Transport equation cavitation models in an unstructured flow solver. Kilian Claramunt, Charles Hirsch

Transcription

1 Transport equation cavitation models in an unstructured flow solver Kilian Claramunt, Charles Hirsch SHF Conference on hydraulic machines and cavitation / air in water pipes June 5-6, 2013, Grenoble, France 1 SHF Conference, June 5-6, 2013, Grenoble

2 Outline FINE/Open with OpenLabs Cavitating flows modeling with FINE TM /Open: Barotropic law cavitation model Thermo-tables cavitation model TEM: Transport-equation based cavitation model Formulation; source terms; numerical aspects Computational results NACA066(MOD) Submerged cylindrical body (45 ) 2D quarter hydrofoil with nitrogen as fluid (thermodynamic effects) Comments and conclusions 2 SHF Conference, June 5-6, 2013, Grenoble

3 FINE/Open simulation environment The Flow Integrated Environment FINE/Open integrates Mesh generation tool Hexpress: full hexahedral unstructured meshes The flow solver of FINE/Open: 3D compressible Navier-Stokes code with an unstructured cell-centered finite volume method Second order central-space discretization with Jameson-type artificial dissipation Explicit time-marching 4-stage Runge-Kutta method for updating the solution in time Preconditioning methods to solve low-speed and incompressible flows Acceleration techniques: local time stepping, multi-grid, implicit residual smoothing and parallelization with automatic load-balancing and domain decomposition Fluid models: incompressible, perfect gas with constant heat capacity, ideal gas with temperature dependent heat capacity, and real fluid modeling with thermodynamic tables Flow models: Euler, laminar or turbulent. State-of-the-art RANS turbulence models available (SA, k-e, k-w and k-w SST and EARSM). LES and DES capabilities also available Multi-physics environment: combustion, radiation, dispersed phase, cavitation, free-surface, OpenLabs for enhanced user-solver interaction: customize and/or add physical models CFView to post-process and visualize the solutions 3 SHF Conference, June 5-6, 2013, Grenoble

4 FINE/Open with OpenLabs OpenLabs Module of FINE/Open to customize / add physical models Users concentrate on physical modeling and not on programming details Users benefit from NUMECA s CFD industrial environment and features: HPC, parallelization, meshing capabilities, advanced numerical methods CFD solutions obtained with identical computing costs (compared to source-coded) OpenLabs in FINE/Open allows adding/customizing: Transport equations (convection-diffusion-source) ; Source terms and diffusion coefficients Initial and boundary conditions Thermal and transport properties Mathematical expressions as discrete profiles Integrated quantities (surface and volume) Implicit relations, 4 SHF Conference, June 5-6, 2013, Grenoble Algebraic relations ψ = f r ( ρ, p, v, T,... )

5 FINE/Open with OpenLabs FINE /Open Set up the test case OpenLabs Add/ customize physical model OpenLabs Create the library with a simple click FINE /Open Launch the flow solver CFView Analyze and post-process the results 5 SHF Conference, June 5-6, 2013, Grenoble

6 Cavitating flows modeling in FINE/Open Model1: Barotropic law cavitation modeling Single-fluid model: Navier-Stokes (plus a turbulence model) solved for a mixture Pure phases are considered incompressible Two-phase: sinusoidal barotropic law (density function of pressure) ρ ρl + ρv ρl ρv p p + sin 2 2 AMIN V = 2 Amin controls de slope of the law ρ Incompressible gas Mixture region Incompressible liquid ρ L AMIN = c ρl ρ V min Transport properties µ = µ α + ( 1 α) V µ L ρl ρ α = ρ ρ Thermal effects negligible (e.g. water) L V p V p int,v p V p int,l c min p 6 SHF Conference, June 5-6, 2013, Grenoble

7 Cavitating flows modeling in FINE/Open Model2: Thermo- tables cavitation modeling (TABGEN) Single-fluid model: Navier-Stokes (plus a turbulence model) solved for a mixture Fluid modeled with (equilibrium) thermo-tables generated with NUMECA s TABGEN NUMECA's TABGEN, based on NIST-REFPROP generates tables using complex EOS and detailed transport properties models (viscosity and conductivity). E.g., water tables Thermal effects captured (e.g. cryogens): Liquid nitrogen: ( dp sat / dt ) 80 K = Liquid hydrogen: ( dp / dt) 20 = SHF Conference, June 5-6, 2013, Grenoble sat K Pa/K Pa/K ( dp sat / dt ) 83 K = Pa/K ( dp sat / dt) 23 = Pa/K K

8 Cavitating flows modeling in FINE/Open Model 3: Transport-equation based cavitation modeling Single fluid model: Navier-Stokes (plus a turbulence model) solved for a mixture and a transport equation for the vapor mass or volume fraction ( ρf ) r r + + ( ρfv) = m / ( ρvα) r r + + ( ρ v) = m / vα t t Pressure in relation to saturation pressure (p s ) determines the source term Source terms based on Rayleigh-Plesset equation for bubble dynamics Effect of turbulence: The saturation pressure is corrected by an estimation of the turbulent pressure fluctuations (k based turbulence models, e.g. k-ε) ps = psatref + 0.5cturb ρk 39 Possibility to add extra transport equations for non-condensable gases (e.g. Helium) Extension of the model to account for thermal effects (e.g. cryogenic applications) without solving the energy equation (correction of the saturation pressure) α ρv L ps = psatref ptherm + 0.5cturb ρk ptherm = Gsat 1 α ρ c 8 SHF Conference, June 5-6, 2013, Grenoble c turb = 0. l p, liq

9 Transport-equation based cavitation modeling. Source Term Rayleigh-Plesset bubble dynamics: growth of vapor bubbles in a liquid [Singhal, 2002] m m + = c e = c k (1 σ c k σ f ) ρ ρ fρ ρ l l l v ps p ρ p p ρ l s l Cavitation Rate Condensation Rate Formulated in terms of mass fraction f Empirical constants: = c c e 0.02, c = 0.01 [Zwart, 2004] m + = F e 3r n R B (1 α)ρ v 2 3 m = F c 3α R B ρ v 2 3 p s p ρ l p s p ρ l Cavitation Rate Condensation Rate F r n Formulated in terms of volume fraction e Empirical constants: = 50, F = 5 10 c 4 = 0.01,, R = 10 B 6 [Tani, 2009] m + = c e (p s p)(1 f )ρ v 2 3 m = c c (p p s ) f ρ l SHF Conference, June 5-6, 2013, Grenoble p p s ρ l p s p ρ l Cavitation Rate Condensation Rate Formulated in terms of mass fraction f Empirical constants: c e = 0.6, c c = 0.3

10 Transport-equation based cavitation modeling. Numerical aspects Two momentum equation formulations are investigated: r v r r r r r r ρ + ρ ( v v) = p + τ + ρg t Non-conservative momentum eq. r ( ρv) t Conservative momentum eq. r r r + ( ρv v) r r r = p + τ + ρg The density update is under-relaxed For water, the minimum density allowed is clipped (e.g. 60 kg/m 3 ) Multigrid and full-multigrid initialization (FMG) are applied In the literature, condensation and evaporation constants are often tuned for specific test cases. In the simulation results presented: [Singhal, 2002] source term with constants from reference: c e =0.02, c c =0.01 [Zwart, 2004] source term calibrated: F e =1.0, F c = [Tani, 2009] source term calibrated: c e =0.06, c c = SHF Conference, June 5-6, 2013, Grenoble

11 11 11 SHF Conference, June 5 SHF Conference, June 5-6, 2013, Grenoble 6, 2013, Grenoble Transport-equation based cavitation modeling. Source term added with OpenLabs Source terms added with OpenLabs: E.g. Singhal with thermal effects l s v l e p p f k c m ρ ρ ρ σ = ) (1 l s l l c p p f k c m ρ ρ ρ σ = 3 2 [Singhal, 2002] source term

12 Cavitating flows. NACA66 (MOD) NACA66(MOD) hydrofoil [Shen&Dimotakis,1989] Incidence 4 0 Re=200000, V inlet =12.2 m/s σ=0.91 Liquid: water Water density : 1000 kg/m 3 Vapor density: kg/m 3 Spalart-Allmaras and k-ε (Extended wall-function) turbulence models Hakimi preconditioning with β* = 3 Central scheme (Jameson) 4 multigrid levels with FMG init. 2D grid, CFL=1 For TEM models, minimum density allowed 60kg/m 3 12 SHF Conference, June 5-6, 2013, Grenoble

13 Cavitating flows. NACA66 (MOD) Non-conservative vs. conservative momentum eq. for TEM models (σ=0.91) Barotropic law modeling with Amin=50 vs. TEM (Singhal, Zwart, Tani) For non-conservative, the size of the cavitation pocket is under-estimated Non-conservative momentum Conservative momentum 13 SHF Conference, June 5-6, 2013, Grenoble

14 Cavitating flows. NACA66 (MOD), σ=0.91 Conservative momentum eq. for TEM models Barotropic Amin50 TEM Singhal TEM Zwart TEM Tani 14 SHF Conference, June 5-6, 2013, Grenoble

15 Transport-equation based cavitation modeling. Under Water body Cavitation of a submerged cylindrical body (45 ) [Rouse&McNown,1948] Re=136000, V inlet =10 m/s σ=0.3 and σ=0.5 Liquid: water Water density : 1000 kg/m 3 Vapor density: kg/m 3 Spalart-Allmaras and k-ε (Extended wallfunction) turbulence models Hakimi preconditioning with β* = 3 Central scheme (Jameson artificial dissipation) 4 multigrid levels with FMG init., CFL=1 For TEM models, minimum density allowed 60kg/m 3 Flow direction Expansion Point Cavitation Point σ=0.3 σ=0.5 Axi-symmetric body Wall 15 SHF Conference, June 5-6, 2013, Grenoble

16 Cavitating flows. Under Water body Conservative momentum eq. for TEM models Barotropic law modeling with Amin=10 vs. TEM (Singhal, Zwart, Tani) σ=0.5 σ = SHF Conference, June 5-6, 2013, Grenoble

17 Cavitating flows. Under Water body, σ=0.5 Conservative momentum eq. for TEM models Barotropic Amin10 TEM Singhal TEM Zwart TEM Tani 17 SHF Conference, June 5-6, 2013, Grenoble

18 Cavitating flows. Under Water body, σ=0.3 Conservative momentum eq. for TEM models Barotropic Amin10 TEM Singhal TEM Zwart TEM Tani 18 SHF Conference, June 5-6, 2013, Grenoble

19 Cavitating flows. 2D quarter hydrofoil 2D quarter hydrofoil [Hord, NASA CR2156, 1973] Run 290C: LN2 at K ( dp / dt ) K = Pa/K V inlet =23.9 m/s, σ in,exp =1.7, Visual cavity 1.90 cm Spalart-Allmaras turbulence model Hakimi preconditioning with β* = 3 Central scheme (Jameson artificial dissipation) 4 multigrid levels with FMG init., CFL=1 Thermo-tables cavitation module vs. TEM TEM source terms which include thermal-effects (no energy equation solved) 2D grid 19 SHF Conference, June 5-6, 2013, Grenoble

20 Cavitating flows. 2D quarter hydrofoil 2D quarter hydrofoil [Hord, NASA CR2156, 1973] Run 290C: LN2 at K, σ in,exp = SHF Conference, June 5-6, 2013, Grenoble

21 Cavitating flows. 2D quarter hydrofoil 2D quarter hydrofoil [Hord, NASA CR2156, 1973] Run 290C: LN2 at K, σ in,exp =1.7 Thermo-tables TEM Tani + Thermal 21 SHF Conference, June 5-6, 2013, Grenoble

22 Comments and conclusions The non-conservative formulation of the momentum equation systematically underpredicts the size of the cavitation pocket The conservative form of the momentum equation produces similar pocket sizes as the barotropic law model and the thermo-tables model. The constants of the source terms are calibrated OpenLabs can be used to add / modify the TEM source terms Thermal effects are taken into account without solving the energy equation Ongoing and future steps: Further improve robustness Energy equation for TEM models 22 SHF Conference, June 5-6, 2013, Grenoble

23 Thank you for your attention!!! 23 SHF Conference, June 5-6, 2013, Grenoble