Solution of the perturbation problem for a shear flow with nonmonotonic velocity profile

1995 ◽  
Vol 36 (5) ◽  
pp. 647-653
Author(s):  
M. M. Sterkhova
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Perturbation Problem

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

scholarly journals The lift on a small sphere in a slow shear flow

1965 ◽  
Vol 22 (2) ◽  
pp. 385-400 ◽  
Author(s):  
P. G. Saffman
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Kinematic Viscosity ◽  
Lift Force ◽  
Flow Direction ◽  
Functional Dependence ◽  
Small Sphere ◽  
Translation Velocity ◽  
Parabolic Velocity Profile ◽  
Direction Opposite

It is shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to the streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamlines moving in the direction opposite to V. Here, a denotes the radius of the sphere, κ the magnitude of the velocity gradient, and μ and v the viscosity and kinematic viscosity, respectively. The relevance of the result to the observations by Segrée & Silberberg (1962) of small spheres in Poiseuille flow is discussed briefly. Comments are also made about the problem of a sphere in a parabolic velocity profile and the functional dependence of the lift upon the parameters is obtained.

Hydraulic control of homogeneous shear flows

2003 ◽  
Vol 475 ◽  
pp. 163-172 ◽  
Author(s):  
CHRIS GARRETT ◽  
FRANK GERDES
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Water Waves ◽  
Flow Speed ◽  
Hydraulic Control ◽  
Mean Flow ◽  
Shallow Water Waves ◽  
Homogeneous Fluid ◽  
Apparent Paradox ◽  
The Mean

If a shear flow of a homogeneous fluid preserves the shape of its velocity profile, a standard formula for the condition for hydraulic control suggests that this is achieved when the depth-averaged flow speed is less than (gh)1/2. On the other hand, shallow-water waves have a speed relative to the mean flow of more than (gh)1/2, suggesting that information could propagate upstream. This apparent paradox is resolved by showing that the internal stress required to maintain a constant velocity profile depends on flow derivatives along the channel, thus altering the wave speed without introducing damping. By contrast, an inviscid shear flow does not maintain the same profile shape, but it can be shown that long waves are stationary at a position of hydraulic control.

An Analysis of Uniform Shear Flow Past a Porous Plate Attached to a Plane Surface

1980 ◽  
Vol 102 (2) ◽  
pp. 160-165 ◽  
Author(s):  
M. Kiya ◽  
M. Arie ◽  
K. Koshikawa
Keyword(s):  
Boundary Layer ◽  
Shear Flow ◽  
Velocity Profile ◽  
Boundary Layer Thickness ◽  
Plane Surface ◽  
Porous Plate ◽  
Source Model ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Layer Velocity

The wake-source model of Koo and James was extended to the case of a two-dimensional porous plate attached to a plane surface along which a turbulent boundary layer developed. The approaching stream is replaced by a uniform shear flow of linearly-varying velocity profile such that the height of the plate is much smaller than the boundary-layer thickness. The theory requires as input the pressure-drop coefficient of the porous plate and the boundary-layer velocity profile at an appropriate location upstream of the plate.

Outline of a theory of turbulent shear flow

1956 ◽  
Vol 1 (5) ◽  
pp. 521-539 ◽  
Author(s):  
W. V. R. Malkus
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Boundary Conditions ◽  
Momentum Transport ◽  
Turbulent Shear ◽  
Spectral Structure ◽  
Turbulent Shear Flow ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

In this paper the spatial variations and spectral structure of steady-state turbulent shear flow in channels are investigated without the introduction of empirical parameters. This is made possible by the assumption that the non-linear momentum transport has only stabilizing effects on the mean field of flow. Two constraints on the possible momentum transport are drawn from this assumption: first, that the mean flow will be statistically stable if an Orr-Sommerfeld type equation is satisfied by fluctuations of the mean; second, that the smallest scale of motion that can be present in the spectrum of the momentum transport is the scale of the marginally stable fluctuations of the mean. Within these two constraints, and for a given mass transport, an upper limit is sought for the rate of dissipation of potential energy into heat. Solutions of the stability equation depend upon the shape of the mean velocity profile. In turn, the mean velocity profile depends upon the spatial spectrum of the momentum transport. A variational technique is used to determine that momentum transport spectrum which is both marginally stable and produces a maximum dissipation rate. The resulting spectrum determines the velocity profile and its dependence on the boundary conditions. Past experimental work has disclosed laminar, ‘transitional’, logarithmic and parabolic regions of the velocity profile. Several experimental laws and their accompanying constants relate the extent of these regions to the boundary conditions. The theoretical profile contains each feature and law that is observed. First approximations to the constants are found, and give, in particular, a value for the logarithmic slope (von Kármán's constant) which is within the experimental error. However, the theoretical boundary constant is smaller than the observed value. Turbulent channel flow seems to achieve the extreme state found here, but a more decisive quantitative comparison of theory and experiment requires improvement in the solutions of the classical laminar stability problem.

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