Index of content:
Volume 9, Issue 1, January 1997

Anomaly of excess pressure drops of the flow through very small orifices
View Description Hide DescriptionExcess pressuredrops are measured for the flow through very small orifices whose diameter ranges from the order of 1 mm to 10 μm using water, silicon oils, and solutions of glycerin in water: For larger orifices it is almost the same as, but for smaller orifices several times higher than, that of the numerical analysis of Newtonian fluid. Water gives the highest among the liquids used. Velocities are also measured along the centerline of a small orifice and found inconsistent with the result of the numerical analysis. A different mechanism may be dominant in the flow through very small orifices.

Consistent initial conditions for the DNS of compressible turbulence
View Description Hide DescriptionRelationships between diverse thermodynamic quantities appropriate to weakly compressible turbulence are derived. It is shown that for turbulence of a finite turbulentMach number there is a finite effect of compressibility. A methodology for generating initial conditions, consistent with finite Mach number, for the fluctuating pressure, density, and dilatational velocity is given. Use of these initial conditions gives rise to a smooth development of the flow, in contrast to cases in which these fields are specified arbitrarily or set to zero. Comparisons of the effect of different types of initial conditions are made using direct numerical simulation of decaying isotropic turbulence.

A theoretical analysis of the onset of surface‐tension‐driven convection in a horizontal liquid layer cooled suddenly from above
View Description Hide DescriptionThe onset of surface‐tension‐driven convection in a horizontal liquid layer cooled suddenly from above is analyzed by using linear stability theory. To obtain the critical condition to mark the onset of instability in the form of regular cellular motion the time domain is divided into two. With small time the basic temperature profile is strongly nonlinear and therefore the propagation theory we have developed is employed. Based on the propagation theory a new set of stability equations are derived and their scale analysis is discussed. It is found that the fluid layer is more stable with decreasing the Biot number and the Prandtl number. With large time the frozen‐time model is applied. In this time domain the stability criteria are independent of the Prandtl number. By connecting the predictions from the above two models the overall stability criteria are constructed. The interesting role of the Biot number on the critical condition is discussed in detail.

Non‐continuum anomalies in the apparent viscosity experienced by a test sphere moving through an otherwise quiescent suspension
View Description Hide DescriptionA comparison is made of the ‘‘apparent viscosity’’ (as defined by Stokes law) between two different cases of a test sphere moving slowly through an unbounded, otherwise quiescent, globally homogeneous, dilute suspension of identical, neutrally buoyant spherical particles dispersed in an incompressible Newtonian liquid. In case I the force on the test sphere is maintained constant for all time (and the torque‐free sphere allowed to rotate) — corresponding to the so‐called ‘‘falling ball’’ case — and its instantaneous velocity allowed to vary with proximity to each suspended sphere encountered during its trajectory; in case II the non‐rotating test sphere is towed with a uniform (instantaneous) velocity through the suspension and the force experienced by it allowed to vary with proximity to each suspended sphere. Allowing for two‐body hydrodynamic interactions between the ball and a suspended particle, the ensemble‐average velocity of the test sphere is calculated in case I and ensemble‐average force in case II, and Stokes law used to calculate the apparent viscosity of the suspension from the ensemble‐averaged, linear force/velocity ratio obtained. In each case the ‘‘apparent suspension viscosity’’ coefficient attains, as expected, the limiting, continuum, Einstein value of 2.5 when the test sphere is much larger than the freely suspended particle. However, in the case of disparate relative sizes, the apparent viscosity is found to be significantly larger in case II than in case I. The difference arises from the locally inhomogeneous nature of the suspension and points up a fundamental non‐continuum aspect of suspension behavior above and beyond the expected test/suspended‐sphere size ratio ‘‘Knudsen’’ non‐continuum effect.

Viscous damping and instabilities in stratified liquid film flowing down a slightly inclined plane
View Description Hide DescriptionA reduced nonlinear model for density stratified viscousfilm flowing down a slightly inclined wall is derived and explored. Under buoyancy stable stratification the system exhibits various long‐wavelength instabilities of noninertial, purely kinetic origin. Unlike many existing models for the filminterface evolution in the present study regularization of the pertinent long‐scale dynamics is provided directly by the filmviscosity rather than surface tension.

Transverse flow and mixing of granular materials in a rotating cylinder
View Description Hide DescriptionThe focus of this work is analysis of mixing in a rotating cylinder—a prototype system for mixing of granular materials—with the objective of understanding and highlighting the role of flow on the dynamics of the process. The analysis is restricted to low speeds of rotation, when the free surface of the granular solids is nearly flat, and when particles are identical so that segregation is unimportant. The flow is divided into two regions: a rapid flow region of the cascading layer at the free surface, and a fixed bed of particles rotating at the angular speed of the cylinder. A continuum model, in which averages are taken across the layer, is used to analyze the flow in the layer. Good agreement is obtained between the predictions of the flow model for the layer thickness profile and experimental results obtained by digital image analysis. The dynamics of the mixing process are studied by advecting tracer particles by the flow and allowing for particle diffusion in the cascading layer. The mixing model predictions for distribution of tracer particles and mixing rates are compared qualitatively and quantitatively to experimental data. Optimal operating conditions, at which mixing rates are maximum, are determined.

Observations of high Reynolds number bubbles interacting with a rigid wall
View Description Hide DescriptionThe behavior of bubbles with radii of 0.5–0.7 mm rising through water in the presence of a solid boundary were observed using a high‐speed video camera. Fluid inertia and surface tension cause a bubble to bounce several times from a horizontal wall before viscosity dissipates the energy. An energy balance involving the kinetic energy of the fluid motion, the surface energy of the air–water interface, and the gravitational potential energy aids in the interpretation of the dynamics of the collision. We also observed the motion of a bubble rising under an oblique wall with an angle of 10°–85° to the horizontal. When the angle was less than about 55° corresponding to We<0.4, the bubble slid steadily along the wall. At steeper angles the bubble was observed to bounce repeatedly from the inclined wall without any apparent loss of amplitude. It was also determined that the critical Weber number of coalescence of a bubble rising toward a stationary bubble is 1.6. At Weber numbers below this critical value, the two bubbles coalesce on impact while bubbles bounce at higher Weber numbers.

Three‐dimensional vortex/wall interaction: Entrainment in numerical simulation and experiment
View Description Hide DescriptionThree‐dimensional interactions between an elliptic vortex ring and two no‐slip parallel walls are visualized in a numerical simulation and an experiment. The vortex ring induces a surfacevorticity layer on the wall which reconnects with the vortex ring. During the interaction core‐area‐varying axial waves are generated and carry the surface layer away from the wall. The vortex ring then becomes two tornado‐like structures with strong upward helical flows near the wall surface, which provides entrainment of the surface fluid into the vortex structure. Similarities between the entrainment mechanisms of vorticity in the simulation and dye in the experiment from the surface layer are identified.

A new instability for finite Prandtl number rotating convection with free‐slip boundary conditions
View Description Hide DescriptionRolls in finite Prandtl number rotating convection with free‐slip top and bottom boundary conditions are shown to be unstable with respect to small angle perturbations for any value of the Crotation rate. This instability is driven by the horizontal mean flow whose estimation requires a special singular perturbation analysis.

Bifurcation phenomena in incompressible sudden expansion flows
View Description Hide DescriptionA numerical study of laminar incompressible flows in symmetric plane sudden expansions was carried out. Computations were performed for various Reynolds number and expansion ratios. The results revealed that the flow remains symmetric up to a certain Reynolds number depending on the expansion ratio, while asymmetries appear at higher Reynolds numbers. The computations indicated that the critical Reynolds number of the symmetry‐breaking bifurcation reduces when increasing the expansion ratio while the flow regains symmetry downstream of an initial channel length. The flow asymmetries were verified by comparing several discretization schemes up to fourth order of accuracy as well as various iterative solvers.

Vortex formation in mixing layers: A weakly nonlinear stability approach
View Description Hide DescriptionThis paper is concerned with the stability of two‐dimensional incompressible mixing layers with small transversal velocity gradients. Using the approach of slight flow divergence, a boundary layer type of approximation of solutions to the steady mixing layer flow is obtained. We derive in the limit of small velocity gradients a velocity profile of an error function type. To gain an insight into the problem of the spatialinstability of this flow we apply a model involving perturbations with only a single frequency component. A generalized approach, however, is outlined in the Appendix. There the interaction of a fundamental mode with its subharmonic, oscillating at one‐third the frequency, is analyzed. First numerical results show that under certain conditions the subharmonic can represent the dominant disturbance. A multiple scales expansion of the disturbance streamfunction is constructed with variables chosen to derive a Landau‐type equation with cubic nonlinearities governing an amplitude function A. Scaling spatial and temporal variables and the Reynolds number we obtain in leading order a generalized Rayleigh equation. We solve the associated eigenvalue problem for spatially growing modes, whereas the calculation of damped modes is beyond the scope of our approach. The solution to this equation can be separated into a shape function and the amplitude A. An investigation of the second‐order terms yields a rederivation of the boundary layer approximation of the steady flow, an equation governing second harmonics of the disturbance and an equation determining the mean flow correction. At third order we have to apply a resonance condition, which demands small linear spatial growth rates. This restriction is consistent with the limit of small velocity gradients and we can thus derive a cubic amplitude equation governing the space–time evolution of A. This equation can be cast into a separated first‐order ODE. The numerical results show that the combined effect of nonlinearity and flow divergence strongly influences the amplitudes such that they reach a maximum and decay farther downstream. The study is based on an essentially inviscid approach. The regime of amplitude decay is therefore restricted and the integrations have to be terminated when the limit of neutral growth is reached. Comparison with experimental data is difficult because the latter are taken for higher values of the velocity gradients. Yet typical experimental trends are predicted by the findings of the present study. We found in particular in the numerical study of vortex contours structural instabilities in terms of breakup of sinusoidal lines to create vortex patches and the phenomenon of cut‐and‐connect of vortices near ‘‘saddle points.’’

The effect of vortex generators on a jet in a cross‐flow
View Description Hide DescriptionThe effect of vortex generators in the form of tabs on the penetration and spreading of a jet in a cross‐flow has been studied experimentally. It is found that the tab has very little effect when placed on the leeward side, i.e., on the downstream edge of the jet nozzle relative to the free‐stream flow. A study of the static pressure distribution reveals significantly lower pressures on the leeward side. Thus, when placed on that side the tab does not produce a ‘‘pressure hill’’ of sufficient magnitude that is the primary source of streamwise vorticity in the flow field over the tab. This qualitatively explains the ineffectiveness. In comparison, there is a significant effect on the flow field when the tab is placed on the windward side. The sense of vorticity generated by the tab in the latter configuration is opposite to that of the bound vortex pair that otherwise characterizes the flow. Thus, the strength of the bound vortex pair is diminished and the jet penetration is reduced.

Horizontal jets in a rotating stratified fluid
View Description Hide DescriptionA horizontal jet emerging continuously from a small round nozzle (concentrated source of momentum) in a rotaing stratified fluid is investigated using laboratory experiments. The jet either (i) deflects from the direction of injection, forming an anticyclonic spiral monopole (monopole regime), or (ii) propagates along the injection direction, forming a dipolar structure (dipole regime). Which of these characteristic flows occurs depends on the system parameters, the Reynolds number Re, and the buoyancy frequency to Coriolis parameter ratio N/f; a flow regime diagram is developed for the parameter ranges 40≲ Re≲200 and 0≲N/f≲35, respectively. A theoretical analysis is advanced to explain the conditions under which the monopole and dipole regimes occur, including the transition curve between the two regimes. The theory is supported by laboratory experiments. Some geophysical examples of the considered flows are discussed.

Gas‐liquid heat transfer in a bubble collapsing near a wall
View Description Hide DescriptionThe collapse of a gas bubble near a solid wall is studied numerically by assuming the liquid to be incompressible and the Mach number of the gas flow to be small. The liquid motion is simulated by a boundary integral method and the gas thermo‐fluid dynamics by finite differences on a boundary‐fitted grid. With the physical properties of a liquid monopropellant, it is found that the liquid heating is essentially localized in the microjet, but is probably not sufficient to cause spontaneous ignition. The reasons for this conclusion — that, while being in general agreement with available experimental evidence, is at variance with deductions from previous spherical collapse calculations — are elucidated.

Renormalization group theory for turbulence: Eddy‐viscosity type model based on an iterative averaging method
View Description Hide DescriptionThe renormalization group (RNG) theory of turbulence is often used for the forced Navier–Stokes equation in order to investigate turbulencemodels in Fourier space. The strong point of this kind of theory is the ability to construct turbulencemodels with the aid of the Kolmogorov −5/3 power law for the energy spectrum. In this paper, we have made use of an iterative averaging method proposed by McComb (1990), which does not have the misleading ε‐expansion technique developed by Yakhot and Orszag (1986), then applied this method to the derivation of an eddy‐viscosity type turbulencemodel. Using the exact Navier–Stokes equation excluding artificial external forces, we have obtained the eddy‐viscosity type turbulencemodel which is equivalent to the Boussinesq postulate, and its model constant C _{μ} is determined from only a Kolmogorov constant α.

Probability density function and Reynolds‐stress modeling of near‐wall turbulent flows
View Description Hide DescriptionProbability density function (pdf) methods are extended to include modeling of wall‐bounded turbulent flows. A pdf near‐wall model is developed in which the generalized Langevin model is combined with an exact model for viscous transport. Then the method of elliptic relaxation is used to incorporate the wall effects without the use of wall functions or damping functions. Information about the proximity of the wall is provided only in the boundary conditions so that the model can be implemented without ad hoc assumptions about the geometry of the flow. A Reynolds‐stress closure is derived from this pdf model, and its predictions are compared with DNS and experimental results for fully developed turbulent channel flow.

Kolmogorov flow in three dimensions
View Description Hide DescriptionA numerical study of the long‐time evolution of incompressible Navier‐Stokes turbulence forced at a single long‐wavelength Fourier mode, i.e., a Kolmogorov flow, has been completed. The boundary conditions are periodic in three dimensions and the forcing is effected by imposing a steady, two‐dimensional, sinusoidal shear velocity which is directed along the x‐direction and varies along the z‐direction. A comparison with experimental data shows agreement with measured cross‐correlations of the turbulent velocity components which lie in the mean‐flow plane. A statistical analysis reveals that the shear‐driven turbulence studied here has significant spectral anisotropy which increases with wave number.

On the decay of two‐dimensional homogeneous turbulence
View Description Hide DescriptionDirect numerical simulations of decaying two‐dimensional turbulence in a fluid of large extent are performed primarily to ascertain the asymptotic decay laws of the energy and enstrophy. It is determined that a critical Reynolds numberR _{ c } exists such that for initial Reynolds numbers with R(0)<R _{ c } final period of decay solutions result, whereas for R(0)>R _{ c } the flow field evolves with increasing Reynolds number. Exactly at R(0)=R _{ c }, the turbulence evolves with constant Reynolds number and the energy decays as t ^{−1} and the enstrophy as t ^{−2}. A t ^{−2} decay law for the enstrophy was originally predicted by Batchelor for large Reynolds numbers [Phys. Fluids Suppl. II, 12, 233 (1969)]. Numerical simulations are then performed for a wide range of initial Reynolds numbers with R(0)>R _{ c } to study whether a universal power‐law decay for the energy and enstrophy exist as t→∞. Different scaling laws are observed for R(0) moderately larger than R _{ c }. When R(0) becomes sufficiently large so that the energy remains essentially constant, the enstrophy decays at large times as approximately t ^{−0.8}.

Scalar intermittency and the ground state of periodic Schrödinger equations
View Description Hide DescriptionRecent studies of a passive scalar diffusing in a rapidly fluctuating Gaussian distributed linear shear layer have demonstrated intermittency in the form of broad tails and non‐symmetric limiting probability distribution functions. In this paper the authors explore similar issues within the context of a large class of rapidly fluctuating bounded periodic shear layers. We compute the evolution of the moments by analogy to an N dimensional quantum mechanics problem. By direct comparison of an appropriate system of interacting and non‐interacting quantum particles, we illustrate that the role of interaction is to induce a lowering of the ground state energy, which implies that the scalar PDF will have broader than Gaussian tails for all large, but finite times. We demonstrate for the case of Gaussian random wave initial data involving a zero spatial mean, that the effect of this energy shift is to induce diverging normalized flatness factors indicative of very broad tails. For the more general case with Gaussian random initial data involving a non‐zero spatial mean, the distribution must approach that of a Gaussian at infinite times, as required by homogenization theory, but we show that the approach is highly non‐uniform. In particular our calculation shows that the time required for the system to approach Gaussian statistics grows like the square of the moment number.

Numerical studies of real‐gas effects on two‐dimensional hypersonic shock‐wave/boundary‐layer interaction
View Description Hide DescriptionNonequilibrium real‐gas effects on surface heating rates, skin friction, and flow field unsteadiness of two‐dimensional hypersonic shock‐wave/boundary‐layer interaction were studied by numerical simulations. The unsteady Navier–Stokes equations with nonequilibrium vibrational and chemical models for five‐species air were solved by a finite‐volume second‐order TVD scheme together with a third‐order semi‐implicit Runge–Kutta scheme. Two cases of high‐enthalpy shock/boundary layer interaction problems were studied in this paper. The freestream enthalpy was high enough to produce vibrational excitation and dissociation/recombination chemistry behind the shock. The first case was a steady two‐dimensional shock/boundary layer interaction on a flat plate with a mixture of N_{2} and O_{2} in the freestream. It was found that the real gas effects reduce the size of the shock induced separation bubble and the magnitude of the surface heating rates. The second case was a self‐sustained unsteady type IV shock–shock interference heating of a pure N_{2}flow over a cylinder. The results showed that type IV shock–shock interference heatingflows with real‐gas effects are inherently unsteady. Vortices are generated and shed off near the jet impingement point. This periodic shedding of the vortices contributes to the self‐sustained oscillations of both the jet and other parts of the flow fields. In addition, the real‐gas effects reduce the level of peak surface heating and peak surface pressure due to endothermic real‐gas effects.