bluuebunny/crossref_metadata_2025 huggingface dataset:
The structure of the Stewartson layers in a gas centrifuge. Part 2. Insulated side wall
Authors: Matsuda Takuya , Takeda Hidenori
Published: April 2006

The Stewartson E½- and E¼-layers in a rapidly rotating compressible fluid are considered within the framework of linearized equations and the boundary-layer method. The fluid is contained in a cylinder made of a thermally insulated side wall and conducting top and bottom end plates. The end plates and the side wall rotate with slightly different angular velocities. The case of an incompressible fluid was discussed by Stewartson, who found that the flow is restricted to the side-wall boundary layers. In the case of a compressible fluid, however, the solutions are strongly dependent upon the thermal boundary conditions assumed on the side wall. In particular, if the wall is insulated the fluid in the inner core is dragged along too since it is coupled strongly to the flow in the side-wall Stewartson layers. The critical parameter governing the solutions is found to be $(\gamma -1) PrG_0 E^{-\frac{1}{3}}/4\gamma$, where γ is the ratio of specific heats, Pr the Prandtl number, G0 the square of the rotational Mach number and E the Ekman number.
Towards an optimum yacht sail
Authors: Wood C. J. , Tan S. H.
Published: April 2006

This paper contains a simple and entirely conventional application of lifting-line theory to a close-hauled, high aspect ratio sail, set upright on a flat water surface. In seeking an optimum spanwise loading distribution however, an unconventional criterion is used, which leads to an unexpected result.

By seeking to maximize the forward thrust for a given rolling moment, rather than maximizing the lift-drag ratio, an optimum loading distribution is obtained which departs dramatically from elliptic loading. It offers the prospect of a significant improvement in extreme yacht performance if the associated mechanical and control problems can ever be solved.
Flow between a stationary and a rotating disk with suction
Authors: Wilson Lynn O. , Schryer N. L.
Published: April 2006

The equations for the viscous flow between two coaxial infinite disks, one stationary and the other rotating, are solved numerically. The effects of applying a uniform suction through the rotating disk are determined. Initially, the fluid and disks are at rest. The angular velocity of one disk and the amount of suction through it are gradually increased to specific values and then held constant. At large Reynolds numbersR, the equilibrium flow approaches an asymptotic state in which thin boundary layers exist near both disks and an interior core rotates with nearly constant angular velocity. We present graphs of the equilibrium flow functions forR= 104and various values of the suction parametera(0 ≤a≤ 2). Whena= 0, the core rotation rate ωcis 0·3131 times that of the disk. Fluid near the rotating disk is thrown centrifugally outwards. Asaincreases, ωcincreases and the centrifugal outflow decreases. Whena> 1·3494, the core rotation rate exceeds that of the disk and the radial flow near the rotating disk is directed inwards. We also present accurate tabular results for two flows of special interest: (i) the flow between a stationary and a rotating disk with no suction (a= 0) and (ii) Bödewadt flow. The latter can be obtained by suitable scaling of the boundary-layer solution near the stationary disk for anya≥ 0. Since several solutions to the steady-state equations of motion have been reported, the question arises as to whether other solutions to the time-dependent equations of motion with zero initial conditions can be found. We exhibit a rotational start-up scheme which leads to an equilibrium solution in which the interior fluid rotates in a direction opposite to that of the disk.
Laser anemometer study of flow development in curved circular pipes
Authors: Agrawal Y. , Talbot L. , Gong K.
Published: April 2006

An experimental investigation was carried out of the development of steady, laminar, incompressible flow of a Newtonian fluid in the entry region of a curved pipe for the entry condition of uniform motion. Two semicircular pipes of radius ratios 1/20 and 1/7 were investigated, covering a Dean number range from 138 to 679. The axial velocity and the component of secondary velocity parallel to the plane of curvature of the pipe were measured using laser anemometry. It was observed that, in the upstream region where the boundary layers are thin compared with the pipe radius, the axial velocity within the irrotational core first develops to form a vortex-like flow. In the downstream region, characterized by viscous layers of thickness comparable with the pipe radius, there appears to be three-dimensional separation at the inner wall. There is also an indication of an additional vortex structure embedded within the Dean-type secondary motion. The experimental axial velocity profiles are compared with those constructed from the theoretical analyses of Singh and Yao & Berger. The quantitative agreement between theory and experiment is found to be poor; however, some of the features observed in the experiment are in qualitative agreement with the theoretical solution of Yao & Berger.
Wind-wave generation by energy and momentum flux to the forced components of a wave field
Authors: Hasselmann Dieter E.
Published: April 2006

The response of surface gravity waves with profile ζ to forcing by the atmospheric pressure p(a) is studied. It is shown that not only the usually considered cross-spectrum 〈p(a)ζ〉 is responsible for wave growth. All higher-order spectra 〈p(a)ζ1,…,ζn〉 also lead to wave growth linear in ε = ρ(a)/ρ,ρ(a) and ρ being the density of air and water respectively. The bispectral contribution is investigated in detail and is calculated for three models of the bispectrum. The results indicate that bispectral contributions are not negligible and may account for typically 30-50%, and possibly more, of the usually considered cross-spectral contribution. The bispectral growth mechanism contains the fluctuating-stress mechanism discussed by Longuet-Higgins (1969) as a special case. As a byproduct new information on the symmetry properties of the hydrodynamic coupling off the dispersion shell is obtained.
The ‘Hurst phenomenon’ in grid turbulence
Authors: Helland K. N. , Atta C. W. Van
Published: April 2006

Measurements of the statistical property called the ‘rescaled range’ in grid-generated turbulence exhibit a Hurst coefficient H = 0·5 for 43 < UT/M < 1850, where M/U is a characteristic time scale associated with the grid size M and mean velocity U. Theory predicts that H = 0·5 for independence of two observations separated by a time interval T, and the deviation from H = 0·5 is referred to as the ‘Hurst phenomenon’. The rescaled range obtained for grid turbulence contains an initial region UT/M < 43 of large H, approaching 1·0, corresponding approximately to the usual region of a finite non-zero autocorrelation of turbulent velocity fluctuations. For UT/M > 1850 the rescaled range breaks from H = 0·5 and rises at a significantly faster rate, H = 0·7-0·8, implying a long-term dependence or possibly non-stationarity at long times. The measured autocorrelations remain indistinguishable from zero for UT/M > 20. The break in the trend H = 0·5 is probably caused by motions on scales comparable to characteristic time scales of the wind-tunnel circulation. Rescaled-range analysis is a powerful statistical tool for determining the time scale separating the grid turbulence from the background wind-tunnel motions.
A universal Strouhal number for the ‘locking-on’ of vortex shedding to the vibrations of bluff cylinders
Authors: Griffin Owen M.
Published: April 2006

It is well known that the vortices shed from a circular cylinder lock on in frequency to the vibrations when the cylinder is forced to vibrate or is naturally excited to sufficient amplitudes by flow-induced forces. This paper presents a model for a universal wake Strouhal number, valid in the subcritical range of Reynolds numbers, for both forced and vortex-excited oscillations in the locking-on regime. The Strouhal numbers thus obtained are constant atSt*= 0·178 over the range of wake Reynolds numbersRe*= 700-5 × 104. This value is in good agreement with the results obtained by Roshko (1954a) and Bearman (1967) for stationary circular cylinders and other bluff bodies in the same range of Reynolds numbers. A correspondence between the amplification of the cylinder base pressure, drag and vortex circulation is demonstrated over a wide range of frequencies and for vibration amplitudes up to a full cylinder diameter (peak to peak). The fraction ε of the shed vorticity in the individual vortices is found to be dependent upon the base-pressure parameter K = (1 −Cpb)½. Consequently, ε is also a function of the amplitude and frequency of the vibrations in the locking-on regime.
Numerical solutions and laser-Doppler measurements of spin-up
Authors: Warn-Varnas Alex , Fowlis William W. , Piacsek Steve , Lee Sang Myung
Published: April 2006

The spin-up flow in a cylinder of homogeneous fluid has been examined both experimentally and numerically. The primary motivation for this work was to check numerical solution schemes by comparing the numerical results with laboratory measurements obtained with a rotating laser-Doppler velocimeter. The laser-Doppler technique is capable of high accuracy with small space and time resolution, and disturbances of the flow are virtually negligible. A series of measurements was made of the zonal flow over a range of Ekman numbers (1·06 × 10−3≤E≤ 3·30 × 10−3) and Rossby numbers (0·10 [les ]|ε| [les ] 0·33) at various locations in the interior of the flow. These measurements exceed previous ones in accuracy. The weak inertial modes excited by the impulsive start are detectable. The numerical simulations used the primitive equations in axisymmetric form and employed finite-difference techniques on both constant and variable grids. The number of grid points necessary to resolve the Ekman layers was determined. A thorough comparison of the simulations and the experimental measurements is made which includes the details of the amplitude and frequency of the inertial modes. Agreement to within the experimental tolerance is achieved. Analytical results for conditions identical to those in the experiments are not available but some similar linear and nonlinear theories are also compared with the experiments.
Experimental study of the velocity field in Rayleigh-Bénard convection
Authors: Dubois M. , Bergé P.
Published: April 2006

Local velocity measurements performed in a convecting layer of fluid show that the velocity field can be described by a dominant fundamental velocity mode mixed with an increasing proportion of second and third harmonics as ε, the reduced distance to the convective thresholdRc, is increased from 0 to ∼ 10. The spatial and thermal dependences of the amplitudes of these different modes are reported and compared with theoretical predictions.
Bubble plumes in stratified environments
Authors: Mcdougall Trevor J.
Published: April 2006

This paper is concerned with the behaviour of buoyant plumes driven by a source of bubbles. It is shown experimentally that, when a bubble plume rises through a stratified environment, fluid can be transported vertically for some distance and then some of this fluid can leave the plume and spread out horizontally at its own density level. A simple plume model which regards the plume as a single entity is discussed in order to make a first assessment of the effects of gas expansion and bubble slip velocity in this stratified case. However, the experiments reveal a more complicated plume structure in which the bubbles remain in the centre part of the plume, and only the outer part of the plume spreads out into the environment at certain levels. On the basis of these observations a double-plume model is proposed which regards the plume as being composed of two parts: an inner circular plume (which contains all the bubbles of gas) and an outer annular plume.
Bulk stresses due to deformation of the electrical double layer around a charged sphere
Authors: Russel William B.
Published: April 2006

A charged particle suspended in an electrolyte solution attracts ions of opposite charge and repels those of like charge. The surface charge and the resulting distributed charge in the fluid comprise an electrical double layer. When a shear flow deforms the diffuse part of the double layer from equilibrium, stresses are generated which make the effective viscosity of the suspension greater than it would be if there were no charges present. In this paper these stresses are calculated for a dilute dispersion of spheres which have small surface charges and which are surrounded by thin double layers. The viscosity is predicted to be Newtonian in extensional flow but shear-thinning with non-zero normal-stress differences in shear flow. For more complex flows a constitutive equation couples the bulk stress directly to the micro-structural deformation responsible for non-Newtonian effects.
On vortex sound at low Mach number
Authors: Möhring W.
Published: April 2006

A transformation is described which relates the sound generated by low Mach number flow to the flow vorticity. For compact flow fields the apparent sound source is of quadrupole type and linear in the vorticity and therefore also linear in the flow velocity. This scheme is applied to the sound generated by the interaction of two identical thin vortex rings. Then a flow field with a number of compact vortices is discussed. It is found that each vortex can be replaced acoustically by a dipole related to the impulse of the vortex, plus the quadrupole just mentioned plus a spherically symmetric sound source related to the energy of the vortex. An application to low Mach number free-space turbulence shows that the generated sound is related to the vorticity correlation tensor.
Effect of free-stream turbulence on large structure in turbulent mixing layers
Authors: Chandrsuda C. , Mehta R. D. , Weir A. D. , Bradshaw P.
Published: April 2006

Flow-visualization investigations and correlation measurements show that the essentially two-dimensional structures which dominated the turbulent mixing layer of Brown & Roshko (1974) are formed only if the free-stream turbulence is low. If free-stream disturbances are significant, as is likely in most practical cases, including a mixing layer entraining ‘still air’ from the surroundings, three-dimensionality develops at an early stage in transition. Other recent experiments strongly suggest that the Brown-Roshko structure will not form if the initial mixing layer is turbulent or subject to instability modes other than spanwise vortices. Therefore the Brown-Roshko structure will be rare in practice. The alternative large structure in a mixing layer, found by several workers, is intense, but fully three-dimensional and thus less orderly than the Brown-Roshko structure.

The balance of evidence suggests that if the Brown-Roshko structure does appear it will eventually relax into the alternative fully three-dimensional form: the Kármán vortex street behind a bluff body provides a precedent for slow development of three-dimensionality. However the Brown-Roshko structure, if formed, may well relax so slowly as to be identifiable for the full length of a practical flow.
The growth of duration-limited wind waves
Authors: Mitsuyasu Hisashi , Rikiishi Kunio
Published: April 2006

Laboratory measurements have been made of the one-dimensional spectra of the duration-limited wind waves which are generated when a wind abruptly begins to blow over a water surface, maintaining a constant speed during the succeeding period of time. The duration dependences of the wave energy E and the spectral peak frequency fm determined from the measured spectra are slightly different from those inferred from the fetch dependences of these quantities. The normalized spectra of the duration-limited wind waves are also slightly different from those of fetch-limited wind waves: the concentration of the normalized spectral energy near the spectral peak frequency is smaller, in many cases, for the duration-limited wind waves than for fetch-limited wind waves. The exponential growth rates β of the duration-limited wind-wave spectra are generally larger than those of fetch-limited wind-wave spectra. Furthermore, both for the duration-limited wind waves and for fetch-limited wind waves the exponential growth rate has a behaviour which is different from the empirical formula of Snyder & Cox (1966). A new empirical formula for the growth rate of the wave spectrum is proposed, from which the empirical formula of Snyder & Cox (1966) can be derived as a special case. Agreement between the new empirical formula and the experimental results is satisfactory for fetch-limited wave spectra, but is confined to the qualitative features for the duration-limited wave spectra.
Slip flow past a tangential flat plate at low Reynolds numbers
Authors: Tamada Ko , Miura Hiroyuki
Published: April 2006

In this paper the slip flow of viscous fluid at low Reynolds numbers past a flat plate aligned with the flow is studied theoretically on the basis of Oseen-Stokes equations of motion. An integral equation for the distribution of fundamental singularities representing the plate is derived and solved approximately in the vicinity of the edge and main portion of the plate. A formula for the local skin friction is obtained and discussed numerically. It is also shown that the slippage of the flow gives rise to reduction of the drag force on the plate by an amount O(K|ln K|), where K is the Knudsen number. The velocity change near the edge of the plate is of particular interest and is found to be logarithmically singular there.
Magnetic instabilities of a rotating gas
Authors: Acheson D. J. , Gibbons M. P.
Published: April 2006

We explore two types of instability which may develop when a highly conducting gas rotates rapidly in the presence of a radial gravitational force and an azimuthal magnetic field. Beyond a critical radius (equal to twice the isothermal scale height) a decrease of magnetic flux (per unit mass) outwards leads to the appearance of eastward-propagating waves by the mechanism of ‘magnetic buoyancy’. Within the critical radius an increase of magnetic flux outwards leads to westward-propagating waves by a totally different mechanism. Provided that the effects of Ohmic dissipation are not too large, either instability may set in for quite modest magnetic flux gradients, even when the magnetic energy of the system is very much smaller than the rotational energy.
Stability of the Stewartson layer in a rapidly rotating gas
Authors: Hashimoto Kiyoshi
Published: April 2006

The linear stability of the Stewartson layer in a compressible fluid is studied. The viscosity and the heat conductivity are shown to be negligible for a special kind of infinitesimal disturbance. The basic equations of the disturbance are shown to reduce to those for a Boussinesq fluid subject to a virtual radial stratification. A Miles-type sufficient condition for stability and a Howard-type semicircle theorem are derived. The growth rates of unstable modes with wavenumber and shear strength are summarized in stability diagrams for typical cases. The results clarify the situation in which the stability of the Stewartson layer is governed by a balance between the shear strength and the temperature stratification in the layer.
Theory of the almost-highest wave. Part 2. Matching and analytic extension
Authors: Longuet-Higgins M. S. , Fox M. J. H.
Published: April 2006

Most methods of calculating steep gravity waves (of less than the maximum height) encounter difficulties when the radius of curvature R at the crest becomes small compared with the wavelength L, or some other typical length scale. This paper describes a new method of calculation valid when R/L is small.

For deep-water waves, a parameter ε is defined as equal to q/2½c0, where q is the particle speed at the wave crest, in a frame of reference moving with the phase speed c. Hence ε is of order (R/L)½. Three zones are distinguished: (1) an inner zone of linear dimensions ε2L near the crest, where the flow is described by the inner solution found previously by Longuet-Higgins & Fox (1977); (2) an outer zone of dimensions O(L) where the flow is given by a perturbed form of Michell's solution for the highest wave; and (3) a matching zone of width O(L). The matching procedure involves complex powers of ε.

The resulting expression for the square of the phase velocity is found to be
\[
c^2 = (g/k)\{1.1931-1.18\epsilon^3\cos(2.143\ln \epsilon + 2.22)\}
\]
(see figures 5a, b), which is in remarkable agreement with independent calculations based on high-order series. In particular, the existence of turning-points in the phase velocity as a function of wave height is confirmed.

Similar expressions, valid to order ε3, are found for the wave height, the potential and kinetic energies and the momentum flux or impulse of the wave.

The velocity field is extended analytically across the free surface, revealing the existence of branch-points of order ½, as predicted by Grant (1973).
Laminar throughflow between closely spaced rotating disks
Authors: Szeri A. Z. , Adams M. L.
Published: April 2006

Laminar throughflow between finite parallel disks, one stationary and the other rotating, can be characterized by four dimensionless parameters in the general case. But, if the ratio of disk spacing to disk radius is small, an approximate definition of the flow may be made with reference to a single parameter, the Ekman number. The equations of motion and the equation of continuity in this ‘thin-film’ approximation are reduced here to an initial-value problem for nonlinear ordinary differential equations, by a Galerkin-type procedure. Hamming's modified predictor-corrector method is employed subsequently to solve for the stream functions. Radial pressure profiles of this solution are compared with published experimental data. The calculated results may be applied to the design of hydrostatic bearings and face seals.
Shock waves at a fast-slow gas interface
Authors: Abd-El-Fattah A. M. , Henderson L. F.
Published: April 2006

This paper presents the results of experiments on plane shock waves refracting at air/SF6and He/CO2interfaces. These are called fast-slow gas combinations because the speed of sound in the incident shock gas is greater than that in the transmitting shock gas. Our work was based on a generalization of the von Neumann (1943) classification of shocks into two classes called weak and strong. We introduced two subclasses of each of these, giving in all four groups of phenomena for study. This is possibly an exhaustive list, at least for conditions where the gases are approximately perfect. We present data on all four groups and study various transition conditions both within and across the groups. Our results appear to conflict with a previously reported irregular refraction; in fact we could apparently completely suppress this wave system by attention to our gas purity and boundary conditions. In its place we found a different system which appears to be a new phenomenon. We found another new system which has the appearance of a Mach-reflexion type of refraction but with its shock dispersed into a band of wavelets. It is interesting that the wavelets remain intense enough to induce identifiable vortex sheets in the flow. Finally we found yet another refraction of the Mach-reflexion type which had no detectable vortex sheet emanating from the triple point: such a system was foreshadowed by von Neumann.
Theory of vortex nutation and amplitude oscillation in an inviscid shear instability
Authors: Miura Akira , Sato Tetsuya
Published: April 2006

In the first part of this paper the nonlinear development of the most unstable mode is numerically studied for a bounded shear layer with a hyperbolic-tangent profile. It is found that the vortex nutation, discovered by Zabusky & Deem (1971) for a jet profile, is a manifestation of strongly coupled oscillations in the vortex amplitude and the phase. In the second part, with the aid of the numerical result we devote ourselves to deriving coupled nonlinear equations that describe the amplitude oscillation, the vortex nutation and the momentum transport. The approximate oscillatory solution for the vortex amplitude and phase in the nonlinear stage is compared with the numerical solution and agreement is found.
The heat/mass transfer to a finite strip at small Péclet numbers
Authors: Ackerberg R. C. , Patel R. D. , Gupta S. K.
Published: April 2006

The problem of heat transfer (or mass transfer at low transfer rates) to a strip of finite length in a uniform shear flow is considered. For small values of the Péclet number (based on wall shear rate and strip length), diffusion in the flow direction cannot be neglected as in the classical Leveque solution. The mathematical problem is solved by the method of matched asymptotic expansions and expressions for the local and overall dimensionless heat-transfer rate from the strip are found. Experimental data on wall mass-transfer rates in a tube at small Péclet numbers have been obtained by the well-known limiting-current method using potassium ferrocyanide and potassium ferricyanide in sodium hydroxide solution. The Schmidt number is large, so that a uniform shear flow can be assumed near the wall. Experimental results are compared with our theoretical predictions and the work of others, and the agreement is found to be excellent.
The fine-scale structure of the turbulent velocity field
Authors: Champagne F. H.
Published: April 2006

The existence of universal similarity of the fine-scale structure of turbulent velocity fields and the validity of the original Kolmogorov local similarity theory and the later reformulations were investigated. Recent studies of the fine-scale velocity field for many different flows, e.g. grid flows, wakes, jets and the atmospheric boundary layer, are shown to provide considerable evidence for the existence of Kolmogorov normalized spectral shapes which are universal in the sense that they describe the high wave-number spectral behaviour of all turbulent flow fields with a similar value of the turbulence Reynolds numberRλ. The normalized spectral shapes vary withRλin a manner consistent with the later reformulations. The Reynolds number dependence of the normalized spectra is demonstrated for theRλrange from about 40 to 13 000. Expressions for the Kolmogorov normalized spectral functions are presented for three values ofRλ. Also revealed in this study is the importance of considering effects on spectra caused by deviations from Taylor's approximation in high intensity turbulent flows. Lumley's (1965) model is used to correct the high frequency portion of the measured one-dimensional spectra for these effects. An analytical solution to Lumley's expression is presented and applied to the data.
Particle motions near the bottom in turbulent flow in an open channel
Authors: Sumer B. Mutlu , Oguz Beyhan
Published: April 2006

A photographic technique has been used to make observations of the motion of heavy suspended particles in an open channel with a smooth bottom. The observations showed that the path of an individual heavy particle consists of an alternation between upward and downward paths traced by the particle as it travels close to the bottom. The path records appear to reveal most of the flow patterns near the wall shown up by earlier visualization observations; the measured kinematical quantities concerning the particle motion are in accord with those of earlier observations (e.g. Nychas, Hershey & Brodkey 1973). Making use of the present observations and following Offen & Kline's (1975) model of the bursting process in turbulent boundary layers, an attempt was made to explain the mechanism of particle suspension close to the wall in turbulent flows.
Extended Stokes series: laminar flow through a loosely coiled pipe
Authors: Dyke Milton Van
Published: April 2006

Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms. Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number. An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers. For curvature ratios no greater than$\frac{1}{250} $, experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers. In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed. This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation.
Experiments on centrifugally driven thermal convection in a rotating cylinder
Authors: Hudson J. L. , Tang Daniel , Abell Steven
Published: April 2006

Heat-transfer measurements have been carried out in a right circular cylinder of fluid which is heated from above and rotated steadily about its vertical axis. Convection is produced relative to solid-body rotation through the coupling of the centrifugal acceleration and density variations in the fluid. Two silicone oils having kinematic viscosities of 350 cS and 0·65 cS were used in the experiments. In the former case viscous forces are important throughout the cylinder whereas in the latter case Ekman layers form and the Coriolis acceleration controls the interior flow.

With the 350 cS oil the Nusselt number for heat transfer from the top to the bottom of the cylinder is a function ofGrωandr0, whereGrωis a Grashof number defined by employing the centrifugal acceleration evaluated at the outer edge of the cylinder in place of the gravitational acceleration, andr0is the cylinder aspect ratio.

The behaviour is quite different for the 0·65 cS oil. Ekman layers form on the horizontal surfaces and heat is convected by Ekman suction. The Nusselt number is given by\[ Nu = 4.16\beta^{0.822}\epsilon^{-0.499}r_0^{0.173},\quad Ac\leqslant 0.025,\quad\sigma\beta\epsilon^{-\frac{1}{2}} > 0.7, \]where β is the thermal Rossby number, ε is the Ekman number, σ is the Prandtl number, andAcis the ratio of gravitational to centrifugal accelerations. This is consistent with previous theories which indicate that the system should depend on the parameters σβε−½andr0in the limit as ε and β approach zero.
Symmetrical flow past a double wedge at high subsonic Mach numbers
Authors: Chattot Jean-Jacques
Published: April 2006

Symmetrical flow past a double wedge is studied for a high subsonic Mach number. The main features of the flow field are discussed in the physical and the hodograph plane. It is shown how a regular solution to the hodograph equation exhibits limit lines in the physical plane, which are eliminated through a ‘cut’ corresponding to the shock wave. In the case of the wedge it is assumed that the shock which is likely to be the most stable is the weakest possible.

The boundary-value problem is solved in the hodograph plane using Telenin's method, which has proved to be successful when dealing with equations of mixed type. The bounded analytic solution which is thus constructed is regular in the hodograph plane but presents three folds in the physical plane.

The computation is carried out for flow past a 4·5° half-angle wedge at a Mach numberM∞= 0·89. These figures are chosen so that the problem may be justifiably treated by potential theory, the entropy gradient behind the shock being negligible. In this case the mapping of the solution into the physical plane gives the pressure distribution along the double wedge, the sonic line and the shock wave. Of particular interest is the point where the sonic line meets the shock. This configuration is in agreement with the hypothesis of Nocilla, according to which the shock terminates in the supersonic domain. Experimental evidence cannot be obtained, however, because of the lack of resolution in this region.
A note on the conservation of the volume flux in free turbulence
Authors: Kotsovinos N. E.
Published: April 2006

The equation usually used to describe the conservation of volume flux in free turbulence (such as mixing layers, jets, etc.) is discussed.
Measurement of high frequency capillary waves on steep gravity waves
Authors: Chang John H. , Wagner Richard N. , Yuen Henry C.
Published: April 2006

The properties of high frequency capillary waves generated by steep gravity waves on deep water have been measured with a high resolution laser optical slope gauge. The results have been compared with the steady theory of Longuet-Higgins (1963). Good qualitative agreement is obtained. However, the quantitative predictions of the capillary wave slopes cannot be verified by the data because the theory requires knowledge of an idealized quantity - the crest curvature of the gravity wave in the absence of surface tension - which cannot be measured experimentally.
Long nonlinear internal waves in channels of arbitrary cross-section
Authors: Grimshaw R. H. J.
Published: April 2006

Using a two-layer fluid model, equations are developed which describe long internal waves propagating along a channel of arbitrary cross-section. Expressions for the phase speed of these waves are derived in terms of geometric properties of the cross-section. When weakly nonlinear effects are balanced by weak dispersion, a Korteweg-de Vries equation is derived to describe the waves. The effects of a slowly varying cross-section are included. Applications of the theory are made to recent observations of internal surges in lakes.
Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation
Authors: Kelly R. E. , Pal D.
Published: April 2006

Thermal convection in a fluid contained between two rigid walls with different mean temperatures is considered when either spatially periodic temperatures are prescribed at the walls or surface corrugations exist. The amplitudes of the spatial non-uniformities are assumed to be small, and the wavelength is set equal to the critical wavelength for the onset of Rayleigh-Bénard convection. For values of the mean Rayleigh number below the classical critical value, the mean Nusselt number and the mean flow are found as functions of Rayleigh number, Prandtl number, and modulation amplitude. For values of the Rayleigh number close to the classical critical value, the effects of the non-uniformities are greatly amplified, and the amplitude of convection is then governed by a cubic equation. This equation yields three supercritical states, but only the state linked to a subcritical state is found to be stable.
Free-surface breakdown in a rapidly rotating liquid
Authors: Scott W. E.
Published: April 2006

It is shown that the wavelets which appear on the inertial wave form of the inner free surface of a fully spun-up cylindrical mass of liquid contained in a vertical, rapidly rotating and gyrating gyrostat are capillary waves. It is further shown that the interaction between these capillary waves and the excited inertial waves is not the mechanism which effects an observed two-period collapse (‘breakdown’) and reappearance of the free-surface inertial wave form. Rather, the two-period breakdown can be explained by the conjecture that it is a beat phenomenon arising from the interaction of two differently structured inertial wave modes, which have the same frequency at small amplitudes of oscillation of the gyrostat but which, owing to the dependence of the inertial mode frequency on the amplitude of the gyrostatic motion, have slightly different frequencies at larger amplitudes of oscillation of the gyrostat.
The uniform distortion of thermal and velocity mixing layers
Authors: Keffer J. F. , Kawall J. G. , Hunt J. C. R. , Maxey M. R.
Published: April 2006

Two similar problems are considered: what is the effect of applying a uniform and constant rate of strain (i) to the two-dimensional thermal mixing region in a homogeneous grid-generated turbulent field, and (ii) to the two-dimensional velocity mixing region formed between two uniform streams moving with different mean velocities? The imposed strain field is orientated so as to compress or separate the isothermal and isokinetic surfaces in the plane of interest.

Two theoretical models are presented; in the first, the profiles of temperature and velocity are assumed to be self-preserving and an assumption is made about the velocity scales; in the second, the statistical, rapid-distortion approach to dispersion due to Hunt & Mulhearn (1973) is applied. The circumstances in which these models differ and those where the simpler self-preserving model can be applied are determined. The measurements presented here indicate that the widths of both mixing layers decrease within the strain field, the width of the thermal mixing layer decreasing at a greater rate than that of the velocity mixing layer. However, the measured length scales were found to be 5% larger than the scales predicted by either of the analyses, which differed from each other by 5%. It is suggested that selective amplification of the energy-containing eddies by the strain field is responsible.
Ground effects on pressure fluctuations in the atmospheric boundary layer
Authors: Gibson M. M. , Launder B. E.
Published: April 2006

Proposals are made for modelling the pressure-containing correlations which appear in the transport equations for Reynolds stress and heat flux in a simple way which accounts for gravitational effects and the modification of the fluctuating pressure field by the presence of a wall. The predicted changes in structure are shown to agree with Young's (1975) measurements in a free stratified shear flow and with the Kansas data on the atmospheric surface layer.
Statistical features of heat transfer in grid-generated turbulence: constant-gradient case
Authors: Venkataramani K. S. , Chevray R.
Published: April 2006

Turbulence produced by a grid which simultaneously imparts a mean temperature profile varying linearly with height was investigated in a specially constructed wind tunnel. While the mean temperature profile is preserved downstream of the grid in accordance with the theory of Corrsin (1952), the downstream evolution of the r.m.s. temperature fluctuation is at variance with his prediction. The reason for this discrepancy is shown to lie in the neglect of molecular diffusivity, which leads to unbounded growth of the fluctuations. Along with conventional correlations and spectra, the filtered heat-transfer correlation is presented. About 60% of the heat transport is accomplished by the low wavenumber components having length scales equal to or larger than the integral scale. An intriguing feature of the present experiments is the presence of an inertial-convective subrange for the temperature field notwithstanding the low Reynolds number and the consequent absence of an inertial subrange for the velocity field. Experimental results show that the temperature has a positive skewness everywhere in contrast to the velocity components, which are symmetrically distributed. Measurements of the joint probability density function of the vertical component of the velocity and the temperature indicate that, while the assumption of joint normality is not uniformly valid, the conditional expectations nearly follow the normal law. Marginal and joint moments of up to fourth order are presented. Odd-order joint moments are clearly sensitive to the skewness of the temperature.
Measurements of turbulent flow downstream of a rearward-facing step
Authors: Etheridge D. W. , Kemp P. H.
Published: April 2006

Measurements have been made in a water channel of the flow in and around the separation region due to a rearward-facing step. Detailed profiles of mean velocities, turbulence intensities and Reynolds shear stress are presented. The turbulence measurements reveal the development of a new shear layer, which splits at reattachment with about one-sixth of the mass flow deflected upstream. The new shear layer is associated with a region of roughly constant values of both the non-dimensional mixing lengthl/xand the shear correlation coefficientK. The mixing-length ratio is larger than that found in plane mixing layers, whereas the shear coefficient is roughly the same. There is strong evidence that near the wall the length scales increase more rapidly with distance from the wall than in an attached boundary layer, and that a local maximum occurs.
Intermittency effects in a numerical simulation of stationary three-dimensional turbulence
Authors: Siggia Eric D. , Patterson G. S.
Published: April 2006

A Navier-Stokes direct spectral simulation code was modified to produce stationary and nearly isotropic turbulence in three dimensions. An approximate$-\frac{5}{3}$energy spectrum was maintained over the entire range of wavenumbers by simultaneously driving the fluid and supplementing the ordinary viscosity with a subgrid-like energy sink in the last 15% of the spectrum. Half-tone and contour plots of the fluctuations in the vorticity, rate-of-strain tensor and helicity show increasing ‘spottiness’ as the system evolves in time. Probability distributions and cross-correlations among these three quantities were also obtained. The flatness factor of the longitudinal velocity derivative, the longitudinal structure functions and the fluctuations in the locally averaged dissipation rate are consistent with some degree of intermittency, but do not unambiguously demonstrate its presence in the simulated flows.
Absorption of sound by homogeneous turbulence
Authors: Noir D. T. , George A. R.
Published: April 2006

The following problem is treated: given a plane acoustic wave propagating through an unbounded field of turbulence, calculate the amount of acoustic energy converted into turbulent kinetic energy. The fluid velocities due to the acoustic waves and the turbulence are assumed to be small compared with the speed of sound. Thus the sound-turbulence interaction is weak and the turbulent field may be considered to be incompressible. The analysis is based on the interaction of two opposite effects: the acoustic distortion of the turbulence (producing anisotropic Reynolds stresses) and the redistribution of the kinetic energy among components (tendency towards isotropy) and among wavenumbers (energy cascade and dissipation). These phenomena are described using semi-empirical turbulence arguments. It is seen that the simplest model for the redistribution among components is not sufficient for unsteady flows. A more complete model is used which is modified to agree with the exact instantaneous distortion analysis of Ribner [map ] Tucker to first order. Owing to the two redistribution effects, the Reynolds stress behaves inelastically and is out of phase with the acoustic field. Thus there is an average production of turbulent energy corresponding to the absorption of acoustic energy and attenuation of the incident wave. For nearly isotropic turbulence, the attenuation coefficient is found to be proportional to the rate of viscous dissipation and independent of the frequency.

In order to compare the theory with experiment several constants involved in the semi-empirical model of the turbulence must be found. Owing to the lack of better information these constants are estimated here by order-of-magnitude considerations. No existing experiments correspond to the homogeneous turbulence assumed by the theory. Comparison with the few reasonably applicable experiments shows qualitative agreement though the importance of the turbulent absorption is generally of nearly the same order as the measurement error. Several discrepancies between jet noise experiments and aerodynamic noise predictions may be roughly explained using the above analysis.
Study of three-dimensional unsteady Oseen flow
Authors: Murata S. , Miyake Y. , Tsujimoto Y. , Yamamoto F.
Published: April 2006

In the present paper, it is intended to give the elementary solutions of three-dimensional unsteady Oseen flow when unsteady concentrated lift and/or drag is applied in the flow field. It is shown that the pressure fields due to concentrated impulsive lift and/or drag can be represented by an impulsive pressure doublet in the direction of the applied force and the corresponding velocity fields by diffusing free doublets in the direction of the external force that are shed from the location of the force application and convected downstream with otherwise uniform velocity. It is also confirmed that combination of the elementary solutions given in the present paper yields the two-dimensional ones.
Non-Newtonian flow characteristics in a steady two-dimensional flow
Authors: Gatski Thomas B. , Lumley John L.
Published: April 2006

The two-dimensional steady flow of a non-Newtonian fluid (a dilute polymer solution) is examined. The flow domain is composed of a parallel-walled inflow region, a contraction region in which the walls are rectangular hyperbolae, and a parallel-walled outflow region. The problem is formulated in terms of the vorticity, stream function and appropriate rheological equation of state, i.e. an Oldroyd-type constitutive equation (with no shear-thinning) for the total shear and normal-stress components. Computational results from the numerical solution of the equations are presented. In particular, the molecular extension and pressure distribution along the centre-line are presented as well as contour plots of the different flow variables. The alignment of the molecules with the principal axes of strain rate is shown by a qualitative comparison of the streamwise normal-stress contours with contours of the eigenvalues of the strain-rate matrix.
Deformation and breakup of a single slender drop in an extensional flow
Authors: Acrivos Andreas , Lo T. S.
Published: April 2006

The deformation and conditions for breakup of a single slender drop placed symmetrically in a uniaxial extensional flow are examined theoretically. For the case of an inviscid drop in zero-Reynolds-number flow, Buckmaster (1972) showed, using slender-body analysis, that the shape of the drop is given byr≡εR(z)=ε(1−|z|ν)/2ν, where εεγ/Gμland γ is the interfacial tension,Gthe strength of the extensional flow, μ the viscosity of the suspending fluid andlthe drop half-length; alsov= ½P− 1, wherePis the unknown constant pressure inside the drop rendered dimensionless with respect toGμ. By requiring thatRbe analytic atz= 0, Buckmaster then concluded thatvhad to be an even integer and thereby obtained a countably infinite set of slender profiles for any (large) value of the flow strengthG. In the present work, the expression forR(z)shown above is obtained readily using the method of inner and outer expansions, the method failing when |z|[les ]O(ε) and ν is not an even integer. Thus, in general, a new solution is needed to describe the shape within the ‘singular’ region |z|[les ]O(ε). The requirement that the two solutions match in their domain of overlap then leads to the conclusion thatvcan be either equal to 2 or greater than or equal to 3. However, a stability analysis reveals that only the solution withv= 2 is stable, and hence a unique shape exists.

Next, drops of low viscosity μi=O(ε2μ) are examined in zero-Reynolds-number flow. Here, again, a unique solution is obtained according to which a steady shape cannot exist if (Gμa/γ) (μi/μ)⅙> 0·148, where$a \equiv(3V4\pi)^{\frac{1}{3}}$andVis the volume of the drop. This breakup criterion is identical to that found by Taylor (1964). A similar analysis for the case of an inviscid drop in a flow with non-zero Reynolds number shows that drop breakup will occur if$(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}} > 0.284$, where ρ is the density of the suspending fluid. Finally, when μi=O(ε2μ) and inertial effects are neglected within the drop but retained in the surrounding fluid, the critical value of$(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}}$required for drop breakup is found as a function of the dimensionless group\[ (\rho a\gamma/\mu^2)^{\frac{1}{5}}(\mu/\mu_i)^{\frac{1}{6}}, \]which depends only on the physical properties of the system and the size of the drop. These last two results are the first which take into account inertial effects in determining the deformation and breakup conditions of a drop placed in a shear field.