C. J. Hwang

Global and local remeshing algorithms for compressible flows

A new adaptive remeshing approach for unstructured meshes, which includes the error indicator, global and local mesh regeneration techniques, has been developed in this paper. In this approach, nodes are first distributed according to the remeshing parameters, and those nodes are connected into a complete mesh. The concepts of side-based and vertex-based fronts are introduced to achieve the triangulation. According to the CPU time and the versatility, the vertex-based triangulation technique is proved to be more efficient. By using vertexbased triangulation approach, a local remeshing method, which regenerates only some regions of the flow domain, is presented. To demonstrate the reliability and availability of the proposed procedure, several compressible flow problems are investigated. The regular/stretched triangles and the mixed type of triangular and quadrilateral stretched elements are used. In this work, the Euler equations are solved by the multi-step Runge-Kutta Galerkin finite element methods. From the numerical results, the approaches, which employ the directionally stretched elements, are effective and suitable for treating the flow problems with one-dimensional features. The development of the local remeshing algorithm for unsteady flows is worthwhile and important.

Adaptive Finite Volume Upwind Approach on Mixed Quadrilateral-Triangular Meshes

An adaptive cell-vertex finite volume upwind approach on unstructured mixed quadrilateral-triangular meshes, where the quadrilaterals are directionally stretched in the flow regions having one-dimensional features, has been developed to solve the unsteady Euler equations. In the present approach, the Runge-Kutta time integration, Roes Riemann solver, MUSCL differencing with characteristic interpolation variables, a new treatment of the rotated extrapolation boundary condition, and a modified technique of the global/local regeneration for directionally stretched meshes are included. By using different combinations of interpolation variables, limiter functions, and numerical implementations of boundary conditions, a systematic study is made to understand the characteristics of the current approach. In this work, the isolated oblique shock problem, shock reflection at a wall, supersonic flow passing through a channel with a 4% circular arc bump, transonic flows around single- and two-element airfoils, as well as the shock propagation in a channel are investigated. It is concluded that the present solution procedure demonstrates good convergence performance and provides accurate and high-resolution results for the steady and unsteady inviscid flows.

Locally implicit hybrid algorithm for steady and unsteady viscous flows

A locally implicit hybrid finite volume algorithm on mixed type of triangular and quadrilateral meshes for the two-dimensional steady and unsteady viscous flows has been developed. The unsteady, full Navier-Stokes equations are solved in the Cartesian coordinate system. A new construction of a total-variation-diminishing formulation on triangles is used to obtain high-resolution results in the convective-dominated flow region. In the viscous-dominated part, a second- and fourth-order dissipative model on quadrilateral grids is employed to minimize the numerical dissipation

Locally implicit total-variation-diminishing schemes on unstructured triangular meshes

A numerical solution procedure that includes locally implicit total-variation-diminishing schemes and adaptive mesh generation techniques has been developed in this work. In a Cartesian coordinate system, the Euler equations are solved by using a cell-centered finite volume algorithm. A new construction of symmetric total-variation-diminishing schemes on unstructured triangular meshes is presented. The validation of the present solution-adaptive methods is confirmed by comparison with related numerical results for inviscid flows around an isolated NACA 0012 airfoil and passing through a channel with a circular arc bump in transonic and supersonic flow regimes. To further prove the feasibility of this approach, a two-element airfoil flow is also investigated. Furthermore, one unsteady transonic channel flow is studied to demonstrate the reliability and capability of the present solution procedure for a time accurate calculation.

Solution-adaptive approach for unsteady flow calculations on quadrilateral-triangular meshes

We develop a solution-adaptive approach for unsteady inviscid flow calculations on quadrilateral-triangular meshes. This approach includes a locally implicit total variation diminishing scheme and an adaptive mesh technique. To evaluate the present interpolation algorithm, sinusoidal, exponential, (x+y) 2 and (x+y) functions are tested. Based on the numerical results for the shock propagation in channel, the present solution-adaptive approach is accurate, reliable, and suitable for studying unsteady inviscid flows.

Inviscid analysis of transonic oscillating cascade flows using a dynamic mesh algorithm

A locally implicit total-variation-diminishing scheme and a rigid-deformable dynamic mesh algorithm are formulated on the quadrilateral-triangular meshes. The unsteady Euler equations with moving domain effects are solved in a Cartesian coordinate system. For transonic flows around an oscillating cascade of four biconvex blades with different oscillation amplitudes, reduced frequencies, and interblade phase angles, the calculated distributions of magnitude and phase angle of the first harmonic dynamic pressure difference coefficient agree better with experimental data than those from linearized theory and related numerical results on triangular meshes in most of the cases. Also, the numerical wiggles of instantaneous blade surface pressure coefficient distributions, which appeared on the triangular meshes, are eliminated. From the instantaneous pressure and Mach number contours, the unsteady flow phenomena, such as periodical characteristics, pressure wave and shock behaviors, and time-variations of zones with high Mach number gradient normal to the blade surfaces, are investigated. Furthermore, the lift coefficient distributions indicate that the oscillation .amplitude, reduced frequency, and interblade phase angle all have significant effects on the transonic oscillating cascade flows.

VISCOUS SOLUTIONS FOR TRANSONIC OSCILLATING CASCADE FLOWS USING DYNAMIC QUADRILATERAL-TRIANGULAR MESHES

The locally implicit scheme and a rigid-deformable dynamic mesh algorithm are introduced to solve the unsteady Navier-Stokes equations with Baldwin-Lomax turbulence model in the Cartesian coordinate system. In this scheme, a new treatment of viscous flux function is developed on the quadrilateral-triangular meshes. To validate the present scheme, the viscous flow over an oscillating flat plate is studied. By using the above solution approach, turbulent flows around transonic oscillating cascade of four biconvex blades with different oscillation amplitudes and interblade phase angles are investigated. From the distributions of magnitude and phase angle of the first harmonic dynamic pressure difference coefficient, which include the experimental data and numerical results obtained by linearized theory, Euler and Navier-Stokes solvers, the present solution approach is reliable and acceptable. The instantaneous pressure and/or Mach number contours indicate the effects of oscillation amplitude and interblade phase angle on the unsteady flow phenomena, such as vortex-shedding, pressure waves, shocks and boundary layer interactions.Copyright

Analysis of Steady and Unsteady Turbine Cascade Flows by a Locally Implicit Hybrid Algorithm

For the two-dimensional steady and unsteady turbine cascade flows, the Euler/Navier-Stokes equations with Baldwin-Lomax turbulence model are solved in the Cartesian coordinate system. A locally implicit hybrid algorithm on the mixed type of meshes is employed, where the convective dominated part in the flowfield is studied by TVD scheme to obtain high-resolution results on the triangular elements, and the second- and fourth-order dissipative model is introduced on the O-typed quadrilateral grid in the viscous dominated region to minimize the numerical dissipation. When the steady subsonic and transonic turbulent flows are investigated, the distributions of isentropic Mach number on the blade surface, exit flow angle and loss coefficient are obtained. Comparing the present results with the experimental data, the accuracy and reliability of the current approach is confirmed. By giving a moving wake-type total pressure profile at the inlet plane in the rotor-relative frame of reference, the unsteady transonic inviscid and turbulent flows calculations are performed to understand the interaction of the upstream wake with a moving blade row. The Mach number contours, perturbation component of the unsteady velocity vectors, shear stress and pressure distributions on the blade surface are presented. The physical phenomena, which include periodical flow separation on the suction side, bowing, chopping and distortion of incoming wake, negative jet, convection of the vortices and wake segments, and vortex shedding at the trailing edge, are observed. It is concluded that the unsteady aerodynamic behaviors are strongly dependent on the wake/shock/boundary layer interactions.Copyright