Volume 7, Issue 12, December 1995
Index of content:

Wavelet based model for small‐scale turbulence
View Description Hide DescriptionUsing a vector wavelet decomposition of the velocity field a simple model for locally isotropic turbulence has been derived from the Navier–Stokes equation (NSE). This model, which involves no empirical or ad hoc parameter, incorporates nonlocal interscale interactions, reveals backscatter and can be applied to represent small‐scale turbulence in LES schemes. The model gives a stationary solution which corresponds to k ^{−5/3} Kolmogorov spectrum. Moreover, the modelequations produce k ^{4}infrared spectrum from an initial energy peak at intermediate scales.

A non‐slip boundary condition for lattice Boltzmann simulations
View Description Hide DescriptionA non‐slip boundary condition at a wall for the lattice Boltzmann method is presented. In the present method unknown distribution functions at the wall are assumed to be an equilibrium distribution function with a counter slip velocity which is determined so that fluid velocity at the wall is equal to the wall velocity. Poiseuille flow and Couette flow are calculated with the nine‐velocity model to demonstrate the accuracy of the present boundary condition.

Fluid motion of monomolecular films in a channel flow geometry
View Description Hide DescriptionSurface pressure‐driven flow of a monolayer in a channel flow geometry is studied and exact representations for the monolayer and subphase velocity fields are given in terms of solutions involving dual integral equations. The monolayer velocity profiles are examined as a function of the viscosity contrast between the monolayer and the subphase and the effects of a finite depth sublayer are investigated also. The calculated velocity profiles may prove useful for determining the effective surface viscosity of monolayer films from comparable experimental measurements.

Investigations of liquid surface rheology of surfactant solutions by droplet shape oscillations: Theory
View Description Hide DescriptionA theoretical analysis is presented for the shape oscillations of drops suspended in air. For drops of surfactantsolution, the oscillation frequency is basically determined by the surface tension; the free‐damping constant depends on the surfaceviscoelasticities. Different types of surfactantmass transfer at the dropletsurface produce different surface rheological behaviors. Analytical approximate solutions for free‐oscillation frequency and damping constant are derived by a perturbation method as functions of surface compositional elasticity,surface dilatational viscosity, and surface shear viscosity. These solutions are verified with exact numerical solutions. The existence of a second oscillation mode due to surfaceelasticity is illustrated. The phase relationships between the external driving forces and the droplet shapes for forced oscillations are discussed. It is found that at the 90° phase shift, the driving frequency and the slope of the phase diagram are equivalent to the free‐oscillation frequency and damping constant.

On contact angles in evaporating liquids
View Description Hide DescriptionIn this paper, the macroscopic contact angle in an evaporating drop is derived, on the assumption that its deviation from the microscopic angle is accounted for by processes taking place in the slip region. This investigation was prompted by a recent paper by Anderson and Davis [Phys. Fluids 7, 248 (1995)], in which an expression for the dynamic macroscopic contact angle for spreading drops is also assumed to hold when evaporation is present.

Recovery of the Rayleigh capillary instability from slender 1‐D inviscid and viscous models
View Description Hide DescriptionFor either inviscid or viscous jets, Rayleigh proved cylindrical jets are linearly unstable due to surface tension of the interface, with instability precisely in all wavelengths greater than the jet circumference. As an alternative to linearized analysis, many past and present studies of surface tension‐driven jet breakup are based on slender asymptotic 1‐D models; here we clarify two issues regarding this approach. First, self‐consistent, leading‐order models of inviscid or viscous slender jets do not have a finite instability cutoff. Indeed, the inviscid 1‐D equations exhibit unbounded exponential growth in the small scale limit, while the viscous counterparts bound the growth rate but remain unstable in all wavelengths. Second, one can recover a finite instability cutoff by extending the asymptotic analysis to higher order. The linearized growth rate corrections at each finite order arise as algebraic approximations to Rayleigh’s exact exponential rate. We explicitly match, at leading and subsequent order, the slender longwave expansion of the exact results with the linearized behavior of 1‐D slender asymptotic equations.

Solutions near the hydraulic control point in a gravity current model
View Description Hide DescriptionA steady‐state gravity currentmodel that incorporates entrainment and friction is used to describe large‐scale gravity currents and channel flows. When the model includes pressure effects from varying current thickness, critical points occur when the current velocity is equal to the phase velocity of waves on the interface. Some solutions have the possibility to pass from super to subcritical flow, or vice versa. These solutions pass through a hydraulic control point and the objective is to analyze the behavior of the solutions in the vicinity of such points. Using a phase space in which the hydraulic control points occur as equilibrium points, and performing Taylor expansion to the first order, the result is a system of autonomous differential equations with constant coefficients that can describe the behavior of the solutions for different parameter regimes near a hydraulic control point. If an equilibrium point in phase space represents a saddle point, it is distinguished between three different solution classes; solutions that approach the critical velocity but never reach it, solutions that reach the critical velocity and obtain infinitely large derivative, and the solutions (one from subcritical and one from supercritical) that reach the critical velocity exactly in the equilibrium point. The method gives a way to tell whether hydraulically controlled solutions exist and in special cases an algorithm for finding these solutions.

Experiments on streamwise vortices in curved wall jet flow
View Description Hide DescriptionHot‐wire anemometry and smoke visualizations are used to study the jet on a concave wall. The measurements were performed in the region 91<Go<198, where Go is the Görtler number. It is found that streamwise vortices are amplified on the concave wall. The growth of vortices is dependent on the initial amplitude, i.e. the highest initial amplitude gives the maximum strength of vortices. The amplification occurs in the laminar region while the time‐averaged strength of vortices decreases further downstream in the transition region. No merging or splitting of vortex pairs was found. Regular oscillations of the time signal occur at higher Görtler numbers before breakdown. Smoke visualization shows that the horseshoe mode appears with a streamwise wave length of the same order as the triggered spanwise wave length. The streamwise velocity fluctuations indicating the secondary instability were found to coincide better with the total shear distribution rather than streamwise velocity gradients in the spanwise or wall normal directions. Power spectra show distinct peaks at different frequencies. At streamwise positions near the start of curvature, the filtered root‐mean‐square (rms) levels of the streamwise velocity component around 95 Hz, corresponding to the horseshoe mode, show maximum amplitude between different vortex pairs. The filtered frequency around 22 Hz, which is the dominating frequency further downstream, shows maximum amplitude further away from the wall, where the velocity profile has an inflectional character. However, as the flow proceeds downstream the maximum at 22 Hz moves against the wall.

Nonlinear spin‐up in a circular cylinder
View Description Hide DescriptionNonlinear spin‐up in a circular cylindrical tank is investigated experimentally and compared with the Wedemeyer model. The experiments were performed with water, using tracer particles floating at the free surface in order to visualize the flow field. The experimentally determined vorticity profiles show differences from the Wedemeyer model that indicate the need for an improved estimation of the Ekman pumping on a finite domain. In particular, the Wedemeyer model appears to be inaccurate in the region close to the sidewall. The vorticity field in a spin‐down experiment can be reproduced very well by using numerical data of Rogers and Lance for the Ekman suction of an unbounded rotating fluid over a nonrotating plate. However, a more general use of the data of Rogers and Lance on a bounded domain is shown to be inadequate because this would lead to a violation of mass conservation of the Ekman layer.

The long‐wave interfacial instability of two liquid layers stratified by thermal conductivity in an inclined channel
View Description Hide DescriptionThe flow of two liquid layers with different thermal conductivities confined in a rigid inclined channel cooled from below is susceptible to a long‐wave interfacialinstability. The results of Yih [Phys. Fluids 29, 1769 (1986)] on this problem are corrected and extended to layers of unequal thickness and to an arbitrary applied pressure gradient. The effects of direct liquid expansion and contraction as described in the conservation of massequation are also included.

Stability of capillary–gravity interfacial waves between two bounded fluids
View Description Hide DescriptionTwo‐dimensional periodic capillary–gravity waves at the interface between two bounded fluids of different densities are considered. Based on a variational formulation, the relation between wave frequency and wave amplitude is obtained through a weakly nonlinear analysis. All classes of space‐periodic waves are studied: traveling and standing waves as well as a degenerate class of mixed waves. As opposed to water waves, mixed interfacial waves exist even for pure gravity waves. The stability of traveling and standing waves with respect to three‐dimensional modulations is then studied. By using the method of multiple scales, Davey–Stewartson‐type equations are obtained. A detailed stability analysis is performed in three cases: pure gravity waves, capillary–gravity waves when one layer is infinitely deep, and capillary–gravity waves when both layers are infinitely deep. The main results for oblique (i.e., combined longitudinal and transverse) modulations reveal a mostly stabilizing effect of the density ratio for traveling waves and a destabilizing effect for standing waves.

An experimental investigation of the instability of a shear flow with multilayered density stratification
View Description Hide DescriptionThe evolution of stratified shear flows with multilayer density distributions is discussed briefly from a theoretical perspective, generalizing the results of Caulfield [J. Fluid Mech. 258, 255 (1994)] to allow for asymmetry. Three distinct types of instability are predicted to occur according to linear theory. In the laboratory, we measure the density profile and the velocity profile continuously, and so are able to identify the flow characteristics that are applicable when each of the different instabilities grow. Knowledge of the bulk Richardson number is insufficient to predict the observed properties of the instabilities of the flow. The parameter that is most determinant of the selection of a particular type of instability is found to be the ratio R of the depth of the intermediate density layer to the depth over which the velocity varies, though any asymmetry in the flow (either in the velocity or density fields) also plays a role. If R is close to 1, and hence the layer of intermediate density occupies a significant portion of the shear layer, overturnings appear in the intermediate layer, which are long lived, and strongly two dimensional. These overturnings are the three layer stratified generalization of the Kelvin–Helmholtz instability first discussed by Taylor [Proc. R. Soc. London Ser. A 132, 499 (1931)]. Such modes inefficiently mix the background flow, and the major mixing mechanisms are found to consist of overturnings in the lower fluid layer (and, to a lesser extent, the upper layer). These overturnings are clearly manifestations of an (asymmetric) three layer generalization of the Holmboe [Geophys. Publ. 24, 67 (1962)] instability. In general, all three instabilities can be observed simultaneously at markedly different wavelengths and phase speeds for extended periods of time, even though linear theory may predict significantly different growth rates.

Growth characteristics downstream of a shallow bump: Computation and experiment
View Description Hide DescriptionMeasurements of the velocity field created by a shallow bump on a wall revealed that an energy peak in the spanwise spectrum associated with the driver decays and an initially small‐amplitude secondary mode rapidly grows with distance downstream of the bump. Linear theories could not provide an explanation for this growing mode. The present Navier–Stokes simulation replicates and confirms the experimental results. Insight into the structure of the flow was obtained from a study of the results of the calculations and is presented.

On the curved wall jet influenced by system rotation and self‐similar suction or blowing
View Description Hide DescriptionIn this paper we use local nonparallel linear stability theory to study the jet on a concave and convex wall with spanwise system rotation and self‐similar suction or blowing. It is found for low negative rotation, i.e. the Coriolis force counteracts the centrifugal force, that the critical Goertler number Go is increased for both the concave and convex wall jet. For the convex wall jet the critical Go is increased up to eight times compared with the nonrotation case. In this region of negative rotation, the principle of exchange of instabilities does not hold for the convex wall jet. For high negative and positive rotation the flow is destabilized on both types of walls. Suction stabilizes the concave wall jet while the convex wall jet is destabilized. For blowing, the concave wall jet is destabilized to a certain limit and then stabilized for increased blowing. The convex wall jet is stabilized for blowing. The combined effects of curvature, system rotation, and self‐similar suction or blowing show that the highest critical Go can be increased for the rotating concave wall jet for both suction and blowing. For the rotating convex wall jet the highest critical Go is increased for suction and decreased for blowing.

Transition in shear flows. Nonlinear normality versus non‐normal linearity
View Description Hide DescriptionA critique is presented of recent works promoting the concept of non‐normal operators and transient growth as the key to understanding transition to turbulence in shear flows. The focus is in particular on a simple model [Baggett et al., Phys. Fluids 7, 883 (1995)] illustrating that view. It is argued that the question of transition is really a question of existence and basin of attraction of nonlinear self‐sustaining solutions that have little contact with the non‐normal linear problem. An alternative nonlinear point of view [Hamilton et al., J. Fluid Mech. 287, 317 (1995)] that seeks to isolate a self‐sustaining nonlinear process, and the critical Reynolds number below which it ceases to exist, is discussed and illustrated by a simple model. Connections with the Navier–Stokes equations and observations are highlighted throughout.

PDFs for velocity and velocity gradients in Burgers’ turbulence
View Description Hide DescriptionWe characterize the tails of the probability distribution functions for the solution of Burgers’ equation with Gaussian initial data and its derivatives ∂^{ kv }(x,t)/∂x ^{ k }, k=0,1,2,... . The tails are ‘‘stretched exponentials’’ of the form P(θ)∝exp[−(Re)^{−p } t ^{ q }θ^{ r }], where Re is the Reynolds number. The exponents p, q, and r depend on the initial spectrum as well as on the order of differentiation, k. These exact results are compared with those obtained using the mapping closure technique.

The higher moments in the Lundgren model conform with Kolmogorov scaling
View Description Hide DescriptionWe calculate the structure functions of the higher moments of the vorticity in the framework of Lundgren’s spiral model of turbulence. We show that they conform to the scaling expected from Kolmogorov’s scaling hypothesis of 1941, and explain the result.

Does fully developed turbulence exist? Reynolds number independence versus asymptotic covariance
View Description Hide DescriptionBy analogy with recent arguments concerning the mean velocity profile of wall‐bounded turbulentshear flows, we suggest that there may exist corrections to the 2/3 law of Kolmogorov, which are proportional to (ln Re)^{−1} at large Re. Such corrections to K41 are the only ones permitted if one insists that the functional form of statistical averages at large Re be invariant under a natural redefinition of Re. The family of curves of the observed longitudinal structure function D _{ LL }(r,Re) for different values of Re is bounded by an envelope. In one generic scenario, close to the envelope, D _{ LL }(r,Re) is of the form assumed by Kolmogorov, with corrections of O[(ln Re)^{−2}]. In an alternative generic scenario, both the Kolmogorov constant C _{K} and corrections to Kolmogorov’s linear relation for the third‐order structure function D _{ LLL }(r) are proportional to (ln Re)^{−1}. Recent experimental data of Praskovsky and Oncley appear to show a definite dependence of C _{K} on Re, which, if confirmed, would be consistent with the arguments given here.

Estimation of the Kolmogorov constant (C _{0}) for the Lagrangian structure function, using a second‐order Lagrangian model of grid turbulence
View Description Hide DescriptionWe review Sawford’s [Phys. Fluids A 3, 1577 (1991)] second‐order Lagrangianstochastic model for particle trajectories in low Reynolds numberturbulence, showing that it satisfies a well‐mixed constraint for the (hypothetical) case of stationary, homogeneous, isotropic turbulence in which the joint probability density function for the fixed‐point velocity and acceleration is Gaussian. We then extend the model to decaying homogeneous turbulence and, by optimizing model agreement with the measured spread of tracers in grid turbulence, estimate that Kolmogorov’s universal constant (C _{0}) for the Lagrangian velocity structure function has the value of 3.0±0.5.

A physical model for merging in two‐dimensional decaying turbulence
View Description Hide DescriptionSome aspects of the simulation by point vortices of two‐dimensional decaying turbulence are discussed, together with their most relevant consequences for the decay of the number of vortices. Two different merging models are proposed to simulate dissipative interactions between corotating vorticity structures within a point vortex simulation. Both models are based on a statistical approach to assign the number of vortices resulting from the merging of two vortices of different size. Probability distributions are defined for the production of one, two, or three vortices, and they are assigned as functions of the circulation ratio for the two interactingvortices. These functions are determined by performing a large number of Contour Dynamics simulations of vortex interactions, each under a given strain field. A simple rule to reset the vorticity field, at the end of the merging process, is discussed in terms of conserved quantities for the formation of one or three vortices. To complete the merging model, a criterion that indicates the onset of a merging event between two close vortices of like sign is required. The first model is defined by adopting a critical distance as merging criterion. In this way, the effect of the strain on the merging process is taken into account only in a statistical manner, i.e., by the probability law for the merging products. The use of this merging model into a point vortex simulation gives a surprisingly good agreement with the results for the vortex number decay obtained by Dritschel via Contour Surgery.
Nevertheless, a merging criterion based on the critical distance appears questionable, since it disregards the effect of the strain on the merging conditions. To this aim, we introduce a more sophisticated merging model that uses the dynamics of elliptical patches under a given strain field to select the merging events. This second model accounts also directly, and not only in a statistical sense, for the strain influence over the merging conditions, revising thoroughly the critical distance concept. The results show that the vortex number decay is not strongly sensitive to a detailed description of each individual merging process and, if the interest is focused on the vortex number decay, further improvements of the merging model are not strictly required.