Modelling compressible turbulent mixing using an improved K-L model

D. Drikakis, I.W. Kokkinakis,

Cranfield University

D.L. Youngs, R .J.R. Williams

AWE

Outline

Modified K-L model Two-fluid model Implicit Large Eddy Simulations

o Eulerian finite volumeo Lagrange-Remap

Assessment of turbulence models vs. ILES for compressible turbulent mixing

o 1D Rayleigh-Tayloro Double Planar Richtmyer-Meshkovo Inverse Chevron Richtmyer-Meshkov

Conclusions and future work

Motivation

Direct 3D simulation of the turbulent mixing zone in real problems is impractical.

Alternative: Engineering turbulence models to represent the average behaviour of the turbulent mixing zone.

Aim: To develop and asses a range of engineering turbulence models for compressible turbulent mixing

K-L model

Favre-averaged Euler multi-component equations:•Fully-Conservative form (4-equation)•Assume turbulent mixing >> molecular diffusion (viscous effects)

i

i

x

u

t

~

j

ji

ij

jii

xx

P

x

uu

t

u

~~~

jF

T

jj

j

x

F

xx

Fu

t

F~~~~

j

jii

jK

T

jje

T

jj

j

j

j

x

u

x

K

xx

e

xx

Pu

x

Eu

t

E

~~~~~~

Additional TermsRequire Modelling

Continuity:

Momentum:

Total Energy:

Mass-Fraction:

K-L model (cont.)

Two-equation turbulent length-scale-based model (K-L):•Originally developed by Dimonte & Tipton (PoF, 2006);•RANS-based linear eddy viscosity model;•Achieves self-similar growth rates for initial linear instability growth.

Kj

iji

jK

T

jj

j Sx

u

x

K

xx

Ku

t

K ~~

tLj

jc

jL

T

jj

j uCx

uLC

x

L

xx

Lu

t

L

~~

Turbulent kineticenergy:

Turbulent lengthscale:

K-L model (cont.)

Additional closure terms:

ij

k

k

j

i

i

jTijji x

u

x

u

x

uK

~

3

1~~

2

12

3

2

LuC TMT

KuT 2

L

uC T

D

3

otherwisegAu

ifgAuCS

iLiT

tiLiTBK ,0max

K

Lt

P

x

P

i

1

ii x

Pg

1

Boussinesq eddy viscosity assumption

Turbulent viscosity

Turbulent velocity

Turbulent dissipation rate

Acceleration of fluids interface due to pressure gradient

Turbulent energy (K) production Mean flow timescale

Turbulent timescale

RM-likeRT-like

Turbulent production source term limiter (SK)

At late time, the model over-predicts the production of the total kinetic energy:

• a posteriori analysis indicates that a threshold in the production of K is reached when the eddy size L exceeds a certain value of the mixing width (at the first interface)

• and the K source production term becomes:

2

,0.1minL

WSS cK

LK

• Limiting the turbulent viscosity affects the terms:

je

T

j x

L

x

je

T

j x

K

x

jF

T

j x

F

x

~

LuCS TMFT 1

Tifj

iF

uusu

uS

~1~

~Tangential velocity to the cell face

s

uus tif

~,1max

Local speed of sound

• Rescale turbulent viscosity (μT) using a limiter, SF :

Limiting the eddy viscosity

Tji

TurbulentShear StressTurbulent Diffusion

x

e

x e

T

j

~

Atwood number calculation

The original K-L model calculates the local cell Atwood number (ALi) based on the van Leer’s Monotonicity principle:

MON

1

11

MON

1

2

2

i

iiiL

i

iiiL

x

xx

x

xx

11

1

2

11

MON

,minsign2ii

ii

ii

iiii

i xxxxx

MON

MONi

iii

iASSi x

xL

LCA

SSiiLi AAA 0

2102100 ,max iii AAA

RR

RRi

LL

LLi

A

A

210

210

Modified K-L, Atwood Number

Uses the average values obtained during the reconstruction phase of the inviscid fluxes to estimate the: local Atwood number; gradients in turbulence model closure and source terms.

Weighted contribution of ASSi and A0i to obtain ALi

Modified K-L, Atwood Number

SSi0iLi AAA LL ww 1

1,minx

Lw i

L

Reconstructed values (F)

LLL

RRR

2

12

1

LR

LR

0iA

xxLR

i

i

iii

iA x

xL

LC

SSiA

Modified K-L: Enthalpy diffusion

Replaced turbulent diffusion of internal energy (qe) with enthalpy (qh), based on suggested physical diffusion mechanism (A.Cook, PoF, 2009)

hq

x

h

x

eq

x

e

x h

T

je

T

j

~~

where:

1~

ii

ii

Pe

i

iii

Peh

~

~

SpeciesofNumber,1

~,~,

~

1

i

heFN

iii

Modified K-L

Summary of modifications introduced to the original K-L (Dimonte & Tipton):Changed the internal energy turbulent diffusion flux to the enthalpy one

Make use of reconstruction values at the cell face to calculate the:o local Atwood number;o turbulence model closure and source terms;o turbulent viscosity for diffusion;

The local Atwood number is calculated using weighted contributions

Introduced an isotropic turbulent diffusion correction for 2D simulations

Reduce late time turbulent kinetic energy production

Young’s Two-Fluid Model

Mass transport:

Momentum transport:

r rp r r rp r rj s sp sr r rp rsj s

m f m f u m V m Vt x

ijr r ri r r rj ri r r rsi r r i

j i j s

Rpf u f u u f m X f g

t x x x

j rr r r r r rj r r r r r

j j j j

s s sr r r rs rs

u ef e f u e h p f D

t x x x x

e V e V f

Internal energy:

jr rj r rj j sr rs r r rs r s

j j js s

uf fu f u u V V h f p p

t x x x

Volume fraction:

Two-Fluid Model (cont.)

ijijsr

rsirisiK eRXuuS ~.

An equation for K is used which is similar to that in the K-L model but with a different source term:

The equation for L includes a source term involving fluid velocity differences and is different to that used in the K-L model:

sr

srsr

risjsrL ffuusffS

Turbulent viscosity is given by:

tt K where ℓt is proportional to L,turbulent diffusion coefficients are proportional to ℓt K1/2.

Two-Fluid Model (cont.)

is the fraction by mass of initial fluid p in phase rrpm

is the rate of transfer of volume from phase r to phase s; determines how rapidly the initial fluids mix at a molecular level.

rsV

is the rate of transfer of momentum from phase s to phase r accounting for drag, added mass and mass exchange.rsiX

• Model coefficients are chosen to give an appropriate value of α for RT mixing (typically 0.05 to 0.06);

• The volume transfer rate ΔVrs is chosen to give the corresponding value of the global mixing parameter for self-similar RT mixing;

• The ratio ℓt /L is chosen so that a fraction of about 0.3 to 0.4 of mixing for self-similar RT is due to turbulent diffusion.

Implicit Large Eddy Simulation CNS3D code

CNS3D code: Finite volume approach in conjunction with the HLLC Riemann solver

Several high-resolution and high-order schemes 2nd-order modified MUSCL (Drikakis et al., 1998, 2004) 5th-order MUSCL (Kim & Kim) and WENO (Shu et al.) 9th-order WENO for ILES (Mosedale & Drikakis, 2007) Specially designed schemes incorporating low Mach corrections

(Thornber et al., JCP, 2008)

5-equation quasi-conservative multi-component model (Allaire et al., JCP, 2007)

3rd-order Runge-Kutta in time

Lagrange-Remap AWE TURMOIL code

TURMOIL code: Lagrange-Remap method (David Youngs) 3rd-order spatial remapping; 2nd-order in time; Mass fraction mixture model.

For the semi-Lagrangian scheme Lagrangian phase:

• Quadratic artificial viscosity;• Negligible dissipation in the absence of shocks.

Remap phase:• 3rd-order monotonic method;• Mass and momentum conserved. The kinetic energy is

dissipated only in regions of non-smooth flow.

Turbulent Mixing Instabilities

Three cases are investigated:

1D Planar RT (1D-RT); 1D Double Planar RM (1D-RM); 2D Inverse Chevron (2D-IC);

Shear at the inclined interface subsequently results in formation of Kelvin-Helmholtz (KH) instabilities.

Validation

The model results are compared against high-resolution ILES:•Profiles of volume fraction (VF);•Profiles of turbulent kinetic energy (K);•Integral quantities such as the Total MIX and Total Turbulent Kinetic Energy (Total TKE) are employed:

dVFFMIX 212 ~~

Total

KdVTKE Total

For comparison with 2D RANS simulations, the 3D ILES results are Favre-averaged to a 2D plane in the homogeneous spanwise direction, and a surface integral is applied instead;

The results need to be multiplied with a spanwise length (Lz) for consistency with the 3D quantities.

Rayleigh-Taylor

FLUID PROPERTIES

Effect of Enthalpy Diffusion

Comparison of static Temperature profiles against Two-Fluid model (TF):

Effect of Enthalpy Diffusion

Comparison of VF and K profiles against Two-Fluid model (TF) and high-resolution ILES (Youngs 2013):

Effect of Enthalpy Diffusion

The modified model gives correct self-similar growth rates of mixing width (W) and maximum turbulent kinetic energy (KMAX):

Richtmyer-Meshkov

FLUID PROPERTIES

Volume fraction

Total MIX

Total TKE – ILES Comparison

Total TKE

VF-profiles

t=1.90ms t=2.22ms

VF-profiles (cont.)

t=2.70ms t=3.82ms

TKE-profiles

t=1.90ms t=2.22ms

TKE-profiles (cont.)

t=2.70ms t=3.82ms

Inverse Chevron

FLUID PROPERTIES

3D High-Resolution ILES EX

PK1

KMIN

2

1.9ms 2.7ms 3.3ms

1280x640x320 resolution (Hahn et. al., PoF, 2011)

K-L model applied to IC

•2D K-L turbulence model on 320x160 cells in x and y-directions;

•Complete on standard multi-core desktop PC within an hour;

•Assumes mean flow is zero in z-direction (only fluctuations).

Challenges:

•Strong anisotropic turbulent effects;

•Late time turbulent energy production;

•De-mixing.

Total MIX

Total TKE

Total TKE (cont.)

Evolution of VF (ILES)

t=0.5ms t=1.3ms t=1.9ms

t=2.2ms t=2.7ms t=3.3ms

VF contours at t=2.7ms

ILES KL

KL modifiedTF

VF contours at t=3.3ms

ILES KL

KL modifiedTF

Evolution of TKE (ILES)

t=0.5ms t=1.3ms t=1.9ms

t=2.2ms t=2.7ms t=3.3ms

TKE contours at t=2.7ms

ILES KL

KL modifiedTF

TKE contours at t=3.3ms

ILES KL

KL modifiedTF

Conclusions

Both models achieve self-similarity.

The correct treatment of the enthalpy flux is required in the K-L model in order to improve the model results.

Modifications in the calculation of the local Atwood number and limiting the turbulent viscosity and production of TKE significantly improve the K-L results.

The TF model overall predicts more accurately the K/Kmax profile.

The TF model gives more accurate results than the KL model at late times, where anisotropy and de-mixing dominates.

A key advantage of the TF model is its capability of representing the degree of molecular mixing in a direct way, by transferring mass between the two phases.