniques are more favorable in practice, due to the limitations of the finite difference method. Finite volume Finite element Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
6 Potential flow v The first tr is the Potential flow, the simples flow equation The velocit is directl derived from the pressure (potential) flow u p u First we specif that per cell no matter can be gained or disappear. ( vel (u,v) ) v vel u v Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
7 Potential flow Replace the? with values X? vel - p -X? u v 3 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
8 Potential flow Replace the? with values 4 vel 3 p -3? u v - Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
9 Potential flow The potential is a Poisson equation, with zero right hand side. The velocit is directl derived from the pressure (potential) flow The different boundar conditions can be implemented through the potential To be the solution uniquel determined, we use a Dirichlet boundar condition for n p p n p p p out Walls Outlet Inlet Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
10 Potential flow Is the velocit still divergence free? Is the following equation still satisfied? vel(u,v) With the following equations: vel p vel p vel p Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
11 Potential flow How does the potential field looks for a channel flow? And the velocit field (u,v) Walls ( u, v) p p p Inlet p- Outlet p u Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
12 Stokes equation There is a complete new second equation: vel vel t p Re vel f et Since we consider onl stationar problems: vel t p vel Re f et Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
13 Stokes equation What does the second equation mean? p vel Re f et This is the so called impulse equation, at each point the sum of the acting forces must equal zero vel Grad(p) Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
14 Stokes equation Reformulate the equation in terms of u,v and p instead of vel,p Vel(u,v) The equations are p vel p vel Re u v Re u u f et, f et p Re v v f et, Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
15 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7, Stokes equation Calculate the parabolic profile of a channel flow: ) ( H u Re c u p 3 Re ) ( c c c u f et u u p, Re f et v v p, Re ) ( c c p ) u( p c
16 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7, Stokes equation Finite difference discretization: Cell wise view of the cont. eq. f et u u p, Re f et v v p, Re v u The operators in the velocit points, since the impulse is point wise satisfied ( and ).
17 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7, Stokes equation f et u u p, Re f et v v p, Re v u
18 Stokes equation Boundar conditions: No-Slip: ( u, v) (,) Free-Slip: Inflow: ( u, v) ( u,) ( u, v) (, v) ( u, v) ( u, v) v a i Outflow: ( u, v) / n v r v Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
19 Stokes equation Driven cavit Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
20 Stokes equation Which are the unknowns? Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
21 Stokes equation What to do at the boundar? Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
22 Stokes equation What to do at the boundar? p u p v p u v p3 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
23 Stokes equation -u p u p -v v v -v p u p3 -u Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
24 Stokes equation -u Continuit equation: u v -v p v u p v -v p u p3 -u v u u v -v - u -v u Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
25 Stokes equation -u p p Re Re u u v v -v p v u p v -v p u p3 -u (/Re)(5v v)p-p (/Re)(5u-u)-pp (/Re) (/Re)(-v 5v)p-p3 (/Re)(-u 5u)-pp3 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
26 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7, Stokes equation v u u v -v - u -v u (/Re)(5v v)p-p (/Re)(5u-u)-pp (/Re) (/Re)(-v 5v)p-p3 (/Re)(-u 5u)-pp p p p p u v u v Write the sstem of equation with Re With unknown vector [v,u,v,u,p,p,p,p3], first cont. eq. then momentum equation
27 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7, Stokes equation Due to the singularit we set p and delete the fourth line from the sstem p p p u v u v We calculate the solution with Octave
28 Stokes equation The solution vector is: [ ] Visualize these data on the grid Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
29 Stokes equation 3D Eample: Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
30 Navier-Stokes equation Is the velocit still divergence free? vel t vel ( vel ) vel p vel fet Re Since we consider onl stationar problems: vel t Re ( vel ) vel p vel fet Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
31 Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7, Navier-Stokes equation The new term in the equation is the so called convective or transport term ( )vel vel v v v u u v u u v u v u Which in more detailed form is (see the non-linearit) The velocit field transports the velocit. The diffusion spreads the velocit in each direction equall This transports in the flow direction velocit Grad(p) vel
32 Navier-Stokes equation Which model to use when? Would ou use Navier-Stokes for slow, viscous flow (diffusion term is dominant)? Would ou use Navier-Stokes for non viscous flow? (Péclet number, is a good indicator for this) Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
33 Navier-Stokes equation We reformulate the equation in terms of u,v and p instead of vel,p Vel(u,v) p vel Re vel ( vel ) vel fet The equations are (the convection term is transformed slightl) p p Re Re u v u u v v uv ( u ) ( uv) f et, Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7, ( ) ( v ) f et,
34 Navier-Stokes equation Convection term, component Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
35 Navier-Stokes equation Convection term, component Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
36 Navier-Stokes equation Outline of the solving method: Coupled approach: u A ( u, v) v p Using non-linear solvers (fi point or Newton method) b Partitioned approach onl for time dependent problem ( u p, v t t ) Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
37 Navier-Stokes equation D Eample: Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
38 Parallelization Shared memor sstems The basics of parallelization on the matri level on distributed memor sstem Distribute the unknown vector to processes A b Distribute the corresponding lines of the matri and the right hand side p p p p p p p p p Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
39 Parallelization Let s think in terms of iterative processes How to divide among processors? Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
40 Parallelization We need additional cells in the same wa as we need boundar cells Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
41 Parallelization (Ghost cells) Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
42 Parallelization Communication needed (?) Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
43 Thank ou for our attention! Janos Benk: Computational fluid dnamics (CFD) www5.in.tum.de/wiki/inde.php/lab_course_computational_fluid_dnamics_-_summer_ 9 th SIMLAB Course, Belgrade, October 7,
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