Perspectives in m odeling w all effects in the RANS approach - PowerPoint PPT Presentation

National I nstitute of Aerospace Future Directions in CFD Research, A Modeling and Sim ulation Conference August 6 -8 , 2 0 1 2 Perspectives in m odeling w all effects in the RANS approach Svetlana V. Poroseva Mechanical Engineering Departm ent University of New Mexico

Outline • Identification of problems with the RANS approach • Possible directions for their solution

RANS approach: identity crisis General belief: RANS models are one- or two-equation turbulence models that require the modeling of wall effects ∂ < > ∂ ∂ ∂ ∂ 2 u u U U 1 P U + + = − + ν i j i i i U ∂ ∂ ∂ ρ ∂ ∂ ∂ j t x x x x x j j i j j   ∂ ∂ U U 2 < > = − ν + − δ   j i u u k   ∂ ∂ i j T ij x x 3   j i − − ω − ε − ξ − ϕ − ε − ω − k l , k , k , k , k , k / k , multi scale ....

RANS approach: reality RANS model is any statistical closure obtained from the infinite set of the Reynolds-Averaged Navier-Stokes equations ∂ < > u u DU = − +⋅⋅⋅ 1 st order i j i ∂ Dt x j < > ∂ < > D u u u u u i j i j k = − + Π − ε + ⋅ ⋅ ⋅ 2 nd order ij ij ∂ Dt x k < > ∂ < > D u u u u u u u i j k i j k l 3 rd order = − + Π − ε + ⋅ ⋅ ⋅ ijk ijk ∂ Dt x l The order of a closure is determined by the order of the moments for which the equations are solved. Higher the order, higher the m odel fidelity.

The infinite set of the Reynolds-Averaged Navier-Stokes equations describes com pletely the turbulent flow physics from the statistics point of view • Two-equation models are the first-order closures. They are the simplest from the RANS family of models with not so much physics left. • The purpose of any corrections in such models is not to bring more physics, but to compensate for its lack. W hy not to increase the closure order instead?

Second-order closures ( RSTMs) ∂ < > ∂ ∂ ∂ ∂ 2 u u U U U 1 P + + = − + ν i j i i i U ∂ ∂ ∂ ρ ∂ ∂ ∂ j t x x x x x j j i j j ∂ < > ∂ < > ∂   ∂ u u u u U U P + = − < > + < > i j i j j i   U u u u u ij ∂ ∂ ∂ ∂ k j k i k   t x x x k k k   ∂ < > ∂ ∂ u u u 1 p p + Π ( ) T − − < > + < > i j k D   u u ij ij ∂ ρ ∂ ∂ j i   x x x   k i j ∂ < > ∂ ∂ 2 u u u u − ε + ν − ν < > i j j ( M ) i 2 D ∂ ∂ ∂ ∂ x x x x ij ij k k k k Algebraic models did not show much potential in flows of practical importance

Current state-of-the art in RSTMs Term s to m odel Modeling directions ∂ < > u u u < > = i j k ( ) T u u u D ∂ i j k ij x k ∂ < > ∂ < > ∂ ∂ pu 1 ( pu 1 ( p p Φ = Π − > + Π = − < > + < > j i ) u u ) ρ ∂ ∂ ρ ∂ ∂ ij ij ij i j x x x x j i j i ε ij

Third-order m om ents m odeling A tensor can only be m odeled as a tensor of the sam e tensor rank, index order, covariance, and sym m etry < > u u u is a symmetric tensor, so its model should be symmetric i k l ∂ < > u u < > < >  i k u u u u u Standard m odel: Daly & Harlow (1970): ∂ i k l l m x m Hanjalić & Launder (1972): Better choice: ∂ < > ∂ < > ∂ < > u u u u u u < > < > + < > + < >  i k k l i l u u u u u u u u u ∂ ∂ ∂ i k l l m i m k m x x x m m m

Relevance to w all effects Rotating pipe flow : = N W / U 0 0 1 . 0 1 . 0 N = 0 2 < u v > < v > > 0 W 2 2 u < v > N = 0 * = 0 W 0 . 5 0 . 5 N = 0 . 6 N = 0 . 6 0 . 0 0 . 0 0 . 0 0 . 5 1 . 0 0 . 0 0 . 5 1 . 0 r / R r / R Daly & Harlow (1970) Kurbatskii & Poroseva, Int. J. Hanjalić & Launder (1972) - - - Heat Fluid Flow , 1999

Pipe Flat plate Back-step Nagano & Tagava, TSFP 7, u 1991

Relevance to w all effects Back-step flow : w all + separation effects Kasagi & Matsunaga, Int. J. Heat Fluid Flow , 1995

Solution for turbulent diffusion Higher-order closures can be a required choice: < > ∂ < > D u u u u u u u i j k i j k l = − + Π − ε + ⋅ ⋅ ⋅ ijk ijk ∂ Dt x l

Pressure-containing correlations m odeling ∂ < > ∂ < > pu pu 1 ( Φ = Π − > + j i ) ρ ∂ ∂ ij ij x x j i Choices for m odeling the pressure diffusion • neglect • absorb in a model for the turbulent diffusion • model separately from the pressure-strain correlations, somehow

Relevance to w all effects Hanjalić & Launder, 2011 cannot be neglected and particularly near a w all

Relevance to w all effects Flat plate, ZPG Spalart, JFM ,1988 ∂ < > ∂ < > ∂ < > ∂ < > ( ) r ( ) s ( w ) 1 pu pu pu pu − ρ × = + + i i i i ∂ ∂ ∂ ∂ x x x x i i i i 1 < > u u u (Lumley, Adv. Appl. Mech .,1978; m m i , i 5 Poroseva, THMT , 2000) cannot be absorbed into a turbulent diffusion m odel

Pressure-strain correlations m odeling ∂ < > ∂ < > pu pu 1 ( Φ = Π − > + Φ + Φ + Φ j ( r ) ( s ) ( w ) i )= ρ ∂ ∂ ij ij ij ij ij x x j i Φ Choices for m odeling ij • linear • non-linear • other approaches ( Q -model)

Relevance to w all effects Rotating pipe flow : IP (Naot et al., 1970), LRR (Launder et al., JFM , 1975), SSG (Speziale et al., JFM , 1991), LSSG (Gatski & Speziale, JFM , 1993), Q-model (Kassinos et al., Int. J. Heat Fluid Flow, 2000) (Poroseva, CTR Ann. Res. Briefs , 2001) Only the sim plest linear m odel required w all corrections

TCL m odel (Craft & Launder, 1996): Φ ( ) r ij

Solution for pressure term s • Non-linear and Q models are too complex, and still need wall corrections • Derived under assumption of turbulence homogeneity. • It is unphysical to model the pressure diffusion separately from the pressure-strain correlations. • Finite boundary conditions for both: pressure-strain correlations and the pressure diffusion. Solution: m odeling Π ij instead

Hanjalić & Launder, 2011 • Π ij are originally present in the RANS equations • easy boundary conditions • No need to model the pressure diffusion separately

Modeling ideas   ∂ ∂ 1 p p Π = − < > + < > = Π + Π + Π   ( r ) ( s ) ( w ) u u   ρ ∂ ∂ ij j i ij ij ij x x   i j ∂ 1 p 1 1 1 1 ' ∫∫∫ ∫∫∫ ′ ′ ′ ′ ′ ′ ′ − < >= −  < >  − < > u  U' u u  dV u u u dV ρ ∂ π π i m n , n i , m m n i mn j , j x 2 r 4 r j ∂ < >   ∂   p' u 1 1 1 ∫∫ − − < > , j i     p' u dS' π ∂ ∂ , j i     4 r n' n' r Π = + M r ( ) ( ) a a U this is NOT an assumption of turbulence homogeneity ij nmij nmji m n , m - j ; ∂ < > 1 u' u 1 ∫∫∫ = − n i if m = n , then a nmji = 0; a dV ' π ∂ ∂ nmji 2 x' x' r if m = j , then a nmji = 2 < u n u i > m j Poroseva, THMT, 2000

Modeling ideas ( Cont.) ∂ < > 1 u' u 1 ∫∫∫ = − I n m odeling Π ( r) n i a dV ' ij : π ∂ ∂ nmji 2 x' x' r m j ∂ ∂ 1 u' u 1 ∫∫∫ = − < > n i a dV ' I n m odeling Φ ( r) ij : π ∂ ∂ nmji 2 x' x r m j Rotta (1951), Launder, Reece, Rodi (1975) To apply the Green’s theorem to this expression, one has to assume the turbulence homogeneity Different integrals have different properties and their analysis results in different m odels

Linear m odel Poroseva, THMT, 2000 M r ( ) Π = + a a ) U ij nmij nmji m n , 1 4 = − < > + < > + < > + < > + ( u u U u u U ) ( u u U u u U ) R i m m j , j m m i , i m j m , j m i m , 1 5 5 1 = + + + − − < > + < > R k C ( C ) ( U U ) ( C C ) ( u u U u u U ) 1 1 2 i j , j i , 1 2 i m m j , j m m i , 2 1 + − − < > + < > + − − < > δ ( C C ) ( u u U u u U ) ( 4 C C ) u u U , 1 2 i m j m , j m i m , 1 2 m n m n ij , 2

Application to lim iting states • Satisfies the exact solution for isotropic turbulence subjected to sudden distortion with any value of C 1 and C 2 . 8 2 = δ δ − δ δ + δ δ a k ( ( )) nmji ni mj nm ji nj mi 15 15 • Transforms to LRR model in homogeneous turbulence: ∂ < > 1 u' u 1 1 5 ∫∫∫ = − − − = n i 2 C C 0 a dV ' 1 2 π ∂ ∂ nmji 5 2 x' x' r m j 6 4 = + − ' C C n i 1 55 11