This paper concerns the flow about a sphere placed in a weak shear flow of an inviscid fluid. The secondary velocity resulting from advection of vorticity by the irrotational component of the flow is computed on the sphere surface, and on the upstream axis. The resulting lift force on the sphere is evaluated, and the result is confirmed by an analytical far-field calculation. The displacement of the stagnation streamline, far upstream of the sphere, is calculated more accurately than in previous papers.

The lift force on a spherical body in a rotational flow

A theoretical study is made of shearing flows bounded by a simple-harmonic wavy surface, the main object being to calculate the normal and tangential stresses on the boundary. The type of flow considered is approximately parallel in the absence of the waves, being exemplified by two-dimensional boundary layers over a plane. Account is taken of viscosity; but, as the Reynolds number is assumed to be large, its effects are seen to be confined within narrow ‘friction layers’, one of which adjoins the wave and another surrounds the ‘critical point’ where the velocity of flow equals the wave velocity. The boundary conditions are made as general as possible by including the three cases where respectively the boundary is rigid, flexible yet still solid, or completely mobile as if it were the interface with a second fluid.The theory is developed on the model of stable laminar flow, although it is proposed that the same theory may usefully be applied also to examples of turbulent flow considered as ‘pseudo-laminar’ with velocity profiles corresponding to the mean-velocity distribution. Use is made of curvilinear co-ordinates which follow the contour of the wave-train. This admits a linearized form of the problem whose validity requires only that the wave amplitude be small in comparison with the wavelength, even when large velocity gradients exist close to the boundary. The analysis is made largely without restriction to particular forms of the velocity profile; but eventually consideration is given to the example of a linear profile and the example of a boundary-layer profile approximated by a quarter-period sinusoid. In § 7 some general methods are set out for the treatment of disturbed boundary-layer proses: these apply with greatest precision to thin boundary layers, but are also useful for the initially very steep but on the whole fairly diffuse profiles which occur in most practical instances of turbulent flow over waves.The phase relationships found between the stresses and the wave elevation are discussed for several examples, and their interest in connexion with problems of wave generation by wind is pointed out. It is shown that in most circumstances the stresses are distributed in much the same way as if the leeward slopes of the waves were sheltered. For instance, the pressure distribution often has a substantial component in phase with the wave slope, just as if a wake were formed behind each wave crest—although of course actual separation effects are outside the scope of the present theory. In this aspect, the analysis amplifies the work of Miles (1957).

Shearing flow over a wavy boundary

Stagnation laminar heat transfer at hypersonic speeds depends on the rate of recombination of the dissociated air behind the detached shock wave. This paper is concerned with the case of a large recombination t ime compared to the t ime of diffusion across the boundary layer. The conditions of existence of such a "frozen flow" and i ts coupling with the dissociation lag behind the shock are discussed. In order to account for finite catalytic recombination rates at the wall, a nonsimilar boundary condition is introduced which can be reduced to similarity for stagnation flow only. In this latter case, Lees' ( l ) 3 and Fay and Ridell's (2) heat transfer solutions are shown to correspond to the l imit ing case of an infinitely fast catalyst. The validity of their solutions is extended to the general case of a wall of finite catalytic efficiency, by introducing a correction factor (p. This factor is a s imple function of the flight condit ion, nose geometry and the wall catalytic recombinat ion rate constant . For a given nose material , the percentage of the heat transfer by catalysis is found to in crease wi th the velocity, the nose diameter and the wall temperature and to decrease with alt itude. Finally, the experimental values obtained for the catalytic recombination rates of oxygen and nitrogen atoms on various surfaces i l lustrate numerically the importance of the nature of the wall on the catalytic heat transfer to a missi le nose . In particular, the superiority of pyrex over metal l ic surfaces stresses the need for more experimental values for glassy and ceramic coatings.

On Catalytic Recombination Rates in Hypersonic Stagnation Heat Transfer

H YPERSONIC flight began in February 1949 when a WAC Corporal rocket was ignited from a U.S.-captured V-2 rocket [1]. In the six decades since this milestone, there have been significant investments in the development of hypersonic vehicle technologies. The NASA X-15 rocket plane in the early 1960s represents early research toward this goal [2,3]. After a lull in activity, the modern era of hypersonic research started in the mid-1980s with the National Aerospace Plane (NASP) program [4], aimed at developing a single-stage-to-orbit reusable launch vehicle (RLV) that used conventional runways. However, it was canceled due mainly to design requirements that exceeded the state of the art [1,5]. A more recent RLV project, the VentureStar program, failed during structural tests, again for lack of the required technology [5]. Despite these unsuccessful programs, the continued need for a low-cost RLV, as well as the desire of the U.S. Air Force (USAF) for unmanned hypersonic vehicles, has reinvigorated hypersonic flight research. An emergence of recent and current research programs [6] demonstrate this renewed interest. Consider, for example, the NASA Hyper-X experimental vehicle program [7], the University of Queensland HyShot program [8], the NASA Fundamental Aeronautics Hypersonics Project [9], the joint U.S. Defense Advanced Research Projects Administration (DARPA)/USAF Force Application andLaunch fromContinentalUnited States (FALCON) program [10], the X-51 Single Engine Demonstrator [11,12], the joint USAF Research Laboratory (AFRL)/Australian Defence Science and Technology Organisation Hypersonic International Flight Research Experimentation project [13], and ongoing basic hypersonic research at the AFRL (e.g., [14–20]). The conditions encountered in hypersonic flows, combined with the need to design hypersonic vehicles, have motivated research in the areas of hypersonic aeroelasticity and aerothermoelasticity. It is evident from Fig. 1 that hypersonic vehicle configurations will consist of long, slender lifting body designs. In general, the body, surface panels, and aerodynamic control surfaces are flexible due to minimum-weight restrictions. Furthermore, as shown in Fig. 2, these

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Aeroelastic and Aerothermoelastic Analysis in Hypersonic Flow: Past, Present, and Future

This paper is a study of the flow of a viscous incompressible fluid in the immediate neighbourhood of a saddle point of attachment, near which the external flow is irrotational with components {ax, by, −(a + b)z}, where a > 0 > b. It is shown that the flow is of a boundary-layer character, and that part of the boundary-layer flow is reversed when b/a < − 0·4294.On the assumption that such flows are physically plausible, the problem may be solved for all values of b/a [ges ] −1. Even in the limiting case b/a = −1, an effect of the boundary layer is everywhere to draw fluid towards the wall, so that vorticity is still convected towards the wall.Numerical solutions have been computed, and some of the results are presented in the tables and diagrams.

Boundary-layer flow at a saddle point of attachment

Atmospheric dissociation will be appreciable in the neighbourhood of projectiles travelling at speeds greater than 2 km/sec. This introductory paper on possible effects of dissociation on the airflow, and hence on the drag, stability and aerodynamic heating of such projectiles, is intended mainly as a source of ideas for later, more comprehensive investigations.

Dynamics of a dissociating gas Part I Equilibrium flow

Four alternative theoretical treatments of ‘displacement thickness’, and, generally, of the influence of boundary layers and wakes on the flow outside them, are set out, first for two-dimensional, and then for three-dimensional, laminar or turbulent, incompressible flow. They may be called the methods of ‘flow reduction’, ‘equivalent sources’, ‘velocity comparison’ and ‘mean vorticity’.The principal expression obtained for the displacement thickness δ1 in three-dimensional flow may be written if, as orthogonal coordinates (x, y) specifying position on the surface, we choose x as the velocity potential of the external flow, and y as a coordinate, constant along the external-flow streamlines, such that hy dy is the distance between (x, y) and (x, y + dy); and if also δx and δy are the streamwise and transverse ‘volume-flow thicknesses’ z is the distance from the surface, u and v are the x and y components of velocity, and u takes the value U just outside the boundary layer.

On displacement thickness

The fundamental solution for small steady three-dimensional disturbances to a two-dimensional parallel shear flow

In an earlier paper the uniform shear flow past a sphere was studied, by investigating how vortex lines are deformed by the ‘primary flow’ (flow in the absence of shear), and deducing the ‘secondary’ vorticity field (first approximation for small shear). In another paper the image system associated with each element of secondary vorticity was found, whence the Biot-Savart law can be used to determine the secondary flow field by integration. The integration is here carried out for the ‘downwash’ (secondary flow component perpendicular to the undisturbed flow, down the velocity gradient) on the dividing streamline. Difficulties due to the infinite domain of integration and singularities of the integrand are overcome by selecting variables of integration carefully and using known analytical properties of the secondary vorticity. From the computation of downwash is inferred the first approximation (for small shear A) to the ‘displacement’ δ (displacement of the dividing streamline, up the velocity gradient, far upstream of the sphere). If U is the upstream flow velocity and a the radius of the sphere, the computed value of lim (Uδ/Aa2) is 0·9. Details of the calculation show that the secondary trailing vorticity is not an important contributor to the displacement, The downwash is due almost entirely to vorticity upstream of the sphere (Hall's earlier simplified theory gave good results, e.g., 1.24 instead of 0.9, because it concentrated on the effect of local vorticity in producing downwash); further, this produces displacement principally through its image vorticity.The relation between theories for a sphere and experimental results on Pitot tubes (beginning with Young & Maas 1936) is discussed. Theoretical evidence on tertiary- and quartary-flow effects is used here in the light of recent work which renders the successive-approximation sequence uniformly valid at infinity. The conclusion is that the theories, taken together, are not inconsistent with the experimental evidence that (i) at values of the ‘shear parameter’ Aa/U at which the displacement is measurable the ratio δ/a seems to have asymptoted to an approximately constant value, and (ii) displacement is greatly reduced in supersonic flow (Johannesen & Mair 1952) or when ‘sharp-lipped’ tubes are used (Livesey 1956).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112057000294

Contributions to the theory of the Pitot-tube displacement effect

Ward's slender-body theory of supersonic flow is applied to bodies terminating in either (i) a single trailing edge at right angles to the oncoming supersonic stream, or (ii) two trailing edges at right angles to one another as well as to the oncoming stream, or (iii) a cylindrical section with two or four identical fins equally spaced round it. The wave drag at zero lift, D , is given by the expression
 $\frac {D}{\frac {1}{2}\rho U^2} &=& \frac {1}{2\pi}\int^l_0 \int^l_0 log\frac{1}{|s-z|}S^{\prime \prime}(s)S^{\prime \prime}(z)dsdz - \\ &-& \frac{S^\prime (l)}{\pi}\int^l_0 log \frac {l}{l-z}S^{\prime \prime}(z)dz + \frac{S^{\prime 2}(l)}{2\pi} \{ log \frac{l}{(M^2-1)^{1|2}b}+k \} $ where l is the length of the body, b the semi-span of the trailing edge (or length of trailing edge of a single fin), and S ( z ) is the cross-sectional area of the body at a distance z behind the apex. The constant k depends on the distribution of trailing-edge angle along the span for each trailing-edge configuration. In case (i) it is 1·5 for a uniform distribution of trailing-edge angle and 1·64 for an elliptic distribution. In case (ii) it is 1·28 for a uniform distribution and 1·44 for an elliptic distribution. Study of case (iii) indicates that interference effects due to the presence of the body reduce the drag of the fins. For example, with a uniform distribution of trailing-edge angle, k for two fins falls from 1·5 in the absence of a body to 1·06 when the body radius equals the trailing-edge semi-span, while k for four fins falls from 1·28 to 0·45 under the same conditions. Where ordinary finite-wing theory is applicable, the present method must agree with it for small $(M^2-1)^{1|2}b|l$ , and this is confirmed by two examples (§3), but within the limit imposed by slenderness the present method is of course more widely applicable, as well as simpler, than finite-wing theory. It is not known experimentally whether slender-body theory gives accurate predictions of drag at zero lift, for the shapes here discussed, under the conditions for which on theoretical grounds it might be expected to do so. It should be noted that, although tests have not yet been made on ideally suitable bodies, no clear the drag is therefore twice that of a wing made up of two of them. The final stages of the process cannot be represented by slender-body theory, but the initial trend may well be indicated fairly accurately.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112056000196

M. J. Lighthill

Papers

Dynamics of a dissociating gas Part I Equilibrium flow

On displacement thickness

The fundamental solution for small steady three-dimensional disturbances to a two-dimensional parallel shear flow

Contributions to the theory of the Pitot-tube displacement effect

The wave drag at zero lift of slender delta wings and similar configurations