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ABSTRACT
Worldwide, computational fluid dynamics (CFD) codes for Navier–Stokes (NS), Reynolds-averaged Navier–Stokes (RaNS), and/or large eddy simulation NS (LES) partial differential equation (PDE) systems are invariably based on second-order discrete numerics. Resulting nonlinear convection term discretizations inject an O(h2) dispersive error mechanism, h the mesh measure, inducing code algebraic destabilization for practical Reynolds numbers (Re). Code universal resolution is PDE discretization augmentation with a (usually) difference algebra derived numerical diffusion scheme to render O(h2) dispersion error destabilization nonpathological. The penalty of such schemes is artificial diffusion compromising of sharp fronts and/or discontinuities and generation of nonmonotone CFD approximations. Such legacy practices are now rendered obsolete by a totally analytical theory that rigorously identifies, in the continuum (!), the O(h2) truncation error terms resident but unspecified in NS/RaNS/LES PDE system second-order CFD spatial discretizations. The theory removes identified O(h2) error terms by alteration of the continuum appearance of NS/RaNS/LES PDE systems with nonlinear vector differential calculus operators. Theory is amenable to any second-order “tri-diagonal stencil” equivalent CFD discretization and, upon implementation, elevates the original second-order numerics code to O(h4) with no further action. This Taylor series error estimate is weak form theory formalized to a regular solution adapted nonuniform mesh refinement O(h4) asymptotic error estimate. Theory implementation in a linear basis optimal Galerkin criterion weak form algorithm CFD code enables a posteriori data generation validating annihilation of O(h2) dispersive error mechanisms for reduced NS, full NS, and RaNS PDE systems. In every instance, theory implementation leads to CFD monotone solution distributions free from artificial diffusion influence on sufficiently refined meshes. Differential definition Galerkin weak forms, code post-processed, quantify theory annihilated O(h2) dispersion error spectra, RaNS state variable member specific.
Nomenclature
| BC | = | boundary condition |
| BL | = | boundary layer |
| CFD | = | computational fluid dynamics |
| d(•) | = | ordinary derivative, differential element |
| D(•) | = | differential definition |
| DOF | = | approximation degrees-of-freedom |
| e | = | element-dependent (subscript) |
| eh | = | discrete approximation error |
| Ec | = | Eckert number = U2/cp (Tmax – Tmin) |
| E | = | energy norm (subscript), total internal energy |
| fj | = | flux vector |
| g | = | gravity magnitude |
| g | = | gravity vector |
| Gr | = | Grashoff number ≡ gβΔTL3/ν2 |
| GWS | = | Galerkin weak statement |
| h | = | mesh measure; spatially discrete (superscript) |
| J | = | algorithm matrix tensor index |
| k | = | trial space basis degree, turbulent kinetic energy |
| ℓ(·) | = | differential operator on ∂Ω |
| L | = | reference length scale |
| LES | = | large eddy simulation Navier–Stokes |
| ℒ(·) | = | differential operator on Ω |
| mi | = | momentum vector |
| [M200] | = | finite element mass matrix |
| M | = | element domain matrix prefix; total elements spanning Ωh |
| n | = | index; normal (subscript); dimension of domain Ω; integer |
| non-D | = | nondimensional |
| = | outward pointing unit vector normal to ∂Ω | |
| NS | = | Navier–Stokes |
| {Nk} | = | finite element trial space basis of degree k |
| O(•) | = | order of argument (•) |
| p | = | pressure |
| P | = | kinematic pressure = p/ρ0 |
| PDE | = | partial differential equation |
| Pr | = | Prandtl number ≡ ρ0νcp/k |
| pr | = | mesh nonuniform progression ratio |
| q | = | generalized dependent variable |
| {q} | = | PDE system state variable |
| Q | = | discrete state variable degree-of-freedom |
| {Q} | = | DOF column matrix |
| Ra | = | Rayleigh number = Gr/Pr |
| RaNS | = | Reynolds-averaged Navier–Stokes |
| Re | = | Reynolds number ≡ UL/ν |
| ReT | = | turbulent Reynolds number ≡ νT/ν |
| ℝn | = | Euclidean space of dimension n |
| s | = | coordinate tangent to ∂Ω, source term |
| Sij | = | Stokes tensor |
| Se | = | weak statement matrix assembly operator |
| Sc | = | Schmidt number ≡ D/ν |
| t | = | time |
| T | = | temperature |
| TS | = | Taylor series |
| u | = | velocity vector |
| u | = | velocity resolution magnitude |
| u+ | = | BL similarity variable = u/uτ |
| = | space filtered NS resolved scale velocity | |
| uτ | = | BL friction velocity |
| = | space filtered NS unresolved scale velocity tensor product | |
| U | = | reference velocity |
| x, xi | = | Cartesian coordinate, system 1 ≤ i ≤ n |
| y+ | = | BL similarity variable = uτy/ν |
| ∇ | = | gradient operator |
| ∇2 | = | Laplacian operator |
| d(·)/dt | = | ordinary derivative |
| ∂(·)/∂xj | = | partial derivative |
| {·} | = | column matrix |
| {·}T | = | row matrix |
| [·] | = | square matrix |
| ‖ · ‖ | = | norm |
| ∪ | = | union (nonoverlapping sum) |
| α | = | regular mesh refinement parameter (subscript) |
| β | = | unsteady TS theory parameter, absolute temperature |
| ε | = | NS viscosity, isotropic dissipation function |
| ϕ | = | velocity potential function |
| ν | = | kinematic viscosity |
| νT | = | turbulent kinematic viscosity |
| θ | = | time TS implicitness factor |
| Θ | = | potential temperature ≡ (T − Tmin)/(Tmax – Tmin) |
| ρ | = | density |
| dσ | = | differential element on ∂Ω |
| τij | = | Reynolds stress tensor |
| Ω | = | domain of differential equation |
| Ωe | = | discretization finite element domain |
| Ωh | = | discretization of Ω |
| ∂Ω | = | boundary segment of Ω |
Nomenclature
| BC | = | boundary condition |
| BL | = | boundary layer |
| CFD | = | computational fluid dynamics |
| d(•) | = | ordinary derivative, differential element |
| D(•) | = | differential definition |
| DOF | = | approximation degrees-of-freedom |
| e | = | element-dependent (subscript) |
| eh | = | discrete approximation error |
| Ec | = | Eckert number = U2/cp (Tmax – Tmin) |
| E | = | energy norm (subscript), total internal energy |
| fj | = | flux vector |
| g | = | gravity magnitude |
| g | = | gravity vector |
| Gr | = | Grashoff number ≡ gβΔTL3/ν2 |
| GWS | = | Galerkin weak statement |
| h | = | mesh measure; spatially discrete (superscript) |
| J | = | algorithm matrix tensor index |
| k | = | trial space basis degree, turbulent kinetic energy |
| ℓ(·) | = | differential operator on ∂Ω |
| L | = | reference length scale |
| LES | = | large eddy simulation Navier–Stokes |
| ℒ(·) | = | differential operator on Ω |
| mi | = | momentum vector |
| [M200] | = | finite element mass matrix |
| M | = | element domain matrix prefix; total elements spanning Ωh |
| n | = | index; normal (subscript); dimension of domain Ω; integer |
| non-D | = | nondimensional |
| = | outward pointing unit vector normal to ∂Ω | |
| NS | = | Navier–Stokes |
| {Nk} | = | finite element trial space basis of degree k |
| O(•) | = | order of argument (•) |
| p | = | pressure |
| P | = | kinematic pressure = p/ρ0 |
| PDE | = | partial differential equation |
| Pr | = | Prandtl number ≡ ρ0νcp/k |
| pr | = | mesh nonuniform progression ratio |
| q | = | generalized dependent variable |
| {q} | = | PDE system state variable |
| Q | = | discrete state variable degree-of-freedom |
| {Q} | = | DOF column matrix |
| Ra | = | Rayleigh number = Gr/Pr |
| RaNS | = | Reynolds-averaged Navier–Stokes |
| Re | = | Reynolds number ≡ UL/ν |
| ReT | = | turbulent Reynolds number ≡ νT/ν |
| ℝn | = | Euclidean space of dimension n |
| s | = | coordinate tangent to ∂Ω, source term |
| Sij | = | Stokes tensor |
| Se | = | weak statement matrix assembly operator |
| Sc | = | Schmidt number ≡ D/ν |
| t | = | time |
| T | = | temperature |
| TS | = | Taylor series |
| u | = | velocity vector |
| u | = | velocity resolution magnitude |
| u+ | = | BL similarity variable = u/uτ |
| = | space filtered NS resolved scale velocity | |
| uτ | = | BL friction velocity |
| = | space filtered NS unresolved scale velocity tensor product | |
| U | = | reference velocity |
| x, xi | = | Cartesian coordinate, system 1 ≤ i ≤ n |
| y+ | = | BL similarity variable = uτy/ν |
| ∇ | = | gradient operator |
| ∇2 | = | Laplacian operator |
| d(·)/dt | = | ordinary derivative |
| ∂(·)/∂xj | = | partial derivative |
| {·} | = | column matrix |
| {·}T | = | row matrix |
| [·] | = | square matrix |
| ‖ · ‖ | = | norm |
| ∪ | = | union (nonoverlapping sum) |
| α | = | regular mesh refinement parameter (subscript) |
| β | = | unsteady TS theory parameter, absolute temperature |
| ε | = | NS viscosity, isotropic dissipation function |
| ϕ | = | velocity potential function |
| ν | = | kinematic viscosity |
| νT | = | turbulent kinematic viscosity |
| θ | = | time TS implicitness factor |
| Θ | = | potential temperature ≡ (T − Tmin)/(Tmax – Tmin) |
| ρ | = | density |
| dσ | = | differential element on ∂Ω |
| τij | = | Reynolds stress tensor |
| Ω | = | domain of differential equation |
| Ωe | = | discretization finite element domain |
| Ωh | = | discretization of Ω |
| ∂Ω | = | boundary segment of Ω |