Large eddy simulations using FR/CPR methods

Large eddy simulations
using FR/CPR methods
Z.J. Wang
Department of Aerospace Engineering,
University of Kansas
Outline
Introduction to large eddy simulations (LES)
A study of sub-grid scale (SGS) models with Burgers’
equation

Numerical method and problem setup
A priori and a posteriori studies
Central difference or high-order upwind schemes for LES?
Wall modeling for LES
Conclusions

Introduction
Approaches to compute turbulent
flows

RANS: model all scales
LES: resolve large scales while model
small scales
DNS: resolve all scales
What is LES

Partition all scales into large scales and
small sub-grid scales with a low pass
filter with width D
Solve the filtered Navier-Stokes
equations with a SGS closure model
Introduction – LES
What Factors Influence LES Results

The filter width D
The computational mesh size h
The spatial discretization
The time marching approach
The SGS model
“Truth” LES solution = filtered DNS solution

Different filter width produces different “truth” LES solution
How to obtain a good LES solution

An accurate numerical scheme with low truncation error
An “accurate” SGS model
Our Venture into LES
Solve the filtered LES equations using

FR/CPR scheme
3 stage SSP Runge-Kutta scheme for time marching
Implemented 3 SGS models

Static Smagorinsky (SS) model
Dynamic Smagorinsky (DS) model
ILES (no model)
Attempted several benchmark problems

FlIsotropic turbulence decay (SS, DS, ILES)
Channel flow (SS, DS, ILES)
LES Results – Isotropic Turbulence Decay
Results – Turbulent Channel Flow
 ILES
solutions converge to the DNS solution.
Results for Turbulent Channel Flow

No benefit observed with either Smagorinsky model
over ILES.
Why ILES Performs Better Consistently
No good explanation!
So we decided to evaluate SGS models using the 1D
Burgers’ equation

High resolution DNS can be easily carried out
True stress can be computed based on DNS data
Both a priori and a posteriori studies can be performed
Yes, the physics of 1D Burger’s equation is vastly simpler than the
Navier-Stokes equations, but if a SGS model has any chance for
3D Navier-Stokes equations, it must perform well for the 1D
Burger’s equation
Filtered Burgers’ Equation
1D Burgers’ equation
¶u
¶u
¶2 u
+u = v 2
¶t
¶x
¶x
Filter the equation with a box filter
where
SGS Models Evaluated
Static Smagorinsky model (SS)
Dynamic Smagorinsky model (DS)
Scale similarity model (SSM)
Mixed model (MM) of SSM and DS
Linear unified RANS-LES model (LUM)
ILES (no model)

Numerical Method and Problem Setup
Numerical method

3rd order FR/CPR scheme
Viscous flux is discretized with BR2
Explicit SSP 3 stage Runge-Kutta scheme
Problem setup

Domain [-1, 1] with periodic boundary condition
The initial solution contains 1,280 Fourier modes satisfying a
prescribed energy spectrum with random phases
The DNS needs 2,560 cells to resolve all the scales
The filter width: D = 32 DxDNS
Various mesh resolutions for LES DxLES/D = 1, 1/2, 1/4, 1/8
Initial Condition
𝟓
−
𝟑
Comparison of SGS Stresses (A Priori)
Correlation Coefficients (A Priori)
Comparison of SGS Stresses (A Posteriori)
Correlation Coefficients (A Posteriori)
Solution Errors
Central Difference vs High-Order Upwind for LES
There have been many conflicting views on this topic in the
LES community
People often judge the quality of LES results based on the
amount of “turbulent structures” in the solution

Bad LES result?
Good LES result?
A 1D Benchmark Case – Burgers’ Equation
Consider the inviscid Burgers’ equation to mimic very high
Reynolds number problems
u
u
u
= 0, x  [1,1]
t
x
with a single Fourier mode as the initial condition and
periodic BC
𝑢 𝑥 = 0.012 sin 𝜋𝑥 + 1.
All higher frequency modes are generated from this single
mode. The simulation is run until t = 26 before a shock wave
forms. The energy spectrums are studied to evaluate the
scheme performance
A 1D Benchmark Case – Energy Spectrum
A 1D Benchmark Case – Numerical Turbulence
Scheme Performance in LES with SGS Models
Wall Modeling for LES
A wall model is developed to obtain the correct wall shear
for high Reynolds LES simulations to relieve the stringent
mesh resolution requirement in all 3 directions
We employ the following ideas:

 Near wall S-A model
t , RANS /  = % f v1 , % =  y  ,
3
3
f v1 = %  / %   cv31 


For zero pressure gradient flow
d    t  du / dy  / dy = 0
Together we obtain a relation between u+ and y+
u  = 3.63975 tan 1  0.927768  0.133902 y    1.33017 log  y   13.8574  y   103.78
2.54968log  y   16.2964  y   122.046   3.59946 tan 1  0.134047 y   1.09224 
6.33792
Wall Model for LES
For FR/CPR schemes, ILES works the best.
This means turbulent viscosity should be zero in ILES
region
A smooth blending function is employed

t ,hybrid = t , RANS g0.5  0.5 tanh  y   25
Results for Turbulent Channel Flow Ret = 640

No wall model: effects of x+ and z+ with the same y+
 Case 1: y+ ~ 12, z+ ~ 30, x+ ~ 60
 Case 2: y+ ~ 12, z+ ~ 15, x+ ~ 30
p=3
Results for Turbulent Channel Flow

With wall model at various Reynolds numbers
 Case 3: Ret = 395, y+ ~ 12
 Case 4: Ret = 640, y+ ~ 12
 Case 5: Ret = 1140, y+ ~ 12
Turbulent Channel Flow – Grid Dependence

With wall model at Ret = 640
 Case 4: y+ ~ 12, 16x24x16 mesh size
 Case 6: y+ ~ 8, 16x24x16 mesh size
 Case 7: y+ ~ 12, 32x24x32 mesh size
Turbulent Flow over Periodic Hill
Mesh size: 32x32x16 (p = 3, 128x28x64 DOFs)
Reb = 2,800 and 10,595

Isosurface of Q criterion colored by streamwise velocity at
Reb=10595 with hybrid approach
Turbulent Flow over Periodic Hill
Reb=2800, ILES
Reb=2800, ILES with wall model
Turbulent Flow over Periodic Hill
Reb=10,595, ILES
Reb=10,595, ILES with wall model
Turbulent Flow over Periodic Hill
x/h=2
x/h=6
Conclusions

In both a priori and a posteriori tests with the 1D Burgers’
equation
 SGS stresses generated by static, dynamic Smagorisky and LUM
models show no correlation with the true stress
SSM (and Mixed model) consistently produces stresses with the
best correlation with the true stresses
When the modeling error is dominant, SSM and MM perform
the best. When the truncation error is dominant, no model
shows any advantage. ILES is preferred
The central flux causes energy to pile up at high frequencies.
For high-order upwind schemes, no energy pileup is
observed.

Conclusions (cont.)
For any methods with dissipation, DO NOT use any SGS
models. For almost all LES simulations, truncation errors
are dominant (D = h), the best choice is ILES.
For non-dissipative central difference schemes, the SGS
model’s role is to stabilize the simulation. I cannot say there
is any true physics captured by the SGS models.
