Subsonic Freestream Research Articles

Calculation of transonic flows by a field integral equation method

A field integral equation method is developed to solve the two-dimensional compressible potential equation for transonic flow over airfoils with embedded shocks at subsonic freestream Mach numbers. The perturbation velocities are induced by linear internal source and vorticity distributions on the mean camber line of the airfoil and nonlinear field source distributions external to the airfoil surface. Derivatives are calculated in the computational plane, and those in the physical plane are determined by using the Jacobian of the transformation

Zonal approach to V/STOL aerodynamics

A zonal method based on the iterative coupling between an in viscid panel method and a Navier-Stokes (NS) method has been developed to analyze the complex flowfield associated with the V/STOL flow environment in order to improve the prediction of aerodynamic characteristics. The low-order panel method, VSAERO, developed by Analytical Methods, Inc. is used to solve for the entire flow region, which includes the V/STOL configuration and jet boundary model. In two separate investigations presented here, ARC3D, a well-known compressible thin layer NS code developed at NASA Ames Research Center, and APPL, a parabolized NS code developed at Amtec Engineering, are used to solve for the details of the jet region subject to the boundary condition provided by VSAERO. Overlapping boundaries are used for the purpose of coupling. Comparisons with the available experimental data for a jet-in-crossflow and a V/STOL configuration with lift-jet in a subsonic freestream show qualitative agreement with experiment and improved correlations over those by inviscid computations alone.

Experimental investigation of vortex formation in the wake of a flat plate for subsonic and supersonic freestream Mach numbers

The wake of a flat plate with thickness H = 6 mm whose surface was either smooth or rough by pasting smooth or rough paper on to it was investigated with optical methods. The studies showed that for supersonic freestream Mach numbers the wake of the smooth flat plate seems to be turbulent in the observation field. In contrast the wake of the rough plate can exhibit a vortex street if certain conditions are met. The most important parameters are the roughness of the paper and the distance from the location where the rough paper ends to the trailing edge, abbreviated with d. It turned out that a vortex street develops in the wake if the roughness exceeds a critical value and if the distance d is of the order of some millimeters. If these conditions are not met the wake of the rough plate is turbulent in the observation field. The dependence of the vortex formation upon the Mach number, the roughness and the distance d was investigated with holographic interferometry, two component laser-Doppler-anemometry and a laser-optics for measuring the vortex shedding frequency.

A second-order approximate method for transonic small-disturbance potential flow and its application to the analysis of flows over airfoils

A transonic small-disturbance equation retaining all second- and some higher-order terms (i.e., the TSDH equation) is proposed for computing transonic flows over blunt leading edge airfoils in subsonic and supersonic free streams. Three difference schemes and relaxation techniques are studied numerically for obtained converged solutions, of which two succeed for both subsonic and supersonic free streams; the other can only be applied to subsonic free streams. Comparisons with reported results and experiments are made for the DSMA 523 and NACA 0012 airfoils. The agreements are generally satisfactory.

有限翼幅の亜音速揚力面理論を検証するに役立つ２次元“ＤＬＭ‐Ｃ”

In order to investigate accuracy and convergence characteristics of any unsteady lifting surface computation programs, it is found that DLM-C scheme for two-dimensional airfoils developed by us is quite useful. This scheme is reliable up to high reduced frequencies k and high subsonic free stream Mach numbers M. It has comparable efficiency with NASA 2-D, within the published k and M values. For a rectangular wing of aspect ratio ten chordwise lift distributions on the midspan section are computed by using some unsteady lifting surface programs, which are compared with DLM-C. The values of error-index parameters E are entered in a k vs M diagram, and constant E curves are drawn. This diagram shows clearly the upper limit of k where the scheme is efficient.

Axisymmetric transonic flow past slender bodies, including perforated wall interference effects

Solutions of the transonic small-perturbation equation for flow past slender bodies of revolution at subsonic freestream Mach numbers are presented in free air as well as in the presence of perforated walls. An artificial viscosity term which enables shock capture is added explicitly to the small-perturbation equation. The modified equation is then converted into an integral equation by the use of Green's theorem. Numerical solution to the integral equation is obtained by discretizing the region of integration into rectangular panels wherein the flow quantities can be considered uniform. Type-dependent operators are introduced in the calculation of the nonlinear source term. The resulting system of algebraic equations is then iteratively solved either by a single- step direct iteration scheme or a quasi-Newton scheme. HE finite difference relaxation procedure has been ap- plied to axisymmetric flow around bodies of revolution by Bailey 1 and Krupp and Mmrman2 and compared with the experimental data of McDevitt and Taylor.3'4 Using the method of split potential Sedin5 has devised a stable iterative scheme for the same problem. Both Bailey and Sedin have included tunnel wall interference effects. They have used the classical homogeneous wall boundary condition developed by Baldwin et al. 6 to model the presence of a perforated wall. Bailey's theoretical calculations seem to confirm the validity of this formulation insofar as the theoretical pressure distribution shows good agreement with the measured values and the shock location is correctly predicted for the ex- perimentally determined value of the porosity parameter. Thus, the theoretical model on which the computation of axisymmetric transonic flow with tunnel wall boundaries is based appears to be reasonably effective in predicting the flow details. It is well known that the axisymmetric transonic small- perturbation equation (TSPE), together with the linearized boundary condition on the body surface and a vanishing far- field disturbance, can be converted into an integral equation by the application of Green's theorem to the flow domain exterior to the body.7 In this formulation, the body boundary condition appears as a line integral on the body contour. Because the line integral for the axisymmetric flow is in- dependent of the unknown nonlinear field source term, iterative readjustment of the boundary condition is not necessary. Since the accuracy with which the tangency con- dition is satisfied depends only on the accuracy of the evaluation of the line integral, choice of the grid details in the vicinity of the body is unimportant to the overall com- putation. Further, since the far-field conditions are implicitly satisfied by the integral equation, no matching is necessary at the far-field boundary as is generally the case in finite difference calculations. The grid boundary is chosen after carrying out numerical experiments to assess its influence on the surface pressure distribution; the number of grid points can thus be minimized. The presence of the wall gives rise to an additional line integral on the wall, with appropriate changes in the domain of integration for the field integral.

SOLUTION OF THE UNSTEADY SUBSONIC THIN AIRFOIL PROBLEM

The problem of a thin airfoil subject to simple harmonic disturbances in a uniform subsonic free stream is solved by extension of a technique developed earlier for a stationary strip vibrating in a uniform fluid. Explicit expressions are given for the lift and moment, acoustic directivity pattern, and total acoustic power for arbitrary upwash and, in particular, for the 'elementary disturbances': plunge, pitch and a stationary transverse gust. Numerical results for a simple skewed gust are presented and compared to the high-frequency asymptotic theory of Martinez and Widnall.

Plane transonic solution with shock by direct iteration

Transonic reduced pressure distributions at thin symmetrical nonlifting profiles, with high subsonic free stream velocity are predicted by determining approximate solutions of the Oswatitsch integral equation by direct iteration scheme [1], which is extended here to include shocks. Converged solution with shock may be obtained if the starting solution contains a shock discontinuity or else, if with a continuous starting solution, the expansion shock which appearts at the accelerating sonic point be excluded at each iteration step. Computational results have been presented for a NACA 0012 profile and a parabolic arc profile and compared with other numerical results, which indicate good agreement, particularly for the shock position. A typical profile shape with shock converges in only 16 iteration steps, correct up to two decimal places requiring about 30 secs. of CPU time on a Burroughs B6700 computer system.

Numerical solution of simplified Oswatitsch equation in transonic flow

HE simplified Oswatitsch equation is a one-dimensional nonlinear singular integral equation, occurring in transonic aerodynamics.l It is an approximate version of Oswatitsch's integral equation, whose kernel has a dipole singularity. According to transonic small perturbation theory, the Oswatitsch equation is the basic equation governing the steady inviscid irrotational flow of a perfect gas past a thin symmetric profile at zero incidence, with subsonic freestream Mach number Mx rstrud 4 and Nixon5 deserve special mention. In the present work, the simplified Oswatitsch equation has been solved by two different numerical procedures, viz., the direct iteration scheme (DIS) proposed by Niyogi and Chakraborty6 and by the recent perturbed iterative scheme (PIS) put forward by Dey7 for solving a system of nonlinear algebraic equations. Further, this nonlinear integral equation was used as a test case for studying the global convergence behavior of PIS. From computational results, it has been found that for a parabolic arc profile there exists a range of values for the reduced thickness ratio r (which is a transonic similarity parameter), where both the procedures lead to the same shock-free supercritical solution, and that in this range PIS converges much faster than DIS. However, for higher values of r beyond this range, there exists another range where, contrary to expectations, DIS converges but PIS fails to converge, indicating that DIS has a wider range of convergence.

Harmonic analysis of unsteady transonic flow

Introduction T time averaged mean flow past an airfoil oscillating harmonically at transonic speeds differs from the steady flow obtained at rest. This difference depends on oscillation frequency and amplitude and arises because both mean and disturbance flowfields interact nonlinearly. For small oscillation amplitudes a time linearized description usually suffices: the mean flow is conveniently solved without reference to the unsteady flow and the harmonic flow is solved without explicit reference to the unsteady disturbance amplitude. For larger amplitudes, however, the wave backinteraction induced by the primary disturbance on the mean flow cannot be discounted because the modified mean flow in turn affects the evolution of the harmonic flowfield. This effect may be significant because transonic flows are inherently nonlinear. Linear theory, which generally assumes small unsteady disturbance amplitudes in comparison to the overall thickness, expands the unsteady solution about the known transonic flow corresponding to the stationary airfoil. For oscillatory motions the mean flowfield is therefore defined by a simplified solution rendered independent of disturbance frequency and amplitude: the model implicitly assumes high-frequency oscillations where the mean flow and mean shock position effectively freeze in space. However, for low-frequency motions where large shock excursions are anticipated, the mean flow couples more strongly with the unsteady flow and linear theory may not apply. Thus the practical need arises for a rational harmonic formulation adaptable to unsteady transonic flutter and aeroelastic analyses yielding unstable amplitude as well as frequency boundaries. In this Note, a nonlinear harmonic approach to general unsteady oscillations in transonic flow is developed for those engineering applications where explicit amplitude and frequency information is required. The extent to which nonlinear feedback is significant is addressed, in particular, for the symmetric NACA 64A006 airfoil, unpitched, with a quarter chord oscillating flap executing large amplitude deflections in a subsonic freestream mildly to strongly supercritical. Details of the numerical algorithm are also outlined.

Comment on "Calculation of Rotor Impedance for Articulated-Rotor Helicopters in Forward Flight"

References Kuhlman, J.M. and Warcup, R.W., Investigation of Jet-Induced Loads on a Flat Plate in Hover Out-of-Ground Effect, NASA CR-159004, Feb. 1979. Margason, R.J., Review of Propulsion-Induced on Aerodynamics of Jet/STOL NASATN D-5617, Feb. 1970. Gentry, G.L. and Margason, R.J., Induced Lift Losses on VTOL Configurations Hovering In and Out of Ground Effect, NASA TN D-3166, Feb. 1966. Kuhlman, J.M. and Warcup R.W., Effects of Jet Decay Rate on Jet-Induced Loads on a Flat Plate, Journal of Aircraft, Vol. 15, May 1978. pp. 293-297. Kuhlman, J.M., Ousterhout, D.S., and Warcup, R.W., Investigation of Effect of Jet Decay Rate on Jet-Induced Pressures on a Flat Plate, NASA CR-2979, April 1978. Kuhlman, J.M., Ousterhout, D.S., and Warcup, R.W., Investigation of of Jet Decay Rate on Jet-Induced Pressures on a Flat Plate: Tabulated Data, NASA CR-158990, Nov. 1978. Keffer, J.F. and Baines, W.D., The Round Turbulent Jet in a Cross-Wind, Journal of Fluid Mechanics, Vol. 15, Pt. 4, April 1963, pp. 481-497. Kamotani, Y. and Greber, I., Experiments on a Turbulent Jet in a Cross Flow, NASA CR-72893, June 1971. Fearn, R.L. and Weston, R.P., The Induced Pressure Distribution of a Jet in a Crossflow, NASA TN D-7916, July 1975. Fearn, R.L., and Weston, R.P., Vorticity Associated with a Jet in a Cross Flow, AIAA Journal, Vol. 12, Dec. 1974, pp. 1666-1671. 11 Ousterhout, D.S., An Experimental Investigation of a Cold Jet Emitting from a Body of Revolution into a Subsonic Free Stream, NASA CR-2089, Aug. 1972. Ziegler, H. and Wooler, P.T., Analysis of Stratified and Closely Spaced Jets Exhausting into a Crossflow, NASA CR132297, Nov. 1973. Moller, P.S. and Elliott, R.L., Base Drag of a Thick Annular Jet, Journal of Aircraft, Vol. 9, July 1972, pp. 451-455.

Unsteady Thin Airfoil Theory for Transonic Flows with Embedded Shocks

A simple modification of classical unsteady thin airfoil theory is presented which accounts for the presence and induced motion of the embedded partial chord shock waves which appear at supercritical transonic Mach numbers. The basic model is found to be divergent at zero frequency because of an unbounded growth of the shock excursion amplitude. This behavior is eliminated by introducing a term associated with the mean flow Mach number gradient near the shock. Numerical results are given for the loads induced by an oscillating flap. LASSICAL unsteady thin airfoil theory fails for low frequencies at the subsonic freestream Mach number at which local supersonic flow first occurs. The main cause of this sudden failure is the formation of a shock wave, which shields the forward region of the airfoil from aft generated disturbances. In the present study,l a simple modification of classical thin airfoil theory is given which accounts for the presence and induced motion of such shocks. Predicted airloads are shown to be in favorable agreement with both experimental observations and finite difference calculations. We assume that the unsteady flow is generated by in- finitesimal harmonic oscillations of an airfoil which is suf- ficiently thin and at sufficiently small mean angles of attack that the transonic small disturbance approximation is valid. The unsteady problem can then be linearized about the steady flow, taking due account of the (small amplitude) displacement of any embedded shocks. The linearized un- steady problem has been formulated for an arbitrary three dimensional planar lifting surface in an earlier paper.2 The linearized equations of motion depend through various coefficients on the steady local Mach number M0 (jc), which in general varies in some complicated way throughout the flowfield. Unless the functional form of M0 is very simple, the boundary value problem for the unsteady flow field must be solved by purely numerical techniques. This has been attempted using finite difference methods by Weatherill et al.3 and Traci et al. 4 However, neither of these studies properly accounted for the between the embedded shocks and the unsteady disturbance field. (Of course, this issue does not arise in nonlinear formulations of the unsteady problem, in which the interaction is implicit; flow with oscillating shocks has been successfully computed from the nonlinear small disturbance equation by, for example, Ballhaus and Goor jian.5) An alternative to direct numerical methods is to ap- proximate the steady Mach number distribution M0(x) in such a way that analytical or semianalytica l methods can be applied. Classical theory, where M0 is set equal to the freestream Mach number M^, is, of course, one example. The

Calculation of Separated Turbulent Flows on Axisymmetric Afterbodies Including Exhaust Plume Effects

A calculative method is presented for turbulent boundary-layer, inviscid flow interactions for axisymmetric configurations of the type used for isolated nozzle afterbody models. The method is applicable to flows with subsonic freestreams, including slightly supercritical flows, and to bodies with high-pressure exhaust plumes or solid plume simulators. Integral boundary-layer and plume entrainment methods are coupled iteratively with a finite-difference inviscid flow method through a boundary-layer plume displacement thickness. Results are presented which indicate that the procedure provides an accurate engineering prediction method for boattail flowfield with moderately underexpanded exhaust flows and for bodies with solid plume simulators.

A Far-Field Matching Method for Transonic Computations

In solving a mixed-type (elliptic-hyperbolic) differential equation in an unbounded region, which is elliptic near infinity, some way must be found to transfer the boundary conditions at infinity to a finite artificial boundary in order to keep the discretized problem finite. The common example of this is transonic flow over an airfoil or wing with subsonic freestream. Here we present an approach which is in many ways analogous to the adaptive wind-tunnel wall concept. Iterative revision of a Dirichlet condition on the common or boundary of the near and far fields results in convergence to a far-field solution that matches the discretized near-field solution in potential and normal derivative across the matching boundary. The far-field equation is either a first-order (FO) Prandtl-Glauert, or a second-order (SO) Poisson-type approximation to the transonic equation. A parameter is easily calculated which gives a good estimate of the accuracy of the far-field solution in either case. Two-dimensional results are given showing the success of the method in reproducing the circulation and Cp for a lifting airfoil. Accurate solutions are given using far-field matching boundaries which are much closer to the airfoil than is permissible with Klunker-type far fields based on multipole expansions. The results are shown to be invariant with the location of the vortex representing the far-field circulation. Thus, we significantly reduce computer time by factors of 3 (FO) and 7 (SO) for mesh density and accuracy equivalent to those of a fixed asymptotic far-field representation. Nonlifting FO calculations for a three-dimensional rectangular wing similarly yield accurate results for a much reduced near field, cutting computer time by more than a factor of 2 in an unoptimized case where the minimum boundary size has not yet been established.

Subsonic loads on wings having sharp leading edges and tips

A vortex-lattice method for predicting the aerodynamics of wings having separation at the sharp edges in incompressible flows is extended to compressible subsonic flows using a modified Prandtl-Glauert transformation. Numerical results showing the effect of freestream Mach number on the aerodynamic coefficients are compared with available experimental data for several planforms. It is shown that the proposed method is suitable for predicting the aerodynamic loads on low-aspect wings at moderate angles of attack for high subsonic freestream Mach number. The method is limited to angles of attack up to 12 deg for high subsonic freestream Mach number and to angles of attack up to 20 deg for Mach number not exceeding 0.5.

Transonic Flow about Two-Dimensional Airfoils by Relaxation Procedures

A successive over-relaxation procedure has been developed to treat the inviscid, fully nonlinear, potential equations governing compressible flow. The procedure is designed to compute embedded shock waves and adjust to the Kutta condition during the course of the iterations. Adjustment to the Kutta condition, convergence rate control and other data handling are accomplished in a practical fashion by means of interacting graphics provided by a cathode-ray tube coupled to the computer. The procedure has been applied to blunt-nosed, lifting airfoils in a subsonic freestream. Calculated results are compared to exact theories and experiment.

Calculation of plane steady transonic flows

Transonic small disturbance theory is used to solve for the flow past thin airfoils including cases with imbedded shock waves. The small disturbance equations and similarity rules are presented, and a boundary value problem is formulated for the case of a subsonic freestream Mach number. The governing transonic potential equation is a mixed (elliptic-hyperbolic) differential equation which is solved numerically using a newly developed mixed finite difference system. Separate difference formulas are used in the elliptic and hyperbolic regions to account properly for the local domain of dependence of the differential equation. An analytical solution derived for the far field is used as a boundary condition for the numerical solution. The difference equations are solved with a line relaxation algorithm. Shock waves, if any, and supersonic zones appear naturally during the iterative process. Results are presented for nonlifting circular arc airfoils and a shock free Nieuwland airfoil. Agreement with experiment for the circular arc airfoils, and exact theory for the Nieuwland airfoil is excellent.

Structure of the laminar ablating air-Teflon boundary layer

Abstract : Radiation profiles in an ablating flat plate air-teflon laminar boundary layer were studied both experimentally and theoretically. The experiments were conducted in a one atmosphere, 3000 - 6000K, subsonic free stream produced by an arc jet. Spatially resolved radiation profiles within the boundary layer were obtained in both the visible and infrared. The major radiation in the visible and near ultraviolet wavelengths was the CN violet. In the infrared, the major radiators were CO, CO2, NO and COF2. The theory which was developed to predict the structure included coupling of the heat and mass transfer at the Teflon surface. A 'partial equilibrium' model for the teflon-air chemistry was utilized, which does not allow the formation of CF3 and CF4 within the boundary layer. For the selected wavelengths in the infrared the theory generally predicted quite well both the spatial location and magnitude of the peak radiation, and also predicted the integrated radiation across the layer to within a factor of two. For the selected wavelength in the visible, the radiation intensity, which comes from CN, is much larger than predicted, indicating that the CN is not in thermodynamic equilibrium. (Author)

Solution of the transonic airfoil problem through parametric differentiation.

The difficulties associated with the nonlinearity of the transonic small disturbance equation are avoided by considering the linear problem governing the rate of change of the flow velocity with respect to the airfoil thickness ratio parameter and integrating solutions of this problem over the thickness ratio. This procedure is equivalent to summing a series of perturbations in airfoil thickness and has the advantage of moving all the nonlinearity to a firstorder ordinary differential equation, where it causes little difficulty. Solutions for nonlifting airfoils exhibit all the commonly observed features of transonic flow, such as the formation and movement of a shock on the airfoil surface and the presence of a detached shock upstream of the airfoil for low supersonic Mach numbers. Numerical results are given for subsonic and supersonic flow and for flows at Mach number unity which are in general agreement with other theoretical results and with experiment. Preliminary results for mixed flows with subsonic freestream Mach numbers indicate a rearward movement of the shock impinging on the airfoil surface with increasing airfoil thickness and exhibit the commonly observed rapid acceleration behind the shock. Extensions to lifting and unsteady flows are also discussed, as well as possible applications of the method to other problems in fluid mechanics.