Vortex formation in a free boundary layer according to stability theory

1965 ◽  
Vol 22 (2) ◽  
pp. 371-383 ◽  
Author(s):  
A. Michalke
Keyword(s):  
Boundary Layer ◽  
Free Boundary ◽  
Stability Theory ◽  
Vortex Formation ◽  
Linear Stability Theory ◽  
Critical Layer ◽  
Disturbed Flow ◽  
Vorticity Distribution ◽  
Linearized Stability ◽  
Non Linear

An attempt is made to explain the formation of vortices in free boundary layers by means of stability theory using a hyperbolic-tangent velocity profile. The vorticity distribution of the disturbed flow, as obtained by the inviscid linearized stability theory, is discussed. The path lines of particles which are initially placed along straight lines parallel to thex-axis are calculated. Lines connecting the positions of these particles give an impression of the instant shape of the disturbed flow. With increasing time the boundary layer becomes thinner in certain regions and thicker in others. A special line—originally positioned at the critical layer—shows in the thicker region a tendency to roll up. Also extrema of the vorticity are located there. Finally, these results are compared with those which can be expected from the non-linear Helmholtz equation. Disagreement is found in the neighbourhood of the critical layer. Using the non-linear stability theory of Stuart up to the third-order terms, the vorticity distribution shows the tendency expected from the non-linear equation.

The development of a two-dimensional wavepacket in a growing boundary layer

1982 ◽  
Vol 384 (1787) ◽  
pp. 317-332 ◽  
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Linear Stability ◽  
Stability Theory ◽  
Linear Stability Theory ◽  
Two Dimensional ◽  
Displacement Thickness ◽  
Power Comparisons ◽  
The One ◽  
Half Power

The evolution of a two-dimensional wavepacket in a growing boundary layer is discussed in terms of linear stability theory. The wavepacket is represented by an integral of periodic wavetrains, each of which is defined as a series in terms of the inverse of the local displacement thickness Reynolds number to the one half power. Comparisons are made between the waveforms computed directly from the integral, a steepest-descent expansion of the integral, and a global expansion about the peak of the wavepacket.

scholarly journals Studies in non-linear stability theory

1966 ◽  
Vol 2 (2) ◽  
pp. 231-231
Author(s):  
M. M. Andrew
Keyword(s):  
Linear Stability ◽  
Stability Theory ◽  
Linear Stability Theory ◽  
Non Linear

On the evolution of a wave packet in a laminar boundary layer

1991 ◽  
Vol 225 ◽  
pp. 575-606 ◽  
Author(s):  
Jacob Cohen ◽  
Kenneth S. Breuer ◽  
Joseph H. Haritonidis
Keyword(s):  
Boundary Layer ◽  
Wave Packet ◽  
Linear Stability ◽  
Laminar Boundary Layer ◽  
Stability Theory ◽  
Free Stream ◽  
Linear Stability Theory ◽  
Stream Velocity ◽  
Turbulent Spot ◽  
Second Stage

The transition process of a small-amplitude wave packet, generated by a controlled short-duration air pulse, to the formation of a turbulent spot is traced experimentally in a laminar boundary layer. The vertical and spanwise structures of the flow field are mapped at several downstream locations. The measurements, which include all three velocity components, show three stages of transition. In the first stage, the wave packet can be treated as a superposition of two- and three-dimensional waves according to linear stability theory, and most of the energy is centred around a mode corresponding to the most amplified wave. In the second stage, most of the energy is transferred to oblique waves which are centred around a wave having half the frequency of the most amplified linear mode. During this stage, the amplitude of the wave packet increases from 0.5 % to 5 % of the free-stream velocity. In the final stage, a turbulent spot develops and the amplitude of the disturbance increases to 27 % of the free-stream velocity.Theoretical aspects of the various stages are considered. The amplitude and phase distributions of various modes of all three velocity components are compared with the solutions provided by linear stability theory. The agreement between the theoretical and measured distributions is very good during the first two stages of transition. Based on linear stability theory, it is shown that the two-dimensional mode of the streamwise velocity component is not necessarily the most energetic wave. While linear stability theory fails to predict the generation of the oblique waves in the second stage of transition, it is demonstrated that this stage appears to be governed by Craik-type subharmonic resonances.

On finite amplitude oscillations in laminar mixing layers

1967 ◽  
Vol 29 (3) ◽  
pp. 417-440 ◽  
Author(s):  
J. T. Stuart
Keyword(s):  
Equilibrium State ◽  
Stability Theory ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Time Dependent ◽  
Present Calculation ◽  
Periodic Perturbations ◽  
Linearized Stability ◽  
Non Linear

In the first part of the paper, a mixing layer of tanhyform is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanhy.Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even iny, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.

Vortex formation in front of a piston moving through a cylinder

2000 ◽  
Vol 416 ◽  
pp. 1-28 ◽  
Author(s):  
J. J. ALLEN ◽  
M. S. CHONG
Keyword(s):  
Boundary Layer ◽  
Vortex Formation ◽  
Field Measurements ◽  
Inviscid Flow ◽  
Cylinder Wall ◽  
Viscous Effects ◽  
Viscous Forces ◽  
Vorticity Distribution ◽  
Piston Speed ◽  
Self Similar

This paper contains the details of an experimental study of the vortex formed in front of a piston as it moves through a cylinder. The mechanism for the formation of this vortex is the removal of the boundary layer forming on the cylinder wall in front of the advancing piston. The trajectory of the vortex core and the vorticity distribution on the developing vortex have been measured for a range of piston velocities. Velocity field measurements indicate that the vortex is essentially an inviscid structure at the Reynolds numbers considered, with viscous effects limited to the immediate corner region. Inviscid flow is defined in this paper as being a region of the flow where inertial forces are significantly larger than viscous forces. Flow visualization and vorticity measurements show that the vortex is composed mainly of material from the boundary layer forming over the cylinder wall. The characteristic dimension of the vortex appears to scale in a self-similar fashion, while it is small in relation to the apparatus length scale. This scaling rate of t0.85+0.7m, where the piston speed is described as a power law Atm, is somewhat faster than the t3/4 scaling predicted by Tabaczynski et al. (1970) and considerably faster than a viscous scaling rate of t1/2. The reason for the structure scaling more rapidly than predicted is the self-induced effect of the secondary vorticity that is generated on the piston face. The vorticity distribution shows a distinct spiral structure that is smoothed by the action of viscosity. The strength of the separated vortex also appears to scale in a self-similar fashion as t2m+1. This rate is the same as suggested from a simple model of the flow that approximates the vorticity being ejected from the corner as being equivalent to the flux of vorticity over a flat plate started from rest. However, the strength of the vorticity on the separated structure is 25% of that suggested by this model, sometimes referred to as the ‘slug’ model. Results show that significant secondary vorticity is generated on the piston face, forming in response to the separating primary vortex. This secondary vorticity grows at the same rate as the primary vorticity and is wrapped around the outside of the primary structure and causes it to advect away from the piston surface.

Studies in Non-Linear Stability Theory

1966 ◽  
Vol 20 (96) ◽  
pp. 633
Author(s):  
Jack K. Hale ◽  
Wiktor Eckhaus
Keyword(s):  
Linear Stability ◽  
Stability Theory ◽  
Linear Stability Theory ◽  
Non Linear
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