Massimo GERMANO Politecnico di Torino

Transcription

1 Hybrid Massimo GERMANO Politecnico di Torino Martín SÁNCHEZ-ROCHA Dassault Systèmes SIMULIA Corporation Suresh MENON Georgia Institute of Technology 64th Annual APS-DFD Meeting Baltimore, Maryland November 20-22, 2011

2 OUTLINE The Hybrid The Mixed subgrid model The RANS-assisted LES The statistical consistency The resolution index Results Conclusions

3 THE HYBRID EQUATIONS Hybrid Filter Formally they are derived by applying to the Navier-Stokes a linear hybrid combination H of a RANS average and a LES filter H = k F F + k E E ; k E = 1 k F where k F is the blending factor They produce the hybrid velocity and pressure fields u i H and p H u i H = k F u i F + k E u i E p H = k F p F + k E p E

4 THE HYBRID EQUATIONS The associated hybrid are given by u i H x i = q u i H t + u i H u j H x j = p H x i + ν 2 u i H x j x j + f i where q and f i are additional terms that explicitly depend on the RANS and LES fields due to the nonlinearity of the and to the non commutivity of the filter with the derivatives when the blending factor k F depends on space and time.

5 THE HYBRID EQUATIONS These additional hybrid terms in general complicate the direct solution of the exact hybrid. However, previous research efforts have demonstrated the importance of these terms in hybrid calculations of wall-bounded flows. (2004) Properties of the hybrid RANS/ LES filter, Theoret. Comput. Fluid Dynamics, 17, M. Sanchez-Rocha and, (2009) The compressible hybrid formulation using an additive operator, J. Comp. Phys., 228, B. Rajamani and J. Kim, (2010) A Hybrid-Filter Approach to Turbulence Simulation, Flow Turbulence Combust., 85, M. Sanchez-Rocha and (2011) An order-of-magnitude approximation for the hybrid terms in the compressible hybrid governing, Journal of Turbulence Vol. 12, No. 16, 1-22

6 THE HYBRID SUBGRID MODEL Hybrid Stresses If the blending factor k F is constant we simply have q = 0 and f i = τ H(u i,u j ) x j. With the unclosed hybrid stress τ H (u i, u j ) = u i u j H u i H u j H = τ B (u i, u j ) + H(u i, u j ) 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 τ B (u i, u j ) = k E τ E (u i, u j ) + k F τ F (u i, u j ) H(u i, u j ) = k E k F ( u i F u i E )( u j F u j E )

7 THE HYBRID SUBGRID MODEL Closure Approach The RANS field can be obtained from the hybrid field u i E = u i H E, while the LES field can be reconstructed as u i F = u i H k E u i H E k F Using RANS and LES closures and the properties of the hybrid additive filter τ B (u i, u j ) = k E M E ( u i H E, u j H E ) + k F M F ( u i F, u j F ) H(u i, u j ) = k E ( u i H u i H E ) ( ) u j H u j H E k F

8 THE STATISTICAL CONSISTENCY If we impose the statistical consistency of the hybrid model, EH = E, we have the following statistical constraints τ E (u i, u j ) = τ H (u i, u j ) E + τ E ( u i H, u j H ) such that the Reynolds stresses can be decomposed in τ(u i, u j ) tot = τ(u i, u j ) res + τ(u i, u j ) mod where the total, resolved, and modeled stresses are τ(u i, u j ) tot = τ E (u i, u j ) τ(u i, u j ) res = τ E ( u i H, u j H ) τ(u i, u j ) mod = τ H (u i, u j ) E = τ B (u i, u j ) E + H(u i, u j ) E

9 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 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.

10 THE RESOLUTION INDEX In the Hybrid Formulation Following Pope we introduce as a measure of the energy partition the unresolved-to-total ratio of kinetic energy α α = K mod K tot = 1 K res K tot which under the hybrid framework yields α HT = 1 k 2 F + τ F (u i, u i ) E k2 F τ E (u i, u i ) if however, the hybrid terms stresses H(u i, u j ) are neglected the resolution index is α NHT = 1 k F + k F τ F (u i, u i ) E τ E (u i, u i )

11 RESULTS Case : Turbulent boundary layer over a flat plate. Re ϑ = Equations : Hybrid with a constant blending functions k F = 0.5 = k E = 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.

12 RESULTS Mean Velocity Profile and Total τ E (u, v) stress U Exp LES GER F0.5 HT F (a) y + <uv> tot Exp LES GER 0.8 F0.5 HT F (b) y + Figure: Flow statistics: (a) Mean velocity profile. (b) Total τ E (u, v) tot = τ E (u, v) mod + τ E (u, v) res stress: Black line LES; Blue line GER; Green line F0.5 HT; Red line F0.5.

13 2<u i u i > tot (a) LES GER F0.5 HT F y <u i u i > mod+ht (c) RESULTS Turbulent kinetic energy, 2<u i u i > res (b) y + LES GER F0.5 HT F y + LES GER F0.5 HT F0.5 Figure: Turbulent Kinetic Energy: (a) Total τ E (u i, u i ); (b) Resolved τ E ( u i H, u i H ); (c) Modeled τ H (u i, u i ) E ;

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

15 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, 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.