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Numerical determination of flux components in vapor flow considers split boundary conditions in both directions to form grid lines for flow continuity equations. Matrix of coefficients obtains direct solution for potential flow problem without iteration.
A multi-grid algorithm has been developed to speed the iterative convergence of calculations for the transonic potential flow past swept wings and wing-fuselage combinations. The method is based upon a fully-conservative, finite-volume approximation to the steady potential equation which is second-order accurate everywhere in the flow field except near shock waves. The multi-grid scheme is incorporated within the framework of an alternating successive-line-overrelaxation (SLOR) solver of the difference equations. Computed results confirm the second-order accuracy of the scheme, and demonstrate the effectiveness of the multi-grid procedure.
The multigrid method is applied to the calculation of transonic, potential flow fields about arbitrary three dimensional wing-body combinations. Numerical results for iterative convergence rate are in good agreement with those predicted by a local mode analysis and show substantial improvement over conventional relaxation algorithms.
The finite volume scheme of Jameson (1977) is used to calculate potential flow around a propeller rotating at high speed. An H-type mesh is generated and used successfully in the calculations. A test calculation with a thick blade cross section shows that the present code is capable of computing the propeller flow at the advance Mach number 0.8. The possible physical mechanisms which may play an important role in the propeller aerodynamics are discussed.
A 3-D, quasi-steady, full potential flow solver was adapted to include realistic wake influence for the aerodynamic analysis of helicopter rotors. The method is based on a finite difference solution of the full potential equation, using an inner and outer domain procedure for the blade flowfield to accommodate wake effects. The nonlinear flow is computed in the inner domain region using a finite difference solution method. The wake is modeled by a vortex lattice using prescribed geometry techniques to allow for the inclusion of realistic rotor wakes. The key feature of the analysis is that vortices contained within the finite difference mesh (inner domain) were treated with a vortex embedding technique while the influence of the remaining portion of the wake (in the outer domain) is impressed as a boundary condition on the outer surface of the finite difference mesh. The solution procedure couples the wake influence with the inner domain solution in a consistent and efficient solution process. The method has been applied to both hover and forward flight conditions. Correlation with subsonic and transonic hover airload data is shown which demonstrates the merits of the approach.
A computer code called NCOREL (for Nonconical Relaxation) has been developed to solve for supersonic full potential flows over complex geometries. The method first solves for the conical at the apex and then marches downstream in a spherical coordinate system. Implicit relaxation techniques are used to numerically solve the full potential equation at each subsequent crossflow plane. Many improvements have been made to the original code including more reliable numerics for computing wing-body flows with multiple embedded shocks, inlet flow through simulation, wake model and entropy corrections. Line relaxation or approximate factorization schemes are optionally available. Improved internal grid generation using analytic conformal mappings, supported by a simple geometric Harris wave drag input that was originally developed for panel methods and internal geometry package are some of the new features.
A free-streamline potential flow model is developed for the flowfield external to a stationary jet or fan inlet in proximity to a ground plane. It is assumed that the flow is steady, incompressible, and irrotational. A two-dimensional flat plate geometry models the inlet configuration, and free streamline theory is used to avoid unrealistic infinite flow velocities at the lips of the inlet, and to account somewhat for flow separation. A water table experiment is used to verify the analytical results. The fact that the solutions are found to be limited to only a certain range of diameter-height ratios implies an overdetermination of the problem by free streamline theory. Also studied are the convective properties of small vortical disturbances superimposed on the ideal flow solution.
PAN AIR is a computer program for predicting subsonic or supersonic linear potential flow about arbitrary configurations. It uses linear source and quadratic doublet strength distributions. These higher-order distributions have been implemented in a manner that greatly reduces the numerical stability problems that have plagued earlier attempts to make surface paneling methods work successfully for supersonic flow. PAN AIR's problem-solving capability, numerical approach, modeling features, and program architecture are described. Numerical results are presented for a variety of geometries at supersonic Mach numbers.
Incompressible potential flow theory corrected for compressibility effects, using the Lieblein-Stockman compressibility correction, was used to predict surface and flow field static pressures for a subsonic inlet at cruise conditions. The calculated internal and external surface static pressures were in good agreement with data at most conditions. The analysis was used to determine the capture stream-tube location and static-pressure distribution. Additive drag coefficients obtained from these results were consistently higher than those obtained using one-dimensional compressible flow theory. Increasing the distance between the inlet and boattail increased the cowl drag force. The effect of the boundary layer on internal and external surface static-pressure distributions was small at the design cruise condition. The analytical results may be used as an aid to data reduction and for predicting inlet mass flow, stagnation point location, and inlet additive drag.
We have solved for the potential flow downstream of the terminal shock of the solar wind in the limit of small departures from a spherical shock due to a latitudinal ram pressure variation in the supersonic solar wind. The solution connects anisotropic streamlines at the shock to uniform streamlines down the heliotail because we use a non-slip boundary condition on the heliopause at large radii. The rotational velocity about the heliotail in the near-field solution decays as the fourth power of distance from the shock. The polar divergence of the streamlines will have consequences for the previously discussed magnetic pressure ridge that may build-up just inside the heliopause.
We have solved for the potential flow downstream of the terminal shock of the solar wind in the limit of small departures from a spherical shock due to a latitudinal ram pressure variation in the supersonic solar wind. The solution connects anisotropic streamlines at the shock to uniform streamlines down the heliotail because we use a non-slip boundary condition on the heliopause at large radii. The rotational velocity about the heliotail in the near-field solution decays as the fourth power of distance from the shock. The polar divergence of the streamlines will have consequences for the previously discussed magnetic pressure ridge that may build-up just inside the heliopause.
An accurate method has been developed for computation of transonic potential flow about a 2-D lifting airfoil or an axisymmetric body in a wind tunnel. The computational mesh is nearly orthogonal everywhere and is generated by a sequence of Schwarz-Christoffel transformations and shearings to obtain an '0' grid near the body. A conservative finite-volume scheme for the full-potential equation and exact boundary conditions is used together with 'retarded' density to solve efficiently transonic flow with embedded shocks and large regions of supersonic flow. Supersonic free-stream flows are also solved with captured bow shocks and embedded subsonic regions.
Analysis of the pressure minimum integral in the calculation of three-dimensional potential flow around wings makes it possible to use non-rectangular mesh networks for distributing the three-dimensional potential into discrete points. The method is comparatively easily expanded to the treatment of realistic airplane configurations. Shock-pressure affected pressure distributions on any wings are determined with accuracy using this method.
A discrete singularity method has been developed for calculating the potential flow around two-dimensional airfoils. The objective was to calculate velocities at any arbitrary point in the flow field, including points that approach the airfoil surface. That objective was achieved and is demonstrated here on a Joukowski airfoil. The method used combined vortices and sources ''submerged'' a small distance below the airfoil surface and incorporated a near-field subvortex technique developed earlier. When a velocity calculation point approached the airfoil surface, the number of discrete singularities effectively increased (but only locally) to keep the point just outside the error region of the submerged singularity discretization. The method could be extended to three dimensions, and should improve nonlinear methods, which calculate interference effects between multiple wings, and which include the effects of force-free trailing vortex sheets. The capability demonstrated here would extend the scope of such calculations to allow the close approach of wings and vortex sheets (or vortices).
A procedure for the finite-difference numerical solution of the lifting potential flow about any number of arbitrarily shaped bodies is given. The solution is based on a technique of automatic numerical generation of a curvilinear coordinate system having coordinate lines coincident with the contours of all bodies in the field, regardless of their shapes and number. The effects of all numerical parameters involved are analyzed and appropriate values are recommended. Comparisons with analytic solutions for single Karman-Trefftz airfoils and a circular cylinder pair show excellent agreement. The technique of application of the boundary-fitted coordinate systems to the numerical solution of partial differential equations is illustrated.
An analytical study was conducted to predict the effect of an oscillating stream on the time dependent sectional pressure and lift coefficients of a model propeller blade. The assumption is that as the blade sections encounter a wake, the actual angles of attack vary in a sinusoidal manner through the wake, thus each blade is exposed to an unsteady stream oscillating about a mean value at a certain reduced frequency. On the other hand, an isolated propeller at some angle of attack can experience periodic changes in the value of the flow angle causing unsteady loads on the blades. Such a flow condition requires the inclusion of new expressions in the formulation of the unsteady potential flow around the blade sections. These expressions account for time variation of angle of attack and total shed vortices in the wake of each airfoil section. It was found that the final expressions for the unsteady pressure distribution on each blade section are periodic and that the unsteady circulation and lift coefficients exhibit a hysteresis loop.
The theoretical basis for PMARC, a low-order potential-flow panel code for modeling complex three-dimensional geometries, is outlined. Several of the advanced features currently included in the code, such as internal flow modeling, a simple jet model, and a time-stepping wake model, are discussed in some detail. The code is written using adjustable size arrays so that it can be easily redimensioned for the size problem being solved and the computer hardware being used. An overview of the program input is presented, with a detailed description of the input available in the appendices. Finally, PMARC results for a generic wing/body configuration are compared with experimental data to demonstrate the accuracy of the code. The input file for this test case is given in the appendices.
A rigorous treatment is given of the problem of determining the two-dimensional potential flow around arbitrary biplane cellules. The analysis involves the use of elliptic functions and is sufficiently general to include the effects of such elements as the section shapes, the chord ratio, gap, stagger, and decalage, which elements may be specified arbitrarily. The flow problem is resolved by making use of the methods of conformal representation. Thus the solution of the problem of transforming conformally two arbitrary contours into two circles is expressed by a pair of simultaneous integral equations, for which a method of numerical solution is outlined. As an example of the numerical process, the pressure distribution over certain arrangements of the NACA 4412 airfoil in biplane combinations is presented and compared with the monoplane pressure distribution.