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.     

The unsteady drag on a spherical bubble at large Reynolds numbers

The flow past a spherical bubble undergoing a rectilinear motion in the unsteady flow of an unbounded liquid medium is investigated. The liquid velocity field at infinity is assumed to be uniform and the Reynolds number to be large. The Strouhal number is taken to be of order unity. The velocity distribution is sought by superposition of a perturbation field on the potential flow past the bubble so that the flow field is divided into four regions, i.e. the external flow field where the potential flow holds, the boundary layer, the rear stagnation point region and the wake. The flow in the rear stagnation point region and the wake is assumed to be essentially inertial. The unsteady drag experienced by the bubble is calculated from the mechanical energy balance of the liquid.     

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.     

EHD flows at large electric Reynolds numbers

The motion of a medium consisting of neutral particles and charged particles of single sign is studied under the assumption that the electric Reynolds number (R_q=u/bE) is large. We calculate the freezingin integral and the Bernoulli and Cauchy-Lagrange integrals, study the fluid motion in a stream tube, and formulate the boundary layer problem.     

LES of incompressible heat and fluid flows past a square cylinder at high Reynolds numbers

This paper presents large eddy simulation (LES) results of incompressible heat and fluid flows around a square cylinder (SC) at zero incident angle at high Reynolds numbers (Re) in the range from 1.25×10⁵ to 3.5×10⁵. LES results are obtained on the basis of swirling strength based sub-grid model, and a higher order upwind scheme developed with respect to the Taylor expansion. It was found that, for the zero incident SC wake flows at a Reynolds number in the range {Re₅ = Re/10⁵ ∈ [1.25, 3.5]}, the Strouhal number equals to 0.1079, completely independent of the Reynolds number; the coefficient of drag is around 1.835 with an uncertainty of about 1.9%, almost non-sensitive to the Re. When Re is beyond 3.0×10⁵, the time-averaged peak value of sub-grid viscosity is over 340, implying that the role of sub-grid model is crucial in some regions where vortex motion is active and vortex interaction is intense. The time–spanwise (t-z) averaged sub-grid viscosity ratio profiles and the profiles of fluctuations of the sub-grid viscosity ratio and velocity components at four locations downstream of the SC are presented. The fields of the t-z averaged sub-grid viscosity ratio, and the instantaneous fields of streamwise and spanwise vorticities are also reported and discussed. The predicted mean Nusselt number is compared with empirical correlations, revealing that swirling strength based LES has its potential in predicting natural and industrial flows.     

Hydrodynamic drag of an ellipsoidal droplet at small Reynolds numbers

At present, there is an absence of the accurate data on the influence of the shape of a droplet on its hydrodynamic drag and mass transfer without which the design of mass transfer apparatus is impossible [1–3]. Most often it is assumed that the drag of an ellipsoidal liquid droplet as it moves along the axis of symmetry is determined by the product of the drag of a spherical liquid droplet and a coefficient which takes into account the shape and is determined from the drag of a solid ellipsoid for which the exact solutions are known. It is shown below that this assumption is not always valid.Translated from Tzvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–8, May–June, 1987.     

Stability of viscoelastic shear flows in the limit of high Weissenberg and Reynolds numbers

We consider a limit of the upper convected Maxwell model where both the Weissenberg and Reynolds numbers are large. The limiting equations have a status analogous to that of the Euler equations for the high Reynolds number limit. These equations admit parallel shear flows with an arbitrary profile of velocity and normal stress. We consider the stability of these flows. An extension of Howard’s semicircle theorem can be used to show that the flow is stabilized if elastic effects are sufficiently strong. We also show how to analyze the long wave limit in a fashion similar to the inviscid case.     

Three-dimensionality of grooved channel flows at intermediate Reynolds numbers

 This paper describes the three-dimensional flow structure in grooved channels with different cavity lengths at intermediate Reynolds numbers. For steady flow, the three-dimensional effects are dominant near the side walls of the channel. However, after the onset of self-sustained oscillatory flow due to Tollmien–Schlichting waves as the primary instability, a secondary instability produces a three-dimensional flow with Taylor–Geortler-like vortical structure, at the bottom of the groove. This trend becomes more significant as the cavity length increases. Furthermore, the reason for three-dimensional flow is discussed using additional numerical analysis, and it is confirmed that the source of three-dimensional instability is the groove vortices due to the presence of side walls, rather than the channel traveling wave. Received: 7 September 1999/Accepted: 11 November 2000     

Annular liquid jets at high Reynolds numbers

The flow of annular liquid jets at high Reynolds numbers is analysed by means of the finite element method and the full‐Newton iteration scheme. Results have been obtained for various values of the inner to the outer diameter ratio and for non‐zero surface tension, using extremely long meshes. The annular film moves far from the symmetry axis at low values of the Reynolds number. At higher Reynolds numbers, the film moves towards the axis of symmetry and appears close to very far downstream, forming a round jet. Asymptotic results for the radius of the resulting round jet are provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Control of vortex shedding behind circular cylinder for flows at low Reynolds numbers

It has been observed by researchers in the past that vortex shedding behind circular cylinders can be altered, and in some cases suppressed, over a limited range of Reynolds numbers by proper placement of a second, much smaller, ‘control’ cylinder in the near wake of the main cylinder. Results are presented for numerical computations of some such situations. A stabilized finite element method is employed to solve the incompressible Navier–Stokes equations in the primitive variables formulation. At low Reynolds numbers, for certain relative positions of the main and control cylinder, the vortex shedding from the main cylinder is completely suppressed. Excellent agreement is observed between the present computations and experimental findings of other researchers. In an effort to explain the mechanism of control of vortex shedding, the streamwise variation of the pressure coefficient close to the shear layer of the main cylinder is compared for various cases, with and without the control cylinder. In the cases where the vortex shedding is suppressed, it is observed that the control cylinder provides a local favorable pressure gradient in the wake region, thereby stabilizing the shear layer locally. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear aspects of high Reynolds number channel flows

This paper considers the flow in a two-dimensional channel at high Reynolds number, with wall deformations which can lead to flow separation. An asymptotic model is proposed by using the successive complementary expansion method with generalized asymptotic expansions. In particular, the model emphasizes the asymmetry of the channel geometry by introducing a change of variables. It is shown that the model is more general than the models developed with the method of matched asymptotic expansions. Comparisons with Navier–Stokes solutions show that the model is well founded and enables us to treat original problems.     

Numerical continuation of high Reynolds number external flows

A numerical continuation method for the compressible Reynolds‐Averaged Navier–Stokes equation with the Spalart–Allmaras turbulence model is presented and applied to the flow around a 2D airfoil. Using continuation methods it is possible to study the steady flow states of a system as a parameter such as angle of attack is varied. This approach allows unstable solutions to be calculated, which are important for understanding the nonlinear dynamics of the system. Furthermore, this method can be used to find any multivalued solutions that exist at a single parameter value. The eigenvalues of the system are calculated using the Cayley transform to precondition the eigenvalue solver ARPACK. The eigenvalues are important as they show the stability of the solutions as well as accurately detect parameter values at which bifurcations take place. Copyright © 2010 John Wiley & Sons, Ltd.     

Deconvolution‐based nonlinear filtering for incompressible flows at moderately large Reynolds numbers

We consider a Leray model with a deconvolution‐based indicator function for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under‐resolved meshes. For the implementation of the model, we adopt a three‐step algorithm called evolve–filter–relax that requires (i) the solution of a Navier–Stokes problem, (ii) the solution of a Stokes‐like problem to filter the Navier–Stokes velocity field, and (iii) a final relaxation step. We take advantage of a reformulation of the evolve–filter–relax algorithm as an operator‐splitting method to analyze the impact of the filter on the final solution versus a direct simulation of the Navier–Stokes equations. In addition, we provide some direction for tuning the parameters involved in the model based on physical and numerical arguments. Our approach is validated against experimental data for fluid flow in an idealized medical device (consisting of a conical convergent, a narrow throat, and a sudden expansion, as recommended by the U.S. Food and Drug Administration). Numerical results are in good quantitative agreement with the measured axial components of the velocity and pressures for two different flow rates corresponding to turbulent regimes, even for meshes with a mesh size more than 40 times larger than the smallest turbulent scale. After several numerical experiments, we perform a preliminary sensitivity analysis of the computed solution to the parameters involved in the model. Copyright © 2015 John Wiley & Sons, Ltd.     

Prediction of drag reduction effect by streamwise traveling wave-like wall deformation in turbulent channel flow at practically high Reynolds numbers

A fully developed turbulent channel flow controlled by traveling wave-like wall deformation under a constant pressure gradient condition is studied numerically and theoretically. First, direct numerical simulation (DNS) at three different friction Reynolds numbers, ${\text{Re}}_{\tau }=90,$ 180, and 360, are performed to investigate the modification in turbulence statistics and their scaling. Unlike the previous study assuming a constant flow rate condition, suppression of the quasi-streamwise vortices is not observed in either drag decrease cases or drag increase cases. It is found in the drag reduction case, however, that the periodic component of the Reynolds shear stress (periodic RSS) is largely negative in the viscous sublayer and the buffer layer. For the maximum drag reduction case, the set of control parameters is found to be identical in wall units regardless of the Reynolds number, and the resulting mean velocity profiles are also observed to be approximately similar even with an additional case of ${\text{Re}}_{\tau }=720$. Based on this scaling, we propose a semi-empirical formula for the mean velocity profile modified by the present control. With this formula, about $20\%-25\%$ drag reduction effect is predicted even at practically high Reynolds numbers, ${\text{Re}}_{\tau }\sim {10}^5-{10}^6$.     

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.     

Packet of gravity surface waves at high Reynolds numbers

Within the framework of the Lagrangian approach a method for describing a wave packet on the surface of an infinitely deep, viscous fluid is developed. The case, in which the inverse Reynolds number is of the order of the wave steepness squared is analyzed. The expressions for fluid particle trajectories are determined, accurate to the third power of the steepness. The conditions, under which the packet envelope evolution is described by the nonlinear Schrödinger equation with a dissipative term linear in the amplitude, are determined. The rule, in accordance with which the term of this type can be correctly added in the evolutionary equation of an arbitrary order is formulated.     

Confined impinging twin air jets at high Reynolds numbers

An experimental study is carried out to investigate flow characteristics of confined twin jets issuing from the lower surface and impinging normally on the upper surface. Pressure distributions on the impingement and confinement plates were obtained for Reynolds numbers ranging from 30,000 to 50,000, nozzle-to-plate spacing (H/D) in the range of 0.5-4 and jet-to-jet spacing (L/D) in the range of 0.5-2. Smoke-wire technique was used to visualize the flow behavior. The effects of Reynolds number, nozzle-to-plate spacing and jet-to-jet spacing on the flow structure are examined. The subatmospheric regions occur on both impingement and confinement plates at the nozzle-to-plate spacing up to 1 for all studied Reynolds numbers and jet-to-jet spacings in consideration. They lie nearly up to the same radial location at both surfaces and move radially outward from the stagnation points with increasing nozzle-to-plate spacing and jet-to-jet spacing. It is concluded that there exists a relation between the subatmospheric regions and peaks in heat transfer coefficients for low spacings in the impinging jets.     

Steady flow with separation past a thin airfoil at high Reynolds numbers

The results of calculating the steady separated flow of a viscous incompressible fluid past a 12% sinusoidal airfoil with and without a splitter plate are presented. The Navier—Stokes equations written in stream-function—vorticity variables are approximated in accordance with a central difference scheme and solved simultaneously by means of the direct method. The results obtained are analyzed from the standpoint of their consistency with the asymptotic theory based on the well-known three-deck scheme. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 155–160, March–April, 1994.     

Experimental study of flow past a square cylinder at high Reynolds numbers

 Measurements of two-components of velocity in the wake of a square cylinder using a hot-wire anemometer are reported. Two Reynolds numbers, namely 8700 and 17,625, have been considered. The measurements were carried out in a low-speed, low-turbulence wind tunnel. Benchmark experiments at much lower Reynolds numbers show good agreement between the present experiments and published results. At higher Reynolds numbers, the experimental data reveal anticipated trends in terms of wake recovery and turbulence decay. Both velocity and velocity fluctuations show symmetry about the wake axis. The experimental data have been compared with the large eddy simulation (LES) calculation reported by Wang et al. [University of Illinois at Urbana – Champaign (1996) Report CFD 96-03] and LDV measurements of Lyn et al. [J Fluid Mech (1995) 304: 285–319]. The agreement among the three sets is generally acceptable in terms of the time-averaged velocity components, but not the velocity fluctuations. The turbulence fluctuations in the present experiments are seen to be lower than in the referred work. The differences have been traced to factors such as the aspect ratio, blockage ratio and upstream turbulence. Experiments with increased upstream turbulence did show a reduction in the discrepancy between the present experiments and the published data. An assessment of the experimental data in terms of physical mechanisms revealed that (a) streamwise normal stresses were correlated with the vortex centers, and (b) the turbulence kinetic energy profiles are similar to the turbulence shear stress. Spectral analysis of the velocity signals was carried out in the present work. Energy transfer from the mean flow to the streamwise velocity fluctuation was confirmed in the near wake. A redistribution of the kinetic energy between the streamwise and transverse components of velocity over a longer distance downstream was subsequently observed. Received: 17 May 1999/Accepted: 29 December 1999