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Research Papers

Time-Derivative Preconditioning Methods for Multicomponent Flows—Part II: Two-Dimensional Applications

[+] Author and Article Information
Jeffrey A. Housman, Mohamed M. Hafez

 University of California Davis, 2132 Bainer Hall, One Shields Avenue, Davis, CA 95616

Cetin C. Kiris

NASA Advanced Supercomputing (NAS) Division, NASA Ames Research Center, Moffett Field, CA 94035

J. Appl. Mech 76(3), 031013 (Mar 13, 2009) (12 pages) doi:10.1115/1.3086592 History: Received January 31, 2008; Revised December 30, 2008; Published March 13, 2009

A time-derivative preconditioned system of equations suitable for the numerical simulation of multicomponent/multiphase inviscid flows at all speeds was described in Part I of this paper. The system was shown to be hyperbolic in time and remain well conditioned in the incompressible limit, allowing time marching numerical methods to remain an efficient solution strategy. Application of conservative numerical methods to multicomponent flows containing sharp fluid interfaces was shown to generate nonphysical pressure and velocity oscillations across the contact surface, which separates the fluid components. It was demonstrated using the one-dimensional Riemann problem that these oscillations may lead to stability problems when the interface separates fluids with large density ratios, such as water and air. The effect of which leads to the requirement of small physical time steps and slow subiteration convergence for the implicit time marching numerical method. Alternatively, the nonconservative and hybrid formulations developed by the present authors were shown to eliminate this nonphysical behavior. While the nonconservative method did not converge to the correct weak solution for flow containing shocks, the hybrid method was able to capture the physically correct entropy solution and converge to the exact solution of the Riemann problem as the grid is refined. In Part II of this paper, the conservative, nonconservative, and hybrid formulations described in Part I are implemented within a two-dimensional structured body-fitted overset grid solver, and a study of two unsteady flow applications is reported. In the first application, a multiphase cavitating flow around a NACA0015 hydrofoil contained in a channel is solved, and sensitivity to the cavitation number and the spatial order of accuracy of the discretization are discussed. Next, the interaction of a shock moving in air with a cylindrical bubble of another fluid is analyzed. In the first case, the cylindrical bubble is filled with helium gas, and both the conservative and hybrid approaches perform similarly. In the second case, the bubble is filled with water and the conservative method fails to maintain numerical stability. The performance of the hybrid method is shown to be unchanged when the gas is replaced with a liquid, demonstrating the robustness and accuracy of the hybrid approach.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Geometry for NACA0015 inside a channel

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Figure 2

Structured overset grid for NACA0015 inside channel

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Figure 3

Plot of the Cp curve on the upper and lower surfaces of the hydrofoil for Pvap=3165 Pa using third-order convective fluxes with TVD limiters for each of the methods

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Figure 4

Convergence history of the maximum residual (top) and the computed maximum/minimum volume fraction for Pvap=3165 Pa using third-order convective fluxes with TVD limiters (bottom)

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Figure 5

Plot of the Cp curve on the upper and lower surfaces of the hydrofoil for Pvap=5372 Pa using first-order convective fluxes for each of the methods

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Figure 24

(Upper) pressure and (Lower) numerical Schlieren for air/water shock bubble interaction at time step 250 using the HYBR method

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Figure 23

(Upper) pressure and (Lower) numerical Schlieren for air/water shock bubble interaction at time step 200 using the HYBR method

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Figure 18

Numerical Schlieren for air/helium shock bubble interaction at time step 200: (Upper) PROE and (Lower) HYBR

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Figure 17

Numerical Schlieren for air/helium shock bubble interaction at time step 150: (Upper) PROE and (Lower) HYBR

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Figure 16

Numerical Schlieren for air/helium shock bubble interaction at time step 100: (Upper) PROE and (Lower) HYBR

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Figure 15

Subiteration convergence history of the PROE and HYBR methods for the MS=1.22 shock air/helium interaction problem

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Figure 14

Configuration for shock/bubble interaction problem

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Figure 13

Volume fraction at t=0.74 to 1.02 s at intervals of 0.02 s

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Figure 12

Volume fraction at t=0.50 to 0.72 s at intervals of 0.02 s

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Figure 11

Volume fraction at t=0.26 to 0.48 s at intervals of 0.02 s

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Figure 10

Volume fraction at t=0.02 to 0.24 s at intervals of 0.02 s

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Figure 7

Plot of the Cp curve on the upper and lower surfaces of the hydrofoil for Pvap=5372 Pa using third-order convective fluxes with TVD limiters for each of the methods

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Figure 6

Convergence history of the maximum residual (top) and the computed maximum/minimum volume fraction for Pvap=5372 Pa using first-order convective fluxes (bottom)

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Figure 22

(Upper) pressure and (Lower) numerical Schlieren for air/water shock bubble interaction at time step 150 using the HYBR method

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Figure 21

(Upper) pressure and (Lower) numerical Schlieren for air/water shock bubble interaction at time step 100 using the HYBR method

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Figure 20

Subiteration convergence history of the PROE and HYBR methods for the MS=1.22 shock air/water interaction problem

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Figure 19

Numerical Schlieren for air/helium shock bubble interaction at time step 250: (Upper) PROE and (Lower) HYBR

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Figure 8

Number of subiterations required to reduce the residual three orders of magnitude as a function of the physical time step

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Figure 9

Maximum pressure as a function of time computed with the PSCM method

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