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A Numerical Method for 3D Barotropic Flows in Turbomachinery Towards Simulation of Cavitation Multiphase05, Porquerolles, April 2005 Edoardo Sinibaldi Scuola Normale Superiore di Pisa François Beux Scuola Normale Superiore di Pisa Maria Vittoria Salvetti Dip. Ing. Aerospaziale, Università di Pisa

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Research Framework (1) Motivation: Numerical investigation of cavitation phenomena occurring in turbopump inducers typical of liquid propellant rocket engines angular velocity INFLOW OUTFLOW 3D CFD tool able to simulate complex cavitating flows in realistic geometries I.C’s B.C’s Performance (e.g. p out - p in )

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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

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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 fluid Thermal Cavitation Model (d’Agostino et al., 2001) Barotropic state law: Pure liquid: Liquid-vapour: 2 model parameters

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Constitutive Law (2) EXAMPLE: MIXTURE SOUND-SPEED CURVE FOR WATER-VAPOUR AT T=20°C Nearly-incompressible:Highly-compressible: validity limit of the model

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Constitutive Law (3) EXAMPLE: BAROTROPIC CURVE FOR WATER-VAPOUR MIXTURE AT T=20°C - Obtained by numerical integration (pre-processing): - Ad-hoc reparameterization, to directly access the constitutive law: Nearly-incompressible! Highly-compressible: Numerical Stiffness!!!

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/ Lsat p / p sat Pre-processing: ( i, p i ) sampling (right) ( piv, p piv ) “pivot” leftright re-parameterization (using geometric sequences) jj j+1 p i = F [ i( ) ] ( analytical! ) Constitutive Law (detail)

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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

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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: x1x1 x2x2 x3x3 Unknown state-vector: Convective fluxes:Constitutive law: Source term: Physical entities: (density)(pressure)(velocity)

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Numerical Discretisation Time and space discretization separate (“line method”) Time discretization Explicit: RK4, low storage Implicit: linearisation based on the 1 st 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 discretization Explicit: 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: OLDOLD NEWNEW

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Space Discretisation (1) FINITE VOLUMES: CjCj x CkCk C k+1 C k-1 (Continuous) measure of the cell C j (numerical unknown) numerical flux between cell l(eft) and cell r(ight) (Semi-discrete) p.w. constant (basis) functions 1D for simplicity q k-1 qkqk q k+1 Φ k-1,k Φ k,k+1 numerical flux between cell k and cell k+1

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Space Discretisation (2) ROE NUMERICAL FLUX FUNCTION: (P. Roe, 1981) where are the eigenvalues-(right) eigenvectors of the Roe matrixsatisfying: is diagonalizable with real eigenvalues(R1) (R2) (R3) andis the n-th coordinate of the vectorw.r.t. the eigenvectors basis. depends on the constitutive law! e.g.: x 1 -sweep FOR A GENERIC BAROTROPIC STATE LAW: (Sinibaldi, Beux & Salvetti 2003) with: classical

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Space Discretisation (3) 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: LOSS OF SPATIAL ACCURACY IN THE SEMI-DISCRETE CASE!!! Continuous eq’ns: 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!

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Space Discretisation (4) PRECONDITIONING for LOW-MACH NUMBERS: ROE, original ROE, preconditioned (Guillard & Viozat, 1999; perfect gas state law) diagonalizable (righ) eigenvectors similarity matrix In conservative variables, for a generic BAROTROPIC state law: where is the preconditioner, originally conceived in primitive variables: andis a constant.

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Space Discretisation (5) PRECONDITIONING for LOW-MACH NUMBERS: By performing the low Mach number asymptotic analysis (M * →0) for the preconditioned semi-discrete eq’ns: SPATIAL ACCURACY RECOVERED AT LOW MACH!!! (Sinibaldi, Beux & Salvetti 2003) The preconditioner preserves time-consistency → unsteady computations! “centered”“upwind” same kind of result as for perfect gas!

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1D Validation (1) 11 5.75.0 22 OUTIN Symmetrical grid, 360 cells, minimum spacing 0.02 (throat) I.C’s:Inlet B.C’s:Outlet B.C’s: Explicit time-advancing (RK4) with constant time-step “Local” preconditioning: STEADY-STATE IN a C-D SYMMETRICAL NOZZLE: cross-sectional area numerical transient source

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1D Validation (2) EXAMPLE: A NON-CAVITATING TEST-CASE AT M * = 3.5e-3 throat NON-PREC. PREC.

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1D Validation (3) EXAMPLE: A CAVITATING TEST-CASE AT M * = 3.5e-3 NON-PREC. PREC. throat

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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. TC13.5e-3Non-cav.1.0e-51.0e-6 TC23.5e-3Cav.1.0e-55.0e-7 TC37.0e-4Non-cav.1.0e-55.0e-7 TC47.0e-5Non-cav.1.0e-55.0e-8

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Time Discretisation (1) (Semi-discrete) (Explicit) (Implicit) “CHEAP” - SMALL TIME-STEP LARGE TIME-STEP – “EXPENSIVE” COMPROMISE: LINEARISED IMPLICIT!!! Assumewhere; then:and

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Time Discretisation (2) …ON THE NUMERICAL FLUX LINEARISATION… - The Roe flux function is not differentiable → - If the convective flux were 1 st -order homogeneous (e.g. for perfect gas state law): 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): 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: lr rl JPPL 1,

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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. TC13.5e-3Non-cav.1.0e-51.0e-6 TC23.5e-3Cav.1.0e-55.0e-7 TC37.0e-4Non-cav.1.0e-55.0e-7 TC47.0e-5Non-cav.1.0e-55.0e-8 New entry NON-CAVITATING TC’s → “NO RESTRICTIONS” ON THE TIME-STEP! CAVITATING TC’s → RECOVER THE EXPLICIT, NON-PRECONDITIONED TIME-STEP Implicit

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3D Numerical Discretisation (1) UNSTRUCTURED GRID (THETAHEDRICAL) Finite volumes built on the dual grid (median planes) nodes in the neighbourhood of node i Semi-discrete conservation for time der.fluxessource

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3D Numerical Discretisation (2) ROE→ with: dot product (formal) Extension of the convective scheme: 1D num. flux associated with the x j -sweep of the gov. eq’ns rotation:

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3D Numerical Discretisation (3) Preconditioner: same as for 1D: heuristic, with k “calibration” parameter Linearised implicit time-advancing: analogous to 1D (the rotation does not affect the linearisation):

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c = 115 mm (hydrofoil chord) INOUT Data: Temperature = 20°C (→a Lsat ≈ 1416 m/s) Inlet speed = 3.12 m/s (→M * ≈ 2.2e-3 ) 3D Validation (1) Test-caseInlet pressure PaCav./Non-cav.δ T /R TC159050Non-cav.0.1 TC27500Cav.0.1 TC37500Cav.0.01 Water flow around a NACA0015 hydrofoil LOMA conference, Porquerolles, June 2004 (no source) IN B.C’s OUT B.C’s

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3D Validation (2) 3 SYMMETRICAL GRIDS GridNo. of nodesNo. of elementsNo. of partitions GR1453792348345 GR2272201377565 GR319322884002 GR1GR2: “optimized” w.r.t. GR1 GR3: similar to GR2; thinner domain (0.1c instead of 0.7c)

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3D Validation (3) TC1 (Non-cav.) NUMERICAL RESULTS vs EXPERIMENTS Pressure coefficient distribution – symmetry plane exp. (CentroSpazio, Pisa) grid GR3 grid GR2 *

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exp. (CentroSpazio, Pisa) TC2 (δ T /R = 0.1), grid GR3 TC3 (δ T /R = 0.01), grid GR3 * Pressure coefficient distribution – symmetry plane 3D Validation (4) TC2 – TC3 (Cav.) NUMERICAL RESULTS vs EXPERIMENTS

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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: β 2 = 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 / GridComputerCFL coeff.Total CPU time TC1 / GR3Intel P4, 2.66GHz 400 7h 30m TC2 / GR3Intel P4 Xeon, 3.06GHz 0.05 ≈ 150h TC3 / GR3Intel P4 Xeon, 3.06GHz 0.05 ≈ 150h

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3D Numerical Discretisation (4) …WELCOME TO THE ROTATING WORLD! Source: 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: with …still heuristic, with k “calibration” parameter (k=1…from the NACA experience) INFLOW OUTFLOW Recast the eq’ns in the rotating frame → the case rotates with - !!! Axisymmetrical case, to keep the grid fixed!!! L IN D L OUT

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3D Validation (6) nose inducer afterbody IN B.C’s (take into account the dragging velocity) OUT B.C’s(gradient w.r.t. the rotating frame) Data: Temperature = 23°C (→a Lsat ≈ 1405 m/s) δ T /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

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3D Validation (7) Details of the inducer geometry INTER-BLADE COVERING (no gap) HUB-BLADE INTERSECTION (detail)

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3D Validation (8) NUMERICAL RESULTS vs EXPERIMENTS A non-cavitating simulation has been performed on a grid having L IN = 1.5 D, L OUT = 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 (p OUT - p IN ) 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…)

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3D Validation (9) PRESSURE CONTOURS ON THE BLADE SURFACE AXIAL BACKFLOW (longitudinal cut plane) NUMERICAL RESULTS

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Conclusions and Perspectives Method robust and quite accurate (1 st 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

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