© british crown copyright 2007/mod numerical simulation using high-resolution methods a. d....
TRANSCRIPT
© 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
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Summary
• The two codes– CNS3D– Turmoil
• Gravity– Rising Bubble
• Rayleigh-Taylor Simulation– Single Mode RT– Multi Mode RT
• Conclusions
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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
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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
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Turmoil
• Staggered grid
• Finite difference
• Lagrange re-map density
internal energy
velocity
velocity
velocity velocity
velocity
velocity
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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] 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)
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High Mach ResultsCNS3D Turmoil
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Low Mach ResultsCNS3D Turmoil
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Rayleigh-Taylor Instability
• Light fluid with higher pressure
• Minor perturbations are unstable
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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/cm3) above a light fluid (=1g/cm3), 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.
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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.
Turmoil VanLeer Superbee 3D WENO
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Results – 2D Slices
Van Leer
WENO3rd Order
3D WENO3rd Order2 x resolution
3D WENO3rd Order
WENO5th Order2 x resolution
Turmoil
Van Leer2 x resolution
Superbee
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High Resolution
Turmoil WENO 5th 3D WENO Van Leer MUSCL5th
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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.
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3D ResultsTurmoil
Van Leer Superbee
5th Order WENO
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Initial Conditions
Wavelengths:
4cells to 16cells
Wavelengths:
8cells to 32cells
Wavelengths:
16cells to 64cells
Van Leer Van Leer Van Leer
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Resolution
Van Leer 1283 Van Leer 2563
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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.
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Results – 2D SlicesVan Leer
Van Leer2 x resolution
Van AlbadaN=20
ENO2nd Order
Van AlbadaN=3
Van Albada
Minbee
MinbeeN=2
MinbeeN=4
Superbee
WENO3rd Order
3D WENO3rd Order2 x resolution
3D WENO3rd Order
WENO5th Order2 x resolution
Turmoil
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3D Results
Van Leer NEPS=20
Turmoil3D WENO
Van Leer NEPS=3
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