LES equations

Hybrid RANS/LES equations M. Germano, M. SánchezRocha, S. Menon

Hybrid RANS/LES equations Massimo GERMANO Politecnico di Torino ´ Mart´ın SANCHEZ-ROCHA Dassault Systèmes SIMULIA Corporation Suresh MENON Georgia Institute of Technology

64th Annual APS-DFD Meeting Baltimore, Maryland November 20-22, 2011

Hybrid RANS/LES equations

OUTLINE

M. Germano, M. SánchezRocha, S. Menon

• The Hybrid RANS/LES equations • The Mixed RANS/LES subgrid model

The RANS-assisted LES The statistical consistency The resolution index Results • Conclusions


THE HYBRID RANS/LES EQUATIONS Hybrid Filter

• Formally they are derived by applying to the

Navier-Stokes equations a linear hybrid combination H of a RANS average and a LES filter H = kF F + kE E

;

kE = 1 − kF

where kF is the blending factor • They produce the hybrid velocity and pressure fields

hui iH and hpiH hui iH = kF hui iF + kE hui iE hpiH = kF hpiF + kE hpiE


THE HYBRID RANS/LES EQUATIONS The associated hybrid RANS/LES equations are given by ∂hui iH =q ∂xi ∂hui iH huj iH ∂hpiH ∂ 2 hui iH ∂hui iH + =− +ν + fi ∂t ∂xj ∂xi ∂xj ∂xj where q and fi are additional terms that explicitly depend on the RANS and LES fields due to the nonlinearity of the equations and to the non commutivity of the filter with the derivatives when the blending factor kF depends on space and time.


THE HYBRID RANS/LES EQUATIONS


These additional hybrid terms in general complicate the direct solution of the exact hybrid equations. However, previous research efforts have demonstrated the importance of these terms in hybrid RANS/LES calculations of wall-bounded flows.

•

M. Germano, (2004) Properties of the hybrid RANS/ LES filter, Theoret. Comput. Fluid Dynamics, 17, 225-231

•

M. Sanchez-Rocha and S. Menon, (2009) The compressible hybrid RANS/LES formulation using an additive operator, J. Comp. Phys., 228, 2037-2062

•

B. Rajamani and J. Kim, (2010) A Hybrid-Filter Approach to Turbulence Simulation, Flow Turbulence Combust., 85, 421-441

•

M. Sanchez-Rocha and S. Menon (2011) An order-of-magnitude approximation for the hybrid terms in the compressible hybrid RANS/LES governing equations, Journal of Turbulence Vol. 12, No. 16, 1-22


THE HYBRID RANS/LES SUBGRID MODEL


Hybrid Stresses

If the blending factor kF is constant we simply have q = 0 and

fi = −

∂τH (ui ,uj ) . ∂xj

With the unclosed hybrid stress

τH (ui , uj ) = hui uj iH − hui iH huj iH = τB (ui , uj ) + H(ui , uj ) The hybrid stresses are composed by two parts a linear combination of the Reynolds and LES subgrid stresses and a hybrid term that includes the RANS and LES fields τB (ui , uj ) = kE τE (ui , uj ) + kF τF (ui , uj ) H(ui , uj ) = kE kF (hui iF − hui iE )(huj iF − huj iE )


THE HYBRID RANS/LES SUBGRID MODEL Closure Approach

• The RANS field can be obtained from the hybrid field

hui iE = hhui iH iE , while the LES field can be reconstructed as hui iF =

hui iH − kE hhui iH iE kF

• Using RANS and LES closures and the properties of the

hybrid additive filter τB (ui , uj ) = kE ME (hhui iH iE , hhuj iH iE ) + kF MF (hui iF , huj iF ) kE (hui iH − hhui iH iE ) huj iH − hhuj iH iE H(ui , uj ) = kF


THE STATISTICAL CONSISTENCY If we impose the statistical consistency of the hybrid model, EH = E, we have the following statistical constraints τE (ui , uj ) = hτH (ui , uj )iE + τE (hui iH , huj iH ) such that the Reynolds stresses can be decomposed in τ (ui , uj )tot = τ (ui , uj )res + τ (ui , uj )mod where the total, resolved, and modeled stresses are τ (ui , uj )tot = τE (ui , uj ) τ (ui , uj )res = τE (hui iH , huj iH ) τ (ui , uj )mod = hτH (ui , uj )iE = hτB (ui , uj )iE + hH(ui , uj )iE


THE RESOLUTION INDEX


A key characteristic of a hybrid approach is the particular way to control the energy partition between the resolved and the modeled scales. S. S. Girimaji, Partially Averaged Navier-Stokes Model for Turbulence: A Reynolds-Averaged Navier-Stokes to Direct Numerical Simulation Bridging Method, Journal of Applied Mechanics, Transactions of the ASME, Vol. 73, (2006), pp. 413-421 R. Manceau, Ch. Friess, and T.B. Gatski, Of the interpretation of DES as a hybrid RANS/Temporal LES method, Proc. 8th ERCOFTAC Int. Symp. on Eng. Turb. Modelling and Measurements, Marseille, France, 2010.


THE RESOLUTION INDEX In the Hybrid RANS/LES Formulation

Following Pope we introduce as a measure of the energy partition the unresolved-to-total ratio of kinetic energy α α=

Kres Kmod =1− Ktot Ktot

which under the hybrid RANS/LES framework yields αHT = 1 − k2F + k2F

hτF (ui , ui )iE τE (ui , ui )

if however, the hybrid terms stresses H(ui , uj ) are neglected the resolution index is αNHT = 1 − kF + kF

hτF (ui , ui )iE τE (ui , ui )


RESULTS • Case : Turbulent boundary layer over a flat plate.

Reϑ = 1400. • Equations : Hybrid RANS/LES with a constant

blending functions kF = 0.5 = kE = 0.5. - Case GER: The hybrid terms are exactly computed reconstructing the RANS and LES fields from the hybrid variables. - Case F0.5 HT: The hybrid terms are computed using a parallel LES simulation, the RANS variables are computed from the hybrid field. - Case F0.5: The hybrid terms are neglected. • Code : Time integration conducted with a five-stage

RungeKutta scheme. Fourth-order scheme in divergence form for space discretization.


RESULTS


Mean Velocity Profile and Total τE (u, v) stress

25

0

Exp LES GER F0.5 HT F0.5

U+

15

−0.2 tot

20

−0.4

10

−0.6

5

−0.8

0

−1 1

10

(a)

y+

100

1000

Exp LES GER F0.5 HT F0.5 1

10

100

1000

y+

(b)

Figure: Flow statistics: (a) Mean velocity profile. (b) Total τE (u, v)tot = τE (u, v)mod + τE (u, v)res stress: Black line LES; Blue line GER; Green line F0.5 HT; Red line F0.5.


RESULTS


Turbulent kinetic energy,

(a)

12

8

10 i i res

2tot

10

LES GER F0.5 HT F0.5

2

12

6 4 2 0


(b)

8 6 4 2

1

10

+

100

0

1000

1

10

y

+

y 2mod+HT

12 10


(c)

8 6 4 2 0

1

10 y

+

100

1000

Figure: Turbulent Kinetic Energy: (a) Total τE (ui , ui ); (b) Resolved τE (hui iH , hui iH ); (c) Modeled hτH (ui , ui )iE ;

100

1000


CONCLUSIONS


Aim of the research is • to simplify in a reasonable way the additive

hybridization of RANS and LES • to reinforce the statistical consistency of the hybrid

simulation with the RANS model • to test the ability of the additive RANS/LES

hybridation to control the partition of turbulent energy between the modelled and the resolved scale


CONCLUSIONS


Based on the old and new results we can say that • the additive hybrid terms associated to the hybrid

stresses include an important fraction of the modeled turbulence • the hybrid terms that depend on gradients of the

blending function do contribute to the governing equations, but their effect appear to be of second order • the linear hybridization of RANS and LES seems an

interesting and reliable approach. It provides a good control of the energy partition in the whole range between RANS and LES.