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NASA NTRS1977

An implicit factored scheme for the compressible Navier-Stokes equations

An implicit finite difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation-law form. The algorithm is second-order-time accurate, noniterative, and spatially factored. In order to obtain an efficient factored algorithm, the spatial cross-derivatives are evaluated explicitly. However, the algorithm is unconditionally stable and, although a three-time-level scheme, requires only two-time-levels of data storage. The algorithm is constructed in a 'delta' form (i.e., increments of the conserved variables and fluxes) that provides a direct derivation of the scheme and leads to an efficient computational algorithm. In addition, the delta form has the advantageous property of a steady-state (if one exists) independent of the size of the time step. Numerical results are presented for a two-dimensional shock boundary-layer interaction problem.

Beam, R. M.↗

NASA NTRS1977

A solution to the Navier-Stokes equations based upon the Newton Kantorovich method

An implicit finite difference scheme based on the Newton-Kantorovich technique was developed for the numerical solution of the nonsteady, incompressible, two-dimensional Navier-Stokes equations in conservation-law form. The algorithm was second-order-time accurate, noniterative with regard to the nonlinear terms in the vorticity transport equation except at the earliest few time steps, and spatially factored. Numerical results were obtained with the technique for a circular cylinder at Reynolds number 15. Results indicate that the technique is in excellent agreement with other numerical techniques for all geometries and Reynolds numbers investigated, and indicates a potential for significant reduction in computation time over current iterative techniques.

Davis, J. E.↗

NASA NTRS1982

Computations of transonic flow over an oscillating airfoil with shock-induced separation

A numerical study is presented on unsteady transonic flow over an airfoil in pitching oscillation at a moderately high incidence. The study was focused on the effects of unsteady shock-wave/boundary-layer interactions and subsequent flow separations. An implicit finite-difference scheme was used to solve the full time-dependent compressible Navier-Stokes equations in conservation-law form. The algorithm adopted for this study is the Beam-Warming spatially factored scheme. It is, however, constructed in a body-fitted curvilinear coordinate system for efficient unsteady flow computations. The computational results are compared with experimental data measured in the Ames 11- by 11-Foot Transonic Wind Tunnel.

Chyu, W. J.↗

NASA NTRS1984

Accelerated convergence for incompressible flow calculations

Two improved algorithms which solve the steady-state Navier-Stokes equations, PISO and SIMPLER, are studied. Computations were carried out on progressively finer grids for the driven cavity and flow over a backward-facing step. The effects of relaxation factor, number of grid nodes and number of sweeps through the pressure equations are studied to evaluate the performance of the PISO and SIMPLER schemes. Results show that these improved schemes accelerate the convergence rate of the solution generally by a factor of two as compared to the SIMPLE method.

Neely, G. M.↗

NASA NTRS1985

Accelerated convergence for incompressible flow calculations

Neely, G. M.↗

NASA NTRS1985

On the use of wall functions as boundary conditions for two-dimensional separated compressible flows

A new and improved wall function method for compressible turbulent flows has been developed and tested. This method is applicable to attached and separated flows, to both high- and low-Reynolds number flows, and to flows with adiabatic and nonadiabatic surfaces. This wall function method has been applied to the Launder-Spalding k-epsilon two-equation model of turbulence. The tests consist of comparisons of calculated and experimental results for: (1) an axisymmetrical transonic shock-wave/boundary-wave interaction flow at low Reynolds number in an adiabatic tube, (2) an axisymmetrical high-Reynolds number transonic flow over a nonadiabatic bump, and (3) a two-dimensional supersonic high-Reynolds number flow on a nonadiabatic deflected flap. Each of these experiments had significant regions of flow separation. The calculations are performed with an implicit algorithm that solves the Reynolds-averaged Navier-Stokes equations. It is shown that the results obtained agree very well with the data for the complex compressible flows tested.

Viegas, J. R.↗

NASA NTRS1986

Numerical simulation of tip vortices of wings in subsonic and transonic flows

A multi block zonal algorithm which solves the thin-layer Navier-Stokes and the Euler equations is used to numerically simulate the formation and roll-up of the tip vortex in both subsonic and transonic flows. Four test cases which used small and large aspect ratio wings have been considered to examine the influence of the tip-cap shape, the tip planform and the free-stream Mach number. It appears that both the tip-planform and the tip-cap shape have some influence on the formation of the tip vortex, but its subsequent roll-up seems to be more influenced by the tip-planform shape. In general, a good definition of the formation and the roll-up of the tip vortex has been observed for all the cases considered here. Comparions of the numerical results with the limited, available experimental data show good agreement with both the surface pressures and the tip-vortex strength.

Srinivasan, G. R.↗

NASA NTRS1987

A numerical investigation of the influence of heating on the excitation of high and low Mach number jet flows

The influence of heating and velocity on the unsteady nature of jet mixing layers is investigated by solving the time-dependent compressible Navier-Stokes equations using MacCormack's explicit finite difference algorithm. The computations are performed for jet Mach numbers of 0.3 and 0.8 with flow total temperatures up to 800 K. The Reynolds number ranges from 3 x 10 to the 5th to 1.3 x 10 to the 6th. Excitation is accomplished by imposing an acoustic pressure signal inside the jet duct. The objective of this effort is to compare the response of heated and unheated jets with acoustic excitation at high and low subsonic Mach numbers. The preliminary results indicate that without acoustic excitation the mixing in the heated jet is greater than in the unheated case. It has also been found that for the low speed heated jets (M = 0.3, Tt = 672K) acoustic excitation causes the production of large scale vortex structures to occur in a regular periodic manner with organized periodic pairing at some Strouhal numbers. These results are in agreement with experimental data.

Scott, J. N.↗

NASA NTRS1989

Upwind Navier-Stokes solutions for leading-edge vortex flows

An incompressible Navier-Stokes solver using an upwind finite-difference algorithm is employed to investigate low-speed, three-dimensional, laminar, leading-edge vortex flows over three round-edged low-aspect-ratio wings. The effects of grid density, Reynolds number, and wing planform on the flowfield structures and integral values are studied. Computed results show good qualitative and quantitative agreement with the available experimental data.

Hsu, C.-H.↗

NASA NTRS1989

Numerical simulation of three-dimensional transonic turbulent projectile aerodynamics by TVD schemes

The two-dimensional symmetric TVD scheme proposed by Yee has been extended to and investigated for three-dimensional thin-layer Navier-Stokes simulation of complex aerodynamic problems. An existing three-dimensional Navier-stokes code based on the beam and warming algorithm is modified to provide an option of using the TVD algorithm and the flow problem considered is a transonic turbulent flow past a projectile with sting at ten-degree angle of attack. Numerical experiments conducted for three flow cases, free-stream Mach numbers of 0.91, 0.96 and 1.20 show that the symmetric TVD algorithm can provide surface pressure distribution in excellent agreement with measured data; moreover, the rate of convergence to attain a steady state solution is about two times faster than the original beam and warming algorithm.

Shiau, Nae-Haur↗

NASA NTRS1989

A fully-coupled implicit method for thermo-chemical nonequilibrium air at sub-orbital flight speeds

A CFD technique is described in which the finite-rate chemistry in thermal and chemical nonequilibrium air is fully and implicitly coupled with the fluid motion. Developed for use in the suborbital hypersonic flight speed range, the method accounts for nonequilibrium vibrational and electronic excitation and dissociation, but not ionization. The steady-state solution to the resulting system of equations is obtained by using a lower-upper factorization and symmetric Gauss-Seidel sweeping technique through Newton iteration. Inversion of the left-hand-side matrices is replaced by scalar multiplications through the use of the diagonal dominance algorithm. The code, named CENS2H (Compressible-Euler-Navier-Stokes Two-Dimensional Hypersonic), is fully vectorized and requires about 8.8 x 10 to the -5th sec per node point per iteration using a Cray X-MP computer. Converged solutions are obtained after about 2400 iterations. Sample calculations are made for a circular cylinder and a 10 percent airfoil at 5 deg angle of attack. The calculated cylinder flow field agrees with that obtained experimentally. The code predicts a 10 percent change in lift, drag, and pitching moment for the airfoil due to the thermochemical phenomena.

Park, Chul↗

NASA NTRS1990

High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows

Hypersonic computations are presently conducted with an extension of a class of high-resolution implicit TVD algorithms suited to transonic multidimensional Euler and Navier-Stokes equations. These conservative shock-capturing schemes, which are spatially second- and third-order, may be first- and second-order accurate in time and suitable for either steady or unsteady calculations. Attention is given to the enhancement of hypersonic flows' convergence rate and stability; accuracy and efficiency is achieved by these means for very complex two-dimensional hypersonic viscous and inviscid shock interactions.

Yee, H. C.↗

NASA NTRS1991

Streamwise Algorithm For Simulation Of Flow

Algorithm for numerical solution of Euler and Navier-Stokes equations of unsteady flow incorporates upwind differencing based on streamwise direction. Generalized to calculate unsteady transonic flows over wings; suitable for analyses of flutter of wings affected by such flows.

Goorjian, Peter M.↗

NASA NTRS1990

Computational analysis of rotor-stator interaction in turbomachinery using zonal techniques

The development of CFD zonal techniques which allow more intensive computational treatment in some regions than in others, in conjunction with robust, high-accuracy algorithms for the numerical solution of the Navier-Stokes equations, is presently shown to have facilitated the investigation of rotor-stator interactions in turbomachinery. Attention is given to integration schemes with two and three spatial dimensions, the conservative 'patched' and the nonconservative zonal boundary schemes, and such natural boundary conditions as those of the endwall, the stator inlet, the airfoil surface, and the rotor exit. Illustrative three-dimensional rotor-stator interaction calculations are presented.

Madavan, Nateri K.↗

NASA NTRS1991

Aerodynamics of powered missile separation from a wing

A 3D dynamic 'chimera' algorithm that solves the thin-layer Navier-Stokes equations over multiple moving bodies was modified to numerically simulate the aerodynamics, missile dynamics, and missile plume of a finless missile separating from a wing in transonic flow. A powered missile separation case was considered to examine the influence of the missile and plume on the wing. The wing and missile is at a two degree angle of attack. The computational results show the details of the flow field.

Shanks, S. P.↗

NASA NTRS1993

Aerodynamics of powered missile separation from F/A-18 aircraft

A 3D dynamic 'chimera' algorithm that solves the thin-layer Navier-Stokes equations over multiple moving bodies was modified to numerically simulate the aerodynamics, missile dynamics, and missile plume interactions of a missile separating from a generic wing and from an F/A-18 aircraft in transonic flow. The missile is mounted below the wing for missile separation from the wing and on the F/A-18 fuselage at the engine inlet side for missile separation from aircraft. Static and powered missile separation cases are considered to examine the influence of the missile and plume on the wing and F/A-18 fuselage and engine inlet. The aircraft and missile are at two degrees angle of attack, Reynolds number of 10 million, freestream Mach number of 1.05 and plume Mach number of 3.0. The computational results show the details of the flow field.

Ahmad, J. U.↗

NASA NTRS1993

Computational techniques for high-speed flows with viscous and chemical effects

Algorithms for solving the Euler and the Navier-Stokes equations in conjunction with chemical kinetic equations are presented. The convective flux is estimated from a quasi one dimensional interpolation procedure. Shock, contact, and expansion waves and thermochemical nonequilibrium phenomena are captured by the Lax-Friedrichs technique. Relaxation techniques were developed to enhance their effectiveness in dealing with spatial and temporal stiffness associated with the physical problems. Both explicit and implicit smoothers were implemented into the standard multigrid time stepping method. Unsteady and steady scalar problems are discussed. A perfect gas and equilibrium air shock tube problem is investigated. Numerical schemes and techniques are compared for the problems of shock and boundary layer interaction and three dimensional viscous, nonequilibrium flow encompassing an aerobrake. The results are comparable in accuracy against other high order non-oscillatory techniques. The multigrid methods are assessed using a Mach 8 flow over a complete planar body, a sphere, and a blunt delta wing at 20 deg incidence. Applying an implicit multigrid method on a nested grid of 128 by 64 nodes the reduction factor is 0.25. The central processing unit reduction factor is 2.2 after both the single and multigrid Runge-Kutta solutions converged to machine zero on a grid of 37 by 41 by 73 nodes.

Li, C. P.↗

NASA NTRS1995

An assessment of viscous effects in computational simulation of benign and burst vortex flows on generic fighter wind-tunnel models using TEAM code

Vortex flows on a twin-tail and a single-tail modular transonic vortex interaction (MTVI) model, representative of a generic fighter configuration, are computationally simulated in this study using the Three-dimensional Euler/Navier-Stokes Aerodynamic Method (TEAM). The primary objective is to provide an assessment of viscous effects on benign (10 deg angle of attack) and burst (35 deg angle of attack) vortex flow solutions. This study was conducted in support of a NASA project aimed at assessing the viability of using Euler technology to predict aerodynamic characteristics of aircraft configurations at moderate-to-high angles of attack in a preliminary design environment. The TEAM code solves the Euler and Reynolds-average Navier-Stokes equations on patched multiblock structured grids. Its algorithm is based on a cell-centered finite-volume formulation with multistage time-stepping scheme. Viscous effects are assessed by comparing the computed inviscid and viscous solutions with each other and experimental data. Also, results of Euler solution sensitivity to grid density and numerical dissipation are presented for the twin-tail model. The results show that proper accounting of viscous effects is necessary for detailed design and optimization but Euler solutions can provide meaningful guidelines for preliminary design of flight vehicles which exhibit vortex flows in parts of their flight envelope.

Kinard, Tim A.↗