Title: A Numerical Method for 3D Barotropic Flows in Turbomachinery
1A Numerical Method for 3D Barotropic Flows in
Turbomachinery
Multiphase05, Porquerolles, April 2005
- Towards Simulation of Cavitation
Edoardo Sinibaldi Scuola Normale Superiore di Pisa
François Beux Scuola Normale Superiore di Pisa
Maria Vittoria Salvetti Dip. Ing. Aerospaziale,
Università di Pisa
2Research Framework (1)
3Research Framework (2)
Development
4Constitutive Law (1)
5Constitutive Law (2)
EXAMPLE MIXTURE SOUND-SPEED CURVE FOR
WATER-VAPOUR AT T20C
6Constitutive Law (3)
EXAMPLE BAROTROPIC CURVE FOR WATER-VAPOUR
MIXTURE AT T20C
7Constitutive Law (detail)
8Governing Equations (1)
9Governing Equations (2)
10Numerical Discretisation
OLD
NEW
11Space Discretisation (1)
1D for simplicity
FINITE VOLUMES
12Space Discretisation (2)
13Space Discretisation (3)
LOW-MACH NUMBER ASYMPTOTIC STUDY (Guillard
Viozat, 1999 perfect gas state law)
same kind of result as for perfect gas!
14Space Discretisation (4)
PRECONDITIONING for LOW-MACH NUMBERS
In conservative variables, for a generic
BAROTROPIC state law
15Space Discretisation (5)
PRECONDITIONING for LOW-MACH NUMBERS
same kind of result as for perfect gas!
161D Validation (1)
171D Validation (2)
181D Validation (3)
191D Validation (4)
20Time Discretisation (1)
21Time Discretisation (2)
ON THE NUMERICAL FLUX LINEARISATION
221D Validation (5)
COMPLETE RESULTS
233D Numerical Discretisation (1)
243D Numerical Discretisation (2)
253D Numerical Discretisation (3)
263D Validation (1)
273D Validation (2)
3 SYMMETRICAL GRIDS
283D Validation (3)
293D Validation (4)
303D 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
ß2 k M2, 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)
313D Numerical Discretisation (4)
WELCOME TO THE ROTATING WORLD!
323D Validation (6)
333D Validation (7)
Details of the inducer geometry
343D Validation (8)
NUMERICAL RESULTS vs EXPERIMENTS
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)
353D Validation (9)
NUMERICAL RESULTS
36Conclusions 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 eqns (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