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NASA NTRS1996

Solution-Adaptive Cartesian Cell Approach for Viscous and Inviscid Flows

A Cartesian cell-based approach for adaptively refined solutions of the Euler and Navier-Stokes equations in two dimensions is presented. Grids about geometrically complicated bodies are generated automatically, by the recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created using modified polygon-clipping algorithms. The grid is stored in a binary tree data structure that provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite volume formulation. The convective terms are upwinded: A linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The results of a study comparing the accuracy and positivity of two classes of cell-centered, viscous gradient reconstruction procedures is briefly summarized. Adaptively refined solutions of the Navier-Stokes equations are shown using the more robust of these gradient reconstruction procedures, where the results computed by the Cartesian approach are compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗

NASA NTRS1996

A mixed volume grid approach for the Euler and Navier-Stokes equations

An approach for solving the compressible Euler and Navier-Stokes equations upon meshes composed of nearly arbitrary polyhedra is described. Each polyhedron is constructed from an arbitrary number of triangular and quadrilateral face elements, allowing the unified treatment of tetrahedral, prismatic, pyramidal, and hexahedral cells, as well the general cut cells produced by Cartesian mesh approaches. The basics behind the numerical approach and the resulting data structures are described. The accuracy of the mixed volume grid approach is assessed by performing a grid refinement study upon a series of hexahedral, tetrahedral, prismatic, and Cartesian meshes for an analytic inviscid problem. A series of laminar validation cases are made, comparing the results upon differing grid topologies to each other, to theory, and experimental data. A computation upon a prismatic/tetrahedral mesh is made simulating the laminar flow over a wall/cylinder combination.

Coirier, William J.↗

NASA NTRS1995

Adaptively Refined Euler and Navier-Stokes Solutions with a Cartesian-Cell Based Scheme

A Cartesian-cell based scheme with adaptive mesh refinement for solving the Euler and Navier-Stokes equations in two dimensions has been developed and tested. Grids about geometrically complicated bodies were generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells were created using polygon-clipping algorithms. The grid was stored in a binary-tree data structure which provided a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations were solved on the resulting grids using an upwind, finite-volume formulation. The inviscid fluxes were found in an upwinded manner using a linear reconstruction of the cell primitives, providing the input states to an approximate Riemann solver. The viscous fluxes were formed using a Green-Gauss type of reconstruction upon a co-volume surrounding the cell interface. Data at the vertices of this co-volume were found in a linearly K-exact manner, which ensured linear K-exactness of the gradients. Adaptively-refined solutions for the inviscid flow about a four-element airfoil (test case 3) were compared to theory. Laminar, adaptively-refined solutions were compared to accepted computational, experimental and theoretical results.

Coirier, William J.↗

NASA NTRS1995

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells are created using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded: A gradient-limited, linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The more robust of a series of viscous flux functions is used to provide the viscous fluxes at the cell interfaces. Adaptively-refined solutions of the Navier-Stokes equations using the Cartesian, cell-based approach are obtained and compared to theory, experiment and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗

NASA NTRS1995

An accuracy assessment of Cartesian-mesh approaches for the Euler equations

A critical assessment of the accuracy of Cartesian-mesh approaches for steady, transonic solutions of the Euler equations of gas dynamics is made. An exact solution of the Euler equations (Ringleb's flow) is used not only to infer the order of the truncation error of the Cartesian-mesh approaches, but also to compare the magnitude of the discrete error directly to that obtained with a structured mesh approach. Uniformly and adaptively refined solutions using a Cartesian-mesh approach are obtained and compared to each other and to uniformly refined structured mesh results. The effect of cell merging is investigated as well as the use of two different K-exact reconstruction procedures. The solution methodology of the schemes is explained and tabulated results are presented to compare the solution accuracies.

Coirier, William J.↗

NASA NTRS1994

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells are created using polygon-clipping algorithms. The grid is stored in a binary-tree structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded: a gradient-limited, linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The more robust of a series of viscous flux functions is used to provide the viscous fluxes at the cell interfaces. Adaptively-refined solutions of the Navier-Stokes equations using the Cartesian, cell-based approach are obtained and compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗

NASA NTRS1993

An accuracy assessment of Cartesian-mesh approaches for the Euler equations

A critical assessment of the accuracy of Cartesian-mesh approaches for solving the Euler equations is made. An exact solution of the Euler equations (Ringleb's flow) is used not only to infer the order of error of the Cartesian mesh approaches, but also to compare the magnitude of the error directly to that obtained with a structured mesh approach. The effect of cell merging is investigated as well as the use of two different K-exact reconstruction procedures. The solution methodology of the schemes is explained and tabulated results are presented to compare the solution accuracies. Adaptive and uniform mesh refinement is evaluated for Ringleb's flow and the supersonic flow through an axisymmetric inlet.

Coirier, William J.↗

NASA NTRS1992

Cartesian based grid generation/adaptive mesh refinement

Grid adaptation has recently received attention in the computational fluid dynamics (CFD) community as a means to capture the salient features of a flowfield by either moving grid points of a structured or by adding cells in an unstructured manner. An approach based on a background cartesian mesh is investigated from which the geometry is 'cut' out of the mesh. Once the mesh is obtained, a solution on this coarse grid is found, that indicates which cells need to be refined. This process of refining/solving continues until the flow is grid refined in terms of a user specified global parameter (such as drag coefficient etc.). The advantages of this approach are twofold: the generation of the base grid is independent of the topology of the bodies or surfaces around/through which the flow is to be computed, and the resulting grid (in uncut regions) is highly isotropic, so that the truncation error is low. The flow solver (which, along with the grid generation is still under development) uses a completely unstructured data base, and is a finite volume, upwinding scheme. Current and future work will address generating Navier-Stokes suitable grids by using locally aligned and normal face/cell refining. The attached plot shows a simple grid about two turbine blades.

Coirier, William J.↗

NASA NTRS1991

Efficient real gas upwinded Navier-Stokes computations of high speed flows

An efficient method to account for the chemically frozen thermodynamic and transport properties of air in three-dimensional Navier-Stokes calculations has been demonstrated. This approach uses an explicitly specified equation of state (EOS) so that the fluid pressure, temperature, and transport properties are directly related to the flow variables. The method is efficient since no subiterations are required to deduce the pressure and temperature from the flux variables and is modular by allowing different equations of state to be easily supplied to the code. The flexibility of the EOS approach is shown by its implementation into a high-order total variation diminishing upwinding scheme as well as a standard central-differencing scheme. The EOS approach is then demonstrated by computing the hypersonic flow through the corner region of two mutually perpendicular flat plates using both the upwind and central-differencing schemes.

Coirier, William J.↗

NASA NTRS1991

Numerical flux formulas for the Euler and Navier-Stokes equations. 2: Progress in flux-vector splitting

The accuracy of various numerical flux functions for the inviscid fluxes when used for Navier-Stokes computations is studied. The flux functions are benchmarked for solutions of the viscous, hypersonic flow past a 10 degree cone at zero angle of attack using first order, upwind spatial differencing. The Harten-Lax/Roe flux is found to give a good boundary layer representation, although its robustness is an issue. Some hybrid flux formulas, where the concepts of flux-vector and flux-difference splitting are combined, are shown to give unsatisfactory pressure distributions; there is still room for improvement. Investigations of low diffusion, pure flux-vector splittings indicate that a pure flux-vector splitting can be developed that eliminates spurious diffusion across the boundary layer. The resulting first-order scheme is marginally stable and not monotone.

Coirier, William J.↗

NASA NTRS1991

Numerical flux formulas for the Euler and Navier-Stokes equations. II - Progress in flux-vector splitting

The accuracy of various numerical flux functions for the inviscid fluxes when used for Navier-Stokes computations is studied. The flux functions are benchmarked for solutions of the viscous, hypersonic flow past a 10 degree cone at zero angle of attack using first order, upwind spatial differencing. The Harten-Lax/Roe flux is found to give a good boundary layer representation, although its robustness is an issue. Some hybrid flux formulas, where the concepts of flux-vector and flux-difference splitting are combined, are shown to give unsatisfactory pressure distributions; there is still room for improvement. Investigations of low diffusion, pure flux-vector splittings indicate that a pure flux-vector splitting can be developed that eliminates spurious diffusion across the boundary layer. The resulting first-order scheme is marginally stable and not monotone.

Coirier, William J.↗

NASA NTRS1990

Efficient real gas Navier-Stokes computations of high speed flows using an LU scheme

An efficient method to account for the chemically frozen thermodynamic and transport properties of air in three dimensional Navier-Stokes calculations was demonstrated. This approach uses an explicitly specified equation of state (EOS) so that the fluid pressure, temperature and transport properties are directly related to the flow variables. Since the pressure is explicitly known as a general function of the flow variables no assumptions are made regarding the pressure derivatives in the construction of the flux Jacobians. The method is efficient since no sub-iterations are required to deduce the pressure and temperature from the flux variables and allows different equations of state to be easily supplied to the code. The flexibility of the EOS approach is demonstrated by implementing a high order TVD upwinding scheme based upon flux differencing and Van Leer's flux vector splitting. The EOS approach is demonstrated by computing the hypersonic flow through the corner region of two mutually perpendicular flat plates and through a simplified model of a scramjet module gap-seal configuration.

Coirier, William J.↗

NASA NTRS1990

Efficient real gas Navier-Stokes computations of high speed flows using an LU scheme

An efficient method to account for the chemically frozen thermodynamic and transport properties of air in three dimensional Navier-Stokes calculations was demonstrated. This approach uses an explicitly specified equation of state (EOS) so that the fluid pressure, temperature and transport properties are directly related to the flow variables. Since the pressure is explicitly known as a general function of the flow variables no assumptions are made regarding the pressure derivatives in the construction of the flux Jacobians. The method is efficient since no sub-iterations are required to deduce the pressure and temperature from the flux variables and allows different equations of state to be easily supplied to the code. The flexibility of the EOS approach is demonstrated by implementing a high order TVD upwinding scheme based upon flux differencing and Van Leer's flux vector splitting. The EOS approach is demonstrated by computing the hypersonic flow through the corner region of two mutually perpendicular flat plates and through a simplified model of a scramjet module gap-seal configuration.

Coirier, William J.↗

NASA NTRS1989

High speed corner and gap-seal computations using an LU-SGS scheme

The hybrid Lower-Upper Symmetric Gauss-Seidel (LU-SGS) algorithm was added to a widely used series of 2D/3D Euler/Navier-Stokes solvers and was demonstrated for a particular class of high-speed flows. A limited study was conducted to compare the hybrid LU-SGS for approximate Newton iteration and diagonalized Beam-Warming (DBW) schemes on a work and convergence history basis. The hybrid LU-SGS algorithm is more efficient and easier to implement than the DBW scheme originally present in the code for the cases considered. The code was validated for the hypersonic flow through two mutually perpendicular flat plates and then used to investigate the flow field in and around a simplified scramjet module gap seal configuration. Due to the similarities, the gap seal flow was compared to hypersonic corner flow at the same freestream conditions and Reynolds number.

Coirier, William J.↗

NASA NTRS1989

High speed corner and gap-seal computations using an LU-SGS scheme

The hybrid Lower-Upper Symmetric Gauss-Seidel (LU-SGS) algorithm was added to a widely used series of 2D/3D Euler/Navier-Stokes solvers and was demonstrated for a particular class of high-speed flows. A limited study was conducted to compare the hybrid LU-SGS for approximate Newton iteration and diagonalized Beam-Warming (DBW) schemes on a work and convergence history basis. The hybrid LU-SGS algorithm is more efficient and easier to implement than the DBW scheme originally present in the code for the cases considered. The code was validated for the hypersonic flow through two mutually perpendicular flat plates and then used to investigate the flow field in and around a simplified scramjet module gap seal configuration. Due to the similarities, the gap seal flow was compared to hypersonic corner flow at the same freestream conditions and Reynolds number.

Coirier, William J.↗

NASA NTRS1988

High speed inlet calculations with real gas effects

A 2-D steady-state Navier-Stokes solver has been upgraded to include the effects of frozen and equilibrium air chemistry for applications to high speed flight vehicles. To provide a computationally economical first order approximation to the high temperature physics, variable thermodynamic data is used for the chemically frozen mode to allow for a variation with temperature of the air specific heats and enthalpy. For calculations involving air in chemical equilibrium, a specially modified version of the NASA Lewis Chemical Equilibrium Code, CEC, is used to compute the chemical composition and resultant thermochemical properties. The upgraded solver is demonstrated by comparing results from calorically perfect (C sub p=constant), thermally perfect (frozen) and equilibrium air calculations for a variety of geometries, and flight Mach numbers.

Coirier, William J.↗

NASA NTRS1988

High speed inlet calculations with real gas effects

A 2-D steady-state Navier-Stokes solver has been upgraded to include the effects of frozen and equilibrium air chemistry for applications to high speed flight vehicles. To provide a computationally economical first order approximation to the high temperature physics, variable thermodynamic data is used for the chemically frozen mode to allow for a variation with temperature of the air specific heats and enthalpy. For calculations involving air in chemical equilibrium, a specially modified version of the NASA Lewis Chemical Equilibrium Code, CEC, is used to compute the chemical composition and resultant thermochemical properties. The upgraded solver is demonstrated by comparing results from calorically perfect (C sub p=constant), thermally perfect (frozen) and equilibrium air calculations for a variety of geometries, and flight Mach numbers.

Coirier, William J.↗