RECIRCULATING FLOW [1] Lars Davidson, www.tfd.chalmers.se/lada QLES - PowerPoint PPT Presentation

H OW TO ESTIMATE THE RESOLUTION OF AN LES OF RECIRCULATING FLOW [1] Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept

H OW T O E STIMATE R ESOLUTION OF AN LES? In boundary layers there are guidelines ` a priori . The cells size in the streamwise and spanwise direction should be approximately 100 and 30 respectively. First wall-adjacent node at y + ≃ 1. No guidelines in free-flow region (shear layers, re-circulation region . . . ) Worse: even after having carried out an LES, it is difficult to know if the resolution is good! I have recently made a similar study for channel flow [2] Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 2 / 34

E NERGY S PECTRUM Energy spectrum -4 10 − 5 z / 3 ww 0 1 10 10 wavenumber Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

E NERGY S PECTRUM AND T WO - POINT C ORRELATION Energy spectrum . . . K O -4 s 10 m − e e s 5 z / 3 ww 0 1 10 10 wavenumber Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

E NERGY S PECTRUM AND T WO - POINT C ORRELATION Energy spectrum Two-point correlation 1 . . . K O 0.8 -4 s 10 m − e e s 5 0.6 z / 3 ww 0.4 0.2 0 0 1 10 10 0 0.1 0.2 0.3 0.4 wavenumber Separation distance in z Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

E NERGY S PECTRUM AND T WO - POINT C ORRELATION Energy spectrum Two-point correlation 1 . . . K O 0.8 -4 s 10 m − e e s 5 0.6 z / 3 ww 0.4 ! D A B 0.2 s i t u b 0 0 1 10 10 0 0.1 0.2 0.3 0.4 wavenumber Separation distance in z Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 3 / 34

E NERGY S PECTRUM VS TIME AND T WO - POINT C ORRELATION Energy spectrum Two-point correlation 1 -5 10 0.8 0.6 E ww ( f ) -6 10 0.4 -7 10 0.2 0 -8 10 -2 -1 0 10 10 10 0 0.1 0.2 0.3 0.4 Separation distance in z frequency, f N x = 256 , N z = 32 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 4 / 34

E NERGY S PECTRUM VS TIME AND T WO - POINT C ORRELATION Energy spectrum Two-point correlation 1 -5 10 0.8 0.6 E ww ( f ) -6 10 0.4 -7 10 0.2 0 -8 10 -2 -1 0 10 10 10 0 0.1 0.2 0.3 0.4 Separation distance in z frequency, f N x = 256 , N z = 32; N x = 512 , N z = 128 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 4 / 34

E NERGY S PECTRUM VS TIME AND T WO - POINT a r t c e p C ORRELATION s y g r e n e t Energy spectrum s Two-point correlation u r t t ’ n 1 o d -5 10 0.8 0.6 E ww ( f ) -6 10 0.4 -7 10 0.2 0 -8 10 -2 -1 0 10 10 10 0 0.1 0.2 0.3 0.4 Separation distance in z frequency, f N x = 256 , N z = 32; N x = 512 , N z = 128 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 4 / 34

P LANE A SYMMETRIC D IFFUSER (N OT TO S CALE ) y H Inlet x 4 . 7 H L 1 L 2 L L 1 = 7 . 9 H , L = 21 H , L 2 = 28 H . The spanwise width is z max = 4 H . • Mesh ( x × y × z ) 258 × 64 × 32, 258 × 64 × 64, 258 × 64 × 128 · 512 × 64 × 32, 512 × 64 × 64, 512 × 64 × 128 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 5 / 34

C OMPUTATIONAL M ETHOD Finite volume with central differencing in space and time (Crank-Nicolson) Fractional step Dynamic Smagorinsky model Inlet fluctuating boundary conditions: synthetic isotropic turbulence [3] All simulations run on a single CPU. Averaging during one week (the finest mesh: two weeks) Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 6 / 34

� ¯ u � / U b , in PROFILES x = 3 6 14 17 20 24 H N x = 256 x = 3 6 14 17 20 24 H N x = 512 N z = 32; N z = 64; N z = 128; ◦ exp. [4]. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 7 / 34

� u ′ v ′ � / U 2 b , in PROFILES x = 3 6 13 16 19 23 H N x = 256 x = 3 6 13 16 19 23 H N x = 512 N z = 32; N z = 64; N z = 128; ◦ exp. [4]. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 8 / 34

� u ′ v ′ � / U 2 b , in PROFILES AT x = − H x = − H Attached flow N x = 256 N x = 512 1 1 0.8 0.8 ∆ x / ∆ z = 0 . 6 , 1 . 2 , 2 . 4 ∆ x / ∆ z = 0 . 3 , 0 . 6 , 1 . 2 0.6 0.6 y / H y / H 0.4 0.4 0.2 0.2 0 0 -2 -1 0 1 2 -2 -1 0 1 2 -3 -3 x 10 x 10 N z = 32; N z = 64; N z = 128 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 9 / 34

� u ′ v ′ � / U 2 b , in PROFILES AT x = 20 H Incipient separation x = 20 H N x = 256 N x = 512 1 1 ∆ x / ∆ z = 2 . 2 , 4 . 4 , 8 . 8 ∆ x / ∆ z = 1 . 1 , 2 . 2 , 4 . 4 0 0 -1 -1 y / H y / H -2 -2 -3 -3 -4 -4 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 2 -3 -3 x 10 x 10 N z = 32; N z = 64; N z = 128; ◦ exp. [4]. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 10 / 34

D IFFERENT W AYS TO E STIMATE R ESOLUTION Energy spectra (both in spanwise direction and time) Two-point correlations Ratio of SGS shear stress � τ sgs , 12 � to resolved � u ′ v ′ � Ratio of SGS viscosity, � ν sgs � to molecular, ν Energy spectra of SGS dissipation i /∂ x j and ∂ � ¯ Comparison of SGS dissipation due to ∂ u ′ u i � /∂ x j • Below we will only analyze results from the N x = 256 meshes Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 11 / 34

E NERGY S PECTRA , T WO -P OINT C ORR . AT x = − H x = − H Attached flow Energy spectrum Two-point correlation 1 -4 10 − 5 / 3 0.8 0.6 z ww -5 10 0.4 0.2 0 -6 10 0 1 2 10 10 10 0 0.1 0.2 0.3 0.4 wavenumber, κ z Separation distance in z N z = 32; N z = 64; N z = 128. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 12 / 34

E NERGY S PECTRA , T WO -P OINT C ORR . AT x = 20 H Incipient separation x = 20 H Energy spectrum Two-point correlation 1 -4 10 0.8 − 5 / 3 0.6 z -6 ww 10 0.4 0.2 -8 10 0 0 1 2 10 10 10 0 0.5 1 1.5 2 wavenumber, κ z Separation distance in z N z = 32; N z = 64; N z = 128. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 13 / 34

E NERGY S PECTRA IN TIME . x = − 1 . x = − H Attached flow Two-point correlation Energy spectra in time 1 − 5 / 3 -5 10 0.8 ww ( ) -6 0.6 10 0.4 -7 10 0.2 0 -8 10 -2 -1 0 10 10 10 0 0.1 0.2 0.3 0.4 frequency, f Separation distance in z N z = 32; N z = 64; N z = 128. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 14 / 34

E NERGY S PECTRA IN TIME . x = 20 . Incipient separation x = 20 H Two-point correlation Energy spectra in time 1 − -4 10 5 / 0.8 3 E ww ( f ) 0.6 -6 10 0.4 -8 10 0.2 0 -10 10 -2 -1 0 10 10 10 0 0.5 1 1.5 2 frequency, f Separation distance in z N z = 32; N z = 64; N z = 128. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 15 / 34

SGS VS . R ESOLVED S HEAR S TRESSES x = − H x = 20 H -0.5 0.4 0.35 -1 0.3 -1.5 y / H 0.25 y / H -2 0.2 -2.5 0.15 0.1 -3 0.05 -3.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.01 0.02 0.03 0.04 0.05 � τ sgs , 12 � / � u ′ v ′ � � τ sgs , 12 � / � u ′ v ′ � N z = 32; N z = 64; N z = 128. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 16 / 34

SGS VS . M OLECULAR V ISCOSITY x = − H x = 20 H 1 1 0.8 0 0.6 -1 y / H y / H 0.4 -2 0.2 -3 0 -4 0 1 2 3 4 0 2 4 6 8 10 12 � ν sgs � /ν � ν sgs � /ν N z = 32; N z = 64; N z = 128. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 17 / 34

SGS VS . M OLECULAR V ISCOSITY , N x = 512 x = − H x = 20 H 1 1 0 0.8 -1 0.6 y / H y / H -2 0.4 -3 0.2 -4 0 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 � ν sgs � /ν � ν sgs � /ν N z = 32; N z = 64; N z = 128. Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 18 / 34

D ISSIPATION E NERGY S PECTRA : T HEORY VS . R EALITY Theory Reality E ( κ ) E ( κ ) ε sgs ,κ ε sgs κ κ κ c κ c � κ c ε sgs = ε sgs ,κ ( κ ) d κ 0 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 19 / 34

A PPROXIMATED D ISSIPATION E NERGY S PECTRA At which wavenumber is the SGS dissipation largest? In the homogeneous direction, z , the SGS dissipation can be analyzed in the wavenumber space ε wz , can — in theory — be obtained from the two-point correlation [5] as N z �� ∂ w ′ � 2 � = 2 ν ∂ 2 B ww (ˆ z ) � � � κ 2 ε wz = 2 ν = 2 ν z E ww ( k z ) � ∂ ˆ z 2 ∂ z � z = 0 ˆ k z = 1 When the equations are discretized, the left side � = the right side z κ − 5 / 3 = κ 1 / 3 The right side gives ε wz ∝ κ 2 z E ww = κ 2 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 20 / 34

E XACT D ISSIPATION E NERGY S PECTRA A discrete Fourier transform of ∂ w ′ /∂ z is formed as N z ∂ w ′ ( n ) D z ( k z ) = 1 ˆ � ∂ z N z (1) n = 1 � � 2 π ( n − 1 )( k z − 1 ) � � 2 π ( n − 1 )( k z − 1 ) �� cos − ı sin N z N z where n is node number in z direction. Power Spectral Density (PSD) �� ∂ w ′ � 2 � N z N z � ∂ w ′ � � ˆ D z ∗ ˆ � � = D ∗ z � = PSD ∂ z ∂ z k z = 1 k z = 1 Lars Davidson, www.tfd.chalmers.se/˜lada QLES 2009, Pisa, 9-11 Sept 21 / 34