Arbitrary Reynolds Number Research Articles

Numerical studies of viscous incompressible flow between two rotating concentric spheres

The problem of determining the induced steady axially symmetric motion of an incompressible viscous fluid confined between two concentric spheres, with the outer sphere rotating with constant angular velocity and the inner sphere fixed, is numerically investigated for large Reynolds number. The governing Navier‐Stokes equations expressed in terms of a stream function‐vorticity formulation are reduced to a set of nonlinear ordinary differential equations in the radial variable, one of second order and the other of fourth order, by expanding the flow variables as an infinite series of orthogonal Gegenbauer functions. The numerical investigation is based on a finite‐difference technique which does not involve iterations and which is valid for arbitrary large Reynolds number. Present calculations are performed for Reynolds numbers as large as 5000. The resulting flow patterns are displayed in the form of level curves. The results show a stable configuration consistent with experimental results with no evidence of any disjoint closed curves.

An exact solution of visco-capillary flow in an inclined channel

An exact solution of visco-capillary flow in an inclined channel is presented. Assuming that the flow is steady and unidirectional the surface shape and the velocity profile are obtained as exact solutions of the Navier-Stokes equations valid for arbitrary Reynolds numbers.¶The contributions of the determining parameters of the problem to the surface shape and to the velocity profile are discussed. A structural change in the velocity profile, the 'velocity overshoot', is found for cases with meniscus elevation higher than the apex height of the flow. The theoretical findings are compared to experimental results.

Statistical mechanical description and modelling of turbulent collision of inertial particles

The collision rate of monodisperse solid particles in a turbulent gas is governed by a wide range of scales of motion in the flow. Recent studies have shown that large-scale energetic eddies are the dominant factor contributing to the relative velocity between two colliding particles (the turbulent transport effect), whereas small-scale dissipative eddies can enhance the collision rate significantly by inducing local non-uniform particle distribution (the accumulation effect). The turbulent transport effect is most noticeable when the particle inertial response time τp is of the order of the flow integral timescale and the accumulation effect is most pronounced when τp is comparable to the flow Kolmogorov time.We study these two contributions separately through direct numerical simulations. The two effects are quantified carefully with a numerical procedure that is independent of the computation of average collision rate. This facilitates the study of not only the statistical description of the collision kernel, but also the relative contributions and modelling of the two physical effects. Simulations at several flow Reynolds numbers were performed to suggest a model for the accumulation effect. The data show that the accumulation effect scales linearly with flow Taylor microscale Reynolds number Rλ, while the theory for fully developed turbulence indicates that the maximum level of the turbulent transport effect scales with R1/2λ. Finally, an integrated model has been developed to predict the collision rate at arbitrary flow Reynolds numbers and particle inertia.

An Oseen-type model for swirling internal separated flows

The viscous, laminar, separated flow downstream of a sudden expansion in a pipe is studied. The flow is modeled by an Oseen-type equation, but with the additional feature that the nonlinearity in the swirl is retained. Exact solutions are obtained for a high-Reynolds-number limit and for arbitrary Reynolds number by use of an eigenfunction-expansion procedure, in the presence of swirl. This leads to a non-standard eigenvalue problem. When the swirl is sufficiently large, a central recirculating region is observed. The effect of the pressure gradients on the velocity profiles and the central recirculating eddy is discussed. The low-Reynolds-number solutions go over smoothly to the large Reynolds number solution as the Reynolds number increases. Good agreement is obtained with the numerically computed value of the reattachment length.

Probabilistic estimates for the two-dimensional stochastic Navier-Stokes equations

We consider the Navier–Stokes equation on a two-dimensional torus with a random force, white noise in time, and analytic in space, for arbitrary Reynolds number R. We prove probabilistic estimates for the long-time behavior of the solutions that imply bounds for the dissipation scale and energy spectrum as R→∞.

Kinetic instabilities of three-layer thermocapillary creeping flows

It is shown that plane viscosity stratified system under longitudinal temperature gradient may exhibit long-wavelength instabilities of a purely kinetic nature which persist at arbitrary small Reynolds number. The first instability is so-called alpha effect, and the second one is a new surface-tension-induced instability. The weakly nonlinear equations for the evolving interfaces are derived and simulated.

Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number

The electrohydrodynamic flow associated with a fluid drop in an electric field is a consequence of the tangential electric stress at the fluid interface. The tangential viscous stress due to the electrohydrodynamic flow arises to just balance the tangential electric stress at the fluid interface so that the traction boundary condition is satisfied. Influenced by both the local electric stress and viscous stress, the drop interface may exhibit various shapes. The presence of fluid flow also leads to charge convection phenomena. The relative significance of the charge convection effect is usually measured in terms of the electric Reynolds number, Re E , defined as the ratio of the timescales of charge convection by flow and that for charge relaxation by ohmic conduction. This work presents a quantitative analysis of the charge convection effects in a framework of the leaky dielectric model at finite Re E , which has not been considered in previous investigations. Axisymmetric steady flows driven by an applied uniform electric field about a deformable fluid drop suspended in an immiscible fluid are studied by computational means of the Galerkin finite–element method with supplementary asymptotic analysis. The results of finite–element computations are in general agreement with the prediction by the asymptotic analysis for spherical drops at vanishingly small Re E . A common effect of charge convection is found to reduce the intensity of electrohydrodynamic flow. As a consequence, oblate drops are predicted to be less deformed in an electric field when charge convection is taken into account. The prolate drops are often associated with an equator–to–pole flow, which convects charges toward the poles to form a charge distribution resembling that in a highly conducting drop immersed in an insulating medium. Therefore, charge convection tends to enhance the prolate drop deformation. In many cases, charge convection effects are found to be significant even at apparently small Re E , corresponding to the charge relaxation time–scale about 10 −3 s, suggesting that many experimental results reported in the previous publications could have been influenced by charge convection effects.

Spatially-periodic steady solutions to the three-dimensional Navier–Stokes equation with the ABC-force

For an appropriately scaled ABC-forcing the ABC-flow is a steady space-periodic solution to the three-dimensional Navier–Stokes equation for an arbitrary Reynolds number R. We are investigating both numerically and analytically different branches of steady solutions in the case A= B= C when the equation has a group of symmetries isomorphic to the rotation group of the cube. Two families of steady flows, each comprised of three mutually symmetric branches, were detected numerically. One was shown to persist for 7.9≤ R≤2000 and another for 149≤ R≤1000. Branches of the first family intersect twice with the ABC-flow in a bifurcation generic to a system with symmetry group D 3. Other possible bifurcations of the ABC-flow and the possibility of existence of branches of steady flows with other symmetries are considered.

Fluid flow through three-dimensional fibrous porous media

A system of self-consistent equations for determining the hydrodynamic resistance of dilute fibrous porous media in the case of arbitrary low Reynolds numbers and arbitrary random packing of the fibers in the media is derived on the basis of a multiple-scattering hydrodynamic theory. The equations obtained are applied to the case of isotropic packing of the fibers and to the anisotropic case when all the fibers are orthogonal to the direction of fluid flow. Equations are derived and analyzed for the velocity correlation function in a random fibrous medium. The longitudinal and transverse diffusion coefficients of a passive impurity embedded in the fluid are calculated.

A theory of three-dimensional interfacial vorticity dynamics

A three-dimensional theory of vorticity dynamics on an incompressible viscous and immiscible fluid–fluid interface, or interfacial vorticity dynamics for short, is presented as a counterpart of the vorticity dynamics 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 vorticity exists on an interface S, (2) how much vorticity is 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 vorticity dynamics 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.

Pressure drop in porous media at arbitrary Reynolds numbers

Pressure drop in porous media at arbitrary Reynolds numbers

Intense vortical structures in grid-generated turbulence

This paper presents a set of experiments aimed at investigating the features and the statistical frequency of intense vortical structures (sometimes called ‘‘filaments’’, or ‘‘worms’’) as manifested by a migrating bubble technique in a mean shear free, homogeneous, isotropic, stationary turbulence generated by oscillating grids in a water tank for Rλ reaching up to 300. It is found that the nucleation of filaments at the surface of the walls of the tank, where boundary layers are liable to destabilize is much more frequent than in the homogeneous bulk of the tank where one filament is typically detected each hundred large scale turnover time. This distinction between the wall surface and the bulk activity, supplemented with the fact that the size of the filaments and their lifetime compare with the length and time-scales of the largest structures of the flow leads us to formulate an elementary model explaining the origin and the geometrical features of these intense vortical structures in turbulent flows for arbitrary Reynolds numbers.

Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers

We consider the stability of steady flows of viscoelastic fluids of Jeffreys type. For sufficiently small Weissenberg numbers, but arbitrary Reynolds numbers, it is proved that the flow is stable to small disturbances if the spectrum of the linearized operator is in the left half plane.

An optimal control problem in exterior hydrodynamics

SynopsisIn this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure. An existence theorem for the speed trajectory which corresponds to the absolute minimum is provided. The results are valid for arbitrary Reynolds numbers.

Renormalization group analysis of anisotropic diffusion in turbulent shear flows

The renormalization group is applied to compute anisotropic corrections to the scalar eddy diffusivity representation of turbulent diffusion of a passive scalar. The corrections are linear in the mean velocity gradients. All model constants are computed theoretically. A form of the theory valid at arbitrary Reynolds number is derived. The theory applies only when convection of the velocity–scalar correlation can be neglected. A ratio of diffusivity components, found experimentally to have a nearly constant value in a variety of shear flows, is computed theoretically for flows in a certain state of equilibrium. The theoretical value is well within the fairly narrow range of experimentally observed values. Theoretical predictions of this diffusivity ratio are also compared with data from experiments and direct numerical simulations of homogeneous shear flows with constant velocity and scalar gradients.

Modelling of developing nearly homogeneous turbulence of velocity and scalar fields

Based on the second-order model suggested by Kolovandin and Martynenko (Heat/mass transfer in homogeneous turbulence, 9th Int. Heat Transfer Conf., Jerusalem (1990)) numerical modelling of the nearly homogeneous turbulence of the velocity and transported passive scalar fields is considered. The work is aimed at a detailed study of the effect exerted by the basic factors of turbulence generation—the gradients of the averaged velocity and scalar values—on the evolution in time of the statistical characteristics of turbulence which, within the framework of the second-order model, determine the processes of turbulent momentum and heat transfer at arbitrary turbulent Reynolds and Peclet numbers.

A numerical study of the turbulent Ekman layer

The three-dimensional time-dependent turbulent flow in a neutrally stratified Ekman layer over a smooth surface is computed numerically by directly solving the Navier–Stokes equations. All the relevant scales of motion are included in the simulation so that no turbulence model is needed. Results of the simulations indicate that the horizontal component of the rotation vector has a significant influence on the turbulence; thus the ‘f-plane’ approximation fails. Differences as large as 20% in the geostrophic drag coefficient, u*/G, and 70% in the angle between the freestream velocity and the surface shear stress are found, depending on the latitude and the direction of the geostrophic wind. At 45° latitude, differences of 6 and 30% are noted in the drag coefficient and the shear angle, respectively, owing to the variation of the wind direction alone. Asymptotic similarity theory and a higher-order correction are first tested for the range of low Reynolds numbers simulated, and then used to predict the friction velocity and stress direction at the surface for flows at arbitrary Reynolds number. A model for the variation of these quantities with latitude and wind angle is also proposed which gives an acceptable fit to the simulation results. No large-scale longitudinal vortices are found in the velocity fields, reinforcing the conjecture that unstable thermal stratification, in addition to inflectional instability, is required to produce and maintain the large-scale rolls observed in the Earth's boundary layer. Comparisons of the Ekman layer with a related three-dimensional boundary layer reveal similarities of the mean profiles, as well as qualitative differences.

Hydrodynamic arrest of a flat body moving towards a parallel surface at arbitrary Reynolds number

The motion of a flat body towards a parallel plane surface in incompressible fluid is considered both in the presence and absence of an applied force for a non-vanishing initial velocity. In the inviscid limit, a first integral of the equations is obtained and analytic solutions are presented for the cases of finite body inertia with zero applied force and finite applied force with negligible body inertia. In the former case when the ratio of body inertia to fluid inertia is large, a singular behaviour is observed in the arrest of the body before impact wherein the time-dependent pressure and radial velocity of the fluid exhibit a sharp peak and there is a large transfer of kinetic energy from the body to the thin fluid layer. For a real fluid, a general procedure is described to obtain solutions at arbitrary Reynolds number for naturally occurring initial velocity conditions. Solutions to the full Navier-Stokes equations are obtained for an arbitrary Reynolds number based on gap height which are valid provided the flow remains laminar and the gap height is small. In general, the equations of motion of the body and fluid are both dynamically and kinematically coupled. The dynamic coupling, however, is removed when the body inertia is neglected. In particular, the cases of hydrodynamic arrest with zero applied force, and draining of the fluid under a constant applied force are considered. The natural initial conditions lead to a new exact similarity solution of the Navier-Stokes equations which is valid for an instantaneous time-dependent Re based on gap height of greater than approximately 100, wherein the top and bottom boundary layers remain distinct. The longer time portions of the motion and the final arrest are described by a numerical calculation for intermediate Reynolds number and a low-Reynolds-number analysis.

Thermocapillary migration of droplets: An exact solution for small marangoni numbers

The thermocapillary motion of immiscible spherical droplets placed in an unbounded fluid with a linear temperature distribution is analyzed. For small Marangoni numbers, an exact solution to the Navier-Stokes equations has been found. The velocity field and the terminal velocity of the droplet are the same as those calculated by previous workers in the small Reynolds number limit. Hence the range of applicability of their results has been extended to arbitrary Reynolds numbers, so long as Ma = PrRe remains small (i.e., the convection of energy can be neglected). The shape of the droplet has also been calculated, when the deformations from the spherical shape are small. The results are qualitatively in agreement with the previous results, in the limit Pr → 0, that droplets of the same density as the host fluid do not deform at all, that bubbles tend to deform oblately, and that droplets with the denser fluid inside tend to elongate in the flow direction.

Film flow on a rotating disk

Unsteady liquid film flow on a rotating disk is analyzed by asymptotic methods for low and high Reynolds numbers. The analysis elucidates how a film of uniform thickness thins when the disk is set in steady rotation. In the low Reynolds number analysis two time scales for the thinning film are identified. The long-time-scale analysis ignores the initial acceleration of the fluid layer and hence is singular at the onset of rotation. The singularity is removed by matching the long-time-scale expansion for the transient film thickness with a short-time-scale expansion that accounts for fluid acceleration during spinup. The leading order term in the long-time-scale solution for the transient film thickness is shown to be a lower bound for film thickness for all time. A short-time analysis that accounts for boundary layer growth at the disk surface is also presented for arbitrary Reynolds number. The analysis becomes invalid either when the boundary layer has a thickness comparable to that of the thinning film, or when nonlinear effects become important.