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NASA NTRS1989

Application of multi-grid methods for solving the Navier-Stokes equations

This paper presents the application of a class of multi-grid methods to the solution of the Navier-Stokes equations for two-dimensional laminar flow problems. The methods consists of combining the full approximation scheme-full multi-grid technique (FAS-FMG) with point-, line- or plane-relaxation routines for solving the Navier-Stokes equations in primitive variables. The performance of the multi-grid methods is compared to those of several single-grid methods. The results show that much faster convergence can be procured through the use of the multi-grid approach than through the various suggestions for improving single-grid methods. The importance of the choice of relaxation scheme for the multi-grid method is illustrated.

Demuren, A. O.↗

NASA NTRS1983

Accuracy of approximations to the Navier-Stokes equations

The systems of truncated differential equations that have been proposed to reduce the complexity and large computational costs of solutions to the full Navier-Stokes equations are considered. These systems are computationally efficient and capture all the physically relevant behavior. The systems follow a certain hierarchy: (1) the classical boundary-layer equations with specified edge properties (usually the streamwise pressure distribution); (2) the coupled boundary-layer/inviscid equations; (3) the so-called thin-layer equations that discard streamwise diffusion; and (4) the Navier-Stokes equations. Consideration is given to each of these approximations applied to an incompressible, laminar-separating flow at low and moderate Reynolds numbers. It is pointed out that for any flow or region of flow for which viscous-inviscid interaction effects are small, classical boundary-layer equations will provide a satisfactory description of the viscous flow at a fraction of the computational cost of any higher approximations.

Murphy, J. D.↗

NASA NTRS1976

A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C deg-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1,000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and asymptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

Cooke, C. H.↗

NASA NTRS1977

A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of the nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The continuous cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

Cooke, C. H.↗

NASA NTRS1984

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗

NASA NTRS1983

Finite element methods for the inhomogeneous Navier-Stokes equations

A conforming mixed finite element method for the stationary Navier-Stokes equations with inhomogeneous boundary conditions is considered. Estimates for the error in the velocity and pressure approximations are derived. A computational example is provided to illustrate the implementation of the method.

Peterson, J. S.↗

NASA NTRS1983

Aspects of a penalty finite element algorithm for the turbulent three-dimensional parabolic Navier-Stokes equations

The formal derivation of the three-dimensional parabolic Navier-Stokes equations for subsonic turbulent flow is reviewed. A penalty finite element algorithm is established for numerical solution of the sixteen dependent variable system. Key numerical results are summarized documenting applications in various problem definitions.

Baker, A. J.↗

NASA NTRS1985

A spectral collocation method for the Navier-Stokes equations

Malik, M. R.↗

NASA NTRS1985

Calculations of two and three-dimensional transonic cascade flow fields using the Navier-Stokes equations

The multidimensional, ensemble-averaged, compressible, time-dependent Navier-Stokes equations have been used to study the turbulent flow field in two and three-dimensional turbine cascades. The viscous regions of the flow were resolved and non-slip boundary conditions were utilized on solid surfaces. The calculations were performed in a constructive 'O'-type grid which allows representation of the blade rounded trailing edge. Converged solutions were obtained in relatively few time steps (about 80-150) and comparisons for both surface pressure and heat transfer showed good agreement with data. The three-dimensional turbine cascade calculation showed many of the expected flow-field features.

Weinberg, B. C.↗

NASA NTRS1987

Streamline-upwind/Petrov-Galerkin procedures for the vorticity-stream function form of the Navier-Stokes equations

The paper presents procedures for the solution of the Navier-Stokes equations in the vorticity-stream function form. The difficulties involved are related to the convection term in the vorticity transport equation and to the lack of boundary conditions for voritcity at no-slip surfaces. The use of a streamline-upwind/Petrov-Galerkin finite element formulation for the solution of the vorticity transport equation. In the present scheme, the weighting functions are dependent on both spatial and temporal discretizations. A proper numerical treatment of the boundary conditions leading to an implicit treatment of the vorticity at no-slip surfaces is presented. These procedures have successfully been employed to simulate various flows of engineering interest.

Tezduyar, T. E.↗

NASA NTRS1991

Newton's method applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations

Finite-difference approximations for steady-state compressible Navier-Stokes equations, whose two spatial dimensions are written in generalized curvilinear coordinates and strong conservation-law form, are presently solved by means of Newton's method in order to obtain a lifting-airfoil flow field under subsonic and transonnic conditions. In addition to ascertaining the computational requirements of an initial guess ensuring convergence and the degree of computational efficiency obtainable via the approximate Newton method's freezing of the Jacobian matrices, attention is given to the need for auxiliary methods assessing the temporal stability of steady-state solutions. It is demonstrated that nonunique solutions of the finite-difference equations are obtainable by Newton's method in conjunction with a continuation method.

Bailey, Harry E.↗

NASA NTRS1992

Dual-Potential Formulation Of Navier-Stokes Equations

Report discusses research on dual-potential formulation of Navier-Stokes equations. Purpose of study to extend capability of dual-potential method to computation of unsteady flows of viscous, compressible fluids.

Gegg, S. G.↗

NASA NTRS1996

The Multigrid-Mask Numerical Method for Solution of Incompressible Navier-Stokes Equations

A multigrid-mask method for solution of incompressible Navier-Stokes equations in primitive variable form has been developed. The main objective is to apply this method in conjunction with the pseudospectral element method solving flow past multiple objects. There are two key steps involved in calculating flow past multiple objects. The first step utilizes only Cartesian grid points. This homogeneous or mask method step permits flow into the interior rectangular elements contained in objects, but with the restriction that the velocity for those Cartesian elements within and on the surface of an object should be small or zero. This step easily produces an approximate flow field on Cartesian grid points covering the entire flow field. The second or heterogeneous step corrects the approximate flow field to account for the actual shape of the objects by solving the flow field based on the local coordinates surrounding each object and adapted to it. The noise occurring in data communication between the global (low frequency) coordinates and the local (high frequency) coordinates is eliminated by the multigrid method when the Schwarz Alternating Procedure (SAP) is implemented. Two dimensional flow past circular and elliptic cylinders will be presented to demonstrate the versatility of the proposed method. An interesting phenomenon is found that when the second elliptic cylinder is placed in the wake of the first elliptic cylinder a traction force results in a negative drag coefficient.

Ku, Hwar-Ching↗

NASA NTRS1996

Solutions to Three-Dimensional Thin-Layer Navier-Stokes Equations in Rotating Coordinates for Flow Through Turbomachinery

The viscous, Navier-Stokes solver for turbomachinery applications, MSUTC has been modified to include the rotating frame formulation. The three-dimensional thin-layer Navier-Stokes equations have been cast in a rotating Cartesian frame enabling the freezing of grid motion. This also allows the flow-field associated with an isolated rotor to be viewed as a steady-state problem. Consequently, local time stepping can be used to accelerate convergence. The formulation is validated by running NASA's Rotor 67 as the test case. results are compared between the rotating frame code and the absolute frame code. The use of the rotating frame approach greatly enhances the performance of the code with respect to savings in computing time, without degradation of the solution.

Ghosh, Amrit Raj↗

NASA NTRS1984

Dynamic response of shock waves in transonic diffuser and supersonic inlet - An analysis with the Navier-Stokes equations and adaptive grid

An existing method which solves the multi-dimensional ensemble-averaged compressible time-dependent Navier-Stokes equations in conjunction with mixing length turbulence model and shock capturing technique has been extended to include the shock-tracking adaptive grid systems. The numerical scheme for solving the governing equations is based on a linearized block implicit approach. The effects of grid-motion and grid-distribution on the calculated flow solutions have been studied in relative detail and this is carried out in the context of physically steady, shocked flows computed with non-stationary grids. Subsequently, the unsteady dynamics of the flows occurring in a supercritically operated transonic diffuser and a mixed compression supersonic inlet have been investigated with the adaptive grid systems by solving the Navier-Stokes equations.

Liu, N.-S.↗

NASA NTRS1993

Calculation of optimum airfoils using direct solutions of the Navier-Stokes equations

A method has been developed which calculates the optimum airfoil shape at a specified operating condition. The Reynolds-averaged Navier-Stokes equations with a turbulence model are used as the governing equations for the fluid flow, and are explicitly introduced into the optimization analysis as constraint equations. The locations of all grid points which define the surface of the airfoil are design variables, eliminating the need for shape functions to describe changes in the airfoil geometry. A generalized reduced-gradient method is used to find the optimum airfoil shape. The use of a direct solution technique (Newton's method) to solve the Navier-Stokes equations allows for the efficient solution of the problem. This paper describes the optimization method and presents sample results.

Felker, Fort F.↗

NASA NTRS1992

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology.

Barth, Timothy J.↗

NASA NTRS1994

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. These algorithms are now used in the practical engineering design of devices such as cars and airplanes as well as theoretical studies of complex phenomena such as fluid turbulence. In past years, limited computer resources have motivated the development of efficient numerical methods in computational fluid dynamics (CFD) utilizing structured meshes. These meshes are comprised of systematic arrays of quadrilateral or hexahedral cells. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Structured meshes also permit the use of highly efficient solution techniques such as alternating direction implicit (ADI) iteration schemes or multigrid. Following the dramatic improvement in computing speed in recent years, emphasis has shifted towards the design of algorithms capable of treating complex geometries. The automatic generation of structured grids about complex geometries is problematic. Unstructured grids offer one promising alternative technique for treating these general geometries. Unstructured meshes have irregular connectivity and usually contain triangles and/or quadrilaterals in two dimensions and tetrahedra and/or hexahedra in three dimensions. The generation and use of unstructured grids poses new challenges in computational fluid dynamics. This is true for both grid generation as well as for the design of algorithms for flow solution. The purpose of these notes is to present recent developments in the unstructured grid generation and flow solution technology.

Barth, T. J.↗