bluuebunny/crossref_metadata_2025 huggingface dataset:
Gravity wave radiation from vortex trains in rotating shallow water
Authors: Ford Rupert
Published: April 2006

Gravity wave radiation by vortical flows in the f-plane shallow-water equations is investigated by direct nonlinear numerical simulation. The flows considered are initially parallel flows, consisting of a single strip in which the potential vorticity differs from the background value. The flows are unstable to the barotropic instability mechanism, and roll up into a train of vortices. During the subsequent evolution of the vortex train, gravity waves are radiated. In the limit of small Froude number, the gravity wave radiation is compared with that predicted by an appropriately modified version of the Lighthill theory of aerodynamic sound generation. It is found that the gravity wave field agrees well with that predicted by the theory, provided typical lengthscales of vortical motions are well within one deformation radius.

It is found that the nutation time for vortices in the train increases rapidly with increasing Froude number in cases where the potential vorticity in the vortices is of the same sign as the background value, whereas the nutation time is almost independent of Froude number in cases where the potential vorticity in the vortices is zero or of opposite sign to the background. Consequently, in the former cases, the unsteadiness of the flow decreases with increasing Froude number, so the effect of the inertial cutoff frequency is increased, leading to an optimal Froude number for gravity wave radiation, above which the intensity of the radiated waves decreases as the Froude number is further increased. It is proposed that the existence of a finite range of interaction within the vortices, for flows with positive vortex potential vorticity, may account for the strong dependence of nutation time on Froude number in those cases. The interaction scale within the vortices becomes infinite in the limit of zero vortex potential vorticity, and so the arguments do not apply in those cases.
Transitional flow between contra-rotating disks
Authors: Kilic M. , Gan X. , Owen J. M.
Published: April 2006

This paper describes a combined computational and experimental study of the flow between contra-rotating disks for – 1 ≤ Γ ≤ 0 and Reϕ = 105, where Γ is the ratio of the speed of the slower disk to that of the faster one and Reϕ is the rotational Reynolds number of the faster disk. For Γ = 0, the rotor-stator case, laminar and turbulent computations and experimental measurements show that laminar Batchelor-type flow occurs: there is radial outflow in a boundary layer on the rotating disk, inflow on the stationary disk and a rotating core of fluid between. For Γ = – 1, the laminar computations produce Batchelor-type flow: there is radial outflow on both disks and inflow in a free shear layer in the mid-plane, on either side of which is a rotating core of fluid. The turbulent computations and the velocity measurements for Γ = – 1 show Stewartson-type flow: radial outflow occurs in laminar boundary layers on the disks and inflow occurs in a non-rotating turbulent core between the boundary layers. For intermediate values of Γ, transition from Batchelor-type flow to Stewartson-type flow is associated with a two-cell structure, the two-cells being separated by a streamline that stagnates on the slower disk; Batchelor-type flow occurs radially outward of the stagnation point and Stewartson-type flow radially inward. The turbulent computations are mainly in good agreement with the measured velocities for Γ = 0 and Γ = – 1, where either Batchelor-type flow or Stewartson-type flow occurs; there is less good agreement at intermediate values of Γ, particularly for Γ = – 0.4 where the double transition of Batchelor-type flow to Stewartson-type flow and laminar to turbulent flow occurs in the two-cell structure.
Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity
Authors: Westin K. J. A. , Boiko A. V. , Klingmann B. G. B. , Kozlov V. V. , Alfredsson P. H.
Published: April 2006

The modification of the mean and fluctuating characteristics of a flat-plate boundary layer subjected to nearly isotropic free stream turbulence (FST) is studied experimentally using hot-wire anemometry. The study is focussed on the region upstream of the transition onset, where the fluctuations inside the boundary layer are dominated by elongated flow structures which grow downstream both in amplitude and length. Their downstream development and scaling are investigated and the results are compared with those obtained by previous authors. This allows some conclusions about the parameters which are relevant for the modelling of the transition process. The mechanisms underlying the transition process and the relative importance of the Tollmien–Schlichting wave instability in this flow are treated in an accompanying paper (part 2 of the present report).
Experiments in a boundary layer subjected to free stream turbulence. Part 2. The role of TS-waves in the transition process
Authors: Boiko A. V. , Westin K. J. A. , Klingmann B. G. B. , Kozlov V. V. , Alfredsson P. H.
Published: April 2006

The natural occurrence of Tollmien-Schlichting (TS) waves has so far only been observed in boundary layers subjected to moderate levels of free stream turbulence (Tu < 1%), owing to the difficulty in detecting small-amplitude waves in highly perturbed boundary layers. By introducing controlled oscillations with a vibrating ribbon, it is possible to study small-amplitude waves using phase-selective filtering techniques. In the present work, the effect of TS-waves on the transition is studied at Tu = 1.5%. It is demonstrated that TS-waves can exist and develop in a similar way as in an undisturbed boundary layer. It is also found that TS-waves with quite small amplitudes are involved in nonlinear interactions which lead to a regeneration of TS-waves in the whole unstable frequency band. This results in a significant increase in the number of turbulent spots, which promote the onset of turbulence.
A study of streamwise vortex structure in a stratified shear layer
Authors: Schowalter David G. , Van Atta Charles W. Van , Lasheras Juan C.
Published: April 2006

The existence of an organized streamwise vortical structure, which is superimposed on the well known coherent spanwise vorticity in nominally two-dimensional free shear layers, has been studied extensively. In the presence of stratification, however, buoyancy forces contribute to an additional mechanism for the generation of streamwise vorticity. As the spanwise vorticity layer rolls up and pulls high-density fluid above low-density fluid, a local instability results. The purpose of the current investigation is to force the three-dimensional instability in the stratified shear layer. In this manner, we experimentally observe the effect of buoyancy on the streamwise vortex tube evolution, the evolution of the buoyancy-induced instability, and the interaction between these two vortical structures. A simple numerical model is proposed which captures the relevant physics of the flow evolution. It is found that, depending on the location, streamwise vortices resulting from vortex stretching may be weakened or enhanced by the stratification. Buoyancy-induced vortex structures are shown to form where the unstable part of the interface is tilted by the streamwise vortex tubes. These vortices strengthen initially, then weaken downstream, the timescale for this process depending upon the degree of stratification. For initial Richardson numbers larger than about 0.03, the baroclinically weakened vortex tubes eventually disappear as the flow evolves downstream and the baroclinically generated vortices dominate the three-dimensional flow structure.
Observations of polymer conformation during flow through a fixed fibre bed
Authors: Evans Anthony R. , Shaqfeh Eric S. G. , Frattini Paul L.
Published: April 2006

Linear birefringence measurements of dilute and semi-dilute polyisobutylene solutions following flow through a disordered fixed fibre bed of 2.47% solids volume fraction provide both transient and steady measurements of chain deformation. Our results indicate that the flexible polyisobutylene polymers undergo a large conformation change, stretching in the direction of the average flow. This occurs even though the average flow in the bed is a plug flow which would not cause any polymer stretch by itself. The polymer stretch or conformation change increases with the number of chain interactions with bed fibres and ultimately reaches a steady-state value that can be correlated with the pore-size Deborah number (i.e. a characteristic polymer relaxation time divided by a characteristic flow time in the bed pore). Large changes in the polymer conformation are noted for values of the Deborah number,De> 5. In addition, the time to steady state scales with the characteristic flow time within a pore over a large range of Deborah numbers. The pressure drop across the fibre bed was also measured simultaneously with the birefringence measurement and was found to be directly proportional to the birefringence throughout the range ofDeinvestigated. Thus, we show empirically, for the first time, that chain elongation, which produces normal stress anisotropy within the fluid, is directly responsible for the increased flow resistance. These findings are then analysed in the light of recent theories for the response of polymer molecules in fixed bed flow fields (Shaqfeh & Koch 1992). It is shown that our results are consistent with the interpretation that these flows are stochastic strong flows, which create an apparent ‘coil-stretch’ transition. After extending the theory of Shaqfeh & Koch to account for the specifics in the experiments, including the bed geometry and statistics as well as the polydispersity of the polymer solutions, it is shown that the theory can predict most of the experimental results both qualitatively and quantitatively.
Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet
Authors: Hussein Hussein J. , Capp Steven P. , George William K.
Published: April 2006

The turbulent flow resulting from a top-hat jet exhausting into a large room was investigated. The Reynolds number based on exit conditions was approximately 105. Velocity moments to third order were obtained using flying and stationary hot-wire and burst-mode laser-Doppler anemometry (LDA) techniques. The entire room was fully seeded for the LDA measurements. The measurements are shown to satisfy the differential and integral momentum equations for a round jet in an infinite environment.

The results differ substantially from those reported by some earlier investigators, both in the level and shape of the profiles. These differences are attributed to the smaller enclosures used in the earlier works and the recirculation within them. Also, the flying hot-wire and burst-mode LDA measurements made here differ from the stationary wire measurements, especially the higher moments and away from the flow centreline. These differences are attributed to the cross-flow and rectification errors on the latter at the high turbulence intensities present in this flow (30% minimum at centreline). The measurements are used, together with recent dissipation measurements, to compute the energy balance for the jet, and an attempt is made to estimate the pressure-velocity and pressure-strain rate correlations.
The effect of hydrodynamic interactions on the tracer and gradient diffusion of integral membrane proteins in lipid bilayers
Authors: Bussell Stuart J. , Hammer Daniel A. , Koch Donald L.
Published: April 2006

Biological membranes can be considered two-dimensional fluids with suspended integral membrane proteins (IMPs). We have calculated the effect of hydrodynamic interactions on the various diffusion coefficients of IMPs in lipid bilayers. The IMPs are modelled as hard cylinders of radius a immersed in a thin sheet of viscosity μ and thickness h bounded by a fluid of low viscosity μ′. We have ensemble averaged the N-body Stokes equations to the pair level and have renormalized them following the methods of Batchelor (1972) and Hinch (1977). The lengthscale for the hydrodynamic interactions is λa = μh / μ′, Which is O (100a), and the slow decay of the interactions introduces new features in the renormalizations compared to the analogous analyses for three-dimensional suspensions of spheres.

We have calculated the asymptotic limits for the short- and long-time tracer diffusivities, Ds and Dl, respectively, and for the gradient diffusivity, Dg, for ϕ [Lt ] 1 and λ [Gt ] 1, where ϕ is the IMP area fraction and λ = μh / (μ′a). The diffusivities are
\begin{eqnarray*}
D_s/D_0 &=& 1-2\phi[1-(1+\ln (2)-9/32)/(\ln(\lambda)-\gamma)],
D_l/D_0 &=& D_s/D_0 - 0.07/(\ln(\lambda)-\gamma),
D_g/D_0 &=& 1+\phi[-7+(6\ln(2)+7/16+0.37)/(\ln(\lambda)-\gamma)],
\end{eqnarray*}
where D0 is the diffusivity in the limit of zero area fraction, and γ = 0.577216 is Euler's constant. The results for Dl and Ds differ only slightly. The decrease in Dg/Do as ϕ increases contrasts with the result for spheres for which Dg/Do > 1.
Nonlinear dynamics of viscous droplets
Authors: Becker E. , Hiller W. J. , Kowalewski T. A.
Published: April 2006

Nonlinear viscous droplet oscillations are analysed by solving the Navier-Stokes equation for an incompressible fluid. The method is based on mode expansions with modified solutions of the corresponding linear problem. A system of ordinary differential equations, including all nonlinear and viscous terms, is obtained by an extended application of the variational principle of Gauss to the underlying hydrodynamic equations. Results presented are in a very good agreement with experimental data up to oscillation amplitudes of 80% of the unperturbed droplet radius. Large-amplitude oscillations are also in a good agreement with the predictions of Lundgren & Mansour (boundary integral method) and Basaran (Galerkin-finite element method). The results show that viscosity has a large effect on mode coupling phenomena and that, in contradiction to the linear approach, the resonant mode interactions remain for asymptotically diminishing amplitudes of the fundamental mode.
A model for characterization of a vortex pair formed by shock passage over a light-gas inhomogeneity
Authors: Yang Joseph , Kubota Toshi , Zukoski Edward E.
Published: April 2006

This work investigates the two-dimensional flow of a shock wave over a circular light-gas inhomogeneity in a channel with finite width. The pressure gradient from the shock wave interacts with the density gradient at the edge of the inhomogeneity to deposit vorticity around the perimeter, and the structure rolls up into a pair of counter-rotating vortices. The aim of this study is to develop an understanding of the scaling laws for the flow field produced by this interaction at times long after the passage of the shock across the inhomogeneity. Numerical simulations are performed for various initial conditions and the results are used to guide the development of relatively simple algebraic models that characterize the dynamics of the vortex pair, and that allow extrapolation of the numerical results to conditions more nearly of interest in practical situations. The models are not derived directly from the equations of motion but depend on these equations and on intuition guided by the numerical results. Agreement between simulations and models is generally good except for a vortex-spacing model which is less satisfactory.

A practical application of this shock-induced vortical flow is rapid and efficient mixing of fuel and oxidizer in a SCRAMJET combustion chamber. One possible injector design uses the interaction of an oblique shock wave with a jet of light fuel to generate vorticity which stirs and mixes the two fluids and lifts the burning jet away from the combustor wall. Marble proposed an analogy between this three-dimensional steady flow and the two-dimensional unsteady problem of the present investigation. Comparison is made between closely corresponding three-dimensional steady and two-dimensional unsteady flows, and a mathematical description of Marble's analogy is proposed.
The dynamics of vorticity tubes in homogeneous turbulence
Authors: Vincent A. , Meneguzzi M.
Published: April 2006

Analysis of data from a direct simulation of statistically steady homogeneous turbulence suggests that the vorticity tubes, which constitute the main structure of the vorticity field, are produced by shear instabilities. This is confirmed by a decay calculation, in which vorticity sheets appear at first, and then roll up to form the first tubes. These instabilities seem to be at least as important as vortex stretching in transferring energy from large to small scales.
The analytical solution of the Riemann problem in relativistic hydrodynamics
Authors: Martí José Ma , Müller Ewald
Published: April 2006

We consider the decay of an initial discontinuity in a polytropic gas in a Minkowski space–time (the special relativistic Riemann problem). In order to get a general analytical solution for this problem, we analyse the properties of the relativistic flow across shock waves and rarefactions. As in classical hydrodynamics, the solution of the Riemann problem is found by solving an implicit algebraic equation which gives the pressure in the intermediate states. The solution presented here contains as a particular case the special relativistic shock-tube problem in which the gas is initially at rest. Finally, we discuss the impact of this result on the development of high-resolution shock-capturing numerical codes to solve the equations of relativistic hydrodynamics.
Numerical simulations of stratified inviscid flow over a smooth obstacle
Authors: Lamb Kevin G.
Published: April 2006

Results of numerical simulations of the flow of a non-rotating, inviscid, Boussinesq fluid over smooth two-dimensional obstacles are described. The fluid has finite depth and a rigid lid. Far upstream of the obstacle the horizontal velocity and buoyancy frequency are uniform. Comparisons with linear theory for small-amplitude obstacles are made and the long-time behaviour is compared with steady-state Long's model solutions. Comparisons with the time-dependent results of Baines (1979) are done. For Froude numbers between ½ and 1 the obstacle amplitude is varied in order to determine the amplitudes needed to initiate wave breaking. These results are compared with the predictions of Long's model and with the experimental results of Baines (1977) showing good agreement with the latter. It is found that wave breaking occurs for amplitudes significantly lower than Long's model predicts for a large range of Froude numbers. This is shown to be the result of the generation of large-amplitude lee waves with wavelengths longer than that of stationary lee waves, but not long enough to propagate upstream. The behaviour of these waves is coupled to the generation of both longer mode-one waves which do propagate upstream from the obstacle and to mode-two waves which propagate against the flow as they are advected downstream. It is also coupled to oscillations in the wave drag. The periods of the wave drag oscillations are compared to experimental results showing good agreement with cases for which oscillations have been observed. The behaviour of these large transient lee waves is compared with the theoretical results contained in Grimshaw & Yi (1991), showing some similarities. As the Froude number approaches 0.5 the breaking behaviour is no longer due to these large waves, with the result that wave breaking occurs much later.
The interaction between the mean flow and coherent structures in turbulent mixing layers
Authors: Cohen J. , Marasli B. , Levinski V.
Published: April 2006

The nonlinear interaction between the mean flow and a coherent disturbance in a two-dimensional turbulent mixing layer is addressed. Based on considerations from stability theory, previous experimental results, in particular the modification of the mean velocity profile, the peculiar growth of the forced shear-layer thickness and the spatial growth of the disturbance amplitude, are explained. A model that assumes a quasi-parallel mean flow having a self-similar mean velocity profile is developed. The model is capable of predicting the downstream evolution of turbulent mixing layers subjected to external excitations.
Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics
Authors: Gama S. , Vergassola M. , Frisch U.
Published: April 2006

The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstrated. Such flows, when subject to a very weak large-scale perturbation of wavenumberkwill amplify it with a rate proportional tok2, independent of the direction.

Specifically, it is assumed that the basic (unperturbed) flow is space-time periodic, possesses a centre of symmetry (parity-invariance) and has three- or six-fold rotational invariance to ensure isotropy of the eddy-viscosity tensor.

The eddy viscosities emerging from the multiscale analysis are calculated by two different methods. First, there is an expansion in powers of the Reynolds number which can be carried out to large orders, and then extended analytically (thanks to a meromorphy property) beyond the disk of convergence. Secondly, there is a spectral method. The two methods typically agree within a fraction of 1%.

A simple example, the ‘decorated hexagonal flow’, of a time-independent flow with isotropic negative eddy viscosity is given. Flows with randomly chosen Fourier components and the required symmetry have typically a 30% chance of developing a negative eddy viscosity when the Reynolds number is increased.

For basic flow driven by a prescribed external force and sufficiently strong largescale flow, the analysis is extended to the nonlinear régime. It is found that the largescale dynamics is governed by a Navier-Stokes or a Navier-Stokes-Kuramoto-Sivashinsky equation, depending on the sign and strength of the eddy viscosity. When the driving force is not mirror-symmetric, a new ‘chiral’ nonlinearity appears. In special cases, the large-scale equation reduces to the Burgers equation. With chiral forcing, circular vortex patches are strongly enhanced or attenuated, depending on their cyclonicity.
The current emitted by highly conducting Taylor cones
Authors: De La Mora J. Fernández , Loscertales I. G.
Published: April 2006

When a liquid meniscus held at the exit of a metallic capillary tube is charged to a high voltage V, the free surface often takes the form of a cone whose apex emits a steady microjet, and thus injects a certain charge I and liquid volume Q per unit time into the surrounding gas. This work deals with liquids with relatively large conductivities K, for which the jet diameter dj is much smaller than the diameter dn of the capillary tube. In the limit dj/dn → 0, the structure of the jet (dj and I, in particular) becomes independent of electrostatic parameters such as V or the electrode configuration, being governed mostly by the liquid properties and flow rate Q. Furthermore, the measured current is given approximately by I = f(ε) (γQK/ε)½ for a wide variety of liquids and conditions (ε, and γ are, respectively, the dielectric constant of the liquid and the coefficient of interfacial tension; f(ε) is shown in figure 11). The following explanation is proposed for this behaviour. Convection associated with the liquid flow Q transports the net surface charge towards the cone tip. This upsets the electrostatic surface charge distribution slightly at distances r from the apex large compared to a certain charge relaxation length λ, but substantially when r ∼ λ. When the fluid motion is modelled as a sink flow, λ is of the order of r* = (Qεε0/K)$\frac13$ (ε0 is the electrical permittivity of vacuum). If, in addition, the surface charge density is described through Taylor's theory, the corresponding surface current convected towards the apex scales as Is ∼ (γQK/ε)½, as observed for the spray current. The sink flow hypothesis is shown to be realistic for sufficiently small jet Reynolds numbers. In a few photographs of ethylene glycol cone jets, we find the rough scaling dj ∼ 0.4r* for the jet diameter, which shows that the jet forms as soon as charge relaxation effects set in. In the limit ε [Gt ] 1, an upper bound is found for the convected current at the virtual cone apex, which accounts for only one-quarter of the total measured spray current. The rest of the charge must accordingly reach the head of the jet by conduction through the bulk.
A numerical and experimental study of transition processes in an obstructed channel flow
Authors: Roberts E. P. L.
Published: April 2006

Incompressible Newtonian flow in a two-dimensional channel with periodically placed sharp edged baffles has been studied both by numerical simulation and by experimental flow visualization. The flow was observed to be steady and symmetric at low Reynolds numbers, with recirculating eddies downstream of each baffle. At a critical Reynolds number (based on channel width and cross-sectional mean velocity) of approximately 100 the flow became asymmetric and unsteady. This transition to unsteadiness led to an eddy shedding regime, with eddies formed and shed successively from each baffle. A stability study suggested that the mechanism for transition to unsteady flow is a Kelvin–Helmholtz instability associated with the shear layer formed downstream of the sharp edged baffles. The frequency of the unsteadiness is, however, dependent on the full flow field, and not only the shear layer characteristics. Experimental observations show that the instability is followed by a secondary transition to three-dimensional disordered flow. Experimentally observed flows in the two-dimensional regime were found to be in close agreement with the numerical simulation for both the steady and unsteady flows.
The stability of statically unstable layers
Authors: Thorpe S. A.
Published: April 2006

We investigate the development of instability in a fluid with density locally of the form ρ0[1 −(N2 / g)z + A sin Kz], composed of an overall stable uniform gradient of buoyancy frequency, N, but with a superimposed sinusoidal variation of vertical wavenumber, K, and amplitude, A [Lt ] 1; g is the acceleration due to gravity and z is the upward vertical coordinate. Layers exist in which the fluid is statically unstable when the parameter r = N2 / gKA, is less than unity.

When r is zero, the density is sinusoidal in z and the problem reduces to one studied by Batchelor & Nitsche (1991). Their solution, which finds a gravest mode of linear instability with terms having vertical motions independent of z and with horizontal scales large in comparison with K−1, is extended to non-zero r. An effect of a small, but finite, r is to stabilize the fluid, increasing the critical Rayleigh number and the corresponding non-dimensional horizontal wavenumber. The vertical scale of the mode which first becomes unstable is reduced as r increases. A small sinusoidal shear destabilizes the fluid.

When r approaches unity, the density field contains regions of static instability which are of thickness small compared to K−1. The problem then approximates to one studied by Matthews (1988). Consistent solutions for the growth of disturbances are obtained by truncated series and, following Matthews, by the solution of a Fourier-transformed equation. A small uniform shear, characterized by a flow Reynolds number, Re > O, is found to stabilize the fluid, in that it increases the critical Rayleigh number of the onset of instability. This suggests that convective Rayleigh–Taylor instability, with constant phase lines parallel to the flow, is then the favoured mode of onset of instability. At very large Rayleigh numbers and at a Prandtl number of 700, however, the growth rate of the most rapidly growing linear disturbances may increase as Re increases from zero, and the form of the evolving flow is then less certain.

The theory is used to estimate the scale and growth rates of instability in overturning internal gravity waves in the laboratory experiment described in a companion paper (Thorpe 1994).
Statically unstable layers produced by overturning internal gravity waves
Authors: Thorpe S. A.
Published: April 2006

Internal waves in a uniformly stratified fluid of sufficiently large amplitude develop tilted layers in which the fluid is statically unstable. To investigate the evolution and subsequent development of this structure, experiments are made in which a horizontal rectangular tube containing a fluid of uniform density gradient is gently rocked at a selected frequency about a horizontal axis normal to the tube length. Large-amplitude standing internal gravity waves of the first mode are generated, and these steepen and overturn, the isopycnal surfaces folding to produce a vertically thin and horizontally extensive layer in which the fluid is statically unstable. In experiments with relatively small forcing, the layer persists for some 6 buoyancy periods, with no detected evidence of secondary instability, and static stability is re-established as the periodic flow reverses. The layer however breaks down, with consequent diapycnal mixing, when greater forcing is applied.

The scale and growth rates of instability in the overturning internal gravity waves are estimated using the theory developed in a companion paper by Thorpe (1994a). For the parameters of the laboratory experiments with relatively small forcing, the growth rates are small, consistent with the absence of signs of secondary instability. Larger growth rates and disturbance amplification factors of about 70 are predicted for the conditions in the experiment in which mixing was observed to occur. The experimental observations are consistent with an instability having a longitudinal structure.

We conclude that the form and development of breaking in internal gravity waves will vary according to the circumstances in which waves break, but depend on the Prandtl number of the fluid and, in particular, on the Rayleigh and Reynolds numbers of regions of static instability which develop as the waves overturn.
Evolution of three-dimensional coherent structures in a flat-plate boundary layer
Authors: Rempfer Dietmar , Fasel Hermann F.
Published: April 2006

Using a data base generated by a numerical simulation, the three-dimensional coherent structures of a transitional, spatially evolving boundary layer are determined and their spatio-temporal behaviour is investigated in detail. The coherent structures are calculated by the proper orthogonal decomposition method (POD), which leads to an expansion of the flow field variables into Karhunen-Loéve eigenfunctions. It is shown that the dynamical coherent structures of the flat-plate boundary layer can be described by pairs of eigenfunctions that contain complete information on the spatial evolution of the structures. It is further demonstrated that first-order coherent structures determined by POD correspond to structures that are observed in experiments. In the region of the boundary layer where the spike signals of transition occur, higher-order coherent structures also play an essential role. By considering these higher-order structures as well as their dynamical behaviour in time, a compact description of the flow phenomena in the boundary layer can be obtained. The description of the events occurring at the spike stages of the transitional boundary layer shows, from a coherent structures point of view, striking similarities to the bursting event of fully turbulent boundary layers.
The evolution equation for the flame surface density in turbulent premixed combustion
Authors: Trouvé Arnaud , Poinsot Thierry
Published: April 2006

One basic effect of turbulence in turbulent premixed combustion is for the fluctuating velocity field to wrinkle the flame and greatly increase its surface area. In the flamelet theory, this effect is described by the flame surface density. An exact evolution equation for the flame surface density, called the Σ-equation, may be written, where basic physical mechanisms like production by hydrodynamic straining and destruction by propagation effects are described explicitly. Direct numerical simulation (DNS) is used in this paper to estimate the different terms appearing in the Σ-equation. The numerical configuration corresponds to three-dimensional premixed flames in isotropic turbulent flow. The simulations are performed for various mixture Lewis numbers in order to modify the strength and nature of the flame-flow coupling. The DNS-based analysis provides much information relevant to flamelet models. In particular, the flame surface density, and the source and sink terms for the flame surface density, are resolved spatially across the turbulent flame brush. The geometry as well as the dynamics of the flame differ quite significantly from one end of the reaction zone to the other. For instance, contrary to the intuitive idea that flame propagation effects merely counteract the wrinkling due to the turbulence, the role of flame propagation is not constant across the turbulent brush and switches from flame surface production at the front to flame surface dissipation at the back. Direct comparisons with flamelet models are also performed. The Bray-Moss-Libby assumption that the flame surface density is proportional to the flamelet crossing frequency, a quantity that can be measured in experiments, is found to be valid. Major uncertainties remain, however, over an appropriate description of the flamelet crossing frequency. In comparison, the coherent flame model of Marble & Broadwell achieves closure at the level of the Σ-equation and provides a more promising physically based description of the flame surface dynamics. Some areas where the model needs improvement are identified.
Expulsion of particles from a buoyant blob in a fluidized bed
Authors: Batchelor G. K. , Nitsche J. M.
Published: April 2006

It is a significant feature of most gas-fluidized beds that they contain rising ‘bubbles’ of almost clear gas. The purpose of this paper is to account plausibly for this remarkable property first by supposing that primary and secondary instabilities of the fluidized bed generate compact regions of above-average or below-average particle concentration, and second by invoking a mechanism for the expulsion of particles from a buoyant compact blob of smaller particle concentration. We postulate that the rising of such an incipient bubble generates a toroidal circulation of the gas in the bubble, roughly like that in a drop of liquid rising through a second liquid of larger density, and that particles in the blob carried round by the fluid move on trajectories which ultimately cross the bubble boundary. Numerical calculations of particle trajectories for practical values of the relevant parameters show that a large percentage of particles, of such small concentration that they move independently, are expelled from a bubble in the time taken by it to rise through a distance of several bubble diameters.

Similar calculations for a liquid-fluidized bed show that the expulsion mechanism is much weaker, as a consequence of the larger density and viscosity of a liquid, which is consistent with the absence of observations of relatively empty bubbles in liquid-fluidized beds.

It is found to be possible, with the help of the Richardson-Zaki correlation, to adjust the results of these calculations so as to allow approximately for the effect of interaction of particles in a bubble in either a gas- or a liquid-fluidized bed. The interaction of particles at volume fractions of 20 or 30 % lengthens the expulsion times, although without changing the qualitative conclusions.
The evolution and bifurcation of a pendant drop
Authors: Schulkes R. M. S. M.
Published: April 2006

In this paper we calculate how a pendant drop evolves at the end of a nozzle when the volume of the drop increases steadily with time. We find that the character of the evolution is strongly dependent on the growth rate of the drop and the radius of the nozzle. Typically we find that once the drop has become unstable, two bifurcations occur shortly after each other when the growth rate of the drop is slow. For large growth rates the bifurcations are well-separated in time. We are able to calculate the volumes of the drops after the bifurcations. A comparison with experimental data shows a satisfactory agreement.
Potential energy in steep and breaking waves
Authors: Schultz William W. , Huh Jin , Griffin Owen M.
Published: April 2006

We find that the RMS wave height (square root of the potential energy) rather than peak-to-peak wave height is a better experimental and analytic criterion for determining when a regular, two-dimensional deep-water wave will break. A spectral algorithm for two-dimensional potential flow is developed and used to compare breaking onset criteria for energy input from (i) converging sidewalls, (ii) a submerged disturbance, and (iii) wave focusing. We also find that wave-breaking criteria (potential energy or the more classical peak-to-peak wave height) are a function of the rate of energy input. Large plunging waves occur when energy input rates are large. As energy input rates become smaller there is a smooth transition to smaller spilling waves. The various energy input methods show similar breaking trends in the limit as the energy input rate becomes small - waves break when the potential energy becomes approximately 52 % of the energy for the most energetic Stokes wave, with the formation of a singularity immediately before the crest. The effects of wave modulation and reflection are briefly discussed and shown not to affect the potential energy breaking criterion significantly. The experimental scatter of the RMS wave height is shown to be half that of wave steepness during incipient breaking in wave packets.
On step approximations for water-wave problems
Authors: Evans D. V. , Linton C. M.
Published: April 2006

The scattering of water waves by a varying bottom topography is considered using two-dimensional linear water-wave theory. A new approach is adopted in which the problem is first transformed into a uniform strip resulting in a variable free-surface boundary condition. This is then approximated by a finite number of sections on which the free-surface boundary condition is assumed to be constant. A transition matrix theory is developed which is used to relate the wave amplitudes at ±∞. The method is checked against examples for which the solution is known, or which can be computed by alternative means. Results show that the method provides a simple accurate technique for scattering problems of this type.
Mixed convection over a cooled horizontal plate: non-uniqueness and numerical instabilities of the boundary-layer equations
Authors: Steinrück Herbert
Published: April 2006

The boundary-layer flow over a cooled horizontal plate is considered. It is shown that the real part of the spectrum of the evolution operator of the linearized equations is not bounded uniformly from above which explains the difficulties encounterd by a numerical solution. Furthermore it is shown that near the leading edge an asymptotic expansion of the solution is not unique. A one-parametric family of asymptotic expansions of solutions can be constructed.
Unsteady flow near a moving cylinder
Authors: Dowling A. P.
Published: April 2006

General representations are derived for both the velocity potential and the surface pressure fluctuations induced by an arbitrary distribution of vorticity near a manoeuvring cylinder. The cylinder is inextensible and in unsteady motion. Its axis may be slightly curved, with radius of curvature large in comparison with the cylinder radius.

Two model problems are considered in detail to investigate the effect of lateral displacements of a cylinder with an established boundary layer. The boundary layer on the flexible cylinder is found to be shed once the lateral displacement of the cylinder axis exceeds the boundary-layer thickness. The unsteady pressures generated by this vortex shedding are investigated.
Parametric instability of the interface between two fluids
Authors: Kumar Krishna , Tuckerman Laurette S.
Published: April 2006

The flat interface between two fluids in a vertically vibrating vessel may be parametrically excited, leading to the generation of standing waves. The equations constituting the stability problem for the interface of two viscous fluids subjected to sinusoidal forcing are derived and a Floquet analysis is presented. The hydrodynamic system in the presence of viscosity cannot be reduced to a system of Mathieu equations with linear damping. For a given driving frequency, the instability occurs only for certain combinations of the wavelength and driving amplitude, leading to tongue-like stability zones. The viscosity has a qualitative effect on the wavelength at onset: at small viscosities, the wavelength decreases with increasing viscosity, while it increases for higher viscosities. The stability threshold is in good agreement with experimental results. Based on the analysis, a method for the measurement of the interfacial tension, and the sum of densities and dynamic viscosities of two phases of a fluid near the liquid-vapour critical point is proposed.
Electrophoresis of slender particles
Authors: Solomentsev Yuri , Anderson John L.
Published: April 2006

The hydrodynamic theory of slender bodies is used to model electrophoretic motion of a slender particle having a charge (zeta potential) that varies with position along its length. The theory is limited to systems where the Debye screening length of the solution is much less than the typical cross-sectional dimension of the particle. A stokeslet representation of the hydrodynamic force is combined with the Lorentz reciprocal theorem for Stokes flow to develop a set of linear equations which must be solved for the components of the translational and angular velocities of the particle. Sample calculations are presented for the electrophoretic motion of straight spheroids and cylinders and a torus in a uniform electric field. The theory is also applied to a straight uniformly charged particle in a spatially varying electric field. The uniformly charged particle rotates into alignment with the principal axes of ∇E∞; we suggest that such alignment can lead to electrophoretic transport of particles through a small aperture in an otherwise impermeable wall. The theory developed here is more general than just for electrophoresis, since the final result is expressed in terms of a general 'slip velocity’ at the surface of the particle. Thus, the results are applicable to diffusiophoresis of slender particles if the proper slip-velocity coefficient is used.
A numerical study of the vorticity field generated by the baroclinic effect due to the propagation of a planar pressure wave through a cylindrical premixed laminar flame
Authors: Batley G. A. , McIntosh A. C. , Brindley J. , Falle S. A. E. G.
Published: April 2006

The importance of vorticity production in combustion systems has been highlighted previously by several authors (Markstein 1964; Picone et al. 1984). The consequent distortion and enlargement of flame surfaces can lead to substantial enhancement of the burning rate which may be beneficial or disastrous depending on the physical context. We describe the results of numerical simulations of an experimental configuration similar to that described by Scarinci & Thomas (1992), who examined the effect of initially planar pressure signals on two-dimensional flame balls. The flame ball is here set-up from ignition using a code, based on the second-order Godunov scheme described by Falle (1991). A simple Arrhenius reaction scheme is adopted in modelling a unimolecular decomposition. As in previous papers (Batley et al. 1993 a, b) the thermal conductivity is assumed to vary linearly with temperature, and the Lewis and Prandtl numbers are taken as unity. A short time after ignition, when the flame ball has reached a radius of approximately 2 cm, a very short-lengthscale pressure step disturbance is introduced, propagating towards the combustion region. As the signal crosses the flame, the interaction of the sharp, misaligned pressure and density gradients, creates a strong vorticity field. The resulting roll-up of the flame eventually divides it into two smaller rotating reacting regions. In order to gauge the effect of the chemical reaction and in particular the viscous diffusion on the evolution of the vorticity field, the results are compared with analogous solutions of the Euler equations.
A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage
Authors: Shariff Karim , Verzicco Roberto , Orlandi Paolo
Published: April 2006

Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss & Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall & Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 – α1(β)/Res, for a large range of Res. Here Res is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then α1(β) varies weakly with core thickness β. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode (n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n. The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted π/n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial modes having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.
The inertial lift on a rigid sphere in a linear shear flow field near a flat wall
Authors: Cherukat Pradeep , Mclaughlin John B.
Published: April 2006

An expression which predicts the inertial lift, to lowest order, on a rigid sphere translating in a linear shear flow field near a flat infinite wall has been derived. This expression may be used when the wall lies within the inner region of the sphere's disturbance flow. It is valid even when the gap is small compared to the radius of the sphere. When the sphere is far from the wall, the lift force predicted by the present analysis converges to the value predicted by earlier analyses which consider the sphere as a point force or a force doublet singularity. The effect of rotation of the sphere on the lift has also been analysed.