Flow in a cylinder driven by rotating split-disk endwalls

Flow of an incompressible viscous fluid contained in a cylindrical vessel (radius R, height H) is considered. Each of the cylinder endwalls is split into two parts which rotate steadily about the central axis with different rotation rates: the inner disk (r < r₁) rotating at Ω₁, and the outer annulus (r₁ < r < R) rotating at Ω₂. Numerical solutions to the axisymmetric Navier-Stokes equations are secured for small system Ekman numbers E ( v/(ΩH²)). In the linear regime, when the Rossby number Ro

Full-size image (1K)

, the numerical results are shown to be compatible with the theoretical prediction as well as the available experimental measurements. Emphasis is placed on the results in the nonlinear regime in which Ro is finite. Details of the structures of azimuthai and meridional flows are presented by the numerical results. For a fixed Ekman number, the gross features of the flow remain qualitatively unchanged as Ro increases. The meridional flows are characterized by two circulation cells. The shear layer is a region of intense axial flow toward the endwall and of vanishing radial velocity. The thicknesses of the shear layer near r = r₁ and the Ekman layer on the endwall scale with E and E , respectively. The numerical results are consistent with these scalings.     

Structure of modulated wavy vortical flows in the circular Couette system

Results of experimental studies are presented on the evolution of the circular Couette flow with the outer cylinder at rest. In particular, several modes of modulated wavy vortical flows have been identified in a consistent way both from flow visualization and from signals of scattered laser beam. The latter technique has been extended to measure the spatial correlation between fluctuations as a function of the azimuthal separation of the points. It has been found that the two-point angular correlation reflects the pattern of modulation, and falls off as (R_d — R)^σ when the ‘disorder Reynolds number’ R_d is approached. The value of the exponent σ has been estimated to be about 1/3 in the experiment.     

Flow measurements inside a heated multiple rotating cavity with axial throughflow

This paper discusses experimental results from a multiple cavity test rig representative of a high pressure compressor internal air system. Measurements of the axial, tangential and radial velocity components are presented. These were made using a two component, laser doppler anemometry (LDA) system for a range of non-dimensional parameters representative of engine conditions (Re up to 4 × 10⁶ and Re_z up to 1.8 × 10⁵). Tests were carried out for two different sizes of annular gap between the (non-rotating) drive shaft and the disc bores.

The axial and radial velocities inside the cavities are virtually zero. The size of the annular gap between disc bore and shaft has a significant effect on the radial distribution of tangential velocity. For the narrow annular gap (d_h/b = 0.092), there is an increase of non-dimensional tangential velocity V/Ωr with radial location from V/Ωr < 1 at the lower radii to solid body rotation V/Ωr = 1 further into the cavity. For the wider annular gap (d_h/b = 0.164), there is a decrease from V/Ωr > 1 at the lower radii to solid body rotation further into the cavity. An analysis of the frequency spectrum obtained from the tangential velocity measurements is consistent with a flow structure in the r– plane consisting of pairs of contra rotating vortices.     

Further consideration of the shape-factor relationship for turbulent boundary layers

The lag-entrainment predictive scheme developed by Green et al. has been modified to include the pressure-gradient parameter Π₁. In the original model suggested by Green et al. the mass-flow shape factor H₁ is related to the common shape factor H, H₁ = f(H). In the present model H₁ is related to H, Reynolds number based on the local momentum thickness θ, and Π₁; thus H₁ = f(H, R_eθ, Π₁). The modified formula for H₁, is introduced into the original lag-entrainment integral model. Calculations are made to examine the present model for the predictions of the development of boundary layers approaching separation studied experimentally by the authors. Slightly improved predictions are obtained using the model developed by El Telbany et al. However, the present model proved to give an improved representation of the development of wall shear stress in cases the two-equation turbulence model proved to be unsuccessful.     

Effect of tube diameter on Preston tube calibration curves for the measurement of wall shear stress

The effect of tube diameter (d) on Preston tube calibration curves for the measurement of wall shear stress (τ_w) in a zero pressure gradient turbulent boundary layer has been investigated. Five different outside diameter tubes of 1.46, 1.82, 3.23, 4.76 and 5.54 mm, corresponding to (d/δ) of 0.022, 0.027, 0.048, 0.071 and 0.082 were used to measure τ_w at Reynolds numbers based on momentum thickness (R_θ) of 2800–4100. The calibration curves of Patel (V.C. Patel, J. Fluid Mech. 23 (part I) (1965) 185–208) and Bechert (D.W. Bechert, AIAA J. 34 (1) (1995) 205–206) are both dependent on the tube diameter. The maximum difference in the τ_w measurements from the different tubes using Patel's calibration is about 8%, while Bechert's calibration gives a maximum difference of approximately 18%.     

Inertial instability of the Stewartson -layer

Inertial stability of a vertical shear layer (Stewartson E^1/4-layer) on the sidewall of a cylindrical tank with respect to stationary axisymmetric perturbations is inverstigated by means of a linear theory. The stability is determined by two non-dimensional parameters, the Rossby number Ro = U/2ΩL and Ekman number E = v/ΩH², where U and L = (E/4)1/4H are the characteristic velocity and width of the shear layer, respectively, Ω the angular velocity of the basic rotation, v the kinematic viscosity and H the depth of the tank.

For a given Ekman number, the flow is more unstable for larger values of the Rossby number. For E = 10⁻⁴, which is a typical value of the Ekman number realized in rotating tank experiments, the critical Rossby number Ro_c for instability and the critical axial wavenumber m_c non-dimensionalized by L⁻¹ are found to be 1.3670 and 8.97, respectively. The value of Ro_c increases and that of m_c decreases with increasing E.     

Fluid flow in a rotating curved rectangular duct

The fluid flowing in a rotating curved duct is subjected to both the Coriolis force due to a rotation and the centrifugal force due to a curvature. In this paper, the combined effects of the two forces on the flows in rotating curved rectangular ducts are examined numerically. According to the aspect ratio of the cross-section, the rectangular ducts are divided into three types: η>1, η=1, η<1, where η is the aspect ratio. The variations of the flow structures with the force ratio F (the ratio of the Corislis force to the centrifugal force) are studied in detail and many hitherto unknown flow patterns are found. The effects of the force ratio and the aspect ratio of the cross-section on the friction factor are also examined. Present results show both the characteristics of the secondary flow, axial flow and the natures of the friction factor.     

The instability of rhombic cell flows

Instability of two-dimensional periodic flows with rhombic cell structure represented by the stream function Ψ=cos kx+cosy is investigated. Stability characteristics are obtained for the Reynolds number R=1, 2, 3 and 4 and the ratio of the diagonals of the cell . Variation of the critical Reynolds number R_c with k is obtained, and the square cell flow (k=1) is found to be most stable (R_c=√2). It is found that R_c → 1 as k → 0, which leads to a finite gap between this limiting R_c and R_c=√2 for K=0 (Ψ=cos y).     

Magnetohydrodynamic flow past a porous rotating disk in a circular magnetic field

This paper studies the effects of a circular magnetic field on the flow of a conducting fluid about a porous rotating disk. Using modern quasi-Newton and globally convergent homotopy methods, numerical solutions are obtained for a wide range of magnetic field strengths, suction and injection velocities and Alfven and disk speeds. Results are presented graphically in terms of three non-dimensional parameters. There is excellent agreement with previous work and asymptotic formulae.     

Axisymmetric magnetohydrodynamic flow of Jeffrey fluid over a rotating disk

This paper examines the magnetohydrodynamic boundary layer flow of Jeffrey fluid due to a rotating disk. The governing partial differential equations are first transformed into the coupled system of ordinary differential equations and then solved by using the homotopy analysis method. The influence of various involved physical parameters on the dimensionless radial and azimuthal velocities is sketched and analyzed. The variation of skin friction coefficients in radial and azimuthal directions is studied for various values of pertinent parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Experimental investigation of three-dimensional flow instabilities in a rotating lid-driven cavity

The swirling flow between a rotating lid and a stationary cylinder is studied experimentally. The flow is governed by two parameters: the ratio of container height to disk radius, h, and the Reynolds number, Re, based on the disk angular velocity, cylinder radius and kinematic viscosity of the working liquid. For the first time, the onset of three-dimensional flow behavior is measured by combining the high spatial resolution of particle image velocimetry and the temporal accuracy of laser Doppler anemometry. A detailed mapping of the transition scenario from steady and axisymmetric flow to unsteady and three-dimensional flow is investigated for 1 ≥ h ≥ 3.5. The flow is characterized by the development of azimuthal modes of different wave numbers. A range of different modes is detected and critical Reynolds numbers and associated frequencies are identified. The results are compared to the numerical stability analysis of Gelfgat et al. (J Fluid Mech 438:363–377, 2001). In most cases, the measured onset of three-dimensionality is in good agreement with the numerical results and disagreements can be explained by bifurcations not accounted for by the numerical stability analysis.     

Pressure drop characteristics in a rotating smooth square channel with a sharp 180° bend

The results of an experimental study to investigate the local pressure drop characteristics in a square cross-sectioned smooth channel with a sharp 180° bend rotating about an axis normal to the free-stream direction are reported here. The sharp 180° turn was obtained by dividing a rectangular passage into two channels using a divider wall with a rounded tip at the location where the flow negotiates the turn. The study was carried out for three ratios of divider wall thickness to hydraulic diameter (W/D), namely, 0.24, 0.37 and 0.73 all with a rounded tip divider wall and only for a bend with a W/D ratio of 0.37, the influence of a sharp tip divider wall was studied. Experiments were conducted for a Reynolds number varying from 10 000 to 17 000 with the rotation number (ωD/V) varying from 0 to 0.38. The pressure drop distribution, normalized with the mainstream fluid dynamic pressure head, is presented for the leading, trailing and the outer surfaces. The results indicate that the local pressure drop characteristics in the bend region are significantly affected by a change in the rotation number but the influence of the Reynolds number is weak. The friction factor is less sensitive to rotation for the bend with a W/D ratio of 0.24 when compared to bends with W/D ratios of 0.37 and 0.73. Friction factor correlations are presented which fit the experimental data within 10% for the range of parameters studied.     

Electromagnetic propagation in periodic porous structures

A variational technique is employed to compute approximate propagation constants for electromagnetic waves in a dielectric structure which is periodic in the X−Y plane and translationally invariant in the Z-direction. The fundamental cell, in the periodic structure, is composed of a pore and the surrounding host media. The pore is a circle of radius R₀ filled with a dielectric ε₁ and the host dielectric characterized by ε₂. The size of the cell is characterized by the length A which is R₀.

Two limiting cases are considered. In the first, the pore size is assumed to be much smaller than the wavelength; this limit is motivated by microwave heating of porous material. The approximate propagation constants are explicitly computed for this case and are shown to depend upon the two dielectric constants, the relative areas of the two regions in the cell, and on a modal number. They are not given by a simple mixture formula.

In the second limit, the pore size is taken to be of the same order as the wavelength; this limit is motivated by the propagation of light in a holey fiber. In this case our argument directly yields the dispersion relationship recently derived by Ferrando et al. [Opt. Lett. 24 (1999) 276], using intuitive and physical reasoning. Thus, our method puts theirs into a mathematical framework from which other approximations might be deduced.     

Stress state of compound polygonal plate

The constructions made of bars and plates with holes, openings and bulges of various forms are widely used in modern industry. By loading these structural elements with different efforts, there appears concentration (accumulation) of stress whose values sometimes exceeds the admissible one. The durability of the given element is defined according to the quantity of these stresses. Since the failure of details and construction itself begins from the place where the stress concentration has the greatest value.

Therefore the exact determination of stress distribution in details (bars, plates, beams) is of great scientific and practical interest and is one of the important problems of the solid fracture.

Compound details (when the nucleus of different material is soldered to the hole) are often used to decrease the stress concentration.

In the present paper, we study a stress–strain state of polygonal plate weakened by a central elliptic hole with two linear cracks info which a rigid nucleus (elliptic cylinder with two linear bulges) of different material was put in (soldered) without preload.

The problem is solved by a complex variable functions theory stated in papers [Theory of Elasticity, Higher School, Moscow, 1976, p. 276; Plane Problem of Elasticity Theory of Plates with Holes, Cuts and Inclusions, Publishing House Highest School, Kiev, 1975, p. 228; Bidimensional Problem of Elasticity Theory, Stroyizdat, Moscow, 1991, p. 352; Science, Moscow (1996) 708; MSB AH USSR OTH 9 (1948) 1371].

Kolosov–Mushkelishvili complex potential (z) and ψ(z) satisfying the definite boundary conditions are sought in the form of sums of functional series.

After making several strict mathematical transformations, the problem is reduced to the solution of a system of linear algebraic equations with respect to the coefficients of expansions of functions (z) and ψ(z).

Determining the values of (z) and ψ(z), we can find the stress components σ_r, σ_θ and τ_rθ at any point of cross-section of the plate and nucleus on the basis of the known formulae. The obtained solution is illustrated by numerical example.

Changing the parameters A₁, m₁, e, A₂, and m₂ we can get the various contour plates.

For example, if we assume m₁=0, A₁=r, then the internal contour of L₁ becomes the circle of radius r with two rectilinear cracks (for the nucleus––a rectilinear bulges).

Further, if we assume a small semi-axis of the ellipse b₁ to be equal to zero (b₁=0), then a linear crack becomes the internal contour of L₁ (and the nucleus becomes the linear rigid inclusion made of other material). For m₂=0; A₂=R, the external contour L₂ turns into the circle of radius R.

The obtained method of solution may be applied and in other similar problems of elasticity theory; tension of compound polygonal plate, torsion and bending of compound prismatic beams, etc.     

Consistency,stability and convergence of the finite-difference equations for flow about a rotating sphere in an axial stream

The finite-difference equations which have previously been developed to solve the problem of laminar boundary layer flow about a rotating sphere in an axial stream are analysed according to the available numerical stability theories. This analysis is necessary to determine the restrictions on velocities and mesh sizes required to obtain a convergent numerical solution. Convergence can be achieved if both consistency and stability of the finite-difference equations are fulfilled. The analysis reported in the present paper shows that the developed finite-difference equations are consistent with their original partial differential equations. Also, the analysis proves that the developed finite-difference procedure is numerically stable for all mesh sizes as long as the downstream meridional velocity is non-negative, i.e.as long as no flow reversals occur within the domain of solution.     

Steady laminar flow through thin curved pipes

Let R, τ denote, respectively, the radius of curvature and radius of torsion of the pipe (centre-line) and let a be a typical cross-sectional diameter.

The major part of the present paper addresses the case of flows through pipes of constant cross-section; (Re)²(a/R), Re(a/τ), (a/R) and (a/τ) all being small. Re is the Reynolds number for the flow. It is found that, even without further specifications of the details of the pipe, many important results can be obtained about the secondary flow which occurs and the pressure losses resulting from it. For example, it is shown that an important feature of such flows is valid for any corss-sectional shape; this was not obvious from previous works which treated only special cases having significant symmetries. Also, a new method for calculating the modified axial pressure gradient is presented which reduces dramatically the amount of work required therefor.

The remainder of the paper presents some results for similar flows through pipes of varying cross-section.     

Dynamic response of a rotating disk submerged and confined. Influence of the axial gap

In this paper, the natural frequencies and mode shapes of a rotating disk submerged and totally confined inside a rigid casing, have been obtained. These have been calculated analytically, numerically and experimentally for different axial gaps disk-casing. A simplified analytical model to analyse the dynamic response of a rotating disk submerged and confined, that has been used and validated in previous researches, is used in this case, generalised for arbitrary axial gaps disk-casing. To use this model, it is necessary to know the averaged rotating speed of the flow with respect to the disk. This parameter is obtained after an analytical discussion of the motion of the flow inside the casing where the disk rotates, and by means of CFD simulations for different axial positions of the disk. The natural frequencies of the rotating disk for the different axial confinements can be calculated following this method. A Finite Element Model has been built up to obtain the natural frequencies by means of computational simulation. The relative velocity of the flow with respect to the disk is also introduced in the simulation model in order to estimate the natural frequencies of the rotating disk. Experimental tests have been performed with a rotating disk test rig. A thin stainless steel disk (thickness of 8 mm, (h/r<5%) and mass of 7.6 kg) rotates inside a rigid casing. The position of the disk can be adjusted at several axial gaps disk-casing. A piezoelectric patch (PZT) attached on the rotating disk is used to excite the structure. Several miniature and submergible accelerometers have measured the response from the rotating frame. Excitation and measured signals are transmitted from the rotating to the stationary frame through a slip ring system. Experimental results are contrasted with the results obtained by the analytical and numerical model. Thereby, the influence of the axial gap disk-casing on the natural frequencies of a rotating disk totally confined and surrounded by a heavy fluid is determined.     

Analysis of functionally graded rotating disks with variable thickness

Elastic solutions for axisymmetric rotating disks made of functionally graded material with variable thickness are presented. The material properties and disk thickness profile are assumed to be represented by two power-law distributions. In the case of hollow disk, based on the form of the power-law distribution for the mechanical properties of the constituent components and the thickness profile function, both analytical and semi-analytical solutions are given under free–free and fixed-free boundary conditions. For the solid disk, only semi-analytical solution is presented. The effects of the material grading index and the geometry of the disk on the stresses and displacements are investigated. It is found that a functionally graded rotating disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared with that of uniform thickness. It is seen that the maximum radial stress for the solid functionally graded disk with parabolic thickness profile is not at the centre like uniform thickness disk. Results of this paper suggest that a rotating functionally graded disk with parabolic concave or hyperbolic convergent thickness profile can be more efficient than the one with uniform thickness.     

Forced convection from an isolated heat source in a channel with porous medium

A numerical study is made of flow and heat transfer characteristics of forced convection in a channel that is partially filled with a porous medium. The flow geometry models convective cooling process in a printed circuit board system with a porous insert.The channel walls are assumed to be adiabatic. Comprehensive numerical solutions are acquired to the governing Navier-Stokes equations, using the Brinkman-Forchheimer-extended Darcy model for the regions of porous media. Details of flow and thermal fields are examined over ranges of the principal parameters; i.e., the Reynolds number Re, the Darcy number Da (≡K/H²), the thickness of the porous substrate S, and the ratio of thermal conductivities R_k (≡k_eff/k). Two types of the location of the porous block are considered. The maximum temperature at the heat source and the associated pressure drop are presented for varying Re, Da, S, and R_k. The results illustrate that as S increases or Da decreases, the fluid flow rate increases. Also, as R_k increases for fixed Da, heat transfer rates are augmented. Explicit influences of Re on the flow and heat transport characteristics are also scrutinized. Assessment is made of the utility of using a porous insert by cross comparing the gain in heat transport against the increase in pressure drop.     

Effusing core at the center of A potential vortex

Experiments were carried out to measure the base pressure distribution of a flow field induced by a potential vortex with its axis normal to a stationary disk. The center base pressure coefficient of the vortex, C₀(0), was found to be proportional to Reynolds number from Re = 2.0 × 10³ to Re > 2.5 × 10⁴, where Re is based on the disk radius and azimuthal velocity at the disk edge. This behavior of C₀(0) is at variance with the experimental results of Phillips (Phys. Fluids, 27, 2215, 1984) and Khoo (M. Eng. Thesis, Natl. Univ. Singapore, 1984), which showed vastly different trends depending on Re. Plausible reasons are suggested for the apparent discrepancies observed. Finally, the extent of the effusing core at the center, r₁ (taken to be the radial position where departure from the outer potential flow took place), was found to be proportional to Re^−1/2 for all Re values considered.