Inlet boundary conditions for two-equation hybrid LES-RANS models [2] Lars Davidson Lars Davidson, www.tfd.chalmers.se/˜lada Go4Hybrid, Final meeting, Berlin, 2015
Research Question 1. I want to use a k − ω DES model 1.1 How do I prescribe inlet values on k and ω ? 1.2 What about the URANS region? Should I prescribe k and ω from a steady RANS solution? 2. The proposed method is to add commutation terms in the k and ω equations. 3. The commutation terms read (∆ goes from ∆ RANS to ∆ LES ) www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 2 / 30
Research Question 1. I want to use a k − ω DES model 1.1 How do I prescribe inlet values on k and ω ? 1.2 What about the URANS region? Should I prescribe k and ω from a steady RANS solution? 2. The proposed method is to add commutation terms in the k and ω equations. 3. The commutation terms read (∆ goes from ∆ RANS to ∆ LES ) ◮ k equation: − ∂ ∆ ∂ ¯ u 1 k ∂ ∆ (sink term) ∂ x 1 www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 2 / 30
Research Question 1. I want to use a k − ω DES model 1.1 How do I prescribe inlet values on k and ω ? 1.2 What about the URANS region? Should I prescribe k and ω from a steady RANS solution? 2. The proposed method is to add commutation terms in the k and ω equations. 3. The commutation terms read (∆ goes from ∆ RANS to ∆ LES ) ◮ k equation: − ∂ ∆ ∂ ¯ u 1 k ∂ ∆ (sink term) ∂ x 1 ◮ ω equation: ω ∂ ∆ ∂ ¯ u 1 k ∂ ∆ (source term) k ∂ x 1 www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 2 / 30
Research Question 1. I want to use a k − ω DES model 1.1 How do I prescribe inlet values on k and ω ? 1.2 What about the URANS region? Should I prescribe k and ω from a steady RANS solution? 2. The proposed method is to add commutation terms in the k and ω equations. 3. The commutation terms read (∆ goes from ∆ RANS to ∆ LES ) ◮ k equation: − ∂ ∆ ∂ ¯ u 1 k ∂ ∆ (sink term) ∂ x 1 ◮ ω equation: ω ∂ ∆ ∂ ¯ u 1 k ∂ ∆ (source term) k ∂ x 1 4. The method can also be used in embedded LES (i.e. at the RANS-LES interface) www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 2 / 30
The Zonal k − ω Hybrid RANS-LES PDH Model ◮ In the LES region, the model reads � ∂ k �� � k 3 / 2 ∂ k ∂ t + ∂ ¯ v i k + ∂ ν + ν t = P k − f k ∂ x i ℓ t ∂ x j σ k ∂ x j � ∂ω �� � ∂ω ∂ t + ∂ ¯ v i ω ω k P k − C ω 2 ω 2 + ∂ ν + ν t ν t ∂ k ∂ω = C ω 1 f ω + C ω ∂ x i ∂ x j σ ω ∂ x j ∂ x j ∂ x j k � ∂ ¯ � ∂ ¯ k u i + ∂ ¯ u j u i P k = ν t ν t = f µ ω, , ℓ t = C LES ∆ dw ∂ x j ∂ x i ∂ x j ∆ dw = min (max [ C dw d w , C w ∆ max , ∆ nstep ] , ∆ max ) www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 3 / 30
The Zonal k − ω Hybrid RANS-LES PDH Model ◮ In the LES region, the model reads � ∂ k �� � k 3 / 2 ∂ k ∂ t + ∂ ¯ v i k + ∂ ν + ν t = P k − f k ∂ x i ℓ t ∂ x j σ k ∂ x j � ∂ω �� � ∂ω ∂ t + ∂ ¯ v i ω ω k P k − C ω 2 ω 2 + ∂ ν + ν t ν t ∂ k ∂ω = C ω 1 f ω + C ω ∂ x i ∂ x j σ ω ∂ x j ∂ x j ∂ x j k � ∂ ¯ � ∂ ¯ k u i + ∂ ¯ u j u i P k = ν t ν t = f µ ω, , ℓ t = C LES ∆ dw ∂ x j ∂ x i ∂ x j ∆ dw = min (max [ C dw d w , C w ∆ max , ∆ nstep ] , ∆ max ) ◮ The length scale, ∆ dw , is taken from the IDDES model [9]. www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 3 / 30
The Zonal k − ω Hybrid RANS-LES PDH Model ◮ In the LES region, the model reads � ∂ k �� � k 3 / 2 ∂ k ∂ t + ∂ ¯ v i k + ∂ ν + ν t = P k − f k ∂ x i ℓ t ∂ x j σ k ∂ x j � ∂ω �� � ∂ω ∂ t + ∂ ¯ v i ω ω k P k − C ω 2 ω 2 + ∂ ν + ν t ν t ∂ k ∂ω = C ω 1 f ω + C ω ∂ x i ∂ x j σ ω ∂ x j ∂ x j ∂ x j k � ∂ ¯ � ∂ ¯ k u i + ∂ ¯ u j u i P k = ν t ν t = f µ ω, , ℓ t = C LES ∆ dw ∂ x j ∂ x i ∂ x j ∆ dw = min (max [ C dw d w , C w ∆ max , ∆ nstep ] , ∆ max ) ◮ The length scale, ∆ dw , is taken from the IDDES model [9]. ◮ In the RANS regions, ℓ t = k 1 / 2 / ( C k ω ). www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 3 / 30
The Zonal k − ω Hybrid RANS-LES PDH Model ◮ In the LES region, the model reads � ∂ k �� � k 3 / 2 ∂ k ∂ t + ∂ ¯ v i k + ∂ ν + ν t = P k − f k ∂ x i ℓ t ∂ x j σ k ∂ x j � ∂ω �� � ∂ω ∂ t + ∂ ¯ v i ω ω k P k − C ω 2 ω 2 + ∂ ν + ν t ν t ∂ k ∂ω = C ω 1 f ω + C ω ∂ x i ∂ x j σ ω ∂ x j ∂ x j ∂ x j k � ∂ ¯ � ∂ ¯ k u i + ∂ ¯ u j u i P k = ν t ν t = f µ ω, , ℓ t = C LES ∆ dw ∂ x j ∂ x i ∂ x j ∆ dw = min (max [ C dw d w , C w ∆ max , ∆ nstep ] , ∆ max ) ◮ The length scale, ∆ dw , is taken from the IDDES model [9]. ◮ In the RANS regions, ℓ t = k 1 / 2 / ( C k ω ). ◮ The interface between LES and RANS regions is chosen at a fixed grid line ( y + ≃ 500) www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 3 / 30
Varying filter size ◮ When filter size in LES varies in space, an additional term appears in the momentum equation. ◮ The reason? the spatial derivatives and the filtering do not commute. ◮ For the convective term in Navier-Stokes, for example, we get ∂ v i v j = ∂ � (∆ x ) 2 � ( v i v j ) + O ∂ x j ∂ x j ◮ Ghosal & Moin [4] showed that the error is proportional to (∆ x ) 2 ; hence it is usually neglected. www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 4 / 30
Commutation error in k equation ◮ In zonal 1 hybrid RANS-LES, the length scale at the RANS-LES interface changes abruptly from a RANS length scale to a LES length scale. ◮ Hamda [5] found that the commutation error at RANS-LES interfaces is large. ◮ For the k equation the commutation term reads ∂ u i k = ∂ ¯ u i k − ∂ ∆ ∂ ¯ u i k ∂ x i ∂ x i ∂ x i ∂ ∆ 1 the interface is chosen at a location where the RANS and LES length scales differ www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 5 / 30
Commutation term: physical meaning ∂ u i k = ∂ ¯ − ∂ ∆ ∂ ¯ u i k u i k ∂ x i ∂ x i ∂ x i ∂ ∆ ◮ Consider a fluid particle in a RANS region moving in the x 1 direction and passing across a RANS-LES interface. www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 6 / 30
Commutation term: physical meaning ∂ u i k = ∂ ¯ − ∂ ∆ ∂ ¯ u i k u i k ∂ x i ∂ x i ∂ x i ∂ ∆ ◮ Consider a fluid particle in a RANS region moving in the x 1 direction and passing across a RANS-LES interface. ◮ The filterwidth decreases across the interface, i.e. ∂ ∆ /∂ x 1 < 0 www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 6 / 30
Commutation term: physical meaning ∂ u i k = ∂ ¯ − ∂ ∆ ∂ ¯ u i k u i k ∂ x i ∂ x i ∂ x i ∂ ∆ ◮ Consider a fluid particle in a RANS region moving in the x 1 direction and passing across a RANS-LES interface. ◮ The filterwidth decreases across the interface, i.e. ∂ ∆ /∂ x 1 < 0 ◮ k decreases when going from RANS to LES ⇒ ∂ ¯ u 1 k /∂ ∆ = ( k LES − k RANS ) / (∆ LES − ∆ RANS ) > 0 � �� � � �� � < 0 < 0 www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 6 / 30
Commutation term: physical meaning ∂ u i k = ∂ ¯ − ∂ ∆ ∂ ¯ u i k u i k ∂ x i ∂ x i ∂ x i ∂ ∆ ◮ Consider a fluid particle in a RANS region moving in the x 1 direction and passing across a RANS-LES interface. ◮ The filterwidth decreases across the interface, i.e. ∂ ∆ /∂ x 1 < 0 ◮ k decreases when going from RANS to LES ⇒ ∂ ¯ u 1 k /∂ ∆ = ( k LES − k RANS ) / (∆ LES − ∆ RANS ) > 0 � �� � � �� � < 0 < 0 ◮ ⇒ The commutation term > 0 www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 6 / 30
Commutation term: physical meaning ∂ u i k = ∂ ¯ − ∂ ∆ ∂ ¯ u i k u i k ∂ x i ∂ x i ∂ x i ∂ ∆ ◮ Consider a fluid particle in a RANS region moving in the x 1 direction and passing across a RANS-LES interface. ◮ The filterwidth decreases across the interface, i.e. ∂ ∆ /∂ x 1 < 0 ◮ k decreases when going from RANS to LES ⇒ ∂ ¯ u 1 k /∂ ∆ = ( k LES − k RANS ) / (∆ LES − ∆ RANS ) > 0 � �� � � �� � < 0 < 0 ◮ ⇒ The commutation term > 0 ◮ ⇒ The commutation term < 0 on the right-side of the k equation. www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 6 / 30
Commutation term: physical meaning ∂ u i k = ∂ ¯ − ∂ ∆ ∂ ¯ u i k u i k ∂ x i ∂ x i ∂ x i ∂ ∆ ◮ Consider a fluid particle in a RANS region moving in the x 1 direction and passing across a RANS-LES interface. ◮ The filterwidth decreases across the interface, i.e. ∂ ∆ /∂ x 1 < 0 ◮ k decreases when going from RANS to LES ⇒ ∂ ¯ u 1 k /∂ ∆ = ( k LES − k RANS ) / (∆ LES − ∆ RANS ) > 0 � �� � � �� � < 0 < 0 ◮ ⇒ The commutation term > 0 ◮ ⇒ The commutation term < 0 on the right-side of the k equation. ◮ Hence, the commutation term at the RANS-LES interface reduces k . www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 6 / 30
Commutation term at the RANS-LES interface u ′ ¯ RANS LES v ′ ¯ 2 δ w ′ ¯ y L x www.tfd.chalmers.se/˜lada Go4Hybrid, Berlin, 2015 7 / 30
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