scholarly journals Instability of the interface of two viscous fluids moving in a channel. Part II a. Nonlinear problem

1989 ◽  
Vol 11 (3) ◽  
pp. 52-59
Author(s):  
Tran Van Tran
Keyword(s):  
Steady State ◽  
Nonlinear Problem ◽  
Hydrodynamic Stability ◽  
Evolution Equations ◽  
Finite Amplitude ◽  
Viscous Fluids ◽  
Weakly Nonlinear ◽  
Concurrent Flow ◽  
Periodic Steady State

In this paper, the method of multiple scaling is used for obtaining the am-altitude evolution equations from the weakly nonlinear problem of hydrodynamic stability of concurrent flow of two viscous fluids in a channel It. is shown that in the case of stability the, interface may evolve to some finite amplitude with periodic steady state.

An asymptotic theory of nonlinear stratified flow of large depth over topography

1993 ◽  
Vol 440 (1910) ◽  
pp. 639-653 ◽  
Keyword(s):  
Steady State ◽  
Asymptotic Theory ◽  
Evolution Equations ◽  
Stratified Flow ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Transient Behaviour ◽  
Large Depth ◽  
Special Cases

The generation of internal-wave disturbances by continuously stratified flow of large depth over topography is studied in the Boussinesq approximation. For constant background velocity and Brunt-Väisälä frequency, the model of Long (1953) predicts a class of two-dimensional nonlinear steady-state wave patterns, under the hypothesis of no upstream effects from the topography. Here, Long’s theory is generalized to describe the dynamics of slightly unsteady nonlinear disturbances near the hydrostatic limit (extended topography), allowing for the presence of weak variations in the background flow. A pair of nonlinear integral-differential amplitude evolution equations are derived for disturbances of finite amplitude that remain valid as long as no flow reversal occurs. Previous weakly nonlinear or steady-state theories follow from this formulation as special cases. The proposed asymptotic theory seems well suited for investigating the long-time transient behaviour of the nonlinear response, and the validity of Long’s hypothesis of no upstream influence.

An investigation of transition to turbulence in bounded oscillatory Stokes flows Part 2. Numerical simulations

1991 ◽  
Vol 225 ◽  
pp. 423-444 ◽  
Author(s):  
R. Akhavan ◽  
R. D. Kamm ◽  
A. H. Shapiro
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Quasi Equilibrium ◽  
Two Dimensional ◽  
Spectral Techniques ◽  
Periodic Steady State

The stability of oscillatory channel flow to different classes of infinitesimal and finite-amplitude two- and three-dimensional disturbances has been investigated by direct numerical simulations of the Navier–Stokes equations using spectral techniques. All infinitesimal disturbances were found to decay monotonically to a periodic steady state, in agreement with earlier Floquet theory calculations. However, before reaching this periodic steady state an infinitesimal disturbance introduced in the boundary layer was seen to experience transient growth in accordance with the predictions of quasi-steady theories for the least stable eigenmodes of the Orr–Sommerfeld equation for instantaneous ‘frozen’ profiles. The reason why this growth is not sustained in the periodic steady state is explained. Two-dimensional infinitesimal disturbances reaching finite amplitudes were found to saturate in an ordered state of two-dimensional quasi-equilibrium waves that decayed on viscous timescales. No finite-amplitude equilibrium waves were found in our cursory study. The secondary instability of these two-dimensional finite-amplitude quasi-equilibrium states to infinitesimal three-dimensional perturbations predicts transitional Reynolds numbers and turbulent flow structures in agreement with experiments.

Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

1999 ◽  
Vol 396 ◽  
pp. 73-108 ◽  
Author(s):  
D. M. MASON ◽  
R. R. KERSWELL
Keyword(s):  
Finite Amplitude ◽  
Complex Conjugate ◽  
Small Scale ◽  
Conjugate Pair ◽  
Weakly Nonlinear ◽  
Elliptical Instability ◽  
Elliptical Flow ◽  
Complex Conjugate Pair ◽  
Generic Instability ◽  
Secondary Instabilities

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

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