SEARCH · Search NASA
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
A time-accurate multiple-grid algorithm is described. The algorithm allows one to take much larger time steps with an explicit time-marching scheme than would otherwise be the case. Sample calculations of a scalar advection equation and the Euler equations for an oscillating airfoil are shown. For the oscillating airfoil, time steps an order of magnitude larger than the single-grid algorithm are possible.
The unsteady form of the full potential equation is solved in conservation form by an implicit method based on approximate factorization. At each time level, internal Newton iterations are performed to achieve time accuracy and computational efficiency. A local time linearization procedure is introduced to provide a good initial guess for the Newton iteration. A novel flux-biasing technique is applied to generate proper forms of the artificial viscosity to treat hyperbolic regions with shocks and sonic lines present. The wake is properly modeled by accounting not only for jumps in phi, but also for jumps in higher derivatives of phi, obtained by imposing the density to be continuous across the wake. The far field is modeled using the Riemann invariants to simulate nonreflecting boundary conditions. The resulting unsteady method performs well which, even at low reduced frequency levels of 0.1 or less, requires fewer than 100 time steps per cycle at transonic Mach numbers. The code is fully vectorized for the CRAY-XMP and the VPS-32 computers.
An investigation is conducted of the time-accurate convergence of representative transonic flows to a steady state under given constraints of time-accuracy. Factored explicit and implicit difference operators are used to accelerate the calculations. Attention is given to flow at a Mach number of 1.35 past a circular cylinder and supersonic flow past a NACA 0012 airfoil for three different Mach numbers. Questions of transonic wave behavior are considered along with the equations of motion and the characteristics of the mesh network.
An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation-law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.
A time-accurate finite-difference Navier-Stokes code has been used to calculate the viscous flow over a severely indented blunt body in a supersonic stream. An algebraic turbulence model is used and the results are compared with experimental data from wind-tunnel tests. Qualitative agreement is obtained for the surface pressure distribution and flow-field structure, including the separated bubble in the indented region. However, uncertainties still exist in the heating calculations, which are attributed to the turbulence model. For both laminar and turbulent calculations, the flow exhibits a fundamental unsteady character at a frequency of about 50 kHz.
Although the Navier-Stokes equations describe most flows of interest in aerodynamics, the inviscid conservation law equations may be used for small regions with viscous forces. Thus, Euler equations and several time-accurate finite difference procedures, explicit and implicit, are discussed. Although implicit techniques require more computational work, they permit larger time steps to be taken without instability. It is noted that the Jacobian matrices for Euler equations in conservation-law form have certain eigenvalue-eigenvector properties which may be used to construct conservative-form coefficient matrices. This reduces the computation time of several implicit and semiimplicit schemes. Extensions of the basic approach to other areas are suggested.
A second-order time-accurate and spatially factored algorithm was used in a finite difference scheme for the numerical solution of the time-dependent, incompressible, two dimensional Navier-Stokes equations in conservation-law form using vorticity and stream function variables. The systems of equations are solved at each time step by an iterative technique. Numerical results were obtained for a circular cylinder at a Reynolds number of 15, and an NACA 0012 airfoil at zero angle of attack at Reynolds numbers of 10 to the third and 10 to the fourth powers. The results are in agreement with another numerical technique, and the computing time required to obtain the steady state solution at the Reynolds number of 10 to the 4th power was 49.7 sec on CDC 7600 computer using a 65 x 84 computational grind.
The problem of the potential incompressible flow about a helicopter rotor blade is solved using an unsteady vortex-panel method where the mutual interaction between the blade and the distorting free wake is taken into account. The present method alleviates the need to rely upon measured-wake geometries or p5escribed-wake models in order to calculate the airloads. A computer program has been developed which is capable of predicting the geometry of the time-dependent three-dimensional (3-D) wake and the instantaneous loadings for a single blade in hover, climb, and forward flight. The solution is obtained by using a time-accurate step-by-step procedure. The complex-wake geometry at any time is presented graphically with a computer graphics system. Calculated results are compared with published data for a rotor blade in both hover and forward flight. The code has also been applied to the study of the effect of changing blade tip geometry.
An implicit, approximate-factorization, finite-difference algorithm has been developed for the computation of unsteady, inviscid transonic flows in two and three dimensions. The computer program solves the full-potential equation in generalized coordinates in conservation-law form in order to properly capture shock-wave position and speed. A body-fitted coordinate system is employed for the simple and accurate treatment of boundary conditions on the body surface. The time-accurate algorithm is modified to a conventional ADI relaxation scheme for steady-state computations. Results from two- and three-dimensional steady and two-dimensional unsteady calculations are compared with existing methods.
Several flow phenomena including flowfield periodicity, rotor shock oscillation, and rotor shock system geometry were investigated in a transonic low aspect ratio fan rotor using laser anemometry. Flow periodicity is found to increase with increasing rotor pressure rise, and to correlate with blade geometry variations. Analysis of time-accurate laser anemometer data indicates that the rotor shock oscillates about its mean location with an amplitude of 3 to 4 percent of rotor chord. The shock surface is nearly two-dimensional or levels of rotor pressure rise at and above the peak efficiency level but becomes more complex for lower levels of pressure rise. Spanwise shock lean generates radial flows due to streamline deflection in the hub-to-shroud streamsurface.
An efficient coordinate transformation technique is presented for constructing grids for unsteady, transonic aerodynamic computations for delta-type wings. The original shearing transformation yielded computations that were numerically unstable and this paper discusses the sources of those instabilities. The new shearing transformation yields computations that are stable, fast, and accurate. Comparisons of those two methods are shown for the flow over the F5 wing that demonstrate the new stability. Also, comparisons are made with experimental data that demonstrate the accuracy of the new method. The computations were made by using a time-accurate, finite-difference, alternating-direction-implicit (ADI) algorithm for the transonic small-disturbance potential equation.
The research during this period continued to expand the class of numerical algorithms that can be accurately and efficiently implemented on overlapping grids. Whereas previous calculations have been used to solve elliptic equations and to find the steady-state solution of parabolic equations, the present work is aimed towards developing time-accurate solution techniques for parabolic and hyperbolic equations. The primary difficulty here is in the correct treatment of the interior boundary nodes that must be updated at each iteration. The implementation of explicit methods is straightforward. However, the common practice of lagging these values when using an implicit methods leads to inconsistencies in the difference equation. One way to avoid this problem is to alternately calculate with an implicit and an explicit method on each subgrid. With this procedure, the explicit method generates boundary values at the next time level which are then used by the implicit step. It can be shown that when a backward implicit method is combined with a forward explicit method, the composite method is second order accurate and unconditionally stable for linear problems. A second area in which progress can be reported is in the distribution of grid points on curves and surfaces.
The flow into an open return wind tunnel inlet was simulated using Euler equations. An explicit predictor-corrector method was employed to solve the system. The calculation is time-accurate and was performed to achieve a steady-state solution. The predictions are in reasonable agreement with the experimental data. Wall pressures are accurately predicted except in a region of recirculating flow. Flow-field surveys agree qualitatively with laser velocimeter measurements. The method can be used in the design process for open return wind tunnels.
A conservative zonal-boundary condition that was used with explicit integration schemes is extended to implicit, upwind, relaxation schemes; in particular to the Osher scheme. The rate of convergence was found to increase considerably with the use of the implicit, relaxation-zonal scheme when compared to the explicit scheme. The relaxation-zonal scheme has also been used in a time-accurate mode. Results demonstrating the time accuracy of the scheme and the feasibility of performing calculations in cases where some parts of the given system move relative to others (for example, rotor-stator configurations) are presented.
The flow into an open return wind tunnel inlet was simulated using Euler equations. An explicit predictor-corrector method was employed to solve the system. The calculation is time-accurate and was performed to achieve a steady-state solution. The predictions are in reasonable agreement with the experimental data. Wall pressures are accurately predicted except in a region of recirculating flow. Flow-field surveys agree qualitatively with laser velocimeter measurements. The method can be used in the design process for open-return wind tunnels.
Several flow phenomena including flowfield periodicity, rotor shock oscillation, and rotor shock system geometry were investigated in a transonic low aspect ratio fan rotor using laser anemometry. Flow periodicity is found to increase with increasing rotor pressure rise, and to correlate with blade geometry variations. Analysis of time-accurate laser anemometer data indicates that the rotor shock oscillates about its mean location with an amplitude of 3 to 4 percent of rotor chord. The shock surface is nearly two-dimensional for levels of rotor pressure rise at and above the peak efficiency level but becomes more complex for lower levels of pressure rise. Spanwise shock lean generates radial flows due to streamline deflection in the hub-to-shroud streamsurface.
The application of both a central-difference and an upwind implicit approximate factorization Navier-Stokes algorithm to highly separated flow is described. Using the thin-layer approximation, both algorithms are employed to solve the low Reynolds number laminar flow around a circular cylinder with periodic shedding. Results agree with experimental Strouhal numbers to within about 10 percent. A grid refinement study is conducted using the upwind code for both the thin-layer and complete Navier-Stokes approximations. The circular cylinder solution is shown to be highly dependent upon both grid density outward from the body and grid extent. Finally, the thin-layer central-difference code is used to predict the time-accurate unsteady flow about two stalled airfoils. The NACA 0012 airfoil turbulent solutions are periodic with Strouhal numbers of about 0.1; a strong vortex is periodically shed from the leading edge. The LS(1)-0417 airfoil turbulent solution shows a high frequency oscillatory trailing edge separation. The 20 degree NACA 0012 airfoil solution is massively separated, but shows no evidence of periodicity.
Methods for efficient computation of numerical algorithms on a wide variety of MIMD machines are proposed. These techniques reorganize the data dependency patterns to improve the processor utilization. The model problem finds the time-accurate solution to a parabolic partial differential equation discretized in space and implicitly marched forward in time. The algorithms are extensions of Jacobi and SOR. The extensions consist of iterating over a window of several timesteps, allowing efficient overlap of computation with communication. The methods increase the degree to which work can be performed while data are communicated between processors. The effect of the window size and of domain partitioning on the system performance is examined both by implementing the algorithm on a simulated multiprocessor system.