• International Journal of Aeronautical and Space Sciences
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• v.3 no.1
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• pp.74-85
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• 2002

The numerical experiment has been conducted to investigate the unsteady shock wave reflecting phenomena. The cell-vertex finite-volume, Roe's upwind flux difference splitting method with unstructured grid is implemented to solve unsteady Euler equations. The $4^{th}$-order Runge-Kutta method is applied for time integration. A linear reconstruction of the flux vector using the least-square method is applied to obtain the $2^{nd}$-order accuracy for the spatial derivatives. For a better resolution of the shock wave and slipline, the dynamic grid adaptation technique is adopted. The new concept of grid adaptation technique, which is much simpler than that of conventional techniques, is introduced for the current study. Three error indicators (divergence and curl of velocity, and gradient of density) are used for the grid adaptation procedure. Considering the quality of the solution and the numerical efficiency, the grid adaptation procedure was updated up to $2^{nd}$ level at every 20 time steps. For the convenience of comparison with other experimental and analytical results, the case of interaction between the straight incoming shock wave and a sharp wedge is simulated for various flow conditions. The numerical results show good agreement with other experimental and analytical results, in the shock wave reflecting structure, slipline, and the trajectory of the triple points. Some critical cases show disagreement with the analytical results, but these cases also have been proven to show hysteresis phenomena.

• Journal of computational fluids engineering
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• v.5 no.1
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• pp.43-59
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• 2000

Characteristic boundary conditions are discussed in conjunction with a flux-difference splitting formulation as modified from Roe's linearization. Details of how one can implement the characteristic boundary conditions which are made compatible with the interior point formulation are described for different types of boundaries including subsonic outflow and adiabatic wall. The validity of boundary conditions are demonstrated through computation of transonic airfoil, supersonic ogive-cylinder, hypersonic cylinder, and S-duct internal flows. The computed wall pressure distributions are compared with published experimental and computed data. Objectives of this paper are thus to give insight of formulation procedure of a flux-difference splitting method and to pave ways for other users to adopt present boundary procedure on their numerical methods.

• 한국전산유체공학회:학술대회논문집
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• 2003.10a
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• pp.181-182
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• 2003

A preconditioned numerical method for gas-liquid to-phase flow is applied to solve cavitating flow. The present method employs a density based finite-difference method of dual time-stepping integration procedure and Roe's flux difference splitting approximation with MUSCL-TVD scheme. A homogeneous equilibrium cavitation model is used. The method permits simple treatment of the whole gas-liquid two-phase flow field including wave propagation, large density changes and incompressible flow characteristics at low Mach number. By this method, two-dimensional internal flows through a venturi tuve and decelerating cascades are computed and discussed.

• 한국전산유체공학회:학술대회논문집
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• 2008.03a
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• pp.215-220
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• 2008

A numerical method for gas-liquid two-phase flow is applied to solve shock-bubble interaction problems. The present method employs a finite-difference Runge-Kutta method and Roe's flux difference splitting approximation with the MUSCL-TVD scheme. A homogeneous equilibrium cavitation model is used. By this method, a Riemann problem for shock tube was computed for validation. Then, shock-bubble interaction problems between cylindrical bubbles located in the liquid and incident liquid shock wave are computed.

• 한국전산유체공학회:학술대회논문집
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• 2008.10a
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• pp.215-220
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• 2008

A numerical method for gas-liquid two-phase flow is applied to solve shock-bubble interaction problems. The present method employs a finite-difference Runge-Kutta method and Roe's flux difference splitting approximation with the MUSCL-TVD scheme. A homogeneous equilibrium cavitation model is used. By this method, a Riemann problem for shock tube was computed for validation. Then, shock-bubble interaction problems between cylindrical bubbles located in the liquid and incident liquid shock wave are computed.

• 한국전산유체공학회:학술대회논문집
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• 1996.05a
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• pp.74-84
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• 1996

Characteristic boundary conditions are discussed in conjunction with a flux-difference splitting formulation as modified from Roe's linearization. Details of how one can implement the characteristic boundary conditions which are compatible with the discrete formulation at interior points are given for different types of boundaries including subsonic outflow and adiabatic wall. The latter conditions are demonstrated through computation of supersonic ogive-cylinder flow at high angle of attack and the computed wall pressure distribution is compared with experiment.

• 한국전산유체공학회:학술대회논문집
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• 2001.05a
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• pp.38-44
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• 2001

An efficient Gauss-Seidel time integration scheme is developed for solving the Euler and Navier-Stokes equations on unstructured meshes. Roe's FDS is used for the explicit residual computations and van Leer's FVS for evaluating implicit flux Jacobian. To reduce the memory requirement to a minimum level, off-diagonal flux Jacobian contributions are repeatedly calculated during the Gauss-Seidel sub-iteration process. Computational results based on the present scheme show that approximately $15\%$ of CPU time reduction is achieved while maintaining the memory requirement level to $50-60\%$ of the original Gauss-Seidel scheme.

• Journal of computational fluids engineering
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• v.2 no.2
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• pp.26-34
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• 1997

An implicit viscous turbulent flow solver is developed for two-dimensional geon unstructured triangular meshes. The flux terms are discretized based on a cell-centered formulation with the Roe's flux-difference splitting. The solution is advanced in time us backward-Euler time-stepping scheme. At each time step, the linear system of equation approximately solved wi th the Gauss-Seidel relaxation scheme. The effect of turbulence is with a standard k-ε two-equation model which is solved separately from the mean flow equation the same backward-Euler time integration scheme. The triangular meshes are generated advancing-front/layer technique. Validations are made for flows over the NACA 0012 airfoil. Douglas 3-element airfoil. Good agreements are obtained between the numerical result experiment.

• Journal of computational fluids engineering
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• v.3 no.2
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• pp.17-26
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• 1998

The three-dimensional incompressible Navier-Stokes equations have been solved by a node-centered finite volume method with unstructured hybrid grids. The pressure-velocity coupling is handled by the artificial compressibility algorithm and convective fluxes are obtained by Roe's flux difference splitting scheme with linear reconstruction of the solutions. Euler implicit method with Jacobi matrix solver is used for the time-integration. The viscous terms are discretised in a manner to handle any kind of grids such as tetragedra, prisms, pyramids, hexahedra, or mixed-element grid. Inviscid bump flow is solved to check the accuracy of high order convective flux discretisation. And viscous flows around a circular cylinder and a sphere are studied to show the efficiency and accuracy of the solver.

• 한국전산유체공학회:학술대회논문집
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• 1997.10a
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• pp.29-37
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• 1997

An implicit viscous turbulent flow solver is developed for two-dimensional geometries on unstructured triangular meshes. The flux terms are discretized based on a cell-centered finite-volume formulation with the Roe's flux-difference splitting. The solution is advanced in time using an implicit backward-Euler time-stepping scheme. At each time step, the linear system of equations is approximately solved with the Gauss-Seidel relaxation scheme. The effect of turbulence effects is approximated with a standard $k-{\varepsilon}$ two-equation model which is solved separately from the mean flow equations using the same backward-Euler time integration scheme. The triangular meshes are generated using an advancing-front/layer technique. Validations are made for flows over the NACA0012 airfoil and the Douglas 3-element airfoil. Good agreements are obtained between the numerical results and the experiment.