Finite Amplitude Stability Research Articles

Transition to Turbulence in Pipe Flow

We review the results of recent experimental investigations into transition to turbulence in fluid flow through a circular straight pipe, at room temperature. The stability of Hagen–Poiseuille flow was investigated using impulsive perturbations by either injecting or sucking small amounts of fluid through holes in the wall of the pipe. The evolution of the induced patches of disturbed flow were observed using flow visualization and laser Doppler velocimetry. The principle result obtained was a finite amplitude stability curve where the critical amplitude of the disturbance required to cause transition is found to be inversely proportional to the Reynolds number. Estimates for the lower threshold value of Reynolds number which is required to sustain turbulence were also measured.

Paradoxes and problems with the longitudinal stress approximation used in glacier mechanics

The biharmonic equation in two dimensions is the governing equation for the slow plane flow of a Newtonian fluid. Where the flow domain is thin, and the aspect ratio small, the biharmonic equation can be expanded using the aspect ratio as the expansion parameter. The one-term approximation to the biharmonic equation is the lubrication approximation. In glaciology, the lubrication approximation is extended to deal with special glaciological features, and is known as the shallow ice approximation. The next order term in glaciology is known as the longitudinal stress approximation, but this is not a higher-order expansion (two-term representation) of the biharmonic equation. The first part of this paper deals with this paradox. It shows how that longitudinal stress approximation relates to the truncated biharmonic equation, and carries out a perturbation expansion in the free surface, comparing the decay spectra for surface perturbations of the shallow ice approximation, the longitudinal stress approximation, the truncated biharmonic expansion and the full system of equations. In most glaciological situations, the longitudinal stress approximation is a better approximation to the truncated biharmonic equation, and even where it is not, the loss in accuracy is relatively small. Moreover, the truncated biharmonic equation introduces spurious instabilities, while the longitudinal stress approximation does not. The second part of this paper deals with the perturbation expansion in terms of the widthto-length ratio of a perfectly slippery streaming flow of ice, where flow is retarded at the lateral margins. The case of a linear rheology is considered. The zeroth order equation for the thickness is a non-linear diffusion equation, whereas the first-order equation is linear, even though a finite amplitude perturbation is being considered. The equation is second-order differential in the flow direction, and integral in the transverse direction. Some numerical examples show finite amplitude stability. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

COMPUTER SIMULATION FOR SUBCRITICAL CONVECTION IN MULTI‐COMPONENT ALLOYS

Subcritical convection with hexagonal flow pattern is registered in 3D computer simulation of convective mass transfer in ternary solution under phase transition conditions. The calculations are evaluated by the classical theory of hydrodynamic stability and display a good agreement with linear and finite amplitude stability analysis. Key words: convective instability, subcritical convection, computer simulation.

On the existence of Arnold-stable barotropic flows on a rotating sphere

Application of two well-known finite-amplitude stability conditions by Arnold to two-dimensional flows on a rotating sphere is discussed, in the context of the barotropic vorticity equation. Attention is focused on zonal basic flows with nonmonotonic profiles of absolute vorticity, since for such flows stability would not descend from the Rayleigh–Kuo inflection point theorem. It is shown that global flows of this kind cannot satisfy either of Arnold’s stability conditions.

THREE-DIMENSIONAL CALCULATION OF FLOW OVER TWO-DIMENSIONAL DUNE

The finite amplitude growth and stability of ripples and dunes is believed to be the result of the interaction between the complex turbulence field created by separation at the crest of the bed feature and bedload sediment transport. Numerical simulation of this turbulence field promises to provide the detailed qualitative and quantitative information of coherent structures that is required to model the sediment transport field over ripples and dunes. We present a high-order Godunov scheme, referred to as the CIP method, for the numerical simulation of turbulence over a 2-dimensional dune. Comparison of the velocity statistics of the numerical simulation with LDV measurements over 2-dimensional dunes in a flume is favorable. Analysis of the vorticity field reveals that the coherent turbulent structures produced in the shear layer are highly three-dimensional, and cross-stream velocities should be considered in future models of bedload transport over dunes.

Finite amplitude stability of attachment line boundary layers

Two-dimensional nonlinear equilibrium solutions for the swept Hiemenz flow attachment line boundary layer are directly computed by solving the full Navier-Stokes equations as a nonlinear eigenvalue problem. The equations are discretized using the two-point fourth order compact scheme and the resulting nonlinear homogeneous equations are solved using the Newton-Raphson iteration technique. It is found that for Reynolds numbers larger than the linear critical Reynolds number of 583, the nonlinear neutral surfaces form open curves. The results showed that the subcritical instability exists near the upper branch neutral curve and supercritical equilibrium solutions exist near the lower branch. These conclusions are in agreement with the weakly nonlinear theory. However, at higher amplitudes away from the linear neutral points the nonlinear neutral surfaces show subcritical instability at lower and higher wave number regions. At Reynolds numbers lower than the critical value, the nonlinear neutral surfaces form closed loops. By reducing the Reynolds number, we found that the nonlinear critical point occurs at a Reynolds number of 511.3, below which all the two-dimensional disturbances will decay. The secondary instability of these equilibrium solutions is investigated using the Floquet theory. The results showed that these two-dimensional finite amplitude neutral solutions are unstable to three-dimensional disturbances.

Stabilité des écoulements de convection thermosolutale en cavité carrée

Bifurcation phenomena in a square enclosure, submitted to horizontal temperature and concentration gradients, is studied when the opposing buoyancy forces due to horizontal thermal and concentration gradients are equal. We perform the linear, weakly non-linear and finite amplitude stability analysis of the equilibrium solution. We verify that the onset of double diffusive convection corresponds to a transcritical bifurcation point. The subcritical solutions are strong attractors beyond a particular value of the thermal Rayleigh number which corresponds to the location of turning point. The structure of subcritical and transcritical steady solutions has been studied.

Finite amplitude stability of the wind-driven ocean circulation

Abstract The stability of the wind-driven ocean circulation to finite amplitude perturbations is explored. Sufficient conditions for stability, in the kinetic energy norm, are determined for steady solutions of the quasi-geostrophic barotropic vorticity equation on a midlatitude β-plane. In addition, the spatial pattern of the perturbation which is able to extract energy from the flow just above criticality is determined (the most energetic disturbance). The energy stability bounds are computed numerically, by using a continuation method to follow branches of steady states in parameter space, energy identities and calculus of variations. It is found that wind-driven single gyre flows are more stable than double gyre flows. For zero bottom friction and slip meridional boundaries, the most energetic disturbance is a single cell basin-wide flow. No-slip conditions at the meridional boundaries stabilize the double gyre flow and modify the structure of the most energetic disturbance to a multicellular one. Bottom friction further stabilizes the flows and its presence may strongly localize the spatial pattern of the most energetic disturbance.

Nonlinear symmetric stability of planetary atmospheres

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.

Chemo-Marangoni Convection: III. Pattern Parameters: Interface Mass Transfer

A nonlinear problem of hydrochemical Marangoni convection and mass transfer across the liquid - liquid interface in the presence of an interfacial chemical second-order reaction has been considered. Basic parameters for convective flows and concentration fields have been found using the finite amplitude stability method for weakly nonlinear structures. The diffusion flux of a surface-inert component across the interface under hydrochemical convection has been calculated. Correlations between the Sherwood number and Marangoni number as well as other physicochemical and regime parameters have been obtained.

Mean flow and turbulence fields over two‐dimensional bed forms

Detailed laser‐Doppler velocity and Reynolds stress measurements over fixed two‐dimensional bed forms are used to investigate the coupling between the mean flow and turbulence and to examine effects that play a role in producing the bed form instability and finite amplitude stability. The coupling between the mean flow and the turbulence is explored in both a spatially averaged sense, by determining the structure of spatially averaged velocity and Reynolds stress profiles, and a local sense, through computation of eddy viscosities and length scales. The measurements show that there is significant interaction between the internal boundary layer and the overlying wake turbulence produced by separation at the bed form crest. The interaction produces relatively low correlation coefficients in the internal boundary layer, which suggests that using local bottom stress to predict bed load flux may not only be erroneous, it may also disregard the essence of the bed form instability mechanism. The measurements also indicate that topographically induced acceleration over the bed form stoss slope has a more significant effect in damping the turbulence over bed forms than was previously supposed, which is hypothesized to play a role in the stabilization of fully developed bed forms.

Chemo-Marangoni Convection: II. Nonlinear Stability Analysis

A nonlinear problem on concentration-capillary instability at the interface of two liquid media in the presence of an interfacial chemical reaction of the second order has been considered. The effect of interfacial chemical reaction on the nonlinear development of interfacial instability has been studied using a finite-amplitude stability method and weak nonlinear convection pattern analysis. As has been ascertained, the incremental perturbations specifically generated by a nonlinear chemical reaction lead either to the establishment of an order space-periodic pattern or to the emergence of a chaotic flow with a sufficiently wide wave spectrum.

Application of the Direct Liapunov Method to the Problem of Symmetric Stability in the Atmosphere

Abstract The problem of symmetric stability is examined within the context of the direct Liapunov method. The sufficient conditions for stability derived by Fjortoft are shown to imply finite-amplitude, normed stability. This finite-amplitude stability theorem is then used to obtain rigorous upper bounds on the saturation amplitude of disturbances to symmetrically unstable flows.By employing a virial functional, the necessary conditions for instability implied by the stability theorem are shown to be in fact sufficient for instability. The results of Ooyama are improved upon insofar as a tight two-sided (upper and lower) estimate is obtained of the growth rate of (modal or nonmodal) symmetric instabilities.The case of moist adiabatic systems is also considered.

Finite-amplitude shear waves in a channel with compliant boundaries

The problem of the finite-amplitude stability of an incompressible fluid flowing in a channel with compliant walls is studied by numerically calculating the traveling wave solutions that bifurcate from plane Poiseuille flow along a neutral stability curve in the Reynolds number, wave-number plane. This curve is the zero-amplitude intersection of a neutral stability surface in amplitude, Reynolds number, and wave-number space. To determine the differences between the rigid and compliant wall flow, solutions are generated for several compliant wall models. For sufficiently compliant walls the finite-amplitude solutions are stable to two-dimensional periodic disturbances in the limit of small amplitude. However, the solutions destabilize as the amplitude increases. Since these stable solutions exist for such a small range of energy and Reynolds number, these calculations indicate that the process of transition for compliant boundaries is (at least for small amplitude) qualitatively the same as that for rigid boundaries.

The initial instability and finite-amplitude stability of alternate bars in straight channels

The initial instability and fully developed stability of alternate bars in straight channels are investigated using linearized and nonlinear analyses. The fundamental instability leading to these features is identified through a linear stability analysis of the equations governing the flow and sediment transport fields. This instability is explained in terms of topographically induced steering of the flow and the associated pattern of erosion and deposition on the bed. While the linear theory is useful for examining the instability mechanism, this approach is shown to yield relatively little information about well-developed alternate bars and, specifically, the linear analysis is shown to yield poor predictions of the fully developed bar wavelength. A fully nonlinear approach is presented that permits computation of the evolution of these bed features from an initial perturbation to their fully developed morphology. This analysis indicates that there is typically substantial elongation of the bar wavelength during the evolution process, a result that is consistent with observations of bar development in flumes and natural channels. The nonlinear approach demonstrates that the eventual stability of these features is a result of the interplay between topographic steering effects, secondary flow production as a result of streamline curvature, and gravitationally induced modifications of sediment fluxes over a sloping bed.

The initial instability and finite-amplitude stability of alternate bars in straight channels

The initial instability and finite-amplitude stability of alternate bars in straight channels

Finite amplitude stability of a plane shear layer

Abstract We analyze the finite amplitude stability of a planar shear layer near the marginal stability point in the limit of large wavelengths and small Reynolds numbers. We find a subcritical bifurcation and therefore instability to finite amplitude perturbations where linear analysis predicts stability. This result is opposite to that found by previous analyses done in the high Reynolds number regime, where a supercritical bifurcation was found.

Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

A numerical study of the nonlinear stability of the eastern ocean circulation

Nonlinear unstable waves in the eastern ocean basin are analyzed using a three‐layer quasi‐geostrophic numerical model. The upper two layers are forced to maintain a mean circulation in the absence of mesoscale motion. The waves are produced by initial small perturbations added to the mean flow. Two experiments, focused on the eastern ocean basin and differing only by mean flow strength, are performed. A region of high eddy kinetic energy is formed in the southern region, but the eastern and northern regions are vacant of eddies because of the energy efflux by westward and southwestward propagating unstable waves. Regional energetics are calculated to highlight the differences between the eastern and southern regions. Two dominant waves are found in the southern region. Waves with a period of 120 days propagate to the west and form several wave trains parallel to the southern boundary. Longer‐period waves with a period of 200 days propagate to the southwest. Comparisons with observed eddies in the northeast Pacific are discussed. Besides baroclinic instability, it is found that the energy flux by propagating waves is very important in the finite amplitude stability of the eastern ocean general circulation.

Finite-amplitude stability analysis of liquid films down a vertical wall with and without interfacial phase change

The generalized kinematic equation for film thickness, taking into account the effect of phase change at the interface, is used to investigate the nonlinear stability of film flow down a vertical wall. The analysis shows that supercritical stability and subcritical instability are both possible for the film flow system. Applications of the result to isothermal, condensate and evaporate film flow show that mass transfer into (away from) the liquid phase will stabilize (destabilize) the film flow. Finally, we find that supercritical filtered waves are always linearly stable with regard to side-band disturbance.