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 volume o Lagrange-Remap  Assessment of turbulence models vs. ILES for compressible turbulent mixing o 1D Rayleigh-Taylor o Double Planar Richtmyer-Meshkov o 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) Additional Terms Require 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. Turbulent kinetic energy: Turbulent length scale:

K-L model (cont.) Additional closure terms: 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-like RT-like

Turbulent production source term limiter (S K ) 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:

Limiting the turbulent viscosity affects the terms: Tangential velocity to the cell face Local speed of sound Rescale turbulent viscosity (μ T ) using a limiter, S F : Limiting the eddy viscosity Turbulent Shear Stress Turbulent Diffusion

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

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 A SSi and A 0i to obtain A Li

Modified K-L, Atwood Number Reconstructed values (F)

Modified K-L: Enthalpy diffusion Replaced turbulent diffusion of internal energy (q e ) with enthalpy (q h ), based on suggested physical diffusion mechanism (A.Cook, PoF, 2009) where:

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:  Internal energy:  Volume fraction:

Two-Fluid Model (cont.)  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:  Turbulent viscosity is given by: where ℓ t is proportional to L, turbulent diffusion coefficients are proportional to ℓ t K 1/2.

Two-Fluid Model (cont.) is the fraction by mass of initial fluid p in phase r is the rate of transfer of volume from phase r to phase s; determines how rapidly the initial fluids mix at a molecular level. is the rate of transfer of momentum from phase s to phase r accounting for drag, added mass and mass exchange. Model coefficients are chosen to give an appropriate value of α for RT mixing (typically 0.05 to 0.06); The volume transfer rate ΔV rs is chosen to give the corresponding value of the global mixing parameter for self-similar RT mixing; The ratio ℓ t /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  2 nd -order modified MUSCL (Drikakis et al., 1998, 2004)  5 th -order MUSCL (Kim & Kim) and WENO (Shu et al.)  9 th -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)  3 rd -order Runge-Kutta in time

Lagrange-Remap AWE TURMOIL code  TURMOIL code: Lagrange-Remap method (David Youngs)  3 rd -order spatial remapping;  2 nd -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: 3 rd -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:  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 (L z ) for consistency with the 3D quantities.

Rayleigh-Taylor FLUID PROPERTIES G=1.105cm 2 /s ρ H =20gr/cm 3 ρ L =1gr/cm 3 P int =1000dyn/cm 2 γ H ≠ γ L Atwood Number ≈ 0.90

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 (K MAX ):

Richtmyer-Meshkov FLUID PROPERTIES P=1bar ρ sf6 =6.34kg/m 3 ρ air =1.184kg/m 3 U * air = m/s P * air =1.675bar ρ * air =1.7047kg/m 3 Atwood Number 0.67

Volume fraction

Total MIX

Total TKE – ILES Comparison

Total TKE

VF-profiles t=1.90mst=2.22ms

VF-profiles (cont.) t=2.70mst=3.82ms

TKE-profiles t=1.90mst=2.22ms

TKE-profiles (cont.) t=2.70mst=3.82ms

Inverse Chevron FLUID PROPERTIES Favre-averaged 3D initial condition to 2D plane for mean flow quantities. ρ sf6 =6.34kg/m 3 ρ air =1.184kg/m 3, ρ * air =1.7264kg/m 3 P * air =1.706bar, P=1bar Shock Mach Number=1.26 Atwood Number 0.67

3D High-Resolution ILES EXP K1 KMIN2 1.9ms2.7ms 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=0.5mst=1.3mst=1.9ms t=2.2mst=2.7ms t=3.3ms

VF contours at t=2.7ms ILES KL KL modified TF

VF contours at t=3.3ms ILES KL KL modified TF

Evolution of TKE (ILES) t=0.5mst=1.3mst=1.9ms t=2.2mst=2.7ms t=3.3ms

TKE contours at t=2.7ms ILES KL KL modified TF

TKE contours at t=3.3ms ILES KL KL modified TF

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.