A numerical method for barotropic
flow simulation with applications to
cavitation
M.V Salvetti, F. Beux, M. Bilanceri (University of Pisa)
E. Sinibaldi (now at Scuola Superiore Sant’Anna, Pisa)
Micro-Macro Modelling and Simulation of Liquid-Vapour Flows
Strasbourg, 23-25 January 2008
Industrial and engineering motivation
Development of a numerical tool for the prediction of the
performance of axial inducers typical of turbopumps for
liquid propellent rockets.
rotating inducer
non rotating cylindrical case
The main role of the inducer is to increase the fluid pressure (velocity
decrease) through rotation.
Fundings from the Italian Space Agency and European Space Agency.
Difficulties
 Complex rotating 3D geometry.
 Severe size limitations which lead to very
high rotational speed  cavitation phenomena
 need of a model to take into account
cavitation.
INFLOW
Choice of the cavitation model
Considering the characteristics of the considered engineering problem
(interest in global performance predictions, short life-time, cryogenic
propellers, distribution of the active cavitation nuclei not known)
Homogeneous-flow model, i.e. liquid/vapour mixture modeled as a singlephase fluid
Cavitation model
Model proposed by L. d’Agostino et al. (2001), which takes into account
(at least approximately) for thermal cavitation effects .
Barotropic flow: the state equation relates pressure and density. In
particular, for the considered model the state equation has the
following form:
liquid
liquid-vapour mixture
p  p sat p  f (  ,  )
p  p sat
d
 f ( p,  ,  )
dp
characteristic fluid
physical parameters
(known)
Starting from this context, the more general scope of the research
activity is to develop a numerical tool for the simulation of barotropic
flows in complex geometry.
Cavitating flow behavior
speed of sound
Difficulty: in the homogenous-flow description, the physical properties
of the flow change dramatically between the zones of pure liquid and
the cavitating regions (fluid/vapour mixture).
high speed of sound (H2O
~ 1400 m/s)
M<<1  incompressible
regime
high M  supersonic-hypersonic flow
density
Modellazione dei flussi cavitanti
 Flows characterized by nearly incompressible zones together with
highly supersonic flow regions.
two possible choices:
Numerical solvers for
incompressible flows suitably
corrected to take into
account compressibility
p unknown
Examples of applications to
cavitating flows: van der Heul et al.,
ECCOMAS 2000, Senocack and
Shyy, JCP 2002, …
Numerical discretization of the
compressible flow equations
 unknown, p from the state
equation
Mathematical model
Because of the barotropic state law the energy equation is decoupled
 only mass and momentum balances are considered
 Equations for 3D laminar viscous compressible flows (no turbulence)
in conservative variables
or
Equations for 3D inviscid compressible flows in conservative variables
+
barotropic state equation (ODE or
analytic laws)
Numerical discretization: outline
 1D inviscid flows

Spatial discretization of 1st order of accuracy (preconditioning)

Linearized implicit time advancing

Extension to 2nd order of accuracy in space (MUSCL)

Time advancing for 2nd order scheme (defect correction)
 3D inviscid flows (in rotating frames)

extension of the previous ingredients to tetrahedral
unstructured grids
 3D laminar viscous flows

P1 finite-element discretization of viscous flows (not shown)
1D inviscid flows:spatial discretization
  
W  
 u 


W  F (W )  0
t
x
i-1,i
 u
F  2
 u 
i,i+1
Wi-1
Wi
Wi+1
Ci-1
Ci
Ci+1
i
d
Wdx  F C C  F C C  0

i
i 1
i 1
i
dt Ci
time discretization


p
p  p
Galerkin projection on the
finite-volume basis functions
(piecewise constant)
x
Numerical fluxes
dWi
i
  i ,i 1   i 1,i  0
dt
Numerical fluxes
Godunov-type flux: the exact solution of the Riemann problem
between two neighboring cells is used.



i ,i 1  F WRP Wi L ,Wi R1,0  i 1,i  F WRP Wi L1,Wi R ,0 
In the present research activity, a procedure has been developed for
the construction of the Riemann problem solution for Euler equations
and a generic convex barotropic law.
Reference: E. Sinibaldi, Implicit preconditioned numerical
schemes for the simulation of three-dimensional
barotropic flows, Pisa, Edizioni della Normale, in press,
ISBN 978-88-7642-310-9.
 Exact 1D benchmark for generic barotropic state laws
 Construction of a Godunov-type scheme

Numerical fluxes
Roe scheme: approximated solution of the Riemann problem
between two neighboring cells is used. .
 l ,r   (Wl ,Wr ) 
F (Wl )  F (Wr ) 1
 A(Wl ,Wr ) (Wl  Wr )
2
2
centered part
Roe matrix
upwind part
numerical viscosity
Contribution of the present research activity  definition of the
Roe matrix for a generic barotropic state law (PhD. Thesis by E.
Sinibaldi or Sinibaldi, Beux & Salvetti, INRIA RR4891, 2003 (available on
line), Sinibaldi, Beux & Salvetti, Flow Turbulence and Combustion 76(4),
2006).
Preconditioning for low-Mach numbers
Problem: the numerical solvers for compressible flows suffer in general of
accuracy problems if applied to low Mach flows. Following Guillard and Viozat
(1999), an asymptotic analysis for M0 (Sinibaldi, Beux & Salvetti, 2003 or
P.H.D. Thesis by E. Sinibaldi) shows that:
 the continous solution is characterized by pressure variations in space of
the order of M2
 the semi-discrete solution is characterized by pressure variations in space
of the order of M
preconditioning
(following Guillard and
Viozat, 1999)
(Wi ,W j ) 
F (Wi )  F (W j ) 1 1
 P PA(Wi ,Wj ) (W j  Wi )
2
2
 the scheme becomes accurate also for M0 (asymptotic analysis)
 preconditioning is applied only to the upwind part  time consistency for
unsteady problems
The preconditioning matrix P is of Turkel-type (for its expression see Sinibaldi,
Beux & Salvetti, Flow Turb. Comb. 2006 or P.H.D. Thesis by E. Sinibaldi).
Time discretization for 1st order schemes
Adopted approach: implicit linearized scheme
 Backward Euler implicit scheme:
Wi n 1 
 t n 1
 i ,i 1   in1,1i   Wi n

i
 We have shown that for the Roe scheme(Sinibaldi et al., 2003 and Sinibaldi
P.h.D. Thesis):



 
ijn1  ijn  A (Wi n ,W jn ) Wi n1  Wi n  A (Wi n ,W jn ) W jn1  W jn  O  t 2 ,  t x


1
 nW j
A A
2
 Thus the implicit scheme can be linearized as follows:
A 
Bi ,1n  nWi 1  B0i ,n  nWi  B1i ,n  nWi 1  ( in,i 1   in1,i )
Bi ,1n   A (Wi n1 ,Wi n )
B0i ,n 
 xi
 nt
B1i ,n  A (Wi n ,Wi n1 )
I  A (Wi n ,Wi n1 )  A (Wi n1 ,Wi n )
linear system (tridiagonal in 1D)
NB: remark that we did not use the homogeneity of the Eulerian fluxes, which
does not hold for generic barotropic state laws.

Space discretization:extension to 2nd order of accuracy
Adopted approach: MUSCL reconstruction
F (Wij )  F (W ji ) 1 1
ij 
 P PAij (W ji  Wij )
2
2
Wij and Wji are the extrapolated values of
the variables at the cell interface
Wi-1,i
Wi-1
Wi,i-1
Wi
Wi+1
Wij  Wi  Wij
Wij  W j  W ji
Gradients can be computed in
different ways, by combining
different approximations (limited
stencil + ad hoc coefficients)
different schemes
2nd order accurate
introducing a numerical viscosity
proportional to 2th, 4th or 6th order
derivatives (Camarri et al., Comp. Fluids
2004).
Time advancing for the 2nd order accurate scheme
Adopted approach: defect correction
 implicit formulation with a BDF method of order q:
q
1
n 1
L p (Wh )  0 with L p (W )   aq ,0W   aq ,iWhn 1i    (hp ) (W )
i 1
t 

 (hp )
p-accurate discretization of the spatial differential operator
 simpler non linear systems are iteratively considered (for p=2):

W 0 given
(e.g . W 0  Whn )

s 1
s
s
s  0,..., M  1
 L1 (W )  L1 (W )  L2 (W )

Whn 1  W M

 s-th DeC iteration after linearization:
First-order
 3 W s  4Whn  Whn 1
 3
(1)
s  s
(2)
s 
I

M
(
W
)

W




(
W
)


h
h

2 t
 2 t



M h(1)
first-order linearized operator
block tridiagonal linear system
second-order
Time advancing for the 2nd order accurate scheme
Adopted approach: defect correction
Full convergence of the DeC iteration is not needed to reach the higher order of
accuracy in space and time (Martin and Guillard, Comput. & Fluids, 1996)
 We have shown that, in our case, one DeC iteration is sufficient to reach 2order (space and time) accuracy:
 Whn  Whn 1
 3
(1)
n 
n 1
n
(2)
n 
2
2
I

M
(
W
)
W

W




(
W

h
h 
h
h
h
h )   O ( x ) ,  x t ,( t )

2 t
 2 t





block tridiagonal linear system:

Bi ,1n  nWi1  B0i ,n  nWi  B1i ,n  nWi1  Sin
Bi ,1n   A (Wi n1 ,Wi n ) B1i ,n  A (Wi n ,Wi n1 )
B0i ,n 

x

Sin    i (Wi n1  Wi n )  ( ni ,i 1   ni 1,i ) 
i I  A (W n ,W n )  A (W n ,W n )
 2 t

i 1
i
i 1
i
x
t
 For comparison, the fully 2-order linearized approach gives a
block pentadiagonal system in 1D
Extension to 2D-3D
Unstructured grids (tetrahedra)
 Easy to build and to refine for 3D complex geometry
With respect to structured grids:
 Larger complexity of implementation of numerical algorithms
 Larger computational requirements for fixed number of nodes.
Extension to 2D-3D
(methodology developed at INRIA Sophia-Antipolis)
Finite-volume dual grid (cells) obtained by using the medians of the tetrahedra faces
nodes in the neighbourhood of
node i
dWi
1

 ij  0
dt Vol (Ci ) j (i )
j   i 
 ij   (Wi ,W j , nij )
normal integrated on the
cell boundary
Roe flux
(Wi ,W j , nij ) 
ij 
F (Wi )  F (W j )
2
F (Wij )  F (W ji )
2
Chk  Ckh
Ck
h
i
k
Ch
1
 nij  P 1 PAij (W j  Wi )
2
1
 nij  P 1 PAij (W ji  Wij )
2
Ch
Ck
1st order
2nd order
Wij  Wi  Wij  ij
W ji  W j  W ji  ij
Rotating frames
Extension to non-inertial frames rotating with a constant rotational speed 
 Incorporation of the non-inertial terms (Coriolis and centrifugal effects) in
a source term (S) in the momentum equation.
 Finite–volume discretization in space of S:
0

Si  Vol  Ci  

  i  2  i ui 
with


1
 i      
xdV 



Vol  Ci  Ci


source term at node i
 Linearized implicit time-discretization (to be incorporated in the scheme)
through the Jacobian:
n 1
i
S
known (RHS term)
 Si  n
 S 
  Wi
 Wi 
n
i
diagonal term in
linear system matrix
independent of time
Quasi-1D water flow in a nozzle
STEADY-STATE IN a C-D SYMMETRICAL NOZZLE:

t
     u 
 dA  u 
  
 2
  
 2 

u

u

p

x
A
dx
 


 u 
A cross-sectional area
  min 2t,1
source
numerical transient
Symmetrical grid, 360 cells, minimum spacing 0.02 (throat)
2
1
IN
OUT
5.7
Inlet B.C’s:
5.0
p  p( 0 ), u  u0
I.C’s:
   0 , u  u0
Outlet B.C’s:
p / x  u / x  0
Water in standard condition  M 10-3
1st order of accuracy in space and Roe numerical flux
Quasi-1D water flow in a nozzle
Effects of preconditioning on the solution accuracy
Pressure distribution along the nozzle axis in non-cavitating conditions
(pure liquid)
non preconditioned
 Preconditioning is actually important and works well in improving the
numerical solution accuracy.
Quasi-1D water flow in a nozzle
Effects of preconditioning and of the time advancing
scheme on the numerical efficiency
Mach
Cav./Noncav.
Time-step
Expl. nonprec.
Time-step
Expl. prec.
Time-step
Impl. prec.
TC1
3.5e-3
Non-cav.
1.0e-5
1.0e-6
∞
TC2
3.5e-3
Cav.
1.0e-5
5.0e-7
1.0e-5
TC3
7.0e-4
Non-cav.
1.0e-5
5.0e-7
∞
TC4
7.0e-5
Non-cav.
1.0e-5
5.0e-8
∞
Test-case
(sample)
 For explicit time advancing, preconditiong significantly decreases the
maximum allowable time step. This reduction becomes more important as the
Mach number decreases  prec=O(M)*noprec (see E. Sinibaldi PhD. Thesis)
 The preconditioned linearized implicit scheme has practically no time step
limitation in non cavitating conditions.
 In cavitating conditions, some improvements with respect to the explicit
scheme are found, but severe limitations on the time-step remain.
Riemann problems
 1D numerical experiments for Riemann problems characterized by:
 different barotropic laws (including the one for cavitating flows)
 different characteristic waves
 different regimes (low Mach, transonic)
 Roe and Godunov fluxes (1st order of accuracy)
 implicit linearized scheme
 No differences between the results obtained with the Godunov scheme and
with the Roe one  the Godunov scheme will not be used in other applications (2D,
3D) because much more computationally demanding
 For non-cavitating barotropic laws the results show:
 accuracy consistent with the used 1st order accurate approximation,
 satisfactory efficiency of time advancing.
 flow regime: low Mach
 barotropic law:
 M  6 1022 
p( )=10 
 shock and rarefaction
p
 t from 5.102 to 5.104 4000 cells
u
blow-up for c(CFL) 100
Riemann problems
 M   0.9 
p( )=
 flow regime: generic Mach
 barotropic law:
 2 shocks

600 cells
 t from 101 to 5. 103

blow-up for c(CFL) 10
stationary
contact
Riemann problem for the cavitation barotropic law
 flow regime: low Mach / high Mach
 barotropic law: LdA model for cavitation
 t from 102 to 104
2 rarefactions
head
p
u
4000 cells
pressure
tail
pressure
(detail)
p’
p
barotropic curve,
for reference
tail !!!
’
Riemann problem for the cavitation barotropic law
 Very fine spatial discretization and small time steps are needed to capture
pressure and density “spikes” in the cavitating region
 t from 102 to 104
head
4000 cells
p
pressure, for
reference

density

density
(detail)
tail
p’
barotropic
curve, for
reference
’
Water flow around a hydrofoil mounted in a tunnel
(Beux et al.,M2AN, 2005)
IN
OUT
Dirichlet
homog. Neumann
 Inviscid flow.
Free-streams (T = 293.16 K):
cavitation number
non-cavitating
cavitating
Grids:
cells
tetrahedra
GR1 (det.)
 1st order of accuracy
and Roe scheme
 Linearized implicit time
advancing
GR2
Water flow around a hydrofoil
mounted in a tunnel
test-section
(Beux et al.,M2AN, 2005)
Pressure distribution over the hydrofoil
effect of 
Centro Spazio, Pisa
almost independent of the grid
 = 0.1
 = 0.01
local preconditioning only
in the cavitating region
 Surprisingly good accuracy.
 Problems of efficiency: noncavitating simulations CFL up to 400,
with cavitation CFLmax = 10-2
Water flow around a hydrofoil mounted in a tunnel
(Beux et al.,M2AN, 2005)
local cavitation number
Mach
sigma
Mach up to 28
 = 0.1
less pronounced Mach variation, OK
Mach up to 11
 = 0.01
more extended cavity, OK
Some Remarks
 First series of test-cases (inviscid flows, 1st order of accuracy in space,
preconditioning, linearized implicit time advancing):
 quasi-1D water flow in a nozzle (non cavitating and cavitating conditions)
 Riemann problems with different barotropic state laws
 water flow around a hydrofoil (non cavitating and cavitating conditions)
water flow in a turbopump inducer in non-cavitating conditions (not shown)
 Satisfactory accuracy (in the limit of the assumptions made) in both noncavitating and cavitating conditions.
 Numerical efficiency problems when cavitating regions are present.
 Additional series of 1D numerical experiments:
 to investigate whether the efficiency problems in cavitating conditions are
due to the adopted linearization technique for time advancing
 to test the efficiency of the defect correction approach for 2nd order
accuracy simulations in non-cavitating conditions
Additional series of 1D numerical experiments:
linearized implicit vs. fully non-linear implicit in cavitating conditions
Test-case: Riemann problem
 2 initial liquid states  2 rarefactions
 LdA cavitating flow state equation
Solution accuracy
Pressure field at t=1s
Discretization:
4000 cells
 t  5.105
-2000
2000
-100
detail
Robustness and computational cost
 No improvement in robustness with the fully implicit formulation
 as for non cavitating flows, the fully implicit simulations blow up
at lower CFL than the ones with the linearized implicit scheme.
 For the same resolution in space and the same time step, the
computational costs are much larger for the fully implicit scheme.
100
Validation of 2-order formulation: 1D numerical experiments
TEST-CASE 1: Quasi-1D water flow in a convergent-divergent nozzle
FO
 flow regime: steady and supersonic
( M   10, M min
Density field
Spatial discretization:
400 cells
 barotropic law:
2)
p( )=2 
Comparison of implicit formulations
FO: first-order (in space and time) linearized
implicit
slope
1
2-order (in space and time) linearized implicit
Velocity error
DeC Defect correction approach
DeC with 1 and 2 inner iterations
log-log scale
SO: fully second-order linearized implicit
FU: fully implicit second-order
(non linear solver (PETSc library) based on a gradientfree Newton-GMRES approach)
slope
2
TEST-CASE 2: Riemann problem (shock and rarefaction)
 flow regime: unsteady and subsonic ( M   0.1)
barotropic law:
p( )=106 
velocity field (t=1 s)
Spatial
refinement
Temporal
discretization:
t=0.0001
40 cells
400 cells
temporal
refinement
Spatial
discretization:
400 cells
DeC2, DeC3
t=0.01
DeC1
t=0.001
Additional series of 1D numerical experiments:
Validation of the second-order formulation
Solution accuracy
 No loss of accuracy with the present formulation:
 neither due to the defect correction comparison DeC/fully secondorder linearized implicit
 nor due to the linearization of the implicit time-advancing
comparison linearized implicit/fully implicit formulations
 In accordance with the theoretical appraisal, one iteration of defect
correction is already sufficient to reach 2nd order accuracy
 Nevertheless, for particular cases (large CFL number), a second inner
iteration can improve the solution (stabilization effect)
Computational cost
 Steady regimes: the steady solution is obtained after very few pseudo-
time iterations for all the linearized implicit approaches while the fully
implicit formulation needs a CFL-like condition (for fine spatial discretization,
i.e. large dimension of the non linear system)).
 For the same grid and time step, DeC1 is approximately two times cheaper
than the fully second-order linearized implicit approach. A larger ratio is
expected for 3D cases due to the increase of complexity and stiffness.
Concluding Remarks and Developments
 For non-cavitating barotropic flows, the proposed numerical methodology shows
satisfactory:
 accuracy (MUSCL reconstruction + preconditioning for low Mach)
 robustness and efficiency (linearized implicit time advancing + defect correction)
 For cavitating flows and the homogeneous flow model:
 severe restriction of the time step are observed  unaffordable CPU requirements
for 3D simulations
 numerical experiments show that this is not due to the adpted linearization of the
implicit time advancing
Application of the numerical set-up (as it is)
to the simulation of problems characterized
by barotropic laws less stiff than the
cavitating one (shallow water, atmosphere…)
 For cavitating flows described through
the homogenous-flow model:
 try more robust numerical fluxes (HLL,
HLLC) and/or
 relaxion techniques in time
 Change cavitation model (two-phases)
pressure
Cavitating flow behavior
liquid: ~ incompressible
liquid-vapour mixture:
highly compressible
density
Time advancing for the 2nd order accurate scheme
Full second-order linearized approach
Bi ,2n  nWi 2  Bi ,1n  nWi 1  B0i ,n  nWi  B1i ,n  nWi 1  B2i ,n  nWi  2  Sin
 i ,n
 B2

 i ,n
 B1





 B0i , n





 Bi ,n
 1




 i ,n
 B2


n
n
3 hi
 Ai1,i    i 2,i 1 
2hi 1

 hi 1
n
n
 Ai,i 1    i1,i  
2hi
 1
n
 Ai1,i    i1,i 
2
n
n 1  
n
n
n
 hi
  i1,i  
  i 2,i 1     Ai1,i 
  Ai1,i  
2hi 1
 2

 x
i I   A  n   A  n    hi 1  A  n     n  hi  A  n     n 

i ,i 1   i 1,i 
i 1,i   i ,i 1  
 i ,i 1 
 i 1,i 
2 t
2  hi 
hi 1 

n
n
n
n
n
n
n
n
1 
 Ai,i 1    i,i 1    Ai1,i    i1,i    Ai1,i    i1,i    Ai,i 1    i,i 1 

2
n
n
n
n
 hi
1 
 Ai1,i    i,i 1  
 Ai,i 1    i,i 1 

2hi 1
2

n   1
n
n
n
 hi 1 
  i,i 1  
  i 1,i  2   +  Ai,i 1 
  Ai,i 1  
2hi  2
 2

n
n
3 hi 1
 Ai,i 1    i1,i  2 

2hi  2

Flusso di acqua in un induttore di turbopompa
(Sinibaldi et al., 2006)
inducer
inter-blade covering: no gap
nose
afterbody
very complex geomety
(detail of hub-blade intersection)
Free-stream (T = 296.16 K):
2.5x106
elements
Flusso di acqua in un induttore di turbopompa
(Sinibaldi et al., 2006)
pressure contours
velocity (longitudinal cut plane)
max (red) 177700 [Pa]
min (blue) 79700 [Pa]
spacing 5000 [Pa]
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)
Homogeneous-flow models
Thermal barotropic model (d’Agostino et al.,
2001)
pure liquid: weakly compressible
fluid
mixture
p  psat
p  psat 
1

ln

 sat
p  psat



d   1 

  pc 





(
1


)
(
1


)

p


g


L
L
L
 p 
dp
p 


   V







In which L, g*, pc, V are constants dependent on the considered
flow
 T 
since   (1   ) lsat
 l   l  ; 
R 

free model parameter
d
1
 2  f (  , p(  ))
dp a