Showing papers in "Physics of Fluids in 2001"

Momentum transfer of a Boltzmann-lattice fluid with boundaries

Abstract: We study the velocity boundary condition for curved boundaries in the lattice Boltzmann equation (LBE). We propose a LBE boundary condition for moving boundaries by combination of the “bounce-back” scheme and spatial interpolations of first or second order. The proposed boundary condition is a simple, robust, efficient, and accurate scheme. Second-order accuracy of the boundary condition is demonstrated for two cases: (1) time-dependent two-dimensional circular Couette flow and (2) two-dimensional steady flow past a periodic array of circular cylinders (flow through the porous media of cylinders). For the former case, the lattice Boltzmann solution is compared with the analytic solution of the Navier–Stokes equation. For the latter case, the lattice Boltzmann solution is compared with a finite-element solution of the Navier–Stokes equation. The lattice Boltzmann solutions for both flows agree very well with the solutions of the Navier–Stokes equations. We also analyze the torque due to the momentum transfer between the fluid and the boundary for two initial conditions: (a) impulsively started cylinder and the fluid at rest, and (b) uniformly rotating fluid and the cylinder at rest.

Electrospinning and electrically forced jets. I. Stability theory

Abstract: Electrospinning is a process in which solid fibers are produced from a polymeric fluid stream (solution or melt) delivered through a millimeter-scale nozzle. The solid fibers are notable for their very small diameters (<1 μm). Recent experiments demonstrate that an essential mechanism of electrospinning is a rapidly whipping fluid jet. This series of papers analyzes the mechanics of this whipping jet by studying the instability of an electrically forced fluid jet with increasing field strength. An asymptotic approximation of the equations of electrohydrodynamics is developed so that quantitative comparisons with experiments can be carried out. The approximation governs both long wavelength axisymmetric distortions of the jet, as well as long wavelength oscillations of the centerline of the jet. Three different instabilities are identified: the classical (axisymmetric) Rayleigh instability, and electric field induced axisymmetric and whipping instabilities. At increasing field strengths, the electrical instabilities are enhanced whereas the Rayleigh instability is suppressed. Which instability dominates depends strongly on the surface charge density and radius of the jet. The physical mechanisms for the instability are discussed in the various possible limits.

Electrospinning and electrically forced jets. II. Applications

Abstract: Electrospinning is a process in which solid fibers are produced from a polymeric fluid stream (solution or melt) delivered through a millimeter-scale nozzle. This article uses the stability theory described in the previous article to develop a quantitative method for predicting when electrospinning occurs. First a method for calculating the shape and charge density of a steady jet as it thins from the nozzle is presented and is shown to capture quantitative features of the experiments. Then, this information is combined with the stability analysis to predict scaling laws for the jet behavior and to produce operating diagrams for when electrospinning occurs, both as a function of experimental parameters. Predictions for how the regime of electrospinning changes as a function of the fluid conductivity and viscosity are presented.

An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows

Abstract: The approximate deconvolution model (ADM) for the large-eddy simulation of incompressible flows is detailed and applied to turbulent channel flow. With this approach an approximation of the unfiltered solution is obtained by repeated filtering. Given a good approximation of the unfiltered solution, the nonlinear terms of the filtered Navier–Stokes equations can be computed directly. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation. Large-eddy simulations are performed for incompressible channel flow at Reynolds numbers based on the friction velocity and the channel half-width of Reτ=180 and Reτ=590. Both simulations compare well with direct numerical simulation (DNS) data and show a significant improvement over results obtained with classical subgrid scale models such as the standard or the dynamic Smagorinsky model. The computational cost of ADM is lower than that of dynamic models or the velocity estimation model.

Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations

Abstract: Of the two-phase mixture models used to study deflagration-to-detonation transition in granular explosives, the Baer–Nunziato model is the most highly developed. It allows for unequal phase velocities and phase pressures, and includes source terms for drag and compaction that strive to erase velocity and pressure disequilibria. Since typical time scales associated with the equilibrating processes are small, source terms are stiff. This stiffness motivates the present work where we derive two reduced models in sequence, one with a single velocity and the other with both a single velocity and a single pressure. These reductions constitute outer solutions in the sense of matched asymptotic expansions, with the corresponding inner layers being just the partly dispersed shocks of the full model. The reduced models are hyperbolic and are mechanically as well as thermodynamically consistent with the parent model. However, they cannot be expressed in conservation form and hence require a regularization in order to fully specify the jump conditions across shock waves. Analysis of the inner layers of the full model provides one such regularization [Kapila et al., Phys. Fluids 9, 3885 (1997)], although other choices are also possible. Dissipation associated with degrees of freedom that have been eliminated is restricted to the thin layers and is accounted for by the jump conditions.

Large eddy simulation of turbulent channel flows by the variational multiscale method

Abstract: The variational multiscale formulation of LES is applied to two-dimensional equilibrium and three-dimensional nonequilibrium channel flows. Simple, constant-coefficient Smagorinsky-type eddy viscosities, without wall damping functions, are used to model the decay of small scales, an approach which is not viable for wall-bounded flows within the traditional LES framework. Nevertheless, very good results are obtained.

The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence

Abstract: The variational multiscale method is applied to the large eddy simulation (LES) of homogeneous, isotropic flows and compared with the classical Smagorinsky model, the dynamic Smagorinsky model, and direct numerical simulation (DNS) data. Overall, the multiscale method is in better agreement with the DNS data than both the Smagorinsky model and the dynamic Smagorinsky model. The results are somewhat remarkable when one realizes that the multiscale method is almost identical to the Smagorinsky model (the least accurate model!) except for removal of the eddy viscosity from a very small percentage of the lowest modes.

Collapse and rebound of a laser-induced cavitation bubble

Abstract: A strong laser pulse that is focused into a liquid produces a vapor cavity, which first expands and then collapses with subsequent rebounds. In this paper a mathematical model of the spherically symmetric motion of a laser-induced bubble is proposed. It describes gas and liquid dynamics including compressibility, heat, and mass transfer effects and nonequilibrium processes of evaporation and condensation on the bubble wall. It accounts also for the occurrence of supercritical conditions at collapse. Numerical investigations of the collapse and first rebound have been carried out for different bubble sizes. The results show a fairly good agreement with experimental measurements of the bubble radius evolution and the intensity of the outgoing shock wave emitted at collapse. Calculations with a small amount of noncondensable gas inside the bubble show its strong influence on the dynamics.

Experimental demonstration of a homogeneous two-scale dynamo

Abstract: It has been shown theoretically that homogeneous kinematic dynamo action is possible for many unconfined and confined velocity fields, but a rigorous experimental validation is still lacking. G. O. Roberts [Philos. Trans. R. Soc. London, Ser. A 266, 535 (1970)] proposed a spatially periodic velocity field capable to generate a dynamo, which Busse [Geophys. J. R. Astron. Soc. 42, 437 (1975)] modified by introducing a second length scale larger in order to obtain a solution for a finite domain. Based on a scale separation approach Busse [Springer Proceedings in Physics, Vol. 69 (Springer Verlag, New York, 1992)] proposed a conceptual design for an experimental homogeneous dynamo. An engineering design was developed and a test facility has been set up. This test facility is described and first experimental results confirming dynamo action are presented.

Direct numerical simulation of decaying compressible turbulence and shocklet statistics

Abstract: We present results from 1283 and 2563 direct numerical simulations (DNS) of decaying compressible, isotropic turbulence at fluctuation Mach numbers of Mt∼0.1–0.5 and at Taylor Reynolds numbers Reλ=O(50–100). The presence or absence of fluctuations of thermodynamic quantities as well as velocity divergence in the initial conditions are found to have a negligible effect on the decay of turbulent kinetic energy. The decay of the turbulent kinetic energy shows no significant effect of Mt and power laws fitted to the timewise decay exhibit exponents n=1.3–1.7 that are similar to those found for decaying incompressible turbulence. The main new phenomenon produced by compressibility is the appearance of random shocklets which form during the main part of the decay. An algorithm is developed to extract and quantify the shocklet statistics from the DNS fields. A model for the probability density function (PDF) of the shocklet strength Mn−1 (Mn is the normal shock Mach number) is derived based on combining weak-sho...

Transient growth: A factor in bypass transition

Abstract: Transient growth arises through the coupling between slightly damped, highly oblique (nearly streamwise) T–S and Squire modes leading to algebraic growth followed by exponential decay in a region that is subcritical with respect to the T–S neutral curve. A weak transient growth can also occur for two dimensional or axisymmetric modes since the Orr–Sommerfeld operator and its compressible counterpart are not self-adjoint, therefore their eigenfunctions are not strictly orthogonal. So transient growth is a candidate mechanism for many examples of bypass transition. The original transient growth theories were all temporal. However spatial growth formulations are now emerging including their extension to compressible flow with pressure gradient and heat transfer. The relevance to bypass transition is examined through several examples including Poiseuille pipe flow, the hypersonic blunt body paradox and distributed roughness effects.

Self-similarity of strongly stratified inviscid flows

Abstract: It is well-known that strongly stratified flows are organized into a layered pancake structure in which motions are mostly horizontal but highly variable in the vertical direction. However, what determines the vertical scale of the motion remains an open question. In this paper, we propose a scaling law for this vertical scale Lv when no vertical lengthscales are imposed by initial or boundary conditions and when the fluid is strongly stratified, i.e., when the horizontal Froude number is small: Fh=U/NLh≪1, where U is the magnitude of the horizontal velocity, N the Brunt–Vaisala frequency and Lh the horizontal lengthscale. Specifically, we show that the vertical scale of the motion is Lv=U/N by demonstrating that the inviscid governing equations in the limit Fh→0, without any a priori assumption on the magnitude of Lv, are self-similar with respect to the variable zN/U, where z is the vertical coordinate. This self-similarity fully accounts for the layer characteristics observed in recent studies reportin...

The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction

Abstract: A formulation of the approximate deconvolution model (ADM) for the large-eddy simulation (LES) of compressible flows in complex geometries is detailed. The model is applied to supersonic compression ramp flow where shock-turbulence interaction occurs. With the ADM approach an approximation to the unfiltered solution is obtained from the filtered solution by a series expansion involving repeated filtering. Given a sufficiently good approximation of the unfiltered solution at a time instant, the flux terms of the underlying filtered transport equations can be computed directly, avoiding the need to explicitly compute subgrid-scale closures. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation and a dynamically estimated relaxation parameter. Results of the large-eddy simulation of the turbulent supersonic boundary layer along a compression ramp compare well with filtered DNS data. The filtered shock solution is correctly predicted by the ADM pr...

The importance of the forces acting on particles in turbulent flows

Abstract: The analysis of the importance of the forces that act over an ensemble of particles in a turbulent field has been carried out by using direct numerical simulation for a wide range of density ratios (2.65<ρ<2650). It has been observed that, compared to the Stokes drag, the added mass is always negligible, the pressure drag is relevant for density ratios O(1), and the Basset force is appreciable for the whole range investigated. However, the effect of these forces on the particle dispersion is about 1% for ρ∼1 as well as for large density ratios.

Turbulent Rayleigh–Bénard convection in gaseous and liquid He

Abstract: In this article we deal with the turbulent regimes of Rayleigh–Benard convection, namely the 2/7 regime and beyond. An experiment with He at low temperature allows us to explore a large Rayleigh number (Ra) range up to 2×1014, under Boussinesq conditions, while the Prandtl number (Pr) is equal to and larger than 0.7. Calorimetric measurements evidence a departure from the 2/7 regime above Ra=1011 toward a new regime where the heat transfer is enhanced. Local measurements with two nearby thermometers allows us to relate this change to a laminar–turbulent transition of the velocity boundary layer induced by the large-scale flow near the walls of the cell. The features of the observed new regime match those of the ultimate regime predicted by R. Kraichnan [Phys. Fluids 5, 1374 (1962)] at moderate Pr; in particular, our experimental data show that the thermal boundary layer lies inside the viscous sublayer of the turbulent boundary layer.

A comparative study of near-wall turbulence in high and low Reynolds number boundary layers

Abstract: The present study explores the effects of Reynolds number, over three orders of magnitude, in the viscous wall region of a turbulent boundary layer. Complementary experiments were conducted both in the boundary layer wind tunnel at the University of Utah and in the atmospheric surface layer which flows over the salt flats of the Great Salt Lake Desert in western Utah. The Reynolds numbers, based on momentum deficit thickness, of the two flows were Rθ=2×103 and Rθ≈5×106, respectively. High-resolution velocity measurements were obtained from a five-element vertical rake of hot-wires spanning the buffer region. In both the low and high Rθ flows, the length of the hot-wires measured less than 6 viscous units. To facilitate reliable comparisons, both the laboratory and field experiments employed the same instrumentation and procedures. Data indicate that, even in the immediate vicinity of the surface, strong influences from low-frequency motions at high Rθ produce noticeable Reynolds number differences in the ...

Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence

Abstract: We derive analytic criteria for the existence of hyperbolic (attracting or repelling), elliptic, and parabolic material lines in two-dimensional turbulence. The criteria use a frame-independent Eulerian partition of the physical space that is based on the sign definiteness of the strain acceleration tensor over directions of zero strain. For Navier–Stokes flows, our hyperbolicity criterion can be reformulated in terms of strain, vorticity, pressure, viscous and body forces. The special material lines we identify allow us to locate different kinds of material structures that enhance or suppress finite-time turbulent mixing: stretching and folding lines, Lagrangian vortex cores, and shear jets. We illustrate the use of our criteria on simulations of two-dimensional barotropic turbulence.

Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime

Abstract: In hypersonic flows about space vehicles in low earth orbits or flows in microchannels of microelectromechanical devices, the local Knudsen number lies in the continuum–transition regime Navier–Stokes equations are not adequate to model these flows since they are based on small deviation from local thermodynamic equilibrium To model these flows, a number of extended hydrodynamics or generalized hydrodynamics models have been proposed over the past fifty years, along with the direct simulation Monte Carlo (DSMC) approach One of these models is the Burnett equations which are obtained from the Chapman–Enskog expansion of the Boltzmann equation [with Knudsen number (Kn) as a small parameter] to O(Kn2) With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with the Burnett model As a result, several improvements to the conventio

Flow structure due to dimple depressions on a channel surface

Abstract: Instantaneous, dynamic and time-averaged characteristics of the vortex structures which are shed from the dimples placed on one wall of a channel are described. The dimpled test surface contains 13 staggered rows of dimples in the streamwise direction, where each dimple has a print diameter of 5.08 cm, and a ratio of depth to print diameter of 0.2. Considered are Reynolds numbers (based on channel height) ReH from 600 to 11 000, and ratios of channel height to dimple print diameter H/D of 0.25, 0.50, and 1.00. For all three H/D, a primary vortex pair is periodically shed from the central portion of each dimple, including a large upwash region. This shedding occurs periodically and continuously, and is followed by inflow advection into the dimple cavity. The frequency of these events appears to scale on time-averaged bulk velocity and dimple print diameter, which gives nondimensional frequencies of 2.2–3.0 for all three H/D values considered. As H/D decreases, (i) the strength of the primary vortex pair in...

Energy amplification in channel flows with stochastic excitation

Abstract: We investigate energy amplification in parallel channel flows, where background noise is modeled as stochastic excitation of the linearized Navier–Stokes equations. We show analytically that the energy of three-dimensional streamwise-constant disturbances achieves O(R3) amplification. Our basic technical tools are explicit analytical calculations of the traces of solutions of operator Lyapunov equations, which yield the covariance operators of the forced random velocity fields. The dependence of these quantities on both the Reynolds number and the spanwise wave number are explicitly computed. We show how the amplification mechanism is due to a coupling between wall-normal velocity and vorticity disturbances, which in turn is due to nonzero mean shear and disturbance spanwise variation. This mechanism is viewed as a consequence of the non-normality of the dynamical operator, and not necessarily due to the existence of near resonances or modes with algebraic growth.

On the role of large-scale structures in wall turbulence

Abstract: Copyright 2001 American Institute of Physics. Publisher PDF version is restricted access in accordance with the American Institute of Physics policy.

Explicit-filtering large-eddy simulation using the tensor-diffusivity model supplemented by a dynamic Smagorinsky term

Abstract: Large-eddy simulation (LES) with regular explicit filtering is investigated. The filtered-scale stress due to the explicit filtering is here partially reconstructed using the tensor-diffusivity model: It provides for backscatter along the stretching direction(s), and for global dissipation, both also attributes of the exact filtered-scale stress. The necessary LES truncations (grid and numerical method) are responsible for an additional subgrid-scale stress. A natural mixed model is then the tensor-diffusivity model supplemented by a dynamic Smagorinsky term. This model is reviewed, together with useful connections to other models, and is tested against direct numerical simulation (DNS) of turbulent isotropic decay starting with Re-lambda=90 (thus moderate Reynolds number): LES started from a 256(3) DNS truncated to 64(3) and Gaussian filtered. The tensor-diffusivity part is first tested alone; the mixed model is tested next. Diagnostics include energy decay, enstrophy decay, and energy spectra. After an initial transient of the dynamic procedure (observed with all models), the mixed model is found to produce good results. However, despite expectations based on favorable a priori tests, the results are similar to those obtained when using the dynamic Smagorinsky model alone in LES without explicit filtering. Nevertheless, the dynamic mixed model appears as a good compromise between partial reconstruction of the filtered-scale stress and modeling of the truncations effects (incomplete reconstruction and subgrid-scale effects). More challenging 48(3) LES are also done: Again, the results of both approaches are found to be similar. The dynamic mixed model is also tested on the turbulent channel flow at Re-tau=395. The tensor-diffusivity part must be damped close to the wall in order to avoid instabilities. Diagnostics are mean profiles of velocity, stress, dissipation, and reconstructed Reynolds stresses. The velocity profile obtained using the damped dynamic mixed model is slightly better than that obtained using the dynamic Smagorinsky model without explicit filtering. The damping used so far is however crude, and this calls for further work. (C) 2001 American Institute of Physics.

Numerical simulation of particle-laden turbulent channel flow

Abstract: This paper presents results for the behavior of particle-laden gases in a small Reynolds number vertical channel down flow. Results will be presented for the effects of particle feedback on the gas-phase turbulence and for the concentration profile of the particles. The effects of density ratio, mass loading, and particle inertia will be discussed. The results were obtained from a numerical simulation that included the effects of particle feedback on the gas phase and particle–particle collisions. The resolution of the simulation was comparable to the smallest scales in the particle-free flow, but the grid spacings were larger than the particle size. Particle mass loadings up to 2 and both elastic and inelastic collisions were considered. Particle feedback causes the turbulent intensities to become more anisotropic as the particle loading is increased. For small mass loadings, the particles cause an increase in the gas flow rate. It will be shown that the particles tend to increase the characteristic length scales of the fluctuations in the streamwise component of velocity and that this reduces the transfer of turbulent energy between the streamwise component of velocity and the components transverse to the flow. Particle–particle collisions greatly reduce the tendency of particles to accumulate at the wall for the range of mass loadings considered. This was true even when the collisions were inelastic.

The recoiling of liquid droplets upon collision with solid surfaces

Abstract: Although the spreading behavior of liquid droplets impacting on solid surfaces has been extensively studied, the mechanism of recoiling which takes place after the droplet reaches its maximum spread diameter has not yet been fully understood. This paper reports the study of the recoiling behavior of different liquid droplets (water, ink, and silicone oil) on different solid surfaces (polycarbonate and silicon oxide). The droplet dynamics are experimentally studied using a high speed video system. Analytical methods using the variational principle, which were originated by Kendall and Rohsenow (MIT Technical Report 85694-100, 1978) and Bechtel et al. [IBM J. Res. Dev. 25, 963 (1981)], are modified to account for wetting and viscous effects. In our model, an empirically determined dissipation factor is used to estimate the viscousfriction. It is shown that the model closely predicts the experimental results obtained for the varying dynamic impact conditions and wetting characteristics. This study shows that droplets recoil fast and vigorously when the Ohnesorge number decreases or the Weber number increases. Droplets with a large equilibrium contact angle are also found to recoil faster. Here the Ohnesorge number scales the resisting force to the recoiling motion, and is shown to play the most important role in characterizing the recoiling motion.

On the development of large-scale structures of a jet normal to a cross flow

Abstract: It is well known that vortex rings are the dominant flow structures in the near field of a free jet, and this has led many researchers to believe that they also occur in a jet in cross flow (JICF). Previous studies have postulated that these vortex rings deform and fold as they convect downstream, which culminates in the formation of vortex loops at both the upstream and the lee-side of the jet column. In this paper, we take a fresh look at the vortical structures of JICF in water by releasing dye at strategic locations around the jet exit. The results show that there is no evidence of ring vortices in JICF, and the postulation that vortex loops are formed from the folding of the vortex rings does not reflect the actual flow behavior. The presence of a counter-rotating vortex pair (CVP) at the jet exit is found to inhibit the formation of the vortex rings. Instead, vortex loops are formed directly from the deformation of the cylindrical vortex sheet or jet column, without going through the vortex rings, in a process similar to the buoyant jet and wake structures studied by Perry and Lim [J. Fluid Mech. 88, 451 (1978)].

Spatial theory of optimal disturbances in boundary layers

Abstract: A spatial theory is proposed for the linear transient growth of disturbances in a parallel boundary layer. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes and Tollmien–Schlichting (TS) like instability modes. Therefore, the problem of optimal disturbances may be considered as an initial value problem on the subset of the decaying eigenmodes and a TS wave, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. The results indicate that the most significant transient growth is associated with stationary streamwise vortices. Numerical examples illustrate that favorable pressure gradient decreases the overall amplification. Effects of compressibility and the wall cooling are investigated as well.

Numerical simulation of pulsating turbulent channel flow

Abstract: Direct and large-eddy simulations of the Navier–Stokes equations are used to study the pulsating flow in a channel. The cases examined span a wide range of frequencies of the driving pressure gradient, and encompass different physical behaviors, from the quasi-Stokes flow observed at high frequencies, to a quasisteady behavior at the lowest ones. The validity of the dynamic Smagorinsky model to study this kind of unsteady flow is established by a posteriori comparison with direct simulations and experimental data. It is shown that the fluctuations generated in the near-wall region by the unsteady pressure gradient do not propagate beyond a certain distance lt from the wall, which can be estimated quite accurately by a simple eddy viscosity argument. No substantial departure from the Stokes regime at very high frequency (ω+ as high as 0.1) is observed. The time-dependent characteristics of the flow are examined in detail, as well as the topology of the coherent structures.

Subgrid scale variance and dissipation of a scalar field in large eddy simulations

Abstract: Subgrid scale (SGS) variance of a scalar field in large eddy simulations is only properly defined in relation to a probability density function. This solves a reported problem in the variance definition [Cook and Riley, Phys. Fluids 6, 2868 (1994); Cook, Riley, and Kosaly, Combust. Flame 109, 332 (1997)] and allows to write a simple evolution equation for the scalar variance. This equation shows that a recently proposed model for scalar dissipation in terms of the large-scale gradients [Pierce and Moin, Phys. Fluids 10, 3041 (1998)] implies dissipation and production canceling out, preventing variance decay and complete mixing at SGS level. An alternative simple model for dissipation in terms of a SGS mixing characteristic time is proposed and tested here.

Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem

Abstract: The classical rectangular lid-driven-cavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tangentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is two-dimensional for small Reynolds numbers. By a linear stability analysis it is shown that this basic flow becomes unstable at higher Reynolds numbers to four different three-dimensional modes depending on the aspect ratio of the cavity’s cross section. For shallow cavities the most dangerous modes are a pair of three-dimensional short waves propagating spanwise in the direction perpendicular to the basic flow. The mode is localized on the strong basic-state eddy that is created at the downstream end of the moving lid when the Reynolds number is increased. In the limit of a vanishing layer depth the critical Reynolds number approaches a finite asymptotic value. When the depth of the cavity is comparable to its width, two different centrifugal-instability modes can appear depending on the exact value of the aspect ratio. One of these modes is stationary, the other one is oscillatory. For unit aspect ratio (square cavity), the critical mode is stationary and has a very short wavelength. Experiments for the square cavity with a large span confirm this instability. It is argued that this three-dimensional mode has not been observed in all previous experiments, because the instability is suppressed by side-wall effects in small-span cavities. For large aspect ratios, i.e., for deep cavities, the critical three-dimensional mode is stationary with a long wavelength. The critical Reynolds number approaches a finite asymptotic value in the limit of an infinitely deep cavity.

Finite Reynolds number effects in the Bretherton problem

Abstract: In this paper we investigate the effect of fluid inertia in the classical Bretherton problem in which a semi-infinite air finger displaces viscous fluid in a two-dimensional channel. The governing free-surface Navier–Stokes equations are discretized by a finite element method and the system’s behavior is studied for capillary and Reynolds numbers in the ranges 0.05