Kelvin-Helmholtz Simulations

Project description:

This paper discusses some aspects of 2D simulations of the Kelvin-Helmholtz instability. Note that the simulations mentioned here are incompressible, while you should run compressible cases. Use the paper to initiate your own simulations, and to make contact with the paper results.

Table of contents

How to run
TODO
MPI-AMRVAC notes
Reference paper notes

How to run

How to run the stuff in SimCode (where you can replace "4" by the number of processors you want to use):

# One time
cd SimCode
mkdir SimData
$AMRVAC_DIR/setup.pl -d=2

# Each time mod_usr.t is updated
make -j 4

# Each simulation
mpirun -np 4 ./amrvac -i test.par

TODO

MPI-AMRVAC notes

Setting parameters: Most of the notes about this can be found in settings.par.
Numerical methods: DONE (see settings.par)
Slope limiters
Adaptive Mesh Refinement
- Module errest
  1. Quantify local error in each gridpoint of each grid block
  2. If any point has an error exceeding refine_threshold(l), the block is refined
  3. If all points have an error below derefine_ratio(l)*refine_threshold(l), the block is coarsened
- For possible values of refine_threshold, see our settings.par
- Formula for the Lohner prescription is also given here.

Reference paper notes

Title: On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem

1 Introduction

Contributions of this paper:

Reference results for 2D Kelvin-Helmholtz up to t = 200
Even with state-of-the-art methods, the time of the final vortex pairing remains very difficult to predict
Application of the theory for self-organization to Kelvin-Helmholtz

2 The Kelvin-Helmholtz instability problem

2.1 Problem statement

Equation (1) is what we need to simulate (but compressible cases, somehow)
Boundary and initial conditions are also defined here
Definition of Reynolds number

2.2 Quantities of interest

This section gives 8 computable quantities which can be used to evaluate a given numerical scheme, check the paper for precise definitions. Kinetic energy (spectra), vorticity (thickness), enstrophy, palinstrophy, and the time intervals of the pairings are of interest.

3 Self-organization and 2D turbulence

The Van Groesen theory behind self-organization in 2D Navier-Stokes (with homogenous Dirichlet BC) is applied to the incompressible 2D Kelvin-Helmholtz problem (with mixed BC). Rather technical, uses some optimization and fancy analysis (didn't really read this section). The most important results are summarized in 2 remarks:

The theory does not make any prediction about the position (in space nor time) of the final vortex
2D flows with high Reynolds numbers are very sensitive to perturbations

4 High-order divergence-free IMEX HDG methods

This section is less relevant to us than it seems. They use some finite element library in Matlab. As far as I know, MPI-AMRVAC only has finite differences and finite volumes.

4.1 Space discretization

Divergence-free HDC (Hybrid Discontinuous Galerkin) method based on H(div) finite elements (rectangular RT8 or triangular BDM8 elements, both using polynomials of order 8).
Some additional fancy stuff (projected jumps etc)
For computing vorticity, enstrophy and palinstrophy: element-wise derivatives
Careful with stabilization mechanisms, they might introduce a lot of numerical dissipation (resulting in lower accuracy) due to the sensitivity of the KH problem.

4.2 Time discretization

Linear multistep IMEX (BDF) method (in MPI-AMRVAC we will probably use IMEX-RK)
Equation (13): result of a spatial discretization of incompressible Navier-Stokes
- $Au$ is the Stokes part, this will be handled implicitly
- $C(u)u$ is the convection part, this will be handled explicitly
For linear system solving: sparse Cholesky with iterative refinement (not sure if we can choose the solver though)

5 Computational studies

Sequence of square meshes with 16² to 256² elements
Some vorticity figures (Figure 3) and energy spectra (Figure 4) are shown and discussed
Then the quantities of interest from 2.2 are shown and discussed for Reynolds numbers 1E2, 1E3 and 1E4; for different mesh sizes:
1. Kinetic energy
2. Enstrophy and vorticity thickness
3. Palinstrophy
4. Numerical dissipation (not a problem quantity, but a general measure for numerical schemes)

6 Computational studies of some possible sources for perturbations

This section tries to find some explanations for why the results in literature vary so wildly, by a sensitivity analysis (i.e. identifying the reasons for small perturbations).

Structured and unstructured triangular meshes: rectangular, structured triangular or unstructured triangular meshes all seem to give different results
Inaccurate solution of linear systems and numerical integration:
- The iterative refinement for solving linear systems more accurately definitely seemed to have an impact on the end result
- Impact of under-integration (because of a too low polynomial order used in the elements)
Different compiler settings (FMA): even rounding modes seemed to have a non-negligible impact, the example of fused multiply-add is given.

7 Summary and conclusions

Contents of each section:

Problem setting, overview of quantities of interest (of which most important one is palinstrophy)
Van Groesen theory of self-organization applied to the Kelvin-Helmholtz instability problem (important: lemma 3.5)
Spatial (div-free HDG) and temporal (BDF2 IMEX) discretization schemes
Study quantities of interest for different Reynolds numbers; reliable reference results up to t = 200
Common sources of perturbations: kind of mesh used, linear system solver, numerical integration, compiler settings

8 Appendix

Reference results are available, can be useful to compare against.

robbedhondt/Kelvin-Helmholtz-CMfAA