Improved finite-difference schemes for transonic potential flow calculations
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Hafez, M.↗
Engineering topics
Whitlow, W., Jr.
Publications and source records attributed to Whitlow, W., Jr..
An entropy correction method for unsteady full potential flows with strong shocks
An entropy correction method for the unsteady full potential equation is presented. The unsteady potential equation is modified to account for entropy jumps across shock waves. The conservative form of the modified equation is solved in generalized coordinates using an implicit, approximate factorization method. A flux-biasing differencing method, which generates the proper amounts of artificial viscosity in supersonic regions, is used to discretize the flow equations in space. Comparisons between the present method and solutions of the Euler equations and between the present method and experimental data are presented. The comparisons show that the present method more accurately models solutions of the Euler equations and experiment than does the isentropic potential formulation.
Whitlow, W., Jr.↗
An entropy correction method for unsteady full potential flows with strong shocks
An entropy correction method for the unsteady full potential equation is presented. The unsteady potential equation is modified to account for entropy jumps across shock waves. The conservative form of the modified equation is solved in generalized coordinates using an implicit, approximate factorization method. A flux-biasing differencing method, which generates the proper amounts of artificial viscosity in supersonic regions, is used to discretize the flow equations in space. Comparisons between the present method and solutions of the Euler equations and between the present method and experimental data are presented. The comparisons show that the present method more accurately models solutions of the Euler equations and experiment than does the isentropic potential formulation.
Whitlow, W., Jr.↗
Nonisentropic unsteady three dimensional small disturbance potential theory
Modifications that allow for more accurate modeling of flow fields when strong shocks are present were made into three dimensional transonic small disturbance (TSD) potential theory. The Engquist-Osher type-dependent differencing was incorporated into the solution algorithm. The modified theory was implemented in the XTRAN3S computer code. Steady flows over a rectangular wing with a constant NACA 0012 airfoil section and an aspect ratio of 12 were calculated for freestream Mach numbers (M) of 0.82, 0.84, and 0.86. The obtained results are compared using the modified and unmodified TSD theories and the results from a three dimensional Euler code are presented. Nonunique solutions in three dimensions are shown to appear for the rectangular wing as aspect ratio increases. Steady and unsteady results are shown for the RAE tailplane model at M = 0.90. Calculations using unmodified theory, modified theory and experimental data are compared.
Gibbons, M. D.↗
Unsteady transonic flows past airfoils using the Euler equations
An explicit finite difference solver for the Euler equations was developed and used to calculate the flow past two of the AGARD standard aeroelastic configurations, an NACA 64A010 and an NLR 7301. The algorithm employed uses a modified four step Runge-Kutta time stepping scheme, boundary conditions determined from the time-dependent theory of characteristics and an unsteady automatic grid generation procedure. In general, the calculated results are in good agreement with available experiment. Moreover, they demonstrate the importance of using the Euler equations for super-critical airfoils and in situations where the transonic small disturbance theory results in poor agreement with experiment.
Smith, G. E.↗
Nonisentropic unsteady three dimensional small disturbance potential theory
Nonisentropic modifications to the three-dimensional transonic small disturbance (TSD) theory, which allows for more accurate modeling of transonic flow fields, are described. The modified flux equation and entropy corrections are presented; the Engquist-Osher differencing (1980) is added to the solution algorithm in order to eliminate the velocity overshoots upstream of shocks. The modified theory is tested in the XTRAN3S finite difference computer code. Steady flows over a rectangular NACA 0012 wing with an aspect ratio of 12 are calculated and compared to Euler equation solutions; good correlation is observed between the data and the modified TSD theory provides more accurate data, particularly for the lift curve slope. The nonisentropic theory is evaluated on an RAE tailplane model for steady and unsteady flows and the modified theory results agree well with the experimental data.
Gibbons, M. D.↗
Nonreflecting boundary conditions for the complete unsteady transonic small-disturbance equation
Nonreflecting far-field boundary conditions that are consistent with the complete transonic small-disturbance (TSD) equations are derived. They are implemented in a new code for solving the complete TSD equation and are tested for a harmonically oscillating NACA 64A010 airfoil in transonic flow and for a flat-plate airfoil with a pulse in the angle of attack. Using the new boundary conditions on a relatively small grid, solutions for the airfoil that are obtained that agree with large-grid calculations, resulting in a 44 percent savings in computer time. Frequency responses for the flat plate show that most of the disturbances incident on the computational boundaries are absorbed by the boundary conditions.
Whitlow, W., Jr.↗
Entropy condition satisfying approximations for the full potential equation of transonic flow
A class of conservative difference approximations for the steady full potential equation was presented. They are, in general, easier to program than the usual density biasing algorithms, and in fact, differ only slightly from them. Rigorous proof indicated that these new schemes satisfied a new discrete entropy inequality, which ruled out expansion shocks, and that they have sharp, steady, discrete shocks. A key tool in the analysis is the construction of a new entropy inequality for the full potential equation itself. Results of some numerical experiments using the new schemes are presented.
Osher, S.↗
Characteristic boundary conditions for three-dimensional transonic unsteady aerodynamics
Characteristic far-field boundary conditions for the three-dimensional unsteady transonic small disturbance potential equation have been developed. The boundary conditions were implemented in the XTRAN3S finite difference code and tested for a flat plate rectangular wing with a pulse in angle of attack; the freestream Mach number was 0.85. The calculated force response shows that the characteristic boundary conditions reduce disturbances that are reflected from the computational boundaries.
Whitlow, W., Jr.↗
Entropy condition satisfying approximations for the full potential equations of transonic flow
A class of conservative difference approximations for the steady full potential equation was presented. They are, in general, easier to program than the usual density biasing algorithms, and in fact, differ only slightly from them. Rigorous proof indicated that these new schemes satisfied a new discrete entropy inequality, which ruled out expansion shocks, and that they have sharp, steady, discrete shocks. A key tool in the analysis is the construction of a new entropy inequality for the full potential equation itself. Results of some numerical experiments using the new schemes are presented.
Osher, S.↗
Improved finite difference schemes for transonic potential calculations
Engquist and Osher (1980) have introduced a finite difference scheme for solving the transonic small disturbance equation, taking into account cases in which only compression shocks are admitted. Osher et al. (1983) studied a class of schemes for the full potential equation. It is proved that these schemes satisfy a new discrete 'entropy inequality' which rules out expansion shocks. However, the conducted analysis is restricted to steady two-dimensional flows. The present investigation is concerned with the adoption of a heuristic approach. The full potential equation in conservation form is solved with the aid of a modified artificial density method, based on flux biasing. It is shown that, with the current scheme, expansion shocks are not possible.
Hafez, M.↗
XTRAN2L: A program for solving the general-frequency unsteady transonic small disturbance equation
A program, XTRAN2L, for solving the general-frequency unsteady transonic small disturbance potential equation was developed. It is a modification of the LTRAN2-NLR code. The alternating-direction-implicit (ADI) method of Rizzetta and Chin is used to advance solutions of the potential equation in time Engquist-Osher monotone spatial differencing is used in the ADI solution algorithm. As a result, the XTRAN2L code is more robust and more efficient than similar codes that use Murman-Cole type-dependent spatial differencing. Nonreflecting boundary conditions that are consistent with the general-frequency equation have been developed and implemented at the far-field boundaries. Use of those conditions allow the computational boundaries to be moved closer to the airfoil with no loss of accuracy. This makes the XTRAN2L code more economical to use.
Whitlow, W., Jr.↗
Time-marching transonic flutter solutions including angle-of-attack effects
Previously cited in issue 13, p. 2109, Accession no. A82-30152
Edwards, J. W.↗
An exploratory study of finite difference grids for transonic unsteady aerodynamics
A pulse-transfer function technique for calculating unsteady aerodynamic forces for a wide range of reduced frequencies is implemented in a finite difference program solving the complete unsteady transonic small perturbation equation. Forces are calculated for a two-dimensional linear flat plate case utilizing the default grids from several currently used finite difference programs. The forces are compared to exact theoretical values and grid generated boundary and internal reflections are demonstrated. Grids designed to alleviate the reflections are presented and forces for a 6% thick parabolic arc airfoil are calculated to investigate non-linear transonic effects.
Seidel, D. A.↗
An exploratory study of finite difference grids for transonic unsteady aerodynamics
Unsteady aerodynamic forces are calculated by the XTRAN2L finite difference program which solves the complete two dimensional unsteady transonic small perturbation equation. The unsteady forces are obtained using a pulse transfer function technique which assumes the flow field behaves in a locally linear fashion about a mean condition. Forces are calculated for a linear flat plate using the default grids from the LTRAN2-NLR, LTRAN2-HI, and XTRAN3S programs. The forces are compared to the exact theoretical values for flat plate, and grid generated boundary and internal numerical reflections are observed to cause significant errors in the unsteady airloads. Grids are presented that alleviate the reflections while reducing computational time up to fifty-three percent and program size up to twenty-eight percent. Forces are presented for a six percent thick parabolic arc airfoil which demonstrate that the transform technique may be successfully applied to nonlinear transonic flows.
Seidel, D. A.↗
Application of a transonic potential flow code to the static aeroelastic analysis of three-dimensional wings
Since the aerodynamic theory is nonlinear, the method requires the coupling of two iterative processes - an aerodynamic analysis and a structural analysis. A full potential analysis code, FLO22, is combined with a linear structural analysis to yield aerodynamic load distributions on and deflections of elastic wings. This method was used to analyze an aeroelastically-scaled wind tunnel model of a proposed executive-jet transport wing and an aeroelastic research wing. The results are compared with the corresponding rigid-wing analyses, and some effects of elasticity on the aerodynamic loading are noted.
Whitlow, W., Jr.↗
Time-marching transonic flutter solutions including angle-of-attack effects
Transonic aeroelastic solutions based upon the transonic small perturbation potential equation were studied. Time-marching transient solutions of plunging and pitching airfoils were analyzed using a complex exponential modal identification technique, and seven alternative integration techniques for the structural equations were evaluated. The HYTRAN2 code was used to determine transonic flutter boundaries versus Mach number and angle-of-attack for NACA 64A010 and MBB A-3 airfoils. In the code, a monotone differencing method, which eliminates leading edge expansion shocks, is used to solve the potential equation. When the effect of static pitching moment upon the angle-of-attack is included, the MBB A-3 airfoil can have multiple flutter speeds at a given Mach number.
Edwards, J. W.↗
Time-marching transonic flutter solutions including angle-of-attack effects
Transonic aeroelastic solutions based upon the transonic small perturbation potential equation are studied. Time-marching transient solutions of plunging and pitching airfoils are analyzed using a complex exponential modal identification technique, and seven alternative integration techniques for the structural equations are evaluated. The HYTRAN2 code is used to determine transonic flutter boundaries versus Mach number and angle-of-attack for NACA 64A010 and MBB A-3 airfoils. In that code, a monotone differencing method, which eliminates leading edge expansion shocks, is used to solve the potential equation. When the effect of static pitching moment upon the angle-of-attack is included, the MBB A-3 airfoil can have multiple flutter speeds at a given Mach number.
Edwards, J. W.↗