A Numerical Method for 3D Barotropic Flows in Turbomachinery Towards Simulation of Cavitation...

A Numerical Method for 3D Barotropic Flows in Turbomachinery

Towards Simulation of Cavitation

Multiphase05, Porquerolles, April 2005

Edoardo SinibaldiScuola Normale Superiore di Pisa

François BeuxScuola Normale Superiore di Pisa

Maria Vittoria SalvettiDip. Ing. Aerospaziale, Università di Pisa

Research Framework (1)

Motivation: Numerical investigation of cavitation phenomena occurring in turbopump inducers typical of liquid propellant rocket engines

angular velocity

INFLOWOUTFLOW

3D CFD tool able to simulate complex cavitating flows in realistic geometries

I.C’s B.C’s

Performance (e.g. pout - pin)

Research Framework (2)

Development:

Governing Eq’ns

Liquid/Cavitation Model

3D Non-Rot. Num. Methods

1D Num. Methods

3D Rot. Num. Methods

Final Solver

valid’n

valid’n

valid’n

Constitutive Law (1)

SIMULATION OBJECTIVES: - Global performance predictions

- Cryogenic propellant flows → thermodynamic effects preponderant; negligible velocity and pressure differences between the two phases

CHARACTERISTIC TIME-SCALE:

UNCERTAINTY on the I.C.:

CHOICE of the CAVITATION MODEL:

- Short life-time → no long-period, local effects such as cavitation erosion

- Distribution of the active cavitation nuclei not known

HOMOGENEOUS-FLOW MODEL: Liquid-vapour mixture described as a homogeneous fluidThermal Cavitation Model

(d’Agostino et al., 2001)

Barotropic state law: Pure liquid:

Liquid-vapour: pp

:Lsat

:Lsat

LLLsatsat Tpppp

L;ln /1

RTpppad

dpTLcavcav /,;,2

2 model parameters


EXAMPLE: MIXTURE SOUND-SPEED CURVE FOR WATER-VAPOUR AT T=20°C

310 Lsatcav aa

]/[ smacav

Lsat

110cava

Nearly-incompressible:

3101 refref Mu

Highly-compressible:

101 refref Mu

validity limit of the model


EXAMPLE: BAROTROPIC CURVE FOR WATER-VAPOUR MIXTURE AT T=20°C

- Obtained by numerical integration (pre-processing):

1,:, kkjj pppp- Ad-hoc reparameterization, to directly access the constitutive law:

jj p,

satp

p

Lsat

]/[1062 smad

dpLsat

Nearly-incompressible!

Highly-compressible: ]/[10 22 smad

dpcav

Numerical Stiffness!!!

Governing Equations (1)

“Incompressible” method suitably corrected to take into account the effects of compressibility within the cavitating region

“Compressible” method suitably preconditioned to deal with the liquid, nearly-incompressible region

?

REQUIREMENTS: - Simultaneous solution of both the nearly-incompressible (pure liquid) and higly-compressible (liquid-vapour mixture) regions

- Ability to cope with a very complex geometry (inducer)

3D Compressible solver :(“AERO”; INRIA, Sophia-Antipolis)

unstructured grids

calorically-perfect gas state law

shock-capturing techniques

preconditioning

parallel implicit time-advancing

…

RESOURCES:

complex geometries

phase transition

Low Mach

TO BE MODIFIED!

efficiency

Governing Equations (2)

Barotropic state law

Viscosity effects negligible w.r.t. the dynamic action of the turbopump

Energy eq’n neglected (decoupled)

Inviscid approximation

MASS + MOMENTUM: qsqfx

qfx

qfx

qt

33

22

11 x1

x2

x3

Unknown state-vector: TT uuuqqqqq 3214321 ,,,,,,

Convective fluxes:T

q

qq

q

qqp

q

qqqf

1

42

1

32

1

2221 ,,,

T

q

qqp

q

qq

q

qqqf

1

43

1

33

1

2332 ,,,

T

pq

qq

q

qq

q

qqqf

1

44

1

34

1

2443 ,,,

Lsatcav

LsatL

qqp

qqpqpp

11

111 ;

;

Constitutive law:

Source term:

s

Tuuuu 321 ,,Physical entities:

p(density) (pressure) (velocity)

Numerical DiscretisationTime and space discretization separate (“line method”)

Time discretizationExplicit: RK4, low storage

Implicit: linearisation based on the 1st order homogeneity of the convective fluxes coupled with the perfect gas state law

Space discretization Finite volumes; Roe scheme based on the perfect gas state law for the convective fluxes, preconditioned for low Mach numbers

Some starting numerical ingredients (AERO):

Time and space discretization separate (“line method”)

Time discretizationExplicit: RK4, low storage

Implicit: new linearisation based only on the algebraic properties of the Roe scheme, valid in particular for a barotropic state law

Space discretization Finite volumes; Roe scheme based on a generic barotropic state law for the convective fluxes, preconditioned for low Mach numbers

Corresponding ingredients of the

derived solver:

OLD

NEW

Space Discretisation (1)

FINITE VOLUMES:

Cj

x

Ck Ck+1Ck-1

0

dxsfx

qt j

(Continuous)

dxtxstsjCj ,

jm measure of the cell Cj

dxtxqm

tqjC

jj

,1 (numerical unknown)

rl , numerical flux between cell l(eft) and cell r(ight)

0,11, jjjjjjj sqdt

dm

(Semi-discrete)

1j

0j 0j

njj ,,1;

p.w. constant (basis) functions

1D for simplicity

qk-1 qk qk+1

Φk-1,k Φk,k+1

numerical flux between cell k and cell k+1


4

1,2

i

ilr

ilrlrirlrl rqqcqfqf ROE NUMERICAL

FLUX FUNCTION:(P. Roe, 1981)

where are the eigenvalues-(right) eigenvectors of the Roe matrix

satisfying: ilr

ilr r,

lrJ

lrJ is diagonalizable with real eigenvalues

(R1)

qq

fJ lr

qqqrl

lim,

(R2)

lrlrlr qfqfqqJ (R3)

and is the n-th coordinate of the vector w.r.t. the eigenvectors basis . wcn w

depends on the constitutive law!

e.g.: x1-sweep

FOR A GENERIC BAROTROPIC STATE LAW:(Sinibaldi, Beux & Salvetti 2003)

lrlrlrlr

lrlrlrlr

lrlrlrlr

uuuu

uuuu

uuaJ

1331

1221

121

2

0

0

002

0010

with:rl

rrlllr

uuu

ˆˆ2

2

lr

lrlr

lr

lr

a

ppa

classical


LOW-MACH NUMBER ASYMPTOTIC STUDY (Guillard & Viozat, 1999; perfect gas state law):

Non-dimensionalization

l* (length)

ρ* (density)

u* (convective speed)

a* (sound speed)

Governing eq’ns

(continuous or semi-discrete)

obtain non-dimensional eq’ns depending on M* = u* / a*

Asymptotic expansion

Expand the unknowns in power of M* and take

the limit for M*→0

obtain an asymptotic eq’n of the form:

001

12

2

MM

012

LOSS OF SPATIAL ACCURACY IN THE SEMI-DISCRETE CASE!!!

txpMtpMtptxp ,,0 22

1012 Continuous eq’ns:

tpMtptp jj 1012 0 Semi-discrete eq’ns:

Pressure asymptotic

behaviour for M*→0

FOR a GENERIC BAROTROPIC STATE LAW:(Sinibaldi, Beux & Salvetti 2003) same kind of result as for perfect gas!


PRECONDITIONING for LOW-MACH NUMBERS:

lrlrrlrl qqJqfqf ,2ROE, original

ROE, preconditioned

(Guillard & Viozat, 1999;

perfect gas state law)

lrlrlrlrrlrl qqJPPqfqf 1

,2

11 ),,( RDiagRM n

M diagonalizable

R (righ) eigenvectors similarity matrix

In conservative variables, for a generic BAROTROPIC state law:

lrPwhere is the preconditioner, originally conceived in primitive variables:

1000

0100

0010

000

,,,

2

321

uuupP

and is a constant. 2

000

000

000

0001

1

3

2

12

u

u

uIqP

000

000

000

0001

1

3

2

12

lr

lr

lrlr

u

u

uIP


PRECONDITIONING for LOW-MACH NUMBERS:

By performing the low Mach number asymptotic analysis (M*→0) for the preconditioned semi-discrete eq’ns:

tpMtpMtptpMO jj 22

1012 0

SPATIAL ACCURACY RECOVERED AT LOW MACH!!!(Sinibaldi, Beux & Salvetti 2003)

The preconditioner preserves time-consistency → unsteady computations!

lrlrlrlrrlrl qqJPPqfqf 1

,2

“centered” “upwind”

same kind of result as for perfect gas!

1D Validation (1)

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

Explicit time-advancing (RK4) with constant time-step

“Local” preconditioning:

otherwise

M Lsatrl

1

,202

STEADY-STATE IN a C-D SYMMETRICAL NOZZLE:

22 u

u

dx

dA

Apu

u

xut

A cross-sectional area

1,2min t numerical transient

source

1D Validation (2)

EXAMPLE: A NON-CAVITATING TEST-CASE AT M* = 3.5e-3

Lsat

satp

p

throat throat

0005.1 0005.1

0004.1 0004.1

480 440

320 350

NON-PREC. PREC.

1D Validation (3)

EXAMPLE: A CAVITATING TEST-CASE AT M* = 3.5e-3

NON-PREC. PREC.

Lsat

throat throat

satp

p

05.1 10.1

70.0 40.0

120 70

0 0

1D Validation (4)

TEMPORARY RESULTS:

The preconditioner does the job!

The preconditioned scheme requires a smaller time-step to remain stable: the smaller the characteristic Mach number, the smaller the time-step required! The reduction is more pronounced for the cavitating test-cases.

(The time-step reduction has been analyzed for the perfect gas state law by Birken, LOMA conference, Porquerolles, June 2004)

…A FASTER TIME-ADVANCING STRATEGY IS NEEDED!

Test-case

(sample)Mach no. Cav./Non-cav.

Time-step

Expl. non-prec.

Time-step

Expl. prec.

TC1 3.5e-3 Non-cav. 1.0e-5 1.0e-6

TC2 3.5e-3 Cav. 1.0e-5 5.0e-7

TC3 7.0e-4 Non-cav. 1.0e-5 5.0e-7

TC4 7.0e-5 Non-cav. 1.0e-5 5.0e-8

Time Discretisation (1)

0,11, jjjjjjj sqdt

dm

(Semi-discrete) 0,11,

1

nj

njj

njj

n

j

n

jj s

t

qqm

011,1

11,

1

nj

njj

njj

n

j

n

jj s

t

qqm

(Explicit)

(Implicit)

“CHEAP” - SMALL TIME-STEP

LARGE TIME-STEP – “EXPENSIVE”

COMPROMISE: LINEARISED IMPLICIT!!!

RHSqMqMqMj

n

j

n

j

n 1)1()0(1)1(

nn 1Assume

r

nrll

nrlrl qRqL ,,, where ; then:

j

njj qDs and

njj

n LM ,1)1( nj

njj

njj

jn DRLIt

mM

,11,)0(

njj

n RM 1,)1(

nj

njj

njj sRHS ,11,

Time Discretisation (2)

…ON THE NUMERICAL FLUX LINEARISATION…

l

rlrl q

L

,,- The Roe flux function is not differentiable →

r

rlrl q

R

,,

- If the convective flux were 1st-order homogeneous (e.g. for perfect gas state law): qq

ff

then classical linearisations for the Roe scheme exist (e.g. Fezoui & Stoufflet, 1989) but the homogeneity property does not hold for the generic barotropic case!

- A new linearisation has been proposed (Sinibaldi, Beux & Salvetti, 2003): lrrl JL , lrrl JR ,

lrlrlr JJJ 2

txtOqJqJr

nlrl

nlrrl ,2

, under some regularity assumptions:

applicable to a wide class of problems (based only on algebraic properties of the Roe scheme)

extended to the preconditioned flux function: lrlrlrrl JPPR 1,

lrlrlrrl JPPL 1,

1D Validation (5)

“COMPLETE” RESULTS:

Time-step

Impl. prec.

∞

1.0e-5

∞

∞

Test-case

(sample)Mach no. Cav./Non-cav.

Time-step

Expl. non-prec.

Time-step

Expl. prec.

TC1 3.5e-3 Non-cav. 1.0e-5 1.0e-6

TC2 3.5e-3 Cav. 1.0e-5 5.0e-7

TC3 7.0e-4 Non-cav. 1.0e-5 5.0e-7

TC4 7.0e-5 Non-cav. 1.0e-5 5.0e-8

New entry

NON-CAVITATING TC’s → “NO RESTRICTIONS” ON THE TIME-STEP!

CAVITATING TC’s → RECOVER THE EXPLICIT, NON-PRECONDITIONED TIME-STEPImplicit

3D Numerical Discretisation (1)

UNSTRUCTURED GRID (THETAHEDRICAL)

Finite volumes built on the dual grid (median planes)

i

ij

h

k

hC kChC

kC

khhk CC

nodes in the neighbourhood of node i

Semi-discrete conservation for

hC

0

h

Chkkhhh sqqq

dt

dm

hk

time der. fluxes source


khkjhhkj

jhkjhk

Ckhqqnqq

hk,,

1,

~~~

0

0

0

0001

~

,,

hkjhkj

ROE→ hkhkhkhkhkkhhk

CkhqqJPPqfqfnqq hk

hk

1,2

with:

nlrlrlrlr

nlrlrlr

lrn

lrlrlrn

lrlrlr

lrlrn

lrlrn

lrlrlrnlr

nununuuan

nununuuan

nununuuan

nnn

J

33231332

3

32221222

2

31211112

1

3210 Tffff 321 ,,

, dot product (formal)

nu lrn

lr ,

Extension of the convective scheme:

h khkn

je

hkhkChk ndSnhk

j 1D num. flux associated with the xj-sweep of the gov. eq’ns

hkj , rotation: hkje ˆ


Preconditioner: same as for 1D:

000

000

000

0001

1

3

2

12

lr

lr

lrlr

u

u

uIP

otherwise

Mk Lsatrl

1

,22 heuristic, with k

“calibration” parameter

Linearised implicit time-advancing: analogous to 1D (the rotation does not affect the linearisation):

RHSqRqDLI

t

mk

hk

nkhh

nh

hk

nkh

h

hk

hkhkhkkh JPPL 1 hk

hkhkhkkh JPPR 1

nh

hk

n

CkhsqqRHS

hk

c = 115 mm (hydrofoil chord)

IN OUT

Data:

• Temperature = 20°C (→aLsat ≈ 1416 m/s)

• Inlet speed = 3.12 m/s (→M* ≈ 2.2e-3 )

3D Validation (1)

Test-caseInlet pressure

PaCav./Non-cav. δT/R

TC1 59050 Non-cav. 0.1

TC2 7500 Cav. 0.1

TC3 7500 Cav. 0.01

Water flow around a NACA0015 hydrofoil

LOMA conference, Porquerolles, June 2004

0hs (no source)

IN B.C’sINpp INuu

0qgrad OUT B.C’s

3D Validation (2)

3 SYMMETRICAL GRIDS

Grid No. of nodes No. of elements No. of partitions

GR1 45379 234834 5

GR2 27220 137756 5

GR3 19322 88400 2

GR1 GR2: “optimized” w.r.t. GR1

GR3: similar to GR2; thinner domain (0.1c instead of 0.7c)

3D Validation (3)TC1 (Non-cav.)

NUMERICAL RESULTS vs EXPERIMENTS

Pressure coefficient distribution – symmetry plane

2

2

inin

inp

u

ppC

5.1

5.1

exp. (CentroSpazio, Pisa)

grid GR3

grid GR2

*

2

2

inin

inp

u

ppC

0.1

5.1

exp. (CentroSpazio, Pisa)

TC2 (δT/R = 0.1), grid GR3

TC3 (δT/R = 0.01), grid GR3

*

Pressure coefficient distribution – symmetry plane

3D Validation (4)TC2 – TC3 (Cav.)


3D Validation (5)

TEMPORARY ACHIEVEMENTS

The PRECONDITIONING strategy effectively counteracts the accuracy problem at low Mach numbers.

Preliminary numerical experiments suggested the value k = 1 for the “calibration” parameter involved in the local preconditioning: β2 = k M*

2, thus confirming the theoretical result.

The LINEARISED IMPLICIT strategy well extends to 3D; indeed, for the non-cavitating test-cases a CFL coefficient as high as ≈400 has been exploited.

When cavitation occurs, however, a significant time-step reduction must be accepted, as already suggested by the 1D validation (efficiency problem):

Test-case / Grid Computer CFL coeff. Total CPU time

TC1 / GR3 Intel P4, 2.66GHz 400 7h 30m

TC2 / GR3 Intel P4 Xeon, 3.06GHz 0.05 ≈ 150h

TC3 / GR3 Intel P4 Xeon, 3.06GHz 0.05 ≈ 150h

3D Numerical Discretisation (4)…WELCOME TO THE ROTATING WORLD!

xus 2Source:

dVxums

hChhhh 2

The local Mach number is significantly affected by the dragging velocity and therefore more locality (and a smooth transition) is needed in the preconditioning strategy:

222 exp1 lrlr MkM with2lr

lrlr

a

uM

…still heuristic, with k “calibration” parameter

(k=1…from the NACA experience)

const INFLOWOUTFLOW

Recast the eq’ns in the rotating frame → the case rotates with -!!!

Axisymmetrical case, to keep the grid fixed!!!LIN

D

LOUT

3D Validation (6)

nose

inducer

afterbody

IN B.C’s INpp ruu IN (take into account the dragging velocity)

OUT B.C’s 0qgrad rel (gradient w.r.t. the rotating frame)

Data:

• Temperature = 23°C (→aLsat ≈ 1405 m/s)

• δT/R = 0.1

• Inlet speed = 4.77e-1 m/s (→M* ≈ 3.4e-4 )

• Inlet pressure = 1.15e5 Pa

• Rotational speed = 2000 rpm (≈ 210 rad/s)

NO TIP CLEARANCE (no gap between the maximum blade tip radius and the case)

Non-cavitating water flow around a real turbopump inducer

3D Validation (7)

Details of the inducer geometry

INTER-BLADE COVERING (no gap)

HUB-BLADE INTERSECTION (detail)

3D Validation (8)


A non-cavitating simulation has been performed on a grid having L IN = 1.5 D, LOUT = 1.5 D, 549139 nodes and 2588501 elements.

The CFL coefficient has been increased during the simulation up to a value of 350.

The simulation has been run on (16 CPUs) IBM POWER4, requiring 1500 hours (total CPU time).

GOOD QUALITATIVE AGREEMENT as for the pressure field (pressure contours on the blade surface); correct prediction of the strong axial backflow (occurring where the volutes are not completely shrouded).

REASONABLE QUANTITATIVE AGREEMENT with the experimental data (CentroSpazio, Pisa, 2004): the static pressure rise (pOUT - pIN) is overestimated by a factor 3.28. Not surprising because:

Inviscid approximation

No tip clearance (→ no secondary flows reducing the pumping effect) A cavitating simulation has been performed but, due to the efficiency problem, it does not seem to converge within “reasonable” cpu times (unless very powerful supercomputing resources are available…)

3D Validation (9)

PRESSURE CONTOURS ON THE BLADE SURFACE

AXIAL BACKFLOW (longitudinal cut plane)

NUMERICAL RESULTS

Conclusions and Perspectives

Method robust and quite accurate (1st order) for non-cavitating flows.

The accuracy has been increased to second order (MUSCL + Defect Correction) for the 1D scheme; application to the 1D shallow-water eq’ns (Sinibaldi & Beux, SIMAI, Venezia, 2004):

For cavitating flows the efficiency problem must be addressed. To the purpose:

investigation of smoother barotropic laws; the one used is significantly stiffer than other common models! (ongoing)

investigation of the entropic character of the scheme (state law) at phase transition; comparison with the exact solution to a 1D Riemann problem for generic barotropic state laws (ongoing)

mixture fractions transport…, relaxation of the density…?

As for physical modelling: inclusion of the effects of viscosity and turbulence

A Numerical Method for 3D Barotropic Flows in Turbomachinery Towards Simulation of Cavitation...

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