© British Crown Copyright 2007/MOD Numerical Simulation Using High-Resolution Methods A. D. Weatherhead, AWE D. Drikakis, Cranfield University

© British Crown Copyright 2007/MOD Aims To validate a well tested code in a new regime. To assess the behaviour of different numerical methods on Rayleigh-Taylor turbulent mix

© British Crown Copyright 2007/MOD Summary The two codes –CNS3D –Turmoil Gravity –Rising Bubble Rayleigh-Taylor Simulation –Single Mode RT –Multi Mode RT Conclusions

© British Crown Copyright 2007/MOD The Two Codes CNS3D Cranfield University’s compressible code developed by D.Drikakis Validated using: –Aerodynamic flows –Wing dynamics –Transonic atmosphere re- entry Turmoil AWE scientific research code developed by D.Youngs Validated using: –Turbulence modeling –Rocket rig experiments –Shock tube experiments

© British Crown Copyright 2007/MOD CNS3D Cell centered finite volume code Range of Riemann solvers: –Eberle –HLLC –Roe Numerous limiting methods: –Van Leer –Superbee –Kim&Kim 5thOrder MUSCL –WENO density momentum total energy

© British Crown Copyright 2007/MOD Turmoil Staggered grid Finite difference Lagrange re-map density internal energy velocity

© British Crown Copyright 2007/MOD Rising Bubble The bubble rise test problem originated from a paper by A S Almgren et al [i] in which they are modelling the rise of hot bubbles in type Ia supernovae. The problem is interesting because it does not have a hard boundary to the bubble and as such in the initial conditions all of the variables are smoothly varying. Having smoothly varying initial conditions should mean that the results are not significantly dependent on the limiter used.[i] [i] A S Almgren, J B Bell, C A Rendleman and M Zingale, “LOW MACH NUMBER MODELING OF TYPE Ia SUPERNOVAE. I. HYDRODYNAMICS”, The Astrophysical Journal, 637:922, 936 (2006) [i]

© British Crown Copyright 2007/MOD High Mach Results CNS3DTurmoil

© British Crown Copyright 2007/MOD Low Mach Results CNS3DTurmoil

© British Crown Copyright 2007/MOD Rayleigh-Taylor Instability Light fluid with higher pressure Minor perturbations are unstable

© British Crown Copyright 2007/MOD Single Mode RT This test problem is a simple single mode Rayleigh-Taylor calculation based on the single mode studies carried out by the  group [1]. The problem consist of a rectangular box with a heavy fluid (  =3g/cm 3 ) above a light fluid (  =1g/cm 3 ), both at rest, in a gravitational field. There is a single mode perturbation on the interface that develops to form a bubble in the centre and spikes at the corners of the box.

© British Crown Copyright 2007/MOD Single Mode 3D Results This test problem is very sensitive to the numerical scheme used. Important points to look out for are the amount of roll up in the spikes and height and shape of the top of the bubble. A dimple or quartering of the bubble often appears with the less diffusive schemes. TurmoilVanLeerSuperbee3D WENO

© British Crown Copyright 2007/MOD Results – 2D Slices Van Leer WENO 3 rd Order 3D WENO 3 rd Order 2 x resolution 3D WENO 3 rd Order WENO 5 th Order 2 x resolution Turmoil Van Leer 2 x resolution Superbee

© British Crown Copyright 2007/MOD High Resolution TurmoilWENO 5th3D WENOVan LeerMUSCL5th

© British Crown Copyright 2007/MOD Multi Mode RT The multimode calculations were carried out using the 128x128x128 initial conditions used by the alpha group [19]. The domain for the multimode calculations was 10.0 x 10.0 x 10.0 and was meshed with a uniform mesh of 128 cells in each direction. The density ratio is initially 3 to 1. Both the density and pressure have been adjusted to give hydrostatic equilibrium. The initial interface has been perturbed using the following equation: h0(x,y) = S ( ak cos(kxx)cos(kyy)+bk cos(kxx)sin(kyy) +ck sin(kxx)cos(kyy)+dk sin(kxx)sin(kyy)) where the sum is over all wavenumbers and spectral amplitudes (ak, bk, ck, dk) are chosen randomly.

© British Crown Copyright 2007/MOD 3D Results Turmoil Van LeerSuperbee 5 th Order WENO

© British Crown Copyright 2007/MOD Initial Conditions Wavelengths: 4cells to 16cells Wavelengths: 8cells to 32cells Wavelengths: 16cells to 64cells Van Leer

© British Crown Copyright 2007/MOD Resolution Van Leer Van Leer 256 3

© British Crown Copyright 2007/MOD Conclusion Different codes and methods agree on the macroscope behaviour. Numerical methods have a significant affect on the details of the calculations. The differences between the methods are more significant at low mach number. Low mach number modifications can significantly improve the behaviour.

© British Crown Copyright 2007/MOD

Results – 2D Slices Van Leer 2 x resolution Van Albada N=20 ENO 2 nd Order Van Albada N=3 Van Albada Minbee N=2 Minbee N=4 Superbee WENO 3 rd Order 3D WENO 3 rd Order 2 x resolution 3D WENO 3 rd Order WENO 5 th Order 2 x resolution Turmoil

© British Crown Copyright 2007/MOD 3D Results Van Leer NEPS=20 Turmoil 3D WENO Van Leer NEPS=3

© British Crown Copyright 2007/MOD