bluuebunny/crossref_metadata_2025 huggingface dataset:
Boundary interactions for two-dimensional granular flows. Part 1. Flat boundaries, asymmetric stresses and couple stresses
Authors: Campbell Charles S.
Published: April 2006

The behaviour of a granular flow at a boundary cannot be specified independently of what is happening in the rest of the flow field. This paper describes a study of two fictitious, but instructive, flat boundary types using a computer simulation of a two-dimensional granular flow with the goal of trying to understand the possible effects of the boundary on the flow. The two boundary conditions, Type A and Type B, differ largely in the way that they apply torques to the flow particles. During a particle–wall collision, the Type A boundary applies the force at the particle surface, thus applying the largest mechanistically possible torque to the particle, while the Type B boundary applies the force directly to the particle centre, resulting in the application of zero torque. Though a small change on continuum scales (i.e. the point at which the force is applied has only been moved by a particle radius) it makes a huge difference to the macroscopic behaviour of the system. Generally, it was found that, near boundaries, large variations in continuum properties occur over distances of a particle diameter, a non-continuum scale, throwing into doubt whether boundaries may be accurately modelled via continuum mechanics. Finally, the large torques applied to the particles by the Type A boundary induce asymmetries in the stress tensor, which, in these steady flows, are balanced by gradients in a couple stress tensor. Thus, near boundaries, a frictional granular material must be modelled as a polar fluid.
Stability of a ridge of fluid
Authors: Hocking Leslie M. , Miksis Michael J.
Published: April 2006

The stability and nonlinear evolution of a ridge of fluid on an inclined plane is investigated. This model was introduced by Hocking (1990). Here we present numerical solutions of the model showing the evolution of the ridge and in some cases the formation of droplets. Also, we investigate the linear stability of the fluid ridge allowing for contact-line motion. We find a preferred wavelength for the linear stability of spanwise disturbances.
Oscillatory flow in a stepped channel
Authors: Tutty O. R. , Pedley T. J.
Published: April 2006

Two-dimensional, unsteady flow of a viscous, incompressible fluid in a stepped channel has been studied by the numerical solution of the Navier–Stokes equation using an accurate finite-difference method.

With a sinusoidal mass flow rate, the problem has three governing parameters: the Reynolds number, the Strouhal number, and the step height. The effects on the flow of varying all three parameters has been investigated systematically. In appropriate parameter regimes, a strong ‘vortex wave’ is generated during the forward phase when the flow is over the step into the expansion. Secondary effects on the wave can result in a complex flow pattern with each major structure of the flow consisting of an eddy with more than one core. No such wave is found during the reverse phase of the flow. The generation and development of the wave is examined in some detail, and our results are compared and contrasted with those of previous studies, both experimental and theoretical, of flow in non-uniform vessels.
Trapped internal gravity waves in a geostrophic boundary current
Authors: Ma Hong
Published: April 2006

The effect of a geostrophic boundary current on internal gravity waves is studied with a reduced-gravity model. We found that the boundary current not only modifies the coastal Kelvin wave, but also forms wave guides for short internal gravity waves. The combined effects of current shear, the boundary, and the slope of the interface create the trapping mechanism. These trapped internal gravity waves appear as groups of discrete zonal modes. They have wavelengths comparable to or shorter than the internal Rossby radius of deformation. Their phase speeds are close to that of the internal Kelvin wave. However, they can propagate both in, or opposite to, the direction of the Kelvin wave. The results of the present work suggest the possibility of finding an energetic internal gravity wave phenomenon with near-inertial frequency in a broad geostrophic boundary current.
Stability of the interface in co-extrusion flow of two viscoelastic fluids through a pipe
Authors: Chen Kang Ping , Zhang Yi
Published: April 2006

The linear stability of co-extrusion flow of two upper convected Maxwell fluids through a pipe at low Reynolds numbers is studied for arbitrary wavelength disturbances. The fluids interface can become unstable due to various mechanisms. It is shown that elasticity of the fluids can stabilize the capillary instability and a linearly stable interface can be achieved by appropriate choice of controlling parameters.
On the ‘δ-equations’ for vortex sheet evolution
Authors: Rottman James W. , Stansby Peter K.
Published: April 2006

We use a set of equations, sometimes referred to as the ‘δ-equations’, to approximate the two-dimensional inviscid motion of an initially circular vortex sheet released from rest in a cross-flow. We present numerical solutions of these equations for the case with δ2 = 0 (for which the equations are exact) and for δ2 > 0. For small values of the smoothing parameter δ a spectral filter must be used to eliminate spurious instabilities due to round-off error. Two singularities appear simultaneously in the vortex sheet when δ2 = 0 at a critical time tc After tc the solutions do not converge as the computational mesh is refined. With δ2 > 0, converged solutions were found for all values of δ2 when t < tc, and for all but the two smallest values of δ2 used when t > tc. Our results show that when δ2 > 0 the vortex sheet deforms into two doubly branched spirals some time after tc The limiting solution as δ→0 clearly exists and equals the δ = 0 solution when t < tc. For t > tc, the limiting solution appears to exist if only the converged solutions are used, but it is unclear what relation this limiting solution has to any δ2 = 0 solution for these times.
Infragravity waves induced by short-wave groups
Authors: Schäffer Hemming A.
Published: April 2006

A theoretical model for infragravity waves generated by incident short-wave groups is developed. Both normal and oblique short-wave incidence is considered. The depth-integrated conservation equations for mass and momentum averaged over a short-wave period are equivalent to the nonlinear shallow-water equations with a forcing term. In linearized form these equations combine to a second-order long-wave equation including forcing, and this is the equation we solve. The forcing term is expressed in terms of the short-wave radiation stress, and the modelling of these short waves in regard to their breaking and dynamic surf zone behaviour is essential. The model takes into account the time-varying position of the initial break point as well as a (partial) transmission of grouping into the surf zone. The former produces a dynamic set-up, while the latter is equivalent to the short-wave forcing that takes place outside the surf zone. These two effects have a mutual dependence which is modelled by a parameter K, and their relative strength is estimated. Before the waves break, the standard assumption of energy conservation leads to a variation of the radiation stress, which causes a bound, long wave, and the shoaling bottom results in a modification of the solution known for constant depth. The respective effects of this incident bound, long wave and of oscillations of the break-point position are shown to be of the same order of magnitude, and they oppose each other to some extent. The transfer of energy from the short waves to waves at infragravity frequencies is analysed using the depth-integrated conservation equation of energy. For the case of normally incident groups a semi-analytical steady-state solution for the infragravity wave motion is given for a plane beach and small primary-wave modulations. Examples of the resulting surface elevation as well as the corresponding particle velocity and mean infragravity-wave energy flux are presented. Also the sensitivity to the variation of input parameters is analysed. The model results are compared with laboratory experiments from the literature. The qualitative agreement is good, but quantitatively the model overestimates the infragravity wave activity. This can, in part, be attributed to the neglect of frictional effects.
The topographic control of planetary-scale flow
Authors: Woods Andrew W.
Published: April 2006

We develop a theory to describe the topographic control of planetary-scale flows resulting from the variation of the Earth's rotation with latitude. We show that on passing over topography, an inertial, zonal current on an equatorial β-plane may pass through a control at which the flow changes from a subcritical to a supercritical solution branch. Downstream of this control, a transition back to the subcritical solution branch may occur, for example, by the generation of planetary eddies or radiating Rossby waves. We calculate the energy dissipated across such a transition and discuss the relevance of this theory for a number of atmospheric and oceanic phenomena. We also show that this phenomenon is analogous to the hydraulic control of a non-rotating, stratified flow passing through a channel of variable width.
Transonic nozzle flow of dense gases
Authors: Kluwick A.
Published: April 2006

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Navier–Stokes equations supplemented with realistic equations of state for gases which have relatively large specific heats a novel form of the viscous transonic small-perturbation equation is derived. Evaluation of the inviscid limit of this equation shows that three sonic points rather than a single sonic point may occur during isentropic expansion of such media, in contrast to the case of perfect gases. As a consequence, a shock-free transition from subsonic to supersonic speeds cannot, in general, be achieved by means of a conventional converging–diverging nozzle. Nozzles leading to shock-free flow fields must have an unusual shape consisting of two throats and an intervening antithroat. Additional new results include the computation of the internal thermoviscous structure of weak shock waves and a phenomenon referred to as impending shock splitting. Finally, the relevance of these results to the description of external transonic flows is discussed briefly.
Dispersed-phase stress tensor in flows of bubbly liquids at large Reynolds numbers
Authors: Sangani A. S. , Didwania A. K.
Published: April 2006

We derive averaged equations for large Reynolds number laminar flows of gas–liquid dispersions accounting for slowly varying spatial and temporal fields. In particular, we obtain an exact expression for the dispersed-phase stress tensor to be used in the force balance equation for gas bubbles and illustrate its application by evaluating the stress tensor for a few special cases. It is shown that the dispersed-phase stress tensor gradient with respect to the mean relative motion or the void fraction for the uniformly random bubbly liquids under conditions of large Reynolds number laminar flows is negative and thus has a destabilizing influence on the dynamics of void fraction waves in bubbly liquids.
Effects of wavelength ratio on wave modelling
Authors: Zhang Jun , Hong Keyyong , Yue Dick K. P.
Published: April 2006

The efficacy of perturbation approaches for short–long wave interactions is examined by considering a simple case of two interacting wave trains with different wavelengths. Frequency-domain solutions are derived up to third order in wave steepness using two different formulations: one employing conventional wave-mode functions only, and the other introducing a modulated wave-mode representation for the short-wavelength wave. For long-wavelength wave steepness and short-to-long wavelength ratio ε1 and ε3 respectively, the two results are shown to be identical for ε1 [Lt ] ε3 < 0.5. As ε1 approaches ε3, the conventional wave-mode approach converges slowly and eventually diverges for ε1 [Gt ] ε3. The loss of convergence is because the linear phase of conventional wave-mode functions is ineffective for modelling the modulated phase of the short wave. As expected, this difficulty can be removed by using a modulated wave-mode function for the short wave. On the other hand, for relatively large ε3 ∼O(1), the conventional wave-mode approach converges rapidly while the slowly varying interaction between the two waves cannot be accurately predicted by the present modulated wave-mode approach. These findings have important implications to (time-domain) numerical simulations of the nonlinear evolution of ocean wave fields, and suggest that a hybrid wave model employing both conventional (for large-ε3 interactions) and modulated (for small-ε3 interactions) wave-mode functions should be particularly effective.
Low-frequency scattering of Kelvin waves by continuous topography
Authors: Johnson E. R.
Published: April 2006

This paper continues the analysis of Johnson (1990, hereinafter referred to as I) of the scattering of Kelvin waves by collections of ridges and valleys. General results, flow patterns and explicit solutions follow by restricting attention to waves whose period is long compared to the inertial period but without the additional further simplification introduced in I of approximating general features by stepped topography. A simple direct method is presented giving explicit formulae for the amplitude of the transmitted Kelvin wave and the scattered topographic long waves. A simple but accurate approximation to the solution is also given. The accuracy and usefulness of the apparently crude method of I are confirmed and a superior method presented for choosing the positions of steps in the approximation of general topography. Inviscid flows, the effects of weak dissipation and weak stratification, the form and relevance of the short-wave field over downslopes, the partition of mass and energy flux between the long-wave and short-wave fields and the size and form of higher-order effects are also discussed.
Feedback control of vortex shedding at low Reynolds numbers
Authors: Roussopoulos Kimon
Published: April 2006

This paper describes experiments undertaken to study in detail the control of vortex shedding from circular cylinders at low Reynolds numbers by using feedback to stabilize the wake instability. Experiments have been performed both in a wind tunnel and in an open water channel with flow visualization. It has been found that feedback control is able to delay the onset of the wake instability, rendering the wake stable at Reynolds numbers about 20% higher than otherwise. At higher flow rates, however, it was not possible to use single-channel feedback to stabilize the wake - although, deceptively, it was possible to reduce the unsteadiness recorded by a near-wake sensor. When control is applied to a long span only the region near the control sensor is controlled. The results presented in this paper generally support the analytical results of other researchers.
Multiple large-scale coherent mode interactions in a developing round jet
Authors: Lee Sang Soo , Liu J. T. C.
Published: April 2006

The integral energy method has been used to study the nonlinear interactions of the large-scale coherent structure in a spatially developing round jet. The streamwise development of a jet is obtained in terms of the mean flow shear-layer momentum thickness, the wave-mode kinetic energy and the wave-mode phase angle. With the energy method, a system of partial differential equations is reduced to a system of ordinary differential equations. The nonlinear differential equations are solved with initial conditions which are given at the nozzle exit. It is shown that the initial wave-mode energy densities as well as the initial phase angles play a significant role in the streamwise evolution of the large-scale coherent wave modes and the mean flow.
Highly accelerated, free-surface flows
Authors: Longuet-Higgins Michael S.
Published: April 2006

Accelerations exceeding 20gin surface waves have been observed both in experiments and in numerically computed flows with a free surface. The present paper describes a family of analytic solutions which display such behaviour. They are expressible in parametric form as z =Fsinh ω + iGcosh ω + γω + iH, whereF, GandHare functions of the timetonly, and γ is linear int. ω is a complex parameter which is real at the free surface. The functionsF(t) andG(t) satisfy two nonlinear, coupled ODEs, which can be solved numerically. Typically the solutions pass through an ‘inertial shock’, or singularity in the time, where the displacements vary ast2/3, the velocities ast1/3 and the accelerations ast-4/3. In this class of solution the free surface develops a cusp ast→ ∞. In a special case,FandGvary ast4/7and the cusp is reached in finite time. Gravity is neglected, but plays a part in setting up the initial conditions for the highly accelerated flow.

In future papers it will be shown that more general solutions exist in which the acceleration is momentarily large but bounded.
Viscous flow normal to a flat plate at moderate Reynolds numbers
Authors: Dennis S. C. R. , Qiang Wang , Coutanceau M. , Launay J.-L.
Published: April 2006

An experimental and numerical investigation of the two-dimensional flow normal to a flat plate is described. In the experiments, the plate is started impulsively from rest in a channel for Reynolds numbers, based on the breadth of the plate, in the range 5 ≤ Re ≤ 20. Over this range of Re the flow remains symmetrical and stable and tends to a steady state but is shown to depend strongly on the ratio λ of the plate to channel breadth. The evolution of the experimental flow with time and Reynolds number is studied and the variation with λ in the range 0.05 ≤ λ ≤ 0.2 is investigated sufficiently to enable an estimate of properties of the flow as λ → 0 to be obtained for the steady-state flow. The numerical results are obtained for steady flow normal to a flat plate in an unbounded fluid for Reynolds numbers up to Re = 100. They supplement and extend results for this flow obtained for values of Re up to 20 by Hudson & Dennis (1985). The present solutions have been found using a vorticity-stream function formulation rather than the primitive-variable approach of Hudson & Dennis and provide an independent check on these results. A comparison of the theoretical results for Re ≤ 20 with the limit λ → 0 of the experimental results is, generally speaking, extremely satisfactory.
Mach reflection of a large-amplitude solitary wave
Authors: Tanaka Mitsuhiro
Published: April 2006

Reflection of an obliquely incident solitary wave by a vertical wall is studied numerically by applying the ‘high-order spectral method’ developed by Dommermuth & Yue (1987). According to the analysis by Miles (1977a, b) which is valid when ai [Lt ] 1, the regular type of reflection gives way to ‘Mach reflection’ when ai/(3ai)½ ≤ 1, Where ai is the amplitude of the incident wave divided by the quiescent water depth d and ψi is the angle of incidence. In Mach reflection, the apex of the incident and the reflected waves moves away from the wall at a constant angle (ψ*, say), and is joined to the wall by a third solitary wave called ‘Mach stem’. Miles model predicts that the amplitude of Mach stem, and so the run-up at the wall, is 4ai when ψi = (3ai)½.

Our numerical results shows, however, that the effect of large amplitude tends to prevent the Mach reflection to occur. Even when the Mach reflection occurs, it is ‘contaminated’ by regular reflection in the sense that all the important quantities that characterize the reflection pattern, such as the stem angle ψ*, the angle of reflection ψr, and the amplitude of the reflected wave ar, are all shifted from the values predicted by Miles’ theory toward those corresponding to the regular reflection, i.e. ψ* = 0, ψr = ψi, and ar = ai. According to our calculations for ai = 0.3, the changeover from Mach reflection to regular reflection happens at ψi ≈ 37.8°, which is much smaller than (3ai)½ = 54.4°, and the highest Mach stem is observed for ψi = 35° (ψi/(3ai)½ = 0.644). Although the ‘four-fold amplification’ is not observed for any value of ψi considered here, it is found that the Mach stem can become higher than the highest two-dimensional steady solitary wave for the prescribed water depth. The numerical result is also compared with the analysis by Johnson (1982) for the oblique interaction between one large and one small solitary wave, which shows much better agreement with the numerical result than the Miles’ analysis does when ψi is sufficiently small and the Mach reflection occurs.
Measurements of the primary instabilities of film flows
Authors: Liu Jun , Paul Jonathan D. , Gollub J. P.
Published: April 2006

We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h(x, y, t) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number Rc for the onset of waves as a function of angle β. The measurements of Rc(β) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noisesustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f). A new phase boundary f*s(R) is measured, which separates the regimes of saturated finite amplitude waves (at high f) from multipeaked solitary waves (at low f). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves.
Excitation of capillary waves by longer waves
Authors: Watson Kenneth M. , Mcbride John B.
Published: April 2006

At low wind speeds the shortest capillary waves appear to be generated hydrodynamically and not by the wind. This phenomenon is investigated using a Hamiltonian representation of the surface wave dynamics. A perturbation technique of Kolmogorov is used to transform away non-resonant, nonlinear interactions. Resonant interactions are treated by the Hasselmann transport equation, applied to the transformed variables. Calculated spectra show reasonable agreement with the observations of Jähne & Riemer (1990).
The mean drift force and yaw moment on marine structures in waves and current
Authors: Grue John , Palm Enok
Published: April 2006

The effect of the steady second-order velocities on the drift forces and moments acting on marine structures in waves and a (small) current is considered. The second-order velocities are found to arise due to first-order evanescent modes and linear body responses. Their contributions to the horizontal drift forces and yaw moment, obtained by pressure integration at the body, and to the yaw drift moment, obtained by integrating the angular momentum flux in the far field, are expressed entirely in terms of the linear first-order solution. The second-order velocities may considerably increase the forward speed part of the mean yaw moment on realistic marine structures, with the most important contribution occurring where the wave spectrum often has its maximal value. The contribution to the horizontal forces obtained by pressure integration is, however, always found to be small. The horizontal drift forces obtained by the linear momentum flux in the far field are independent of the second-order velocities, provided that there is no velocity circulation in the fluid.
Microstructure suspended in three-dimensional flows
Authors: Szeri Andrew J. , Leal L. Gary
Published: April 2006

The dynamical behaviour of stretchable, orientable microstructure suspended in a general three-dimensional fluid flow is investigated. Model equations given by Olbricht, Rallison & Leal (1982) are examined in the case of microstructure travelling through arbitrarily complicated flows of the carrier fluid. As in the two-dimensional analysis of Szeri, Wiggins & Leal (1991), one must first treat the orientation dynamics problem; only then can the equation for stretch of the microstructure be analyzed rationally. In three-dimensional flows that are steady in the Lagrangian frame, attractors for the orientation dynamics are shown to be equilibria or limit cycles; this asymptotic behaviour was first deduced by Bretherton (1962). In three-dimensional flows that are time periodic in the Lagrangian frame (e.g. recirculating flows), the orientation dynamics may be characterized by periodic or quasi-periodic attractors. Thus, robust (generic) behaviour in these cases is always characterized by a single global attractor; there is no asymptotic dependence of orientation dynamics on the initial orientation. The type of asymptotic orientation dynamics – steady, periodic, or quasi-periodic - is signified by a simple criterion. Details of the relevant bifurcations, as well as history-dependent strong flow criteria are developed. Examples which illustrate the various types of behaviour are given.
Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number
Authors: Zhang Keke , Gubbins David
Published: April 2006

We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.
Mixing by a turbulent plume in a confined stratified region
Authors: Cardoso Silvana S. S. , Woods Andrew W.
Published: April 2006

An experimental and theoretical study of the mixing produced by a plume rising in a confined stratified environment is presented. As a result of the pre-existing stable stratification, the plume penetrates only part way into the region; at an intermediate level it intrudes laterally forming a horizontal layer. As time evolves, this layer of mixed fluid is observed to increase in thickness. The bottom front advects downward in a way analogous to the first front in the filling box of Baines & Turner (1969), while the lateral spreading of the plume occurs at an ever-increasing level and an ascending top front results. We develop a model of this stratified filling box; the model predicts the rate at which the two fronts advance into the environment.

It is found that stratification in the environment, when smooth, has no significant influence on the dynamics of the descending front. We show that the rate of rise of the ascending front is determined by the turbulent mixing occurring at the spreading level. Entrainment of environmental fluid from above into the overshooting plume is significant; as a result, a density interface develops at this level. Asymptotically, the system reaches a state in which a bottom convecting layer, with an almost homogeneous density, deepens in a stratified background. The model proposed for this large-time behaviour is based on the simple energetic formulation that a constant fraction of the kinetic energy supplied by the plume, for mixing across the interface, is converted into potential energy of the convective layer. Our experimental results suggest an efficiency of approximately 50 % for this conversion.

We discuss our results in the light of previous studies on turbulent penetrative convection and conclude that the theory developed should be valid for an intermediate range of values of the Richardson number characterizing the dynamic conditions at the interface. The model is applied quantitatively to the process of cooling of a room wherein stratification is relevant. The geological problem of replenishment of a magma chamber by a light input of magma is also analysed.
Dynamic simulations of flows of bubbly liquids at large Reynolds numbers
Authors: Sangani A. S. , Didwania A. K.
Published: April 2006

Results of dynamic simulations of bubbles rising through a liquid are presented. The Reynolds number of the flow based on the radius and the terminal speed of bubbles is large compared to unity, and the Weber number, which is the ratio of inertial to surface tension forces, is small. It is assumed that the bubbles do not coalesce when they approach each other but rather bounce instantaneously, conserving the momentum and the kinetic energy of the system. The flow of the liquid is assumed to be irrotational and is determined by solving the many-bubble interaction problem exactly. The viscous force on the bubbles is estimated from the rate of viscous energy dissipation. It is shown that the random state of bubbly liquids under these conditions is unstable and that the bubbles form aggregates in planes transverse to gravity. These aggregates form even when the size distribution of the bubbles is non-uniform. While the instability results primarily from the nature of inertial interaction among pairs of bubbles, which causes them to be attracted toward each other when they are aligned in the plane perpendicular to gravity, it is shown that the presence of viscous forces facilitates the process.
Particle-driven gravity currents
Authors: Bonnecaze Roger T. , Huppert Herbert E. , Lister John R.
Published: April 2006

Gravity currents created by the release of a fixed volume of a suspension into a lighter ambient fluid are studied theoretically and experimentally. The greater density of the current and the buoyancy force driving its motion arise primarily from dense particles suspended in the interstitial fluid of the current. The dynamics of the current are assumed to be dominated by a balance between inertial and buoyancy forces; viscous forces are assumed negligible. The currents considered are two-dimensional and flow over a rigid horizontal surface. The flow is modelled by either the single- or the two-layer shallow-water equations, the two-layer equations being necessary to include the effects of the overlying fluid, which are important when the depth of the current is comparable to the depth of the overlying fluid. Because the local density of the gravity current depends on the concentration of particles, the buoyancy contribution to the momentum balance depends on the variation of the particle concentration. A transport equation for the particle concentration is derived by assuming that the particles are vertically well-mixed by the turbulence in the current, are advected by the mean flow and settle out through the viscous sublayer at the bottom of the current. The boundary condition at the moving front of the current relates the velocity and the pressure head at that point. The resulting equations are solved numerically, which reveals that two types of shock can occur in the current. In the late stages of all particle-driven gravity currents, an internal bore develops that separates a particle-free jet-like flow in the rear from a dense gravity-current flow near the front. The second type of bore occurs if the initial height of the current is comparable to the depth of the ambient fluid. This bore develops during the early lock-exchange flow between the two fluids and strongly changes the structure of the current and its transport of particles from those of a current in very deep surroundings. To test the theory, several experiments were performed to measure the length of particle-driven gravity currents as a function of time and their deposition patterns for a variety of particle sizes and initial masses of sediment. The comparison between the theoretical predictions, which have no adjustable parameters, and the experimental results are very good.
The eddy structure in Stokes flow in a cavity
Authors: Shankar P. N.
Published: April 2006

Stokes flow in a two-dimensional cavity of rectangular section, induced by the motion of one of the walls, is considered. A direct, efficient calculational procedure, based on an eigenfunction expansion, is used to study the eddy structure in the cavity. It is shown that some of the results of earlier studies are quantitatively in error. More importantly, two interesting questions, namely the extent of the symmetry of the corner eddies and their relationship to the large-eddy structure are settled. By carefully examining the rather sudden change in the main eddy structure for cavities of depth around 1.629, it is shown that the main eddies are formed by the merger of the primary corner eddies; the secondary corner eddies then become the primary corner eddies and so on. Thus, in the evolution of the large-eddy structure the corner eddies, in some sense, play the role of progenitors. This explicit prediction should be experimentally verifiable.
Dynamics of liquid jets revisited
Authors: Schulkes R. M. S. M.
Published: April 2006

In this paper we investigate the long-wavelength approximations of the equations governing the motion of an inviscid liquid jet. Using a formal perturbation expansion it will be shown that the one-dimensional equations presented by Lee (1974) are inconsistent. The inconsistency arises from the fact that terms which have been retained in the boundary conditions should have been rejected in view of the approximations made in the momentum equations. With the correct equations a number of anomalies between Lee's model and other models are eliminated. An explicit periodic solution to the nonlinear evolution equations we have derived is presented. However, it turns out that the wavenumbers for which this solution is valid lie outside the range in which the long-wavelength approximations are applicable. In addition we present numerical solutions to the nonlinear equations we have derived. In the unstable regime we find that, as disturbances grow, the characteristic axial lengthscales of the major features are typically of the order of the radius of the jet. This casts some doubt on the validity of the long-wavelength approximations in the study of nonlinear liquid jet dynamics.
Momentum and heat transport in the turbulent intermediate wake of a circular cylinder
Authors: Matsumura M. , Antonia R. A.
Published: April 2006

Hot-wire anemometry has been used in the intermediate wake (x/d= 10 to 40) of a slightly heated circular cylinder in order to quantify the contribution from the coherent motion to various conventionally averaged quantities, in particular the average momentum and heat fluxes. The overall contribution to the lateral heat flux is always greater than that to the Reynolds shear stress, indicating that the turbulent Kármán vortex street transports heat more effectively than momentum. The difference in these contributions is reflected in the topologies of the velocity and temperature fields. There is significant streamwise evolution of these topologies throughout the intermediate wake. At x/d = 10, the net heat transport associated with the vortical motion occurs in the downstream region of each vortex. At other downstream stations, the net heat transport is equally distributed between the upstream and downstream regions of individual vortices.
Hydrodynamic forces on a submerged circular cylinder undergoing large-amplitude motion
Authors: Wu G. X.
Published: April 2006

The hydrodynamic problem of a circular cylinder submerged below a free surface and undergoing large-amplitude oscillation is investigated based on the velocity potential theory. The body-surface boundary condition is satisfied on its instantaneous position while the free-surface condition is linearized. The solution is obtained by writing the potential in terms of the multipole expansion. Various interesting results associated with the circular cylinder are obtained.
On the motion of bubbles in a periodic box
Authors: Smereka Peter
Published: April 2006

The motion of spherical bubbles in a box containing an incompressible and irrotational liquid with periodic boundary conditions is studied. Equations of motion are deduced using a variational principle. When the bubbles have approximately the same velocity their configuration is Lyapunov stable, provided they are arranged in such a way so as to minimize the effective conductivity of a composite material where the bubbles are treated as insulators. The minimizing configurations become asymptotically stable with the addition of gravity and liquid viscosity. This suggests that a randomly arranged configuration of bubbles, all with approximately the same velocity, cannot be stable. Explicit equations of motion for two bubbles are deduced using Rayleigh's method for solving Laplace's equation in a periodic domain. The results are extended for more than two bubbles by considering pairwise interactions. Numerical simulations with gravity and liquid viscosity ignored show the bubbles form clusters if initially they are randomly arranged with identical velocities. The clustering is inhibited, however, if the initial velocities are sufficiently different. The variance of the bubbles’ velocities is observed to act like a temperature; a lesser amount of clustering occurs with increasing temperature. Finally, in considering the effects of gravity and liquid viscosity, the long time behaviour is found to result in the formation of bubble clusters aggregated in a plane perpendicular to the direction of gravity.
A numerical nonlinear analysis of the flow around two- and three-dimensional partially cavitating hydrofoils
Authors: Kinnas Spyros A. , Fine Neal E.
Published: April 2006

The partially cavitating two-dimensional hydrofoil problem is treated using nonlinear theory by employing a low-order potential-based boundary-element method. The cavity shape is determined in the framework of two independent boundary-value problems; in the first, the cavity length is specified and the cavitation number is unknown, and in the second the cavitation number is known and the cavity length is to be determined. In each case, the position of the cavity surface is determined in an iterative manner until both a prescribed pressure condition and a zero normal velocity condition are satisfied on the cavity. An initial approximation to the nonlinear cavity shape, which is determined by satisfying the boundary conditions on the hydrofoil surface rather than on the exact cavity surface, is found to differ only slightly from the converged nonlinear result.

The boundary element method is then extended to treat the partially cavitating three-dimensional hydrofoil problem. The three-dimensional kinematic and dynamic boundary conditions are applied on the hydrofoil surface underneath the cavity. The cavity planform at a given cavitation number is determined via an iterative process until the thickness at the end of the cavity at all spanwise locations becomes equal to a prescribed value (in our case, zero). Cavity shapes predicted by the present method for some three-dimensional hydrofoil geometries are shown to satisfy the dynamic boundary condition to within acceptable accuracy. The method is also shown to predict the expected effect of foil thickness on the cavity size. Finally, cavity planforms predicted from the present method are shown to be in good agreement to those measured in a cavitating three-dimensional hydrofoil experiment, performed in MIT's cavitation tunnel.
Finite-amplitude instability of mixed convection
Authors: Rogers B. B. , Moulic S. Ghosh , Yao L. S.
Published: April 2006

The finite-amplitude instability of mixed convection of air in a vertical concentric annulus with each cylinder maintained at a different temperature is studied by use of weakly nonlinear instability theory and by direct numerical simulation. A strictly shear instability and two thermally induced instabilities exist in the parameter space of Reynolds and Grashof numbers. The first thermal instability occurs at low Reynolds numbers as the rate of heating increases, and is called a thermal-shear instability because it is a shear-driven instability induced by thermal effects. The second thermal instability occurs at larger Reynolds number as heating increases, and is also a thermally induced shear instability called the interactive instability. The weakly nonlinear results demonstrate that the thermal-shear instability is supercritical at all wavenumbers. With the shear and interactive instabilities, however, both subcritical and supercritical branches appear on the neutral curves. The validity of the weakly nonlinear calculations are verified by comparison with a direct simulation. The results for subcritical instabilities show that the weakly nonlinear calculations are accurate when the magnitude of the amplification rate is small, but the accuracy deteriorates for large amplification rates. However, the trends predicted by the weakly nonlinear theory agree with those predicted by the direct simulations for a large portion of the parameter space. Analyses of the energy sources for the disturbance show that subcritical instability of the shear and interactive modes occurs at larger wavenumbers because of increased gradient production of disturbance kinetic energy. This is because, at shorter wavelengths, the growth of the wave causes the shape of the fundamental disturbance to change from that predicted by linear instability theory to a shape more favourable for shear-energy production. The results also show that many possibly unstable modes may be present simultaneously. Consequently, all of these modes, as well as all of the possible wave interactions among the modes, must be considered to obtain a complete picture of mixed-convection instability.
On the solution near the critical frequency for an oscillating and translating body in or near a free surface
Authors: Liu Yuming , Yue Dick K. P.
Published: April 2006

We consider a floating or submerged body in deep water translating parallel to the undisturbed free surface with a steady velocity U while undergoing small oscillations at frequency ω. It is known that for a single source, the solution becomes singular at the resonant frequency given by τ ≡ Uω/g=¼, where g is the gravitational acceleration. In this paper, we show that for a general body, a finite solution exists as τ → ¼ if and only if a certain geometric condition (which depends only on the frequency ω but not on U) is satisfied. For a submerged body, a necessary and sufficient condition is that the body must have non-zero volume. For a surface-piercing body, a sufficient condition is derived which has a geometric interpretation similar to that of John (1950). As an illustration, we provide an analytic (closed-form) solution for the case of a submerged circular cylinder oscillating near τ = ¼. Finally, we identify the underlying difficulties of existing approximate theories and numerical computations near τ = ¼, and offer a simple remedy for the latter.
Direct numerical calculation of acoustics: solution evaluation through energy analysis
Authors: Brentner Kenneth S.
Published: April 2006

The propagation of acoustic energy from a sound source to the far field is a fundamental problem of acoustics. In this paper the use of computational fluid dynamics (CFD) to directly calculate the acoustic field is investigated. The two-dimensional, compressible, inviscid flow about an accelerating circular cylinder is used as a model problem. The time evolution of the energy transfer from the cylinder surface to the fluid, as the cylinder is moved from rest to some non-negligible velocity, is shown. Energy is the quantity of interest in the calculations since various components of energy have physical meaning. By examining the temporal and spatial characteristics of the numerical solution, a distinction can be made between the propagating acoustic energy, the convecting energy associated with the entropy change in the fluid, and the energy following the body. In the calculations, entropy generation is due to a combination of physical mechanisms and numerical error. In the case of propagating acoustic waves, entropy generation seems to be a measure of numerical damping associated with the discrete flow solver.
The instability of the steady flow past spheres and disks
Authors: Natarajan Ramesh , Acrivos Andreas
Published: April 2006

We consider the instability of the steady, axisymmetric base flow past a sphere, and a circular disk (oriented broadside-on to the incoming flow). Finite-element methods are used to compute the steady axisymmetric base flows, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for the sphere and the circular disk, the first instability of the base flow is through a regular bifurcation, and the critical Reynolds number (based on the body radius) is 105 for the sphere, and 58.25 for the circular disk. In both cases, the unstable mode is non-axisymmetric with azimuthal wavenumber m = 1. These computational results are consistent with previous experimental observations (Magarvey & Bishop 1961 a, b; Nakamura 1976; Willmarth, Hawk & Harvey 1964).
Observations on transition in plane bubble plumes
Authors: Alam M. , Arakeri V. H.
Published: April 2006

Flow visualization studies of plane laminar bubble plumes have been conducted to yield quantitative data on transition height, wavelength and wave velocity of the most unstable disturbance leading to transition. These are believed to be the first results of this kind. Most earlier studies are restricted to turbulent bubble plumes. In the present study, the bubble plumes were generated by electrolysis of water and hence very fine control over bubble size distribution and gas flow rate was possible to enable studies with laminar bubble plumes. Present observations show that (a) the dominant mode of instability in plane bubble plumes is the sinuous mode, (b) transition height and wavelength are related linearly with the proportionality constant being about 4, (c) wave velocity is about 40% of the mean plume velocity, and (d) normalized transition height data correlate very well with a source Grashof number. Some agreement and some differences in transition characteristics of bubble plumes have been observed compared to those for similar single-phase flows.