New results in rotating Hagen–Poiseuille flow

2000 ◽  
Vol 417 ◽  
pp. 103-126 ◽  
Author(s):  
D. R. BARNES ◽  
R. R. KERSWELL
Keyword(s):  
Hopf Bifurcation ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Axial Rotation ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Rotation Rates ◽  
Supercritical Hopf Bifurcation

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ωa, Ωp]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ωa and at Ωp about a perpendicular diameter. For purely axial rotation (RHPF[Ωa, 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ωp] which connects smoothly to Hagen–Poiseuille flow as Ωp → 0 is found to be stable at all Reynolds numbers below 104. At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ωa, 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.

Finite-amplitude equilibrium states in plane Couette flow

1997 ◽  
Vol 342 ◽  
pp. 159-177 ◽  
Author(s):  
A. CHERHABILI ◽  
U. EHRENSTEIN
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Equilibrium States ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Plane Couette Flow

A numerical bifurcation study in plane Couette flow is performed by computing successive finite-amplitude equilibrium states, solutions of the Navier–Stokes equations. Plane Couette flow being linearly stable for all Reynolds numbers, first two-dimensional equilibrium states are computed by extending nonlinear travelling waves in plane Poiseuille flow through the Poiseuille–Couette flow family to the plane Couette flow limit. The resulting nonlinear states are stationary with a spatially localized structure; they are subject to two-dimensional and three-dimensional secondary disturbances. Three-dimensional disturbances dominate the dynamics and three-dimensional stationary equilibrium states bifurcating at criticality on the two-dimensional equilibrium surface are computed. These nonlinear states, periodic in the spanwise direction and spatially localized in the streamwise direction, are computed for Reynolds numbers (based on half the velocity difference between the walls and channel half-width) close to 1000. While a possible relationship between the computed solutions and experimentally observed coherent structures in turbulent plane Couette flow has to be assessed, the present findings reinforce the idea that subcritical transition may be related to the existence of finite-amplitude states which are (unstable) fixed points in a dynamical systems formulation of the Navier–Stokes system.

Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

1995 ◽  
Vol 291 ◽  
pp. 57-81 ◽  
Author(s):  
S. M. Churilov ◽  
I. G. Shukhman
Keyword(s):  
Travelling Waves ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Layer ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
New Type ◽  
Two Stages

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

The anatomy of the mixing transition in homogeneous and stratified free shear layers

2000 ◽  
Vol 413 ◽  
pp. 1-47 ◽  
Author(s):  
C. P. CAULFIELD ◽  
W. R. PELTIER
Keyword(s):  
Shear Layer ◽  
Three Dimensional ◽  
Streamwise Vortex ◽  
Shear Layers ◽  
Finite Amplitude ◽  
Small Scale ◽  
Two Dimensional ◽  
Streamwise Vortices ◽  
Free Shear Layers ◽  
Nonlinear Amplification

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity

1990 ◽  
Vol 217 ◽  
pp. 519-527 ◽  
Author(s):  
M. Nagata
Keyword(s):  
Couette Flow ◽  
Rotation Rate ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Narrow Gap ◽  
Plane Couette Flow ◽  
Bifurcation Problem ◽  
Bifurcation From Infinity ◽  
Flow Bifurcation ◽  
Average Rotation

Finite-amplitude solutions of plane Couette flow are discovered. They take a steady three-dimensional form. The solutions are obtained numerically by extending the bifurcation problem of a circular Couette system between co-rotating cylinders with a narrow gap to the case with zero average rotation rate.

The wall jet in a rotating fluid

1997 ◽  
Vol 335 ◽  
pp. 1-28 ◽  
Author(s):  
MELVIN E. STERN ◽  
ERIC P. CHASSIGNET ◽  
J. A. WHITEHEAD
Keyword(s):  
Vertical Wall ◽  
Three Dimensional ◽  
Point Vortex ◽  
Separation Distance ◽  
Finite Amplitude ◽  
Large Reynolds Number ◽  
Spatial Evolution ◽  
Two Dimensional ◽  
Rotating Fluid ◽  
Coriolis Parameter

The previously observed spatial evolution of the two-dimensional turbulent flow from a source on the vertical wall of a shallow layer of rapidly rotating fluid is strikingly different from the non-rotating three-dimensional counterpart, insofar as the instability eddies generated in the former case cause the flow to separate completely from the wall at a finite downstream distance. In seeking an explanation of this, we first compute the temporal evolution of two-dimensional finite-amplitude waves on an unstable laminar jet using a finite difference calculation at large Reynolds number. This yields a dipolar vorticity pattern which propagates normal to the wall, while leaving some of the near-wall vorticity (negative) of the basic flow behind. The residual far-field eddy therefore contains a net positive circulation and this property is incorporated in a heuristic point-vortex model of the spatial evolution of the instability eddies observed in a laboratory experiment of a flow emerging from a source on a vertical wall in a rotating tank. The model parameterizes the effect of Ekman bottom friction in decreasing the circulation of eddies which are periodically emitted from the source flow on the wall. Further downstream, the point vortices of the model merge and separate abruptly from the wall; the statistics suggest that the downstream separation distance scales with the Ekman spin-up time (inversely proportional to the square root of the Coriolis parameter f) and with the mean source velocity. When the latter is small and f is large, qualitative support is obtained from laboratory experiments.

scholarly journals Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ruiqing Shi ◽  
Junmei Qi ◽  
Sanyi Tang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Three Dimensional ◽  
Distributed Delay ◽  
Dimensional System ◽  
Two Dimensional ◽  
Normal Form Theory ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory

We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delayτas a bifurcation parameter, we show that Hopf bifurcation can occur as the time delayτpasses through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.

Upper bounds for turbulent Couette flow incorporating the poloidal power constraint

1996 ◽  
Vol 328 ◽  
pp. 161-176 ◽  
Author(s):  
R. R. Kerswell ◽  
A. M. Soward
Keyword(s):  
Couette Flow ◽  
Three Dimensional ◽  
Optimal Solution ◽  
Momentum Equation ◽  
Momentum Transport ◽  
Large Reynolds Number ◽  
Main Stream ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Lagrange Equations

The upper bound on momentum transport in the turbulent regime of plane Couette flow is considered. Busse (1970) obtained a bound from a variational formulation based on total energy conservation and the mean momentum equation. Two-dimensional asymptotic solutions of the resulting Euler-Lagrange equations for the system were obtained in the large-Reynolds-number limit. Here we make a toroidal poloidal decomposition of the flow and impose an additional power integral constraint, which cannot be satisfied by two-dimensional flows. Nevertheless, we show that the additional constraint can be met by only small modifications to Busse's solution, which leaves his momentum transport bound unaltered at lowest order. On the one hand, the result suggests that the addition of further integral constraints will not significantly improve bound estimates. On the other, our optimal solution, which possesses a weak spanwise roll in the outermost of Busse's nested boundary layers, appears to explain the three-dimensional structures observed in experiments. Only in the outermost boundary layer and in the main stream is the solution three-dimensional. Motion in the thinner layers remains two-dimensional characterized by streamwise rolls.

Development of finite-amplitude two-dimensional and three-dimensional disturbances in jet flows

1986 ◽  
Vol 20 (5) ◽  
pp. 668-679
Author(s):  
S. Ya. Gertsenshtein ◽  
I. I. Olaru ◽  
A. Ya. Hudnitokii ◽  
A. N. Sukhorukov
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Jet Flows ◽  
Two Dimensional
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