Volume 7, Issue 10, October 1995
Index of content:

Turbulent vortex breakdown
View Description Hide DescriptionReported herein is a cone‐shaped turbulentvortex breakdown in noncavitatingswirling flows at high Reynolds numbers in a slightly diverging cylindrical tube. The turbulent conical form is in addition to the well‐known double‐helix, spiral, and nearly axisymmetric or ‘‘bubble’’‐type breakdowns.

Disturbance growth triggered by steady heating of a jet’s nozzle exit boundary layer
View Description Hide DescriptionExperiments were conducted to investigate how heating only the nozzle exit boundary layer of an axisymmetric jet, with an unheated potential core, affects disturbance growth in the initial shear layer. The exit boundary layer had a minimum density ratio of 0.74, was laminar, and had a constant momentum thickness (θ) for all levels of nozzle heating used in this study. The fluctuating velocity (u′) and temperature (t′) in the exit boundary layer increased monotonically with increasing nozzle temperature. Low‐amplitude acoustic excitation produced a more rapid growth of coherent velocity fluctuations for the heated case than for the unheated.

Scaling of streamwise vortices in wakes
View Description Hide DescriptionIn this Letter, we demonstrate the coexistence of two distinct systems of streamwise vortices in a bluff body wake. It appears that there exist conditions to amplify streamwise vorticity in bluff body wakes, by vortex stretching, in both the separating shear layers from the sides of the body and also in the vortex street wake. The length scale governing the streamwise vortices in the shear layer has a 1/√Re dependence, whereas the scale of such structures in the wake is independent of Reynolds number, Re (over a large range of Re). The proposition that there should exist two distinct, and possibly disparate, spanwise length scales in the cylinder wake is well supported by compiled measurements, particularly those of Williams and co‐workers (Mansy et al. [J. Fluid Mech. 270, 277 (1994)]), as well as those from Chyu and Rockwell (submitted to J. Fluid Mech.).

Collision rates of spherical drops or particles in a shear flow at arbitrary Péclet numbers
View Description Hide DescriptionCollision rates of two nondeformable, freely suspended drops (or particles) subject to Brownian motion in a simple shear at low Reynolds number are calculated from the solution of the full Fokker–Plank equation for the pair distribution function. Unlike previous studies on shear‐induced collisions, the solution is presented for arbitrary Péclet number (Pe), thus covering a broad range of drop sizes. An efficient numerical technique includes a mixed Galerkin/finite‐difference approximation and the ideas of analytical continuation, to represent the solution of the discrete problem as a convergent series for all real Pe. The mobility functions are provided from exact two‐drop hydrodynamics and near‐contact asymptotics. Extensive calculations are presented for the collision efficiency as a function of the size ratio, drop‐to‐medium viscosity ratio (μ̂), and Pe≤O(10^{2}), for the case of no interdroplet forces. For μ̂≳0, the correction to the collision efficiency for Pe≫1 is O(Pe^{−1/2}). For bubbles (μ̂=0), there is also an O(Pe^{−2/3}) correction of opposite sign, resulting in a local minimum for the collision efficiency. The asymptotic analysis for the opposite limit of Pe≪1 is in excellent agreement with the numerical calculations. For intermediate Pe, the exact numerical solution is compared with different ‘‘additive approximations.’’ The simple two‐term additivity approximation is generally unsuccessful, whereas a modified, three‐term approximation provides reasonable results except at small size ratios and large viscosity ratios. The effect of the van der Waals attractions on the collision efficiency for typical emulsion drops of 1–10 micron size with μ̂=O(1) is relatively small, of the order 10% in the Brownian regime. As a limiting case of drops, the collision efficiency for equal‐sized solid spheres with van der Waals attractions is calculated for Pe≤200; this limit shows a stronger dependence on the Hamaker constant and the retardation parameter. The solution for solid spheres is in excellent agreement with reported experimental data on flocculation dynamics for suspensions with moderate Péclet numbers.

The role of surface tension in the dominant balance in the die swell singularity
View Description Hide DescriptionThe two‐dimensional, free‐surface flow of a Newtonian fluid exiting from a planar die is computed by finite element analysis using quasiorthogonal mesh generation and local mesh refinement with irregular, embedded elements to obtain extreme resolution of the velocity and pressure fields near the die edge, where the fluid sheet attaches to the solid boundary. Calculations for the limit of large surface tension, the stick‐slip problem, reproduce the singular behavior near the die edge expected from asymptotic analysis using a self‐similar form for the velocity field. Results for finite capillary number (Ca) predict that the meniscus separates from the die at a finite contact angle and suggest that the capillary force enters the dominant normal stress balance at the die edge through an infinite curvature, as previously suggested by Schultz and Gervasio. The size of this region with large positive curvature increases with increasing Ca, and the strength of the singularity is in good agreement with theoretical predictions for a straight meniscus attached to the die at the appropriate contact angle predicted by the simulations. The contact angle appears to be determined from matching of the inner solution structure valid near the singularity with the bulk flow, in agreement with arguments made by Ramalingam; increasing the Reynolds number decreases the contact angle, corroborating this effect. Introducing fluid slip along the surface of the die changes the structure of the singularity in the pressure and stresses, but does not alleviate the singular behavior. In fact, the calculations with slip coefficients small enough not to change the bulk solution are more difficult than calculations with the no‐slip boundary condition.

Stability analysis of source and sink flows
View Description Hide DescriptionA linear stability analysis is performed for two‐ and three‐dimensional steady source and sink flows. Cases studied include inviscid compressible and incompressible fluids. For two‐dimensional flowsviscous incompressible fluid is also examined. The one‐dimensional nature of the unperturbed base flow suggested taking the vorticity as a perturbation in order to reduce the number of variables and to simplify the analysis. It is shown that source flows are always unstable. Sink flows are found unstable for inviscid compressible fluid and also for two‐dimensional flow of viscous incompressible fluid for low Reynolds numbers. The different modes of instability existing in perturbed flow are obtained.

Boundary layer instability over compliant walls: Comparison between theory and experiment
View Description Hide DescriptionTheoretical studies have shown that compliant walls are able to attenuate the Tollmien–Schlichting waves that lead to conventional two‐dimensional boundary‐layer transition. This phenomenon was demonstrated in towing‐tank tests conducted by Gaster et al. The results of these experiments also featured a different and very dramatic form of boundary‐layer breakdown. We contend that this type of breakdown was due to a hydroelastic mode of instability, namely traveling‐wave flutter. In this paper we model the two‐layer viscoelastic compliant wall of Gaster et al. and its interaction with the boundary‐layer flow using the asymptotic theory of Carpenter and Gajjar; e ^{ n }‐type calculations are carried out for the traveling‐wave flutter. Excellent agreement is found between the stability characteristics of the TWF mode and the measurements of the new form of breakdown found in the experiments; thus a complete understanding of the physical features found in the experiments is now available. Such understanding is essential for progress to be made in the technological development of compliant panels for transition delay.

An experimental observation of low‐dimensional dynamics in an open channel flow
View Description Hide DescriptionWe present the results of an experimental study of dynamical phenomena in the fluid flow through a symmetric nominally two‐dimensional channel expansion. The particular phenomena of interest are observed as a result of modulating the flow rate through the system. In the absence of modulation, the flow downstream of the expansion is found to be steady at low Reynolds numbers. It then loses stability to a high‐dimensional dynamical state for Reynolds numbers above a critical value. However, when a small periodic modulation is added to the flow rate new low‐dimensional dynamical phenomena emerge. In some parameter ranges this novel temporal behavior arises at Reynolds numbers below those at which the transition to irregular fluid flow occurs in the unforced system. In other regions of parameter space the low‐dimensional dynamics suppress the irregular flow of the unforced system. Moreover, the dynamics are organized by the underlying solution structure and show evidence for a subharmonic resonance, quasiperiodicity, and homoclinicity.

A theory of three‐dimensional interfacial vorticity dynamics
View Description Hide DescriptionA three‐dimensional theory of vorticitydynamics on an incompressible viscous and immiscible fluid–fluid interface, or interfacial vorticity dynamics for short, is presented as a counterpart of the vorticitydynamics on an arbitrarily curved rigid wall [J. Fluid Mech. 254, 183 (1993)]. General formulas with arbitrary Reynolds numbers Re are derived for determining (1) how much vorticityexists on an interfaceS, (2) how much vorticityis created from S and sent into the fluid per unit area in per unit time, and (3) the force and moment acted on a closed interface by the created vorticity thereon. The common feature and fundamental difference between interfacial vorticitydynamics and its rigid‐wall counterpart are analyzed. In particular, on a free surface, the primary driving mechanism of vorticity creation is the balance between the shear stress (measured by tangent vorticity) and the tangent components of the surface‐deformation stress alone, which results in a weak creation rate of O (Re^{−1/2}) at large Re. Therefore, the exact form of the theory with its full complexity is of importance mainly at low Reynolds numbers, especially in understanding the small‐scale coherent structures of interfacial turbulence. The vorticity creation rate at high‐Re approximations, including an interfacial boundary layer of finite thickness and the limit of Re→∞ (the so‐called Euler limit), is also studied, both allowing for a rotational inviscid outer flow. While for the former this leads to a generalization of Lundgren’s theory [in Mathematic Aspects of Vortex Dynamics, edited by R. E. Caflish (SIAM, Philadelphia, PA, 1989), pp. 68–79] and amounts to solving a linear boundary‐layer problem, for the latter the creation rate can be directly obtained from an inviscid solution, leading to a dynamic evolution equation of interfacial vortex sheet. In three dimensions, a vortex sheet may bifurcate into a normal vorticity field, upon which the dependence of the sheet velocity is determined. A few examples are examined to illustrate different aspects and approximation levels of the general theory.

A numerical coupled model for studying air–sea–wave interaction
View Description Hide DescriptionA numerical coupled model of air–sea–wave interaction is developed to study the influence of ocean wind waves on dynamical, turbulent structures of the air–sea system and their impact on coupled modeling. The modelequations for both atmospheric and oceanic boundary layers include equations for: (1) momentum, (2) a k‐ε turbulence scheme, and (3) stratification in the atmospheric and oceanic boundary layers. The modelequations are written in the same form for both the atmosphere and ocean. In this model, wind waves are considered as another source of turbulentenergy in the upper layer of the ocean besides turbulentenergy from shear production. The dissipation ε at the oceansurface is written as a linear combination of terms representing dissipation from mean flow and breaking waves. The ε from breaking waves is estimated by using similarity theory and observed data. It is written in terms of wave parameters such as wave phase speed, height, and length, which are then expressed in terms of friction velocity. Numerical experiments are designed for various geostrophic winds, wave heights, and wave ages, to study the influence of waves on the air–sea system. The numerical simulations show that the vertical profiles of ε in the atmospheric and oceanic boundary layers (AOBL) are similar. The magnitudes of ε in the oceanic surface zone are much larger than those in the atmospheric surface zone and in the interior of the oceanic boundary layer (OBL). The model predicts ε distributions with a surface zone of large dissipation which was not expected from similarity scaling based on observed wind stress and surface buoyancy. The simulations also show that waves have a strong influence on eddy viscosity coefficients (EVC) and momentum fluxes, and have a dominated effect on the component of fluxes in the direction of the wind. The depth of large changes in flux magnitudes and EVC in the ocean can reach to 10–20 m. The simulations of surface drift currents confirm that the currents are overestimated if the surface waves are not considered.

Nonlinear sound–vortex interactions in an inviscid isentropic fluid: A two‐fluid model
View Description Hide DescriptionA new two‐fluid model is developed to describe the nonlinear interaction of acoustic waves and vortices. Analytical and computational results are presented for a sound pulse interacting with and being modified by a vortex. A novel numerical method based on a particle‐in‐cell discretization of the acoustic field is developed and used to study the nonlinear scattering of sound by a cylindrical vortex. Equations for the sound wave packet propagating in an axially symmetric mean flow are integrated analytically. Nonlinear modification of the vortex flow by the high‐frequency sound is found to be mediated by growing pressure disturbances generated by the radiative forcing on the high gradient regions of the acoustic pulse. The total energy of the vortex mean flow grows monotonically, as the acoustic component loses its energy. The changes in the kinetic and internal energies of the vortex are greater than the changes in its total energy, although these changes are reversible in lowest order of the nonlinear vortex–acoustic interaction.

Chemically reactive turbulent vortex rings
View Description Hide DescriptionEmploying an aqueous acid‐base reaction, the minimum mixing rate of turbulent vortex rings was investigated in a water tank. Vortex rings were generated by a simple apparatus with a cylindrical geometry. The released fluid surrounding the vortex core mixed very rapidly when compared with the fluid in the toroidal core. Moreover, the fluid within the core did not mix uniformly in the azimuthal direction. The normalized distance a vortex ring must travel, in order to completely mix with the ambient fluid to a specific volumetric ratio, depends on the aspect ratio of the generating cylinder. Scaling arguments are presented that relate the above distance to the spreading rate and the generating apparatus parameters. Due to the very small net entrainment rate of vortex rings, the detrainment of core material cannot be ignored when the mixing rate of the core is considered.

Hydrostatics and oscillatory flows of magnetic fluid under a nonuniform magnetic field
View Description Hide DescriptionA nonuniform magnetic field induces an inhomogeneous ferroparticle distribution in a magnetic fluid (MF). As a consequence the magnetic field influences MF flows. In particular, we consider an oscillatory pipe flow in a stationary nonuniform magnetic field subject to the quasielastic magnetic force. Concentration inhomogeneities can be formed from both single particles and many‐particle drops, which appear as a result of the field‐induced MF phase separation.Kinematic considerations show that only the drops contribute to the magnetic force. Our theory is in good agreement with recent experimental results.

Nonuniversality of sublayer streaks in turbulent flow
View Description Hide DescriptionCoefficients of series expansions of turbulent velocity fluctuations in the viscous wall region are used to generate an arbitrary but quantitative measure of the time‐average strength of the near‐wall quasi‐streamwise vortices, which appear as ‘‘streaks’’ in flow visualization. Existing databases from direct numerical simulations of wall bounded turbulence are used to compute some estimates. The results show that the strength of the streaks is Reynolds‐number‐dependent, even in simple flows, as well as flow‐dependent, contrary to traditional law‐of‐the‐wall arguments.

On the streak spacing and vortex roll size in a turbulent channel flow
View Description Hide DescriptionStreamwise high vorticity rolls and streaks in the turbulent channel flows have been the subject of many studies due to their important role in turbulence production, as a result of sweeping, ejection, and bursting of these structures. Understanding the physics of these streamwise structures is important in controlling drag producing events. Investigations of the average streak spacing of the low‐speed streaks have resulted in the generally accepted range of λ^{+}=λ̄u _{τ}/ν=100±20, where λ̄ is the mean spanwise spacing between streaks, normalized to the viscous length ν/u _{τ}. It is also reported, for y ^{+}≤30, that the streak spacing grows nearly linearly with distance from the wall. The previous studies mostly have focused on distances close to the wall. Here we report on correlation measurements extended into the log layer, which show that the linear growth of the vortex diameter and the streak spacing extends well in the log layer. Arguments are presented to distinguish these two measures.

Experimental investigation of turbulence properties in the interface region of variable density jets
View Description Hide DescriptionThe global evolution of variable density turbulent jets is now quite well documented, showing that the entrainment of external fluid into these jets is considerably modified by density variations. But, to our knowledge, no specific study has so far been devoted to the intermittent region of such flows. For constant density flows, the radial evolutions of the velocity variances are known to follow the so‐called Phillips’ relations in this interface region. The main objective of the present work is to investigate whether density variations affect properties of the interface. It is found that Phillips’ relations are also valid in the presence of large‐density variations, albeit their extent is slightly different. In relation to this, the structure of turbulence is almost unchanged, even in the outer region where large‐scale structures are dominating the flow mixing properties, and departure from isotropy for the Reynolds stresses is rather similar for all jets within the range of density ratios considered here. Therefore, except in the very near‐field region, most of the influence of density variations can be taken into account by considering only the different evolutions of the flow characteristics on the jet axis. The implication of our results for various aspects associated with these flows, such as in modeling, is also discussed.

Velocity autocorrelations of decaying isotropic homogeneous turbulence
View Description Hide DescriptionVelocity autocorrelations and the mean‐square displacements of fluid particles are obtained for decaying, isotropic homogeneous turbulence by numerical simulation of the flow field, using 128^{3} and 256^{3} grids, and tracking several tens of thousands of fluid particles, using a third‐order interpolation scheme. A self‐preserving Lagrangian velocity autocorrelation coefficient is found in terms of a dimensionless time variable s, defined by ds=dt/T_{ s }(t), under the observation of a power‐law energy decay and the assumption that T_{ s }(t) is proportional to the Lagrangian integral timescale T_{L}. This timescale is in turn assumed to be proportional to the length scale of the energy‐containing eddiesL_{ e }∼K ^{3/2}/ε divided by the turbulent velocity u ^{′}, where K=3/2u ^{′2} is turbulent energy and ε is the energy dissipation rate.

Effect of concentrated wall suction on a turbulent boundary layer
View Description Hide DescriptionThe effect of suction, applied through a short porous wall strip, on a low Reynolds number self‐preserving turbulent boundary layer has been quantified by measuring the local wall shear stress and the main Reynolds stresses downstream of the strip. When the suction rate is sufficiently high, pseudo‐relaminarization occurs almost immediately downstream of the strip. Farther downstream, transition occurs followed by a slow return to a fully turbulent self‐preserving state. During relaminarization, the measured skin friction coefficient c _{ f } falls below the level corresponding to the no suction value, reaching a minimum where transition begins. An empirical c _{ f } distribution is proposed that groups together results obtained at different streamwise stations and different suction rates. Of all the measuredReynolds stresses, the longitudinal turbulence intensity recovers relatively quickly from the change in boundary conditions while the wall‐normal turbulence intensity and the Reynolds shear stress are significantly affected by the suction. The Reynolds shear stress, which is negligible during relaminarization, has the slowest recovery.

The numerical simulation of shock bifurcation near the end wall of a shock tube
View Description Hide DescriptionThe reflection of a normal shock wave from the end wall of a two‐dimensional channel has been numerically simulated to investigate the unsteady, viscousinteraction aspects of shock bifurcation. The numerical simulation implements a data‐parallel version of the Flux‐Corrected Transport algorithm that has been coupled to the viscous transport terms of the Navier–Stokes equations. All numerical simulations were performed on the Connection Machine, the CM‐5. The results indicate that the shear layer in the bifurcation zone is unstable, and the large and small scale vortices lead to complex flow patterns. In addition, the high‐speed, essentially inviscid flow, which is adjacent to the shear layer, is deflected over this region. As a result, weak shock and expansions waves are generated and a reattachment shock is formed at the trailing edge of the interaction region. The impact of heat transfer,Reynolds number, and incident shock strength on the viscousinteraction is also investigated. Heat transfer to the walls weakens the interaction between the boundary layer and the reflected shock. However, the decreased Reynolds number and increased shock strength enhances the interaction.

Instability of wake‐dominated compressible mixing layers
View Description Hide DescriptionThe instability of supersonic mixing layers, with velocity profiles possessing a wake component, is investigated using linear, inviscid, spatial theory. The mean‐velocity profile is represented by a hyperbolic‐tangent profile plus a wake component. Such profiles are encountered in the initial region of experimental supersonic shear‐layer flows, as well as in envisaged hypersonic propulsion systems in which ingested boundary layers generate substantial wake components. Shear‐layer and wake instability modes previously found in incompressible mixing layers are also found in compressible mixing layers. The existence of a wake component in the velocity profile renders the mixing layer more unstable at all free‐stream Mach numbers. For convective Mach numbers exceeding unity, the shear‐layer mode splits into two supersonic modes, and the mixing layer becoming more unstable with increasing wake deficit. The wake mode becomes less unstable and eventually stable with increasing compressibility, i.e., increasing convective Mach numbers.