Showing papers on "Turbulence published in 1971"

On the nature of turbulence

Abstract: A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.

Orderly Structure in Jet Turbulence

Abstract: Past evidence suggests that a large-scale orderly pattern may exist in the noiseproducing region of a jet. Using several methods to visualize the flow of round subsonic jets, we watched the evolution of orderly flow with advancing Reynolds number. As the Reynolds number increases from order 102 to 103, the instability of the jet evolves from a sinusoid to a helix, and finally to a train of axisymmetric waves. At a Reynolds number around 104, the boundary layer of the jet is thin, and two kinds of axisymmetric structure can be discerned: surface ripples on the jet column, thoroughly studied by previous workers, and a more tenuous train of large-scale vortex puffs. The surface ripples scale on the boundary-layer thickness and shorten as the Reynolds number increases toward 105. The structure of the puffs, by contrast, remains much the same: they form at an average Strouhal number of about 0·3 based on frequency, exit speed, and diameter.To isolate the large-scale pattern at Reynolds numbers around 105, we destroyed the surface ripples by tripping the boundary layer inside the nozzle. We imposed a periodic surging of controllable frequency and amplitude at the jet exit, and studied the response downstream by hot-wire anemometry and schlieren photography. The forcing generates a fundamental wave, whose phase velocity accords with the linear theory of temporally growing instabilities. The fundamental grows in amplitude downstream until non-linearity generates a harmonic. The harmonic retards the growth of the fundamental, and the two attain saturation intensities roughly independent of forcing amplitude. The saturation amplitude depends on the Strouhal number of the imposed surging and reaches a maximum at a Strouhal number of 0·30. A root-mean-square sinusoidal surging only 2% of the mean exit speed brings the preferred mode to saturation four diameters downstream from the nozzle, at which point the entrained volume flow has increased 32% over the unforced case. When forced at a Strouhal number of 0·60, the jet seems to act as a compound amplifier, forming a violent 0·30 subharmonic and suffering a large increase of spreading angle. We conclude with the conjecture that the preferred mode having a Strouhal number of 0·30 is in some sense the most dispersive wave on a jet column, the wave least capable of generating a harmonic, and therefore the wave most capable of reaching a large amplitude before saturating.

The production of turbulence near a smooth wall in a turbulent boundary layer

Abstract: The structure of the flat plate incompressible smooth-surface boundary layer in a low-speed water flow is examined using hydrogen-bubble measurements and also hot-wire measurements with dye visualization. Particular emphasis is placed on the details of the process of turbulence production near the wall. In the zone 0 < y+ < 100, the data show that essentially all turbulence production occurs during intermittent ‘bursting’ periods. ‘Bursts’ are described in some detail.The uncertainties in the bubble data are large, but they have the distinct advantage of providing velocity profiles as a function of time and the time sequences of events. These data show that the velocity profiles during bursting periods assume a shape which is qualitatively distinct from the well-known mean profiles. The observations are also used as the basis for a discussion of possible appropriate mathematical models for turbulence production.

Simple Eulerian time correlation of full-and narrow-band velocity signals in grid-generated, ‘isotropic’ turbulence

Abstract: Space-time correlation measurements in the roughly isotropic turbulence behind a regular grid spanning a uniform airstream give the simplest Eulerian time correlation if we choose for the upstream probe signal a time delay which just ‘cancels’ the mean flow displacement. The correlation coefficient of turbulent velocities passed through matched narrow-band niters shows a strong dependence on nominal filter frequency (∼ wave-number at these small turbulence levels). With plausible scaling of the time separations, a scaling dependent on both wave-number and time, it is possible to effect a good collapse of the correlation functions corresponding to wave-numbers from 0·5 cm−1, the location of the peak in the three-dimensional spectrum, to 10 cm−1, about half the Kolmogorov wave-number. The spectrally local time-scaling factor is a ‘parallel’ combination of the times characterizing (i) gross strain distortion by larger eddies, (ii) wrinkling distortion by smaller eddies, (iii) convection by larger eddies and (iv) gross rotation by larger eddies.

Structural features of turbulent flow over smooth and rough boundaries

Abstract: An experimental study of boundary-layer turbulence in a free surface channel flow is described. Attention is concentrated on the effects of different surface roughness conditions on the turbulence structure in the boundary region. Hydrogen bubble flow tracers and medium high-speed motion photography were used to obtain an instantaneous visual and quantitative description of the flow field. In particular it proved possible to record instantaneous longitudinal and vertical velocity profiles from which distributions of the instantaneous Reynolds stress contribution were computed.Two well-defined intermittent features of the flow structure were visually identified close to the boundary. These consisted of fluid ejection phases, previously reported by Kline et al. (1967) for smooth boundary flow, and fluid inrush phases. Conditional averaging of the instantaneous velocity data yielded quantitative confirmation that ejection phases corresponded with ejection of low momentum fluid outwards from the boundary whilst inrush phases were associated with the transport of high momentum fluid inwards towards the boundary. Inrush and ejection events were present irrespective of the surface roughness condition.Conditional averaging also indicated that both inrush and ejection sequences correlate with an extremely high contribution to Reynolds stress and hence turbulence production close to the boundary. Indeed the present results, taken with those from previous studies, suggest that turbulence production is dominated by the joint contribution from the inrush and ejection events. It is emphasized that these structural features are intermittent, forming important linked elements of a randomly repeating cycle of wall-region turbulence production which is apparently driven by some violent three-dimensional instability mechanism.Whilst the most coherent effects of the observed inrush phases appear to be mainly confined to a region close to the boundary, the influence of the ejection phases is far more extensive. The ejected low momentum fluid elements, drawn from the viscous sublayer and from between the interstices of the roughness elements, travel outwards from the boundary into the body of the flow and give rise to very large positive contributions to Reynolds stress at points remote from the boundary. This effect is sufficiently strong to prompt the suggestion that the ejection process could represent a universal and dominant mode of momentum transport outside the immediate wall region and possibly extending across the entire thickness of the boundary layer.A structural model based on the present observations is seen to exhibit consistency with many commonly visualized features and recorded average properties of turbulent boundary-layer flows in general.

Momentum absorption by vegetation

Abstract: Measurements were made in a wind-tunnel of the drag on elements of a simply-structured artificial crop, and of the wind profiles above and within the crop. Analysis demonstrates (i)that the drag force on an element of such an array can be calculated from the profile of the turbulent shear flow within the array, using the known (and unmodified), wind-tunnel value of the drag coefficient of the individual element; (ii)that the zero-plane displacement (d) of an aerodynamically rough surface can be identified with the level of action of the drag on its elements; and (iii)that von Karman's constant = 0.41 ± 0.03. The relation z0 = 0.36 (h – d) is suggested for the roughness parameter of vegetation of height h. Calculated values of the drag force, f, on unit column of a real stand of beans in the field, using individual-element drag coefficients (Cd) and measured wind speeds, give f = 3.5 τ0 where τ0 is the downward momentum flux derived from the shape of the wind profile above. On the evidence of conclusion (i) and the dense and complex nature of the bean canopy, the factor 3.5 is attributed to mutual sheltering of neighbouring canopy elements rather than as evidence that the Cd – values are modified by turbulent shear flow. For the artificial crop, and for the real crop, recognition of the wind-speed dependence of the individual-element drag coefficients gives values of eddy viscosity, KM, almost constant in the height range h/3 < z ⩽ h and significantly larger than those found when constant drag coefficients are assumed. Constant KM within a crop canopy is consistent with the wind profile u(z)/u(h) = {1 + α(1 – z/h)}−2: an explicit expression is given for the parameter α.

The Budgets of Turbulent Kinetic Energy and Temperature Variance in the Atmospheric Surface Layer

Abstract: Measurements of the shear production, buoyant production, turbulent transport (flux divergence) and dissipation terms in the budget of turbulent kinetic energy, and production and turbulent transport terms in the temperature variance budget are presented. Direct observations of the surface stress and heat flux over a horizontally uniform site enable presentation of the data in terms of surface layer similarity theory. The dissipation term, obtained from differentiated hot-wire anemometer signals, agrees with estimates made from the inertial subrange levels of longitudinal velocity spectra with a value of 0.5 for the spectral constant. Under stable conditions dissipation essentially balances shear production, while turbulent transport and buoyant production are of secondary importance. Under unstable conditions, dissipation slightly exceeds the total production, and energy is also lost at a substantial rate due to upward export by the turbulence. The large imbalance among the measured terms in the...

Local Free Convection, Similarity, and the Budgets of Shear Stress and Heat Flux.

Abstract: Equations for the conservation of Reynolds shear stress and the two components of heat flux (velocity-temperature covariance) in the homogeneous atmospheric surface layer are derived. The behavior of the production and turbulent transport (flux divergence) terms in each budget is determined directly from measurements obtained over a wide range of stability conditions during the 1968 Kansas field program of AFCRL. The data are presented in the dimensionless form suggested by Monin-Obukhoy similarity theory, and follow universal functions quite well. The theory is extended to the “local free convection” regime which exists under very unstable conditions, and specific power law forms are predicted. Several of these are verified and values are given for the proportionality factors in the power laws. The flux divergence terms are small, implying that in each budget the local production and destruction rates are in balance. The third moments which represent the vertical fluxes of stress and heat flux a...

Some measurements of particle velocity autocorrelation functions in a turbulent flow

Abstract: Particle velocity autocorrelations of single spherical beads (46·5 μhollow glass, 87 μ glass, 87 μ corn pollen, and 46·5 μ copper) were measured in a grid-generated turbulence. The hollow glass beads were small and light enough to behave like fluid points; the other types had significant inertia and ‘crossing trajectories’ effects. The autocorrelations decreased much faster for heavier particles, in contradiction to previous experimental results. The integral scale for the copper beads was 1/3 of that for the hollow glass beads. The particle velocity correlations and the Eulerian spatial correlation were coincident within experimental error when the separation was non-dimensionalized by the respective integral scale. The data generated by the hollow glass beads can be used to estimate Lagrangian fluid properities. The Lagrangian time integral scale is approximated by L/u′, where L is the Eulerian integral scale and u′ is the turbulence intensity.

Turbulence in an atmosphere with a non-uniform temperature

Abstract: In this work the effect of static stability on the development of atmospheric turbulence is investigated. This influence is considered quantitatively by generalizing Prandtl's semi-empirical theory, i.e., by using a correcting factor in the form of a universal function of the Richardson number. An evaluation of the thickness of the layer of dynamic turbulence under various external conditions is successfully achieved quantitatively.

Atmospheric Predictability and Two-Dimensional Turbulence

Abstract: Recent observations indicate that the planetary-scale motions of the atmosphere obey some of the laws of two-dimensional turbulence. The eddy-damped Markovian approximation to two-dimensional turbulence is applied to these motions to predict for an observed energy spectrum the nonlinear transfer rates, characteristic error spectra, and the rate of error growth. In this way estimates are derived of the predictability of the atmosphere and of the errors inherent in numerical models. The use of stochastic models for turbulence approximations is described in an Appendix.

The ‘bursting’ phenomenon in a turbulent boundary layer

Abstract: Using a hot wire in a turbulent boundary layer in air, an experimental study has been made of the frequent periods of activity (to be called ‘bursts’) noticed in a turbulent signal that has been passed through a narrow band-pass filter. Although definitive identification of bursts presents difficulties, it is found that a reasonable characteristic value for the mean interval between such bursts is consistent, at the same Reynolds number, with the mean burst periods measured by Kline et al. (1967), using hydrogen-bubble techniques in water. However, data over the wider Reynolds number range covered here show that, even in the wall or inner layer, the mean burst period scales with outer rather than inner variables; and that the intervals are distributed according to the log normal law. It is suggested that these ‘bursts’ are to be identified with the ‘spottiness’ of Landau & Kolmogorov, and the high-frequency intermittency observed by Batchelor & Townsend. It is also concluded that the dynamics of the energy balance in a turbulent boundary layer can be understood only on the basis of a coupling between the inner and outer layers.

Influence of surface roughness on the cross-flow around a circular cylinder

Abstract: The influence of surface roughness on the cross-flow around a circular cylinder is the subject of the present experimental work. The investigations were carried out in a high-pressure wind tunnel, thus high Reynolds numbers up to Re = 3 × 106 could be obtained. Local pressure and skin friction distributions were measured. These quantities were evaluated to determine the total drag coefficient and the percentage of friction as functions of Reynolds number and roughness parameter. In addition the local skin friction distribution yields the angular position of boundary-layer transition from laminar to turbulent flow and the location of boundary-layer separation.

Stability of the flow in a differentially heated inclined box

Abstract: The effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical. Like its two limiting cases, Benard convection and convection in the vertical slot, a basic state which exists for low Rayleigh numbers becomes unstable as this parameter is increased. The types of instability and indeed the manner in which the motions become turbulent depend crucially on δ. In our work with water the following general picture of the primary instabilities applies: For 90° > δ > 10° with the bottom plate hotter, the instabilities are stationary longitudinal convectively driven rolls with axes oriented up the slope. Near δ = 10° there is an upper and lower Rayleigh number cut off. If the Rayleigh number is too small diffusion damps the instabilities, but if it is too large they are damped by the development of a stable upslope temperature gradient in the mean flow.For 10° > δ > −10° (negative angles imply a hotter upper plate), transverse travelling waves oriented across the slope are the first instabilities of the mean flow. They obtain their kinetic energy via the working of the upslope buoyancy force.For - 10° > δ > −85° longitudinal modes are again observed. These are rather curious in that they may exist when the stratification is everywhere positive. The necessary energy for these modes comes out of the mean velocity field and out of the mean available potential energy.Agreement between the stability theory and the experiments is generally quite good over the whole range of δ, considering the approximations involved in finding a suitable basic flow solution.For Rayleigh numbers less than ∼ 106 turbulence is only possible for positive angles. For 85° > δ > 20° the development of unsteadiness involves the occurrence and the breaking of wavy longitudinal vortices in a manner reminiscent of the development of turbulence in cylindrical Couette flow.

The structure of turbulent shear flow

Abstract: A theoretical investigation is made of the mixing layer between two streams. The work is divided into four sections. The first involves the solution of the mean problem of laminar and turbulent mixing. The equations of motion are written in terms of a similarity variable. An eddy viscosity hypothesis is made to describe the shear stresses. The similarity equations for both laminar and turbulent problems are solved numerically by an iterative scheme. The second section examines the stability of the mixing layer. The Orr-Sommerfeld equation of hydrodynamic stability is solved numerically. Doth cases of spatial and temporal amplification are examined. The shear layer is shown to be unstable to both spatially and temporally growing disturbances at all Reynolds numbers. A correction is made for the divergence of the mean flow and leads to a value of critical Reynolds number of 12.3. The third section presents a model for the turbulent mixing layer. A set of partial differential equations describing the flow are obtained. A Fourier transform technique is employed to reduce this set to an ordinary differential equation for the fluctuating flow field. The homogeneous form of this equation is solved numerically. The resulting predictions of fluctuating velocity, pressure and their correlations are compared with measured values. The agreement is good in certain cases and this serves as a guide to components of the flow governed by non-linear processes. The final section examines the non-linear growth of the mixing layer through transition from laminar to turbulent flow. A set of integral momentum and energy equations are written in terms of a number of shape parameters of mean and fluctuating flow. The amplitude of disturbances is shown to grow rapidly and reach a limiting value. A sudden growth of the mixing layer thickness through transition is also predicted. Comparison is made with experimental results.

Wake characteristics of two-dimensional perforated plates normal to an air-stream

Abstract: The flow in the wakes behind two-dimensional perforated plates has been investigated in the Reynolds number range 2·5 × 104 to 9·0 × 104.Measurements of drag and shedding frequency were made and a pulsed hotwire anemometer was used to measure the mean velocity and turbulent intensity variations in the highly turbulent regions immediately behind the plates.The results indicate the existence of two distinct types of flows: one appropriate to high and the other to low values of plate porosity.

The determination of the rate of dissipation in turbulent pipe flow

Abstract: Measurements of turbulence energy diffusion and the spectral distributions of stress components in the core of turbulent pipe flow are presented. The results tend to confirm the proposal of Bradshaw (1967a, b) that an inertial subrange in the spectra can exist at quite modest laboratory Reynolds numbers. They also illuminate the inconsistencies in Laufer's (1954) measurements of dissipation and suggest that the fitting of a −5/3 power law to the spectra may well provide the most accurate method of determining dissipation for Re [gsim ] 105.

Hydraulic Jumps in Turbidity Currents

Abstract: If uniform flow conditions are approached, turbidity currents should be supercritical (Froude number greater than unity) while passing through submarine canyons, and subcritical (Froude number less than unity) in the upper submarine fan channel. The change in flow regime requires that the current pass through a hydraulic jump in the vicinity of the canyon mouth. During the jump, the velocity of the flow would be reduced and its thickness markedly increased. The aspects of such hydraulic jumps are examined. The quantitative analysis of hydraulic jumps in turbidity currents is based upon solutions of the momentum flux equation and the equation resulting from the assumption of a continuity of sediment flux. It is found that for the average submarine canyon-channel system, the flow would more than double in thickness and its velocity would be halved during a hydraulic jump. The entrainment of water through the interface of the flow during the jump is significant in reducing the density of the current. The higher the initial Froude number, the greater the expected density reduction. However, the density reduction due to the jump is probably not sufficient to convert a high-density slide into a low-density turbidity current. There must be an initial density reduction by entrainment while the slide is passing through the canyon at high Froude number. Turbulence generated within the hydraulic jump would stir the flow and help to insure suspension of its sediment.

Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid

Abstract: Spatial ‘intermittency’ in the velocity field fine-structure of fully turbulent flow regions, first observed by Batchelor & Townsend (1949), is studied further here in grid-generated nearly isotropic turbulence and on the axis of a round jet. At large enough Reynolds numbers, appropriately filtered hot-wire anemometer signals appear intermittent as the turbulent patterns are convected past the hot wire by the mean flow. Measurements show that there is a decrease in the relative fluid volume (equal to the ‘intermittency factor’) occupied by fine-structure of given size as the turbulence Reynolds number is increased. They show also that, for a fixed Reynolds number, the relative volume is smaller for smaller fine-structure. The average linear dimension of the fine-structure regions turns out to be much larger than the sizes of fine-structure therein. At Rλ, = 110, for example, the ratio ranges from 15 to 30, decreasing with decreasing ‘eddy’ size. It appears to be approaching an asymptote with increasing Rλ.The flatness factors and probability distributions of the first derivative, the second derivative, band-passed and high-passed velocity fluctuation signals were also measured. The turbulence Reynolds numbers Rλ ranged from 12 to 830. The flatness factors of the first and the second derivatives increase monotonically with Rλ. Those of the second derivative vary with Rλ0.25 for Rλ 300. No indication of asymptotic constant values was observed for Rλ up to the order of one thousand.The probability distributions of velocity fluctuations and large-scale signals are nearly normal, while the small-scale signals are not. The flatness factor of the filtered band-pass velocity signal increases with increasing frequency.At the larger Reynolds numbers, the square of the signal associated with large wave-numbers may be approximated by a log-normal probability distribution for amplitudes when probabilities fall between 0·3 and 0·95, in limited agreement with the theory of Kolmogorov (1962), Oboukhov (1962), Gurvich & Yaglom (1967).

A study of free jet impingement. Part 2. Free jet turbulent structure and impingement heat transfer

Abstract: An experimental study of jet impingement is completed with the presentation of the measured turbulent characteristics of the circular subsonic jet and the heat transfer rates measured when this jet impinges normal to a flat plate. The data suggest that for impingement very close to the stagnation point, the heat transfer can be computed by applying a turbulent correction factor to the laminar value calculated for a flow having the same pressure distribution as that present in the impingement region. The correction factor is found to be a function of the axial distance and not of Reynolds number. Farther away, the measurements agree well with the heat transfer estimated using the method of Rosenbaum & Donaldson (1967). At large distances from the stagnation point, the heat transfer falls off in inverse proportion with the distance.The documentation of the turbulent jet flow field includes measurements of the radial and axial velocity fluctuations and their spectra, as well as the radial distribution of turbulent shear . In addition, measurements of the turbulence near the stagnation point and the total pressure fluctuation at the stagnation point are presented.

Survey of viscous interactions associated with high Mach number flight

Abstract: T most severe problems of atmospheric flight at high Mach numbers are associated with viscous-inviscid interactions. Cruise vehicles for Mach numbers above four and lifting re-entry vehicles have highly complex three-dimensional configurations in which exist many regions of high compression that can cause boundary layers to separate. Although separation can result in loss of control effectiveness or flow degradation in an engine inlet, flow reattachment gives rise to heat rates that can far exceed those for an attached boundary layer. A further, and possibly far more severe viscous interaction problem is the impingement of shock waves generated by the forebody and other external components of a vehicle on aft sections resulting in local heat rates that may be many times larger than stagnation point values. Peak heating conditions may be laminar for lifting re-entry configurations, though our knowledge of boundary layertransition is far from adequate so that transitional and turbulent flows cannot be ruled out. However, Reynolds num, bers of potential high Mach number cruise vehicles are high— 10 to 10—so that viscous interactions will be predominately associated with turbulent boundary layers and their attendant higher heat rates. The high local heat rates resulting from viscous interactions cause "hot spots" that could lead to catastrophic failure. Vivid examples of damage resulting from viscous interactions are given in Figs. 1 and 2. A ventral pylon on the X-15 airplane, shown in Fig. 1, caused high^local heating of the fuselage around its root, and developed large holes near its tip due to the impingement of the shock wave from a dummy ramjet it supported, during a flight at Mach 6.7 in 1967. A study of the flowfield of the pylon-mounted dummy ramjet configuration is reported in Ref. 1. Figure 2 shows considerable damage due to interaction heating to the underside of a sled and its supporting slipper as a result of a run at 7000 fps on the 7-mile test track at Holloman Air Force Base. Unlike stagnation-point heating where the location is obvious, the problem with complex configurations is to determine "where" high heat rates are likely to occur, as well as their magnitude. It is the purpose of this paper to identify some potentially critical areas of viscous interactions associated with high Mach number vehicles and briefly review our state of knowledge in these areas with emphasis on the basic flow phenomena.

A pulsed-wire technique for velocity measurements in highly turbulent flows

Abstract: The velocity measuring technique described in this paper consists of measuring the time of flight of a tracer of heated air from an electrically pulsed wire to one of two sensor wires which are operated as resistance thermometers. These sensor wires are at right angles to the pulsed wire and are placed one on either side of the pulsed wire. The instrument may be used in highly turbulent flows including regions in which flow reversals occur. The paper discusses the theoretical behaviour of the probe and the results of some calibration experiments.

On the Spectrum and Decay of Random Two-Dimensional Vorticity Distributions at Large Reynolds Number

Abstract: The hypothesis is made that the vorticity distribution in a field of random two-dimensional vorticity (two-dimensional turbulence) develops discontinuities. The energy spectrum is then calculated and shown to behave like k^(−4) for large values of the wave-number k. The rates of decay of mean square vorticity and velocity are estimated. An expression for the growth of length scale is obtained and it is noted that the size of turbulent trailing vortices is apparently well fitted by the formula.

Experimental Study on Structure of Gas Flow in Tube Banks with Tube Axes Normal to Flow : Part 1, Karman Vortex Flow from Two Tubes at Various Spacings

Abstract: A gas flow in and behind tube banks with tuber axes normal to the gas flow is highly turbulent containing numerous vortices of different sizes and intensities. Each vortex in this complicated wake is the result of a periodic generation, and the periodicity sometimes induces serious vibration problems in steam generators and heat exchangers. The structure of the flow may be treated ass a complex synthesized result of Karman vortex flow. This paper is the first report of a series of studies carried out form the above mentioned viewpoint, and presents the experimental results about a Karman vortex flow from two tubes at various spacings. The vortex formation region was visualized by means of the Schlieren method and was recorded by a 16mm movie camera. The vortex shedding frequency was systematically studied. The frequency for two tubes is entirely different from that for a single tube. It was found that the size of the vortex formation domain has a dominant effect on the frequency.

Calculation of boundary-layer development using the turbulent energy equation: compressible flow on adiabatic walls

Abstract: The basic method described by Bradshaw, Ferriss & Atwell (1967) is extended to compressible flow in two-dimensional boundary layers in arbitrary pressure gradient (excluding shock waves and expansion fans) by invoking Morkovin's hypothesis (Favre 1964) that the turbulence structure is unaffected by compressibility Using the same empirical functions as in incompressible flow, skin friction in zero pressure gradient is predicted to within 3% of Spalding & Chi's (1964) correlation for free-stream Mach numbers less than 5 Comparisons with experiments in pressure gradient are restricted by the lack of data, but, since Morkovin's hypothesis does not depend on pressure gradient, methods which use it (of which the present method seems to be the first) can be checked fairly adequately by comparisons with data in zero pressure gradientNo ‘compressibility transformations’ are needed, although the Crocco relation is used, provisionally, for the temperature: since the calculations take only about 20% longer than in incompressible flow, Morkovin's hypothesis does as much as any transformation could do It is pointed out that, in supersonic flow, surface curvature which is large enough to induce a significant longitudinal pressure gradient is also large enough to have a very significant effect on the turbulence structure