Volume 6, Issue 4, April 1994
Index of content:

Bottleneck phenomenon in developed turbulence
View Description Hide DescriptionIt is shown how viscosity increases turbulence level in the inertial interval by suppressing turbulent transfer.

Thermocapillary flow near a cold wall
View Description Hide DescriptionThe thermocapillary feedback mechanism important at the edge of weld pools and other materials processes is examined through a model problem. A pool of liquid with a flat horizontal free surface is bounded on one side by a vertical solid wall, which is maintained at a cold temperature to unit depth, and at a warmer temperature below; far away the fluid is at the warmer temperature. Surface tension is a decreasing function of temperature, so that the surface thermal gradient drives flow toward the corner. When convection is vigorous, the flow compresses the thermal gradient which is driving the flow; this positive feedback results in small local length scales and high velocities near the corner. This problem is examined through a detailed scaling analysis and through numerical simulation for a range of parameters. The results show that for vigorous convection, the flow in the cold corner is locally determined.

Comments on the numerical investigation of Rayleigh and Marangoni convection in a vertical circular cylinder
View Description Hide DescriptionConvection in a cylindrical container was simulated with a three‐dimensional, time‐dependent code. For the case of purely Rayleigh convection, a completely rigid cylinder with adiabatic vertical walls and conducting horizontal walls was considered. The calculations showed that the individual velocity components could be in a transient state, while the total heat transfer was steady. This occurred in cases where the maximum azimuthal component of velocity was very small in magnitude, in comparison to the other two components, and this component decreased in time. On the other hand, the total kinetic energy along with the heat transfer reached a steady value. Consequently the present results have been shown to be at variance with the calculations of Neumann [J. Fluid Mech. 214, 559 (1990)]. Marangoni convection was modeled with a free flat surface on the upper side, assuming the superimposed second layer to be passive. The numerically obtained critical Marangoni numbers and flow patterns were compared favorably to earlier results from linearized stability. In addition, flow structural changes for supercritical Marangoni numbers were illustrated. Interestingly, axisymmetric disturbances led to nonsymmetric bifurcation diagrams, but three‐dimensional disturbance calculations led to symmetry in the bifurcation plots. Another very interesting result was the observed transition from three‐dimensional to two‐dimensional patterns as the Marangoni number was increased. Large computational requirements precluded a detailed parametric study. The special case of Prandtl number equal to 6.7 (corresponding to water), and in the case of Marangoni convection, a surface Biot number of unity was assumed.

Computer simulation study of the effective viscosity in Brinkman’s equation
View Description Hide DescriptionBrinkman’s equation is often used to match solutions of Stokes’ equation to solutions of Darcy’s law at free‐fluid:porous medium interfaces by the introduction of an effective viscosity parameter, μ_{ e }. Theoretical predictions of the dependence of μ_{ e } on the porosity of the porous medium have given conflicting results. A finite difference solution of Stokes’ equation in three dimensions was used to study fluid flow near the interface between a free fluid and a porous medium. It was found that in order to match solutions of Brinkman’s equation to the numerical solutions, the value of μ_{ e } had to be greater than the free‐fluid viscosity. Within numerical precision, the effective viscosity μ_{ e } was monotonically increasing with decreasing porosity. Good fits to the numerical fluid velocity profiles were obtained for porosities ranging from 50% to 80%.

Rectified flow of a rotating fluid along a vertical sidewall
View Description Hide DescriptionThe motion field resulting from an oscillatory, rotating flow parallel to a vertical wall and bounded from below by a horizontal plane along which there is no slip and from above by a horizontal frictionless lid is investigated numerically by using primitive equations. The model is barotropic and forced by oscillatory, spatially uniform, along‐wall, and lateral pressure gradients. The pertinent parameters are the Rossby, temporal Rossby, and Ekman numbers. By means of simulated particle tracking, including a statistical analysis, and a consideration of momentum balances, the physical mechanisms leading to flowrectification are identified; i.e., the exchange of along‐wall momentum of fluid parcels communicating between the bottom Ekman layer, the vertical shear layer, and the fluid interior. A rectified flow is generated along the wall for all subinertial, inertial, and superinertial oscillation frequencies investigated. The direction of the rectified current is such that the sidewall is on the right, facing downstream, for a vertically upward or Northern Hemisphere rotation. The mean transports and widths of the rectified currents are determined as functions of the appropriate system parameters. The current width is much larger than the classical E ^{1/4} Stewartson layer thickness. The numerical results are in good agreement with recent laboratory experiments.

Stability loss and sensitivity in hollow fiber drawing
View Description Hide DescriptionStability of hollow fiber drawing is studied, based on the quasi‐one‐dimensional equations of thin film dynamics. It is shown that the model isothermal drawing process is unstable when the draw ratio E (that of output to input velocities) exceeds the critical value E _{*}=20.22 when the viscosity force is dominating. Under stable regimes with E<E _{*}, the response of the as‐drawn fiber to external perturbations is studied (the sensitivity problem). In unstable situations with E≳E _{*}, onset of the so‐called draw resonance regime with self‐sustained oscillations is predicted by using numerical simulation. The effects of the inertia, gravity, surface tension, and gas pressure differential are considered.

Experimental observations and marginal stability calculations for counterflowing streams with swirl
View Description Hide DescriptionIn this paper reports are given on laser‐Doppler velocimeter measurements on counterflowing water streams with swirl. The experiments lend qualitative support to previous theory and computations [Goddard, Didwania, and Wu, J. Fluid Mech. 251, 149 (1993)] of the effects of swirl and confinement on the hydrodynamic stability of this flow in the high‐Re regime. In the present paper a linear stability analysis and computations of the marginal stability curve for self‐similar, radially unconfined flows are also given. This and the above‐cited work serve to define the practical operating limits on an associated flow device for generating a uniformly accessible interface of mass or heat transfer between counterflowing fluid streams.

The streaming flow initiated by oscillating cascades of cylinders and their stability
View Description Hide DescriptionThe steady fluid flow that is initiated by a cascade of cylinders, which oscillates harmonically in an unbounded, incompressible, viscous fluid that is otherwise at rest, is investigated, both numerically and experimentally. Finite‐difference techniques are used to obtain the numerical solutions, and the streaming flow results show a reasonable agreement with the experimental data. It is found, both numerically and experimentally, that for large values of the streaming Reynolds number,R _{ s }, the flow is not symmetrical about the axis of the oscillation, i.e., there exists a critical value of R _{ s }, R _{ s0} say, such that the streaming flows for R _{ s }≳R _{ s0} are unstable and those for R _{ s }<R _{ s0} are stable. In order to understand this asymmetry, a stability analysis is performed, and the agreement between the theory and the experimental results is encouraging.

Effect of surface tension on the stability of a binary fluid layer under reduced gravity
View Description Hide DescriptionA temperature gradient imposed across a binary fluid layer with a nonzero Soret coefficient will induce a solute concentration gradient. The ratio of these two gradients is proportional to the separation ratio χ, a property of the fluid. Similarly, the ratio of the thermal and solutal Marangoni numbers, which are nondimensional increments in surface tension due to changes in temperature and concentration, is also proportional to the separation ratio. As a consequence, the stability of a given binary fluid layer with a free surface under zero gravity depends only on the temperature difference, ΔT, imposed across the layer or, equivalently, on the thermal Marangoni number, M, albeit the dependence, is rather complicated. When the gravity is nonzero but of small magnitude, such that the buoyancy effects are not dominant, the stability characteristics of the layer are functions of two parameters, M and R, the thermal Rayleigh number. In this paper, the stability of such a binary layer under zero and reduced gravity by means of linear stability analysis is studied. Results show that the nature of the instability depends on the product χK, where K is a material constant=(α/α_{ S })(γ_{ S }/γ), with α and α_{ S } denoting the volumetric expansion coefficient due to temperature and solute concentration, respectively, and γ and γ_{ S } the rate of change of surface tension with respect to temperature and solute concentration, respectively. Both χ and K can assume positive and negative values. Under zero gravity, instability at the critical value of M onsets in steady convection if χK<0 and in oscillating convection if χK≳0. For a layer that is being heated from below and K≳0, the steady instability in the case of χK<0 can be rendered stable by subjecting the layer to a gravity of small magnitude. But for χK≳0, the effect of gravity is always destabilizing.

An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non‐Newtonian fluids
View Description Hide DescriptionAn experimental study on the onset and evolution of centrifugally driven rivulets is presented, which aims to investigate the influence on the instability of various experimental conditions (drop volume and rotational frequency), the wetting properties of the liquid (surface tension and contact angle), and fluid viscoelasticity. The apparatus allows continuous observation of the drop shapes following an impulsive spin‐up of the substrate, and these are analyzed by digital image analysis. The flows exhibit an onset time, or, equivalently, a critical radius, before which the drop spreads axisymmetrically. Data on drop spreading are compared with simple predictions of lubrication theory. The measured azimuthal wave number and growth rate of the instability are in good agreement with the linear stability analysis of Troian et al. [Europhys. Lett. 10, 25 (1989)], as long as the critical radius is taken from the experiment itself. The most unstable wavelength is found to be independent of both drop size and rotation speed in the range of parameters investigated, as observed previously by Melo et al. [Phys. Rev. Lett. 63, 1958 (1989)]. On the other hand, a change in the wetting properties of the liquid significantly modifies the critical radius, which, in turn, affects the number of fingers, with the nonwetting fluid exhibiting a smaller critical radius. This trend is in agreement with the mechanism of instability that is linked to the presence of a capillary ridge near the edge of the drop. No qualitative nor quantitative difference in behavior has been observed between a Boger fluid having a relaxation time of about 1 s, and its Newtonian solvent, in the experimental conditions considered.

Numerical simulation of a viscous vortex ring interaction with a density interface
View Description Hide DescriptionThe incompressible, variable‐density Navier–Stokes equations in axisymmetric coordinates are solved for the interaction of a vortex ring with a density interface. The effect of progressively weakening stratification and variation in Reynolds number are examined. Secondary and tertiary ring formation, vortex rebound, and backflow jets are some of the features observed in the interactions.

On the practical application of vortex breakdown theory to axially symmetrical and three‐dimensional compressible flows
View Description Hide DescriptionA perfect‐fluid theory for axially symmetrical vortex breakdownflows by Keller et al. [Z. Angew. Math. Phys. 36, 854 (1985); 39, 404 (1988)] is extended to three‐dimensional compressible flows. In a first step, an approximate theory for thin vortex cores is compared to the exact theory for axially symmetrical flows of inviscid, incompressible fluids. In subsequent steps, the approximation for thin vortex cores is extended to flows of compressible fluids that are not axially symmetrical. Finally, the various types of vortex breakdownflows and the shear layer instabilities at the surface of vortex breakdown bubbles are briefly discussed.

On tornado‐like vortex flows
View Description Hide DescriptionIn a first step, the problem of inviscid axisymmetric flow with buoyancy is investigated. It is found that both supercritical and subcritical vortex flows depart from their critical flow states if they are dominated by buoyancy. In a second step, the effects of entrainment are also taken into account. However, the investigation is restricted to nearly self‐similar flows. Both supercritical and subcritical vortex flows approach their critical flow states if they are dominated by entrainment. However, the superimposed effects of buoyancy and entrainment may lead to an equilibrium close to the critical state. It is argued that the existence of such an equilibrium is the direct reason for the appearance of certain violent vortex flows in nature, including tornadoes and fire storms. The weather conditions that may lead to the appearance of tornadoes and the different forms of appearance are also discussed.

Generation of ring vortices in axisymmetric spin‐down: A numerical investigation
View Description Hide DescriptionIn this paper the sequence of finite‐amplitude events that occurs prior to the onset of Taylor–Görtler (TG) vortices in axisymmetric spin‐down to rest is described. It is conjectured that the TG vortices are induced by these events, which include internal waves that propagate along the axis of rotation and transient vortex breakdown. The cascade of events is led by an internal ‘‘solitary wave’’ from each end wall. These waves are induced by the end‐wall (Bödewadt) boundary layers with concomitant Ekman ‘‘blowing.’’ The passage of the solitary wave causes a transient vortex breakdown followed by a train of internal waves. The combined effect of these phenomena induces disturbances in the sidewall boundary layer that grow and lead to the formation of TG vortices. The TG vortices that originate adjacent to the midplane of the cylinder in the sidewall (Rayleigh) boundary layer dominate the flow field at an intermediate time interval following the onset of spin‐down. Favorable comparisons with computationally determined onset times and subsequent evolution of TG vortices reported in the literature support the prediction that naturally occurring finite‐amplitude phenomena induce these vortices.

Corrections to Taylor’s hypothesis in a turbulent circular jet
View Description Hide DescriptionThe relation between mean square values of the time derivative θ_{,t }(≡∂θ/∂t) and the spatial derivatives θ_{,i }(≡∂θ/∂x _{ i }; i=1, 2, 3) of the temperature fluctuation θ, obtained with the assumptions of homogeneity and independence between small scales and large scales, is experimentally verified in the self‐preserving region of a circular jet where the turbulence intensity levels are relatively high. Local isotropy is approximately satisfied in the fully turbulent region of this flow and Wyngaard and Clifford’s [J. Atmos. Sci. 34, 922 (1977)] simplified relation between θ_{,t } ^{2} and θ_{,1} ^{2} is shown to be quite adequate in this region.

A numerical study of the viscous supersonic flow past a flat plate at large angles of incidence
View Description Hide DescriptionA numerical study of the viscoussupersonic flow past a flat plate is presented. The objective is to investigate the supersonic flow at high angles of incidence where large flow gradients occur. The effect of the angle of incidence and the Reynolds number (Re) in the flow structure especially in the formation of the separation region is investigated. The study is based on the solution of the full Navier–Stokes equations by high resolution schemes, and it focuses on the supersonic flow over the plate at Re≤10^{5}. Results on fine computational grids are presented for flow angles up to 20°. The calculations reveal that the flow remain attached for angles of incidence less than a=5°. For a=5° and Re=10^{5}, separation of the flow at the trailing edge appeared. Increasing the flow angle (a≳5°) moves the separation point upstream while a reverse flow region forms for the entire range of the Reynolds numbers used in this study. The results reveal that for large angles of incidence, the variation of the Reynolds number has significant effects on the variation of the flow variables. The flow behind the trailing edge is also affected from the flow angle as well as from the Reynolds number. Comparisons are also presented between viscous and inviscid solutions. The comparisons show that the viscous effects are dominant on the upper surface of the plate as well as behind the trailing edge. These effects become stronger when the flow angle is a=20°.

Observations regarding ‘‘Coherence and chaos in a model of turbulent boundary layer’’ by X. Zhou and L. Sirovich [Phys. Fluids A 4, 2855 (1992)]
View Description Hide DescriptionIn the following, the present authors comment on a paper by Zhou and Sirovich (ZS) [Phys. Fluids A 4, 2855 (1992)] which contained a critical appraisal of the models developed by Aubry et al. (AHLS) [J. Fluid Mech. 192, 115 (1988)]. It is found that ZS’s suggestion to use ‘‘a full channel interpretation of wall eigenfunctions,’’ thereby avoiding boundary terms, while attractive mathematically, is questionable in fluid mechanical terms. A major point of AHLS’s study was precisely to isolate the wall region and use the boundary terms to investigate the interaction with the outer flow. Also, it is demonstrated that certain instances of ‘‘irregular’’ bursting, as reported by ZS, are probably numerical effects.

Reply to ‘‘Observations regarding ‘Coherence and chaos in a model of turbulent boundary layer’ by X. Zhou and L. Sirovich [Phys. Fluids A 4, 2855 (1992)]’’
View Description Hide DescriptionIn view of the ‘‘observations’’ of the Cornell group, reassessment of their and our models for wall‐bounded turbulence has been made. Wide ranging evidence is presented for the existence and key role of propagating modes (streamwise dependent modes), absent in the original Cornell model but present in some of their later models. Evidence that the heteroclinic orbit (the bursting mechanisms) found in the original Cornell model is most likely an artifact of their Galerkin projection is presented. A thorough discussion detailing the physical and mathematical soundness, as well as the universality, of our models is presented.

Energy transfer in numerically simulated wall‐bounded turbulent flows
View Description Hide DescriptionUsing velocity fields obtained in direct numerical simulations of turbulent convection and of turbulent channel flow, the energy transfer process among lateral scales of motion in these low Reynolds number flows is analyzed. In all cases the energy is transferred most effectively between scales of similar size. As a result, the subgrid‐scale energy transfer is caused almost exclusively by interactions between resolved scales and subgrid scales characterized by wave numbers not greater than twice the cutoff wave number. The scale dependence of forward and inverse energy transfers contributing to the total subgrid‐scale eddyviscosity is discussed. The local energy transfer between small scales is strongly affected by the nonlocal interactions characterized by a scale separation greater than a factor of 2 in wave number. However, the direct energy transfer between scales satisfying this condition is one order of magnitude less than the local energy transfer between scales of similar size.

Detonation structures behind oblique shocks
View Description Hide DescriptionDetonation structures generated by wedge‐induced, oblique shocks in hydrogen–oxygen–nitrogen mixtures were investigated by time‐dependent numerical simulations. The simulations show a multidimensional detonation structure consisting of the following elements: (1) a nonreactive, oblique shock, (2) an induction zone, (3) a set of deflagration waves, and (4) a ‘‘reactive shock,’’ in which the shock front is closely coupled with the energy release. In a wide range of flow and mixture conditions, this structure is stable and very resilient to disturbances in the flow. The entire detonation structure is steady on the wedge when the flow behind the structure is completely supersonic. If a part of the flow behind the structure is subsonic, the entire structure may become detached from the wedge and move upstream continuously.