scholarly journals Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone

1999 ◽  
Vol 384 ◽  
pp. 339-374 ◽  
Author(s):  
R. E. HEWITT ◽  
P. W. DUCK ◽  
M. R. FOSTER
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Swirling Flow ◽  
Rotating Disk ◽  
Stratified Fluid ◽  
Steady States ◽  
General Description ◽  
Homogeneous Fluid ◽  
Axisymmetric Solution ◽  
Steady Solutions

We consider a set of nonlinear boundary-layer equations that have been derived by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified fluid in a conical container. In contrast to the unsteady analysis of DFH, we restrict attention to steady solutions and extend the previous discussion further by allowing the container to both co-rotate and counter-rotate relative to the contained swirling fluid. The system is governed by three parameters, which are essentially non-dimensional measures of the rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; however, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to this system proves to be rather complicated and shows many interesting features, including non-uniqueness, singular solutions and bifurcation phenomena.We present a broad description of the steady states with particular emphasis on boundaries in parameter space beyond which steady states cannot be continued.A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to non-axisymmetric boundary-layer states appearing as bifurcations of the axisymmetric solutions. In an Appendix we give details of an exact, non-axisymmetric solution to the Navier–Stokes equations (with axisymmetric boundary conditions) corresponding to the flow of homogeneous fluid above a rotating disk.

scholarly journals Non-axisymmetric rotating-disk flows: nonlinear travelling-wave states

2000 ◽  
Vol 413 ◽  
pp. 287-316 ◽  
Author(s):  
R. E. HEWITT ◽  
P. W. DUCK
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Rotating Disk ◽  
Travelling Wave ◽  
Numerical Results ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Navier Stokes Equations ◽  
Axisymmetric Solution ◽  
Wave States

We consider the classical problem of the laminar flow of an incompressible rotating fluid above a rotating, impermeable, infinite disk. There is a well-known class of solutions to this configuration in the form of an exact axisymmetric solution to the Navier–Stokes equations. However, the radial self-similarity that leads to the ‘rotating- disk equations’ can also be used to obtain solutions that are non-axisymmetric in nature, although (in general) this requires a boundary-layer approximation. In this manner, we locate several new solution branches, which are non-axisymmetric travelling-wave states that satisfy axisymmetric boundary conditions at infinity and at the disk. These states are shown to appear as symmetry-breaking bifurcations of the well-known axisymmetric solution branches of the rotating-disk equations. Numerical results are presented, which suggest that an infinity of such travelling states exist in some parameter regimes. The numerical results are also presented in a manner that allows their application to the analogous flow in a conical geometry.Two of the many states described are of particular interest. The first is an exact, nonlinear, non-axisymmetric, stationary state for a rotating disk in a counter-rotating fluid; this solution was first presented by Hewitt, Duck & Foster (1999) and here we provide further details. The second state corresponds to a new boundary-layer-type approximation to the Navier–Stokes equations in the form of azimuthally propagating waves in a rotating fluid above a stationary disk. This second state is a new non-axisymmetric alternative to the classical axisymmetric Bödewadt solution.

scholarly journals On the boundary layer arising in the spin-up of a stratified fluid in a container with sloping walls

1997 ◽  
Vol 335 ◽  
pp. 233-259 ◽  
Author(s):  
P. W. DUCK ◽  
M. R. FOSTER ◽  
R. E. HEWITT
Keyword(s):  
Boundary Layer ◽  
Layer Structure ◽  
Outer Region ◽  
Stratified Fluid ◽  
Main Body ◽  
Similarity Type ◽  
Cone Apex ◽  
Uniform Solution ◽  
Steady Solutions ◽  
Finite Time Singularity

In this paper we consider the boundary layer that forms on the sloping walls of a rotating container (notably a conical container), filled with a stratified fluid, when flow conditions are changed abruptly from some initial (uniform) state. The structure of the solution valid away from the cone apex is derived, and it is shown that a similarity-type solution is appropriate. This system, which is inherently nonlinear in nature, is solved numerically for several flow regimes, and the results reveal a number of interesting and diverse features.In one case, a steady state is attained at large times inside the boundary layer. In a second case, a finite-time singularity occurs, which is fully analysed. A third scenario involves a double boundary-layer structure developing at large times, most significantly including an outer region that grows in thickness as the square-root of time.We also consider directly the nonlinear fully steady solutions to the problem, and map out in parameter space the likely ultimate flow behaviour. Intriguingly, we find cases where, when the rotation rate of the container is equal to that of the main body of the fluid, an alternative nonlinear state is preferred, rather than the trivial (uniform) solution.Finally, utilizing Laplace transforms, we re-investigate the linear initial-value problem for small differential spin-up studied by MacCready & Rhines (1991), recovering the growing-layer solution they found. However, in contrast to earlier work, we find a critical value of the buoyancy parameter beyond which the solution grows exponentially in time, consistent with our nonlinear results.

Numerical Integration of the Navier-Stokes Equations for a Disk Rotating in a Housing

1985 ◽  
Vol 40 (8) ◽  
pp. 789-799 ◽  
Author(s):  
A. F. Borghesani
Keyword(s):  
Boundary Layer ◽  
Cylindrical Cavity ◽  
Boundary Layer Thickness ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Rotating Disk ◽  
Infinite Medium ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Torque

The Navier-Stokes equations for the fluid motion induced by a disk rotating inside a cylindrical cavity have been integrated for several values of the boundary layer thickness d. The equivalence of such a device to a rotating disk immersed in an infinite medium has been shown in the limit as d → 0. From that solution and taking into account edge effect corrections an equation for the viscous torque acting on the disk has been derived, which depends only on d. Moreover, these results justify the use of a rotating disk to perform accurate viscosity measurements.

An experimental study of the instability of the laminar Ekman boundary layer

1963 ◽  
Vol 15 (4) ◽  
pp. 560-576 ◽  
Author(s):  
Alan J. Faller
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Rotating Disk ◽  
Homogeneous Fluid ◽  
Viscous Boundary Layer ◽  
Ekman Boundary Layer ◽  
Characteristic Angle ◽  
Layer Flow ◽  
The Stability ◽  
Viscous Boundary

This study concerns the stability of the steady laminar boundary-layer flow of a homogeneous fluid which occurs in a rotating system when the relative flow is slow compared to the basic speed of rotation. Such a flow is called an Ekman boundary-layer flow after V. W. Ekman who considered the theory of such flows with application to the wind-induced drift of the surface waters of the ocean.Ekman flow was produced in a large cylindrical rotating tank by withdrawing water from the centre and introducing it at the rim. This created a steady-state symmetrical vortex in which the flow from the rim to the centre took place entirely in the shallow viscous boundary layer at the bottom. This boundary-layer flow became unstable above the critical Reynolds number$Re_c = vD|v = 125 \pm 5$wherevis the tangential speed of flow,$D = (v| \Omega)^{\frac {1}{2}}$is the characteristic depth of the boundary layer,vis the kinematic viscosity, and Ω is the basic speed of rotation. The initial instability was similar to that which occurs in the boundary layer on a rotating disk, having a banded form with a characteristic angle to the basic flow and with the band spacing proportional to the depth of the boundary layer.

scholarly journals Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices

2017 ◽  
Vol 836 ◽  
pp. 43-71 ◽  
Author(s):  
E. Appelquist ◽  
P. Schlatter ◽  
P. H. Alfredsson ◽  
R. J. Lingwood
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Low Frequency ◽  
Rotating Disk ◽  
High Amplitude ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Absolute Instability ◽  
Global Instability ◽  
Roughness Elements

This paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotating-disk boundary layer. It also draws some comparison with swept-wing flows. Direct numerical simulations based on the incompressible Navier–Stokes equations of the flow over the surface of a rotating disk with modelled roughness elements are presented. The rotating-disk flow has been of particular interest for stability and transition research since the work by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) where an absolute instability was found. Here stationary disturbances develop from roughness elements on the disk and are followed from the linear stage, growing to saturation and finally transitioning to turbulence. Several simulations are presented with varying disturbance amplitudes. The lowest amplitude corresponds approximately to the experiment by Imayama et al. (J. Fluid Mech., vol. 745, 2014a, pp. 132–163). For all cases, the primary instability was found to be convectively unstable, and secondary modes were found to be triggered spontaneously while the flow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the low-amplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the high-amplitude roughness cases the transition scenario gives a turbulent flow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Engng Maths, vol. 57, 2007, pp. 237–251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large amplification rate where the secondary global instability occurred. For smaller radial positions, low-frequency secondary instabilities were observed, tripped by the global instability.

SWIRLING FLOW OF FIELD DEPENDENT VISCOUS FERROFLUID OVER A POROUS ROTATING DISK WITH HEAT TRANSFER

2014 ◽  
Vol 06 (04) ◽  
pp. 1450033 ◽  
Author(s):  
PARAS RAM ◽  
VIKAS KUMAR
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Boundary Layer ◽  
Swirling Flow ◽  
Flow Characteristics ◽  
Rotating Disk ◽  
Displacement Thickness ◽  
Field Dependent ◽  
Magnetic Field Dependent Viscosity ◽  
Dependent Viscosity

The purpose of present study is to investigate the effects of field dependent viscosity on swirling flow of an incompressible electrically non-conducting ferrofluid over a porous rotating disk with suction and heat transfer at the wall. Karman's similarity transformations are used to convert the governing boundary layer equations involved in the problem to a system of nonlinear coupled differential equations. The solution of this system is obtained by using a second-order numerical scheme which combines the features of Finite Difference method and Newton's zero finding algorithms. The flow characteristics including velocity and temperature profiles and boundary layer displacement thickness are studied for various values of MFD (magnetic field dependent) viscosity and suction parameter. Beside these, skin friction coefficients and the rate of heat transfer are also calculated on the surface of the disk. Magnetic field dependent viscosity and suction at the surface of porous rotating disk affect significantly the velocity and temperatures fields, rate of heat transfer and other flow characteristics in the generated ferrofluid boundary layer.

EXPERIMENTAL INVESTIGATION OF FLOW RESPONSE TO STATIONARY DISTURBANCE IN ROTATING-DISK FLOW

2019 ◽  
Vol XVI (2) ◽  
pp. 13-22
Author(s):  
Muhammad Ehtisham Siddiqui
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Three Dimensional ◽  
Rotating Disk ◽  
Careful Analysis ◽  
Laboratory Frame ◽  
Transition To Turbulence ◽  
Turbulent Regime ◽  
Mean Velocity ◽  
Layer Flow

Three-dimensional boundary-layer flow is well known for its abrupt and sharp transition from laminar to turbulent regime. The presented study is a first attempt to achieve the target of delaying the natural transition to turbulence. The behaviour of two different shaped and sized stationary disturbances (in the laboratory frame) on the rotating-disk boundary layer flow is investigated. These disturbances are placed at dimensionless radial location (Rf = 340) which lies within the convectively unstable zone over a rotating-disk. Mean velocity profiles were measured using constant-temperature hot-wire anemometry. By careful analysis of experimental data, the instability of these disturbance wakes and its estimated orientation within the boundary-layer were investigated.

Direct simulation of transition in an oscillatory boundary layer

1998 ◽  
Vol 371 ◽  
pp. 207-232 ◽  
Author(s):  
G. VITTORI ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Temporal Development ◽  
Direct Simulation ◽  
Navier Stokes Equations ◽  
Oscillatory Boundary Layer

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

Essential Ingredients for the Computation of Steady and Unsteady Blade Boundary Layers

1991 ◽  
Vol 113 (4) ◽  
pp. 608-616 ◽  
Author(s):  
H. M. Jang ◽  
J. A. Ekaterinaris ◽  
M. F. Platzer ◽  
T. Cebeci
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Transitional Region ◽  
Low Reynolds Numbers ◽  
Flow Equations ◽  
Low Reynolds

Two methods are described for calculating pressure distributions and boundary layers on blades subjected to low Reynolds numbers and ramp-type motion. The first is based on an interactive scheme in which the inviscid flow is computed by a panel method and the boundary layer flow by an inverse method that makes use of the Hilbert integral to couple the solutions of the inviscid and viscous flow equations. The second method is based on the solution of the compressible Navier–Stokes equations with an embedded grid technique that permits accurate calculation of boundary layer flows. Studies for the Eppler-387 and NACA-0012 airfoils indicate that both methods can be used to calculate the behavior of unsteady blade boundary layers at low Reynolds numbers provided that the location of transition is computed with the en method and the transitional region is modeled properly.

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