bluuebunny/crossref_metadata_2025 huggingface dataset:
Response of Pitot probes in turbulent streams
Authors: Becker H. A. , Brown A. P. G.
Published: March 2006

The response of a Pitot probe in a uniform laminar stream is commonly expressed in the form\[ P_s = P+{\textstyle\frac{1}{2}}C_{\rho} U^2, \]wherePsis the probe signal pressure,Pis the stream static pressure,Uis the stream speed and ρ is the fluid density. It has been found that for ordinary sphere-nosed, round-nosed and square-nosed probes\[ 1-C = K(\sin^2\theta)^m = K(U^2_n/U^2)^m, \]where θ is the angle between the velocity vector and the probe axis, andUn≡Usin θ is the transverse velocity component. The parametersmandKare functions of the probe geometry. These formulae also describe the performance in a turbulent stream when the probe is small compared with the turbulence scale. The evaluation of the time-averaged response is treated, and an answer is developed to the question of what it is that a Pitot probe measures in a turbulent stream. In a turbulent shear flow having the properties of a boundary layer, the reference pressure is best taken to be the static pressure at the shear-layer edge. It is shown that round-nosed probes withDi/D≃0·45 and square-nosed probes withDi/D≃0·15 then detect${\textstyle\frac{1}{2}}\rho\overline{U}^2_x$with good accuracy, whereDi/Dis the ratio of the inside and outside diameters of the Pitot tube. When measurements are made with two probes of dissimilar geometry, the differential response can be used to find the mean-square level of the transverse velocity fluctuations. Turbulence levels so measured agree closely with results from hot-wire anemometry.
Measurements of the nearly isotropic turbulence behind a uniform jet grid
Authors: Gad-El-Hak Mohamed , Corrsin Stanley
Published: March 2006

Wind-tunnel turbulence behind a parallel-rod grid with jets evenly distributed along each rod is nearly isotropic. Homogeneity improvement over prior related experiments was attained by the use of controllable nozzles. Compared with the ‘passive’ case, the downwind-jet ‘active’ grid has a smaller static pressure drop across it and gives a smaller turbulence level at a prescribed distance from it, while the upwind-jet grid gives a larger static pressure drop and larger turbulence level. ‘Counterflow injection’ generates larger turbulence energy and larger scales, both events being evidently associated with instability of the jet system. This behaviour is much like that commonly observed behind passive grids of higher solidities.

If the turbulent kinetic energy is approximated as an inverse power law in distance, the (positive) exponent decreases with increasing (downwind or upwind) jet strength, corresponding to slower absolute decay rates. No peculiar decay behaviour occurs when the jet grid is ‘self-propelled’ (zero net average force), or when the static pressure drop across it is zero.

The injection does not change the general behaviour of the energy spectra, although the absolute spectra change inasmuch as the turbulence kinetic energy changes.
Distorted gas bubbles at large Reynolds number
Authors: Sawi M. El
Published: March 2006

The distortion of a gas bubble rising steadily in an inviscid incompressible liquid of infinite extent under the action of surface tension forces is investigated theoretically using an appropriate extension of the tensor virial theorem. A convenient parameter for distinguishing the bubble shape is the Weber numberW. The virial method leads to an expression relatingWand the axis ratio χ, of the transverse and longitudinal axes of the bubble. To first order inW, this relation agrees with the linear theory established by Moore (1959). Also, comparison of the results with his (1965) approximate theory reveals similar features and excellent agreement up to χ = 2. In particular, it confirms his prediction of the existence of a maximum Weber number. Although the present work does not consider the stability of these bubbles, it is interesting to note that the maximum value of 3.271 attained byWdiffers only by about 2.8% from the critical Weber number obtained by Hartunian & Sears (1957) for the onset of instability.

An approximate method for the study of slightly distorted spheroidal gas bubbles is also formulated and the resulting boundary-value problem solved numerically. The theory is then extended to include gravity. The joint effect of surface tension as well as gravitational forces has not been included in earlier theories. The shapes of the bubbles are traced and compared with the unperturbed spheroids. Comparisons for the velocity of bubble rise are made between the present predictions and some experimental results. In particular the results are compared with recent experimental data for the motion of gas bubbles in liquid metals.
The transverse force on a drop in an unbounded parabolic flow
Authors: Wohl Philip R. , Rubinow S. I.
Published: March 2006

The steady flow in and around a deformable liquid sphere moving in an unbounded viscous parabolic flow and subject to an external body force is calculated for small values of the ratio of the Weber number to the Reynolds number in the creeping-flow regime. It is found that, in addition to the drag force, the drop experiences a force orthogonal to the undisturbed flow direction. When the body force is absent (neutrally buoyant drop), this lift force tends to drive the drop inwards to the axis, where the undisturbed flow velocity is maximum, i.e., towards a position of lower velocity gradient. In the case for which the parabolic flow profile is a Poiseuille flow profile, the lift force is given by the expression.
\[
{\bf F}_1 =-6\pi\mu\epsilon U_0\frac{\alpha +\frac{2}{3}}{\alpha + 1}\bigg(\frac{a}{R_0}\bigg)^4{\bf b}F[1+o(\epsilon)].
\]
Here a is the radius of the undeformed sphere, R0 is the radial distance from the position of maximum undisturbed flow U0 at the profile axis to the position of zero flow, ε is the ratio of the Weber number to the Reynolds number, given by ε=μU0T−1, where μ is the external fluid viscosity and T is the surface tension of the drop, α is the ratio of the drop and external fluid viscosities, b is the radial vector from the flow axis to the centre of mass of the drop, and F is a function of α and a dimensionless parameter dependent on the body force that is determined in the analysis. Reasonable agreement is found between the observations by Goldsmith & Mason (1962) of the axial drift of liquid drops in Poiseuille flow and the predictions of the theory herein.
Vortex shedding from spheres
Authors: Achenbach Elmar
Published: March 2006

Vortex shedding from spheres has been studied in the Reynolds number range 400 < Re < 5 × 106. At low Reynolds numbers, i.e. up to Re = 3 × 103, the values of the Strouhal number as a function of Reynolds number measured by Möller (1938) have been confirmed using water flow. The lower critical Reynolds number, first reported by Cometta (1957), was found to be Re = 6 × 103. Here a discontinuity in the relationship between the Strouhal and Reynolds numbers is obvious. From Re = 6 × 103 to Re = 3 × 105 strong periodic fluctuations in the wake flow were observed. Beyond the upper critical Reynolds number (Re = 3.7 × 105) periodic vortex shedding could not be detected by the present measurement techniques.

The hot-wire measurements indicate that the signals recorded simultaneously at different positions on the 75° circle (normal to the flow) show a phase shift. Thus it appears that the vortex separation point rotates around the sphere. An attempt is made to interpret this experimental evidence.
The locomotion of elongated bodies in pipes
Authors: Lavie A. M.
Published: March 2006

The theory describing the swimming mechanism of an elongated body is extended to cover the cases of locomotion through unsteady streams in pipes. Such an extension is essential for the artificial fish ‘Pod’, a medical device which swims in the patient's blood vessel. Two approaches are considered. First, potential theory is considered, and the results achieved show that the main influence of the pipe is on evaluation of the proper virtual mass. Next the flow is assumed to be viscous. The consideration of viscosity is obviously necessary for flows in pipes. In that case the virtual mass is replaced by another equivalent mass depending on the viscosity and on the angular frequency of the lateral motion and in addition new terms appear in the local lift expressions. These are recognized as the viscous damping force.
A measurement of Lagrangian velocity autocorrelation in approximately isotropic turbulence
Authors: Shlien D. J. , Corrsin S.
Published: March 2006

By measuring the heat dispersion behind a heated wire stretched across a wind tunnel (Taylor 1921, 1935), the Lagrangian velocity autocorrelation was determined in an approximately isotropic, grid-generated turbulent flow. The techniques were similar to previous ones, but the scatter is less. Assuming self-preservation of the Lagrangian velocity statistics in a form consistent with recent measurements of decay in this flow (Comte-Bellot & Corrsin 1966, 1971), a stationary and an approximately self-preserving form for the dispersion were derived and approximately verified over the range of the experiment.

Possibly the most important aspect of this experiment is that data were available in the same flow on the simplest Eulerian velocity autocorrelation in time, the correlation at a fixed spatial point translating with the mean flow (Comte-Bellot & Corrsin 1971). Thus, the Lagrangian velocity autocorrelation coefficient function calculated from the dispersion data could be compared with this corresponding Eulerian function. It was found that the Lagrangian Taylor micro-scale is very much larger than the analogous Eulerian microscale (76 ms compared with 6.2ms), contrary to an estimate of Corrsin (1963). The Lagrangian integral time scale is roughly equal to the Eulerian one, being larger by about 25 %.
Two-dimensional turbulence near the viscous limit
Authors: Green J. S. A.
Published: March 2006

Two-dimensional incompressible motion is generated by a steady external body force varying sinusoidally with a transverse co-ordinate. Such flow is found to be unstable for Reynolds numbers greater than 2½, and under these conditions evolves towards a new steady state. This ‘steady-eddy’ state is itself unstable in a sense, and its breakdown suggests the catastrophic onset of a cascade of turbulence. The mechanics of this cascade can be represented by a kind of recursion system in which the turbulence dynamics of one scale is repeated in the next, and a law of turbulent stress results. The spectrum of kinetic energy generated by a steady input of momentum at a discrete wavelength shows a rapid decrease (as k−5) towards shorter wavelengths but a much slower decrease (as k) towards longer wavelengths.
Plate-injection into a separated supersonic boundary layer. Part 2. The transition regions
Authors: Stewartson K.
Published: March 2006

The model proposed by Smith & Stewartson (1973), to describe the separated boundary layer induced by strong injection over a finite length of a flat plate in a supersonic mainstream, is shown to provide the basis for a fully consistent solution of the Navier–Stokes equations for this problem, valid in the limit of infinite Reynolds number. The solution takes the form of asymptotic expansions in each of a large number of overlapping regions of the flow field, which are consistently matched across areas of common validity.
On Kolmogorov's inertial-range theories
Authors: Kraichnan Robert H.
Published: March 2006

Consistency and uniqueness questions raised by both the 1941 and 1962 Kolmogorov inertial-range theories are examined. The 1941 theory, although unlikely from the viewpoint of vortex-stretching physics, is not ruled out just because the dissipation fluctuates; but self-consistency requires that dissipation fluctuations be confined to dissipation-range scales by a spacewise mixing mechanism. The basic idea of the 1962 theory is a self-similar cascade mechanism which produces systematically increasing intermittency with a decrease of scale size. This concept in itself requires neither the third Kolmogorov hypothesis (log-normality of locally averaged dissipation) nor the first hypothesis (universality of small-scale statistics as functions of scale-size ratios and locally averaged dissipation). It does not even imply that the inertial range exhibits power laws. A central role for dissipation seems arbitrary since conservation alone yields no simple relation between the local dissipation rate and the corresponding proper inertial-range quantity: the local rate of energy transfer. A model rate equation for the evolution of probability densities is used to illustrate that even scalar nonlinear cascade processes need not yield asymptotic log-normality. The approximate experimental support for Kolmogorov's hypothesis takes on added significance in view of the wide variety ofa prioriadmissible alternatives.

If the Kolmogorov law$E(k) \propto k^{-\frac{5}{3}-\mu}$is asymptotically valid, it is argued that the value of μ depends on the details of the nonlinear interaction embodied in the Navier–Stokes equation and cannot be deduced from overall symmetries, invariances and dimensionality. A dynamical equation is exhibited which has the same essential invariances, symmetries, dimensionality and equilibrium statistical ensembles as the Navier–Stokes equation but which has radically different inertial-range behaviour.
A separated flow in mixed convection
Authors: Wilks Graham
Published: March 2006

A numerical solution is presented for the flow of a uniform stream past a semi-infinite heated flat plate at whose surface the heat flux remains constant. The buoyancy forces oppose the free-stream motion and separation occurs. An examination of the singularities in the skin-friction and heat-transfer coefficients suggests, rather surprisingly, a behaviour as (ξs−ξ)⅗ at separation.
Ageostrophic effects in rotating stratified flow
Authors: Huppert Herbert E. , Stern Melvin E.
Published: March 2006

We consider the low Rossby number (R) flow of a stratified fluid in a long rotating channel, for which the bottom elevation varies in the downstream direction. The quasi-geostrophic response is shown to be singular at the side walls of the channel, and thus an ageostrophic analysis is necessary even for vanishingly small R. Part of the ageostrophic steady-state response is a modified quasi-geostrophic perturbation trapped near the bottom. A second component which is present even as R approaches zero is an internal Kelvin wave whose vertical wavelength adjusts so that the wave remains stationary with respect to the channel bottom and which propagates energy and momentum to infinite heights in an unbounded channel. The case of a bounded layer of fluid is also considered, and the resonance conditions are given. We also calculate the flow field when the bottom elevation varies in the cross-stream direction. We conclude that stagnation or flow reversal can be caused either by the modified quasi-geostrophic component or by the Kelvin wave and estimate the critical condition by an extrapolation of the perturbation velocity computed from linear theory.
Global stability of time-dependent flows. Part 2. Modulated fluid layers
Authors: Homsy George M.
Published: March 2006

The method of energy is used to develop two stability criteria for a large class of modulated Bénard problems. Both criteria give stability limits which hold for disturbances of arbitrary amplitude. The first of these, designated as strong global stability, requires the energy of all disturbances to decay monotonically and exponentially in time. Application of this criterion results in a prediction of Rayleigh numbers below which the diffusive stagnant solution to the Bous-sinesq equations is unique. The second criterion requires only that disturbances decay asymptotically to zero over many cycles of modulation, and is a weaker concept of stability. Computational results using both criteria are given for a wide range of specific cases for which linear asymptotic stability results are available, and it is seen that the energy and linear limits often lie close to one another.
Application of an inner expansion method to plane, inviscid, compressible flow stability studies
Authors: Beckemeyer Roy J.
Published: March 2006

The inviscid compressible flow stability problem is mathematically similar to that of sound propagation in a sheared flow field. This similarity has been exploited by applying an inner expansion technique to study the effect of finite shear gradients on free parallel flow instabilities. This technique had previously been used to investigate the effect of thin boundary layers on sound propagation in ducts. The expansion, which is applicable to flow profiles involving thin, but finite, shear layers separating regions of uniform flow, offers a significant computational advantage over the numerical methods commonly employed to determine the stability of continuous mean flow profiles. Although equally applicable to three-dimensional and to spatially growing hydrodynamic instabilities, the procedure is demonstrated by application to the eigenvalue problem for temporal instabilities of shear layers and jets in plane inviscid compressible flow.

For the case of vanishingly thin shear layers, the eigenvalue equations derived here reduce to those obtained by Miles (1958) for parallel flows bounded by vortex sheets. The series solution of Graham & Graham (1969), valid for linear shear-layer profiles of arbitrary thickness, provides a basis of comparison for the expansion-method results. Unstable-mode eigenvalues obtained using the two methods are found to be in good agreement for a significant range of values of the ratio of shear-layer thickness to axial wavelength.
Phase and amplitude discrepancies in the surface wave due to a wedge-ended hull form
Authors: Standing R. G.
Published: March 2006

A deep surface-piercing wedge-ended hull model was towed through still water. Measurements of the surface wave pattern confirmed earlier findings for ship models, that the measured bow-wave cusp line often lies well forward of the position predicted by thin-ship theory, and that this shift increases with bow water-line angle and with decreasing model speed. Two possible explanations are considered here in terms of changes of wave phase speed with wave convection and steepness. Calculations based on a transformation method due to Guilloton predict more realistic wave profiles than linear theory, but account for less than half the observed shift. Some tentative conclusions are drawn.

The singularity in the Green's function double integral is removed by an improved method, which simplifies the numerical integration. The new integrand decays within one oscillation.
Generalized Eady waves
Authors: Williams Gareth P.
Published: March 2006

Solutions are obtained for the baroclinic instability problem for situations in which the static stability and mean shear vary geminately with height. The simple solution given by Eady is shown to be a special limiting case of a class of exact solutions for flows whose basic states have a vanishing interior potential vorticity gradient. The generalized solutions show that the temperature amplitude distribution is particularly sensitive to vertical variations in static stability but that phases and other amplitudes are only slightly influenced by such variations. When the static stability and shear increase (decrease) with height an enhanced temperature maximum occurs at the upper (lower) surface in comparison with the standard Eady solution.

The generalized solutions also help to explain the character of annulus waves and predict a short-wave cut-off that is the same as that given by Eady's theory provided that it is based on the vertically averaged gravitational frequency.
Linear stability theory of oscillatory Stokes layers
Authors: Kerczek Christian Von , Davis Stephen H.
Published: March 2006

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to bemore stablethan that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.
Edge waves: a long-wave theory for oceans of finite depth
Authors: Grimshaw R.
Published: March 2006

Edge waves travelling along a straight coastline are examined in the case when the water depth approaches a constant value at large distances from the coast. Only the fundamental mode is examined in the limit as the ratio of the water depth at infinity to the edge-wave wavelength tends to zero. Two comparison theorems are used to obtain upper and lower bounds for the dispersion relation. A long-wave approximation procedure is used to obtain the leading terms in the dispersion relation for a wide class of bottom topographies. The results obtained are compared with an exact result for the case when the bottom topography is a rectangular step.
On the flow field of a rapidly oscillating airfoil in a supersonic flow
Authors: Kurosaka M.
Published: March 2006

This paper examines the features of the flow field off the surface of an oscillating flat-plate airfoil immersed in a two-dimensional supersonic flow Although the exact linearized solution for a supersonic unsteady airfoil has been known for a long time, its expression in the form of an integral is not convenient for a physical interpretation. In the present paper, the quintessential features of the flow field are extracted from the exact solution by obtaining an asymptotic expansion in descending powers of a frequency parameter through the repeated use of the stationary-phase and steepest descent methods. It is found that the flow field consists of two dominant and competing signals: one is the acoustic ray or that component arising from Lighthill's ‘convecting slab’ and the other is the leading-edge disturbance propagating as a convecting wavelet. The flow field is found to be divided into several identifiable regions defined by the relative magnitude of the signals, and the asymptotic expansions appropriate for each flow region are derived along with their parametric restrictions. Such intimate knowledge of the flow field in unsteady, supersonic flow is of interest for interference aerodynamics and related acoustic problems.
Mean velocity and longitudinal dispersion of heavy particles in turbulent open-channel flow
Authors: Sumer B. Mutlu
Published: March 2006

This paper deals with the motion of a heavy particle in a turbulent flow in an open channel with a smooth bottom. For the case when the particle stays in suspension in the main body of the flow almost all the time, (a) the probability density function of the projection on a cross-sectional plane of the particle position at any instant, and (b) the mean velocity and longitudinal dispersion coefficient of particles are determined analytically by employing the Eulerian formulation and applying the Aris moment transformations. It is found that the mean particle velocity decreases and the longitudinal dispersion coefficient of particles increases with the fall velocity.
Topological pumping of magnetic flux by three-dimensional convection
Authors: Drobyshevski E. M. , Yuferev V. S.
Published: March 2006

Attention is drawn to a principal difference between the transfer of a horizontal magnetic field by turbulence and by three-dimensional cell convection.

If the motion of the conducting medium in the cells is such that the heated material ascends at the centre while descending along the sides of the cells, then the magnetic tubes of force will be carried downwards by peripheral flows. Discrete ascending flows separated from one another by descending material carry only closed magnetic field loops. Such loops do not transfer net magnetic flux. As a result, the magnetic flux becomes blocked at the base of the convective layer.
The structure of turbulent density interfaces
Authors: Crapper P. F. , Linden P. F.
Published: March 2006

The structure of density interfaces upon which turbulent motions have been imposed is investigated in the laboratory. Particular attention is paid to the profiles of density across such interfaces, and the temporal response of the interface to the imposed turbulence. Interfaces over a range of Péclet numbers and Richardson numbers are examined. It is found that the interface thickness h/l (non-dimensionalized with respect to the length scale l of the turbulence) is a function of the Péclet number but independent of the Richardson number. At low Péclet numbers (Pe [lsim ] 200) molecular diffusion is important in the determination of the interface structure and a diffusive core (across which all transport occurs by molecular diffusion) is formed in the centre of the interface. At higher Péclet numbers the interface structure appears to be determined by non-diffusive processes and h/l becomes approximately constant at a value of about 1·5. Some information concerning the intermittent nature of the mixing process at high Péclet number is obtained from records of salinity fluctuations measured at a fixed point. Finally, the implications of these data concerning the interpretations of the measurements of entrainment across an interface made by Turner (1968) are discussed.
On the motion of a fluid-fluid interface along a solid surface
Authors: Dussan V. Elizabeth B. , Davis Stephen H.
Published: March 2006

A fluid-fluid interface that joins a solid surface forms a common line. If the common line moves along the solid, a mutual displacement process is involved and is studied here. Some simple experiments motivate the formulation of the basic assumption of the analysis. The basic assumption is a formalization of the idea that the fluid-fluid interface rolls on or unrolls off the solid. This forms an axiom for the mostly kinematical analysis that follows. The predictions are tested through a series of qualitative experiments. The role of the no-slip boundary condition at the solid surface is discussed.
Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers
Authors: Nishioka Michio , Sato Hiroshi
Published: March 2006

Velocity measurements were made in the flow field behind a circular cylinder at Reynolds numbers from 10 to 80 and results compared with existing numerical solutions. Takami & Keller's solution for the velocity distribution in the wake shows good agreement at low Reynolds numbers and fair agreement at high Reynolds numbers. The drag coefficient of the cylinder and the size of the standing eddies behind the cylinder were also determined. They are compatible with existing experimental and numerical results. Details of the velocity distribution in the standing eddies are clarified.
The effects of surface roughness and tunnel blockage on the flow past spheres
Authors: Achenbach Elmar
Published: March 2006

The effect of surface roughness on the flow past spheres has been investigated over the Reynolds number range 5 × 104 < Re < 6 × 106. The drag coefficient has been determined as a function of the Reynolds number for five surface roughnesses. With increasing roughness parameter the critical Reynolds number decreases. At the same time the transcritical drag coefficient rises, having a maximum value of 0·4.

The vortex shedding frequency has been measured under subcritical flow conditions. It was found that the Strouhal number for each of the various roughness conditions was equal to its value for a smooth sphere. Beyond the critical Reynolds number no prevailing shedding frequency could be detected by the measurement techniques employed.

The drag coefficient of a sphere under the blockage conditions 0·5 < ds/dt < 0·92 has been determined over the Reynolds number range 3 × 104 < Re < 2 × 106. Increasing blockage causes an increase in both the drag coefficient and the critical Reynolds number. The characteristic quantities were referred to the flow conditions in the smallest cross-section between sphere and tube. In addition the effect of the turbulence level on the flow past a sphere under various blockage conditions was studied.
Shock-wave structure and intermolecular collision laws
Authors: Yen Shee-Mang , Ng Winnie
Published: March 2006

The nonlinear Boltzmann equation has been solved for shock waves in a Max-wellian gas for eight upstream Mach numbers M1 ranging from 1·1 to 10. The numerical solutions were obtained by using Nordsieck's method, which was revised for use with the differential cross-section corresponding to an intermolecular force potential following an inverse fifth-power law. The accuracy of the calculations of microscopic and macroscopic properties for this collision law is comparable with that for elastic spheres published earlier (Hicks, Yen & Reilly 1972).

We have made comparisons of the detailed characteristics of the internal shock structure in a Maxwellian gas with those in a gas of elastic spheres. The purpose of this comparative study is to find the shock properties that are sensitive as well as those which are insensitive to the change in collision law and to find effective ways to study them.

The variation of thermodynamic and transport properties of interest with respect to density and to each other was found to depend only weakly on the change in collision law. The principal effect on the macroscopic shock structure due to the change in intermolecular potential is in the spatial variation of the macroscopic properties. The spatial variation of macroscopic properties may be determined accurately from the corresponding moments of the collision integral, especially in the upstream and downstream wings of the shock wave. The results for the velocity distribution function exhibit the microscopic shock characteristics influenced by a difference in intermolecular collisions, in particular the departure from equilibrium in the upstream wing of the shock and the relaxation towards equilibrium in the downstream wing. The departure of several characteristics of weak shock waves from those of the Chapman-Enskog linearized theory and the Navier-Stokes shock is also insensitive to the change in collision law. The deviation of the half-width of the function ∫fdvyduz from the Chapman-Enskog first iterate at M1 = 1·59 is in agreement with an experiment (Muntz & Harnett 1969).
A Markovian random coupling model for turbulence
Authors: Frisch U. , Lesieur M. , Brissaud A.
Published: March 2006

The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.

The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.
Mixing layers for highly underexpanded supersonic jets in hypersonic streams
Authors: Moran James P.
Published: March 2006

The mixing layer bounding the exhaust plume associated with hypersonic highaltitude rockets is analysed as a laminar binary mixture of perfect gases with Lewis and Prandtl numbers of unity. A far-field approximation to the undisturbed jet core and a Newtonian pressure balance between the jet and ambient gases are used to construct the mixing-layer location. Longitudinal pressure variations are neglected and resultant errors are evaluated. Boundary conditions at the edge of the mixing layer are evaluated by streamline tracing to shock entry points. The sensitivity of properties in the mixing layer to variations in the plume angle of attack, engine nozzle efficiency and engine thrust are examined, and an approximate density and thrust scaling of mixing-layer overall properties is developed.
Natural convection in a shallow cavity with differentially heated end walls. Part 1. Asymptotic theory
Authors: Cormack D. E. , Leal L. G. , Imberger J.
Published: March 2006

The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes: a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. A solution valid at all orders in the aspect ratio A is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A → 0, are derived.
Natural convection in a shallow cavity with differentially heated end walls. Part 2. Numerical solutions
Authors: Cormack D. E. , Leal L. G. , Seinfeld J. H.
Published: March 2006

Numerical solutions of the full Navier-Stokes equations are obtained for the problem of natural convection in closed cavities of small aspect ratio with differentially heated end walls. These solutions cover the parameter range Pr = 6·983, 10 ≤ Gr 2 × 104 and 0·05 [les ] A [les ] 1. A comparison with the asymptotic theory of part 1 shows excellent agreement between the analytical and numerical solutions provided that A [lsim ] 0·1 and Gr2A3Pr2 [lsim ] 105. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit of part 1 and the boundary-layer limit; A fixed, Gr → ∞.
Natural convection in a shallow cavity with differentially heated end walls. Part 3. Experimental results
Authors: Imberger Jorg
Published: March 2006

The steady motion of water in an enclosed rectangular cavity with differentially heated vertical end walls was studied experimentally, and the results are compared with the findings of parts 1 and 2. The depth-to-length ratios of the cavities were 102 and 1·9 × 102, and the Rayleigh number was allowed to vary sufficiently to enable a study to be made of the transition from a flow driven by the vertical wall boundary layers to one sustained by a longitudinal temperature gradient in the central sections of the cavity.
The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments
Authors: Hammack Joseph L. , Segur Harvey
Published: March 2006

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.
An internal wave in a viscous heat-conducting isothermal atmosphere
Authors: Stevenson T. N. , Bearon J. N. , Thomas N. H.
Published: March 2006

The effects of viscosity and heat conduction on the propagation of an internal wave generated by a simple harmonic localized disturbance are considered for the case of an isothermal compressible atmosphere. A similarity solution of the linearized equations shows that the velocities decay and the wave width increases away from the disturbance. Superpositions of this solution show how a few waves of small wavelength attenuate rapidly whereas waves of larger wavelength can increase in amplitude as they propagate upwards before eventually attenuating.
Laser anemometer measurements of trailing vortices in water
Authors: Baker G. R. , Barker S. J. , Bofah K. K. , Saffman P. G.
Published: March 2006

A series of measurements of trailing vortices behind lifting hydrofoils is described. These measurements were made in the Caltech Free-Surface Water Tunnel, using a laser-Doppler velocimeter to measure two components of velocity in the vortex wake. Two different model planforms were tested, and measurements were made at several free-stream velocities and angles of attack for each. Velocity profiles were measured at distances downstream of the model of from five to sixty chord lengths. These measurements are the first results of a continuing experimental programme.

In § 3 of this paper, the theory of trailing vortices is discussed. The effects of ‘vortex wandering’ upon the measurements are computed, and the corrected results are seen to be in reasonable agreement with the theory.
Reflexion of a weak shock wave with vibrational relaxation
Authors: Hung Ching-Mao , Seebass Richard
Published: March 2006

The structure of a shock wave in a vibrationally relaxing gas undergoing reflexion from a plane wall is examined. The shock wave is assumed to be weak, and departures from thermodynamic equilibrium are assumed small; both an adiabatic and an isothermal wall are considered. The flow field is divided into three regions: a far-field region, an interaction region, and, for the isothermal-wall case, a thermal boundary layer. Different asymptotic expansions are determined for the various regions through the method of matched asymptotic expansions. In the region far from the wall, a non-equilibrium Burgers equation governs the motion and the incident and the reflected shock wave structures. During reflexion, a non-equilibrium wave equation applies; its first-order terms are equivalent to an acoustic approximation. Heat conduction to the wall is modelled by an isothermal wall boundary condition which requires the introduction of a thermal boundary layer adjacent to the wall. This thermal boundary layer is thin and the adiabatic-wall result provides the outer solution for treating this layer. This thermal layer affects the structure of the reflected wave.
Inertial migration of rigid spheres in two-dimensional unidirectional flows
Authors: Ho B. P. , Leal L. G.
Published: March 2006

The familiar Segré-Silberberg effect of inertia-induced lateral migration of a neutrally buoyant rigid sphere in a Newtonian fluid is studied theoretically for simple shear flow and for two-dimensional Poiseuille flow. It is shown that the spheres reach a stable lateral equilibrium position independent of the initial position of release. For simple shear flow, this position is midway between the walls, whereas for Poiseuille flow, it is 0·6 of the channel half-width from the centre-line. Particle trajectories are calculated in both cases and compared with available experimental data. Implications for the measurement of the rheological properties of a dilute suspension of spheres are discussed.
Sphere drag coefficient for subsonic speeds in continuum and free-molecule flows
Authors: Bailey A. B.
Published: March 2006

An extensive series of measurements of sphere drag coefficients has been made in an aeroballistic range for a broad range of Reynolds and Mach numbers. These measurements have been compared with those obtained in other test facilities. As a result of this comparison it has been possible to suggest reasons for many of the inconsistencies in the earlier measurements and to establish more accurate values of the sphere drag coefficient for M∞ [lsim ] 0·2 and 10−2 [lsim ] Re∞ [lsim ] 107.
The structure of the viscous sublayer and the adjacent wall region in a turbulent channel flow
Authors: Eckelmann Helmut
Published: March 2006

Hot-film anemometer measurements have been carried out in a fully developed turbulent channel flow. An oil channel with a thick viscous sublayer was used, which permitted measurements very close to the wall. In the viscous sublayer betweeny+≃ 0·1 andy+= 5, the streamwise velocity fluctuations decreased at a higher rate than the mean velocity; in the regiony+[lsim ] 0·1, these fluctuations vanished at the same rate as the mean velocity.

The streamwise velocity fluctuationsuobserved in the viscous sublayer and the fluctuations (∂u/∂y)0of the gradient at the wall were almost identical in form, but the fluctuations of the gradient at the wall were found to lag behind the velocity fluctuations with a lag time proportional to the distance from the wall. Probability density distributions of the streamwise velocity fluctuations were measured. Furthermore, measurements of the skewness and flatness factors made by Kreplin (1973) in the same flow channel are discussed. Measurements of the normal velocity fluctuationsvat the wall and of the instantaneous Reynolds stress −ρuvwere also made. Periods of quiescence in the − ρuvsignal were observed in the viscous sublayer as well as very active periods where ratios of peak to mean values as high as 30:1 occurred.
Symmetrical flow past a uniformly accelerated circular cylinder
Authors: Collins W. M. , Dennis S. C. R.
Published: March 2006

The flow normal to an infinite circular cylinder which is uniformly accelerated from rest in a viscous fluid is considered. The flow is assumed to remain symmetrical about the direction of motion of the cylinder. Two types of solution are presented. In the first an expansion in powers of the time from the start of the motion is given which extends the results of boundary-layer theory by taking into account corrections for finite Reynolds numbers. Physical properties of the flow for small times and finite but large Reynolds numbers are calculated from this expansion. In the second method of solution the Navier-Stokes equations are integrated by an accurate procedure which is a logical extension of the solution in powers of the time. Results are obtained forR2= 97·5, 5850, 122 × 103and ∞, whereRis the Reynolds number. This is defined asR= 2a(ab)½/v, whereais the radius of the cylinder,bthe uniform acceleration andvthe kinematic viscosity of the fluid. The methods are in good agreement for small times.

The numerical method of integration has been carried to moderate times and various flow properties have been calculated. The growth of the length of the separated wake behind the cylinder forR2= 97·5, 5850 and 122 × 103is compared with the results of recent experimental measurements. The agreement is only moderate forR2= 97·5 but it improves greatly asRincreases. The numerical integrations were continued in each case until the implicit method of integration failed to converge, which terminated the procedure. A secondary vortex appeared on the surface of the cylinder for the caseR2= 122 × 103.
Anomalous drag-reducing phenomenon at a water/fish-mucus or polymer interface
Authors: Ling S. C. , Ling T. Y. J.
Published: March 2006

The sublayer of a turbulent boundary-layer flow with and without polymer additives was measured by a simple optical technique having a high degree of spatial resolution. The reduction of skin friction with polymer additives was found to be due to a decrease in the effective viscosity of water at the interface between the mucus (glycoprotein) or polymer (polyethylene oxide) and the water and a thickening of the laminar sublayer. The rate at which mucus diffuses away from a medium-sized fish is almost constant for all swimming speeds.
On the numerical computation of the optimum profile in Stokes flow
Authors: Bourot J.-M.
Published: March 2006

The body of given volume with the smallest drag in Stokes flow is obtained by making use of theoretical results due to Pironneau. A suitable family of solutions of the Stokes equations is used and the no-slip condition is expressed numerically by a technique of quadratic minimization. We find that the drag on this optimal body is 0·95425 times the drag on the sphere of equal volume.
Universal similarity at high grid Reynolds numbers
Authors: Schedvin J. , Stegen G. R. , Gibson C. H.
Published: March 2006

Kolmogorov's second hypothesis has been examined for the case of turbulence generated behind a very large grid. The turbulent Reynolds number RΛ = 280 was sufficient to obtain a short inertial subrange. The one-dimensional subrange constant a1 = 0·48 ± 0·06 is in agreement with recent determinations made in geophysical flows. Isotropy was tested by comparing the transverse velocity spectrum with the transverse spectrum predicted from the longitudinal spectrum using the isotropic relations. The comparison showed the flow to be isotropic everywhere except at the largest scales.

It was observed that at high wavenumbers the spectra were attenuated by effects of finite wire length. The wire-length corrections suggested by Wyngaard (1968) were found to be inadequate. New corrections based on experimentally determined universal spectra are proposed.
One-dimensional shock turbulence in a compressible fluid
Authors: Tatsumi Tomomasa , Tokunaga Hiroshi
Published: March 2006

The interactions of weak nonlinear disturbances in a compressible fluid including shocks, expansion waves and contact surfaces are investigated by making use of the reductive perturbation method. It is found that the nonlinear waves belonging to different families of characteristics behave almost independently of each other, while those belonging to the same family are governed by either the Burgers equation or the equation of heat conduction. Thus the statistical properties of one-dimensional shock turbulence in a compressible fluid are reduced to those of the solutions of the Burgers equation. In particular, the law of energy decay of shock turbulence is shown to be identical to that of Burgers turbulence.
Transition to time-dependent convection
Authors: Clever R. M. , Busse F. H.
Published: March 2006

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.