Phys. Fluids A (1989-1993) - Physics of Fluids

Select this Article		The 1992 François Naftali Frenkiel Award for Fluid Mechanics Phys. Fluids A 5, 1 (1993); http://dx.doi.org/10.1063/1.3480472 (1 page) Full Text: \| Download PDF
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Precise measurement of particle–wall hydrodynamic interactions at low Reynolds number using laser interferometry

N. Lecoq, F. Feuillebois, N. Anthore, R. Anthore, F. Bostel, and C. Petipas

Phys. Fluids A 5, 3 (1993); http://dx.doi.org/10.1063/1.858787 (10 pages) | Cited 11 times

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The motion of a spherical particle (with radius 1 to 6 mm) in a viscous fluid is measured using laser interferometry. The typical sensitivity on the measured displacement of the sphere is of the order of 50 nm. The particle is moving on the axis of a closed cylinder. The hydrodynamic interactions between the particle and the walls of the cylinder are compared with the theoretical result of Sano [J. Phys. Soc. Jpn. 56, 2713 (1987)] valid for a very small sphere. The agreement is excellent for the smallest sphere used in the experiment. The experiment also agrees with the result from the theory of lubrication when the sphere is close to a plane end wall. The effect of the particle roughness appears at small distances. Laser interferometry appears as a useful tool to study particle–wall hydrodynamic interactions when the geometry is cumbersome.

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47.15.G-	Low-Reynolds-number (creeping) flows
47.15.-x	Laminar flows
47.80.-v	Instrumentation and measurement methods in fluid dynamics

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Measurement of the translational and rotational velocities of a noncolloidal sphere rolling down a smooth inclined plane at low Reynolds number

Jeffrey R. Smart, Sean Beimfohr, and David T. Leighton

Phys. Fluids A 5, 13 (1993); http://dx.doi.org/10.1063/1.858799 (12 pages) | Cited 25 times

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The translational and rotational velocities of a sphere rolling down an inclined plane at low Reynolds number were measured as a function of the angle of inclination of the plane. Both 140 μm glass and 6350 μm diam acrylic particles were used. A theoretical model including the effects of particle surface roughness was developed, and is in quantitative agreement with the measurements. In addition, statistically significant velocity fluctuations were measured and are explained by variation in the coefficient of friction of the sphere. The disagreement between the measurements of Carty (B.S. thesis, Massachusetts Institute of Technology, 1957) and the theory of Goldman et al. [Chem. Eng. Sci. 22, 637 (1967)] is explained.

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47.15.G-	Low-Reynolds-number (creeping) flows
47.15.-x	Laminar flows
47.55.Kf	Particle-laden flows
47.90.+a	Other topics in fluid dynamics (restricted to new topics in section 47)

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End effects for the falling cylinder viscometer

Elias G. Wehbeh, T. J. Ui, and R. G. Hussey

Phys. Fluids A 5, 25 (1993); http://dx.doi.org/10.1063/1.858781 (9 pages) | Cited 8 times

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Experimental results are presented for the Stokes velocity of a circular cylinder of radius a and length L moving axially through a viscous fluid contained within a coaxial closed cylindrical tube of radius b and length H. The results cover the intermediate range 0.21 ≤a/b≤0.70 and are found to be consistent with theoretical results for the narrow gap case (a≂b) and with previous experimental results for the wide gap case (a≪b). The fractional difference between the observed drag and the theoretical drag is represented well by 1.67x−0.017, where x=(b−a)(a/b)^1/2/L. The negative term is interpreted as representing an effective cylinder length of 0.983L, and the 1.67x term is interpreted qualitatively as a measure of the degree to which the fluid is displaced radially by the falling cylinder. The limitations of our expression for end effects are explored experimentally. Additional experimental results are presented for two cases in which the ends of the cylindrical tube are not closed.

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47.15.G-

Low-Reynolds-number (creeping) flows

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Kinetics of a one‐dimensional granular medium in the quasielastic limit

Sean McNamara and W. R. Young

Phys. Fluids A 5, 34 (1993); http://dx.doi.org/10.1063/1.858896 (12 pages) | Cited 86 times

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The dynamics of a one‐dimensional granular medium has a finite time singularity if the number of particles in the medium is greater than a certain critical value. The singularity (‘‘inelastic collapse’’) occurs when a group of particles collides infinitely often in a finite time so that the separations and relative velocities vanish. To avoid the finite time singularity, a double limit in which the coefficient of restitution r approaches 1 and the number of particles N becomes large, but is always below the critical number needed to trigger collapse, is considered. Specifically, r→1 with N∼(1−r)⁻¹. This procedure is called the ‘‘quasielastic’’ limit. Using a combination of direct simulation and kinetic theory, it is shown that a bimodal velocity distribution develops from random initial conditions. The bimodal distribution is the basis for a ‘‘two‐stream’’ continuum model in which each stream represents one of the velocity modes. This two‐stream model qualitatively explains some of the unusual phenomena seen in the simulations, such as the growth of large‐scale instabilities in a medium that is excited with statistically homogeneous initial conditions. These instabilities can be either direct or oscillatory, depending on the domain size, and their finite‐amplitude development results in the formation of clusters of particles.

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51.10.+y	Kinetic and transport theory of gases
47.90.+a	Other topics in fluid dynamics (restricted to new topics in section 47)
81.05.Rm	Porous materials; granular materials

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Tracer dispersion in porous media with a double porosity

P. Magnico, C. Leroy, J. P. Bouchaud, C. Gauthier, and J. P. Hulin

Phys. Fluids A 5, 46 (1993); http://dx.doi.org/10.1063/1.858788 (12 pages) | Cited 5 times

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Tracer dispersion in double porosity packings of porous grains obtained by grinding sintered glass samples of various porosities and permeabilities is studied experimentally. The dispersivity l_D increases faster with velocity than for single porosity packings: this variation is predicted by modeling the medium as a one‐dimensional (1‐D) sequence of identical cells with one slow and one fast path in parallel. The grain size and the permeability contrast between the inside and the outside of the grains can also be determined. At low grain porosities, the transit time τ_i inside individual grains becomes of the order of the mean transit time across the whole sample and non‐Gaussian dispersion curves are observed.

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47.56.+r	Flows through porous media
66.10.C-	Diffusion and thermal diffusion

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The spreading of heat or soluble surfactant along a thin liquid film

O. E. Jensen and J. B. Grotberg

Phys. Fluids A 5, 58 (1993); http://dx.doi.org/10.1063/1.858789 (11 pages) | Cited 49 times

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The spreading of a localized distribution of surfactant on a thin viscous film is considered, in the situation in which the surfactant is soluble in the bulk layer and the boundary beneath the fluid is impermeable to surfactant. The surfactant distribution is controlled by advection and diffusion both at the surface of the film, where the surfactant forms a monolayer, and in the bulk. The bulk and surface surfactant concentrations are related by linearized sorption kinetics. The surfactant diffuses rapidly across the thin fluid layer, and lubrication theory is used to derive evolution equations for the film height and the surface and cross‐sectionally averaged bulk surfactant concentrations. A special case of the governing equations describes the Marangoni flow induced by a locally hot region of the layer. It is shown that in comparison to the spreading of insoluble surfactant, transient desorption of surfactant from the monolayer to the bulk causes the spreading rate to diminish, although once the bulk and surface concentrations are locally in equilibrium, film deformations are more severe, with a sharp pulse in the film height created just upstream of the leading edge of the surfactant distribution.

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47.15.G-	Low-Reynolds-number (creeping) flows
47.55.Kf	Particle-laden flows

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The effect of surfactant on the transient motion of Newtonian drops

W. J. Milliken, H. A. Stone, and L. G. Leal

Phys. Fluids A 5, 69 (1993); http://dx.doi.org/10.1063/1.858790 (11 pages) | Cited 50 times

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The effect of dilute, insoluble surfactant on the deformation and breakup of a viscous drop is examined. Two cases are considered: the deformation and stretching of a drop in a uniaxial extensional flow and the surface‐tension‐driven motion of an elongated drop in a quiescent fluid. Aside from rescaling the mean capillary force through an average decrease in the interfacial tension, surfactants alter the motion of a viscous drop through gradients in interfacial tension. The effects of surfactants are found to be most pronounced for small viscosity ratios, where Marangoni stresses substantially retard the interfacial velocity and cause the drop to behave as though it were more viscous. Surfactants are found to facilitate the formation of pointed ends during drop stretching, and this may explain the observation of tip streaming in experiments with viscoelastic drops. Surfactant gradients also allow drops to be elongated to a larger degree without producing end pinching.

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47.15.-x	Laminar flows
47.55.Kf	Particle-laden flows

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Droplet patterns from capillary stream breakup

Melissa Orme, Keeney Willis, and Tuong‐Van Nguyen

Phys. Fluids A 5, 80 (1993); http://dx.doi.org/10.1063/1.858791 (11 pages) | Cited 13 times

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Droplet patterns have been generated in a vacuum environment by capillary stream breakup initiated by a new form of applied disturbances. The method of producing droplet streams with flexibly controllable patterns is presented. The patterns are periodic sequences of droplets, characterized by specific drop‐to‐drop separations and diameters. The pattern repeats until the disturbance to the capillary stream is removed or altered. An important feature of this work is that the patterns are deterministic given knowledge of the characteristics of the disturbance waveform. A model has been developed that is based on conservation of momentum and linear theory, and predicts droplet stream configurations with excellent agreement with experiment. Use of spectral analysis demonstrates long‐term periodicity and stability of the droplet patterns as well as the excellent agreement between the predicted droplet patterns and those experimentally obtained.

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47.55.D-

Drops and bubbles

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Capillary instability of thin liquid film on a cylinder

Alexander L. Yarin, Alexander Oron, and Philip Rosenau

Phys. Fluids A 5, 91 (1993); http://dx.doi.org/10.1063/1.858792 (8 pages) | Cited 16 times

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The capillary instability of a thin liquid layer on a cylinder is studied. Using an integral approach and lubrication approximation, transport equations governing the spatiotemporal evolution of a film thickness and the temperature along the film are obtained. Evolution of the system under both isothermal and nonisothermal conditions is studied numerically. It is shown that nonlinear interaction of the linearly unstable modes begets an additional mode with a wavelength equal to that of the fastest growing wave. This, in turn, causes the formation of satellite drops along with the main ones. Application of these results in a possible continuous technology of high‐temperature superconductor wire fabrication is discussed.

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47.20.Dr	Surface-tension-driven instability
68.15.+e	Liquid thin films

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Convection in a fluid layer with asymmetric boundary conditions

R. M. Clever and F. H. Busse

Phys. Fluids A 5, 99 (1993); http://dx.doi.org/10.1063/1.858793 (9 pages) | Cited 2 times

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Steady convection rolls in a horizontal fluid layer heated from below are described numerically with a Galerkin method. A rigid lower and stress‐free upper boundary are assumed, while the temperature is fixed at both boundaries. The stability of the steady solutions with respect to arbitrary three‐dimensional infinitesimal disturbances is analyzed and the stability boundaries in the Rayleigh number–wave‐number plane are determined for selected Prandtl numbers. It is found that results of the analysis correspond more closely to the case of two rigid boundaries than to the case of two stress‐free boundaries. The domains of stability in the case of asymmetric boundaries are larger at high Prandtl numbers than in the case of two rigid boundaries, but smaller for low Prandtl numbers. Some of the asymmetric properties of convection rolls are discussed.

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47.20.Bp	Buoyancy-driven instabilities (e.g., Rayleigh-Benard)
47.20.Ky	Nonlinearity, bifurcation, and symmetry breaking
47.27.T-	Turbulent transport processes

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Linear‐stability theory of thermocapillary convection in a model of the float‐zone crystal‐growth process

G. P. Neitzel, K.‐T. Chang, D. F. Jankowski, and H. D. Mittelmann

Phys. Fluids A 5, 108 (1993); http://dx.doi.org/10.1063/1.858796 (7 pages) | Cited 30 times

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Linear‐stability theory has been applied to a basic state of thermocapillary convection in a model half‐zone to determine values of the Marangoni number above which instability is guaranteed. The basic state must be determined numerically since the half‐zone is of finite, O(1) aspect ratio with two‐dimensional flow and temperature fields. This, in turn, means that the governing equations for disturbance quantities are nonseparable partial differential equations. The disturbance equations are treated by a staggered‐grid discretization scheme. Results are presented for a variety of parameters of interest in the problem, including both terrestrial and microgravity cases; they complement recent calculations of the corresponding energy‐stability limits.

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47.20.-k	Flow instabilities
47.20.Dr	Surface-tension-driven instability

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Direct numerical simulation of three‐dimensional open‐channel flow with zero‐shear gas–liquid interface

Satoru Komori, Ryuichi Nagaosa, Yasuhiro Murakami, Satoshi Chiba, Katsuya Ishii, and Kunio Kuwahara

Phys. Fluids A 5, 115 (1993); http://dx.doi.org/10.1063/1.858797 (11 pages) | Cited 39 times

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Turbulence structure in an open‐channel flow with a zero‐shear gas–liquid interface was numerically investigated by a three‐dimensional direct numerical simulation (DNS) based on a fifth‐order finite‐difference formulation, and the relationship between scalar transfer across a zero‐shear gas–liquid interface and organized motion near the interface was discussed. The numerical predictions of turbulence quantities were also compared with the measurements by means of a two‐color laser Doppler velocimeter. The results by the DNS show that the vertical motion is restrained in the interfacial region and there the turbulence energy is redistributed from the vertical direction to the streamwise and spanwise directions through the pressure fluctuation. The large‐scale eddies are generated by bursting phenomena in the wall region and they are lifted up toward the interfacial region. Then, the eddies renew the interface and promote the scalar transfer across the gas–liquid interface. Both the damping effect and the generation process of the surface‐renewal motions predicted by the DNS explain well the experimental results deduced in previously published studies. Furthermore, the predicted bursting frequency and mass transfer coefficient are in good agreement with the measurements.

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06.30.Dr	Mass and density
47.27.-i	Turbulent flows

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On the Lundgren–Townsend model of turbulent fine scales

D. I. Pullin and P. G. Saffman

Phys. Fluids A 5, 126 (1993); http://dx.doi.org/10.1063/1.858798 (20 pages) | Cited 27 times

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The strained‐spiral vortex model of turbulent fines scales given by Lundgren [Phys. Fluids 25, 2193 (1982)] is used to calculate vorticity and velocity‐derivative moments for homogeneous isotropic turbulence. A specific form of the relaxing spiral vortex is proposed modeled by a rolling‐up vortex layer embedded in a background containing opposite signed vorticity and with zero total circulation at infinity. The numerical values of two dimensionless groups are fixed in order to give a Kolmogorov constant and skewness which are within the range of experiment. This gives the result that the ratio of the ensemble average hyperskewness S_2p+1≡ (∂u/∂x)^2p+1/[(∂u/∂x)²]^(2p+1)/2 to the hyperflatness F_2p≡(∂u/∂x)^2p/[(∂u/∂x)²] ^p, p=2,3,..., is constant independent of Taylor–Reynolds number R_λ, as is the ratio of the 2pth moment of one component of the vorticity Ω_2p≡ω^2p_x/(ω²_x)^p to F_2p. A cutoff in a relevant time integration is then used to eliminate vortex‐sheet‐induced divergences in the integrals corresponding to ω^2p_x, p=2,3,..., and an assumption is made that the lateral scale of the spiral vortex in the model is the geometric mean of the Taylor and the Kolmogorov microscales. This gives Ω_2p=Ω_2pR_λ^p/2−3/4, F_2p= math

_2pR_λ^p/2−3/4 and S_2p+1= math

_2p+1R_λ^p/2−3/4, p=2,3,..., with explicit calculation of the numbers Ω_2p, math

_2p, and

_2p+1. The results of the model are compared with experimental compilation of Van Atta and Antonia [Phys. Fluids 23, 252 (1980)] for F₄ and with the isotropic turbulence calculations of Kerr [J. Fluid Mech. 153, 31 (1985)] and of Vincent and Meneguzzi [J. Fluid Mech. 225, 1 (1991)].

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47.27.-i	Turbulent flows
47.27.Gs	Isotropic turbulence; homogeneous turbulence

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A proper velocity scale for modeling subgrid‐scale eddy viscosities in large eddy simulation

Kiyosi Horiuti

Phys. Fluids A 5, 146 (1993); http://dx.doi.org/10.1063/1.858800 (12 pages) | Cited 24 times

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The limitations of the commonly used Smagorinsky subgrid‐scale (SGS) eddy viscosity model in large eddy simulation (LES) of turbulent flows are that the model’s eddy viscosity constant must be optimized in different flows, and also that a damping function must be used to account for near‐wall effects. A new SGS model which mitigates these drawbacks is proposed, i.e., a more proper eddy viscosity velocity scale was determined by utilizing the third‐order terms in an anisotropic representation model of the Reynolds stresses [K. Horiuti, Phys. Fluids A 2, 1708 (1990)]. This method utilizes the direct numerical simulation (DNS) database for fully developed turbulent channel flow to show these drawbacks to be inherent in the use of an improper velocity scale, i.e., the total SGS energy component adopted in the Smagorinsky model. As a result, the SGS normal shear stress was alternatively employed as the velocity scale, thereby significantly improving the correlation with DNS data. Methods to correlate the SGS normal shear stress to the grid scale quantities are proposed and compared, and the resultant high accuracy of the scale‐similarity model to represent the SGS turbulence fluctuations is shown. The proposed SGS model was also tested in actual LES computations of turbulent channel flow, where it was found that the SGS eddy viscosity in the near‐wall region similarly acted as the conventionally used Van Driest damping function. This result is consistent with previous reports which assert that in the Reynolds averaged models, the rapid reduction of the Reynolds shear stress as the wall is approached is due to the preferential damping of the normal shear stress. It is shown that three eddy viscosity parameters contained in the proposed SGS model can be practically reduced to a single parameter, which is subsequently shown to be more universal and independent of the flow field than the Smagorinsky model constant. A qualitative interpretation for the variance of the Smagorinsky model constant in different flows is also provided via a correlation with the anisotropy of SGS turbulence intensities.

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47.27.N-	Wall-bounded shear flow turbulence
47.27.tb	Turbulent diffusion
47.60.-i	Flow phenomena in quasi-one-dimensional systems

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Spatial correlations in turbulence: Predictions from the multifractal formalism and comparison with experiments

John O’Neil and Charles Meneveau

Phys. Fluids A 5, 158 (1993); http://dx.doi.org/10.1063/1.858801 (15 pages) | Cited 28 times

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Questions about applicability of multiplicative cascade models for turbulent small‐scale intermittency (such as lognormal, random curdling, β, α, p models, etc.) are addressed by using the multifractal formalism to predict new properties of two‐point moments. These predictions are compared with experimental data. Measurements are performed in the wake of a cylinder and grid turbulence. Data at high Reynolds number in the atmospheric surface layer are also considered. The autocorrelation function of the local singularity strength α(x), as well as mixed moments of the form <ϵ_r(x)^qϵ_r(x+s)^−q≳ are computed from the kinetic energy dissipation obtained from single‐component, single‐probe measurements using Taylor’s hypothesis. For flows at high‐enough Reynolds number, the α(x) autocorrelation function exhibits logarithmic decay with distance, as predicted from a random multiplicative cascade process. Some discrepancies exist in the quantitative details, implying enhanced randomization. The mixed moments are found to exhibit a scaling transition, also in agreement with the multiplicative models. The results illustrate the usefulness of the two‐point multifractal formalism in characterizing intermittency and, as far as two‐point statistics is concerned, lend further (qualified) support to the multiplicative cascade models.

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47.27.-i	Turbulent flows
47.27.Gs	Isotropic turbulence; homogeneous turbulence
47.27.wb	Turbulent wakes
02.50.-r	Probability theory, stochastic processes, and statistics

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A fast contour dynamics method for many‐vortex calculations in two‐dimensional flows

David G. Dritschel

Phys. Fluids A 5, 173 (1993); http://dx.doi.org/10.1063/1.858802 (14 pages) | Cited 17 times

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A new computational method extending the contour dynamics/surgery (CS) algorithm is announced which gives typical speed‐up factors of two orders of magnitude in calculations of flows involving many interacting vortices. The method makes use of an alternative expression for the velocity field in the exterior of a vortex that takes the form of a rapidly convergent series. Each term in this series can be expressed as a complex coefficient divided by the complex distance x+iy from the vortex center. The complex coefficient, or moment, is a real number pair that describes shape characteristics of the vortex (e.g., circulation, eccentricity, etc.). In numerical calculations, where accuracy is necessarily limited, it is frequently sufficient to retain only the leading‐order terms in this series, particularly for a gas of well‐separated vortices. The real computational gain is made, however, by reexpanding the series of all vortices that are sufficiently separated from a given vortex as a single, truncated series in positive powers of the complex distance from this vortex’s center. The coefficients of this series involve only the moments of the other vortices and their centroid separation from the given vortex. The leading‐order truncation, for instance, simply gives point vortex dynamics, except that self‐ or close‐range interactions are computed using the full contour integral expression of contour dynamics (hence, all vortices retain nontrivial spatial structure, vital to a proper dynamical description of close‐range interactions). In general, the optimal truncation depends on a dynamic balance between the cost of all moment computations and the cost of all contour integrations. This method, called ‘‘moment‐accelerated contour surgery,’’ which is briefly outlined above for the planar case, has a direct analog in spherical geometry. There are also extensions to generalized two‐dimensional (2‐D) flows having more general linear operator relationships between streamfunction and vorticity. Details are provided for quasigeostrophic flow.

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47.15.ki	Inviscid flows with vorticity
47.27.-i	Turbulent flows

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On streamwise vortices in high Reynolds number supersonic axisymmetric jets

S. A. Arnette, M. Samimy, and G. S. Elliott

Phys. Fluids A 5, 187 (1993); http://dx.doi.org/10.1063/1.858803 (16 pages) | Cited 8 times

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Pitot pressure measurements and flow visualizations were used to investigate streamwise vortices previously observed in underexpanded jets. A simple model was developed, which gives reasonable agreement with the pressure measurements. A converging nozzle and converging–diverging nozzle of design Mach number 1.5 were used to generate jet flows of equivalent Mach numbers up to 2.5 (stagnation to ambient pressure ratios up to 17.1). By operating the nozzles fully expanded, overexpanded, and underexpanded, insight was gained into both the occurrence and cause for formation of the vortices. Spatially stationary streamwise vortices were found to exist in the near‐field region around the circumference of underexpanded jets in the vicinity of the jet boundary. Short exposure visualizations show the vortices persist much farther downstream with a loss of spatial organization. Visualizations suggest adjacent vortices have streamwise vorticity of opposite sign, so the action of adjacent vortices is to either pump jet fluid radially outward or entrain ambient fluid radially inward toward the jet. The downstream extent, strength, and number of vortices around the jet circumference increase with degree of underexpansion. A large number of vortices is found near the nozzle exit. Fewer vortices of larger scale are found farther downstream, indicative of a merging process. The absence of the vortices in fully expanded and overexpanded jets suggests the vortices are a consequence of a Taylor–Goertler‐type instability.

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47.27.wg	Turbulent jets
47.40.Ki	Supersonic and hypersonic flows

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Reynolds stress description of opposed and impinging turbulent jets. Part I: Closely spaced opposed jets

Michel Champion and Paul A. Libby

Phys. Fluids A 5, 203 (1993); http://dx.doi.org/10.1063/1.858776 (14 pages) | Cited 18 times

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The flow arising from two, closely spaced turbulent jets flowing counter to one another is analyzed for both two‐dimensional and axisymmetric configurations. By closely spaced it is indicated that the diameter of the jets is large compared to their separation distance. Two parameters, one a measure of the integral scale of the turbulence compared with half the separation distance of the jets, and a second, a measure of the intensity of the turbulence issuing from the jets, are assumed small and form the basis of an asymptotic analysis. As a consequence, the mean velocity components are given by the mean Euler equations, except in a thin layer that is centered about the plane containing either the stagnation line or point and within which discontinuities in the flow from each jet are adjusted. Thus, outside of this layer, the turbulence characteristics in a known mean velocity field are determined, in the present study, in terms of a Reynolds stress description. The rate of strain field associated with the stagnating flow results in anisotropy of the turbulence as the plane containing the stagnation line or point is approached. Differences in the mean velocities in two‐dimensional and axisymmetric configurations result in significant differences in the evolution of the gradient of the Reynolds shear stress from the exit planes of the jets and thus in the thin layer at the stagnation plane. For two‐dimensional flows, the velocity characteristics in the thin layer satisfy to lowest order in an expansion parameter the requisite symmetry conditions, whereas this is not the case for axisymmetric flows. The temperature in the two streams is assumed to be slightly different but uniform so that there is also a thermal layer at the stagnation plane. Comparison is made with the applicable experimental data for the mean velocity and the turbulence intensities with good and reasonable agreement, respectively.

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47.27.wg

Turbulent jets

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Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard‐sphere molecules

Taku Ohwada

Phys. Fluids A 5, 217 (1993); http://dx.doi.org/10.1063/1.858777 (18 pages) | Cited 36 times

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The structure of normal shock waves is investigated on the basis of the standard Boltzmann equation for hard‐sphere molecules. This fundamental nonlinear problem in rarefied gas dynamics is analyzed numerically by a newly developed finite‐difference method, where the Boltzmann collision integral is computed directly without using the Monte Carlo method. The velocity distribution function, as well as the macroscopic quantities, is accurately obtained. The numerical results are compared with the Mott‐Smith and the direct simulation Monte Carlo results in detail. The analytical solution for a weak shock wave based on the standard Boltzmann equation is also presented up to the second order of the shock strength together with its explicit numerical data for hard‐sphere molecules.

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47.40.Nm	Shock wave interactions and shock effects
47.45.-n	Rarefied gas dynamics
51.10.+y	Kinetic and transport theory of gases

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Numerical simulation of rarefied flow through a slit. Part I: Direct simulation Monte Carlo results

D. C. Wadsworth and D. A. Erwin

Phys. Fluids A 5, 235 (1993); http://dx.doi.org/10.1063/1.858778 (8 pages) | Cited 9 times

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The pressure‐driven flow of a rarefied monatomic gas through a two‐dimensional slit is simulated using the direct simulation Monte Carlo technique. Of particular interest is the change in flow field structure as pressure ratio and Knudsen number are varied. Comparisons are made to quantify the limits of validity of free‐molecular theory and approximate, nearly free‐molecular iterative methods. Also addressed is the sensitivity of the numerical solutions to grid structure and boundary conditions. The free‐molecular theory is found to predict quantitative flow field properties (e.g., centerline velocities or downstream flux profiles) reasonably well for large finite Knudsen number with the error dependent on the pressure ratio. The nearly free‐molecular corrections are shown to have limited range of applicability. A previously derived parameter is found to correlate total mass flux well as a function of pressure ratio and Knudsen number over a large portion of the transitional regime.

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47.45.-n	Rarefied gas dynamics
47.40.-x	Compressible flows; shock waves
47.60.-i	Flow phenomena in quasi-one-dimensional systems

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Monte Carlo study of Knudsen layers in evaporation from elemental and binary media

Dieter Sibold and Herbert M. Urbassek

Phys. Fluids A 5, 243 (1993); http://dx.doi.org/10.1063/1.858779 (14 pages) | Cited 26 times

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By Monte Carlo simulation the Knudsen layer in front of a surface from which atoms evaporate is studied. Evaporation into a vacuum is simulated by means of an evaporation–condensation geometry. Hard sphere interaction cross sections are employed. With the help of the present simulation data, the Knudsen layer is defined as that region adjacent to the evaporating surface, where the temperature of the flow parallel and perpendicular to the flow direction deviate by at least a given resolution δ. Taking δ=1%, it is found that the Knudsen layer is established after 800 mean‐free flight times; it has an extension of 20 mean‐free paths. It takes 60 monolayers to desorb before a Knudsen layer is formed. The data are generally in good agreement with predictions of analytical theory, where available. The differences observed in the case of evaporation from a binary target are discussed.

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47.40.-x	Compressible flows; shock waves
82.80.Ms	Mass spectrometry (including SIMS, multiphoton ionization and resonance ionization mass spectrometry, MALDI)
68.03.Fg	Evaporation and condensation of liquids
68.43.Mn	Adsorption kinetics

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Nearly incompressible fluids. II: Magnetohydrodynamics, turbulence, and waves

G. P. Zank and W. H. Matthaeus

Phys. Fluids A 5, 257 (1993); http://dx.doi.org/10.1063/1.858780 (17 pages) | Cited 70 times

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The theory of nearly incompressible (NI) fluid dynamics developed previously for hydrodynamics is extended to magnetohydrodynamics (MHD). On the basis of a singular expansion technique, modified systems of fluid equations are derived for which the effects of compressibility are admitted only weakly in terms of the different possible incompressible solutions (thus ‘‘nearly incompressible MHD’’). NI MHD represents the interface between the compressible and incompressible magnetofluid descriptions in the subsonic regime. The theory developed here does not hold in the presence of very large thermal, gravitational, or field gradients. It is found that there exist three distinct NI descriptions corresponding to each of the three possible plasma beta (β ≡ the ratio of thermal to magnetic pressure) regimes (β≪1, β∼1, β≫1). In the β≫1 regime, the compressible MHD description converges in the low Mach number limit to the equations of classical incompressible three‐dimensional (3‐D) MHD. However, for the remaining plasma beta regimes, the imposition of a large dc magnetic field forces the equations of fully compressible 3‐D MHD to converge to the equations of 2‐D incompressible MHD in the low Mach number limit. The ‘‘collapse in dimensionality’’ corresponding to the different plasma beta regimes clarifies the distinction between the 3‐D and 2‐D incompressible MHD descriptions (and also that of 21/2‐D incompressible MHD). The collapse in dimensionality that occurs as a result of a decreased plasma beta can carry over to the weakly compressible corrections. For a β∼1 plasma, Alfvén waves propagate parallel to the applied magnetic field (reminiscent of reduced MHD), while for a β≪1 magnetofluid, quasi‐1‐D long‐wavelength acoustic modes propagate parallel to the applied magnetic field. The detailed theory of weakly compressible corrections to the various incompressible MHD descriptions is presented and the implications for the solar wind emphasized.

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47.65.-d	Magnetohydrodynamics and electrohydrodynamics
52.35.Ra	Plasma turbulence
52.35.Bj	Magnetohydrodynamic waves (e.g., Alfven waves)
96.20.Br	Origin and evolution

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The kinematics of the four‐roll mill

J. J. L. Higdon

Phys. Fluids A 5, 274 (1993); http://dx.doi.org/10.1063/1.858782 (3 pages) | Cited 13 times

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Numerical computations are employed to study the flow field produced by a four‐roll mill. The radius of the cylinders a, the cylinder spacing 2b, and the size 2l of the square container are varied to assess the effects on the kinematics of the flow field. It is found that a ratio of a/b=0.625 with l/b≥3.0 produces the best approximation to a pure extensional flow. With these parameter values, the extension rate remains constant with an error of less than 1% over an axial region x/b≤0.5. By contrast, the commonly accepted design a/b=0.772 suggested by Fuller and Leal [J. Polym. Sci. Polym. Phys. 19, 557 (1981)] produces a variation in extension rate of 50% over the same region. Streamline patterns and velocity gradient error contours are presented for these two designs.

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47.15.G-	Low-Reynolds-number (creeping) flows
47.80.-v	Instrumentation and measurement methods in fluid dynamics

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A steady breaking wave

Frédéric Dias and E. O. Tuck

Phys. Fluids A 5, 277 (1993); http://dx.doi.org/10.1063/1.858783 (3 pages) | Cited 4 times

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The problem of a supercritical uniform stream meeting a stagnant mass of water is considered. The whole oncoming flow is forced to overturn and to fall back and down forever like a waterfall into a bottomless chasm. The interaction between this waterfall and the oncoming flow is neglected. It is shown that the solution occurs only at a unique Froude number equal to 2.994.

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