Stability of Horizontal Boundary Layers

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Mathematical Proof ◽  
Vertical Component ◽  
Ekman Layer ◽  
Numerical Results ◽  
Energy Estimates ◽  
Navier Stokes ◽  
The 1960S ◽  
The Stability

Let us now detail the stability properties of an Ekman layer introduced in Part I, page 11. First we will recall how to compute the critical Reynolds number. Then we will describe briefly what happens at larger Reynolds numbers. The first step in the study of the stability of the Ekman layer is to consider the linear stability of a pure Ekman spiral of the form where U∞ is the velocity away from the layer and ζ is the rescaled vertical component ζ = x3/√εν. The corresponding Reynolds number is Let us consider the Navier–Stokes–Coriolis equations, linearized around uE The problem is now to study the (linear) stability of the 0 solution of the system (LNSCε). If u=0 is stable we say that uE is linearly stable, if not we say that it is linearly unstable. Numerical results show that u=0 is stable if and only if Re<Rec where Rec can be evaluated numerically. Up to now there is no mathematical proof of this fact, and it is only possible to prove that 0 is linearly stable for Re<Re1 and unstable for Re>Re2 with Re1<Rec<Re2, Re1 being obtained by energy estimates and Re2 by a perturbative analysis of the case Re=∞. We would like to emphasize that the numerical results are very reliable and can be considered as definitive results, since as we will see below, the stability analysis can be reduced to the study of a system of ordinary differential equations posed on the half-space, with boundary conditions on both ends, a system which can be studied arbitrarily precisely, even on desktop computers (first computations were done in the 1960s by Lilly).

scholarly journals On the Importance of the Influence of both Velocity and Aspect Ratios on the Occurrence of Bifurcation Phenomena within a Two-Sided Lid-Driven Enclosure

2015 ◽  
Vol 55 ◽  
pp. 160-172
Author(s):  
Fayçal Hammami ◽  
Nader Ben Cheikh ◽  
Brahim Ben Beya
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Numerical Results ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Left Wall ◽  
Driven Cavity ◽  
Aspect Ratios ◽  
The Stability ◽  
The Right

This paper deals with the numerical study of bifurcations in a two-sided lid driven cavity flow. The flow is generated by moving the upper wall to the right while moving the left wall downwards. Numerical simulations are performed by solving the unsteady two dimensional Navier-Stokes equations using the finite volume method and multigrid acceleration. In this problem, the ratio of the height to the width of the cavity are ranged from H/L = 0.25 to 1.5. The code for this cavity is presented using rectangular cavity with the grids 144 × 36, 144 × 72, 144 × 104, 144 × 136, 144 × 176 and 144 × 216. Numerous comparisons with the results available in the literature are given. Very good agreements are found between current numerical results and published numerical results. Various velocity ratios ranged in 0.01≤ α ≤ 0.99 at a fixed aspect ratios (A = 0.5, 0.75, 1.25 and 1.5) were considered. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. The stability analysis depending on the aspect ratio, velocity ratios α and the Reynolds number when transition phenomenon occurs is considered in this paper.

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.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Non-parallel linear stability analysis of the vertical boundary layer in a differentially heated cavity

1997 ◽  
Vol 352 ◽  
pp. 265-281 ◽  
Author(s):  
A. M. H. BROOKER ◽  
J. C. PATTERSON ◽  
S. W. ARMFIELD
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Navier Stokes ◽  
Parabolized Stability Equations ◽  
The Stability ◽  
Flow Quantity ◽  
Energy Equations ◽  
Made In

A non-parallel linear stability analysis which utilizes the assumptions made in the parabolized stability equations is applied to the buoyancy-driven flow in a differentially heated cavity. Numerical integration of the complete Navier–Stokes and energy equations is used to validate the non-parallel theory by introducing an oscillatory heat input at the upstream end of the boundary layer. In this way the stability properties are obtained by analysing the evolution of the resulting disturbances. The solutions show that the spatial growth rate and wavenumber are highly dependent on the transverse location and the disturbance flow quantity under consideration. The local solution to the parabolized stability equations accurately predicts the wave properties observed in the direct simulation whereas conventional parallel stability analysis overpredicts the spatial amplification and the wavenumber.

Linear stability theory of oscillatory Stokes layers

1974 ◽  
Vol 62 (4) ◽  
pp. 753-773 ◽  
Author(s):  
Christian Von Kerczek ◽  
Stephen H. Davis
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Stability Theory ◽  
Absolute Stability ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Full Time ◽  
Full Theory ◽  
The Stability

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to bemore stablethan that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.

Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids

2007 ◽  
Vol 592 ◽  
pp. 177-194 ◽  
Author(s):  
C. NOUAR ◽  
A. BOTTARO ◽  
J. P. BRANCHER
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Channel Flow ◽  
Shear Thinning ◽  
Transition To Turbulence ◽  
Shear Thinning Fluids ◽  
Viable Approach ◽  
The Stability ◽  
Definition Of ◽  
Do So

A viscosity stratification is considered as a possible mean to postpone the onset of transition to turbulence in channel flow. As a prototype problem, we focus on the linear stability of shear-thinning fluids modelled by the Carreau rheological law. To assess whether there is stabilization and by how much, it is important both to account for a viscosity disturbance in the perturbation equations, and to employ an appropriate viscosity scale in the definition of the Reynolds number. Failure to do so can yield qualitatively and quantitatively incorrect conclusions. Results are obtained for both exponentially and algebraically growing disturbances, demonstrating that a viscous stratification is a viable approach to maintain laminarity.

scholarly journals A SEMI-IMPLICIT SCHEME FOR SOLVING INCOMPRESSIBLE VISCOUS FREE SURFACE FLOWS

2005 ◽  
Vol 4 (2) ◽  
Author(s):  
C. M. Oishi ◽  
J. A. Cuminato ◽  
V. G. Ferreira ◽  
M. F. Tomé ◽  
A. Castelo ◽  
...  
Keyword(s):  
Free Surface ◽  
Numerical Method ◽  
Stokes Equations ◽  
Free Surface Flows ◽  
Numerical Results ◽  
Navier Stokes ◽  
Variable Order ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
The Stability

The present work is concerned with a numerical method for solving the two-dimensional time-dependent incompressible Navier-Stokes equations in the primitive variables formulation. The diffusive terms are treated by Implicit Backward and Crank-Nicolson methods, and the non-linear convection terms are, explicitly, approximated by the high order upwind VONOS (Variable-Order Non-Oscillatory Scheme) scheme. The boundary conditions for the pressure field at the free surface are treated implicitly, and for the velocity field explicitly. The numerical method is then applied to the simulation of free surface and confined flows. The numerical results show that the present technique eliminates the stability restriction in the original explicit method. For low Reynolds number flow dynamics, the method is robust and produces numerical results that compare very well with the analytical solutions.

On the stability of flow in an elliptic pipe which is nearly circular

1978 ◽  
Vol 87 (2) ◽  
pp. 233-241 ◽  
Author(s):  
A. Davey
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Major Axis ◽  
Reynolds Numbers ◽  
Minor Axis ◽  
Critical Reynolds Numbers ◽  
The Stability

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

Linear Stability of Flow in a Concentric Annulus

1977 ◽  
Vol 44 (2) ◽  
pp. 338-340
Author(s):  
A. K. Agarwal ◽  
V. K. Garg
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Annular Flow ◽  
Parallel Flow ◽  
Concentric Annulus ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of fully developed, parallel flow in a rigid, concentric annulus to infinitesimal disturbances is analyzed. It is found that the annular flow is spatially stable to axisymmetric disturbances upto a modified Reynolds number of 10,000.

scholarly journals A SEMI-IMPLICIT SCHEME FOR SOLVING INCOMPRESSIBLE VISCOUS FREE SURFACE FLOWS

2005 ◽  
Vol 4 (2) ◽  
pp. 106
Author(s):  
C. M. Oishi ◽  
J. A. Cuminato ◽  
V. G. Ferreira ◽  
M. F. Tomé ◽  
A. Castelo ◽  
...  
Keyword(s):  
Free Surface ◽  
Numerical Method ◽  
Stokes Equations ◽  
Free Surface Flows ◽  
Numerical Results ◽  
Navier Stokes ◽  
Variable Order ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
The Stability

The present work is concerned with a numerical method for solving the two-dimensional time-dependent incompressible Navier-Stokes equations in the primitive variables formulation. The diffusive terms are treated by Implicit Backward and Crank-Nicolson methods, and the non-linear convection terms are, explicitly, approximated by the high order upwind VONOS (Variable-Order Non-Oscillatory Scheme) scheme. The boundary conditions for the pressure field at the free surface are treated implicitly, and for the velocity field explicitly. The numerical method is then applied to the simulation of free surface and confined flows. The numerical results show that the present technique eliminates the stability restriction in the original explicit method. For low Reynolds number flow dynamics, the method is robust and produces numerical results that compare very well with the analytical solutions.

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