Showing papers in "Journal of the Aeronautical Sciences in 1956"

On Turbulent Flow Near a Wall

Abstract: y* h k* a* A* = = = = = = = = = = = = = = = = = = = = = = = = = = = = distance from wall pipe radius pipe diameter mean local velocity parallel to wall velocity fluctuations parallel and normal to flow mass density coefficient of viscosity shear stress velocity correlation coefficient mixing length universal constant in I = Ky modified universal constant eddy viscosity size of roughness friction factor = 8rw/p V 2

Piston Theory-A New Aerodynamic Tool for the Aeroelastician

Abstract: Representative applications are described which illustrate the extent to which simplifications in the solutions of high-speed unsteady aeroelastic problems can be achieved through the use of certain aerodynamic techniques known collectively as "piston theory." Based on a physical model originally proposed by Hayes and Lighthill, piston theor}^ for airfoils and finite wings has been systematically developed by Landahl, utilizing expansions in powers of the thickness ratio 8 and the inverse of the flight Mach Number M. When contributions of orders 8/M and 8/M are negligible, the theory predicts a point-function relationship between the local pressure on the surface of a wing and the normal component of fluid velocity produced by the wing's motion. The computation of generalized forces in aeroelastic equations, such as the flutter determinant, is then always reduced to elementary integrations of the assumed modes of motion.

System Mode Shapes in the Flutter of Compressor Blade Rows

Abstract: The problems associated with the prohibitive number of possible system modes for a fluttering compressor or turbine blade row are eliminated by the development that comprises the present report. The existence and uniqueness of extremely simple system flutter modes are proved for blade rows consisting of identical blades equally spaced about a common rotor. These simple system modes, if properly interpreted, have the effect of reducing by a factor of n the number of degrees of freedom necessary to analyze an w-bladed configuration. Stated differently, the system of n blades may be considered, with no loss of generality whatsoever, in terms of a single "equivalent blade." The proof holds under any type of flow and any and all types of interblade coupling, so long as a linear analysis is permissible. Moreover, since it is the flutter-inception point that is of interest in predicting critical velocity or rotational speed, it may well be that the conclusions developed apply even to the onset of stall flutter. Practical application of the method to stall-flutter calculations would, of course, require the availability of aerodynamic stallflutter coefficients. The development is carried out first under the assumption of infinite rotor inertia or, in other words, constant rotor velocity. This restriction is then relaxed, and the treatment is expanded to permit torsional oscillations of the rotor itself. I t is proved that under certain conditions the assumption of infinite rotor inertia introduces no error whatsoever.

Some Recent Developments in Airfoil Theory

Abstract: Several recent developments in airfoil and wing theory have as thjeir goals the extension of classical methods to account for characteristically viscous phenomena. Airfoil theory has always recognized the existence of such phenomena as explanations of the presence of circulation and vortex wakes; these new investigations are attempts to include detailed descriptions in theoretical models or to extend classical models into areas of strong viscous effects, such as separation and stalling. Some of these studies follow directly from suggestions made by von Karman, and others are reminiscent of his earlier research. This review is concerned with investigations in four categories: (1) the theory of profiles with boundary layers in steady flow, (2) the theory of profiles with boundary layers in unsteady flow, including extensions of unsteady airfoil theory, (3) the theory of wings with leading-edge separation, and (4) Prandtl wing theory applied to partially stalled wings.

Effect of Wind-Tunnel Contraction on Free-Stream Turbulence

Abstract: The effect of wind-tunnel contraction on free-stream turbulence is determined by passing a well-defined turbulence through three contractions of ratios 4 : 1 , 9 :1 , and 16:1. Turbulent velocity measurements show that, in absolute magnitudes, the longitudinal component decreases and the lateral component increases as the flow accelerates through the contraction. Postand precontraction spectra are measured to get an idea about the distortion of turbulence structure.

An Investigation on Fully Developed Turbulent Flows in a Curved Channel

Abstract: Measurements of turbulent properties have been made on a fully developed, plane incompressible flow in a curved channel between circular, concentric walls Comparison with the flow in a straight channel of the same Reynolds Number shows strong influence of curvature of the mean flow on mean velocity, turbulence intensity, scale, and spectrum The influence of curvature is discussed on the basis of turbulent energy equations

The Determination of Temperature, Stresses, and Deflections in Two-Dimensional Thermoelastic Problems

Abstract: An analytical successive-approximation method for the solution of linear partial differential equations is presented first in general terms, and then applied to the solution of two-dimensional heat and thermal stress problems. The method is applicable when solutions are desired for bars or plates—i.e., for bodies with one dimension small compared to the others. The final expressions given by this procedure for example for the stress aconsist of a number of terms (a= So-^), where the term at is proportional to the quantity [/3 d r/(d:x;)~']; (3 is the ratio of height to length of the bar, and x measures the distance along the span. The first term of the series in that corresponding to the assumption that sections plane before heating remain plane after heating; the solution obtained thus shows that this term provides a good approximation for thermal loadings varying smoothly along the span, and for thin bars. Similar results are obtained for the temperature and the deflections. Explicit formulas for the calculation of stresses and deflections are given. The validity of the Bernoulli-Euler hypothesis of beam-theory is examined. Illustrative examples are presented for all the above developments. The use of the method in problems in which the material properties are functions of the temperature is outlined.

Spectral Theory of Buffeting and Gust Response: Unification and Extension

Abstract: The statistical approach to the dynamic response of aircraft in turbulent air is generalized and extended. The basic element is the response of the airplane to flight through an inclined sinusoidal upwash pattern in (a) three, (b) two, or (c)—without inclination-—one dimension. Case (a) is general; in case (b) the airplane is idealized as a lifting surface or a lifting line; in case (c), as a lifting point. Earlier results of Liepmann are recovered in the cases of a lifting line and a lifting point. An extension of the case of a lifting line is made for the sweptback lifting line, and an extension of the case of a lifting point to provide rolling moments due to spanwise upwash gradients. The general case (a) and the lifting surface in case (b) are new.

On Heat Transfer in Laminar Boundary Layers at High Prandtl Number

Abstract: A study of the asymptotic behavior of the heat transfer in laminary boundary layers for large Prandtl Number is presented. The analysis is based on the observation that, for large Prandtl Number, the conduction term in the energy equation must be important only in a region that is very narrow compared with the velocity boundary layer, but that, in that region, both convection and conduction are essential. A transformation (based on the Prandtl Number) of the coordinate normal to the body surface leads to a form of the energy equation in which the appropriate convection and conduction terms can be balanced with respect to their asymptotic dependence on the Prandtl Number, and the behavior of the local heat-transfer coefficient can be deduced immediately. The general method is applied to the well-known problems of forced convection over a body and of the plate thermometer, where it confirms results previously obtained by other methods. The problems of natural convection over a vertical body, of the flow above a rotating disc, and of the Hamel converging channel flow are then treated; new theoretical results are obtained for these cases.

Heat Transfer in Turbulent Shear Flow

Abstract: A new and relatively simple description is proposed for the velocity profile in turbulent flow close to a smooth wall. Heat transfer coefficients are calculated from the description and are shown to agree better with experiment than other theories. The analysis is extended to transport processes in liquids where the viscosity has a large variation close to the wall.

A Creep Buckling Hypothesis

Abstract: Published work on creep buckling has implied that failure of columns after a critical time is caused by initial imperfections. Such analyses are relatively complex and ultimately leave the choice of selecting the proper value of the initial imperfection to the designer. Furthermore, recent test results on creep buckling of columns have indicated that there is a random and relatively unimportant effect of small initial imperfections on the critical time. To avoid the difficulties associated with initial imperfections, a formulation of the creep buckling phenomenon in terms of classical stability theory is presented. The theorj^ permits the extension of known solutions for plastic buckling of certain thin plates and shells to creep buckling problems.

The Development of Turbulent Boundary Layers

Abstract: The structure of several recently measured sets of velocity profiles in quasi-two-dimensional turbulent boundary layers is examined. I t is shown that the inner one-fifth of each profile is well represented, independently of pressure gradient, by the universal logarithmic law that was observed by Ludwieg and Tillmann. An adequate expression for the velocity in the outer four-fifths of the layer is provided by the well-known power law u/U = (y/8). The significance of the composite type of profile obtained from the two laws is discussed in the light of Townsend's work on the eddy structure of the boundary layer on a flat plate. The logarithmic expression for velocity is substituted into the equation of motion at height y = 0 from the surface, at which height it is still valid. (0 — momentum thickness.) With an approximation suggested by Schubauer and Klebanoff's measurements for the term that involves shear stress, this leads to an equation for the variation of the form parameter 7 = (n/U)y = o. 7 is a simple function of the usual parameter H = 8*/0, and the equation could be put into the form

Boundary-Layer Measurements in Hypersonic Flow

Abstract: Experimental data are presented on boundary-layer formation, heat transfer, and skin-friction coefficient at Mach Numbers of 8.25 and 9.0. The boundary-layer measurements were made on the wall of a conical nozzle in the presence of a favorable pressure gradient and several rates of heat transfer. The Reynolds Number based on momentum thickness varied from 1,500 to 3,500. Comparison is made with data at lower Mach Numbers and with the semiempirical theory of von Karman. The existing data up to Mach Numbers of nine indicate agreement to within 5 per cent when compared with a form of the Wilson theory, but it is clear that the effects of heat transfer and pressure gradients present problems which require extensive study and experiment in the future.

Viscous Flow Along a Flat Plate Moving at High Supersonic Speeds

Abstract: By the distortion of coordinates, it is shown that, in the case of supersonic viscous flow past a flat plate, the boundary-layer and simple wave theories can be combined to give a complete representation of the velocity and pressure fields. Consistent first-order solutions are considered. An expression for the in duced pressure on the plate, correct to the second order, is obtained. At high Mach Numbers the important parameter satisfies the hypersonic similarity law; and for arbitrary Mach and Reynolds Numbers and for different gases, the theoretical curve correlates closely the experimental data. Asymptotic shock curve and skin-friction coefficient are also deduced, but the experimental verifications are yet to be made.

The Dynamic Response of a Large Airplane to Continuous Random Atmospheric Disturbances

Abstract: The statistical approach to the gust-loads problem, which consists in considering flight though turbulent air to be a stationary random process, is extended by including the effect of lateral variations of the instantaneous gust intensity on the aerodynamic forces and on the resultant motions and stresses of rigid and flexible airplanes. By means of some calculations of normal and rolling accelerations, as well as of the root bending moment, it is shown that these effects may be significant for large airplanes.

The Behavior of Transverse Cylindrical and Forward Facing Total Pressure Probes in Transverse Total Pressure Gradients

Abstract: When a total pressure probe is used for measuring flows with transverse total pressure gradients, a displacement of the effective center of the probe toward the higher total pressure is sometimes observed. This paper gives the results of an investigation of the effect for certain transverse cylindrical and forward facing (or pitot type) total pressure probes. An estimate of the error is given for the transverse cylindrical type of probe, and the errors due to position of hole, depth of hole, and wall proximity with this type of probe are considered incidentally. A design of a probe of the forward facing type possessing negligible displacement error is suggested.

Heat transfer to an incompressible turbulent boundary layer and estimation of heat-transfer coefficients at supersonic nozzle throats

Abstract: An approximate solution is obtained for the heat transfer to an incompressible turbulent boundary layer with arbitrary freestream-velocity and surface-temperature distributions. The analysis is based on the simultaneous solution of the boundarylayer momentum and energy integral equations and employs several simplifying assumptions, the most important of which is the approximation that the boundary-layer velocity and totaltemperature profiles and the skin-friction coefficient are independent of pressure gradient. Solutions are given for twodimensional and axisymmetric flows. Although the flow in supersonic nozzles is not incompressible, an adaptation of the aforementioned result, termed the quasiincompressible solution, is suggested for calculating the heat transfer to such nozzles. In order to estimate rapidly the maximum rate of heat transfer which occurs near the throat of a supersonic nozzle, a throat approximation is developed which involves only the nozzle supply conditions, throat opening, and throat radius of curvature. The application of these solutions to a sample nozzle gave the following results using the quasi-incompressible analysis: (1) The heat-transfer coefficients based on solution of both the momentum and energy equations were between 0 and 15 per cent lower than the values based on a solution of the momentum equation alone, and (2) the heat-transfer coefficients that included the effects of variable wall temperature were between 0 and 15 per cent lower than the values obtained assuming constant wall temperature. The throat-approximation, heat-transfer coefficients were within 10 per cent of the corresponding quasiincompressible values.

Thermal Buckling of Supersonic Wing Panels

Abstract: WH E N THE SURFACE of a multicellular supersonic wing is heated by the air flowing in the boundary layer (see Fig. 1), the temperature of the cover plates rises and the cover plates expand. This expansion is resisted by the shear webs which remain comparatively cool as they are not in contact with the boundary layer and are heated only by the cover plates through conduction and possibly through convection and radiation. T h e resistance of the shear webs sets up compressive stresses in the cover plates and tensile stresses in the shear webs. Under the compressive stresses so arising, the cover plates have been observed to buckle. Buckling of the cover plates is objectionable not only for structural reasons bu t also because the buckled, wavy surfaces alter the flow of air, lead to losses in aerodynamic efficiency, and may even cause aerodynamic disturbances resulting in failure. I t is important , therefore, for the airplane designer to be able to predict the onset of thermal buckling—that is, buckling caused by stresses arising from a nonuniform heating of the structure. In multicellular wing structures, the variation of the temperature through the thickness of the cover plates is usually not pronounced, a t least not when the speed of flight does not exceed Mach Numbers of 3 or 4. Similarly, with a wing of normal aspect ratio, the variation in the spanwise direction may be disregarded in order to simplify the analysis. However, the heating rate generally varies noticeably in the chordwise direction and with time. As all possible variations in flight pa th and construction cannot be considered simultaneously when formulas to be used in practical work are developed, i t is assumed in the present analysis that , over one cell of the wing and during the period of time considered, the heat inpu t q from the air to the cover plates is constant. In an actual problem when q is a variable, substitution of its average value into the formulas here derived should give an indication of the danger of buckling. Hea t transfer from cover plate to web through the interior of

The flow over a body in a choked wind tunnel and in a sonic free-jet

Abstract: The pressure distribution over a double wedge airfoil under free flight conditions with Mach number one is compared with the pressure distribution over the same airfoil in a choked closed wind tunnel and in a sonic free-jet. The computation is carried out as a development with respect to a parameter which indicates the deviation from free flight conditions with Mach number one. The results are of interest for the question of wind tunnel wall influences. It is found that the deviatiors of the pressure distribution for a sonic free-jet from the distribution in an infinite air flow are somewhat larger than the deviations in a closed wind tunnel under choked flow conditions. For a specific exam of a wedge of a length of 13% of the tunnel height and a thickness ratio of 10%, the deviation of the pressure distribution does not go much beyond the usual experimental scatter. The results are quite encouraging for the application of closed throat wind tunnels in transonic testing although the axial symmetric case may not show entirely the same desirable behavior.