On Oseen flows for large Reynolds numbers

This investigation offers a detailed analysis of solutions to the two-dimensional Oseen problem in the exterior of an obstacle for large Reynolds numbers. It is motivated by mathematical results highlighting the important role played by the Oseen flows in characterizing the asymptotic structure of steady solutions to the Navier–Stokes problem at large distances from the obstacle. We compute solutions of the Oseen problem based on the series representation discovered by Tomotika and Aoi (Q J Mech Appl Math 3:140–161, 1950) where the expansion coefficients are determined numerically. Since the resulting algebraic problem suffers from very poor conditioning, the solution process involves the use of very high arithmetic precision. The effect of different numerical parameters on the accuracy of the computed solutions is studied in detail. While the corresponding inviscid problem admits many different solutions, we show that the inviscid flow proposed by Stewartson (Philos Mag 1:345–354, 1956) is the limit that the viscous Oseen flows converge to as Re → ∞. We also draw some comparisons with the steady Navier–Stokes flows for large Reynolds numbers.     

Spin-down of a fluid in a low cylinder at large Reynolds numbers

The problem of the axisymmetric motion of a fluid between infinite disks is solved by the method of matched asymptotic expansions without introducing model assumptions. For the strongly nonlinear stage of spin-down solutions are found that correspond to initial states different from rigid-body rotation, when the boundary layer is not a Kármán layer. The experimental results obtained are in qualitative and quantitative agreement with the theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–46, May–June, 1986.The authors wish to thank A. M. Obukhov and F. V. Dolzhanskii for formulating the problem and for constructive discussion.     

Approximate solutions for power-law fluid flow past a particle at low Reynolds numbers

A simple perturbation approximation is proposed for describing flow behaviour of particles immersed in a uniform flow and an extensional flow of power-law fluids. The present solution for particles in a uniform flow field is in good agreement with the numerical solution in the literature. Theoretical predictions indicate that the effect of pseudoplasticity on flow around particles in an extensional flow field is small compared with that for particles in a uniform flow field.From the viewpoint of perturbation techniques, existing analytical solutions based on linearization of the equations of motion for particle in a power-law fluid are re-examined. Mass transfer to a power-law fluid from a particle is also discussed.     

Linear stability of plane couette flow of an upper convected maxwell fluid

We investigate the linear stability of plane Couette flow of an upper convected Maxwell fluid using a spectral method to compute the eigenvalues. No instabilities are found. This is in agreement with the results of Ho and Denn [1] and Lee and Finlayson [2], but contradicts “proofs” of instability by Gorodtsov and Leonov [3] and Akbay and Frischmann [4,5]. The errors in those arguments are pointed out. We also find that the numerical discretization can generate artificial instabilities (see also [1,6]). The qualitative behavior of the eigenvalue spectrum as well as the artificial instabilities is discussed.     

Linear stability of plane creeping Couette flow for Burgers fluid

It is well known that plane creeping Couette flow of UCM and Oldroy-B fluids are linearly stable. However, for Burges fluid, which includes UCM and Oldroyd-B fluids as special cases, unstable modes are detected in the present work. The wave speed, critical parameters and perturbation mode are studied for neutral waves. Energy analysis shows that the sustaining of perturbation energy in Poiseuille flow and Couette flow is completely different. At low Reynolds number limit, analytical solutions are obtained for simplified perturbation equations. The essential difference between Burgers fluid and Oldroyd-B fluid is revealed to be the fact that neutral mode exists only in the former.     

Linear stability diagrams for the shear flow of an Oldroyd-B fluid with slip along the fixed wall

We consider the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. Slip is allowed by means of a generic slip equation predicting that the shear stress is a non-monotonic function of the velocity at the wall. The complete one-dimensional stability analysis to one-dimensional disturbances is carried out and the corresponding neutral stability diagrams are constructed. Asymptotic results for large values of the elasticity number and finite element calculations are also presented. The instability regimes are within or coincide with the negative-slope regime of the slip equation. The numerical calculations agree with the linear stability results when the size of the initial perturbation is small. Large perturbations may destabilize a linearly stable steady state, leading to a periodic solution. The period and the amplitude of the periodic solutions increase with elasticity. Received: 19 June 1997 Accepted: 22 September 1997     

Nearly singular two-dimensional Kolmogorov flows for large Reynolds numbers

We study the convergence of two-dimensional stationary Kolmogorov flows as the Reynolds number increases to infinity. Since the flows considered are stationary solutions of Navier-Stokes equations, they are smooth whatever the Reynolds number may be. However, in the limit of an infinite Reynolds number, they can, at least theoretically, converge to a nonsmooth function. Through numerical experiments, we show that, under a certain condition, some smooth solutions of the Navier-Stokes equations converge to a nonsmooth solution of the Euler equations and develop internal layers. Therefore the Navier-Stokes flows are nearly singular for large Reynolds numbers. In view of this nearly singular solution, we propose a possible scenario of turbulence, which is of an intermediate nature between Leray's and Ruelle-Taken's scenarios.     

Boundary element analysis of driven cavity flow for low and moderate Reynolds numbers

A boundary element method for steady two‐dimensional low‐to‐moderate‐Reynolds number flows of incompressible fluids, using primitive variables, is presented. The velocity gradients in the Navier–Stokes equations are evaluated using the alternatives of upwind and central finite difference approximations, and derivatives of finite element shape functions. A direct iterative scheme is used to cope with the non‐linear character of the integral equations. In order to achieve convergence, an underrelaxation technique is employed at relatively high Reynolds numbers. Driven cavity flow in a square domain is considered to validate the proposed method by comparison with other published data. Copyright © 2001 John Wiley & Sons, Ltd.     

Steady flow of a viscous incompressible fluid past two particles at moderate Reynolds numbers

A study is made of axisymmetric flow past two particles of spherical shape at Reynolds numbers 1 R 80. The flow patterns, pressure distributions, and values of the drag are investigated for different distances between the spheres. It is found that the drag depends nonmonotonically on the parameters that determine the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 167–171, January–February, 1982.     

Linear stability of a plane poiseuille flow of oldroyd fluid

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 5–10, January–February, 1988.     

Asymptotic analysis of stability problem of plane poiseuille flow for high Reynolds number

A study of the stability of plane Poiseuille flow at higher Reynolds number is made. Within a “triple-deck” structural framework, the qualitative behavior of the eingenvalue of Orr-Sommerfeld equation is analytically obtained. The corresponding eigenfunction is formulated approximately.     

Linear stability analysis of coupled parallel flexible plates in an axial flow

We study here the linear stability of N identical flexible plates with clamped–free boundary conditions forced by a uniform parallel flow. Flow viscosity and elastic damping are neglected, and the flow around the plates is assumed potential. The shedding of vorticity from the plates’ trailing edges is accounted for by introducing a force-free wake behind each plate. A Galerkin method is used to compute the eigenmodes of the system. We are interested in the effects of the number of plates and their relative distance on the stability property of the state of rest, as well as in the nature and structure of the coupled states. Detailed results are presented for the cases N=2, N=3 and N1.     

Visualizations of the flow inside an open cavity at medium range Reynolds numbers

The interaction between a laminar boundary layer and an open cavity is investigated experimentally for medium range Reynolds numbers. Flow visualizations are carried out for three different observation directions in order to understand the spatial development of dynamical structures. In particular, synchronized visualizations in two parallel planes picture the transverse development of the flow. The study is conducted by changing the cavity aspect ratio, the Reynolds number and therefore the flow patterns inside the cavity. The issue is to emphasize the 3-D development of the flow. In particular, we show that the dynamical structures are not due to secondary shear layer instabilities.     

Straight flow in an elastoviscoplastic cylindrical layer with possible two-sided slip

The solution of the boundary value problem of the theory of large strains dealing with the viscoplastic flow of an elastoviscoplasticmaterial in the gap between two rigid coaxial cylindrical surfaces in the case of motion of the inner surface is obtained under the assumption that the material can slip on both surfaces. The reversible deformation, the development of a viscoplastic flow in the cases of uniformly accelerated and constant-velocity motions of the surface, the flow deceleration in the case of uniformly decelerated motion of the surface until full stop, and the unloading of the medium are considered.     

Investigation of supersonic flow around a blunt body with injection for high Reynolds numbers

The paper discusses the supersonic flow around a blunt smooth body by a stream of viscous gas with subsonic injection from the surface of the body. The effect of various injection cycles on the physical flow characteristics ahead of the body are studied in [1, 2]; the problem is considered in the approximation of a boundary layer. The nonuniform composition of the gas ahead of the body, chemical reactions between the various components, and the effect of radiation are taken into account. For a number of flow cycles, which are of practical importance, it will be of interest to consider higher approximations in powers of [=1/Re, see Eq. (1.1) below] in the shock layer ahead of the body and, in particular, to explain the action of the displacement effect and also the limits of applicability of the boundary-layer approximation assumed in [1, 2]. Extensive literature has been devoted to the asymptotics of the problem of flow around a blunt body of a viscous gas at high Reynolds numbers (see, for example, Van Dyke's book [3]). An investigation of the problem, based on the method of M. I. Vishik and L. A. Lyusternik, is contained in [4–6]. (The advantage of the use of Vishlik and Lyusternik's method in comparison with the method of internal and external expansion is discussed in [4].) The effect of injection on the flow has not been considered in the papers listed. In this paper, approximate solutions are constructed with an error of order and ² which take into account the effect of the injectionf on the flow . The approximate solutions are compiled from a more accurate nonviscous flow (external solution) and boundary-layer corrections. The boundary-layer corrections are constructed on a shock wave and a contact boundary in such a way that the solution would be continuous and quite smooth. For the external solution at the contact boundary, conditions are obtained which take into account the effect of viscosity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 69–77, January–February, 1974.     

Swirling-strength based large eddy simulation of turbulent flow around single square cylinder at low Reynolds numbers

In view of the fact that large scale vortices play the substantial role of momentum transport in turbulent flows, large eddy simulation(LES) is considered as a better simulation model. However, the sub-grid scale(SGS) models reported so far have not ascertained under what flow conditions the LES can lapse into the direct numerical simulation. To overcome this discrepancy, this paper develops a swirling strength based the SGS model to properly model the turbulence intermittency, with the primary characteristics that when the local swirling strength is zero, the local sub-grid viscosity will be vanished. In this paper, the model is used to investigate the flow characteristics of zero-incident incompressible turbulent flows around a single square cylinder(SC)at a low Reynolds number range Re ∈ [103, 104]. The flow characteristics investigated include the Reynolds number dependence of lift and drag coefficients, the distributions of time-spanwise averaged variables such as the sub-grid viscosity and the logarithm of Kolmogorov micro-scale to the base of 10 at Re = 2 500 and 104, the contours of spanwise and streamwise vorticity components at t = 170. It is revealed that the peak value of sub-grid viscosity ratio and its root mean square(RMS) values grow with the Reynolds number. The dissipation rate of turbulent kinetic energy is larger near the SC solid walls.The instantaneous factor of swirling strength intermittency(FSI) exhibits some laminated structure involved with vortex shedding.     

Stability analysis of loaded coaxial cylindrical shells with internal fluid flow

We present numerical results for dynamical stability of loaded coaxial shells of revolution interacting with the internal fluid flow. The motion of the incompressible fluid is described in the framework of the theory of frictionless potential flow, whereas the static load acting on the shells is caused by the steady forces of viscous drag arising in the viscous turbulent flow in a closed channel. For shells with different boundary conditions, we study how the stability boundary is affected by the value of the gap between the shells for different versions of the outer shell rigidity and fluid flow. We show that, as in the case of unloaded coaxial shells, there is a significant deviation from the previous numerical and analytical results.     

The Hall effect in an electrically conducting fluid for variable magnetic field and high magnetic Reynolds numbers

Hall phenomena in an electrically conducting fluid with a variable magnetic field were considered in [1]. In that paper the basic characteristics of the above-mentioned phenomena are determined, with certain unimportant constraints, for the case of fluid motion along a channel of rectangular cross section in a traveling magnetic field. The magnetic Reynolds number was assumed to be small, and a solution was given for the induction field in the form of a series in powers of the indicated parameter. Quantitative estimates based on the data of [1] are impossible in the case of relatively high electrical conductivity of the fluid, although certain conclusions of a qualitative nature remain valid. There is thus reason to consider the case of high magnetic Reynolds numbers. This will also allow a fuller picture of the characteristic Hall effect phenomena to be constructed for a variable magnetic field.