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ABSTRACT
To provide an effective numerical method for the large eddy simulation (LES) of turbulent flows with shocks, a hybrid scheme is developed in a finite volume framework based on the fourth-order central scheme and the third-order weighted essentially non-oscillatory (WENO) scheme. A total of six easy-to-implement and promising switch functions (SFs) are examined in the hybrid central–WENO scheme for the LES of compressible turbulent flows. Both the dissipation and dispersion of the developed hybrid central–WENO scheme are theoretically confirmed using the Fourier technique. Then, the effectiveness and accuracy of this scheme and the SFs are numerically tested by three problems: decaying compressible isotropic turbulence, inviscid, and turbulent transonic flow over a bump. The numerical results show the developed hybrid scheme, coupled with the SF based on local velocity divergence and pressure gradient, has excellent capabilities of capturing shocks and resolving turbulence.
Nomenclature
| ds | = | control parameter of switch function 4 |
| e0 | = | specific total energy |
| E | = | energy spectrum |
| EK | = | turbulent kinetic energy |
| Euu | = | frequency spectrum of x-velocity |
| Evv | = | frequency spectrum of y-velocity |
| Eww | = | frequency spectrum of z-velocity |
| Ec | = | inviscid flux vector in x-direction |
| Ev | = | viscid flux vector in x-direction |
| Fc | = | inviscid flux vector in y-direction |
| Fv | = | viscid flux vector in y-direction |
| Gc | = | inviscid flux vector in z-direction |
| Gv | = | viscid flux vector in z-direction |
| = | inviscid numerical flux vector at cell face | |
| = | viscid numerical flux vector at cell face | |
| k | = | wave number |
| ki | = | imaginary part of modified wave number |
| kr | = | real part of modified wave number |
| LI | = | integral length scale |
| M | = | Mach number |
| p | = | static pressure |
| p0 | = | total pressure |
| ps | = | control parameter of switch function 1 |
| Pr | = | Prandtl number |
| q | = | primitive variable |
| = | candidate stencil in WENO scheme | |
| q | = | primitive variable vector |
| Q | = | conservative variable vector |
| rc | = | control parameter of switch function 2 |
| rd | = | control parameter of switch function 5 |
| Re | = | Reynolds number |
| Reλ | = | Taylor Reynolds number |
| Sij | = | strain rate tensor |
| S | = | area vector of cell face |
| t | = | time |
| T | = | temperature |
| u, v, w | = | velocities in x-, y- and z-directions |
| u | = | velocity vector |
| y+ | = | distance to wall in wall unit |
| α | = | wave number times space step |
| = | smoothness indicator in WENO scheme | |
| χ | = | control parameter in filter operator |
| Δ | = | filter width |
| Δx | = | space step |
| Δx+ | = | streamwise grid size in wall unit |
| Δy+ | = | wall-normal grid size in wall unit |
| Δz+ | = | spanwise grid size in wall unit |
| γ | = | specific heat ratio |
| λ | = | Taylor scale |
| λc | = | control parameter of switch function 3 |
| μ | = | dynamic viscosity |
| Θj | = | heat flux vector |
| = | subgrid-scale (SGS) heat flux vector | |
| ρ | = | density |
| σij | = | viscid stress tensor |
| σj + 1/2 | = | switch function |
| τ | = | initial large-eddy-turnover time |
| = | subgrid-scale (SGS) viscid stress tensor | |
| τs | = | control parameter of switch function 6 |
| = | weight of candidate stencil in WENO scheme | |
| ω | = | vorticity |
| Ω | = | grid cell volume |
| Subscripts | = | |
| i, j, k | = | grid cell index or Cartesian components |
| t | = | turbulence value |
| ref | = | reference |
| Superscripts | = | |
| L | = | left side of cell face |
| Overbars | = | |
| − | = | filter operator |
| ∼ | = | Favre-average filter operator |
| Additional symbols are defined in the text. | = |
Nomenclature
| ds | = | control parameter of switch function 4 |
| e0 | = | specific total energy |
| E | = | energy spectrum |
| EK | = | turbulent kinetic energy |
| Euu | = | frequency spectrum of x-velocity |
| Evv | = | frequency spectrum of y-velocity |
| Eww | = | frequency spectrum of z-velocity |
| Ec | = | inviscid flux vector in x-direction |
| Ev | = | viscid flux vector in x-direction |
| Fc | = | inviscid flux vector in y-direction |
| Fv | = | viscid flux vector in y-direction |
| Gc | = | inviscid flux vector in z-direction |
| Gv | = | viscid flux vector in z-direction |
| = | inviscid numerical flux vector at cell face | |
| = | viscid numerical flux vector at cell face | |
| k | = | wave number |
| ki | = | imaginary part of modified wave number |
| kr | = | real part of modified wave number |
| LI | = | integral length scale |
| M | = | Mach number |
| p | = | static pressure |
| p0 | = | total pressure |
| ps | = | control parameter of switch function 1 |
| Pr | = | Prandtl number |
| q | = | primitive variable |
| = | candidate stencil in WENO scheme | |
| q | = | primitive variable vector |
| Q | = | conservative variable vector |
| rc | = | control parameter of switch function 2 |
| rd | = | control parameter of switch function 5 |
| Re | = | Reynolds number |
| Reλ | = | Taylor Reynolds number |
| Sij | = | strain rate tensor |
| S | = | area vector of cell face |
| t | = | time |
| T | = | temperature |
| u, v, w | = | velocities in x-, y- and z-directions |
| u | = | velocity vector |
| y+ | = | distance to wall in wall unit |
| α | = | wave number times space step |
| = | smoothness indicator in WENO scheme | |
| χ | = | control parameter in filter operator |
| Δ | = | filter width |
| Δx | = | space step |
| Δx+ | = | streamwise grid size in wall unit |
| Δy+ | = | wall-normal grid size in wall unit |
| Δz+ | = | spanwise grid size in wall unit |
| γ | = | specific heat ratio |
| λ | = | Taylor scale |
| λc | = | control parameter of switch function 3 |
| μ | = | dynamic viscosity |
| Θj | = | heat flux vector |
| = | subgrid-scale (SGS) heat flux vector | |
| ρ | = | density |
| σij | = | viscid stress tensor |
| σj + 1/2 | = | switch function |
| τ | = | initial large-eddy-turnover time |
| = | subgrid-scale (SGS) viscid stress tensor | |
| τs | = | control parameter of switch function 6 |
| = | weight of candidate stencil in WENO scheme | |
| ω | = | vorticity |
| Ω | = | grid cell volume |
| Subscripts | = | |
| i, j, k | = | grid cell index or Cartesian components |
| t | = | turbulence value |
| ref | = | reference |
| Superscripts | = | |
| L | = | left side of cell face |
| Overbars | = | |
| − | = | filter operator |
| ∼ | = | Favre-average filter operator |
| Additional symbols are defined in the text. | = |