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Aug 1998

Volume 10, Issue 8, pp. 1781-2116

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Drop breakup in three-dimensional viscous flows

Vittorio Cristini, Jerzy Bławzdziewicz, and Michael Loewenberg

Phys. Fluids 10, 1781 (1998); http://dx.doi.org/10.1063/1.869697 (3 pages) | Cited 63 times

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A new three-dimensional boundary integral algorithm is presented that is capable of simulating the process of drop breakup in viscous flows. The surface discretization is fully adaptive, thus providing accurate resolution of the highly deformed drop shapes that are characteristic of breakup events. Our algorithm is used to study drop breakup in shear flow and in buoyancy; the predictions are compared with experimental observations. © 1998 American Institute of Physics.
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47.55.D- Drops and bubbles
47.11.-j Computational methods in fluid dynamics

The compressible inviscid algebraic instability for streamwise independent disturbances

Ardeshir Hanifi and Dan S. Henningson

Phys. Fluids 10, 1784 (1998); http://dx.doi.org/10.1063/1.869698 (3 pages) | Cited 7 times

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An inviscid algebraic instability for streamwise independent disturbances in compressible flow is found to be related to Ellingsen and Palm’s [Phys. Fluids 18, 487 (1975)] solution for incompressible flow. For compressible flow, the streamwise disturbance velocity, the density, as well as temperature perturbations grow linearly with time. The effect of viscosity on the inviscid algebraic growth is clarified using a rescaling of the viscous disturbance equations, showing the dependence of the viscous transient growth on the Reynolds number. © 1998 American Institute of Physics.
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47.20.-k Flow instabilities
47.40.-x Compressible flows; shock waves
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The mechanism of surface wave suppression in film flow down a vertical plane

S. P. Lin and J. N. Chen

Phys. Fluids 10, 1787 (1998); http://dx.doi.org/10.1063/1.869699 (6 pages) | Cited 5 times

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The equation relating the average rate of change of disturbance kinetic energy to the rates of work done by the surface tension, the shear stress, the Reynolds stress, and the rate of mechanical energy dissipation through viscosity in a falling liquid film flow down an oscillating vertical plate is obtained. Each term in the equation is evaluated in various regions of parameter space to elucidate the physical mechanism of stabilizing an inherently unstable vertical film flow by use of plate oscillations [Lin, Chen, and Woods, Phys. Fluids 8, 3247 (1996); Lin and Chen, Phys. Fluids 9, 3926 (1997)]. © 1998 American Institute of Physics.
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47.35.-i Hydrodynamic waves
68.15.+e Liquid thin films
68.03.Cd Surface tension and related phenomena

The effects of thin films on the hydrodynamics near moving contact lines

K. Stoev, E. Ramé, T. Leonhardt, and S. Garoff

Phys. Fluids 10, 1793 (1998); http://dx.doi.org/10.1063/1.869700 (11 pages) | Cited 15 times

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We explore the effects of thin films on the hydrodynamics of macroscopic fluid bodies spreading over solid surfaces. To examine these effects, we measure the interface shape within microns of moving contact lines and compare those measurements to two asymptotic models in the limit of small capillary number, Ca. One model requires that the films affect the hydrodynamics only in a microscopic region near the contact line and allows the macroscopic meniscus to exhibit a nonzero effective contact angle. The other model describes the film as containing mobile fluid and specifically models the flow as fluid moves into or out of the film as the contact line moves. We examine fluids advancing and receding on wetting and nonwetting surfaces with spontaneously forming (molecular scale) and pre-existing (micron scale) films. Our results emphasize the importance of the mobility of the molecules in these very thin films in determining the hydrodynamics governing the moving contact line. The first model, which describes fluids advancing over dry surfaces, also accounts for the hydrodynamics of liquids advancing over very thin, immobile films. Surprisingly, the same model fails when fluid recedes on a nonwetting surface and no film is present. For mobile pre-existing films, the second model, based on Landau and Levich’s theory, accounts for the hydrodynamics in the limit of small Ca. © 1998 American Institute of Physics.
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68.15.+e Liquid thin films
68.03.-g Gas-liquid and vacuum-liquid interfaces
68.05.-n Liquid-liquid interfaces
68.08.Bc Wetting
68.03.Cd Surface tension and related phenomena
47.11.-j Computational methods in fluid dynamics

Pattern formation in thin liquid films with insoluble surfactants

E. Ramos de Souza and D. Gallez

Phys. Fluids 10, 1804 (1998); http://dx.doi.org/10.1063/1.869701 (11 pages) | Cited 14 times

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The problem of pattern formation in thin liquid films with insoluble surfactants under attractive and repulsive forces is addressed. A thin fluid film bounded by a wall is modeled by a set of two nonlinear evolution equations for the film thickness and surfactant concentration on the free interface. We perform a bifurcation analysis valid for the general case of apolar and polar forces and predict a supercritical bifurcation to new stationary and periodic structures. Numerical simulations for the particular case of a negative apolar spreading coefficient (attractive van der Waals forces) and a positive polar spreading coefficient (repulsive hydration pressure) are discussed in terms of the analytical predictions. Nonlinearities in the competition between attractive and repulsive forces can lead to formation of periodic patterns for the film thickness with homogeneously distributed surfactants. Due to diffusion and Marangoni effects, insoluble surfactants alter the time required for pattern formation but do not alter the final pattern profile itself. Bifurcation analysis allows us then to predict the ranges of film parameters in which pattern formation, rupture, or total film spreading is possible. © 1998 American Institute of Physics.
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68.15.+e Liquid thin films
81.15.Lm Liquid phase epitaxy; deposition from liquid phases (melts, solutions, and surface layers on liquids)

Time-dependent equations governing the shape of a three-dimensional liquid curtain

Steven J. Weinstein, Joseph W. Hoff, and David S. Ross

Phys. Fluids 10, 1815 (1998); http://dx.doi.org/10.1063/1.869730 (4 pages) | Cited 3 times

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In a previous paper, Weinstein et al. derive time-dependent equations that govern the response of a planar liquid curtain falling under the influence of gravity and subjected to ambient pressure disturbances. In the previous study, disturbances to the curtain are assumed to be two-dimensional, and thus, the curtain response is independent of widthwise location in the curtain. In this paper, we generalize the previous equations to incorporate the widthwise dimension. The validity of these equations is demonstrated by their ability to predict standing wave shapes in agreement with those studied by Lin and Roberts. © 1998 American Institute of Physics.
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47.10.-g General theory in fluid dynamics
68.15.+e Liquid thin films

Elastic membranes in viscous shear flow

Yiftah Navot

Phys. Fluids 10, 1819 (1998); http://dx.doi.org/10.1063/1.869702 (15 pages) | Cited 19 times

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Liquid capsules with an elastic membrane surface in simple shear flow are considered beyond the regime of small deformations. A simple model is introduced to give a mathematical description of an elastic membrane in viscous flow. The question of whether there is a steady state of a system of a single membrane in external shear flow is studied. It is found by analytical considerations that the possible steady-state flows are restricted by symmetry. The evolution of the membrane in time is found by numerical calculations. For a single spherically symmetric membrane, I find by numerical simulations, a steady-state shape and its dependence on the shear strength. For nonspherically symmetric membranes we see that there is no steady-state shape in general, but by numerical simulations I find that the shape changes can become periodic. This leads to a new alternative explanation of previous experimental results. The stress tensor of the membranes and the effective viscosity of a dilute system of elastic membranes immersed in a liquid is calculated. I find an agreement with former analytical calculations for small shear, and obtain shear thinning behavior when the shear rate is increased. © 1998 American Institute of Physics.
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47.55.Kf Particle-laden flows

Large deformation of red blood cell ghosts in a simple shear flow

C. D. Eggleton and A. S. Popel

Phys. Fluids 10, 1834 (1998); http://dx.doi.org/10.1063/1.869703 (12 pages) | Cited 115 times

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Red blood cells are known to change shape in response to local flow conditions. Deformability affects red blood cell physiological function and the hydrodynamic properties of blood. The immersed boundary method is used to simulate three-dimensional membrane–fluid flow interactions for cells with the same internal and external fluid viscosities. The method has been validated for small deformations of an initially spherical capsule in simple shear flow for both neo-Hookean and the Evans-Skalak membrane models. Initially oblate spheroidal capsules are simulated and it is shown that the red blood cell membrane exhibits asymptotic behavior as the ratio of the dilation modulus to the extensional modulus is increased and a good approximation of local area conservation is obtained. Tank treading behavior is observed and its period calculated. © 1998 American Institute of Physics.
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87.17.Aa Modeling, computer simulation of cell processes
87.19.U- Hemodynamics
87.19.Wx Pneumodyamics, respiration
47.11.-j Computational methods in fluid dynamics

The influence of surfactant on two-phase flow in a flexible-walled channel under bulk equilibrium conditions

Darren Y. K. Yap and Donald P. Gaver

Phys. Fluids 10, 1846 (1998); http://dx.doi.org/10.1063/1.869792 (18 pages) | Cited 21 times

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A preliminary model of pulmonary airway reopening is developed that includes the physicochemical influence of surfactant under bulk-equilibrium conditions. The airway is modeled following Gaver et al. [J. Fluid. Mech. 319, 25–65 (1996)] as a flexible-walled channel, where walls are membranes under longitudinal tension T, and supported with elasticity E with a stress-free separation distance 2H. The lining fluid has viscosity μ and surface tension γ∗. Airway reopening occurs when a semi-infinite bubble of air with pressure Pb∗ progresses steadily at velocity U and separates the walls. Surfactant exists in the lining fluid (C∗) and at the air–liquid interface (Γ∗). Bulk equilibrium is assumed (C∗ = C0) and the kinetic transfer of surfactant between the bulk and interface occurs with a rate k. The equilibrium relationship between Γ∗ and C∗ is based upon Henry’s isotherm (Γeq = KC0). The surface tension equation of state, a relationship between γ∗ and Γ∗, is assumed to be linear near Γeq. Marangoni stresses develop from the transport of surfactant at the interface, leading to interfacial rigidification and changes in the airway reopening behavior. The behavior is governed by the following dimensionless parameters: the capillary number Ca = μU/γeq, the surface elasticity number El = −(dγ∗/dΓ∗)(Γeq/γeq), the modified Stanton number Stλ = (k/K)/(U/H), the wall elastance parameter β = EH2/γeq, the wall tension ratio η = T/γeq, and the surface Pèclet number Peint = UH/Dint. The results indicate that El can have a dual, contrasting influence on the airway reopening behavior. By increasing El through an increase in dγ∗/dΓ∗ (method 1), larger Pb∗ are predicted from the resulting interfacial surfactant gradients and interfacial rigidification. In contrast, increasing Γeq (method 2) increases El but reduces Pb∗ due to the global reduction of γ∗; however, the reduction in Pb∗ is augmented by the increasing importance of viscous, elastic, tension and Marangoni stresses. Furthermore, for Stλ>10 the interface remains mobile due to rapid surfactant adsorption and the elimination of Marangoni stresses, which minimizes Pb∗. This behavior may be important in the development of improved exogenous surfactants for the treatment of a variety of pulmonary diseases. © 1998 American Institute of Physics.
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47.55.Kf Particle-laden flows
87.16.D- Membranes, bilayers, and vesicles
47.60.-i Flow phenomena in quasi-one-dimensional systems
87.19.U- Hemodynamics
87.19.Wx Pneumodyamics, respiration
47.55.D- Drops and bubbles

Linear bioconvection in a suspension of randomly swimming, gyrotactic micro-organisms

M. A. Bees and N. A. Hill

Phys. Fluids 10, 1864 (1998); http://dx.doi.org/10.1063/1.869704 (18 pages) | Cited 19 times

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We have analyzed the initiation of pattern formation in a layer of finite depth for Pedley and Kessler’s new model [J. Fluid Mech. 212, 155 (1990)] of bioconvection. This is the first analysis of bioconvection in a realistic geometry using a model that deals with random swimming in a rational manner. We have considered the effects of a distribution of swimming speeds, which has not previously received attention in theoretical papers and find that it is important in calculating the diffusivity. Our predictions of initial pattern wavelengths are reasonably close to the observed ones but better experimental measurements of key parameters are needed for a proper comparison. © 1998 American Institute of Physics.
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87.19.-j Properties of higher organisms
87.17.-d Cell processes
47.27.T- Turbulent transport processes
47.55.Kf Particle-laden flows
82.70.Kj Emulsions and suspensions
83.80.Hj Suspensions, dispersions, pastes, slurries, colloids
83.80.Iz Emulsions and foams

Asymmetric salt fingers induced by a nonlinear equation of state

Tamay M. Özgökmen and Oleg E. Esenkov

Phys. Fluids 10, 1882 (1998); http://dx.doi.org/10.1063/1.869705 (9 pages) | Cited 3 times

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The impact of the nonlinearity in the equation of state associated with the change in the thermal expansion coefficient with temperature on the structure of fingers growing from an interface between two mixed layers is investigated using a numerical model. It is shown that the nonlinearity acts to enhance the buoyancy force acting on the descending fingers with respect to that acting on the ascending fingers, resulting in narrower and faster-growing descending fingers than ascending fingers. The results are discussed with emphasis on the vertical variability of properties along the fingers. © 1998 American Institute of Physics.
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47.20.-k Flow instabilities
64.30.-t Equations of state of specific substances
92.05.Hj Physical and chemical properties of seawater (salinity, density, temperature)

Steady buoyant droplets with circulation

Shin-Shin Kao and Russel E. Caflisch

Phys. Fluids 10, 1891 (1998); http://dx.doi.org/10.1063/1.869706 (12 pages)

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Numerical solutions are presented for the steady flow corresponding to a two-dimensional moving droplet with circulation. Differences in the density of the droplet and surrounding fluid result in a buoyancy force which is balanced by a lift force due to the Magnus effect. The droplet is assumed to have constant vorticity in its interior, and its boundary may be a vortex sheet, as in a Prandtl–Batchelor flow. Only symmetric solutions are calculated. For Atwood number A = 0 (no density difference) the droplet is a circle. As the Atwood number is increased, the droplet shape begins to resemble a circular cap with a dimpled base. There is a critical Atwood number Alim at which the droplet develops two corners. For 0 ⩽ A<Alim, the solution is smooth; while for Alim<A, we do not find a solution. © 1998 American Institute of Physics.
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47.55.D- Drops and bubbles
47.32.C- Vortex dynamics
47.55.Kf Particle-laden flows

Buoyant flows with low-frequency jitter

P. Grassia and G. M. Homsy

Phys. Fluids 10, 1903 (1998); http://dx.doi.org/10.1063/1.869707 (21 pages) | Cited 2 times

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A slot with applied temperature stratification is considered when mean gravity is directed along its length and weak quasistatic jitter is applied in the spanwise direction, but when there is no component of gravity in the vertical. The behavior of the slot is governed by a number of factors: The sense of the mean gravity with respect to the applied stratification, the spanwise and lengthwise Rayleigh numbers, the Prandtl and Biot numbers, and the spanwise–lengthwise aspect ratio of the slot. A perturbation expansion of the governing equations is performed for weak spanwise jitter. At the first order of perturbation there is a circulation around the slot, producing an advected temperature field with spanwise gradients. At second order there are inflows or outflows in both the spanwise and lengthwise directions, along with a vertical redistribution of fluid. There is also a temperature field with lengthwise gradients, which typically competes with the applied temperature gradient. Equations are derived governing the vertical structure of all these fields and are solved in terms of a set of special basis functions. A parametric study is performed for the solutions. When lengthwise buoyancy forces are absent (the lengthwise Rayleigh number is zero), it is comparatively easy to deduce the required fields. However, finite lengthwise Rayleigh numbers couple the momentum and thermal equations thereby affecting the structure of the fields. Interesting behavior is predicted for small Biot numbers, when convected heat is effectively trapped in the slot: Infinitessimal flows can produce finite advected temperatures. The limits of small Biot number and small lengthwise Rayleigh number are found to be noninterchangeable. At large lengthwise Rayleigh number, boundary layers occur for stable applied stratification and layered cellular structures occur for unstable stratification. For the stable case at moderately small Biot number, the temperature jump across the boundary layer is small compared with the depth independent temperature in the bulk. Then by exploiting the boundary layer nature of the solutions, it becomes simple to predict the bulk fluid temperatures, interfacial heat fluxes and the circulations associated with the buoyant flows. Turning to the unstably stratified case, it is demonstrated that runaways can occur at first order in the spanwise jitter, and these correspond to resonant excitation of three-dimensional, stationary, long wave Rayleigh–Bénard modes. It is demonstrated how the Biot number and the spanwise–lengthwise aspect ratio of the slot influence the lengthwise Rayleigh number at which these resonances occur. There is in addition a set of two-dimensional Rayleigh–Bénard modes, which can potentially become excited at second order. When the Biot number and the spanwise–lengthwise aspect ratio are not too large, the Rayleigh numbers corresponding to the two sets of modes are nearly coincident. The second-order system will then be strongly forced near resonance, causing it to have a disproportionately large response. © 1998 American Institute of Physics.
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47.55.Hd Stratified flows
47.20.Bp Buoyancy-driven instabilities (e.g., Rayleigh-Benard)

Flow patterns of natural convection in an air-filled vertical cavity

Shunichi Wakitani

Phys. Fluids 10, 1924 (1998); http://dx.doi.org/10.1063/1.869708 (5 pages) | Cited 16 times

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Flow patterns of two-dimensional natural convection in a vertical air-filled tall cavity with differentially heated sidewalls are investigated. Numerical simulations based on a finite difference method are carried out for a wide range of Rayleigh numbers and aspect ratios from the onset of the steady multicellular flow, through the reverse transition to the unicellular pattern, to the unsteady multicellular flow. For aspect ratios (height/width) from 10 to 24, the various cellular structures characterized by the number of secondary cells are clarified from the simulations by means of gradually increasing Rayleigh number to 106. Unsteady multicellular solutions are found in some region of Rayleigh numbers less than those at which the reverse transition has occurred. © 1998 American Institute of Physics.
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47.27.T- Turbulent transport processes
47.60.-i Flow phenomena in quasi-one-dimensional systems
47.11.-j Computational methods in fluid dynamics
02.70.Bf Finite-difference methods
47.20.-k Flow instabilities

Lagrangian dynamics in high-dimensional point-vortex systems

Jeffrey B. Weiss, Antonello Provenzale, and James C. McWilliams

Phys. Fluids 10, 1929 (1998); http://dx.doi.org/10.1063/1.869709 (13 pages) | Cited 34 times

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We study the Lagrangian dynamics of systems of N point vortices and passive particles in a two-dimensional, doubly periodic domain. The probability distribution function of vortex velocity, pN, has a slow-velocity Gaussian component and a significant high-velocity tail caused by close vortex pairs. In the limit for N→∞, pN tends to a Gaussian. However, the form of the single-vortex velocity causes very slow convergence with N; for N ≈ 106 the non-Gaussian high-velocity tails still play a significant role. At finite N, the Gaussian component is well modeled by an Ornstein-Uhlenbeck (OU) stochastic process with variance σN = math. Considering in detail the case N = 100, we show that at short times the velocity autocorrelation is dominated by the Gaussian component and displays an exponential decay with a short Lagrangian decorrelation time. The close pairs have a long correlation time and cause nonergodicity over at least the time of the integration. Due to close vortex dipoles the absolute dispersion differs significantly from the OU prediction, and shows evidence of long-time anomalous dispersion. We discuss the mathematical form of a new stochastic model for the Lagrangian dynamics, consisting of an OU model combined with long-lived close same-sign vortices engaged in rapid rotation and long-lived close dipoles engaged in ballistic motion. From a dynamical-systems perspective this work indicates that systems of dimension O(100) can have behavior which is a combination of both low-dimensional behavior, i.e., close pairs, and extremely high-dimensional behavior described by traditional stochastic processes. © 1998 American Institute of Physics.
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47.32.C- Vortex dynamics
02.50.Cw Probability theory
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.50.Ey Stochastic processes

The curvature of material lines in a three-dimensional chaotic flow

D. M. Hobbs and F. J. Muzzio

Phys. Fluids 10, 1942 (1998); http://dx.doi.org/10.1063/1.869710 (11 pages) | Cited 7 times

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Folding of material filaments was examined computationally in the three-dimensional flow in a cylindrical duct with helical deflectors by tracking the curvature of line elements in the flow. Two geometries were analyzed: a configuration in which the flow is globally chaotic, and an alternative geometry which has a mixture of chaotic and regular motion. The behavior of the curvature field in this complex flow geometry was in agreement with that previously observed for much simpler two-dimensional model flows [Phys. Fluids 8, 75 (1996)]. Curvature profiles along individual element trajectories indicate that an inverse relationship exists between the rates of stretching and curvature. Material elements are compressed when they are folded. After an initial transient, the mean curvature oscillates within a finite range with a periodicity matching that of the flow geometry. The spatial structure of the curvature field becomes period-independent, and the probability density functions of curvature computed for different numbers of periods collapse to an invariant, self-similar distribution without the need for scaling. © 1998 American Institute of Physics.
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47.52.+j Chaos in fluid dynamics
47.60.-i Flow phenomena in quasi-one-dimensional systems

A low-order model for vortex shedding patterns behind vibrating flexible cables

D. J. Olinger

Phys. Fluids 10, 1953 (1998); http://dx.doi.org/10.1063/1.869711 (9 pages) | Cited 2 times

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A recent focus in studies of vortex shedding behind circular cylinders has been on the use of low-order dynamical systems such as circle maps to predict wake dynamics. These purely temporal models have been limited by their inability to describe three-dimensional spatial flow variations along the cylinder span, a hallmark of transitional flows such as the cylinder wake. In the present work this limitation is overcome through development of a spatial-temporal map lattice which utilizes a series of coupled circle map oscillators along the cylinder span. This model allows for the study of vortex shedding patterns and wake dynamics behind vibrating flexible cables. Required input for the model includes the forcing frequency, amplitude, mode shape, aspect ratio and wavelength of the cable, Reynolds number, vortex convection velocity, and various phase angles. Model output parameters studied in this work include vortex shedding patterns and wake response frequency. Standing wave mode shapes and traveling waves along the cable span are modeled. Lacelike vortex patterns are observed for the standing wave case. A physical mechanism for the lacelike patterns is postulated. For traveling waves oblique shedding patterns are confirmed. Nonharmonic forcing outside the classical lock-on region yields vortex dislocation patterns in the wake. Honeycomb patterns are also observed for higher-order mode shapes at large forcing amplitudes. The current work establishes a new class of models based on circle maps for modeling spatially varying cylinder wakes. © 1998 American Institute of Physics.
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47.32.C- Vortex dynamics
47.27.wb Turbulent wakes
47.35.-i Hydrodynamic waves

Chaotic transitions in magnetic fluids

H. K. Khosla and S. K. Malik

Phys. Fluids 10, 1962 (1998); http://dx.doi.org/10.1063/1.869712 (10 pages) | Cited 2 times

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Parametric excitation of surface waves in a magnetic fluid container is investigated under the influence of an externally applied magnetic field. With a view to explain the experimental observation the dynamical equations governing the evolution of the amplitude of instability are obtained using asymptotic methods. The solution of dynamical equations yields a period doubling sequence, orbits of odd periods, and windows of chaos. Also observed is the phenomenon of strange attractor. © 1998 American Institute of Physics.
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47.52.+j Chaos in fluid dynamics
47.65.-d Magnetohydrodynamics and electrohydrodynamics
47.35.-i Hydrodynamic waves

A weakly nonlinear theory of the spatial evolution of disturbances in a shear flow with a parallel magnetic field

I. G. Shukhman

Phys. Fluids 10, 1972 (1998); http://dx.doi.org/10.1063/1.869713 (15 pages)

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A study is made of the spatial downstream evolution of a weakly unstable disturbance excited by an external source of a frequency ω close to the frequency of a marginally stable (neutral) mode, in a mixing layer of conducting, nearly inviscid fluid with a uniform parallel magnetic field. The nonlinear dynamics of such a disturbance is governed largely by two critical layers, i.e., by two narrow regions on the flow profile vx = u(y) near critical levels y = yj (j = ±1), on which the resonance condition u(y±1) = c±cA is satisfied (c being the wave’s phase velocity corresponding to the neutral mode, and cA is the Alfven velocity). A nonlinear integrodifferential evolution equation is derived for a complex amplitude of the disturbance, and its solutions are analyzed (numerically and analytically) with different relationships between problem parameters. It is shown that the nonlinearity can play both a stabilizing and destabilizing role depending on the magnitude of the magnetic field and on the degree of supercriticality of the wave. With not too large a magnetic field (cA<cA∗), the future of a disturbance depends only on the relationship between the reciprocal of the Reynolds number ν and the wave’s linear spatial growth rate γL (provided that the magnetic Prandtl number is of order unity). With a sufficiently small growth rate (γL≲ν1/3), the critical layer regime throughout the evolution remains a viscous one, and the wave amplitude, upon reaching a very smooth maximum, goes very slowly to zero. At larger values of γL(γL≳ν1/3), when the nonlinearity threshold is attained, a peculiar kind of self-maintaining unsteady regime of critical layers is established, and the amplitude grows explosively, ∣A∣∝(x0−x)−2 (and ultimately reaches values where weakly nonlinear theory becomes invalid). With a sufficiently large magnetic field (cA>cA∗, but cA<½Δu, where Δu is the velocity difference), the nonlinearity leads to an “explosion” with any relationship between γL and ν1/3. © 1998 American Institute of Physics.
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47.20.Ft Instability of shear flows (e.g., Kelvin-Helmholtz)
02.30.-f Function theory, analysis
02.60.Nm Integral and integrodifferential equations
47.65.-d Magnetohydrodynamics and electrohydrodynamics

Development of a Reynolds stress closure for modeling of homogeneous MHD turbulence

Ola Widlund, Said Zahrai, and Fritz H. Bark

Phys. Fluids 10, 1987 (1998); http://dx.doi.org/10.1063/1.869714 (10 pages) | Cited 10 times

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A Reynolds stress closure is developed for homogeneous shear-free turbulence subjected to a strong magnetic field at low magnetic Reynolds numbers. A scalar dimensionality anisotropy parameter is introduced to carry information about the distribution of energy in spectral space. This information is vital in modeling MHD turbulence, as it determines both magnitude and anisotropy of the Joule dissipation tensor. The Joule dissipation tensor is modeled by a tensor function, which is bilinear in the Reynolds stress anisotropy and the unit direction vector of the magnetic field. The tensor function coefficients are second-order in the scalar dimensionality parameter. A phenomenological transport equation for the dimensionality parameter is proposed. The model is closed using the pressure–strain model of Sarkar, Speziale and Gatski and a magnetic destruction term in the standard dissipation equation. The purely magnetic linear problem contains no undetermined constants, while the complete model contains two constants. Model predictions for the case of decaying turbulence show very good agreement with direct numerical simulations. © 1998 American Institute of Physics.
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47.27.-i Turbulent flows
47.65.-d Magnetohydrodynamics and electrohydrodynamics

Control of streamwise vortices with uniform magnetic fluxes

Junwoo Lim, Haecheon Choi, and John Kim

Phys. Fluids 10, 1997 (1998); http://dx.doi.org/10.1063/1.869715 (9 pages) | Cited 4 times

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Numerical experiments are conducted to investigate the effect of uniform magnetic fluxes on streamwise vortices in conducting fluids. A simple flow configuration, in which a pair of two-dimensional vortices interact with a wall, is used to study the effect of magnetic forcing on the vortices. Effects of a uniform magnetic flux applied in three directions—streamwise, wall-normal, and spanwise—are investigated. The electromagnetic force induced from either the wall-normal or spanwise magnetic flux inhibits the induced motion by the streamwise vortices and reduces their strength, while the streamwise magnetic flux does not affect the flow in the present flow configuration. It is also shown that, in the case of a closely interacting pair of vortices, the spanwise magnetic flux is more effective than the wall-normal magnetic flux in reducing the strength of the streamwise vortices. © 1998 American Institute of Physics.
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47.32.C- Vortex dynamics
47.27.nb Boundary layer turbulence
47.65.-d Magnetohydrodynamics and electrohydrodynamics

Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation

Robert G. Jacobs and Paul A. Durbin

Phys. Fluids 10, 2006 (1998); http://dx.doi.org/10.1063/1.869716 (6 pages) | Cited 26 times

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The expansion into eigenfunctions of a general disturbance in a viscous flow is possible only when both the discrete and continuous modes of the Orr–Sommerfeld equation are employed. Proper implementation of the boundary conditions and a method for computation of the continuous modes are developed. The unique phenomenon known as shear sheltering is discussed and illustrated. It is shown that the penetration depth of disturbances into the boundary layer has a dependence on frequency and Reynolds number similar to that of a Stokes layer. A simple model that captures this dependence is developed. © 1998 American Institute of Physics.
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47.27.nb Boundary layer turbulence
47.15.Cb Laminar boundary layers
47.20.Ft Instability of shear flows (e.g., Kelvin-Helmholtz)

Dynamics of velocity gradient invariants in turbulence: Restricted Euler and linear diffusion models

Jesús Martín, César Dopazo, and Luis Valiño

Phys. Fluids 10, 2012 (1998); http://dx.doi.org/10.1063/1.869717 (14 pages) | Cited 16 times

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A complete system of dynamical equations for the invariants of the velocity gradient, the strain rate, and the rate-of-rotation tensors is deduced for an incompressible flow. The equations for the velocity gradient invariants R and Q were first deduced by Cantwell [Phys. Fluids A 4, 782 (1992)] in terms of Hij, the tensor containing the anisotropic part of the pressure Hessian and the viscous diffusion term in the velocity gradient equation. These equations are extended here for the strain rate tensor invariants, RS and QS, and for the rate-of-rotation tensor invariant, QW, using HijS and HijW, the symmetric and the skew-symmetric parts of Hij, respectively. In order to obtain a complete system, an equation for the square of the vortex stretching vector, Vi ≡ Sijωj, is required. The resulting dynamical system of invariants is closed using a simple model for the velocity gradient evolution: an isotropic approximation for the pressure term and a linear model for the viscous diffusion term. The local topology and the resulting statistics implied by this model reproduce a number of trends similar to known results from numerical experiments for the small scales of turbulence. © 1998 American Institute of Physics.
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47.27.-i Turbulent flows
47.32.C- Vortex dynamics

Direct excitation of small-scale motions in free shear flows

John M. Wiltse and Ari Glezer

Phys. Fluids 10, 2026 (1998); http://dx.doi.org/10.1063/1.869718 (11 pages) | Cited 39 times

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The conventional approach to small-scale mixing enhancement in free shear flows by the manipulation of global flow instabilities and the ensuing large-scale vortical structures depends on the classical cascading mechanism to transfer the control influence to the scales at which molecular mixing takes place. Thus the manipulation of mixing at the smallest scales is indirect and only weakly coupled to the control input. The present work focuses on direct excitation of the small scales within the dissipation range of a free shear flow. This approach is demonstrated in a shear layer segment of an air jet emanating from a square conduit. The flow is forced at a frequency that is approximately an order of magnitude lower than the passage frequency of eddies at the Kolmogorov scale using cantilevered piezoelectric actuators. Cross-stream distributions of the streamwise velocity component are measured at a number of streamwise stations downstream of the actuator using hot wire anemometry. Direct small scale excitation results in enhanced energy transfer from the large to the small scales and in a substantial increase in the dissipation and in the decay rate of turbulent kinetic energy. © 1998 American Institute of Physics.
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47.20.Ft Instability of shear flows (e.g., Kelvin-Helmholtz)
47.27.nb Boundary layer turbulence

Shock induced turbulence in composite materials at moderate Reynolds numbers

Alexei D. Kotelnikov and David C. Montgomery

Phys. Fluids 10, 2037 (1998); http://dx.doi.org/10.1063/1.869719 (18 pages) | Cited 13 times

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Numerical simulation is used to study the turbulence generated by the passage of strong shocks (typical Mach number 7.3) through an inhomogeneous fluid at moderate Reynolds numbers. Before passage of the shock, the material consists of mass-density inhomogeneities embedded in a background fluid. The entire system is initially at uniform temperature, pressure, and number density, with the nonuniform mass density resulting from differing mass species in different regions. In the present application, the substances are treated as ideal gases, though in the motivating physical problems they are more complex materials. The shock retains its identity and a sharp front, but leaves behind it a turbulent state whose locally averaged properties only slowly become spatially uniform. The shock acquires a turbulent “thickness” (the linear dimension of the nonuniform region behind the shock front) that seems ultimately damped by viscous and thermally conducting properties that are dependent on transport coefficients and (highly uncertain) Reynolds numbers. Typically, the turbulence is highly compressible, with comparable mean divergences and curls in the velocity field, and fractional rms density fluctuations of the order of 0.25 in the parameter ranges studied. The rms vorticity generated can be estimated reasonably well from dimensional considerations. The effect of the high density inhomogeneities is primarily to create a wide region of compressible turbulence behind the shock. The inhomogeneities create both a succession of reflected shocks and considerable vorticity. © 1998 American Institute of Physics.
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47.27.-i Turbulent flows
47.40.-x Compressible flows; shock waves
47.40.Nm Shock wave interactions and shock effects
47.40.Ki Supersonic and hypersonic flows
47.11.-j Computational methods in fluid dynamics
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