Stability Of Laminar Flow Research Articles

Fluid Motion Induced by Surface-Tension Variation

Fluid motion in a long straight channel induced by longitudinally varying surface tension has been discussed by Levich. This problem is re-examined and a different solution is given. In addition, the stability of laminar flows involving surface-tension variation is briefly discussed, and a correction of a previous result [C.-S. Yih, J. Fluid Mech. 28, 493 (1967)] is made.

Transition to turbulence for flow through a dumped bed of spheres

A transition from laminar to turbulent flow in a dumped bed of spheres occurs over the Reynolds number range 110 to 150. (Reynolds number is based on the diameter of the particle and the fluid velocity in the empty column.) The transition was detected by measuring the local rates of mass transfer around the circumference of one of the packings. The change from a steady to an unsteady mass transfer rate occurred over the whole surface of the test sphere at approximately the same Reynolds number. These results are supported by visual studies similar to those performed by Reynolds in his studies of the stability of laminar flow in a pipe. Time averaged mass transfer rates in a packed bed do not change dramatically in the transition range.

Apparent Departures from Newtonian Behavior in Liquids Caused by Viscous Heating

As part of a general study of the mechanical behavior of materials with temperature-dependent properties, idealized Couette flow is considered here. The model fluid is Newtonian and incompressible. Inertial terms are neglected in the analysis. Time-dependent temperature and velocity distributions in the fluid are computed for: (1) the constant stress case, (2) the constant rate of stress increase case, and (3) the constant boundary velocity case. In each case, a range of thermal boundary conditions is also considered. The results are compared with closed form analytical solutions that can be obtained in special cases. The temperature distributions can influence the stability of laminar flow and cavitation in liquids. The derived velocity distributions influence the drag and protective action of lubricants. Furthermore, it appears that some apparently non-Newtonian flows can be described more simply when the deviation is attributed to heating than when it is attributed to intrinsic, non-Newtonian behavior. A thermistor circuit analog is demonstrated as an aid to the solution of the Couette flow equations with thermal feedback.

Modified Hankel Functions and their Integrals to Argument 10

Abstract : For analyses of the stability of laminar flows, solutions of Stokes' equation are used to construct asymptotic solutions of the Orr-Sommerfeld equation. The pertinent solutions are the second integrals of the modified Hankel functions of order one-third. Tables of these functions and their derivatives for arguments bounded in absolute value by 6 were published by the Harvard Computation Laboratory in 1945; no tables of the integrals existed. In 1959 R. Betchov, starting from the Harvard tables, computed the first two integrals for use in a study of boundary layer oscillation, but his computations were not published. Kirtan Singh has extended the Harvard values of the modified Hankel functions to an argument of =10 (on the imaginary axis), recomputing the values beyond 5.0, and has determined the first and second integrals in the same range. These new values, together with Betchov's integrals to an argument of =5, are published here for the first time. The values of the functions themselves to an argument of =5 are reproduced by permission of the Harvard Computation Laboratory. (Author)

Stability of laminar flow between parallel planes for two- and three-dimensional finite disturbances

Stability of laminar flow between parallel planes is considered for two- and three-dimensional finite disturbances. The critical Reynolds number obtained is close toR=1000; three-dimensional vibrations that could generate turbulence are less stable than two-dimensional ones, particularly with the increase of Reynolds number.

On the stability of laminar flow of a dusty gas

The equations describing the motion of a gas carrying small dust particles are given and the equations satisfied by small disturbances of a steady laminar flow are derived. The effect of the dust is described by two parameters; the concentration of dust and a relaxation time τ which measures the rate at which the velocity of a dust particle adjusts to changes in the gas velocity and depends upon the size of the individual particles. It is shown that if the dust is fine enough for τ to be small compared with a characteristic time scale associated with the flow, then the addition of dust destabilizes a gas flow; whereas if the dust is coarse so that τ is relatively large, then the dust has a stabilizing action.For plane parallel flow, it is shown that the stability characteristics for a dusty gas are still determined by solutions of the Orr-Sommerfeld equation, but with the basic velocity profile replaced by a modified profile which is in general complex. A simple, although unrealistic, example is used to illustrate some features of the action of dust. It is intended to describe the solution of the modified Orr-Sommerfeld equation for plane Poiseuille flow in a later paper.

Chapter VI. Magneto-Fluid Dynamical Stability and Turbulence

The effect of the magnetic field on the stability of laminar flow is considered. It is shown that laminar flows are always stabilized by a uniform magnetic field parallel to the flow, but they may be destabilized by other types of fields. Evidence is presented to show that the conductivity of the fluid acts as a stabilizing agent so long as it is small and reversely if it is large. For large values of the conductivity there exists a critical intensity of the magnetic field above which the flows are completely stabilized against all infinitesinal disturbances. Magnetohydrodynamic turbulence in which the fluctuations in the velocity and the magnetic field are not small in comparison with their mean values are also investigated. It is shown that the effect of the uniform magnetic field is stabilizing and, turbulence decays more rapidly in the presence of the magnetic field. Homogeneous magnetohydrodynamic turbulence in the absence of a mean magnetic field is discussed with emphasis on energy balance between the kinetic and magnetic components of turbulence. (C.E.S.)

On the non-linear mechanics of hydrodynamic stability

In most work on the theory of stability of laminar flow, infinitesimal disturbances only have been considered, so that only the initial growth of the disturbance has been determined. It is the object of the present paper to extend the theory to larger amplitudes and to study the mechanics of disturbance growth with the inherent non-linearity of the hydrodynamical system taken into account.The Reynolds stress (where averages are taken with respect to some suitable space coordinate) is the fundamental consequence of the non-linearity, and its effects can be anticipated as follows. Initially a disturbance grows exponentially with time according to the linear theory, but eventually it reaches such a size that the transport of momentum by the finite fluctuations is appreciable and the associated mean stress (the Reynolds stress) then has an appreciable effect on the mean flow. This distortion of the mean flow modifies the rate of transfer of energy from the mean flow to the disturbance and, since this energy transfer is the cause of the growth of the disturbance, there is a modification of the rate of growth of the latter.It is suggested that, in many cases, an equilibrium state may be possible in which the rate of transfer of energy from the (distorted) mean flow to the disturbance balances precisely the rate of viscous dissipation of the energy of disturbance. A theory based on certain assumptions about the energy flow is given to describe both the growth of the disturbance and the final equilibrium state, and application is made to the cases of Poiseuille flow between parallel planes and flow between rotating cylinders. The distorted mean flow in the equilibrium state can be calculated and from this, in the latter case, the torque required to maintain the cylinders in motion. Good agreement is obtained with G. I. Taylor's measurements of the torque for the case when the inner cylinder rotates and the outer cylinder is at rest.

Stability of laminar flow in curved channels

Abstract Occurrence of secondary flow of a viscous fluid flowing under a pressure gradient in curved channels has often been explained by invoking the existence of boundary layers before the primary flow is established. In this paper it is shown that instability can occur even after the full establishment of the primary flow. Chandrasekhar's method is used in the analysis and the relationship between the stability parameter and the wave number of the disturbance for neutral stability is obtained.

ON A MECHANISM FOR THE RELEASE OF POTENTIAL ENERGY IN THE ATMOSPHERE

It is shown that solitary atmospheric vortices with vertical axes can not remain in equilibrium in any arbitrary latitude but are subjected to horizontal forces which seek to displace them across the latitude circles, in such a manner as to diminish the kinetic and potential energy of the vortex motion. Hence anticyclonic vortices are driven towards the equator, cyclonic vortices towards the pole. In particular, anticyclonic cold domes tend to move southward and in so doing they sink and spread out along the surface of the earth. If the surrounding air is homogeneous the energy released through the weakening of the vortex motion is converted into translational energy, but if the environment is statically stable a fraction of the energy of the anticyclonic vortex goes into the creation of a cyclonic circulation above the sinking cold dome. Thus the southward movement of anticyclonic cold domes in the lower troposphere gives rise to forced displacements southward of cyclonic vortices in the upper troposphere and at the tropopause level. Ultimately such composite vortices reach an equilibrium latitude and may even slowly return towards higher latitudes. The stability of vortices centered over the pole is investigated. Circumpolar cyclonic vortices are stable but anticyclonic vortices are unstable, unless surrounded by a strong outer belt of cyclonic, i.e. westerly winds. The statistical effect of the behavior of sinking cold domes upon the zonal motion in the upper troposphere is discussed, particularly with reference to the maintenance of the jet stream. It is finally shown that the results of the force calculations in this paper may be obtained also by means of a method developed by C. C. Lin in connection with a study of the stability of laminar flows. A qualitative application of this method to perturbations on barocline zonal currents with easterly winds near the surface of the earth and moderate westerly winds aloft suggests a mechanism for the formation of isolated cold domes, in areas and longitudes prescribed by the long-wave pattern of the upper westerlies.

Note on stability of laminar viscous flow between parallel planes

The problem of stability of laminar flow in two dimensions is reducible, as is well known, to the integration of a fourth-order differential equation containing a large parameter. The equation has two slowly varying and two rapidly varying integrals; and because of the presence of a large parameter it is necessary to find the asymptotic expansions of these integrals. The main difficulty of the problem consists in finding the transformations of these integrals through the critical points, particularly of the slowly varying integrals. This problem was first attacked by Heisenberg (1924), and the solution improved by Tollmien (1929); their result was tested numerically by Schlichting (1935, p. 59), and the question was recently reconsidered by Lin (1945,1946) and Tollmien (1947). The application of Tollmien’s theory to the problem of stability of the laminar boundary layer on a plane, worked out in great detail by Schlichting (1933, 1935), was recently tested experimentally by Schubauer & Skramstad (1947), and a good agreement was obtained.

Instability in Pipe Flow

WHEN considering the conditions necessary for the stability of laminar flow in a cylindrical pipe, it may be assumed that the following quantities are involved : transverse velocity gradient (dv/dx) at the point considered; distance (x) of the point from the wall ; and kinematic viscosity (v) of the liquid.

Laminar Boundary-Layer Oscillations and Stability of Laminar Flow

Laminar Boundary-Layer Oscillations and Stability of Laminar Flow