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ABSTRACT
Rigorously space filtering the thermal, multispecies Navier–Stokes (NS) conservation principle partial differential equation (PDE) system embeds a priori undefined tensor and vector quadruples. Large eddy simulation (LES) computational fluid dynamics algorithm resolutions replace the tensor quadruple with a single tensor then secures closure through “physics-based” modeling, assuming the velocity field is turbulent, i.e., the Reynolds number (Re) is large. In complete distinction, a totally analytical closure is derived for the rigorously generated tensor/vector quadruples, achieved totally absent any modeling component or Re assumption. For Gaussian filter of uniform measure δ, derived analytical filtered Navier–Stokes (aFNS) theory PDE system state variable is significance scaled O(1; δ2; δ3) through classic fluid mechanics perturbation theory. That uniform measure δ filter penetrates domain boundaries requires O(1) resolved scale PDE system inclusion of boundary commutation error (BCE) integrals, (unfiltered) NS state variable extension in the sense of distributions, and domain enlargement to encompass all surfaces with Dirichlet boundary condition (DBC) specification. Theory-derived O(δ2) resolved–unresolved scale interaction PDE system, also the O(1) system, is rendered bounded domain, well posed through a priori identification of O(1; δ2) state variable nonhomogeneous DBCs. BCE and DBC resolution algorithm derivations use O(δ4) approximate deconvolution (AD) differential definition Galerkin weak forms. Theory analytically derived unresolved scale O(δ3) state variable annihilates discretization-induced O(h2) dispersion error at unresolved scale threshold δ, h the mesh measure. Net is an analytical theory closing rigorously space-filtered NS exhibiting potential for first principles prediction of viscous laminar–turbulent transition, separation, and relaminarization.
Nomenclature
| AD | = | approximate deconvolution |
| aFNS | = | analytical filtered Navier–Stokes |
| Aδ(∇ (•)) | = | boundary commutation error integral |
| BC | = | boundary condition |
| BL | = | boundary layer |
| cij | = | |
| CS | = | Smagorinsky constant, generalizations |
| CFD | = | computational fluid dynamics |
| d(•) | = | ordinary derivative, differential element |
| D | = | dimensional |
| D(•) | = | differential definition |
| D(u, P) | = | NS total stress tensor ≡− ∇ P + (2/Re) ∇ · S(u) |
| diag[•] | = | diagonal matrix |
| DNS | = | direct numerical simulation |
| DOF | = | approximation degrees of freedom |
| e | = | element-dependent (subscript) |
| eh | = | semidiscrete approximation error |
| EBV | = | elliptic boundary value |
| Ec | = | Eckert number = U2/cp (Tmax − Tmin) |
| E | = | energy norm (subscript) |
| fj | = | flux vector |
| F(•) | = | Fourier transform |
| f | = | source term |
| g | = | gravity magnitude |
| g | = | gravity vector |
| Gr | = | Grashof number ≡ gβΔTL3/ν2 |
| GWS | = | Galerkin weak statement |
| h | = | mesh measure; spatially semidiscrete (superscript) |
| H | = | Hilbert space |
| I | = | identity matrix |
| I-EBV | = | initial-elliptic boundary value |
| J | = | algorithm matrix tensor index |
| k | = | trial space basis degree |
| ℓ(·) | = | differential operator on ∂Ω |
| L | = | reference length scale |
| ℒ(·) | = | differential operation on Ω |
| LES | = | large eddy simulation |
| [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 | |
| N | = | boundary domain matrix prefix |
| N | = | continuum approximation (superscript) |
| NWR | = | near-wall resolution |
| NS | = | Navier–Stokes |
| {Nk} | = | trial space (finite element) 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 | = | discrete-dependent variable degrees of freedom |
| {Q} | = | DOF column matrix |
| Ra | = | Rayleigh number =Gr/Pr |
| RaNS | = | Reynolds-averaged Navier–Stokes |
| Re | = | Reynolds number ≡ UL/ν |
| = | Euclidean space of dimension n | |
| s | = | coordinate tangent to ∂Ω |
| S | = | Stokes tensor dyadic |
| Sij | = | Stokes tensor |
| Se | = | weak statement matrix assembly operator |
| Sc | = | Schmidt number ≡ D/ν |
| SFS | = | subfilter scale (tensor, vector) |
| SGS | = | subgrid scale (tensor) |
| sym | = | symmetric |
| t | = | time |
| T | = | temperature |
| TS | = | Taylor series |
| u | = | velocity vector |
| u | = | velocity resolution magnitude |
| = | subfilter scale stress tensor | |
| = | subfilter scale thermal vector | |
| = | subfilter scale mass fraction vector | |
| = | space-filtered velocity | |
| u+ | = | BL similarity variable =u/uτ |
| U | = | reference velocity |
| x, xi | = | Cartesian coordinate, system 1 ≤i ≤ n |
| y+ | = | BL similarity variable =uτy/ν |
| Yα | = | mass fraction |
| ∇ | = | gradient operator |
| ∇2 | = | Laplacian operator |
| d(·)/dt | = | ordinary derivative |
| ∂(·)/∂xj | = | partial derivative |
| {·} | = | column matrix |
| {·}T | = | row matrix |
| [·] | = | square matrix |
| diag[·] | = | diagonal square matrix |
| || · || | = | norm |
| ∪ | = | union (nonoverlapping sum) |
| ∀ | = | denotes for all |
| ∈ | = | denotes inclusion |
| ⊂ | = | denotes belongs to |
| * | = | convolution |
| = | space-filtered variable | |
| α | = | mass fraction member (subscript) |
| β | = | absolute temperature |
| γ | = | Gaussian filter shape factor |
| δ | = | Gaussian filter measure (diameter) |
| ϕ | = | velocity potential function |
| Ψα(x) | = | continuum trial space |
| ν | = | 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
| AD | = | approximate deconvolution |
| aFNS | = | analytical filtered Navier–Stokes |
| Aδ(∇ (•)) | = | boundary commutation error integral |
| BC | = | boundary condition |
| BL | = | boundary layer |
| cij | = | |
| CS | = | Smagorinsky constant, generalizations |
| CFD | = | computational fluid dynamics |
| d(•) | = | ordinary derivative, differential element |
| D | = | dimensional |
| D(•) | = | differential definition |
| D(u, P) | = | NS total stress tensor ≡− ∇ P + (2/Re) ∇ · S(u) |
| diag[•] | = | diagonal matrix |
| DNS | = | direct numerical simulation |
| DOF | = | approximation degrees of freedom |
| e | = | element-dependent (subscript) |
| eh | = | semidiscrete approximation error |
| EBV | = | elliptic boundary value |
| Ec | = | Eckert number = U2/cp (Tmax − Tmin) |
| E | = | energy norm (subscript) |
| fj | = | flux vector |
| F(•) | = | Fourier transform |
| f | = | source term |
| g | = | gravity magnitude |
| g | = | gravity vector |
| Gr | = | Grashof number ≡ gβΔTL3/ν2 |
| GWS | = | Galerkin weak statement |
| h | = | mesh measure; spatially semidiscrete (superscript) |
| H | = | Hilbert space |
| I | = | identity matrix |
| I-EBV | = | initial-elliptic boundary value |
| J | = | algorithm matrix tensor index |
| k | = | trial space basis degree |
| ℓ(·) | = | differential operator on ∂Ω |
| L | = | reference length scale |
| ℒ(·) | = | differential operation on Ω |
| LES | = | large eddy simulation |
| [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 | |
| N | = | boundary domain matrix prefix |
| N | = | continuum approximation (superscript) |
| NWR | = | near-wall resolution |
| NS | = | Navier–Stokes |
| {Nk} | = | trial space (finite element) 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 | = | discrete-dependent variable degrees of freedom |
| {Q} | = | DOF column matrix |
| Ra | = | Rayleigh number =Gr/Pr |
| RaNS | = | Reynolds-averaged Navier–Stokes |
| Re | = | Reynolds number ≡ UL/ν |
| = | Euclidean space of dimension n | |
| s | = | coordinate tangent to ∂Ω |
| S | = | Stokes tensor dyadic |
| Sij | = | Stokes tensor |
| Se | = | weak statement matrix assembly operator |
| Sc | = | Schmidt number ≡ D/ν |
| SFS | = | subfilter scale (tensor, vector) |
| SGS | = | subgrid scale (tensor) |
| sym | = | symmetric |
| t | = | time |
| T | = | temperature |
| TS | = | Taylor series |
| u | = | velocity vector |
| u | = | velocity resolution magnitude |
| = | subfilter scale stress tensor | |
| = | subfilter scale thermal vector | |
| = | subfilter scale mass fraction vector | |
| = | space-filtered velocity | |
| u+ | = | BL similarity variable =u/uτ |
| U | = | reference velocity |
| x, xi | = | Cartesian coordinate, system 1 ≤i ≤ n |
| y+ | = | BL similarity variable =uτy/ν |
| Yα | = | mass fraction |
| ∇ | = | gradient operator |
| ∇2 | = | Laplacian operator |
| d(·)/dt | = | ordinary derivative |
| ∂(·)/∂xj | = | partial derivative |
| {·} | = | column matrix |
| {·}T | = | row matrix |
| [·] | = | square matrix |
| diag[·] | = | diagonal square matrix |
| || · || | = | norm |
| ∪ | = | union (nonoverlapping sum) |
| ∀ | = | denotes for all |
| ∈ | = | denotes inclusion |
| ⊂ | = | denotes belongs to |
| * | = | convolution |
| = | space-filtered variable | |
| α | = | mass fraction member (subscript) |
| β | = | absolute temperature |
| γ | = | Gaussian filter shape factor |
| δ | = | Gaussian filter measure (diameter) |
| ϕ | = | velocity potential function |
| Ψα(x) | = | continuum trial space |
| ν | = | 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 |
Acknowledgments
During completion of this dissertation project, the first author served as HPC Graduate Assistant in the Joint Institute for Computational Sciences (JICS), a collaboration between the USA DOE Oak Ridge National Laboratory and the University of Tennessee/Knoxville (UTK). Dissertation coordination was through the UTK College of Engineering CFD Laboratory.