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Computational method for surface pressures on blunt bodies of revolution at small angles of attack in supersonic flow
Mathematical model for determining total drag of pointed body of revolution in supersonic flow with turbulent boundary layer
A computational procedure is presented which is capable of determining the supersonic flow field surrounding three-dimensional wing-body configurations such as a delta-wing space shuttle. The governing equations in conservation-law form are solved by a finite difference method using a second-order noncentered algorithm between the body and the outermost shock wave, which is treated as a sharp discontinuity. Secondary shocks which form between these boundaries are captured automatically, and the intersection of these shocks with the bow shock posed no difficulty. Resulting flow fields about typical blunt nose shuttle-like configurations at angle of attack are presented. The differences between perfect and real gas effects for high Mach number flows are shown.
In order to provide experimental data for comparison with newly developed finite difference methods for computing supersonic flows over aircraft configurations, wind tunnel tests were conducted on four arrow wing models. The models were machined under numeric control to precisely duplicate analytically defined shapes. They were heavily instrumented with pressure orifices at several cross sections ahead of and in the region where there is a gap between the body and the wing trailing edge. The test Mach numbers were 2.36, 2.96, and 4.63. Tabulated pressure data for the complete test series are presented along with selected oil flow photographs. Comparisons of some preliminary numerical results at zero angle of attack show good to excellent agreement with the experimental pressure distributions.
The three-dimensional supersonic flow over passive, that is, nonablating, indented nosetips of reentry vehicles is determined using an unsteady implicit numerical algorithm which solves either the inviscid Euler equations or the 'thin-layer' Navier-Stokes equations. A nonorthogonal independent variable transformation is used to map the distorted physical domain, containing multiple zones of embedded subsonic flow and separated flow regions into a rectangular computational volume at whose boundaries the required permeable or impermeable boundary conditions are simulated. Use of the implicit algorithm results in faster convergence to the steady state because of a larger allowable time step over conventional explicit schemes. The numerical results obtained compare favorably with existing numerical solutions and experimental data for simple spheres which validates the program. Results are also presented for analytically defined indented bodies for both laminar and turbulent flow conditions that demonstrate the program's capability for computing such flows.
The parabolized Navier-Stokes (PNS) equations are used to calculate the viscous, supersonic flow fields about a six-finned projectile and a generic four-finned missile at angles of attack. Since current computer speeds and storage preclude a fully three-dimensional calculation using the unsteady, Reynolds-averaged, Navier-Stokes equations, the applicability of the PNS equations to the above flow fields is of considerable interest. Two important aspects of the calculation are grid generation and the type of smoothing used to prevent nonphysical solutions. This paper includes a description of the grid-generation process. Results in the form of density contours and velocity vector plots are presented for the two configurations. The applicability of the PNS equations to the complicated flow fields considered is successfully demonstrated.
A vector study of the partial-differential equation of steady linearized supersonic flow is presented. General expressions which relate the velocity potential in the stream to the conditions on the disturbing surfaces, are derived. In connection with these general expressions the concept of the finite part of an integral is discussed. A discussion of problems dealing with planar bodies is given and the conditions for the solution to be unique are investigated. Problems concerning nonplanar systems are investigated, and methods are derived for the solution of some simple nonplanar bodies. The surface pressure distribution and the damping in roll are found for rolling tails consisting of four, six, and eight rectangular fins for the Mach number range where the region of interference between adjacent fins does not affect the fin tips.
Small perturbation technique solves the problem of an oscillating cone in supersonic flow. The logic of the program is straightforward, as reflected in the actual instructions for solving the problem.
Effect of angle of attack and injection pressure on jet penetration and spreading from delta wing in supersonic flow
The results of the calculation of supersonic flow of helium about right circular cones at zero angle of attack are presented in tabular form. The calculations were performed using the Taylor-Maccoll theory. Numerical integrations were performed using a Runge-Kutta method for second-order differential equations. Results were obtained for cone angles from 2.5 to 30 degrees in regular increments of 2.5 degrees. In all calculations the desired free-stream Mach number was obtained to five or more significant figures.
A higher order panel method using linearly varying source and quadratically varying doublet distributions for computing linearized supersonic flow over arbitrary wings and bodies is described. With tangential mass flux boundary conditions on circular cones at angles of attack, the source panel method gave pressure distributions which are in excellent agreement with exact nonlinear theory. The numerical source panel method was demonstrated to be very stable by randomly panelling an axially symmetric body. For wings with thickness and camber, both the source and doublet panel methods yield excellent agreement with exact linearized solutions. Computational experiments indicate that the use of exact surface paneling in lieu of linearized boundary conditions for wings provides better agreement with higher order theory and with experiment.
The performance of a constant-temperature normal hot-wire in a supersonic flow is critically examined. It is shown that this instrument is inherently unsuitable for measuring turbulent temperature correlations because of the highly nonlinear response to temperature fluctuations, particularly at low overheat ratios. The instrument is therefore limited to measurements of mean and fluctuating mass-flow rates. Suitable calibration procedures, as well as the limits on spatial and temporal resolution are discussed, and corrections for mean stagnation temperature changes are suggested.
Boundary layer transition along the leeward symmetry plane of axisymmetric bodies at nonzero angle of attack in supersonic flow was investigated experimentally and numerically as part of joint research between the Japan Aerospace Exploration Agency (JAXA) and National Aeronautics and Space Administration (NASA). Transition over four axisymmetric bodies (namely, Sears-Haack body, semi-Sears-Haack body, straight cone and flared cone) with different axial pressure gradients was measured in two different facilities with different unit Reynolds numbers. The semi-Sears-Haack body and flared cone were designed at JAXA to broaden the range of axial pressure distributions. For a body shape with an adverse pressure gradient (i.e., flared cone), the experimentally measured transition patterns show an earlier transition location along the leeward symmetry plane in comparison with the neighboring azimuthal locations. For nearly zero pressure gradient (i.e.,straight cone), this feature is only observed at the larger unit Reynolds number. Later transition along the leeward plane was observed for the remaining two body shapes with a favorable pressure gradient. The observed transition patterns are only partially consistent with the numerical predictions based on linear stability analysis. Additional measurements are used in conjunction with the stability computations to explore the phenomenon of leeward line transition and the underlying transition mechanism in further detail.
An asymptotic description is proposed for supersonic laminar flow over a wedge or a backward-facing step, for large Reynolds number and for a base or step height which is small compared with the boundary-layer length. The analysis is carried out for adiabatic wall conditions and a viscosity coefficient proportional to temperature. In a particular limit corresponding to a very thick boundary layer, a similarity law is obtained for the base pressure. For a thinner boundary layer an asymptotic form for the base pressure is obtained which shows the dependence on the parameters explicitly and which permits good agreement with experiment. This latter result is based on an inviscid-flow approximation for the corner expansion and for reattachment with viscous forces important primarily in a thin sublayer about the dividing streamline. A prediction of the pressure distribution at reattachment is given and the result is compared with experimental pressure distributions.
A method has been developed for two- and three-dimensional computations of viscous supersonic jet flows interacting with an external flow. The approach employs a reduced form of the Navier-Stokes equations which allows solution as an initial-boundary value problem in space, using an efficient noniterative forward marching algorithm. Numerical instability associated with forward marching algorithms for flows with embedded subsonic regions is avoided by approximation of the reduced form of the Navier-Stokes equations in the subsonic regions of the boundary layers. Supersonic and subsonic portions of the flow field are simultaneously calculated by a consistently split linearized block implicit computational algorithm. The results of computations for a series of test cases associated with supersonic jet flow is presented and compared with other calculations for axisymmetric cases. Demonstration calculations indicate that the computational technique has great promise as a tool for calculating a wide range of supersonic flow problems including jet flow. Finally, a User's Manual is presented for the computer code used to perform the calculations.
An investigation of the effects of mistuning on flutter and forced response of a cascade in subsonic and supersonic flows is presented. The aerodynamic and structural coupling between the bending and torsional motions and the aerodynamic coupling between the blades are included. It is shown that frequency mistuning always has a beneficial effect on flutter. Additionally, the results indicate that frequency mistuning may have either a beneficial or an adverse effect on forced response, depending on the engine order of the excitation and Mach number.
A finite difference code was developed for modeling inviscid, unsteady supersonic flow by solution of the compressible Euler equations. The code uses a deforming grid technique to capture the motion of the airfoils and can model oscillating cascades with any arbitrary interblade phase angle. A flat plate cascade is analyzed, and results are compared with results from a small-perturbation theory. The results show very good agreement for both the unsteady pressure distributions and the integrated force predictions. The reason for using the numerical Euler code over a small-perturbation theory is the ability to model real airfoils that have thickness and camber. Sample predictions are presented for a section of the rotor on a supersonic throughflow compressor designed at NASA Lewis Research Center. Preliminary results indicate that two-dimensional, flat plate analysis predicts conservative flutter boundaries.
A finite difference code was developed for modeling inviscid, unsteady supersonic flow by solution of the compressible Euler equations. The code uses a deforming grid technique to capture the motion of the airfoils and can model oscillating cascades with any arbitrary interblade phase angle. A flat plate cascade is analyzed, and results are compared with results from a small-perturbation theory. The results show very good agreement for both the unsteady pressure distributions and the integrated force predictions. The reason for using the numerical Euler code over a small-perturbation theory is the ability to model real airfoils that have thickness and camber. Sample predictions are presented for a section of the rotor on a supersonic throughflow compressor designed at NASA Lewis Research Center. Preliminary results indicate that two-dimensional, flat plate analysis predicts conservative flutter boundaries.