Volume 8, Issue 3, March 1996
Index of content:

Sweeping and straining effects in sound generation by high Reynolds number isotropic turbulence
View Description Hide DescriptionThe sound radiated by isotropic turbulence is computed using inertial range scaling expressions for the relevant two‐time and two‐point correlations. The result depends on whether the decay of Eulerian time correlations is dominated by large scale sweeping or by local straining: the straining hypothesis leads to an expression for total acoustic power given originally by Proudman, whereas the sweeping hypothesis leads to a more recent result due to Lilley.

Capillary effects during droplet impact on a solid surface
View Description Hide DescriptionImpact of water droplets on a flat, solid surface was studied using both experiments and numerical simulation. Liquid–solid contact angle was varied in experiments by adding traces of a surfactant to water. Impacting droplets were photographed and liquid–solid contact diameters and contact angles were measured from photographs. A numerical solution of the Navier–Stokes equation using a modified SOLA‐VOF method was used to modeldroplet deformation. Measured values of dynamic contact angles were used as a boundary condition for the numerical model. Impacting droplets spread on the surface until liquid surface tension and viscosity overcame inertial forces, after which they recoiled off the surface. Adding a surfactant did not affect droplet shape during the initial stages of impact, but did increase maximum spread diameter and reduce recoil height. Comparison of computer generated images of impacting droplets with photographs showed that the numerical model modeled droplet shape evolution correctly. Accurate predictions were obtained for droplet contact diameter during spreading and at equilibrium. The model overpredicted droplet contact diameters during recoil. Assuming that dynamic surface tension of surfactant solutions is constant, equaling that of pure water, gave predicted droplet shapes that best agreed with experimental observations. When the contact angle was assumed constant in the model, equal to the measured equilibrium value, predictions were less accurate. A simple analytical model was developed to predict maximum droplet diameter after impact. Model predictions agreed well with experimental measurements reported in the literature. Capillary effects were shown to be negligible during droplet impact when We≫Re^{1/2}.

Coarse‐grained description of thermo‐capillary flow
View Description Hide DescriptionA mesoscopic or coarse‐grained approach is presented to study thermo‐capillary induced flows. An order parameter representation of a two‐phase binary fluid is used in which the interfacial region separating the phases naturally occupies a transition zone of small width. The order parameter satisfies the Cahn–Hilliard equation with advective transport. A modified Navier–Stokes equation that incorporates an explicit coupling to the order parameter field governs fluid flow. It reduces, in the limit of an infinitely thin interface, to the Navier–Stokes equation within the bulk phases and to two interfacial forces: a normal capillary force proportional to the surface tension and the mean curvature of the surface, and a tangential force proportional to the tangential derivative of the surface tension. The method is illustrated in two cases: thermo‐capillary migration of drops and phase separation via spinodal decomposition, both in an externally imposed temperature gradient.

The instability of a liquid layer heated from the side when the upper surface is open to air
View Description Hide DescriptionWhen a liquid layer is heated from the side, a monocellular flow develops immediately, no matter how small the temperature difference is. If the temperature gradient between the side walls is increased, this flow becomes unstable. Laser Doppler velocimetrymeasurements are reported here in an attempt to describe the main features of both the basic flow and the instability modes. It is found that before the appearance of traveling waves (the most dangerous mode as predicted by the theory), stable rolls with their axes perpendicular to the temperature gradient, span over the whole liquid layer, starting from the hot side, even if the aspect ratio (the length of the layer divided by its thickness) is very high. This unexpected situation modifies the basic flow. A further increase of the temperature gradient leads to the appearance of a time periodic motion.

Chaotic mixing and heat transfer between confocal ellipses: Experimental and numerical results
View Description Hide DescriptionThe annular region between two concentric, confocal ellipses is a new geometry which is particularly effective for either mixing of viscous fluids or heat transfer enhancement in the important limit of a high Péclet number. This geometry is in many respects similar to the annular space between two eccentric, rotating cylinders although it possesses two (instead of one) axes of symmetry. The recently obtained analytical solution of the Stokes flowequations in this geometry shows that at steady state and for counter‐rotation of the inner and outer ellipses, two opposite saddle points (connected by two different streamlines) appear in the region of minimum gap. This flow characteristic is also exhibited by the eccentric cylinder system for some cases of co‐rotation. The Poincaré sections obtained when the inner and outer ellipses are displaced using a discontinuous velocity protocol show that a more effective long term mixing is obtained for the counter‐rotating case, this is confirmed by the experimental data we have obtained. The opposite conclusions (more effective mixing for co‐rotation) have been given in the eccentric cylinder geometry.
Photographs of the fluorescent dye after 5 periods are compared with remarkable success to numerical blob deformation experiments. Experimental results also confirm previous results based on an analysis of Poincaré sections. In particular, better mixing is obtained when the inner ellipse displacement per period increases. Finally, this geometry is shown to be particularly effective as a heat exchanger. For steady, counter‐rotation of the two boundaries, the recirculation zones can lead to a heat transfer rate increase of 80% over that of pure conduction at high Péclet numbers, and, by an appropriate sinusoidal modulation of the angular velocity of one boundary, the heat transfer rate can be more than double that of pure conduction. Since an analysis of the experimental data also suggests that the mixing rate for a sinusoidal modulation of the angular velocity of the boundaries is better than for a discontinuous velocity protocol, we propose that the average Nusselt number per period could be one of the several useful tools in the global optimization of the mixing protocol.

Instabilities of the sidewall boundary layer in a differentially driven rotating cylinder
View Description Hide DescriptionWe report experiments on the instability of the sidewall Stewartson layer that arises in the fluid next to the vertical boundary of a cylinder that is rotating about its axis at rate Ω, and is driven by slow differential rotation of a contact lid. There are three primary instabilities that come in sequentially as the driving is increased: vertically traveling axisymmetric rolls, quasistationary diagonal rolls inclined backward (retrograde, or opposite to the direction of the basic rotation Ω) with height, and prograde‐traveling waves that tilt forward with height. These instabilities are markedly different from those observed in laboratory studies or computational simulations of pressure‐driven flow down a rotating channel. The instability mechanisms appear to be related to our particular method of driving these flows.

Internal structure of extreme standing waves on deep water
View Description Hide DescriptionA special procedure is applied to remove the non‐physical distortion that always appears when numerically calculating velocity and related quantities in the close vicinity of a discretized vortex sheet. The procedure is used in conjunction with a semi‐Lagrangian boundary integral method to study the flowfield beneath free surface standing water waves. Pressure, velocity and acceleration are calculated below the wave and some aspects of crest formation are investigated. At large depths results confirm the existence of an unattenuated oscillatory pressure component, giving behavior consistent with previously known theory.

Unsteady separation from the leading edge of a thin airfoil
View Description Hide DescriptionAt high Reynolds numbers, the process leading to dynamic stall on airfoils initiates in the leading‐edge region. For thin airfoils, the local motion near rounded leading edges can be represented as flow past a parabola and when the mainstream flow is at an angle of attack to the airfoil, a portion of the boundary layer will be exposed to an adverse pressure gradient. Once the angle of attack exceeds a certain critical value, it is demonstrated that unsteady boundary‐layer separation will occur in the leading‐edge region in the form of an abrupt focused boundary‐layer eruption. This process is believed to initiate the formation of the dynamic stall vortex. For impulsively‐started incompressible flow past a parabola, a generic behavior is found to occur over a range of angles of attack, and a limit solution corresponding to relatively large angles is found. The separation in the leading‐edge region develops in a zone of relatively limited streamwise extent over a wide range of angles of attack. This suggests that localized control measures (such as suction) may possibly be effective at inhibiting separation.

The effect of streamwise braid vortices on the particle dispersion in a plane mixing layer. I. Equilibrium points and their stability
View Description Hide DescriptionThe dynamics of small, heavy, spherical particles are investigated in an analytical model of the stretched counterrotating streamwise braid vortices commonly found in three‐dimensionally evolving mixing layers. The flow field consists of two superimposed rows of Stuart vortices of opposite sign, with an additional two‐dimensional strain field. The particle dynamics are determined by a balance of inertial, gravitational, and viscous drag forces, i.e., the dimensionless Stokes and Froude numbers, St and Fr, as well as by the dimensionless strain rate, and the Stuart vortex family parameter. Equilibrium points for the particles, as well as their stability criteria, are determined analytically, both in the absence and in the presence of gravity, and for different orientations of the gravity vector. In the absence of gravity, accumulation of low St particles can occur at the center of the braid vortices. An analytical expression for the critical particle diameter, below which accumulation is possible, is derived. The presence of gravity can lead to the emergence of multiple equilibrium points, whose stability properties depend on their locations. For a horizontal mixing layer flow and strong gravity effects, unconditional accumulation can occur midway between the streamwise braid vortices in the upwelling regions. Conditionally stable accumulation regions exist a short horizontal distance away from the centers of the braid vortices. If the gravity vector lies within the plane of the mixing layer, accumulation points exist only for moderate strengths of gravity. Under these circumstances, conditional accumulation is possible near the streamwise vortex centers.

The effect of streamwise braid vortices on the particle dispersion in a plane mixing layer. II. Nonlinear particle dynamics
View Description Hide DescriptionA computational investigation of the nonlinear dynamics of heavy particles in a row of counterrotating strained vortices is presented. By tracking the particles numerically in the quasi‐two‐dimensional fluid velocity field, information is obtained on the nature of their trajectories, as well as on probability distribution functions and potential accumulation regions. The particle behavior is discussed as a function of the dimensionless strain rate, the particle Stokes number St, and the dimensionless gravity parameter Fr. Only for very low values of the St can the particles accumulate at the vortex centers. For moderate values of St, they remain trapped on closed trajectories around the vortex centers. Increasing St leads to periodic open trajectories that allow for spanwise transport of the particles. Further bifurcations lead to the generation of multiple trajectories, as well as to subharmonic solutions. Eventually, intermittent and chaotic particle dynamics are observed. In the chaotic regime, a simplified flow model is employed in order to derive various scaling laws for the particle concentration field. For strong levels of gravity, the accumulation of large numbers of particles is observed in the upwelling regions predicted in part I of the present investigation [B. Marcu and E. Meiburg, Phys. Fluids 8, 715 (1996)].

The bifurcation of circular jets in crossflow
View Description Hide DescriptionAn experimental study of an incompressible circular jet in a crossflow and theoretical analysis based on inviscid flowmodels are described. The jet exits from a rigidly mounted pipe projecting distant from the floor of a tunnel carrying a steady stream of water; density of the jet and the stream are the same. The results of scalar and velocity measurements and visualizations showed that the jet bifurcated into two separated, counterrotating arms for values of ε=U _{∞}/U _{JET}, the ratio of the mean crossflow velocity U _{∞} to the mean jet discharge velocity U _{JET}, less than or equal to 0.25. The angle of separation between the two arms of the bifurcated jet was found to vary inversely with ε. For higher values of ε the jet does not bifurcate but is dominated by a different mode of instability. The structure of the flow field, which is different for bifurcated and nonbifurcated jets, comprised a variety of vortical structures which survived for very long distances x beyond x/2a≳400, where a is the radius of the jet exit and x is distance downstream from the jet axis. The location of the point of bifurcation is predicted from consideration of potential flowmodels and the characteristics of bifurcating elliptical jets. The location of the point of bifurcation is more distant from the jet exit for smaller values of ε, and experimental results were in good agreement with the theoretical predictions. The initial jet trajectory is shown to be associated with the presence in the wake of vorticity shed from the pipe. The near‐field geometry and centerline trajectory of the jet are also found to be in accord with predictions in that it is observed that z∼x ^{1/2} and z∼ε^{−1}. Dilutions of bifurcated jets are found to be greater than for nonbifurcated jets.

Energy balance for turbulent flow around a surface mounted cube placed in a channel
View Description Hide DescriptionResults of an experimental investigation of the inhomogeneous, three‐dimensional flow around a surface mounted cube in a channel are presented. LDAmeasurements of single‐point velocity correlations are used to determine the production, convection and transport of the turbulence kinetic energy, k, in the obstacle wake. The turbulence dissipation rate is obtained as a closing term to the balance of the k‐transport equation. The results provide some insight to the evolution of the turbulence dissipation rate from the near field recirculation zone to the asymptotic wake. Also presented is a comparison between measured and modeled transport terms.

On the consistency of Reynolds stress turbulence closures with hydrodynamic stability theory
View Description Hide DescriptionThe consistency of second‐order closure models with results from hydrodynamic stability theory is analyzed for the simplified case of homogeneous turbulence. In a recent study, Speziale, Gatski, and Mac Giolla Mhuiris [Phys. Fluids A 2, 1678 (1990)] showed that second‐order closures are capable of yielding results that are consistent with linear stability theory for the case of homogeneous shear flow in a rotating frame. It is demonstrated in this paper that this success is due to the fact that the stability boundaries for rotating homogeneous shear flow are not dependent on the details of the spatial structure of the disturbances. For those instances where they are—such as in the case of elliptical flows where the instability mechanism is more subtle—the results are not so favorable. The origins and extent of this modeling problem are examined in detail along with a possible resolution based on Rapid Distortion Theory (RDT) and its implications for turbulence modeling.

A low‐shear turbulent boundary layer
View Description Hide DescriptionThis paper describes experimental results measured in a low‐shear turbulent boundary layer. The low‐shear condition is exerted after the boundary layer reaches Re_{θ}≂2000 and has the effect of removing the inner layer; thus, these are the first results to show the behavior of an outer‐layer‐only turbulent boundary layer. The removal of the inner layer causes the gradual decay of the turbulent stresses over eleven boundary‐layer thicknesses (roughly 20 large‐eddy length scales) of streamwise distance, with the decay beginning at the wall and propagating into the outer flow with increasing downstream distance. However, the structure of the outer layer is little affected by the perturbation, as demonstrated by stress (anisotropy) ratios, quadrant analysis, and spectral measurements. Although the lack of near‐wall production implies this flow must eventually decay into isotropic turbulence, this decay occurs relatively slowly because the dissipation is also greatly reduced with the decrease in near‐wall shear. In addition, the outer‐layer production is significant in maintaining the turbulence level. These results show that, once formed, the outer‐layer characteristics are not explicitly dependent on the presence of the inner layer. These results are compared with similar studies of isotropic turbulence near a shear‐free wall. Very close to the wall the two flows both show that the normal‐stress components respond differently to the presence of the wall. However, away from the wall the isotropic results underpredict the distance to which the tangential stresses are damped by the impermeability condition at the wall. Finally, the results show general similarities to those in a boundary layer just downstream of reattachment, after a similar low‐shear condition over the separation bubble. This raises the possibility that many of the important features of the reattaching flow can be captured by the present, simpler experiment.

Interaction of wake turbulence with a free surface
View Description Hide DescriptionThe turbulent wake of a flat plate aligned with a uniform water flow and extending through the free surface was investigated experimentally. Laser‐Doppler velocimetry (LDV)measurements show good agreement with published data for a two‐dimensional wake, except in a shallow layer near the free surface. In this surface layer, the wake width is observed to double while the wake centerline velocity remains essentially unchanged from its value at depth, resulting in a wake momentum deficit that is twice that at depth. Instantaneous, full‐field measurements of the velocity were made using digital particle image velocimetry (DPIV) to elucidate the role of vortical structures in the development of the surface layer. DPIV measurements reveal that in the deep wake, vortexstructures predominantly of opposite sign exist on opposite sides of the wake centerline and contribute to a velocity and vorticity field that is two‐dimensional in the mean. However, near the surface,vortexstructures tend to become either surface‐normal or surface‐parallel and contribute to a velocity and vorticity field that is highly three‐dimensional in the mean. The resulting surface layer is characterized by surface–parallel structures interacting with their images above the surface to retard and widen the surfaceflow to a depth comparable to the size of the vortexstructures. Histograms taken from many independent DPIV realizations of the flow characterize the distribution of vorticity in the wake and verify a mean flow consistent with the LDVmeasurements. The existence of these structures in the surface layer is further confirmed by flow visualization using laser‐induced fluorescence.

A two‐phase flow model of the Rayleigh–Taylor mixing zone
View Description Hide DescriptionThe Rayleigh–Taylor instability of an interface separating fluids of distinct density is driven by an acceleration across the interface. Low order statistical moments of fluctuating fluid quantities characterize the hydrodynamics of the mixing zone. A new model is proposed for the momentum coupling between the two phases. This model is validated against computational data for compressible flows, including flows near the incompressible limit. Our main result is a zero parameter first order closure for ensemble averaged two phase flow equations. We do not, however, fully solve the closure problem, as the equations we derive are missing an (internal) boundary condition along any surface for which either phase goes to zero volume fraction. In this sense, the closure problem is reduced from a volume to a surface condition, rather than being solved completely. We compare two formulations of the statistical moments, one based on two phase flow and the other on turbulencemodels. These formulations describe different aspects of the mixing process. For the problem considered, the two phase flow moments appear to be preferable, in that they subsume the turbulence moments but not conversely.

Transient growth in compressible boundary layer flow
View Description Hide DescriptionThe potential for transient growth in compressible boundary layers is studied. Transient amplification is mathematically associated with a non‐orthogonal eigenvector basis, and can amplify disturbances although the spectrum of the linearized evolution operator is entirely confined to the stable half‐plane. Compressible boundary layer flow shows a large amount of transient growth over a wide range of parameter values. The disturbance size is here measured by a positive definite energy like quantity that has been derived such that pressure‐related transfer terms in its evolution equation mutually cancel. The maximum of the transient growth is found for structures which are independent of the streamwise direction and is found to scale with R ^{2}. This suggests that the transient growth originates from the same lift‐up mechanism found to give large growth in incompressible shear flows. The maximum growth is also found to increase with Mach number. In compressible flow, disturbances that experience optimal transient growth can be excited naturally by a non‐linear interaction of oblique unstable first mode waves. Thus, a triggering of transient growth may account for the difference in timescales between the fast oblique breakdown process and traditional secondary instability.

Two‐point velocity and vorticity correlations for axisymmetric turbulence
View Description Hide DescriptionDirect numerical simulation data for two‐point velocity and vorticity correlations at small separations near the centerline of a fully developed turbulent channel flow are more closely approximated by axisymmetry than isotropy.