Book contents
- Frontmatter
- Contents
- Foreword by John Miles
- Preface
- 1 INTRODUCTION
- 2 THERMAL INSTABILITY
- 3 CENTRIFUGAL INSTABILITY
- 4 PARALLEL SHEAR FLOWS
- 5 UNIFORM ASYMPTOTIC APPROXIMATIONS
- 6 ADDITIONAL TOPICS IN LINEAR STABILITY THEORY
- 7 NONLINEAR STABILITY
- APPENDIX: A CLASS OF GENERALIZED AIRY FUNCTIONS
- Addendum: Weakly non-parallel theories for the Blasius boundary layer
- Solutions
- Bibliography and author index
- Motion picture index
- Subject index
4 - PARALLEL SHEAR FLOWS
Published online by Cambridge University Press: 06 August 2010
- Frontmatter
- Contents
- Foreword by John Miles
- Preface
- 1 INTRODUCTION
- 2 THERMAL INSTABILITY
- 3 CENTRIFUGAL INSTABILITY
- 4 PARALLEL SHEAR FLOWS
- 5 UNIFORM ASYMPTOTIC APPROXIMATIONS
- 6 ADDITIONAL TOPICS IN LINEAR STABILITY THEORY
- 7 NONLINEAR STABILITY
- APPENDIX: A CLASS OF GENERALIZED AIRY FUNCTIONS
- Addendum: Weakly non-parallel theories for the Blasius boundary layer
- Solutions
- Bibliography and author index
- Motion picture index
- Subject index
Summary
The transition of laminar flow, with its clean layers of flow tubes, to strongly mixed, irregular turbulent flow is one of the principal problems of modern hydrodynamics. It is certain that this fundamental change in type of motion of the fluid is traceable to an instability in the laminar flow, for laminar flows of themselves would always be possible solutions of the hydrodynamic equations.
– W. Tollmien (1935)Introduction
In this chapter we wish to consider the stability of steady two-dimensional or axisymmetric flows with parallel streamlines. Flows of this type were first studied experimentally by Reynolds (1883), who observed that instability could occur in quite different ways depending on the form of the basic velocity distribution. Thus, when the velocity profile is of the form shown in Fig. 4.1(a) he observed that ‘eddies showed themselves reluctantly and irregularly’ whereas when the profile is as shown in Fig. 4.1(b) the ‘eddies appeared in the middle regularly and readily’. From these observations he was led to consider the role of viscosity in flows of this type. By comparing the flow of a viscous fluid with that of an inviscid fluid, both flows being assumed to have the same basic velocity distribution, he was led to formulate two fundamental hypotheses which can be stated as follows:
First Hypothesis. The inviscid fluid may be unstable and the viscous fluid stable. The effect of viscosity is then purely stabilizing. […]
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- Hydrodynamic Stability , pp. 124 - 250Publisher: Cambridge University PressPrint publication year: 2004
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