M.V Salvetti,M.V Salvetti, F. Beux, M. Bilanceri ( F. Beux, M. Bilanceri (University of PisaUniversity of Pisa))E. Sinibaldi (now at E. Sinibaldi (now at Scuola Superiore Sant’Anna, Scuola Superiore Sant’Anna,

PisaPisa))

A numerical method for A numerical method for barotropic flow simulation with barotropic flow simulation with

applications to cavitationapplications to cavitation

Micro-Macro Modelling and Simulation of Liquid-Vapour Micro-Macro Modelling and Simulation of Liquid-Vapour FlowsFlows Strasbourg, 23-25 January 2008Strasbourg, 23-25 January 2008

Industrial and engineering Industrial and engineering motivationmotivation

Development of a Development of a numerical toolnumerical tool for the prediction of for the prediction of the performance of axial inducers typical of turbopumps the performance of axial inducers typical of turbopumps for liquid propellent rockets. for liquid propellent rockets.

rotating inducerrotating inducer

non rotating cylindrical non rotating cylindrical casecase

The main role of the inducer is to increase the The main role of the inducer is to increase the fluid pressurefluid pressure (velocity decrease) through rotation. (velocity decrease) through rotation. Fundings from the Italian Space Agency and European Space Fundings from the Italian Space Agency and European Space Agency.Agency.

Difficulties Difficulties Complex rotating 3D geometry.Complex rotating 3D geometry. Severe size limitations which lead to Severe size limitations which lead to very high rotational speed very high rotational speed cavitation cavitation phenomena phenomena need of a need of a model model to take into to take into account cavitation.account cavitation.

INFLOW

Choice of the cavitation Choice of the cavitation modelmodel

Considering the characteristics of the considered engineering Considering the characteristics of the considered engineering problem problem ((interest in global performance predictions, short life-interest in global performance predictions, short life-time, cryogenic propellers, distribution of the active cavitation time, cryogenic propellers, distribution of the active cavitation nuclei not known)nuclei not known)

Homogeneous-flow modelHomogeneous-flow model, i.e. liquid/vapour mixture modeled as a , i.e. liquid/vapour mixture modeled as a single-phase fluid single-phase fluid

Cavitation modelCavitation modelModel proposed by L. d’Agostino et al. (2001), which takes into Model proposed by L. d’Agostino et al. (2001), which takes into account (at least approximately) for account (at least approximately) for thermal cavitation effectsthermal cavitation effects . .

Barotropic flowBarotropic flow: the state equation relates : the state equation relates pressurepressure and and densitydensity. In . In particular, for the considered model the state equation has the particular, for the considered model the state equation has the following form:following form:

( , , )d f pdp satpp

( , )p f satppliquidliquid

liquid-vapour mixtureliquid-vapour mixture

characteristic fluid characteristic fluid physical physical

parameters parameters (known)(known)

Starting from this context, the more general scope of the research Starting from this context, the more general scope of the research activity is activity is to develop a numerical tool for the simulation of to develop a numerical tool for the simulation of barotropic flowsbarotropic flows in complex geometry. in complex geometry.

Cavitating flow behaviorCavitating flow behaviorDifficultyDifficulty: in the homogenous-flow description, the physical : in the homogenous-flow description, the physical properties of the flow change dramatically between the zones of properties of the flow change dramatically between the zones of pure liquid and the cavitating regions (fluid/vapour mixture).pure liquid and the cavitating regions (fluid/vapour mixture).

densitdensityy

spee

d of

soun

dsp

eed

of so

und

high speed of sound high speed of sound (H(H22O ~ 1400 m/s)O ~ 1400 m/s)

high M high M supersonic-hypersonic flow supersonic-hypersonic flow

M<<1 M<<1 incompressible regimeincompressible regime

Modellazione dei flussi cavitantiModellazione dei flussi cavitanti Flows characterized by Flows characterized by nearly incompressiblenearly incompressible zones together zones together

with with highly supersonichighly supersonic flow regions. flow regions.

two possible two possible choices:choices:

Numerical solvers for Numerical solvers for incompressible flowsincompressible flows suitably corrected to take suitably corrected to take into account into account compressibilitycompressibility

Numerical discretization of the Numerical discretization of the compressible flowcompressible flow equations equations

p unknownp unknown unknown, p from the state unknown, p from the state equationequation

Examples of applications to Examples of applications to cavitating flows: van der Heul et cavitating flows: van der Heul et al., ECCOMAS 2000, Senocack al., ECCOMAS 2000, Senocack and Shyy, JCP 2002, …and Shyy, JCP 2002, …

Mathematical modelMathematical modelBecause of the barotropic state law the energy equation is Because of the barotropic state law the energy equation is decoupled decoupled only mass and momentum balances are only mass and momentum balances are consideredconsidered

Equations for 3D Equations for 3D laminar viscouslaminar viscous compressible flows ( compressible flows (no no turbulenceturbulence) in conservative variables) in conservative variables

oror Equations for 3D Equations for 3D inviscid inviscid compressible flows in conservative compressible flows in conservative

variablesvariables ++barotropic state equation barotropic state equation (ODE or (ODE or

analytic laws)analytic laws)

Numerical discretization: outlineNumerical discretization: outline 1D inviscid flows1D inviscid flows

Spatial discretization of 1Spatial discretization of 1stst order of accuracy order of accuracy (preconditioning)(preconditioning)

Linearized implicit time advancing Linearized implicit time advancing Extension to 2Extension to 2ndnd order of accuracy in space (MUSCL) order of accuracy in space (MUSCL) Time advancing for 2Time advancing for 2ndnd order scheme (defect correction) order scheme (defect correction)

3D inviscid flows (in rotating frames)3D inviscid flows (in rotating frames) extension of the previous ingredients to tetrahedral extension of the previous ingredients to tetrahedral

unstructured gridsunstructured grids 3D laminar viscous flows3D laminar viscous flows

P1 finite-element discretization of viscous flows (not P1 finite-element discretization of viscous flows (not shown)shown)

1D inviscid flows:spatial 1D inviscid flows:spatial discretizationdiscretization

( ) 0W F Wt x

Wu

2

uF

u p

p p

1 10

i i i iiC C C CC

d Wdx F Fdt

Wi+1Wi-1 Wi

i,i+1i-1,i

, 1 1, 0ii i i i idWdt

Ci Ci+1Ci-1

x

iNumerical Numerical

fluxesfluxes

Galerkin projection on the Galerkin projection on the finite-volumefinite-volume basis functions basis functions

(piecewise constant)(piecewise constant)

time discretizationtime discretization

Numerical fluxesNumerical fluxesGodunov-type fluxGodunov-type flux: the exact solution of the Riemann : the exact solution of the Riemann problem between two neighboring cells is used. problem between two neighboring cells is used.

, 1 1, ,0L Ri i RP i iF W W W 1, 1, ,0L R

i i RP i iF W W W

In the present research activity, a procedure has been developed In the present research activity, a procedure has been developed for the construction of the Riemann problem solution for Euler for the construction of the Riemann problem solution for Euler equations and a equations and a generic convex barotropic lawgeneric convex barotropic law. .

Reference:Reference: E. Sinibaldi, E. Sinibaldi, Implicit preconditioned Implicit preconditioned numerical schemes for the simulation of three-numerical schemes for the simulation of three-dimensional barotropic flowsdimensional barotropic flows, , Pisa, Edizioni della Pisa, Edizioni della Normale, in press, Normale, in press, ISBN 978-88-7642-310-9.ISBN 978-88-7642-310-9.

Exact 1D benchmark for generic barotropic state lawsExact 1D benchmark for generic barotropic state laws Construction of a Godunov-type schemeConstruction of a Godunov-type scheme

Numerical fluxesNumerical fluxesRoe schemeRoe scheme: approximated solution of the Riemann problem : approximated solution of the Riemann problem between two neighboring cells is used.between two neighboring cells is used. . .

,( ) ( ) 1( , ) ( , ) ( )

2 2l r

l r l r l r l rF W F WW W A W W W W

centered partcentered part upwind partupwind part

numerical numerical viscosityviscosity

Roe matrixRoe matrix

Contribution of the present research activity Contribution of the present research activity definition of definition of the Roe matrix for a the Roe matrix for a generic barotropic state lawgeneric barotropic state law (PhD. Thesis (PhD. Thesis by E. Sinibaldi or Sinibaldi, Beux & Salvetti, INRIA RR4891, 2003 by E. Sinibaldi or Sinibaldi, Beux & Salvetti, INRIA RR4891, 2003 (available on line), Sinibaldi, Beux & Salvetti, Flow Turbulence and (available on line), Sinibaldi, Beux & Salvetti, Flow Turbulence and Combustion 76(4), 2006).Combustion 76(4), 2006).

Preconditioning for low-Mach Preconditioning for low-Mach numbersnumbers

ProblemProblem: the numerical solvers for compressible flows suffer in general : the numerical solvers for compressible flows suffer in general of accuracy problems if applied to low Mach flows. Following Guillard of accuracy problems if applied to low Mach flows. Following Guillard and Viozat (1999), an asymptotic analysis for Mand Viozat (1999), an asymptotic analysis for M0 0 (Sinibaldi, Beux & (Sinibaldi, Beux & Salvetti, 2003 or P.H.D. Thesis by E. Sinibaldi) Salvetti, 2003 or P.H.D. Thesis by E. Sinibaldi) shows that:shows that: the continous solution is characterized by pressure variations in space the continous solution is characterized by pressure variations in space of the order of of the order of MM22

the semi-discrete solution is characterized by pressure variations in the semi-discrete solution is characterized by pressure variations in space of the order ofspace of the order of M M

preconditioning preconditioning (following Guillard (following Guillard and Viozat, 1999)and Viozat, 1999)

1( ) ( ) 1( , ) ( , ) ( )2 2

i j

i j i j iF W F W

W W P PA W Wj W W

the scheme becomes the scheme becomes accurateaccurate also for M also for M0 (asymptotic analysis)0 (asymptotic analysis) preconditioning is applied only to the upwind part preconditioning is applied only to the upwind part time consistency time consistency for for unsteady problemsunsteady problems

The The preconditioning matrixpreconditioning matrix P is of P is of Turkel-typeTurkel-type (for its expression see (for its expression see Sinibaldi, Beux & Salvetti, Flow Turb. Comb. 2006 or P.H.D. Thesis by E. Sinibaldi, Beux & Salvetti, Flow Turb. Comb. 2006 or P.H.D. Thesis by E. Sinibaldi).Sinibaldi).

Time discretization for 1Time discretization for 1stst order order schemesschemes

Adopted approach: implicit linearized schemeAdopted approach: implicit linearized scheme

We have shown that for the Roe schemeWe have shown that for the Roe scheme(Sinibaldi et al., 2003 and (Sinibaldi et al., 2003 and Sinibaldi P.h.D. Thesis):Sinibaldi P.h.D. Thesis):

1 1 1 2( , ) ( , ) , n n n n n n n n n nij ij i j i i i j j jA W W W W A W W W W O t t x

Thus the implicit scheme can be Thus the implicit scheme can be linearizedlinearized as follows: as follows:

12

A A A

, , ,1 1 0 1 1 , 1 1,( ) i n i n i n n n

i i i i i i in n nB W B W B W

,1 1( , )

i n n ni iB A W W

,1 1( , )

i n n ni iB A W W

,0 11( , ) ( , )

i ni

n n n nii i in

xB I A W W A W Wt

njW

linear system linear system (tridiagonal in (tridiagonal in 1D)1D)

Backward Euler implicit Backward Euler implicit scheme:scheme: 1 1 1

, 1 1,

n n n n

i i i i i ii

tW W

NB:NB: remark that we did not use the homogeneity of the Eulerian fluxes, remark that we did not use the homogeneity of the Eulerian fluxes, which does not hold for generic barotropic state laws. which does not hold for generic barotropic state laws.

Space discretization:extension to 2Space discretization:extension to 2ndnd order of order of accuracyaccuracy

Adopted approach: MUSCL reconstructionAdopted approach: MUSCL reconstruction

1( ) ( ) 1 ( )2 2

ij i

ij jiji ijj

F FWA

WWP

WP

WWijij and and WWjiji are theare the extrapolated extrapolated values values of the variables at the of the variables at the cell interfacecell interface

ij i ij

ij j ji

W W W

W W W

Gradients can be computed in Gradients can be computed in different ways, by combining different ways, by combining different approximations (limited different approximations (limited stencil + ad hoc coefficients)stencil + ad hoc coefficients)

different schemesdifferent schemes2nd order accurate2nd order accurateintroducing a numerical viscosity introducing a numerical viscosity

proportional to 2proportional to 2thth, 4, 4thth or 6 or 6thth order order derivatives (Camarri et al., Comp. derivatives (Camarri et al., Comp. Fluids 2004).Fluids 2004).

Wi+1Wi-1 Wi

WWi-1,ii-1,i WWi,i-1i,i-1

Time advancing for the 2Time advancing for the 2ndnd order accurate order accurate schemescheme

Adopted approach: defect correctionAdopted approach: defect correction

1 1 ( ),0 ,

1

1( ) 0 with ( ) ( )qn n i p

p h p q q i h hi

L W L a a Wt

W W W

11 1 2

1

given ( . . )( ) ( ) ( ) 0,..., 1s s s

M

nh

nh

e g WL L L s M

W

0 0W WW W W

W

implicit formulation with a BDF method of order implicit formulation with a BDF method of order q:q:

( )ph pp-accurate discretization of the spatial differential operator-accurate discretization of the spatial differential operator

simpler non linear systems are iteratively considered (for simpler non linear systems are iteratively considered (for p=2):p=2):

s-th DeC iteration after linearization:s-th DeC iteration after linearization:1

(1) (2)43 ( ) ( )2 2

s n ns s sh h

h hW WI M

t t

3 WW W W

first-order linearized operatorfirst-order linearized operator (1)hM

block tridiagonal linear block tridiagonal linear systemsystem

First-orderFirst-order second-ordersecond-order

Time advancing for the 2Time advancing for the 2ndnd order accurate order accurate schemescheme

Adopted approach: defect correctionAdopted approach: defect correction

1

(1) 1 (2) 2 23 ( ) ( ) ( ) , ,( )2 2

n nn n n nh h

h h h h h hW WI M W W W W O x x t t

t t

Full convergence of the DeC iteration is not needed to reach the higher Full convergence of the DeC iteration is not needed to reach the higher order of accuracyorder of accuracy in space and timein space and time ( (Martin and Guillard, Comput. & Fluids, Martin and Guillard, Comput. & Fluids, 19961996))

block tridiagonal linear block tridiagonal linear system:system:

We have shown that, in our case, one DeC iteration is sufficient to We have shown that, in our case, one DeC iteration is sufficient to reach 2-order (space and time) accuracy:reach 2-order (space and time) accuracy:

, , ,1 1 0 1 1i n i n i n n

i i i in n nB W B W B W S

,1 1( , )

i n n ni iB A W W ,

1 1( , ) i n n n

i iB A W W

,0 11( , ) ( , )i n

in n n ni i i

xiB I A W W A W Wt

, 1 1,

1( ) ( )2

ni i i i i

n n n nii i

xS W Wt

For comparison, the fully 2-order linearized approach gives a For comparison, the fully 2-order linearized approach gives a block pentadiagonal system in 1Dblock pentadiagonal system in 1D

Extension to 2D-3DExtension to 2D-3DUnstructured grids Unstructured grids (tetrahedra)(tetrahedra)

Easy to build and to refine for 3D complex geometryEasy to build and to refine for 3D complex geometry

With respect to structured grids:With respect to structured grids: Larger complexity of implementation of numerical algorithms Larger complexity of implementation of numerical algorithms Larger computational requirements for fixed number of nodes.Larger computational requirements for fixed number of nodes.

Extension to 2D-3D Extension to 2D-3D (methodology developed at INRIA Sophia-(methodology developed at INRIA Sophia-

Antipolis)Antipolis)

i

ij

hk

hCkC

hC

kC

khhk CC

nodes in the neighbourhood of node i

( )

1 0( )

iij

j ii

dWdt Vol C

( , , ) ijij i jW W n

normal integrated on the cell boundary

1( ) ( ) 1( , , ) ( )2 2

F F

i jij iji j ij j i

W WW W n n P PA W W

Roe fluxRoe flux

Finite-volume dual gridFinite-volume dual grid (cells) obtained by using the medians of the tetrahedra (cells) obtained by using the medians of the tetrahedra facesfaces

1( ) ( ) 1 ( )2 2

ij jiiji ijj jj ii

F Fn WPA WP

W W

11stst order order

2nd order2nd order

ij i ij

ji j ji

W W W ij

W W W ij

Rotating frames Rotating frames Extension to non-inertial frames rotating with a constant rotational Extension to non-inertial frames rotating with a constant rotational speed speed

Incorporation of the non-inertial terms (Coriolis and centrifugal effects) Incorporation of the non-inertial terms (Coriolis and centrifugal effects) in a source term (S) in the momentum equationin a source term (S) in the momentum equation. . Finite–volumeFinite–volume discretization in space of S: discretization in space of S:

0

2

i i

i i i

S Vol Cu withwith

source term at node source term at node ii

1

i

ii C

xdVVol C

Linearized implicit time-discretization (to be incorporated in the Linearized implicit time-discretization (to be incorporated in the scheme) through the Jacobian: scheme) through the Jacobian:

1

n n nii i i

i

SS S WW

known (RHS term)known (RHS term)independent of timeindependent of time

diagonal term in diagonal term in linear system linear system

matrixmatrix

Quasi-1D water flow in a nozzleQuasi-1D water flow in a nozzle

1

5.7 5.0

2

OUTIN

Symmetrical grid, 360 cells, minimum spacing 0.02 (throat)

I.C’s: 00 , uu Inlet B.C’s: 00 ),( uupp Outlet B.C’s: 0// xuxp

STEADY-STATE IN a C-D SYMMETRICAL NOZZLE:

22 uu

dxdA

Apuu

xut

A cross-sectional area

1,2min t numerical transient

source

Water in standard condition Water in standard condition M M 10 10--33

11stst order of accuracy in space and Roe numerical flux order of accuracy in space and Roe numerical flux

Quasi-1D water flow in a nozzleQuasi-1D water flow in a nozzle

non preconditionednon preconditioned

Pressure distribution along the nozzle axis in Pressure distribution along the nozzle axis in non-cavitating non-cavitating conditionsconditions (pure liquid) (pure liquid)

Effects of preconditioning on the solution accuracyEffects of preconditioning on the solution accuracy

Preconditioning is actually important and works well in Preconditioning is actually important and works well in improving the numerical solution accuracy. improving the numerical solution accuracy.

Quasi-1D water flow in a nozzleQuasi-1D water flow in a nozzleEffects of preconditioning and of the time Effects of preconditioning and of the time

advancing scheme on the numerical efficiencyadvancing scheme on the numerical efficiency

Time-stepTime-stepImpl. prec.Impl. prec.

∞∞1.0e-51.0e-5

∞∞∞∞

Test-caseTest-case(sample)(sample) MachMach Cav./Non-Cav./Non-

cav.cav.Time-stepTime-stepExpl. non-Expl. non-

prec.prec.Time-stepTime-stepExpl. prec.Expl. prec.

TC1TC1 3.5e-33.5e-3 Non-cav.Non-cav. 1.0e-51.0e-5 1.0e-61.0e-6TC2TC2 3.5e-33.5e-3 Cav.Cav. 1.0e-51.0e-5 5.0e-75.0e-7TC3TC3 7.0e-47.0e-4 Non-cav.Non-cav. 1.0e-51.0e-5 5.0e-75.0e-7TC4TC4 7.0e-57.0e-5 Non-cav.Non-cav. 1.0e-51.0e-5 5.0e-85.0e-8

For For explicit time advancingexplicit time advancing, preconditiong significantly decreases the , preconditiong significantly decreases the maximum allowable time step. This reduction becomes more important maximum allowable time step. This reduction becomes more important as the Mach number decreases as the Mach number decreases precprec=O(M)*=O(M)*noprec noprec (see E. Sinibaldi PhD. (see E. Sinibaldi PhD. Thesis)Thesis) The The preconditioned linearized implicit schemepreconditioned linearized implicit scheme has practically has practically no time no time stepstep limitation in non cavitating conditionslimitation in non cavitating conditions.. In In cavitating conditionscavitating conditions, some improvements with respect to the , some improvements with respect to the explicit scheme are found, but explicit scheme are found, but severe limitations on the time-stepsevere limitations on the time-step remain.remain.

Riemann problemsRiemann problems 1D numerical experiments for Riemann problems characterized by:1D numerical experiments for Riemann problems characterized by:

different barotropic laws (including the one for cavitating flows)different barotropic laws (including the one for cavitating flows) different characteristic wavesdifferent characteristic waves different regimes (low Mach, transonic)different regimes (low Mach, transonic) Roe and Godunov fluxes (1Roe and Godunov fluxes (1stst order of accuracy) order of accuracy) implicit linearized scheme implicit linearized scheme

No differences between the results obtained with the Godunov scheme No differences between the results obtained with the Godunov scheme and with the Roe one and with the Roe one the Godunov scheme will not be used in other the Godunov scheme will not be used in other applications (2D, 3D) because much more computationally demandingapplications (2D, 3D) because much more computationally demanding For For non-cavitating barotropic lawsnon-cavitating barotropic laws the results show: the results show:

accuracy consistent with the used 1accuracy consistent with the used 1stst order accurate approximation, order accurate approximation, satisfactory efficiency of time advancing.satisfactory efficiency of time advancing.

p

u

stationary contact

blow-up for c(CFL) 10blow-up for c(CFL) 100

Riemann problemsRiemann problems flow regime: flow regime: low Machlow Mach flow regime: flow regime: generic Machgeneric Mach barotropic law:barotropic law: shock and rarefactionshock and rarefaction

barotropic law:barotropic law: 2 shocks2 shocks

p( )= 210M

0.9M 6 2p( )=10

600 cells600 cells4000 cells4000 cells2 4from 5.10 to 5.10t 1 3from 10 to 5. 10 t

u

Riemann problem for the cavitation barotropic Riemann problem for the cavitation barotropic lawlaw flow regime: flow regime: low Machlow Mach / / high Machhigh Mach

barotropic law: LdA model for cavitationbarotropic law: LdA model for cavitation

2 rarefactions2 rarefactions

’

head

tail

u p

p

tail !!!

p’

pressure(detail)

barotropic curve, for reference

pressure

2 4from 10 to 10 t 4000 cells4000 cells

Riemann problem for the cavitation barotropic Riemann problem for the cavitation barotropic lawlaw

’

head

tail

p

p’

barotropic curve, for reference

pressure, for reference

density

density (detail)

Very fine spatial discretization and small time steps are needed to Very fine spatial discretization and small time steps are needed to capture pressure and density “spikes” in the cavitating regioncapture pressure and density “spikes” in the cavitating region

4000 cells4000 cells

2 4from 10 to 10 t

IN OUT

Free-streams (T = 293.16 K):

non-cavitatingcavitating

cavitation number

Grids: cells

tetrahedra

GR2GR1 (det.)

Dirichlet homog. Neumann

Water flow around a hydrofoil mounted in a Water flow around a hydrofoil mounted in a tunnel tunnel

(Beux et al.,M(Beux et al.,M22AN, 2005)AN, 2005)

Inviscid flow.Inviscid flow. 11stst order of accuracy order of accuracy and Roe schemeand Roe scheme Linearized implicit Linearized implicit time advancingtime advancing

almost independent of the grid

test-section

Centro Spazio, Pisa

effect of

= 0.1

= 0.01

local preconditioning local preconditioning only in the cavitating only in the cavitating

region region

Water flow around a Water flow around a hydrofoil mounted in a hydrofoil mounted in a

tunnel tunnel (Beux et al.,M(Beux et al.,M22AN, 2005)AN, 2005)

Surprisingly good accuracy.Surprisingly good accuracy. Problems of efficiencyProblems of efficiency: non-: non-cavitating simulations CFL up to cavitating simulations CFL up to 400, with cavitation CFL400, with cavitation CFLmaxmax = 10 = 10-2-2

Pressure distribution over the Pressure distribution over the hydrofoilhydrofoil

= 0.1

= 0.01

Mach sigma

less pronounced Mach variation, OK more extended cavity, OK

local cavitation numberlocal cavitation number

Water flow around a hydrofoil mounted in a Water flow around a hydrofoil mounted in a tunnel tunnel

(Beux et al.,M(Beux et al.,M22AN, 2005)AN, 2005)

Mach up to 28Mach up to 28

Mach up to 11Mach up to 11

Some RemarksSome Remarks First series of test-cases (inviscid flows, 1First series of test-cases (inviscid flows, 1stst order of accuracy in space, order of accuracy in space, preconditioning, linearized implicit time advancing):preconditioning, linearized implicit time advancing):

quasi-1D water flow in a nozzle (non cavitating and cavitating quasi-1D water flow in a nozzle (non cavitating and cavitating conditions)conditions) Riemann problems with different barotropic state lawsRiemann problems with different barotropic state laws water flow around a hydrofoil (non cavitating and cavitating water flow around a hydrofoil (non cavitating and cavitating conditions)conditions)water flow in a turbopump inducer in water flow in a turbopump inducer in non-cavitating conditionsnon-cavitating conditions (not (not shown)shown) Satisfactory accuracy (in the limit of the assumptions made) in both non-Satisfactory accuracy (in the limit of the assumptions made) in both non-

cavitating and cavitating conditions.cavitating and cavitating conditions. Numerical efficiency problems when cavitating regions are present.Numerical efficiency problems when cavitating regions are present.

Additional series of 1D numerical experiments:Additional series of 1D numerical experiments: to investigate whether the to investigate whether the efficiency problemsefficiency problems in cavitating in cavitating conditions are due to the conditions are due to the adopted linearization techniqueadopted linearization technique for time for time advancing advancing to test the to test the efficiencyefficiency of the of the defect correctiondefect correction approach for approach for 22ndnd order order accuracy simulationsaccuracy simulations in non-cavitating conditions in non-cavitating conditions

linearized implicit vs. fully non-linear implicit in cavitating conditionslinearized implicit vs. fully non-linear implicit in cavitating conditions

No improvementNo improvement in in robustnessrobustness with the fully implicit with the fully implicit formulation formulation as for non cavitating flows, the fully implicit as for non cavitating flows, the fully implicit simulations blow up at lower CFL than the ones with the simulations blow up at lower CFL than the ones with the linearized implicit scheme.linearized implicit scheme. For the same resolution in space and the same time step, For the same resolution in space and the same time step, the the computational costscomputational costs are are much largermuch larger for the fully implicit for the fully implicit scheme.scheme.

Solution accuracySolution accuracy

Robustness and computational costRobustness and computational cost

Additional series of 1D numerical experiments:Additional series of 1D numerical experiments:

2 initial liquid states 2 initial liquid states 2 rarefactions 2 rarefactions LdA cavitating flow state equationLdA cavitating flow state equation

Test-case: Riemann problemTest-case: Riemann problem

Pressure field at t=1sPressure field at t=1s

4000 cells4000 cells55.10t

Discretization:Discretization:

detaildetail 100100-100-10020002000-2000-2000

Density field Density field

Velocity errorVelocity error

Spatial discretization: Spatial discretization: 400 cells400 cells

log-log scalelog-log scale

FOFO

TEST-CASE 1: Quasi-1D water flow in a convergent-divergent TEST-CASE 1: Quasi-1D water flow in a convergent-divergent nozzlenozzle

flow regime: steady and flow regime: steady and supersonic supersonic

barotropic law:barotropic law:

Validation of 2-order formulation: 1D numerical experimentsValidation of 2-order formulation: 1D numerical experiments

min( 10, 2)M M

p( )=2

Comparison of implicit formulationsComparison of implicit formulations

FO:FO: first-order (in space and time) first-order (in space and time) linearized implicit linearized implicit

slope slope 1 1 2-order (in space and time) linearized 2-order (in space and time) linearized implicitimplicit DeCDeC Defect correction approachDefect correction approach DeC with 1 and 2 inner iterationsDeC with 1 and 2 inner iterations

SOSO: : fully second-order linearized implicitfully second-order linearized implicit FU:FU: fully implicit second-orderfully implicit second-order (non linear solver (PETSc library) (non linear solver (PETSc library) based on a gradient-free Newton-GMRES based on a gradient-free Newton-GMRES approach)approach)

slope slope 2 2

TemporalTemporaldiscretizationdiscretization: :

t=0.0001t=0.0001

t=0.01 t=0.01

400 cells400 cells40 cells40 cells

Spatial Spatial refinemenrefinemen

tt

DeC2, DeC2, DeC3DeC3

DeC1DeC1

TEST-CASE 2: Riemann problem (shock and TEST-CASE 2: Riemann problem (shock and rarefaction)rarefaction)

flow regime: unsteady and subsonic barotropic law:flow regime: unsteady and subsonic barotropic law:( 0.1)M 6p( )=10

Spatial Spatial discretizationdiscretization

: : 400 cells400 cells

temporal temporal refinemenrefinemen

tt

t=0.001 t=0.001

velocity field (t=1 velocity field (t=1 s)s)

Validation of the second-order formulationValidation of the second-order formulation

Steady regimes: the steady solution is obtained after very few Steady regimes: the steady solution is obtained after very few pseudo-time iterations for all the linearized implicit approaches while pseudo-time iterations for all the linearized implicit approaches while the fully implicit formulation needs a CFL-like condition (for fine spatial the fully implicit formulation needs a CFL-like condition (for fine spatial discretization, i.e. large dimension of the non linear system)). discretization, i.e. large dimension of the non linear system)). For the same grid and time step, For the same grid and time step, DeC1DeC1 is approximately is approximately two times two times cheapercheaper than the than the fully second-order linearized implicitfully second-order linearized implicit approach. A approach. A larger ratio is expected for 3D cases due to the increase of complexity larger ratio is expected for 3D cases due to the increase of complexity and stiffness.and stiffness.

No loss of accuracyNo loss of accuracy with the present formulation: with the present formulation: neither due to the neither due to the defect correctiondefect correction comparison DeC/fully comparison DeC/fully second-order linearized implicitsecond-order linearized implicit nor due to the nor due to the linearization linearization of the implicit time-advancing of the implicit time-advancing comparison linearized implicit/fully implicit formulations comparison linearized implicit/fully implicit formulations

In accordance with the theoretical appraisal, one iteration of defect In accordance with the theoretical appraisal, one iteration of defect correction is already sufficient to reach 2correction is already sufficient to reach 2ndnd order accuracy order accuracy Nevertheless, for particular cases (large CFL number), a second inner Nevertheless, for particular cases (large CFL number), a second inner iteration can improve the solution (stabilization effect)iteration can improve the solution (stabilization effect)

Solution accuracySolution accuracy

Computational costComputational cost

Additional series of 1D numerical experiments:Additional series of 1D numerical experiments:

Concluding Remarks and Concluding Remarks and DevelopmentsDevelopments

For For non-cavitating barotropic flowsnon-cavitating barotropic flows, the proposed numerical methodology , the proposed numerical methodology shows satisfactory:shows satisfactory:

accuracy (MUSCL reconstruction + preconditioning for low Mach)accuracy (MUSCL reconstruction + preconditioning for low Mach) robustness and efficiency (linearized implicit time advancing + defect robustness and efficiency (linearized implicit time advancing + defect correction)correction)

For For cavitating flowscavitating flows and the homogeneous flow model: and the homogeneous flow model: severe restriction of the time step are observed severe restriction of the time step are observed unaffordable CPU unaffordable CPU requirements for 3D simulationsrequirements for 3D simulations numerical experiments show that this is not due to the adpted linearization of numerical experiments show that this is not due to the adpted linearization of the implicit time advancingthe implicit time advancing

Application of the numerical set-up (as it Application of the numerical set-up (as it is) to the simulation of problems is) to the simulation of problems characterized by barotropic laws less characterized by barotropic laws less stiff than the cavitating one (shallow stiff than the cavitating one (shallow water, atmosphere…)water, atmosphere…)

For For cavitating flowscavitating flows described through described through the the homogenous-flow modelhomogenous-flow model::

try more robust numerical fluxes try more robust numerical fluxes (HLL, HLLC) and/or(HLL, HLLC) and/or relaxion techniques in timerelaxion techniques in time

Change cavitation model (two-phases)Change cavitation model (two-phases)

Cavitating flow behaviorCavitating flow behavior

densitdensityy

pres

sure

pres

sure

liquid: ~ incompressibleliquid: ~ incompressible

liquid-vapour mixture: liquid-vapour mixture: highly compressiblehighly compressible

Time advancing for the 2Time advancing for the 2ndnd order accurate order accurate schemescheme

Full second-order linearized approachFull second-order linearized approach

, , , , ,2 2 1 1 0 1 1 2 2i n i n i n i n i n n

i i i i i in n n n nB W B W B W B W B W S

,2

,1

1, 2, 11

1, 1 1, 1, 1,

1, 1, 2, 1 1,1

32

12 2

1 2 2

i n

i n

n nii i i i

i

n n n nii i i i i i i i

i

n n n nii i i i i i i i

i

hB AhhB A Ah

hA Ah

B

,0

1, 1 1, , 1 1, 1, , 1

1

, 1 , 1 1, 1, 1, 1,

2 21

2

i n n n n n n ni ii i i i i i i i i i i i

i i

n n n n ni i i i i i i i i i i i

x h hi I A A A At h h

A A A

,

1

,2

, 1 , 1

1, , 1 , 1 , 11

1, 1 , 1 1, 2 , 1

2

1

12 2

1 +2 2

32

i n

i n

n n ni i i i

n n n nii i i i i i i i

i

n n n nii i i i i i i i

i

i

A

hB A Ah

hA Ah

hB

, 1 1, 22

n ni i i i

i

Ah

nose

inducer

afterbody

inter-blade covering: no gap

very complex geomety(detail of hub-blade

intersection)

Free-stream (T = 296.16 K):

2.5x106 elements

Flusso di acqua in un induttore di Flusso di acqua in un induttore di turbopompaturbopompa

(Sinibaldi et al., 2006)(Sinibaldi et al., 2006)

max (red) 177700 [Pa]min (blue) 79700 [Pa]spacing 5000 [Pa]

pressure contours velocity (longitudinal cut plane)

axial back-flow correctly described!

pretty nice results… it seems a promising scheme!

(cavitating simulation not affordable -at a “reasonable” cost- due to the efficiency issue)

Flusso di acqua in un induttore di Flusso di acqua in un induttore di turbopompaturbopompa

(Sinibaldi et al., 2006)(Sinibaldi et al., 2006)

Homogeneous-flow modelsHomogeneous-flow modelsThermal barotropic model Thermal barotropic model (d’Agostino et (d’Agostino et

al., 2001)al., 2001) pure liquid:pure liquid: weakly compressible

fluid satsatpp

ln1

satpp

mixturemixture

V

cLLL p

pgp

pdpd

)1()1(1

satp p

In which L, g*, pc, V are constants dependent on the considered flow

RT

ll ;

free model parameterfree model parameter

since lsat )1(

))(,(12 pfadp

d