Physics of turbulent flow

Transcription

1 ECL-MOD 3A & MSc. Physics of turbulent flow Christophe Bailly Université de Lyon, Ecole Centrale de Lyon & LMFA - UMR CNRS

2 Outline of the course A short introduction to turbulent flow Statistical description of turbulent flow Wall-bounded turbulent flow Anatomy of a RANS model : the k-epsilon model Dynamics of vorticity Homogeneous and isotropic turbulence - Kolmogorov s theory An overview of numerical simulation and experimental techniques 107 Physic of turbulent flow Christophe Bailly ECL 17

3 Anatomy of a RANS model Numerical simulation of turbulent flows Turbulence models (statistical approach) based on Reynolds Averaged Navier-Stokes equations (RANS) Direct Numerical Simulation (DNS) Time-dependent Navier-Stokes equations are solved for all the scales (i.e. turbulent structures) Large Eddy Simulation (LES) Time-dependent filtered Navier-Stokes equations are solved, effects of smaller scales are modeled 108 Physic of turbulent flow Christophe Bailly ECL 17

4 Reynolds Averaged Navier-Stokes (RANS) equations U i x i = 0 (incompressible flow) (ρu i ) t + (ρu i U j ) = P x i + ( τij ρu i u j ) Closure of the Reynolds stress tensor through a turbulent viscosity µ t, Boussinesq s hypothesis ρu i u j = 2µ ts ij 2 3 ρk tδ ij k t u i u i 2 S ij = 1 2 ( Ui + U ) j x i (ρu i ) t + (ρu i U j ) = ( P + 2 ) x i 3 ρk t + [ 2(µ + µt )S ij ] 109 Physic of turbulent flow Christophe Bailly ECL 17

5 Reynolds Averaged Navier-Stokes (RANS) equations Almost all the physics is now contained in ν t [ Dimensional argument, ν t length velocity m 2.s 1] Two scales are required to evaluate ν t (or ad-hoc transport equation on ν t ) k t ε turbulence model length kt 3/2 /ε velocity kt 1/2 ν t C µ k 2 t ε (C µ = cst) Need of the transport equations of k t (okay!) and ε (tricky problem) Usually, high-reynolds number assumption ρε = τ ij u i ρε h = µ u i u i as Re (that will be demonstrated later in the course) ε h is the dissipation rate for homogeneous turbulence 110 Physic of turbulent flow Christophe Bailly ECL 17

6 Standard k t ε turbulence model (high-reynolds number form) Transport equation for the turbulent kinetic energy k t (refer to Chap. 2) (ρk t ) t + (ρk tu j ) = ρu i u j U i τ u i ij 1 ρu i 2 x u i u j p u i + ( u i j x i x τ ij ) j By noting that τ ij u i + ( u i x τ ij ) = k t µ 2 j xi 2 µ u i u i x }{{} j ρε h d dt (ρk t) = [( µ + µ t σ k ) ] kt + P ρε h (k t equation) 111 Physic of turbulent flow Christophe Bailly ECL 17

7 Standard k t ε turbulence model (high-reynolds number form) Transport equation for ε h (ε h ε as Re ) (ρε h ) t + U k (ρε h ) x k = 2µ U ( k u i u i + u u ) k j 2µu u i 2 U i k x k x i x }{{ i x } j x }{{ k } (i) (ii) 2µ u i u k u i µ ( u u i u ) i k 2ν ( ) u i p x }{{ k x } k x }{{ j x } i x }{{ j } (iii) (iv) (v) +µ 2 ε h 2ρν 2 2 u i 2 u i x k x }{{ kj x } k x k x }{{} j (vi) (vii) The ε h equation is finally derived by mimicking the structure of the k t equation 112 Physic of turbulent flow Christophe Bailly ECL 17

8 Standard k t ε turbulence model (high-reynolds number form) Jones & Launder (1972), Launder & Spalding (1974) d dt (ρk t) = d dt (ρεh ) = [( µ + µ t σ k [( µ + µ t σ ε ) ] kt + P ρε h ) ε h ] + εh k t ( Cε1 P C ε2 ρε h) P = ρu i u j U i µ t = C µ ρ k2 t ε h Constants are determined from canonical experiments C µ = 0.09, C ε1 = 1.44, C ε2 = 1.92, σ k = 1.0, σ ε = 1.3 Refer to lecture notes : low-reynolds number form, Favre average & compressible form, near-wall treatment 113 Physic of turbulent flow Christophe Bailly ECL 17

9 Turbulent mean flow through a diaphragm (standard k t ε model) Lafon (LaMSID - UMR EDF/CNRS 2832, 1999) See also Gloerfelt & Lafon, Comput. & Fluids (2008) for LES results 114 Physic of turbulent flow Christophe Bailly ECL 17

10 Turbulent mean flow in a diaphragm (standard k t ε model) k t ε turbulence model Duct height 2.29h & width w = 2.86h, h = 35 mm obstruction height, Re h = U m h/ν = Streamlines in xy-planes computed with the U 1 and U 2 components of the 3-D mean velocity field at three spanwise locations z/w = 0.1, 0.5 and 0.9. Mélanie Piellard (Ph.D. Thesis, ECL - DELPHI, 2008) 115 Physic of turbulent flow Christophe Bailly ECL 17

11 Realizability conditions Galilean invariance & thermodynamics Realizability conditions for the Reynolds stress tensor (no summation for Greek indices) u αu α 0 u αu β2 u α u α u β u β det(u αu β ) 0 De Vachat, Phys. Fluids (1977) Schumann, Phys. Fluids (1977) A turbulent closure should guarantee realizable conditions for the computed Reynolds stress tensor Renormalization group approach : RNG k t ε model (rather weak impact on turbulence closure) Yakhot, Orszag, Thangam, Gatski & Speziale (1992) 116 Physic of turbulent flow Christophe Bailly ECL 17

12 The k t ε turbulence model Anomalously growth of k t near a stagnation point (spurious production) Contours of k t /U 2 (a) with realizability (b) without constraint Durbin, 1996, Int. J. Heat and Fluid Flow ν t = C µ k t k t ε T k ( t ε = min kt ε, 1 ) 3 C µ 8S 2 S 2S ij S ij 117 Physic of turbulent flow Christophe Bailly ECL 17

13 The k t ε turbulence model Behnia, Parneix & Durbin, Int. J. Heat Mass Transfer (1998) 118 Physic of turbulent flow Christophe Bailly ECL 17

14 How interprete Unsteady RANS (URANS) simulations? obtained through time-marching algorithms, t U + F(U) = 0 Let s consider a mean shear flow U 1 = U 1 (x 2 ) U 2 = U 3 = 0 The Schwarz s inequality u 1 u 2 provides for the k t ε model, 2 u 2 1 u 2 2 9C 2 µs 2 12 ε2 k 2 t frequency 2 Implicit low-pass filter imposed by the mean shear : no development of energy cascade with grid rafinement. The constant C µ (and thus ν t ) is reduced in practice, 3-D is required. Semi-deterministic modelling (Ha Minh, 1991) Bastin, Lafon & Candel, J. Fluid Mech. (1997) Spalart, Int. J. Heat and Fluid Flow (2000) Iaccarino et al. Int. Journal Heat Fluid Flow (2003) 119 Physic of turbulent flow Christophe Bailly ECL 17

15 k t ω SST model The k t ω SST (shear-stress transport) model Formal change of variable ε = C µ ωk t, and thus ν t = k t /ω, k t ε model transformed into a k t ω model dω dt = + ω k t [( µ + µ t σ ε [( µt σ ε µ t σ k ) ] ω + α P βρω 2 ν t ) ] kt + 2 k t (µ + σ ω µ t ) k t ω From standard k t ε model, α = C ε1 1, β = C µ (C ε2 1), σ ω = 1/σ ε, the transport term in gray is neglected. Without the cross-diffusion term in blue and specific values of α and β, Wilcox s model (1988, 2008, AIAA Journal) - better calibrated, no damping function near the wall - too sensitive to freestream values (Menter, 1991 ; Spalart & Rumsey, 2007 ; AIAA Journal ) 120 Physic of turbulent flow Christophe Bailly ECL 17

16 k t ω SST model The k t ω SST (shear-stress transport) model One of the most widely used model Should be automatically used as default model in CFD codes Ad hoc modifications (constant calibration, F 1 blending function,...) Menter (1994) AIAA Journal dω dt = [( µ + µ t σ ε ) ] ω + α P βρω 2 ρ + 2(1 F 1 ) ν t σ ω2 ω k t ω Compressibility corrections Dezitter et al. AIAA Paper Rumsey J. Spacecraft & Rockets (2010) 121 Physic of turbulent flow Christophe Bailly ECL 17

17 k t ω SST model The k t ω SST (shear-stress transport) model Supersonic underexpanded jet with flight effects M j = 1.35 (NPR = 2.97, convergent nozzle), M f = 0.4 elsa solver, k t ω SST (ONERA) U 1 k t Cyprien Henry, Ph.D. Thesis, SNECMA - ECL (2012) SNECMA / AIRBUS 122 Physic of turbulent flow Christophe Bailly ECL 17

18 k t ω SST model The k t ω SST (shear-stress transport) model Dezitter et al., AIAA Paper (VITAL project) 123 Physic of turbulent flow Christophe Bailly ECL 17

19 k t ω SST model The k t ω SST (shear-stress transport) model Dezitter et al., AIAA Paper (VITAL project) 124 Physic of turbulent flow Christophe Bailly ECL 17

20 Turbulence models Other strategies for solving RANS equations Reynolds stress models : RSM or R ij ε or R ij ε ij (not so often used in industrial context) Hanjalić & Launder (1972, 1976), Gatski & Speziale (1993, 1997, 2004), Gerolymos et al. Spalart-Allmaras (SA) model (1992, 1994) Ad-hoc transport equation on the turbulent viscosity d ν dt = c b 1 S ν + 1 σ { [(ν + ν) ν] + cb2 ( ν) 2} c w1 f w ( ν d w ) 2 ν related to the turbulent viscosity ν t = νf ν1 S vorticity magnitude sensor d w distance to the wall f w and f ν1 are damping functions (near-wall corrections) 125 Physic of turbulent flow Christophe Bailly ECL 17

21 Turbulence models Advantages Simple and easy to use and inplement May handle complex geometries, many developments to model complex flows (reactive flows, combustion, two-phase flows, transport & pollution) Various (ad-hoc) models! Shortcomings Strong assumptions regarding the behaviour of the mean flow (concept of turbulent viscosity) Modeling of the laminar-turbulent transition prediction Boundary layer with a mean pressure gradient, detached flows,... No information (or poor/bad information) about the time-dependence 126 Physic of turbulent flow Christophe Bailly ECL 17

22 Compressible Reynolds Averaged Navier-Stokes (RANS) equations Formulation dramatically simplified by introducing the Favre average (1965) or density-weighted average for compressible flows ρ = ρ + ρ ρ conventional Reynolds-averaged density Favre decomposition of u i = Ũi + u i Ũ i = ρu i ρ ρu i = ρ ũ i = 0 u i 0 Equation of state P = ρr T or c p ρ T = c v ρ T + P Introducing ρ = ρ + ρ u i = Ũi + u i p = P + p T = T + T Favre-averaged equations for compressible flows 127 Physic of turbulent flow Christophe Bailly ECL 17

23 Favre-Averaged Navier-Stokes equations ρ t + ( ρũj) = 0 ( ρũi) t ρẽt t + ( ρũiũj + Pδ ij τ ij T ij ) = 0 + [ ( ρẽt + P)Ũj τ ij Ũ i + Q j E j P j + D j ] = 0 Ũ Ẽ t = c v T 2 + i 2 + k t k t ũ 2 i u i T ij = ρu i u j + ρũiũj = ρu i u j τ τ Q Q E j = ρe t u j + ρẽtũj = ρe u j ŨjT ij 1 2 ρu 2 i u j P j = pu j + PŨj = p u j D j = τ ij u i + τ ij Ũ i = τ ij u i 128 Physic of turbulent flow Christophe Bailly ECL 17

24 Favre-Averaged Navier-Stokes equations Closure based on the introduction of a turbulent viscosity µ t (generalization of Boussinesq s hypothesis) T ij = ρu i u j = 2µ t Sij 2 3 ρk tδ ij Constant turbulent Prandtl number σ t for the turbulent heat flux, Reynolds analogy (1874) between momentum and heat transfert ρe u j p u j = c p ρt u j = c p ν T θ ρ = µ tc p T σ t σ t ν t ν θ 129 Physic of turbulent flow Christophe Bailly ECL 17

25 k t ε turbulence model with compressible corrections Supersonic jet M j = 1.33, T j /T = 1 (estet - EDF) k t M Bailly, Lafon & Candel, J. Sound Vib. (1996) 130 Physic of turbulent flow Christophe Bailly ECL 17

26 k t ε turbulence model with compressible corrections Supersonic jet M j = 2.0, T j /T = 1 (estet - EDF) k t M Bailly, Lafon & Candel, J. Sound Vib. (1996) 131 Physic of turbulent flow Christophe Bailly ECL 17