LES, H YBRID LES-RANS AND S CALE -A DAPTIVE S IMULATIONS (SAS) Lars Davidson, www.tfd.chalmers.se/˜lada
L ARGE E DDY S IMULATIONS SGS SGS GS In LES, large (Grid) Scales (GS) are resolved and the small (Sub-Grid) Scales (SGS) are modelled. LES is suitable for bluff body flows where the flow is governed by large turbulent scales www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 2 / 58
BLUFF-BODY FLOW: S URFACE -M OUNTED C UBE [1] Krajnovi´ c & Davidson (AIAA J., 2002) Snapshots of large turbulent scales illustrated by Q = − ∂ ¯ ∂ ¯ u j u i ∂ x j ∂ x i www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 3 / 58
BLUFF-BODY FLOW: F LOW A ROUND A B US [2] Krajnovi´ c & Davidson (JFE, 2003) www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 4 / 58
BLUFF-BODY FLOW: F LOW A ROUND A C AR [3] www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 5 / 58
BLUFF-BODY FLOW: F LOW A ROUND A T RAIN [4] c, 2006 Hemida & Krajnovi´ www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 6 / 58
S EPARATING F LOWS Wall TIME-AVERAGED flow and INSTANTANEOUS flow www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 7 / 58
S EPARATING F LOWS Wall TIME-AVERAGED flow and INSTANTANEOUS flow In average there is backflow (negative velocities). Instantaneous, the negative velocities are often positive. www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 7 / 58
S EPARATING F LOWS Wall TIME-AVERAGED flow and INSTANTANEOUS flow In average there is backflow (negative velocities). Instantaneous, the negative velocities are often positive. How easy is it to model fluctuations that are as large as the mean flow? www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 7 / 58
S EPARATING F LOWS Wall TIME-AVERAGED flow and INSTANTANEOUS flow In average there is backflow (negative velocities). Instantaneous, the negative velocities are often positive. How easy is it to model fluctuations that are as large as the mean flow? Is it reasonable to require a turbulence model to fix this? www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 7 / 58
S EPARATING F LOWS Wall TIME-AVERAGED flow and INSTANTANEOUS flow In average there is backflow (negative velocities). Instantaneous, the negative velocities are often positive. How easy is it to model fluctuations that are as large as the mean flow? Is it reasonable to require a turbulence model to fix this? Isn’t it better to RESOLVE the large fluctuations? www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 7 / 58
N EAR -W ALL T REATMENT Biggest problem with LES: near walls, it requires very fine mesh in all directions, not only in the near-wall direction. www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 8 / 58
N EAR -W ALL T REATMENT Biggest problem with LES: near walls, it requires very fine mesh in all directions, not only in the near-wall direction. The reason: violent violent low-speed outward ejections and high-speed in-rushes must be resolved (often called streaks). www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 8 / 58
N EAR -W ALL T REATMENT Biggest problem with LES: near walls, it requires very fine mesh in all directions, not only in the near-wall direction. The reason: violent violent low-speed outward ejections and high-speed in-rushes must be resolved (often called streaks). A resolved these structures in LES requires ∆ x + ≃ 100, ∆ y + min ≃ 1 and ∆ z + ≃ 30 www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 8 / 58
N EAR -W ALL T REATMENT Biggest problem with LES: near walls, it requires very fine mesh in all directions, not only in the near-wall direction. The reason: violent violent low-speed outward ejections and high-speed in-rushes must be resolved (often called streaks). A resolved these structures in LES requires ∆ x + ≃ 100, ∆ y + min ≃ 1 and ∆ z + ≃ 30 The object is to develop a near-wall treatment which models the streaks (URANS) ⇒ much larger ∆ x and ∆ z www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 8 / 58
N EAR -W ALL T REATMENT Biggest problem with LES: near walls, it requires very fine mesh in all directions, not only in the near-wall direction. The reason: violent violent low-speed outward ejections and high-speed in-rushes must be resolved (often called streaks). A resolved these structures in LES requires ∆ x + ≃ 100, ∆ y + min ≃ 1 and ∆ z + ≃ 30 The object is to develop a near-wall treatment which models the streaks (URANS) ⇒ much larger ∆ x and ∆ z In the presentation we use Hybrid LES-RANS for which the grid requirements are much smaller than for LES www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 8 / 58
N EAR -W ALL T REATMENT from Hinze (1975) www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 9 / 58
N EAR -W ALL T REATMENT 1.5 1 z 0.5 0 0 1 2 3 4 5 6 x Fluctuating streamwise velocity at y + = 5. DNS of channel flow. We find that the structures in the spanwise direction are very small which requires a very fine mesh in z direction. www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 10 / 58
H YBRID LES-RANS Near walls: a RANS one-eq. k or a k − ω model. In core region: a LES one-eq. k SGS model. wall URANS Interface LES URANS y y + ml wall x • Location of interface either pre-defined or automatically computed www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 11 / 58
M OMENTUM E QUATIONS • The Navier-Stokes, time-averaged in the near-wall regions and filtered in the core region, reads � � ∂ ¯ ∂ ¯ ( ν + ν T ) ∂ ¯ ∂ t + ∂ u i = βδ 1 i − 1 p + ∂ u i � ¯ � u i ¯ u j ∂ x j ρ ∂ x i ∂ x j ∂ x j ν T = ν t , y ≤ y ml ν T = ν sgs , y ≥ y ml • The equation above: URANS or LES? Same boundary conditions ⇒ same solution! www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 12 / 58
T URBULENCE M ODEL • Use one-equation model in both URANS region and LES region. k 3 / 2 � � ∂ k T ∂ t + ∂ u j k T ) = ∂ ( ν + ν T ) ∂ k T T (¯ + P k T − C ε ∂ x j ∂ x j ∂ x j ℓ P k T = 2 ν T ¯ S ij ¯ S ij , ν T = C k ℓ k 1 / 2 T LES-region: k T = k sgs , ν T = ν sgs , ℓ = ∆ = ( δ V ) 1 / 3 URANS-region: k T = k , ν T = ν t , ℓ ≡ ℓ RANS = 2 . 5 n [ 1 − exp( − Ak 1 / 2 y /ν )] , Chen-Patel model (AIAA J. 1988) Location of interface can be defined by min( 0 . 65 ∆ , y ) , ∆ = max(∆ x , ∆ y , ∆ z ) www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 13 / 58
S TANDARD H YBRID LES-RANS • Coarse mesh: ∆ x + = 2 ∆ z + = 785. δ/ ∆ x ≃ 2 . 5, δ/ ∆ z ≃ 5. 1 30 0.8 25 20 0.6 B ( x ) U + 15 0.4 10 0.2 5 0 0 1 2 3 0 0.5 1 1.5 2 10 10 10 y + x standard LES-RANS; B ( x ) = � u ( x 0 ) u ( x − x 0 ) � DNS; LES u rms u rms ◦ 0 . 4 ln( y + ) + 5 . 2 • Too high velocity because too low shear stress www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 14 / 58
W AYS TO I MPROVE THE RANS-LES M ETHOD [5, 6, 7] The reason is that LES region is supplied with bad boundary (i.e. interface) conditions by the URANS region. The flow going from the RANS region into the LES region has no proper turbulent length or time scales New approach: Synthesized isotropic turbulent fluctuations are added as momentum sources at the interface. The superimposed fluctuations should be regarded as forcing functions rather than boundary conditions. www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 15 / 58
F ORCING F LUCTUATIONS A DDED AT THE I NTERFACE • Object: to trig the momentum equations into resolving large-scale turbulence interface LES region u ′ f , v ′ f , w ′ f URANS region y + y ml wall x • For more info, see Davidson at al. [5, 7] www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 16 / 58
I MPLEMENTATION v ′ f Control Volume u ′ f LES Interface URANS A n Fluctuations u ′ f , v ′ f , w ′ f are added as sources in all three momentum equations. The source is − γρ u ′ i , f u ′ 2 , f A n = − γρ u ′ i , f u ′ 2 , f V / ∆ y ( A n =area, V =volume of the C.V.) The source is scaled with γ = k T / k synt www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 17 / 58
I NLET B OUNDARY C ONDITIONS U inlet constant in time; u inlet function of time. 1 U in ( y ) u in ( y , t 0 ) 0.8 0.6 y u ( x , y 0 , t 0 ) 0.4 0.2 0 x E 0 5 10 15 20 25 0 20 40 60 80 100 U x Left: Inlet boundary profiles Right: Evolution of u velocity depending of type of inlet B.C. • With steady inlet B.C., u gets turbulent first at x = x E . www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 18 / 58
E MBEDDED LES (B LUFF B ODY F LOWS ) Steady RANS U in + u ′ i ( t ) U out LES U out Steady RANS U in + u ′ i ( t ) used as B.C. for LES in the inner region. Examples of inner region: external mirror of a car; a flap/slat; a detail of a landing gear. Often in connection with aero-acoustics. www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 19 / 58
I NLET B OUNDARY C ONDITIONS VS . F ORCING Inlet U b ( y ) LES region u ′ ( x i , t ) u ′ ( y , t ) U b ( x i , t ) y x URANS region www.tfd.chalmers.se/˜lada LES course, 19-21 Oct 2009 20 / 58
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