A New Approach to Treat the RANS-LES interface in PANS [1] Lars - PowerPoint PPT Presentation

A New Approach to Treat the RANS-LES interface in PANS [1] Lars Davidson Lars Davidson, www.tfd.chalmers.se/˜lada

PANS Low Reynolds Number Model [4] � ∂ k ∂ k ∂ t + ∂ ( kU j ) = ∂ �� ν + ν t � + ( P − ε ) ∂ x j ∂ x j σ ku ∂ x j � ∂ε ε 2 ∂ε ∂ t + ∂ ( ε U j ) = ∂ �� ν + ν t � + C ε 1 P ε k − C ∗ ε 2 ∂ x j ∂ x j σ ε u ∂ x j k k 2 f 2 f 2 ε 2 = C ε 1 + f k k k ν t = C µ f µ ε , C ∗ ( C ε 2 f 2 − C ε 1 ) , σ ku ≡ σ k , σ ε u ≡ σ ε f ε f ε f ε LRN Damping functions, f 2 , f µ as in [4] f ε = 1 . 0 LES region: f k = 0 . 4 RANS region: f k = 1 . 0 www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 2 / 19

PANS Low Reynolds Number Model [4] � ∂ k ∂ k ∂ t + ∂ ( kU j ) = ∂ �� ν + ν t � + ( P − ε ) ∂ x j ∂ x j σ ku ∂ x j � ∂ε ε 2 ∂ε ∂ t + ∂ ( ε U j ) = ∂ �� ν + ν t � + C ε 1 P ε k − C ∗ ε 2 ∂ x j ∂ x j σ ε u ∂ x j k k 2 f 2 f 2 ε 2 = C ε 1 + f k k k ν t = C µ f µ ε , C ∗ ( C ε 2 f 2 − C ε 1 ) , σ ku ≡ σ k , σ ε u ≡ σ ε f ε f ε f ε LRN Damping functions, f 2 , f µ as in [4] f ε = 1 . 0 LES region: f k = 0 . 4 RANS region: f k = 1 . 0 ◮ Zonal RANS-LES: f k has a large gradient at the RANS-LES interface www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 2 / 19

PANS: derivation The PANS k equation is derived by multiplying the RANS k equation by f k . The left hand reads Dk tot f k (1) Dt where D / Dt = ∂/∂ t + ¯ u i ∂/∂ x i , k tot = k + k res (modeled plus resolved). If it is assumed that f k constant, Eq. 1 can be re-written as Dk tot = Df k k tot = Dk k Dt , f k = (2) f k Dt Dt k tot If f k is not constant, Eq. 2 must be written as (Girimaji & Wallin [2]) Dk tot = Df k k tot Df k Dt = Dk Df k − k tot Dt − k tot f k Dt Dt Dt This work presents models for the boxed term at RANS-LES interfaces, i.e. www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 3 / 19

PANS: derivation The PANS k equation is derived by multiplying the RANS k equation by f k . The left hand reads Dk tot f k (1) Dt where D / Dt = ∂/∂ t + ¯ u i ∂/∂ x i , k tot = k + k res (modeled plus resolved). If it is assumed that f k constant, Eq. 1 can be re-written as Dk tot = Df k k tot = Dk k Dt , f k = (2) f k Dt Dt k tot If f k is not constant, Eq. 2 must be written as (Girimaji & Wallin [2]) Dk tot = Df k k tot Df k Dt = Dk Df k − k tot Dt − k tot f k Dt Dt Dt This work presents models for the boxed term at RANS-LES interfaces, i.e. ◮ horizontal RANS-LES interface in boundary layer (channel flow) www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 3 / 19

PANS: derivation The PANS k equation is derived by multiplying the RANS k equation by f k . The left hand reads Dk tot f k (1) Dt where D / Dt = ∂/∂ t + ¯ u i ∂/∂ x i , k tot = k + k res (modeled plus resolved). If it is assumed that f k constant, Eq. 1 can be re-written as Dk tot = Df k k tot = Dk k Dt , f k = (2) f k Dt Dt k tot If f k is not constant, Eq. 2 must be written as (Girimaji & Wallin [2]) Dk tot = Df k k tot Df k Dt = Dk Df k − k tot Dt − k tot f k Dt Dt Dt This work presents models for the boxed term at RANS-LES interfaces, i.e. ◮ horizontal RANS-LES interface in boundary layer (channel flow) ◮ vertical RANS-LES interface in embedded LES (channel flow) www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 3 / 19

Interface Model 1 In [2], k tot Df k / Dt is represented by introducing an additional turbulent viscosity, ν tr , in the momentum equation � ∂ ¯ � ∂ s ij = 1 u i + ∂ ¯ u j ( ν tr ¯ s ij ) , ¯ ∂ x j 2 ∂ x j ∂ x i where ν tr = P k tr Df k Dt = k Df k s | 2 = k tot s | 2 , P k tr = ν tr | ¯ | ¯ f k Dt The object of P k tr is to decrease ν t and facilitate growth of resolved turbulence on the LES side of an interface Hence, only ν tr < 0 is used which corresponds to Df k / Dt < 0 (from RANS to LES). ν t + ν tr > 0 in the momentum equation (but not in the k equation) www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 4 / 19

Interface Model 2 This model is identical to Model 1 except that k / f k is replaced by k tot i.e. P k tr = k Df k Df k P k tr = k tot f k Dt Dt Model 2 Model 1 k tot = k + 1 u ′ u ′ 2 � ¯ i ¯ i � r . a where subscript r . a . denotes running average. In PANS, f k is defined as f k = k / k tot In post-processing it is usually found that f k > k / k tot (approx. a factor 4 larger) ⇒ | P k tr | model 2 > | P k tr | model 1 www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 5 / 19

Interface Model 3 s | 2 which may cause numerical problems. In Models 1 & 2, ν tr = P k tr / | ¯ Model 3 does not involve ν tr . The original term k tot Df k / Dt is used in the k equation Adding the term Df k k ¯ u ′ Df k i − 0 . 5¯ u ′ Dt − i � ¯ u ′ m ¯ u ′ m � Dt in the momenum equation corresponds to the time-averaged term � ¯ u ′ i ¯ u ′ � − k � ¯ u ′ i ¯ u ′ i � Df k Df k Dt = −� k tot � Df k i − 2 Dt � ¯ u ′ m ¯ u ′ m � Dt in the k res equation. However, this term causes numerical instability. Hence it is not used. Only the term in the k equation, k tot Df k / Dt < 0, is used. www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 6 / 19

Interface Models: summary s | 2 < 0, in Models 1 & 2: additional turbulent viscosity, ν tr = P k tr / | ¯ P k and momentum equations ◮ Limit in momentum equations: ν t + ν tr > 0 ◮ Model 1: P k tr = k Df k f k Dt Df k ◮ Model 2: P k tr = � k tot � r . a Dt Model 3: additional production term, P k tr , in k equation without use of ν tr Df k P k tr = � k tot � r . a Dt < 0 Models 1-3 correspond to the non-commutivity in DES beteen filtering and spatial derivative at RANS-LES interfaces (Hamba [3]) = ∂ ¯ ∂ ¯ ∂ f − ∂ ∆ f f ∂ x i ∂ x i ∂ x i ∂ ∆ www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 7 / 19

Fully Developed Channel Flow The URANS and the LES regions. LES, f k = 0 . 4 URANS, f k = 1 y int y wall x Re τ = u τ δ/ν = 2 000, Re = 4 000 and Re = 8 000 x max = 3 . 2, y max = 2 and z max = 1 . 6. 32 × 32 cells in the x − z plane N y = 80 cells ( Re τ = 2 000 and 4 000) or N y = 96 ( Re τ = 8 000) www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 8 / 19

Fully Developed Channel Flow: Results Velocity Turbulent viscosity 30 150 ( ν t + ν tr ) /ν 100 20 U + 50 10 0 0 100 1000 0 500 1000 y + y + Df k Df k : Model k : Model k tot Dt . : no interface model. f k Dt +: U + = ln( y + ) / 0 . 4 + 5 . 2 • : location of the computational cell centers. www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 9 / 19

Fully Developed Channel Flow: Results Production terms in k eq modeled & resolved stresses 1 −� u ′ v ′ � resolved 40 0.5 P k + P k tr 20 0 12 − τ 12 , 0 −0.5 modeled − τ ν −20 −1 0 500 1000 0 0.5 1 y y + Df k Df k : Model k : Model k tot Dt . : no interface model. f k Dt • : the location of the computational cell centers. www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 10 / 19

Embedded Channel Flow Interface LES, f k = 0 . 4 RANS f k = 1 2 y x 0 . 95 5 . 45 Re τ = u τ δ/ν = 950 The domain size is 6 . 4 × 2 × 1 . 6 ( x , y , z ) 128 × 80 × 64 cells. www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 11 / 19

Synthetic Fluctuations at the Interface 2 1 0.8 1.5 z ) 0.6 B ww (ˆ 1 y 0.4 0.2 0.5 0 0 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 ˆ z : u 2 rms / u 2 τ : v 2 rms / u 2 τ : w 2 rms / u 2 Two-point correlation of synthetic in- τ : � u ′ v ′ � / u 2 let fluctuations τ www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 12 / 19

Velocity and Skin Friction Friction velocity Velocity at x = 5 . 5 20 1.1 u � / u τ, in ∗ 15 1 u τ 10 � ¯ ◦ 0.9 5 0 0.8 0 1 2 0 2 4 6 10 10 10 y + x : Model k Df k f k Dt Df k : Model k tot Dt : no interface model ∗ : u τ , RANS when x → ∞ ◦ : U + = ln( y + ) / 0 . 4 + 5 . 2 ◦ : u τ , PANS when x → ∞ www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 13 / 19

Resolved and Modeled Turbulence Streamwise fluctuation Turbulent viscosities 100 y (( ν tot ( x , y ) /ν rms ( x , y )) 15 80 10 60 y ( u 2 40 max 5 max 20 0 0 0 2 4 6 0 2 4 6 x x Df k : Model k f k Dt Df k : Model k tot Dt : no interface model ν tot = ν + ν t + ν tr www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 14 / 19

Production at Interface x = 1 1 0.8 0.6 y 0.4 0.2 0 −1000 −500 0 500 Production of k : P k + P k tr , Model k Df k Dt . f k Df k : P k + P k tr , Model k tot Dt . : P k , no interface model; www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 15 / 19

Conclusions Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented: www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19

Conclusions Three interface models for horizontal (wall-parallel) and vertical interfaces (embedded LES) have been presented: k Df k Dt added via ν t tr to the k eq and mom eq ( ν t tr + ν t > 0): Model 1 ◮ f k www.tfd.chalmers.se/˜lada ETMM10, Marbella, 2014 16 / 19