Flow over compliant rotating disks

Flow over compliant materials and compliant coatings has been studied for decades because of the potential of using such materials for laminar-flow control. Since boundary layers in most flows of engineering interest are three-dimensional the classic rotating-disk flow geometry, the paradigm for studying three-dimensional boundary layers, has been adapted to investigate flow over compliant rotating disks. This paper reviews the literature on the existing experimental and theoretical research on flow over compliant rotating disks. The article concludes by evaluating the status of the available results and their implications as regards future research routes to investigate the capabilities of compliant materials for laminar-flow control.     

Vortical flow between rotating ellipsoids

Summary. Experimental results on incompressible viscous flows in the gap between rotating spheroids are presented. Oblate ellipsoids are discussed as a connecting link between a sphere and a disk, while a prolate ellipsoid represents the geometrical intermediate stage between a sphere and a cylinder. The axes ratios of the ellipsoids are A : B=2 : 1 and A : B=1 : 2, respectively. The basic flow field and the occurring instabilities in form of Taylor- and cross-flow vortices have been studied by flow visualization. The common properties and their differences regarding the various flow patterns of the disks, spheres, cylinders as well as of the oblate and prolate ellipsoids are treated. The location of the generation of the different vortex systems at the various rotating bodies is determined. A comparison of the friction torque for these bodies of rotation is given by diagrams. Finally, the theoretically derived curves for the friction torque of the disks, spheres and cylinders are compared with the experimentally obtained data for the oblate and prolate ellipsoids.Dedicated to Prof. Dr.-Ing. Dr. techn. E. h. J. Zierep on the occasion of his 75th birthday.     

Instabilities and constitutive modelling

The plastics industry today sees huge wastage through product defects caused by unstable flows during the manufacturing process. In addition, many production lines are throughput-limited by a flow speed threshold above which the process becomes unstable. Therefore, it is critically important to understand the mechanisms behind these instabilities.In order to investigate the flow of a molten plastic, the first step is a model of the liquid itself, a relation between its current stress and its flow history called a constitutive relation. These are derived in many ways and tested on several benchmark flows, but rarely is the stability of the model used as a criterion for selection. The relationship between the constitutive model and the stability properties of even simple flows is not yet well understood. We show that in one case a small change to the model, which does not affect the steady flow behaviour, entirely removes a known instability. In another, a change that makes a qualitative difference to the steady flow makes only tiny changes to the stability.The long-term vision of this research is to exactly quantify what are the important properties of a constitutive relation as far as stability is concerned. If we could understand that, not only could very simple stability experiments be used to choose the best constitutive models for a particular material, but our ability to predict and avoid wasteful industrial instabilities would also be vastly improved.     

Transition from Taylor vortices to cross-flow instabilities

Summary We present experimental results of fluid flow instabilities between different rotating surfaces. We start withcounter-rotating Taylor vortices between two coaxial cylinders. We go over to rotating cones with increasing apex angle. Due to the growing cross-flow we finally end up with spiral vortices allrotating in the same direction between a rotating disk and a housing. Figure 13 gives a complete survey of these results. We discuss the transition from one vortex system to the other in detail.Dedicated to Prof. Dr. Dr. h. c. mult. S. Wittig on the occasion of his 60th birthday     

Global stability of the rotating-disk boundary layer

The global stability of the von Kármán boundary layer on the rotating disk is reviewed. For the genuine, radially inhomogeneous base flow, linearized numerical simulations indicate that convectively propagating forms of disturbance are predominant at all radii. The presence of absolute instability does not lead to the formation of any unstable linear global mode, even though the temporal growth rate of the absolute instability increases along the radial direction. Analogous behaviour can be found in the impulse solutions of a model amplitude equation, namely the linearized complex Ginzburg–Landau equation. These solutions show that, depending on the precise balance between spatial variations in the temporal growth rate and the corresponding shifts in the temporal frequency, globally stable behaviour can be obtained even in the presence of a strengthening absolute instability. The radial dependency of the absolute temporal frequency is sufficient to detune the disturbance oscillations at different radial positions, thus overcoming the radially increasing absolute growth, thereby giving rise to a stable global response. The origin of this form of behaviour can be traced to the fact that the cylindrical geometry of the rotating-disk flow dictates a choice of a globally valid time non-dimensionalization that, when properly employed, leads to a significant radial variation in the frequency for the absolute instability.     

Stability of two-layered fluid flows in an inclined channel

A linear stability analysis of two-layer fluid flows in an inclined channel geometry has been carried out. The onset of flow transitions and the spatio-temporal characteristics of secondary flows produced by the flow instabilities have been examined. The effects of density and viscosity stratifications and surface tension on flow structures also have been investigated at various values of Froude numbers (channel inclinations). Multi-domain Chebyshev–Tau spectral methods along with MATLAB QZ eigenvalue solver are used to determine the whole spectrum of the eigenvalues and associated eigenfunctions. The neutral stability diagrams and stability boundaries are constructed for various values of flow parameters. The onset of flow transitions and flow structures predicted by linear stability analysis are compared against experimental results and they agree reasonably well. The results presented in the present paper imply that the shear mode of flow transitions is the one likely to be identified in experiments.     

Circle theorems for inviscid steady flows

General circle theorems which localize the complex eigenfrequencies arising in the linear stability analysis of conservative steady flows are given. Howard's circle theorem for incompressible plane parallel flow is contained as a special case. Two applications are considered: swirling flow of an inviscid incompressible fluid, and rotating flow of an inviscid, incompressible, perfectly conducting magnetofluid with an axial magnetic field. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.     

Midplane-symmetry breaking in the flow between two counter-rotating disks

This paper considers the axisymmetric steady flow driven by exact counter rotation of two co-axial disks of finite radius. At the edges of the rotating disks one of three conditions is (typically) imposed: (i) zero velocity, corresponding to a stationary, impermeable, cylindrical shroud (ii) zero normal velocity and zero tangential fluid traction, corresponding to a (confined) free surface and (iii) an edge constraint that is consistent with a similarity solution of von Kármán form. The similarity solution is valid in an infinite geometry and possesses a pitchfork bifurcation that breaks the midplane symmetry at a critical Reynolds number. In this paper, similar bifurcations of the global (finite-domain) flow are sought and comparisons are made between the resulting bifurcation structure and that found for the similarity solution. The aim is to assess the validity of the nonlinear similarity solutions in finite domains and to explore the sensitivity of the solution structure to edge conditions that are implicitly neglected when assuming a self-similar flow. It is found that, whilst the symmetric similarity solution can be quantitatively useful for a range of boundary conditions, the bifurcated structure of the finite-domain flow is rather different for each boundary condition and bears little resemblance to the self-similar flow.     

Stability of Bödewadt flow

The stability of the flow produced over an infinite stationary plane in a fluid rotating with uniform angular velocity at an infinite distance from the plane is considered. The basic flow is an exact solution of the Navier-Stokes equations making it amenable to theoretical study. An asymptotic investigation is presented in the limit of large Reynolds number. It is shown that the stationary spiral instabilities observed experimentally can be described by a linear inviscid stability analysis. The prediction obtained for the wave angle of the disturbances is found to agree well with the available experimental and numerical results.     

Stability of the boundary layer on a sphere rotating in still fluid

Summary A theoretical study of the transition of a three-dimensional boundary layer on a sphere rotating in still fluid is carried out by a linear stability analysis. A set of perturbation equations governing the instability of the flow field is derived assuming the perturbations to be consisting of spiral vortices. It is shown that the critical Reynolds numbers obtained in the present analytical study are close to those observed in experiments. It has been found that the streamline-curvature instability appears in the rotating sphere flow. It is also shown that the cross-flow instability is dominant near the poles of a sphere while the streamline-curvature instability overtakes near the equator.     

Flow rate measurement in a pipe flow by vortex shedding

The flow rate measurement of liquid, steam, and gas is one of the most important areas of application for today’s field instrumentation. Vortex meters are used in numerous branches of industry to measure the volumetric flow by exploiting the unsteady vortex flow behind a blunt body. The classical Kármán vortex street behind a cylinder shows a decrease in Strouhal number with decreasing Reynolds number. Considering the flow behind a vortex shedding device in a pipe the Strouhal-Reynolds number dependence shows a different behaviour for turbulent flows: a decrease in Reynolds number leads to an increase in Strouhal number. This phenomenon was found in the experimental investigations as well as in the numerical results and has been confirmed theoretically by a stability analysis.     

A statistical study of spots in torsional Couette flow

This article presents some results on the statistical behavior of localized structures—called “spots”—that propagate in the flow between a rotating and a stationary disk when those are very close one to the other. Under these conditions the rotating-disk flow belongs to the Couette-flow family and is called the torsional Couette flow. Some visualizations of its transition to turbulence have already revealed the propagation of these spots (Schouveiler et al., J Fluid Mech 443:329–350, 2001) from the rim of the disk towards its center. Using flow visualizations and an original image analysis, the present study aims to better describe the characteristics of the spots whose number continuously increases with the Reynolds number until they invade the whole flow. Moreover, we propose a statistical model that predicts an error-function shape for the probability to observe a spot at a given radial position. This prediction is confirmed by an image analysis of the flow and the stability curve of torsional Couette flow is deduced from these observations.     

Nonaxisymmetric unsteady motion over a rotating disk in the presence of free convection and magnetic field

The nonaxisymmetric unsteady motion produced by a buoyancy-induced cross-flow of an electrically conducting fluid over an infinite rotating disk in a vertical plane and in the presence of an applied magnetic field normal to the disk has been studied. Both constant wall and constant heat flux conditions have been considered. It has been found that if the angular velocity of the disk and the applied magnetic field squared vary inversely as a linear function of time (i.e. as (1−λt*)⁻¹, the governing Navier-Stokes equation and the energy equation admit a locally self-similar solution. The resulting set of ordinary differential equations has been solved using a shooting method with a generalized Newton's correction procedure for guessed boundary conditions. It is observed that in a certain region near the disk the buoyancy induced cross-flow dominates the primary von Karman flow. The shear stresses induced by the cross-flow are found to be more than these of the primary flow and they increase with magnetic parameter or the parameter λ characterizing the unsteadiness. The velocity profiles in the x- and y-directions for the primary flow at any two values of the unsteady parameter λ cross each other towards the edge of the boundary layer. The heat transfer increases with the Prandtl number but reduces with the magnetic parameter.     

Instabilities with reduced frictional drag in the rotating channel flow of dilute polymer solutions

Summary A numerical investigation is conducted on secondary flows and roll-cell instabilities in the laminar channel flow of dilute polymer solutions subjected to a steady spanwise rotation. Finite difference calculations of the full nonlinear equations of motion for a Maxwell fluid and a Rivlin-Ericksen fluid of the second grade are presented which indicate that there is a double-vortex secondary flow at weak and rapid rotation rates with an instability in the form of longitudinal roll cells at intermediate rotation rates (regimes analogous to those for a Newtonian fluid). However, for a given physical pressure gradient and rotation rate, the introduction of a minute amount of a polymeric additive to a Newtonian fluid so that the Weissenberg number is of the order of 10^–5 has a stabilizing effect on rotating channel flow and gives rise to secondary flows with a substantially reduced frictional drag. Comparisions with previously conducted experimental and analytical studies are made along with a brief discussion of potential applications to the field of polymeric drag reduction.With 14 Figures     

A fast response superconducting thin-film probe for detection of second-sound shock waves in superfluid helium

Fast response probes are needed for studying the formation and propagation of second-sound shock waves (and for applying such waves to special measuring purposes) in superfluid helium. Newly developed superconducting thin-film probes enable shock-front rise times of down to 0.3 μs to be detected at signal-to-noise ratios higher than about 100. Using high vacuum evaporation techniques, such probes are relatively easy to produce. Their main body consists of a cylindrical quartz glass rod 1.5 mm in diameter with one end face polished to a plane of optical quality. The sensor strip is deposited onto this plane face as a two-component film of 0.02 mm width and 1 mm length. The temperature variations due to second sound cause changes in the resistance of the film and thus, at constant bias current, variations of the voltage drop across it. The temperature where the film undergoes its steep transition to superconductance and where, therefore, the probe works at its greatest sensitivity, is primarily fixed by the ratio of the two components (tin and gold) of the film, but can be adjusted to special values via the magnetic field produced by the adjustable bias current. The high resolution in time which is achievable by this probe makes it useful for accurate measurement of even small variations of the running time of second-sound shock waves. Such variations may be caused by flows, as is shown in the case of a flow produced by a rotating vane; their measurement may, therefore, serve as a tool for flow investigation.     

On the magnetohydrodynamic flow between eccentrically rotating disks

The steady flow of an incompressible, viscous, electrically conducting fluid between two parallel, infinite, insulated disks rotating with different angular velocities about two noncoincident axes has been investigated; under the application of a uniform magnetic field in the axial direction. The solutions for the symmetric and asymmetric velocities are presented. The interesting feature arising due to the magnetic field is that in the central region the flow attains a uniform rotation with mean angular velocity at all rotation speeds for sufficiently large Hartmann number. In this case the flow adjusts to the rotational velocities of the disks mainly in the boundary layers near the disks. The forces on the disks are found to increase due to the presence of the applied magnetic field.     

Kinematic instabilities in two-layer eccentric annular flows,part 2: shear-thinning and yield-stress effects

This paper investigates the possibility of kinematic interfacial instabilities occurring during the industrial process of primary cementing of oil and gas wells. This process involves flows in narrow eccentric annuli that are modelled via a Hele-Shaw approach. The fluids present in primary cementing are strongly non-Newtonian, usually exhibiting shear-thinning behaviour and often with a yield stress. The study is a sequel to Moyers-González and Frigaard (J Eng Math, DOI , 2007), in which the base analysis has been developed for the case of two Newtonian fluids. The occurrence of static mud channels in primary cementing has been known of since the 1960s, (see McLean et al. 1966; SPE 1488), and is a major cause of process failure. This phenomenon is quantified, which provides a simple semi-analytic expression for the maximal volume of residual fluid left behind in the annulus, f _static, and illustrate the dependency of f _static on its five dimensionless parameters. It is shown that three of the four different types of static channel flows are linearly stable. Via dimensional analysis, it is shown that the base flows depend on a minimal set of eight dimensionless parameters and the stability problem depends on an additional two dimensionless parameters. This large dimensional parameter space precludes use of the full numerical solution to the stability problem as a predictive tool or for studying the various stability regimes. Instead a semi-analytical approach has been developed based on solution of the long-wavelength limit. This prediction of instability can be evaluated via simple quadrature from the base flow and is suitable for use in process optimisation.     

Periodic flow of a second-grade fluid induced by non-torsional oscillations of eccentric rotating disks

This paper deals with the periodic flow of a second-grade fluid caused by non-torsional oscillations of two disks rotating about non-coincident axes. While the two parallel disks are initially rotating with the same angular velocity about distinct axes, they start to execute non-torsional oscillations in their own planes and in the opposite directions. An exact solution is obtained for the components of the horizontal force per unit area exerted by the top and bottom disks on the fluid in the periodic state. The results are graphically displayed and the influence of the second-grade fluid parameter, the ratio of the frequency of oscillation to the angular velocity of the disks, the Reynolds number and the dimensionless velocity amplitudes of oscillation is discussed. It is observed that the change in the \( x \)-component of the mentioned force gets larger when the second-grade fluid parameter increases. However, an opposite effect is seen for the change in the \( y \)-component.     

How disorder can diminish avalanche risks: effect of size distribution

We study the behavior of disk assemblies with a variable disorder distribution. The packing is first consolidated and then continuously tilted very slowly. The amount of displaced disks for each tilted angle is recorded. Large displacements of the disks can occur due to some local or global mechanical instabilities. The definition of neighboring disks is based on radical (extension of Voronoï) tessellation rules to decompose, in a unique and perfectly defined manner, the two-dimensional space for polydisperse disks. In this way, by comparing the characteristics of stability for one disk to the neighboring ones for local ordered cluster, we can predict the global amount of displaced disks. Some tilting cycles have been performed to check the correlation between the instability of the packing structure (collective displacements) with micro and macro order parameters.     

Analysis and computation of non-equilibrium two-phase flows

Non-equilibrium fluid mechanics and thermodynamics of two-phase vapour-droplet and gas-particle flow are considered. Formation of the droplets as well as their subsequent interaction with the vapour are discussed. A new theory of nucleation in steam turbines is developed that reproduces many aspects of measured droplet size spectra which cannot be explained by any available steady-flow theories. (Steam turbines are responsible for 80% of global electricity production and the presence of moisture significantly reduces turbine efficiency costing 50 million pounds per annum in UK alone.) Fluid dynamic interactions discussed include flow instabilities induced by condensation, condensation wave theory, relaxation gas dynamics for vapour-droplet flow, thermal choking due to non-equilibrium condensation, the structure of shock waves and their development through unsteady processes, and jump conditions and the interpretation of total pressure in two-phase flows.