Backscatter from a scale-similarity model: embedded LES of channel flow, developing boundary layer flow and backstep flow [2] Lars Davidson Lars Davidson, www.tfd.chalmers.se/˜lada
Embedded LES: Problem Formulation Interface RANS LES u ′ , v ′ , w ′ y x At the interface between RANS and LES, turbulent fluctuations, u ′ , v ′ , w ′ , are imposed to stimulate growth of resolved fluctuations www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 2 / 22
Embedded LES: Problem Formulation Interface RANS LES u ′ , v ′ , w ′ forcing y x At the interface between RANS and LES, turbulent fluctuations, u ′ , v ′ , w ′ , are imposed to stimulate growth of resolved fluctuations To promote transition from RANS to LES (reducing the gray area), additional forcing may be used in the LES region www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 2 / 22
Embedded LES: Problem Formulation Interface RANS LES u ′ , v ′ , w ′ forcing y x At the interface between RANS and LES, turbulent fluctuations, u ′ , v ′ , w ′ , are imposed to stimulate growth of resolved fluctuations To promote transition from RANS to LES (reducing the gray area), additional forcing may be used in the LES region In the present work, forcing is added using a scale-similarity model www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 2 / 22
Momentum Equation The momentum equations for LES read � � D ¯ Dt + 1 u i ∂ ¯ p ∂ ( ν + ν SGS ) ∂ ¯ u i − ∂τ ik = ρ ∂ x i ∂ x k ∂ x k ∂ x k where D / Dt denotes material derivative. The stress tensor, τ ik , is obtained from the scale-similarity model u k − ¯ u i ¯ τ ik = ¯ u i ¯ ¯ ¯ u k www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 3 / 22
Turbulent Kinetic Energy Eq Let us take a closer look at the equation for the resolved, turbulent kinetic energy, K = � ¯ u ′ i ¯ u ′ i � / 2, which reads ( � . � denotes averaging in time) ∂ � ¯ p ′ ¯ u ′ i � ∂ � ¯ u ′ k ¯ u ′ i ¯ u ′ i � DK i � ∂ � ¯ u i � 1 + 1 u ′ u ′ Dt + � ¯ k ¯ + = ∂ x k ρ ∂ x i 2 ∂ x k � ∂ 2 ¯ � �� ∂τ ik � ∂τ ik �� � u ′ i u ′ u ′ ν ¯ − − ¯ i i ∂ x k ∂ x k ∂ x k ∂ x k The second line is simply the ¯ u ′ i eq. multiplied by ¯ u ′ i www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 4 / 22
Turbulent Kinetic Energy Eq (cont’d) The right side can be re-written as � ∂ 2 ¯ � � ∂τ ik � u ′ i u ′ u ′ ν ¯ − ¯ = i i ∂ x k ∂ x k ∂ x k � �� � ε non � ∂ ¯ � � ∂τ ik � ν ∂ 2 K u ′ ∂ ¯ u ′ i i u ′ − ν − ¯ i ∂ x k ∂ x k ∂ x k ∂ x k ∂ x k � �� � � �� � ε ε SGS The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, ε non ; this is predominately negative. www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22
Turbulent Kinetic Energy Eq (cont’d) The right side can be re-written as � ∂ 2 ¯ � � ∂τ ik � u ′ i u ′ u ′ ν ¯ − ¯ = i i ∂ x k ∂ x k ∂ x k � �� � ε non � ∂ ¯ � � ∂τ ik � ν ∂ 2 K u ′ ∂ ¯ u ′ i i u ′ − ν − ¯ i ∂ x k ∂ x k ∂ x k ∂ x k ∂ x k � �� � � �� � ε ε SGS The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, ε non ; this is predominately negative. The first term on the right side is the viscous diffusion www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22
Turbulent Kinetic Energy Eq (cont’d) The right side can be re-written as � ∂ 2 ¯ � � ∂τ ik � u ′ i u ′ u ′ ν ¯ − ¯ = i i ∂ x k ∂ x k ∂ x k � �� � ε non � ∂ ¯ � � ∂τ ik � ν ∂ 2 K u ′ ∂ ¯ u ′ i i u ′ − ν − ¯ i ∂ x k ∂ x k ∂ x k ∂ x k ∂ x k � �� � � �� � ε ε SGS The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, ε non ; this is predominately negative. The first term on the right side is the viscous diffusion the second term, ε , is the (isotropic) dissipation which is positive www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22
Turbulent Kinetic Energy Eq (cont’d) The right side can be re-written as � ∂ 2 ¯ � � ∂τ ik � u ′ i u ′ u ′ ν ¯ − ¯ = i i ∂ x k ∂ x k ∂ x k � �� � ε non � ∂ ¯ � � ∂τ ik � ν ∂ 2 K u ′ ∂ ¯ u ′ i i u ′ − ν − ¯ i ∂ x k ∂ x k ∂ x k ∂ x k ∂ x k � �� � � �� � ε ε SGS The first term on the left side is the non-isotropic (i.e. the true) viscous dissipation, ε non ; this is predominately negative. The first term on the right side is the viscous diffusion the second term, ε , is the (isotropic) dissipation which is positive The last term, ε SGS , can be positive (forward scattering=dissipation) or negative (backward scattering=forcing). www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 5 / 22
Physical Interpretation The SGS term � ∂τ ik � u ′ ε SGS = ¯ i ∂ x k consists of a net SGS force vector, T SGS , (per unit mass), multiplied by a i velocity fluctuation vector, ¯ u ′ i i.e. � � T SGS u ′ ε SGS = ¯ i i When the SGS vector, T SGS , opposes the fluctuation, ¯ u ′ i , it is i damping the fluctuation, i.e. it is dissipative www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 6 / 22
Select Forward or Backscatter We want to be able to make the term ε SGS dissipative or forcing � ∂ 2 ¯ � � ∂τ ik � u ′ i u ′ u ′ ν ¯ − ¯ = i i ∂ x k ∂ x k ∂ x k � �� � ε non � ∂ ¯ � � ∂τ ik � ν ∂ 2 K u ′ ∂ ¯ u ′ i i u ′ − ν − ¯ i ∂ x k ∂ x k ∂ x k ∂ x k ∂ x k � �� � � �� � ε ε SGS � ∂ 2 ¯ � u ′ i The viscous term in the mom. eq., ν , is dissipative ∂ x k ∂ x k ∂ 2 ¯ If − ∂τ ik u ′ i has the same sign as , then ε SGS is dissipative ∂ x k ∂ x k ∂ x k Otherwise, it is a forcing term (backscatter) www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 7 / 22
Select Backscatter Events We want the SGS stress tensor to act as backscatter in the K equation. Hence we add − ∂τ ik /∂ x k to the momentum equation only when its sign is opposite to that of the viscous diffusion term. i.e. [1] � ∂τ ik � � ∂τ ik � − ∂ 2 ¯ u ′ ∂τ ik � = − � i M ik = sign , M ik = max( M ik , 0) , M ik ∂ x k ∂ x k ∂ x k ∂ x k ∂ x k www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 8 / 22
∂ 2 ¯ u ′ ∂ 2 ¯ u i i vs. ∂ x k ∂ x k ∂ x k ∂ x k � ∂τ ik � � ∂τ ik � − ∂ 2 ¯ u ′ ∂τ ik � = − � i M ik = sign , M ik = max( M ik , 0) , M ik ∂ x k ∂ x k ∂ x k ∂ x k ∂ x k u ′ ¯ i , is not known at run-time. It could be computed as ¯ u ′ i = ¯ u i − � ¯ u i � ra , where � ¯ u i � ra denotes the running-time average of ¯ u i . It was shown in [1] that, for y + � 20 in channel flow, the second derivative of ¯ u ′ i is almost 100% correlated with that of ¯ u i Hence, in the present work, the relation at the top-left is replaced by � ∂τ ik � ∂ 2 ¯ u i M ik = sign ∂ x k ∂ x k ∂ x k www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 9 / 22
Stability The forcing has a positive feedback, i.e. the more the momentum eq is destabilized, the larger the velocity gradients, the larger the forcing Hence, the forcing term has to be limited � � � � � � � ∂ 2 ¯ � � − ∂τ ik u i � � � � � ≤ β ( ν + ν SGS ) � � ∂ x k ∂ x k ∂ x k The baseline value is β = 2. www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 10 / 22
PANS Low Reynolds Number Model [3] � ∂ 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 ε , C ∗ ν t = C µ f µ ( 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 [3] RANS region: f k = 1 . 0 (∆ / L t ) 2 / 3 , L t = ( k res + k ) 3 / 2 /ε 1 LES region: i) f k = 0 . 4 or ii) f k = c 1 / 2 µ Option i and ii give same results. but Option ii unstable in backstep flow with forcing www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 11 / 22
Test Case I: Channel Flow Interface RANS LES u ′ , v ′ , w ′ y x www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 12 / 22
Test Case I: Channel Flow Interface RANS LES u ′ , v ′ , w ′ forcing y x www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 12 / 22
Test Case I: Channel Flow Interface RANS LES u ′ , v ′ , w ′ forcing y x Interface Re τ = 950 based on u τ LES, f k = 0 . 4 RANS 2 128 × 80 × 32 ( x , y , z ) cells f k = 1 . 0 y z max = 1 . 6 x 0 . 95 5 . 45 www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 12 / 22
Test Case II: Boundary Layer Flow LES u ′ , v ′ , w ′ y x www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 13 / 22
Test Case II: Boundary Layer Flow forcing LES u ′ , v ′ , w ′ y x www.tfd.chalmers.se/˜lada TSFP8, Poitiers, 2013 13 / 22
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