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NASA NTRS1982

Error norms for the adaptive solution of the Navier-Stokes equations

The adaptive solution of the Navier-Stokes equations depends upon the successful interaction of three key elements: (1) the ability to flexibly select grid length scales in composite grids, (2) the ability to efficiently control residual error in composite grids, and (3) the ability to define reliable, convenient error norms to guide the grid adjustment and optimize the residual levels relative to the local truncation errors. An initial investigation was conducted to explore how to approach developing these key elements. Conventional error assessment methods were defined and defect and deferred correction methods were surveyed. The one dimensional potential equation was used as a multigrid test bed to investigate how to achieve successful interaction of these three key elements.

Forester, C. K.↗

NASA NTRS1987

Evaluation of a research circulation control airfoil using Navier-Stokes methods

The compressible Reynolds time averaged Navier-Stokes equations were used to obtain solutions for flows about a two dimensional circulation control airfoil. The governing equations were written in conservation form for a body-fitted coordinate system and solved using an Alternating Direction Implicit (ADI) procedure. A modified algebraic eddy viscosity model was used to define the turbulent characteristics of the flow, including the wall jet flow over the Coanda surface at the trailing edge. Numerical results are compared to experimental data obtained for a research circulation control airfoil geometry. Excellent agreement with the experimental results was obtained.

Shrewsbury, George D.↗

NASA NTRS1988

Three-dimensional incompressible Navier-Stokes computations of internal flows

Several incompressible Navier-Stokes solution methods for obtaining steady and unsteady solutions are discussed. Special attention is given to internal flows which involve distinctly different features from external flows. The characterisitcs of the flow solvers employing the method of pseudocompressibility and a fractional step method are briefly described. This discussion is limited to a primitive variable formulation in generalized curvilinear coordinates. Computed results include simple test cases and internal flow in the Space Shuttle main engine hot-gas manifold.

Kwak, D.↗

NASA NTRS1988

An upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner in using the method of pseudocompressibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme based on flux-difference splitting is used to compute the convective terms. The upwind differencing is biased based on the sign of the local eigenvalue of the Jacobian matrix. Third-order or fifth-order spatial accuracy is maintained throughout the interior grid points. The equations are solved using an implicit line-relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudotime, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at -90 deg angle-of-attack is also computed.

Rogers, Stuart E.↗

NASA NTRS1989

Multigrid solution of the Navier-Stokes equations on triangular meshes

A new Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite-element Galerkin approximation, which can be shown to be equivalent to a finite-volume approximation for regular equilateral triangular meshes. Integration to steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. The convergence efficiency of the present method is shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.

Mavriplis, D. J.↗

NASA NTRS1989

Time dependent viscous incompressible Navier-Stokes equations

Information on time dependent incompressible Navier-Stokes equations is given in viewgraph form. Information is given on streamfunction equations for unsteady incompressible flow, the streamfunction algorithm for unsteady incompressible flow, and a multigrid solver for the laminar implicit equations.

Goodrich, John W.↗

NASA NTRS1991

The prediction of viscous hypersonic flows about complex configurations using an upwind parabolized Navier-Stokes code

A three-dimensional parabolized Navier-Stokes computer code which employs an upwind algorithm is used to conduct a numerical study of an advanced maneuvering reentry vehicle configuration. Comparisons between numerical solutions and experimental data are presented for surface pressure, wall heat flux, and overall forces and moments. The effects of angle of attack, angle of yaw, and surface mass injection are investigated. Good agreement is observed between the calculated and measured data. The results of this investigation demonstrate the accuracy and efficiency of an upwind scheme in predicting the hypersonic flow field characteristics about a complex configuration.

Narain, J. P.↗

NASA NTRS1992

Navier-Stokes computations for oscillating control surfaces

Unsteady Navier-Stokes computations have been performed for simulating transonic flows over wings with oscillating control surfaces using a locally moving grid and a stationary-mismatched zoning scheme. An F-5 wing and a clipped delta wing are chosen for the present study. The computed unsteady pressures and the response characteristics to the control surface motions are compared with experimental data. The results successfully predict main features of the unsteady pressure profiles, such as the double peaks at the shock wave and at the hinge line.

Obayashi, Shigeru↗

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 NTRS1992

Navier-Stokes computations of a viscous optimized waverider

The performance of a Mach 6 viscous optimized waverider was calculated using the 3-D Navier-Stokes equations. The Mach 6 viscous optimized waverider was generated using MAXWARP, a code developed at the University of Maryland. The computations were performed using CFL3D, an implicit upwind-biased finite-volume algorithm developed at NASA Langley. Results show that good agreement was found between the calculated performance by MAXWARP and results from the Mach 6 Navier-Stokes computation. Furthermore, off-design performance of the Mach 6 optimized waverider was computed at Mach 4 and 8. The performance at these Mach numbers compared well with the performance of the viscous optimized waveriders specifically designed for these Mach numbers. Finally, contours of different flow parameters in the cross-flow plane were examined for the three calculations. The results indicate that the flow gradients are relatively small within the captured flow, and the variation itself is well behaved; thus, making the waverider configuration a promising choice for an engine/airframe design, especially for cruise-type applications.

Takashima, Naruhisa↗

NASA NTRS1986

Transonic Navier-Stokes wing solutions using a zonal approach. Part 2: High angle-of-attack simulation

A computer code is under development whereby the thin-layer Reynolds-averaged Navier-Stokes equations are to be applied to realistic fighter aircraft configurations. This transonic Navier-Stokes code (TNS) utilizes a zonal approach in order to treat complex geometries and satisfy in-core computer memory constraints. The zonal approach was applied to isolated wing geometries in order to facilitate code development. The TNS finite difference algorithm, zonal methodology, and code validation with experimental data is addressed. Also addressed are some numerical issues such as code robustness, efficiency, and accuracy at high angles of attack. Special free-stream-preserving metrics proved an effective way to treat H-mesh singularities over a large range of severe flow conditions, including strong leading edge flow gradients, massive shock induced separation, and stall. Furthermore, lift and drag coefficients were computed for a wing up through CLmax. Numerical oil flow patterns and particle trajectories are presented both for subcritical and transonic flow. These flow simulations are rich with complex separated flow physics and demonstrate the efficiency and robustness of the zonal approach.

Chaderjian, Neal M.↗

NASA NTRS1991

Navier-Stokes solutions about the F/A-18 wing-LEX-fuselage configuration with multi-block structured grids

Three-dimensional thin-layer Navier-Stokes computations are presented for the F/A-18 configuration. The modeled configuration includes an accurate surface representation of the fuselage, leading-edge-extension, as well as the wing with and without leading-edge-flap deflection. A multi-block structured volume grid with various topologies is generated using transfinite interpolation technique. The flowfield domain is divided into twenty blocks, each representing a particular geometrical complexity of the configuration. The results are obtained from an algorithm for solving the compressible Navier-Stokes equations that incorporates an upwind-biased, flux-difference-splitting approach. In addition, a newly developed capability that allows for generalized surface patching among blocks is employed. Turbulent results are presented for flow conditions that correspond to recent NASA F/A-18 High Alpha Research Vehicle flight experiments. Good correlations between the computations and the flight test results are disclosed for both surface flow patterns as well as surface pressure distributions.

Ghaffari, Farhad↗

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 NTRS1993

Virtual zone Navier-Stokes computations for oscillating control surfaces

A new zoning method called 'virtual zones' has been developed for application to an unsteady finite difference Navier-Stokes code. The virtual zoning method simplifies the zoning and gridding of complex configurations for use with patched multi-zone flow codes. An existing interpolation method has been extensively modified to bring the run time for the interpolation procedure down to the same level as for the flow solver. Unsteady Navier-Stokes computations have been performed for transonic flow over a clipped delta wing with an oscillating control surface. The computed unsteady pressure and response characteristics of the control-surface motion compare well with experimental data.

Klopfer, G. H.↗

NASA NTRS1986

Transonic Navier-Stokes wing solutions using a zonal approach. Part 2: High angle-of-attack simulation

A computer code is under development whereby the thin-layer Reynolds-averaged Navier-Stokes equations are to be applied to realistic fighter-aircraft configurations. This transonic Navier-Stokes code (TNS) utilizes a zonal approach in order to treat complex geometries and satisfy in-core computer memory constraints. The zonal approach has been applied to isolated wing geometries in order to facilitate code development. Part 1 of this paper addresses the TNS finite-difference algorithm, zonal methodology, and code validation with experimental data. Part 2 of this paper addresses some numerical issues such as code robustness, efficiency, and accuracy at high angles of attack. Special free-stream-preserving metrics proved an effective way to treat H-mesh singularities over a large range of severe flow conditions, including strong leading-edge flow gradients, massive shock-induced separation, and stall. Furthermore, lift and drag coefficients have been computed for a wing up through CLmax. Numerical oil flow patterns and particle trajectories are presented both for subcritical and transonic flow. These flow simulations are rich with complex separated flow physics and demonstrate the efficiency and robustness of the zonal approach.

Chaderjian, N. M.↗

NASA NTRS1999

A Robust Locally Preconditioned Semi-Coarsening Multigrid Algorithm for the 2-D Navier-Stokes Equations

The goal of this thesis is to develop an efficient and robust locally preconditioned semi-coarsening multigrid algorithm for the two-dimensional Navier-Stokes equations. This thesis examines the performance of the multigrid algorithm with local preconditioning for an upwind-discretization of the Navier-Stokes equations. A block Jacobi iterative scheme is used because of its high frequency error mode damping ability. At low Mach numbers, the performance of a flux preconditioner is investigated. The flux preconditioner utilizes a new limiting technique based on local information that was developed by Siu. Full-coarsening and-semi-coarsening are examined as well as the multigrid V-cycle and full multigrid. The numerical tests were performed on a NACA 0012 airfoil at a range of Mach numbers. The tests show that semi-coarsening with flux preconditioning is the most efficient and robust combination of coarsening strategy, and iterative scheme - especially at low Mach numbers.

Cain, Michael D.↗

NASA NTRS1996

Further Development of a New, Flux-Conserving Newton Scheme for the Navier-Stokes Equations

This paper is one of a series of papers describing the development of a new numerical approach for solving the steady Navier-Stokes equations. The key features in the current development are (1) the discrete representation of the dependent variables by way of high order polynomial expansions, (2) the retention of all derivatives in the expansions as unknowns to be explicitly solved for, (3) the automatic balancing of fluxes at cell interfaces, and (4) the discrete simulation of both the integral and differential forms of the governing equations. The main purpose of this paper is, first, to provide a systematic and rigorous derivation of the conditions that are used to simulate the differential form of the Navier-Stokes equations, and second, to extend our previously-presented internal flow scheme to external flows and nonuniform grids. Numerical results are presented for high Reynolds number flow (Re = 100,000) around a finite flat plate, and detailed comparisons are made with the Blasius flat plate solution and Goldstein wake solution. It is shown that the error in the streamwise velocity decreases like r(sup alpha)(Delta)y(exp 2), where alpha approx. 0.25 and r = delta(y)/delta(x) is the grid aspect ratio.

Scott, James R.↗

NASA NTRS1991

A multi-block multigrid method for the solution of the Euler and Navier-Stokes equations for three-dimensional flows

A multi-block multigrid method for the solution of the three-dimensional Euler and Navier-Stokes equations is presented. The basic flow solver is a cell-vertex method which employs central-difference spatial approximations and Runge-Kutta time stepping. The use of local time stepping, implicit residual smoothing, multigrid techniques, and variable-coefficient numerical smoothing results in an efficient and robust scheme. The multi-block strategy places the block loop within the Runge-Kutta loop such that accuracy and convergence are not affected by block boundaries. This has been verified by comparing the results of one- and two-block calculations in which the two-block grid is generated by splitting the one-block grid. Results are presented for both Euler and Navier-Stokes computations of wings and wing-fuselage combinations.

Atkins, H. L.↗