Freie Konvektion in einem flachen Behälter mit und ohne Rotation

For a horizontal water layer with isothermal vertical end walls keept at a constant temperature difference of 0 ? ΔT ? 50 K and rotating arround its vertical axis, the temperature and velocity field was experimentally investigated at angular velocities of Ω=0, 0.934, 1.321 and 1.868 1/s. The test facilities are discribed. The flow patterns found experimentally can be arranged in a Gr/Ta² ?Ta² diagram. For Ω=0 a stability analysis results in the prediction of the velocity profiles and their dependence on the temperature difference. For Ω ≠ 0 a stability analysis leads to qualitative results.     

Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus

This paper presents results obtained by employing a modified Galerkin finite element method to analyse the steady state flow of a fluid contained between two concentric, rotating spheres. The spheres are assumed to be rigid and the cavity region between the spheres is filled with an incompressible, viscous, Newtonian fluid. The inner sphere is constrained to rotate about a vertical axis with a prescribed angular velocity, while the outer sphere is fixed. Results for the circumferential function Ω, streamfunction ψ, vorticity function ζ and inner boundary torque T₁ are presented for Reynolds numbers Re ? 2000 and radius ratios 0.1 ? α ? 0.9. The method proved effective for obtaining results for a wide range of radius ratios (0.1 ? α ? 0.9) and Reynolds numbers (0 ? Re ? 2000). Previous investigators who employed the finite difference method experienced difficulties in obtaining results for cases with radius ratios α ? 0.2, except for small Reynolds numbers (Re ? 100). Results for Ω, Ψ, ζ and T₁ obtained in this study for radius ratios 0.8 ≤ α ≤ 0.9 verified the development of Taylor vortices reported by other investigators. The research indicates that the method may be useful for analysing other non-linear fluid flow problems.     

Numerical studies of slow viscous rotating flow past a sphere—1

The Navier-Stokes equations for a steady, viscous rotating fluid, rotating about the z-axis with angular velocity ω are linearized using the Stokes approximation. The linearized Navier-Stokes equations governing the axisymmetric flow can be written as three coupled partial differential equations for the stream function, vorticity and rotational velocity components. One parameter, R_eω = 2ωa²/v, enters the resulting equations. For R_eω « 1, the coupled equations are solved by the Peaceman-Rachford A.D.I. (Alternating Direction Implicit) method and the resulting algebraic equations are solved by the ‘method of sweeps’. Stream lines for ψ = 0·05, 0·2, 0·5 and magnitude of the vorticity vector z = 0·2 are plotted for R_eω = 0·1, 0·3, 0·5. Correction to the Stokes drag due to the rotation of fluid is calculated.     

Drift of spheres in a rotating fluid

The drift of spheres in a rotating fluid is investigated. The problem is studied experimentally and numerically using the Galerkin method. It is shown that for small angular velocities of the fluid Ω the drift velocity of the spheres is almost independent of Ω, but once a certain threshold value Ω_* is attained the drift velocity rapidly decreases. The experimental dependence of the translational velocity of the sphere on the fluid angular velocity is explained on the basis of a theoretical analysis.     

Viscous slow-down motion between coaxial cylinders

An analysis of the viscous tangential motion of a Newtonian fluid in the gap between coaxial cylinders is given for the particular situation in which the inner cylinder, rotating initially with a constant angular velocity, is suddenly left to move with its own inertia and viscous retarding couple alone. The exact solution for the transient angular velocity distribution in the annular space is given in terms of a rapidly converging series in which the use of the sole first term seems to be of sufficient accuracy for most practical calculations. For discrete values of the relevant parameters of the problem, the eigenvaluesα _n<41 were obtained numerically on a digital computer.     

Dependence of the aerodynamic forces on the Reynolds number with the three-dimensional flow of a vortical stream around bodies

The article discusses the dependence of the viscous stresses on the Reynolds number Re in three-dimensional flows around bodies of arbitrary form. It is shown that, with an infinite growth of a vortex with the approach to a body, singular terms appear in an asymptotic expansion in terms of ε=Re^?1/2. The infinite values of the derivatives of the velocity in flows of an incompressible liquid are due to the initial vorticity; in a supersonic flow, they can be connected with the absence of a maximum of the entropy at the critical flow line behind a leading shock wave. The singularity in the tangential stresses brings about the appearance of additional terms in the total aerodynamic forces and moments acting on the body.     

Three-dimensional gas flow in a rotating cylinder with a retarding cover

The study of rotating flows is of interest due to both the development of the centrifugal method of separation of gas and isotope mixtures and the possibility of astrophysical applications. An analytical nonlinear model for calculating the hydrodynamic characteristics of the viscous incompressible fluid flow in a rotating cylinder in the presence of a retarding cover is presented. The cases of stationary and rotating covers are considered. The analysis is performed on the basis of the system of hydrodynamic Navier-Stokes equations. The flow domain is divided up into the main flow and end boundary layers at the cylinder bottom and at the rotating cover. In its turn, the main flow is divided up into an inviscid quasi-rigid core and a lateral layer within which almost the entire upward circulatory flow is concentrated. The equations of the boundary layers at the end surfaces are analyzed by the approximate Slezkin-Targ method. The solutions in the boundary and lateral layers are “stitched” together with the velocity distribution in the main flow core. The unknown angular velocity ω ₁ and radial boundary R ₁ of the core are determined from the balance of the moments of the friction forces acting on the main rotating flow and the continuity condition for the circulatory flow. The experimental and calculated data are compared.     

CONTROL OF FLOW PAST BLUFF BODIES USING ROTATING CONTROL CYLINDERS

Computational results for control of flow past a circular cylinder using small rotating cylinders are presented. A well-proven stabilized finite-element method, that has been applied to various flow problems earlier, is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The formulation is first applied to study flow past an isolated rotating cylinder. Excellent match with experimental results, reported earlier, is observed. It is found that in purely two-dimensional flows, very high lift coefficients can be realized. However, it is observed, via three-dimensional Navier–Stokes simulations, that the end-effects and centrifugal instabilities along the cylinder span lead to a loss of lift and increase in drag. The aspect ratio of the cylinder plays an important role. The flow past a bluff body with two rotating control cylinders is studied using 2-D numerical simulations. The effect of the Reynolds number is studied by carrying out simulations for Re=10²and 10⁴. Finite element meshes with an adequate number of grid points are employed to resolve the flow in the gap between the main and control cylinders. Two values of the gap are considered: 0·01D and 0·075 D, where D is the diameter of the main cylinder. It is observed that when the control cylinders rotate at high speed, such that the tip speed is 5 times the free-stream speed, the flow at Re=100 achieves a steady state. For Re=10⁴, even though the flow remains unsteady, the wake is highly organized and narrower compared to the one without control. The results are in good agreement with the flow-visualization studies conducted by other researchers for bluff bodies using similar control concepts. In all the cases, a significant reduction in the overall drag coefficient and the unsteady aerodynamic forces acting on the main cylinder is observed. Results are also presented for the power requirements of the system for translation and rotation. It is found that the coefficient of power required for the rotation of control cylinders is significant for Re=100 but negligible for Re=10⁴flow. The size of the gap is found to be more critical for the Re=10⁴flows. This study brings out the relevance of the gap as a design parameter for such flow control devices.     

Near-wall hot-wire measurements

This work continues the studies of Khoo et al. (Exp. Fluids 29: 448–460, 2001), where experiments were performed in turbulent-channel and flat-plate boundary-layer flows using near-wall hot-wire probes. The probability density function (pdf) of the wall-shear stress and streamwise velocity fluctuations in the viscous sublayer, buffer region and beyond were compared and analyzed. The convective velocity U _c of the streamwise velocity fluctuations in the very near-wall region was obtained using a two-point correlation technique. It was found that in the viscous sublayer, U _c is approximately constant at 13u _τ and 15u _τ , respectively, for the channel and boundary-layer flows. Spectra data for the viscous sublayer are presented for the first time, and the normalized spectral plots for different flow conditions collapse at high frequencies or wavenumbers, thus indicating the possible presence of small-scale universality at different Reynolds numbers. The integral time scale corresponding to the streamwise velocity fluctuations in the viscous sublayer is also presented. Received: 18 October 2000/Accepted: 2 April 2001     

Fluid trapping by two partially shrouded rotating cylinders

In this paper we examine the prospect of using localized flow control for biomimetic fluid trapping. The problem is of interest for applications that call for guided transport of fluid volumes.The study shows that trapping can be achieved with the help of two partially shrouded rotating cylinders in a side-by-side arrangement. Secondary flows that manifest successful trapping resemble recirculation zones forming under the crests of peristaltic deformation waves, in particular with respect to their response to increasing incident flow velocity. Varying the rotation speed of the cylinders provides means to control the amount of trapped fluid.Numerical calculations to support these conclusions are presented in the paper for 0≤Re≤100 and h≈2, where Re and h are, respectively, the Reynolds number and the center-to-center distance between two cylinders divided by the cylinder diameter. Experimental validation of numerical results is performed for 0≤Re≤4.     

Reynolds number, thickness and camber effects on flapping airfoil propulsion

The effect of varying airfoil thickness and camber on plunging and combined pitching and plunging airfoil propulsion at Reynolds number Re=200, 2000, 20 000 and 2×10⁶ was studied by numerical simulations for fully laminar and fully turbulent flow regimes. The thickness study was performed on 2-D NACA symmetric airfoils with 6-50% thick sections undergoing pure plunging motion at reduced frequency k=2 and amplitudes h=0.25 and 0.5, and for combined pitching and plunging motion at k=2, h=0.5, phase ?=90°, pitch angle θ_o=15° and 30° and the pitch axis was located at 1/3 of chord from leading edge. At Re=200 for motions where positive thrust is generated, thin airfoils outperform thick airfoils. At higher Re significant gains could be achieved both in thrust generation and propulsive efficiency by using a thicker airfoil section for plunging and combined motion with low pitch amplitude. The camber study was performed on 2-D NACA airfoils with varying camber locations undergoing pure plunging motion at k=2, h=0.5 and Re=20 000. Little variation in thrust performance was found with camber. The underlying physics behind the alteration in propulsive performance between low and high Reynolds numbers has been explored by comparing viscous Navier-Stokes and inviscid panel method results. The role of leading edge vortices was found to be key to the observed performance variation.     

Effect of latitudinal temperature gradient on spherical Couette flow

The motion of a viscous thermally stratified liquid in a spherical layer due to rotation of the boundaries of the layer at different angular velocities with nonuniform (latitudinal) heating of the outer boundary is investigated. The investigated problem can be regarded as a very simple qualitative model of some astrophysical and geophysical effects, e.g., flows in planetary atmospheres. The characteristic parameters of the problem are the Reynolds number Re=r_i ²_i/, the ratios of the angular velocities of rotation =_e/_i and of the radiia=r_e/r_i of the outer and inner spheres, and the heating nonuniformity parameter . The Boussinesq approximation is adopted in the investigation: It is assumed that the Reynolds numbers Re are fairly small and the solution is sought in the form of a series of whole positive powers of Re. The first two terms of the series are found in analytical form for arbitrary values of the other characteristic parameters. Possible types of meridional flows are established and the regions of the parameters in which a particular type of circulation takes place are determined. It is shown that a latitudinal temperature gradient on the outer boundary leads to the appearance of new (in comparison with the isothermal case) types of meridional flows. The asymptotic form of the stream function of meridional flow in very thin and very thick layers is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 15–23, March–April, 1975.     

Numerical studies of slow viscous rotating flow past a sphere. III

The flow of steady incompressible viscous fluid rotating about the z-axis with angular velocity ω and moving with velocity u past a sphere of radius a which is kept fixed at the origin is investigated by means of a numerical method for small values of the Reynolds number Re_ω. The Navier–Stokes equations governing the axisymmetric flow can be written as three coupled non-linear partial differential equations for the streamfunction, vorticity and rotational velocity component. Central differences are applied to the partial differential equations for solution by the Peaceman–Rachford ADI method, and the resulting algebraic equations are solved by the ‘method of sweeps’. The results obtained by solving the non-linear partial differential equations are compared with the results obtained by linearizing the equations for very small values of Re_ω. Streamlines are plotted for Ψ = 0·05, 0·2, 0·5 for both linear and non-linear cases. The magnitude of the vorticity vector near the body, i.e. at z = 0·2, is plotted for Re_ω = 0·05, 0·24, 0·5. The correction to the Stokes drag as a result of rotation of the fluid is calculated.     

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.     

Stationärer Stoff- und Wärmeübergang an stationär quer angeströmten Zylindern

The steady forced convection mass and heat transfer from circular cylinders has been investigated. The full mass transport differential equation has been integrated numerically. The employed velocity distributions are known [1]. The most important result is reproduced in a correlation for the mass transfer, which regards the turbulence intensity in the flow of the cylinders. This mass transfer law is proofed theoretically and experimentally in the range of Schmidt numbers from Sc=0.73 up to S=3.3×10⁴; however it is valid for 0≤Sc∞. It can be used for all values of Re Sc greater than Re Sc=7.3×10^?5 and for all values of the Reynolds number less than the critical value, Re_kr. The critical Reynolds number, Re_kr, is a known function of the turbulence intensity [1]. For values of Re Sc less than Re Sc=7.3 x10^?5 the mass transfer can be predicted by an analytical equation that based on Oseen type linearization of the differential equation. The conditions are illustrated, which allow to calculate the quantities for heat transfer by means of the correlations for the mass transfer.     

Numerical studies of slow viscous rotating flow past a sphere. II

The Navier–Stokes equations, which are the governing equations for a steady, viscous, incompressible fluid rotating about the z-axis with angular velocity ω, are linearized using the Oseen approximation. Two parameters, namely the Reynolds number Re = Ua/v and Re_ω = 2ωa²/v (the Reynolds number w.r.t. rotation), enter the linearized equations. These equations are solved by the Peaceman–Rachford ADI method and the resulting algebraic equations are solved by the SOR method. Streamlines are plotted and compared with the Oseen solution for the non-rotating case. The magnitude of the vorticity vector with increasing θ is also plotted.     

Influence of the aspect ratio on boiling flows in rectangular mini-channels

This article presents experiments conducted with two single rectangular mini-channels of same hydraulic diameter (1.4 mm) and different aspect ratios for conditions of horizontal boiling flow. The Forane® 365 HX used was subcooled (ΔT_sub = 15 °C) for all the boiling curves presented in the paper. Local heat transfer coefficients were measured for heat flux ranging from 25 to 62 kW m⁻² and mass flux from 200 kg m⁻² s⁻¹ to 400 kg m⁻² s⁻¹. The boiling flows were observed with two different cameras (depending on the flow velocity) through a visualization window. The flow patterns in the two channels were compared for similar conditions. The results show that the boiling heat transfer coefficient and the pressure drop values are different for the two single mini-channels. For low heat flux condition, the channel with lowest aspect ratio (H/W = 0.143) has a higher heat transfer coefficient. On the other hand, for high heat flux condition, the opposite situation occurs, namely the heat transfer coefficient becomes higher for the channel with highest aspect ratio (H/W = 0.43). This is probably due to the earlier onset of dryout in the channel with lowest aspect ratio. For the two cases of heating, the pressure drop for the two-phase flow remains lower for the channel with lowest aspect ratio. These results show that the aspect ratio plays a substantial role for boiling flows in rectangular channels. As for single-phase flows, the heat transfer characteristics are significantly influenced (even though the hydraulic diameter remains the same) by this parameter.     

Flows in the initial sections of channels with permeable walls

Calculations of two types of flows in the initial sections of channels with permeable walls are carried out on the basis of semiempirical turbulence theories during fluid injection only through the walls and during interaction of the external flow with the injected fluid. Experimental studies of the first type [1–3] show that at least within the limits of the lengths L/h<30 and L/a< 50 (2h is the distance between permeable walls of a flat channel anda is the tube radius) the velocity distributions in the laminar and turbulent flow regimes differ little and are nearly self-similar for solutions obtained in [4]. For sufficiently large Reynolds numbers, Re₀>100 (Re₀=v₀h/ or Re₀=v₀ a/, where v₀ is the injection velocity), and small fluid compressibility, the axial velocity component is described by the relations for ideal eddying motion: u=(/2)x× cos (y/2) in a flat channel and u=x cos (y²/2) in atube (the characteristic values for the coordinates are, respectively, h anda). Measurements indicate the existence of a segment of laminar flow; its length depends on the Reynolds number of the injection [3]. In the turbulent regime the maximum generation of turbulent energy occurs significantly farther from the wall than in parallel flow. Flows of the second type in tubes were studied in [5–7]. These studies disclosed that for Reynolds numbers of the flow at the entrance to the porous part of the tube Re=u₀ a/<3.10³ fluid injection with v₀/u₀>0.01 leads to suppression of turbu lence in the initial section of the tube. An analogous phenomenon was observed in the boundary layer with v₀/u₀>0.023 [8, 9]. Laminar-turbulent transition in flows with injection was explained in [10, 11] on the basis of hydrodynamic instability theory, taking into account the non-parallel character of these flows. The mechanisms for the development of turbulence and reverse transition in channels with permeable walls are not theoretically explained. Simple semiempirical turbulence theories apparently are insufficient for this purpose. In the present work results are given of calculations with two-parameter turbulence models proposed in [12, 13] for describing complex flows. Due to the sharp changes of turbulent energy along the channel length, a numerical solution of the complete system of equations of motion was carried out by the finite-difference method [14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 43–48, September–October, 1976.     

Two- and three-dimensional laminar flows between disks co-rotating in a fixed cylindrical enclosure

A numerical investigation is performed for the constant property laminar flow of air in the space between a pair of disks clamped co-axially on a central hub and co-rotating in a stationary cylindrical enclosure. Both two- and three-dimensional flow conditions are examined in relation to the interdisk spacing, H, and the disk angular velocity, Ω. Two interdisk spacings are considered, corresponding to aspect ratios Γ = 0.186 and 0.279 (with Γ = H/(R₂+a−R), where R₂ is the disk radius, a is the disk rim–enclosure wall clearance, and R is the hub radius). A range of rotational speeds encompassing the transition from axisymmetric two-dimensional steady flow to non-axisymmetric three-dimensional unsteady flow are considered for various values of the Reynolds number, Re (with $ Re=\Omega R_2^2/v $, where v is the kinematic viscosity of air). Axisymmetric calculations are first performed for both aspect ratios in the range 3858≤Re≤23 150. Fully three-dimensional calculations are then performed for the configuration with Γ = 0.186 and Re = 23 150, and for the configuration with Γ = 0.279 and Re = 7715, 15 430 and 23 150. The axisymmetric calculations performed with Γ = 0.186 confirm many known features of the flow, including the transition from a steady flow to an oscillatory periodic regime. This occurs at ≈Re = 23 150 for a configuration with a/H = 0, and at ≈Re = 14 670 for one with a/H = 0.28 and a finite disk thickness (b/H = 0.2). Three-dimensional calculations performed for Γ = 0.186 with a/H = 0 and Re = 23 150 reveal a circumferentially periodic flow pattern with eight foci of intensified axial component of vorticity. The axisymmetric calculations performed with Γ = 0.279 and Re > 7715 yield a novel, non-unique steady solution for the velocity field that is asymmetric with respect to the interdisk mid-plane. No experimental verification of this finding exists to date, but similar situations are known to arise in the context of anomalous modes of the Taylor–Couette flow. Relaxing the axisymmetry constraint allows this flow to evolve to an oscillatory three-dimensional regime of increasing irregularity with increasing rotational speed. In this case, the number of foci of intensified axial vorticity varies with time, ranging from six at Re = 7715 to between six and eight at Re = 23 150. © 1998 John Wiley & Sons, Ltd.     

Direct Numerical Simulation of Transitional Turbulent Flow in a Closed Rotor-Stator Cavity

The transitional turbulent regime in confined flow between a rotating and a stationary disc is studied using direct numerical simulation. Besides its fundamental importance as a three-dimensional prototype flow, such flows frequently arise in many industrial devices, especially in turbomachinary applications. The present contribution extends the DNS simulation into the turbulent flow regime, to a rotational Reynolds number Re =3 × 10⁵. An annular rotor-stator cavity of radial extension ΔR and height H, is considered with L = 4.72(L = ΔR/H) and Rm = 2.33 (Rm = (R ₁+ R ₀)/ΔR). The direct numerical simulation is performed by integrating the time-dependent Navier–Stokes equations until a statistically steady state is reached. A three-dimensional spectral method is used with the aim of providing both very accurate instantaneous fields and reliable statistical data. The instantaneous quantities are analysed in order to enhance our knowledge of the physics of turbulent rotating flows. Also, the results have been averaged so as to provide target turbulence data for any subsequent modelling attempts at reproducing the flow. This revised version was published online in July 2006 with corrections to the Cover Date.