Turbulence modelling for rotating flows

Transcription

1 Turbulence modelling for rotating flows

2 Presentation plan. Thesis context, goals and roadmap 2. Effects of rotation on a turbulent flow 3. RANS modelling of rotating flows 4. Conclusion and perspectives 2

3 Presentation plan. Thesis context, goals and roadmap 2. Effects of rotation on a turbulent flow 3. RANS modelling of rotating flows 4. Conclusion and perspectives 3

4 Thesis context Complex industrial applications : Thermal fatigue caused by hot water/cold water mixing in reactor coolant pumps Unsteady flow, rotation dominated dynamic, various convection regimes Which turbulence model one can use? A new numeric tool strategy : Code_Saturne selected for hydraulic machinery computations (pumps, marine turbines, ) A turbomachinery module in Code_Saturne Extend turbulence models for rotating flows 4

5 Thesis goals and roadmap Goal : a methodology for turbulence modelling in turbomachinery applications Rotation dominated flows Unsteady thermal fluctations captured Computational cost not prohibitive Solution : the researches focus on : Accurate turbulence models for forced and mixed convection in rotating flows RANS/LES coupling for thermal applications RANS/LES coupled model must be validated on an academic test case Rotating heated channel flow 5

6 Presentation plan. Thesis context, goals and roadmap 2. Effects of rotation on a turbulent flow 3. RANS modelling of rotating flows 4. Conclusion and perspectives 6

7 Rotating flows description Navier-Stokes equations in a rotating (non inertial) frame : ρ + div( ρu) = 0 t ( ρu) + div ( ρu) u) = P + µ u 2ρΩ u ρω ( Ω r) t fe = φ Coriolis force f c centrifugal force f e 2 2 Ω r Centrifugal force for incompressible flows = body force from potential Φ : φ = ρ 2 => combined with the pressure gradient («effective pressure») = P φ Coriolis force normale to (Ω,u) plane : P eff Energy distribution but no production neither destruction : f c dx = 0 Direct interaction with the flow (trajectory seems curved) Effect of strong rotation on a flow : two-dimensionnal state (Taylor-Proudman theorem, viscous and advective terms neglected) ( Ω ) u = 0 «geostrophic» flow Strong analogy beetwen rotation and curvature 7

8 Effects of rotation on turbulence (/2) When there is only rotation acting on the mean flow : Experimental studies (Wigeland & Nagib, 978 ; Jacquin et al., 990) and DNS (Rogallo, 98) Integral length scale in the transverse direction becomes stronger : v v r x Lv L < '( x) '( x + e ) > = 22, = dr 0 < v'( x) v'( x) > turbulent sturctures are more and more coherent, Taylor-Proudman organisation Energy deacreasing is slower than in the non rotating case (spectral transfer inhibition) Non linear mecanisms, should be explained with two-points correlations in spectral space (Waleffe, 993) 8

9 Effects of rotation on turbulence (2/2) When rotation Ω is combinated with shear S on the mean flow : Displaced particule analysis (Bradshaw, 969 ; Tritton, 992), linear stability analysis (Johnston et al, 972 ; Speziale, 996) B = R( R + ) > 0 2Ω R = S R = «planetary vorticity» on vorticity ratio > 0 => stabilising effect of rotation Bradshaw criterion => stable flow (particular case of Rayleigh criterion) RDT computations : maximum destabilisation occurs for Ω/S=0.25 but Ω/S=0.5 case Ω=0 case (Bertoglio, 982 ; Cambon et al, 994) With walls : spanwise rotating channel flow (Johnston et al, 972 ; Kristoffersen & Andersson, 993) Consistency with Bradshaw criterion Turbulent kinetic energy k decreases near the suction side Near the pressure side, k first increases and then decreases for strong rotation rate Ω (complete laminarisation of the flow) At the center of the channel, we have S 2Ω => neutral stability (R=-) Longitudinal roll cells (Taylor-Görtler vortices) in the center of the channel link with neutral stability of azimutal flows (Taylor-Couette instability) 9

10 Turbulence model equations in noninertial frames RANS modelling => Reynolds tensor Objective tensor (Reynolds «stress») but Reynolds stress transport equation is not frame invariant : Du ' u «Coriolis production» term : Second order closure (RSM) : Coriolis term explicit, no more modelling requirement than in the inertial case Eddy viscosity closure (EVM) => Boussinesq hypothesis : G ii = 0 => no Coriolis term in the k (neitherε) equation : Strain rate tensor S is objective i Dt Eddy viscosity hypothesis completely failed to reproduced direct anisotropic (distributive) effects of Coriolis force on turbulence Turbulence only indirectly affected by the mean flow j ' = P + D + D + φ ε + G ν T G = 2Ωk ( eikmu j ' um' + e jkmui ' um') Dk Dt u ' u ' = i = P + D j ν kδ 2ν S T D ε t 0

11 Presentation plan. Thesis context, goals and roadmap 2. Effects of rotation on a turbulent flow 3. RANS modelling of rotating flows 4. Conclusion and perspectives

12 Selected approach We want to simulate turbulent rotating flows with the models implemented in Code_Saturne We focus on the test case «spanwise rotating channel flow» Idealisation of a radial turbomachinery channel (no curvature, no transverse pressure gradient) Very sensitive to turbulence modelling (no Coriolis force on the mean flow) DNS data available (Kristoffersen & Andersson, 994 ; Lamballais et al, 996 ; Grundestam et al, 2008) Bibliographic study and tests with Code_Saturne Which is the best RSM? LRR and SSG tested How can one sensitize EVM to rotation? Three corrections tested : Spalart & Shur (997), Pettersson Reif et al (999) and Cazalbou et al (2005) 2

13 Test case preparation DNS data : 65 Re τ 200 => low Reynolds/near wall models needed LS = Launder-Sharma model ; SST = k-ω SST model ; Φ-f from Laurence et al, 2004 ; EB-SSG for last version of elliptic blending RSM (Manceau & Hanjalić, 2002) Static channel flow : Moser, Kim, Mansour (999) data Mesh convergence : y+ 0.2 convenient for all models Large difference beetween models for computed bulk mean velocity 3 Bulk velocity imposed rather than friction velocity imposed for rotating case

14 Second order modelling of turbulent rotating flows (/2) Coriolis production term G must be take into account Two models in Code_Saturne : φ SSG : calibrated for several flows including rotating shear flow LRR : extended for rotating flows in Launder et al (987) objective tensor but models don t satisfy «invariance of constitutive law» constraint SSG : Φ = F(b,S,ω) => Φ* = F(b*,S*,W*) with the «absolute mean vorticity» tensor LRR : production P becomes «absolute production» P+½G in rotating frame Equilibrium states ( ε / Sk) of RSM show a bifurcation with respect to Ω/S (Speziale & Mac Giolla Mhuiris, 989) Useful for evaluate models performance Bifurcation diagram of RSM : u = 2 u i j W + x j x i e mji Ω m 4

15 Second order modelling of turbulent rotating flows (2/2) Bifurcation diagram : Ω/S=B => S Ω/B for neutral stability SSG : B = 0.53 => S.9Ω LRR : B = 0.38 => S 2.6Ω 5

16 First order modelling : the Spalart & Shur correction (/2) An attempt to provide a 3D-applicable measure of stabilising/destabilising effect of rotation for empirical correction of EVM Unifies rotation and curvature by considering variations of S tensor principal axes Following the Bradshaw criterion, rotation/curvature must be compare with vorticity Generarized Bradshaw number P = f r P avec fr = (+ c ~ r = 2W iks r jk 2r ) + r DS Dt c + ( e imn r 3 S tan jn + e ( c jmn r 2 S ~ r ) c in ) Ω m r / D 4 r = S / W D² = ( S² + W ²) / 2 S² = 2S S, W ² = 2W W Initially proposed with Spalart-Allmaras model but could be use with other EVM Original model is recovered for non-rotating/curved flows Implemented in CFX (slight modifications) and tested on the Radiver (=centrifugal impeller) test case (Dufour et al, 2008) 6

17 First order modelling : the Spalart & Shur correction (2/2) Lamballais case (DNS) Good predictions for U ( uv) but energy k profile is coarse (at least qualitatively) Numerical stiffness of υ / k terms with Φ-f model υ t k could be changed to υ /( T ) t ε but don t avoid stiffness at high Ω t / 7

18 8 First order modelling : the Pettersson Reif et al correction (/3) First order modelling : the Pettersson Reif et al correction (/3) Problem : equilibrium state of EVM can t bifurcate : Solution : Linearised EARSM in order to «mimic» the bifurcation diagram of SSG model The original model is recovered for non-rotating flows Initially proposed for v²-f model but could be use with other EVM Weak equilibrium assumption of EARSM is false when streamline curvature occurs Curvature corrected equilibrium assumption can be add (Duraisamy & Iaccarino, 2005) ), ( 2 η η µ C = η η η α η α η α η α η α η α C µ C µ η η η η η = = = W W S S m jim i j j i i j j i Ω e C x u x u T W x u x u T S ω + = + = = ε ε µ ε C C C Sk

19 First order modelling : the Pettersson Reif et al correction (2/3) Good results except for k at the highest Ω Big «artefact» appears on the computed du/dy (Code_Saturne problem or model problem?) very short t iterations at the end of a computation to smooth the solution 9

20 First order modelling : the Pettersson Reif et al correction (3/3) Others elliptic EVM (Lien & Durbin, 996 and Φ-α, 2008) don t seem better than the Φ-f model 20

21 First order modelling : the Cazalbou et al correction (/2) Goal : design a model for both the «spectral transfer inhibition» and «Coriolis-shear instability» effects of rotation on turbulence C ε2 is the only available coefficient 0 ~ C Sk ~ 0 ε 2 0 C ε 2 = Cε C CSC tanh( bbr+ c) d ~ 3 / 2 ε ε a 4Ro 443 HS Ω HI Ω Cε 2 RC ~ ε Ro = ~ Ωk ~ k BR = ~ 2 WikS 3 S ε jk DS Dt + ( e imn S jn + e jmn S in ) Ω m ~ Ω = ~ S = ( WW / 2) ( 2S S ) / 2 / 2 HI Ω C ε 2 : Park & Chung (999) model for «spectral transfer inhibition» HS Ω RC : based on Howard et al (980) model but removes its mathematical difficiencies (non-realisability, possible blowup at finite time) Use the generalised Bradshaw number of Spalart & Shur (997) => unify rotation and curvature The original model (Launder-Sharma in Cazalbou et al (2005)) is not recovered for non rotating/curved flows Tested on the industrial Radiver test case 2

22 First order modelling : the Cazalbou et al correction (2/2) Not tested with k-ω Numerical stiffness with Φ-f Coarse results for k profil 22

23 For higher rotation rate? Turbomachinery involves strong rotation rate up to Ro=2ΩL/U=6 Grundestam et al (2008) data : complete laminarisation at Ro=3 Elliptic blending not optimal Unrealistic k with «Pettersson Reif» (PR) but allowsφ-f computation up to the highest Ω Empirical corrections «Spalart» (SS) and «Cazalbou» (Caz.) quantitatively good for all Ω Association with Φ-f model prohibited Cazalbou solution is «shaking» sometime 23

24 Presentation plan. Thesis context, goals and roadmap 2. Effects of rotation on a turbulent flow 3. RANS modelling of rotating flows 4. Conclusion and perspectives 24

25 Development budget and conclusion Rotation has important effects on turbulence Some of them are naturally taken into account with a second order modelling Rotation/curvature corrected version of Code_Saturne should offer various model : For fine results at all rotation rate : SSG For coarser but quantitatively good (at a cheaper cost) results at all rotation rate : k-ω SST + Spalart & Shur (997) correction or k-ε + Cazalbou et al (2005) correction For fine results at moderate rotation rate : Φ-f + Pettersson Reif et al (999) correction Perspectives : we ll now focus on LES and RANS/LES coupling for thermal transfer. 25

26 Some bibliography Rotation/curvature corrections for eddy-viscosity closures : ) P.R. Spalart, M. Shur ; On the sensitization of turbulence models to rotation and curvature ; Aerospace Science and Technology, 5 (997) 2) B.A. Pettersson Reif, P. Durbin and A. Ooi ; Modeling rotationnal effects using eddy-viscosity closures ; Int. J. Heat Fluid Flow, 20 (999) 3) J.-B. Cazalbou, P. Chassaing, G. Durfour and X. Carbonneau ; Twoequation modeling of turbulent rotating flows ; Phys. Fluids, 7 (2005) 26