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Turbulence modelling in cavitating flows
Eric Goncalvès da Silva
To cite this version:
Eric Goncalvès da Silva. Turbulence modelling in cavitating flows. Doctoral. Udine, France. 2014. �cel-01420694�
Turbulence modelling in cavitating flows
Eric Goncalves
LEGI, University of Grenoble-Alpes, France
7-11 July 2014
Outline of the speech
Outline
1 General considerations 2 One-fluid RANS equations 3 Limitation of eddy viscosity 4 Compressibility terms 5 Wall models
6 Improved models - hybrid RANS/LES 7 One-fluid filtered equations and LES 8 Conclusions
General considerations
A large quantity of opened questions
Questions
Kolmogorov spectrum, slope 6= -5/3
Kolmogorov scale / size of two-phase structures. Induced turbulence or pseudo-turbulence.
Compressibility effects on turbulence. Anisotropy of the Reynolds tensor.
Increase or decrease of the turbulence intensity. Cavitation-turbulence interaction at small scales.
Remarks
no DNS data or turbulent quantities measurements. non universality of two-phase flows.
Example of flows
Cavitation pockets
Cavitation in vortex
Multiple scales
Turbulence scales
DNS : all scales are resolved
LES : large scale are resolved and small scales are modelled RANS : mean flow is resolved and turbulence is modelled Hybrid RANS/LES : model adapts to the mesh resolution
Phase scales
DNS : resolution of each fluid + interface
pseudo-DNS : resolution of each fluid + tracking of the interface (VoF, level set...)
filtered (LES) and averaged (RANS) models : two-fluid model,
reduced models, one-fluid model
Averaged approach - homogeneous mixture models
Averaged one-fluid equations (RANS)
Temporal averaged or ensemble averaged equations
The same operator for two-phase structure and turbulent structure Boussinesq analogy used similarly to single-phase flows
Transport-equation models : k − ε, k − ω,...
Filtered one-fluid equations (LES)
First works in incompressible flow without phase transition Lots of sub-grid terms - problem of modelling
Link with the DNS
Averaged equations
Average operator
Temporal phase average
Averaging over a time T : φk = 1
T Z
T
φk(x, τ ) d τ
Averaging over the time Tk of presence of the phase k :
φk = 1 Tk Z Tk φk(x, τ ) d τ Void fraction : α(x, t) = Tk T = φk φk
Decomposition of variable
Mass-weighted average (Favre average)
For a variable ρφ, a mass-weighted average : f φk = ρkφk ρk = ρkφk ρk and φ =˜ P αkρkφfk P αkρk = P ρkφfk P ρk Mass-weighted decomposition
An average part and a fluctuation (two-phase and turbulent contribution) : ρk = ρk + ρ′k, uk =uek + u′′k, φk = fφk + φ′′k
ρ′
k = 0 , ρku′′k = 0 , ρkφ′′k = 0
Averaged equations (1)
Conservative equations for the phase k
Mass conservation : ∂αkρk ∂t + div (αkρkeuk) = Γk Momentum conservation : ∂αkρkuk ∂t + div (αkρkeuk⊗euk + Pk) = div αk(τk + τ t k) + Mk Total energy conservation :
∂αkρk e Ek+ kk ∂t + div αkρk(eEk+ kk)euk = div (αkPkeuk) + div αk(τk+ τkt)euk − div αk(qk+ qtk) + Qk
New unknowns : Reynolds stress tensor τt
k and turbulent heat flux qkt
Averaged equations (2) Fluctuating fields
The velocity fluctuating field is not divergence free (even with incompressible phases) : ∂uk,l′ ∂xl = −∂uk,l ∂xl = 1 αk uk′.nkδI
It leads to supplementary unknowns, called "compressible terms".
Equation for the momentum fluctuation :
ρk ∂u′′k,i ∂t + ρkeuk,l ∂uk,i′′ ∂xl + ρku ′′ k,l ∂euk,i ∂xl + ρku ′′ k,l ∂u′′k,i ∂t = ∂Tk,il ∂xl + ρkeuk,i∂e uk,l ∂xl + ρk ρkαk " −∂αkTk,il ∂xl − M k+ ∂ ∂xl αkρku^ ′′ k,iu ′′ k,l # + ρk ρkαk e uk,i ∂αkρk ∂t + euk,l ∂αkρk ∂xl
Turbulent kinetic energy equation for the phase k
it is possible to write the equation for the phasic Reynolds stress and the phasic TKE kk
∂αkρkkk ∂t + div (αkρkkkeuk) = αkρk (Pk− εk+ Πk+ Mk+ Dk) + ΓkK Γ with : ρkαkPk = −αkρku^′′k,iu ′′ k,l ∂euk,i ∂xl Production term ρkαkǫk = αkτk,il′ ∂u′′ k,i ∂xl Dissipation rate ρkαkΠk = αkp′k ∂u′′ k,i ∂xi Pressure-dilatation term ρkαkMk = αku′′k,i " −∂pk ∂xi +∂σk,il ∂xl #
Mass flux term
ρkαkDk = − ∂ ∂xl αkPk′u ′′ k,iδil− αkτk,il′ u ′′ k,i −∂αkρk u′′ k,iu ′′ k,i 2 u ′′ k,l ∂xl Diffusion term
ΓkKΓ Mass transfer term
Homogeneous mixture equations or one-fluid model (1) Mixture mean quantities
mixture density and presure : ρm =Pαkρk, Pm =PαkPk
mixture internal energy : ρmem=Pαkρkeek
mass center velocity : ρmum,i =Pαkρkeuk,i
mixture viscosity : µm =Pαkµk
mixture viscous stress tensor : τm,ij =Pαkτk,ij
mixture heat flux : qm =Pαkqk Mixture turbulent quantities
mixture turbulent kinetic energy : km =Pαkkk =Pαkgu
′′
2 k,i/2
mixture Reynolds stress tensor : τm,ijt = −Pαkρku
′′
k,iu
′′
k,j
mixture eddy viscosity : µtm=Pαkµtk
mixture turbulent heat flux : qt
m =
P αkqkt
mixture dissipation rate : εm, mixture specific dissipation ωm...
Homogeneous mixture equations (2) Conservative equations Mass equation : ∂ρm ∂t + div (ρmum) = 0 Momentum equation : ∂ρmum ∂t + div (ρmum⊗ um+ Pm) = div (τm+ τ t m) Energy equation : ∂ρm(Em+ km)
∂t + div (ρm(Em+ km)um) = div (−Pmum) − div (qm− q
t m)
+ div(τm+ τmt)um
Homogeneous mixture equations (3) Mixture turbulent kinetic energy equation
∂ρmkm
∂t + div (ρmkmum) = ρmPm+ ρmΠm− ρmǫm+ ρmMm+ ρmDm+ ΓmK
Γ m
It is assumed that each phase shares the same fluctuating velocity u′′i , the same fluctuating pressure P′ and the same fluctuating viscous stress τij′ :
ρmPm= τm,ilt ∂um,i ∂xl ; ρmΠm= P′ ∂u′′ i ∂xi ρmǫm= τil′ ∂u′′ i ∂xl ; ΓmKmΓ = 0 ρmMm = u′′i −∂pm ∂xi + u′′ i " ∂σm,il ∂xl − X k σk,il ∂αk ∂xl # ρmDm = − ∂ ∂xl h p′u′′ i δil− σ′ilu ′′ i i −∂ρm ^ ui′′u′′i 2 u ′′ l ∂xl
Turbulence modelling (1) Main features
A fluctuation is associated to a mixture quantity !
The fluctuating velocity field is not divergence free → supplementary terms, difficult to model.
The pressure-dilatation term ρmΠm is null in mean but not
instantaneously.
The mixture dissipation ε is not only solenoidal.
Inhomogeneous and dilatational (or compressible) contributions. ρǫ ≈ 2µ ω′ ikω ′ ik | {z } ρεs + 2 µ ∂ ∂xk " ∂u′ ku ′ l ∂xl −2 u′ ks ′ ll # | {z } ρεinh +4 3 µ s ′ kks ′ ll | {z } ρεd
The diffusion term is modelled with a gradient formulation.
The mixture dissipation equation ε is completely modelled following the single-phase formulation.
Turbulence modelling (2) Boussinesq assumption
Boussinesq analogy and mixture eddy viscosity assumption µtm :
τm,ijt = µtm ∂um,i ∂xj +∂um,j ∂xi −2 3div umδij − 2 3ρmkmδij
Evaluation of the mixture eddy viscosity with transport-equation models.
Turbulent Fourier law
Fourier law analogy and mixture thermal conductivity λtm :
qmt = −λtmgrad Tm
Assumption of constant turbulent Prandtl number Prt :
λtm = Xαkλtk = X αk µt kCpk Prt approximed by λ t m≃ µtmCpm Prt
Turbulence modelling (3)
Usual model, k − ε model for a mixture
transport-equation models equivalent to single-phase turbulent models. all supplementary terms are neglected.
only the solenoidal dissipation is taken into account.
∂ρ k ∂t +div ρ u k − µ + µt σk grad k = ρPk− ρε ∂ρε ∂t +div ρ u ε − µ +µt σε grad ε = cε1ε kρPk− ρcε2f2 ε2 k µt≈ ρ k2 εs Remarks
a large quantity of assumptions.
introduction of wall treatment (damping functions or wall functions) a large quantity of problems !
Limitation of the eddy viscosity
Turbulent eddy corrections The Reboud correction
µt = f(ρ)Cµ k2 ǫ f(ρ) = ρv+ ρv− ρ ρv− ρl n (ρl− ρv) nis usually set to 10
SST correction - Bradshaw’s assumption for 2D boundary layer µt= ρk/ω max1,ΩF2 a1ω ; a1= 0.3 ; Ω = q 2ΩijΩij withΩij = 1 2 ∂˜ui ∂xl − ∂˜ul ∂xi
Realisability constrains of Durbin
µt= min Cµ0; c s√3 ρk2 ǫ ; 0 ≤ c ≤ 1 ; C 0 µ= 0.09 ; s = k ǫS with S= 2SijSij− 2 3S 2 kk
The Venturi 4◦ Experimental conditions Operating point : Uinlet= 10.8 m/s σinlet= Pinlet−Pvap 0.5 ρU2 inlet ≈ 0.55 Observations :
A quasi stable cavitation of 0.70 to 0.85 m length
An unsteady closure region with vapour cloud shedding and a liquid re-entrant jet
Measurements
Time-averaged longitudinal velocity and void ratio
Time-averaged wall pressure evolution and RMS fluctuations
Reboud limiter
Results using the Spalart-Allmaras model, density gradient
SA SA + Reboud limiter
Limitation of the eddy viscosity µt
mut / mu Y (m ) 0 25 50 75 0 0.005 0.01 0.015 SA Reboud SA mut / mu Y (m ) 0 25 50 75 100 0 0.005 0.01 0.015 KE KE Reboud KE Realizable station 3
Eddy viscosity limiter, profiles at stations 3 and 4 alpha Y (m ) 0 0.2 0.4 0.6 0.8 1 0 0.002 0.004 0.006 0.008 0.01 EXPERIMENT SA Reboud KL Reboud KE Reboud KE Realizable KWSST alpha Y (m ) 0 0.2 0.4 0.6 0.8 1 0 0.002 0.004 0.006 0.008 0.01 EXPERIMENT SA Reboud KL Reboud KE Reboud KE Realizable KWSST u (m/s) Y (m ) 0 5 10 0 0.002 0.004 0.006 0.008 0.01 EXPERIMENT SA Reboud KL Reboud KE Reboud KE Realizable KWSST u (m/s) Y (m ) 0 5 10 0 0.002 0.004 0.006 0.008 0.01 EXPERIMENT SA Reboud KL Reboud KE Reboud KE Realizable KWSST
Time-averaged void fraction (left) and velocity (right) profiles
The k − ℓ model + SST correction alpha y (m ) 0 0.2 0.4 0.6 0.8 1 0 0.002 0.004 0.006 0.008 0.01 EXP KL-SST : c = 0.3 KL-SST : c = 0.2 KL-SST : c = 0.1 KL-Realizable : c = 0.2 u (m/s) y (m ) -4 -2 0 2 4 6 8 10 12 14 0 0.002 0.004 0.006 0.008 0.01 EXP KL-SST : c = 0.3 KL-SST : c = 0.2 KL-SST : c = 0.1 KL-Realizable : c = 0.2
Time-averaged void fraction (left) and velocity (right) profiles
Contour of the density gradient : a1 = 0.2 (left) and a1 = 0.1 (right)
Compressibility corrections
Closure relations for turbulence compressible terms Pressure-dilatation, Sarkar formulation
ρ Π = P′∂u ′′
i
∂xi
= −α2ρPMt+ α3ρεsMt2
α2, α3 are constants to calibrate. Mt = √
k
2c is the turbulent Mach number. Dilatational dissipation, Sarkar formulation
εd = 4 3 µ s ′ kks ′ ll = α1εsMt2 with α1to calibrate Mass flux, Jones formulation
ρM = ρ ′ u′′ i ρ ∂P ∂xi −∂σil ∂xl = − µt ρ2σ p ∂ρ ∂xi ∂P ∂xi
σp is a turbulent Schmidt number, which has to be calibrated.
k −ε compressible model x (m) y (m ) 0 0.05 0.1 0.15 0 0.02 0.04 0.06 0.08 x (m) y (m ) 0 0.025 0.05 0.075 0.1 0 0.01 0.02 0.03 0.04 0.05
k −ε + Π, α2=0.15, α3 = 0.001 (left) and α3= 0.025 (right)
x (m) y (m ) 0 0.025 0.05 0.075 0.1 0 0.01 0.02 0.03 0.04 0.05 x (m) y (m ) 0 0.025 0.05 0.075 0.1 0 0.01 0.02 0.03 0.04 0.05
k −ǫ + mass flux term, σp= 1 (left) and σp = 0.0001 (right)
k −ε compressible model (2) x (m) y (m ) 0 0.025 0.05 0.075 0.1 0 0.01 0.02 0.03 0.04 0.05 x (m) y (m ) 0 0.025 0.05 0.075 0.1 0 0.01 0.02 0.03 0.04 0.05
k −ǫ + Π + M + ǫd, Sarkar values, at 2 instants
x-xi(m) (P -P v )/ P v 0.1 0.15 0.2 0.25 0.3 0 5 10 15 EXPE KEcompressible KE pd KE epsd KE Reboud x-xi(m) P ’ rm s / P a v 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 EXPE KE compressible KE pd KE epsd KE Reboud
Time-averaged wall pressure (left) and RMS fluctuations (right)
Wall functions
Turbulent boundary layer
Opened questions
Existence of an universal velocity profile. Instantaneous logarithmic area.
Cavitating law of the wall (κ function of α). Turbulence damping functions.
Modifications of turbulent properties downstream a pocket. Compressibility effects.
Numerical study
Computations using various meshes : y+ from 1 to 50.
Comparison of wall functions : two-layer model versus TBLE model.
Two-layer wall model
Formulation - similar to single-phase flows
u+ = y+ if y+< 11.13
u+ = 1
κln y
++ 5.25 if y+> 11.13
where κ = 0.41 is the von Karman constant
For unsteady flows, validity of the velocity profile at each instant. Turbulent quantities : the production of k or directly k is fixed following the formulation by Viegas and Rubesin :
Pk = 1 y Z y 0 τxyt ∂u ∂y dy
The second variable is computed through a length scale.
Thin boundary layer equations TBLE Formulation
Simplified momentum equation : ∂ui ∂t + ∂uiuj ∂xj + 1 ρ dP dxi = ∂ ∂y (µ + µt) ∂ui ∂y
Use of an embedded grid between the first grid point and the wall.
Discretization and integration of TBL equations in the embedded mesh. Iterative solving (Newton algorithm) for the variable τw. The number of
nodes in the embedded grid is N =30.
Venturi simulations u (m/s) y (m ) 0 5 10 0 0.001 0.002 0.003 0.004 0.005 0.006 EXPE 251x77 251x62 251x61 251x59 Station 3 SA u (m/s) y (m ) 0 5 10 0 0.001 0.002 0.003 0.004 0.005 0.006 EXPE 251x77 251x62 251x61 251x59 Station 3 KE u (m/s) y (m ) 0 5 10 0 0.001 0.002 0.003 0.004 0.005 0.006 EXPE 251x77 251x62 251x61 251x59 Station 3 KL u (m/s) y (m ) 0 5 10 0 0.001 0.002 0.003 0.004 0.005 0.006 EXPE 251x77 251x62 251x61 251x59 Station 3 KWSST Mesh influence near wall, station 3
alpha y (m ) 0 0.2 0.4 0.6 0.8 1 0 0.002 0.004 0.006 0.008 EXPE standard wall law TBLE station 3 u (m/s) y (m ) 0 5 10 0 0.001 0.002 0.003 0.004 0.005 0.006 EXPE
standard wall law TBLE
station 3
Two-layer model versus TBLE model
Periodic self-oscillating Venturi simulations
Frequency and CPU cost
The experimental frequency is around 45 Hz. The inlet cavitation parameter σinlet ∼2.15.
mesh σinlet frequency (Hz) cost for 100 ite. (s) ratio
174 × 77 2.13 30 525.6 1 2.18 no frequency 174 × 62 2.13 35 429.6 0.817 2.17 40 174 × 60 2.13 35 419.4 0.80 2.19 43.5 174 × 59 2.14 44 408 0.776 174 × 57 2.145 46 393 0.747 174 × 56 2.14 46 387 0.736
Improved model and hybrid
RANS/LES
Scale Adaptive Simulation (1)
Framework
Starting point : the k − kℓ model of Rotta (1972) where kℓ =163
R∞
−∞Rii(~x, ry) dry
Correlation tensor Rij(~x, ry) = Ui(~x) Uj(~x + ry)
velocity correlation between a fixed point and a moving point in direction y Transport equation for Ψ = kℓ involving the integral quantity I :
I= − 3 16 Z ∞ −∞ ∂U (~x + ry) ∂y R12dry
Expansion in Taylor series gives : I ≈∂U (~x) ∂y Z∞ −∞ R12dry+∂ 2U(~x) ∂y2 Z ∞ −∞ R12rydry+1 2 ∂3U(~x) ∂y3 Z ∞ −∞ R12ry2dry+ ...
The second order derivative is neglected (assumption of homogeneous flow). The third order derivative is difficult to model and Rotta neglected also this term.
Scale Adaptive Simulation (2)
Menter modelling
Menter proposed a model using the second derivative of the velocity (indicator of the heterogeneity of the flow) and the von Karman scale Lvk
∂2U ∂y2 Z ∞ −∞ R12rydry≈ Pk kℓ k ℓ Lvk 2 and Lvk= κ U′ U′′ New term in the transport equation for Ψ (S-A term), driven by a constant ξ. ξ = 1.47 following the calibration of Menter.
Scale-adaptive model
The characteristic length scale is self-adaptive, function of the von Karman scale for a standard model, the length scale is proportional to δ
It allows to adjust the solve of turbulent structures → behaviour close to a LES BUT : problem in the near-wall area, the log zone is not respected
→ the S-A term is not activated in the near-wall area calibration of ξ in cavitating flows.
Venturi 4o simulations
Comparison between Reboud and S-A k − ℓ models
x-xi (P -P v )/ P v 0.1 0.15 0.2 0.25 0.3 -2 0 2 4 6 8 10 12 14 16 EXP KL KL-Reboud KL-SAS x-xi P ’ rm s / P a v 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 EXPKL KL-Reboud KL-SAS
• time-averaged profiles quasi similar
• better pressure fluctuations with S-A model
Density gradient, Schlieren visualization
k −ℓ SAS x (m) y (m ) 0 0.025 0.05 0.075 0.1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 k −ℓ Reboud x (m) y (m ) 0 0.025 0.05 0.075 0.1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Hybrid RANS/LES turbulence models : DES Detached Eddy Simulation of Spalart
∂ρ˜ν ∂t + div ~ Vρ˜ν −σ1(µ + ρ˜ν) ~gradν˜ = cb1(1 − ft2) ˜Sρ˜ν + cb2 σ grad~ ρ˜ν. ~gradν˜ − cω1fω− cb1 κ2ft2 ρν˜ 2 ˜ d2
with ˜d= min (d , CDES∆) and ∆ = max(∆x, ∆y , ∆z).
CDES is a constant evaluated for the decay of isotropic turbulence =0.65
In equilibrium area : ˜ν = C2 DES∆2S
Drawbacks
Grid induced separation.
Transition between RANS-mode and LES-mode : "grey" zone.
Calibration of the constant Cdes.
2D simulations of the Venturi 4o
Influence of CDES. RANS regions (black) and LES (white)
x (m) y ( m ) 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 Cdes= 0.65 x (m) y ( m ) 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 Cdes= 0.9
Wall pressure and RMS fluctuations
x-xi (P -P v )/ P v 0.1 0.15 0.2 0.25 0.3 -2 0 2 4 6 8 10 12 14 16 EXP SA-DES-c065-sigma065 SA-DES-c08-sigma06 SA-DES-c09-sigma0588 SA-DES-c09-sigma0579 x-xi P ’ rm s / P a v 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 EXP SA-DES-c065-sigma065 SA-DES-c08-sigma06 SA-DES-c09-sigma0588 SA-DES-c09-sigma0579
3D simulations of the Venturi 4o
Comparison between k − ℓ S-A and DES simulations
x (m) z (m ) 0 0.01 0.02 0.03 0.04 0.05 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Grad-rho:300006000090000120000 150000 180000 210000 240000 270000 300000 x (m) z (m ) 0 0.01 0.02 0.03 0.04 0.05 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Grad-rho:300006000090000120000 150000 180000 210000 240000 270000 300000 x (m) z (m ) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Grad-rho:300006000090000120000 150000 180000 210000 240000 270000 300000 x (m) z (m ) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Grad-rho:300006000090000120000 150000 180000 210000 240000 270000 300000
gradient density visualization at two different instants : k − ℓ S-A (left) and DES (right)
3D simulations of the Venturi 4o (2)
Comparison between k − ℓ Scale-Adaptive and DES simulations
X Y Z X Y Z X Y Z X Y Z
Iso-surface of the void fraction for the value of 60% at two different instants : k − ℓ S-A (left) versus DES (right)
3D simulations on the Venturi 4o (3)
Comparison between 2D and 3D simulations
Good agreement between models and experiment (mid-span profiles) 3D computations provides void fraction lower than 2D computations
alpha y (m ) 0 0.2 0.4 0.6 0.8 1 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 EXPE SA-SAS 3D SA-SAS 2D KL-SAS 3D KL-SAS 2D alpha y (m ) 0 0.2 0.4 0.6 0.8 1 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 EXPE SA-SAS 3D SA-SAS 2D KL-SAS 3D KL-SAS 2D u (m/s) y (m ) -4 -2 0 2 4 6 8 10 12 14 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 EXPE SA-SAS 3D SA-SAS 2D KL-SAS 3D KL-SAS 2D u (m/s) y (m ) -6 -4 -2 0 2 4 6 8 10 12 14 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 EXPE SA-SAS 3D SA-SAS 2D KL-SAS 3D KL-SAS 2D
3D simulations on the Venturi 4o (4)
Comparison between 2D and 3D simulations
The level of RMS pressure fluctuations is largely overestimated by the 3D computations x-xi(m) (P -P v )/ P v 0.1 0.15 0.2 0.25 0.3 -2 0 2 4 6 8 10 12 14 16 EXP SA-SAS 3D SA-SAS 2D KL-SAS 3D KL-SAS 2D x-xinlet(m) P ’ rm s / Pme a n 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 EXP SA-SAS 3D SA-SAS 2D KL-SAS 3D KL-SAS 2D
Time-averaged wall pressure and RMS fluctuations
3D simulations on the Venturi 4o (5)
Dynamic behaviour - oblique mode using S-A simulations
x (m) z (m ) 0 0.01 0.02 0.03 0.04 0.05 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06Grad-rho: 30000 60000 90000120000 150000 180000 210000 240000 270000 300000
Density gradient visualization → a transversal instability at a low frequency : 6 Hz.
One-fluid filtered equations and
LES
One-fluid filtered equations
Scales
Micro-scales, scales which are small enough to describe individual bubble shapes.
Meso-scales, which are comparable to bubble sizes. Macro-scales, which entail enough bubbles for statistical representation.
When LES is applied at a micro-scale, combination with interface tracking methods (see Lakehal).
When LES is applied at a macro-scale, the interface resolution is not considered.
The scale separation is mathematically obtained by applying a convolution product using a large-scale-pass filter (function G ).
One-fluid filtered equations
Filter
For a quantity φ, the filtered variable is defined as : φ = G ◦ φ The Favre filtered variable : ˜φ = ρφ/ρ
The filtered phase indicator function Xk = G ◦ Xk = αk can be
interpreted as a filtered volume fraction of phase k.
Phase indicator function
The phase indicator function is defined as : Xk(M(x, t)) =
1 if phase k is present in point M(x, t) at t 0 otherwise
One-fluid filtered equations Equations
Assumptions : the filtering operator commutes with time and spatial derivatives. The mass transfer is assuming to be proportional to the velocity divergence through a constant C .
∂ρ ∂t + ∂ (ρ˜ui) ∂xj = 0 ∂ρ˜ui ∂t + ∂ ρ˜uiu˜j + Pδij ∂xj = div2µ˜S+ 2τµS−2τρS−τρuu ∂ρ ˜E ∂t + ∂ρ ˜Hu˜i + Q v i ∂xj
= div2τµSu−2µ ˜VτρS−2µ˜Sτρu
+ div4µτρuτρS+ ρ ˜Hτρu−τρHu
∂α
∂t + ˜V.∇α − C ∂˜ui
∂xj
= −τuα− Cdiv (τρu)
One-fluid filtered equations Subgrid terms τuα = V .∇Xv− ˜V.∇α ; τρu = ˜V − V τµS = µS − µS ; τρS = ˜S − S τρHu = ρHV − ρ ˜H ˜V ; τµSu = µSV − µSV τρuu = ρ ^ V ⊗ V − ˜V ⊗ ˜V
The subgrid term τuα is specific to two-phase flows.
The influence and the hierarchy of all these terms have never been investigated in cavitating flows.
The magnitude of the different subgrid terms was a priori evaluated in the case of phase separation flows and turbulence bubble interaction (Labourasse,Vincent) → the influence of τuα is highly dependent on
the flow configurations and/or on the chosen two-phase description.
Conclusion
Modelling difficulties
Lots of assumption in models. Lack of experimental data or DNS.
Compressibility turbulent closure due to the no divergence free fluctuating velocity field
Advanced turbulence models and hybrid turbulence model to improve the level of resolved scales
A challenge : LES in cavitating flows.
