The linear stability of viscous compressible plane Couette flow

1989 ◽  
Vol 202 ◽  
pp. 515-541 ◽  
Author(s):  
W. Glatzel
Keyword(s):  
Phase Velocity ◽  
Linear Stability ◽  
Couette Flow ◽  
Normal Modes ◽  
Reynolds Numbers ◽  
Plane Couette Flow ◽  
Phase Velocities ◽  
Viscous Instability ◽  
Supersonic Regime ◽  
Critical Reynolds Numbers

The structure of normal modes in viscous compressible plane Couette flow is investigated. The spectrum is found to consist of two types of modes: the viscous modes, which obtain finite phase velocities by the mechanism of mode pairing; and the sonic modes, whose phase velocity becomes distorted in the supersonic regime. This leads to mode crossings which unfold, depending on the type of crossing modes, either into purely sonic or viscous-sonic instability bands. The latter provide a new example for viscous instability. Both mode pairing of viscous modes and distortion of the phase velocity of sonic modes is caused by the shear. Critical Reynolds numbers for the instabilities are derived.

scholarly journals On the linear stability of compressible plane Couette flow

1994 ◽  
Vol 258 ◽  
pp. 131-165 ◽  
Author(s):  
Peter W. Duck ◽  
Gordon Erlebacher ◽  
M. Yousuff Hussaini
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Small Amplitude ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Plane Couette Flow ◽  
Moving Wall ◽  
Type Boundary ◽  
The Stability ◽  
Radiation Type

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

The stability of plane Couette flow with viscosity stratification

1969 ◽  
Vol 36 (4) ◽  
pp. 685-693 ◽  
Author(s):  
Alex D. D. Craik
Keyword(s):  
Phase Velocity ◽  
Couette Flow ◽  
Liquid Velocity ◽  
Reynolds Numbers ◽  
Critical Layer ◽  
Plane Couette Flow ◽  
Small Reynolds Numbers ◽  
Liquid Depth ◽  
The Stability

The stability of plane Couette flow is examined for liquids in which the viscosity varies with depth. Under suitable conditions, the flow may be stable or unstable at small Reynolds numbers for disturbances with wavelengths long compared with the liquid depth.A mechanism which may be stabilizing or destabilizing is found to derive from the role of diffusion in the neighbourhood of a ‘critical layer’, where the liquid velocity equals the phase velocity of a wave-like disturbance. This mechanism, which requires a viscosity gradient, is quite distinct from that found by Yih (1967) for the case of a viscosity discontinuity. An example is considered, for which these two mechanisms may be of comparable importance.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

scholarly journals Linear stability of boundary layers under solitary waves

2014 ◽  
Vol 761 ◽  
pp. 62-104 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen
Keyword(s):  
Linear Stability ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Linear Instability ◽  
Navier Stokes ◽  
Acceleration Region ◽  
Critical Reynolds Numbers ◽  
Tollmien Schlichting ◽  
The Stability

AbstractIn the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.

An investigation of turbulent plane Couette flow at low Reynolds numbers

1995 ◽  
Vol 286 ◽  
pp. 291-325 ◽  
Author(s):  
Knut H. Bech ◽  
Nils Tillmark ◽  
P. Henrik Alfredsson ◽  
Helge I. Andersson
Keyword(s):  
Couette Flow ◽  
Reynolds Stress ◽  
Reynolds Numbers ◽  
Wall Turbulence ◽  
Turbulence Structure ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
Low Reynolds Numbers ◽  
Low Reynolds ◽  
Near Wall

The turbulent structure in plane Couette flow at low Reynolds numbers is studied using data obtained both from numerical simulation and physical experiments. It is shown that the near-wall turbulence structure is quite similar to what has earlier been found in plane Poiseuille flow; however, there are also some large differences especially regarding Reynolds stress production. The commonly held view that the maximum in Reynolds stress close to the wall in Poiseuille and boundary layer flows is due to the turbulence-generating events must be modified as plane Couette flow does not exhibit such a maximum, although the near-wall coherent structures are quite similar. For two-dimensional mean flow, turbulence production occurs only for the streamwise fluctuations, and the present study shows the importance of the pressure—strain redistribution in connection with the near-wall coherent events.

scholarly journals Dynamics of spatially localized states in transitional plane Couette flow

2019 ◽  
Vol 867 ◽  
pp. 414-437 ◽  
Author(s):  
Anton Pershin ◽  
Cédric Beaume ◽  
Steven M. Tobias
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Initial Conditions ◽  
Reynolds Numbers ◽  
Localized States ◽  
Transition To Turbulence ◽  
Dynamical Structure ◽  
Plane Couette Flow ◽  
Homoclinic Snaking ◽  
Chaotic Transients

Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider et al., Phys. Rev. Lett., vol. 104, 2010, 104501). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed ($Re<175$) to transitional ones ($Re\approx 325$), these spatially localized states traverse various regimes where their relaminarization time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterized in this paper for a $4\unicode[STIX]{x03C0}$-periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarizing. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarizing. Very long chaotic transients ($t>10^{4}$) can be observed without difficulty for relatively low values of the Reynolds number ($Re\approx 250$).

On the behaviour of small disturbances in plane Couette flow with a temperature gradient

1965 ◽  
Vol 286 (1404) ◽  
pp. 117-128 ◽  
Keyword(s):  
Rayleigh Number ◽  
Couette Flow ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Plane Couette Flow ◽  
Infinitesimal Disturbance ◽  
Prandtl Numbers ◽  
The Stability ◽  
Small Disturbances

The stability of plane Couette flow with a heated lower plate is considered with respect to a two-dimensional infinitesimal disturbance. The eigenvalues are found with the aid of a digital computer as the latent roots of a matrix. Neutral stability curves for various Prandtl numbers at Reynolds numbers up to 150 are obtained by a second method. It is found that the principle of the exchange of stabilities does not hold for this problem. With the aid of Squire’s transformation the conclusion is drawn that all fluids will become unstable at the same value of the Rayleigh number irrespective of whether shear is present or not.

On the stability of viscous plane Couette flow

1963 ◽  
Vol 15 (4) ◽  
pp. 623-630 ◽  
Author(s):  
J. W. Deardorff
Keyword(s):  
Boundary Layer ◽  
Couette Flow ◽  
Reynolds Numbers ◽  
Plane Couette Flow ◽  
Flow Channels ◽  
Boundary Layer Instability ◽  
The Stability

The problem of the stability of plane Couette flow to infinitesimal disturbances is carried numerically to larger Reynolds numbers than heretofore. The flow is definitely stable up to R = 1430, although at Reynolds numbers in this vicinity plane Couette flow has been observed experimentally to be turbulent. Observations on boundary-layer instability suggest this to be the source of turbulence in plane Couette flow channels.

scholarly journals Benchmark solution for the stability of plane Couette flow with net throughflow

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
B. M. Shankar ◽  
I. S. Shivakumara
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Couette Flow ◽  
Normal Modes ◽  
Base Flow ◽  
Parallel Plates ◽  
Plane Couette Flow ◽  
Benchmark Solution ◽  
Vertical Throughflow ◽  
The Stability

AbstractThis paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow. A linear stability analysis has been performed by employing the method of normal modes and the resulting stability equation is solved numerically using the Chebyshev collocation method. Contrary to the stability of plane Couette flow (PCF) to small disturbances for all values of the Reynolds number in the absence of vertical throughflow, it is found that PCF becomes unstable owing to the change in the sign of growth rate depending on the magnitude of throughflow. The critical Reynolds number triggering the instability is computed for different values of throughflow dependent Reynolds number and it is shown that throughflow instills both stabilizing and destabilizing effect on the base flow. It is seen that the direction of throughflow has no influence on the stability of fluid flow. A comparative study between plane Poiseuille flow and PCF has also been carried out and the similarities and differences are highlighted.

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