An algorithmic approach to the linear stability of the Ekman layer

1983 ◽  
Vol 132 ◽  
pp. 283-293 ◽  
Author(s):  
Mogens V. Melander
Keyword(s):  
Linear Stability ◽  
Maximum Growth ◽  
Ekman Layer ◽  
Plane Boundary ◽  
Neutral Stability ◽  
Nonlinear Stability Analysis ◽  
Algorithmic Approach ◽  
Analytical Formulas ◽  
Boundary Layer Problems ◽  
Layer Flow

The linear stability of the stationary Ekman-layer flow near a plane boundary is considered. Analytical formulas for the eigenfunctions are derived by a spectral analysis. Standard optimization algorithms are used to calculate critical points, maximum growth rates and neutral-stability curves. The near approach provides a better basis for both a linear and a nonlinear stability analysis than the well-known methods have done. The method may also be applied to other boundary-layer problems.

scholarly journals Nonlinear stability analysis of Darcy’s flow with viscous heating

2016 ◽  
Vol 472 (2189) ◽  
pp. 20160036 ◽  
Author(s):  
Michele Celli ◽  
Leonardo S. de B. Alves ◽  
Antonio Barletta
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Viscous Dissipation ◽  
Nonlinear Stability ◽  
Linear Stability Analysis ◽  
Integral Transform ◽  
Internal Heat ◽  
Neutral Stability ◽  
Nonlinear Stability Analysis ◽  
Rate Analysis

The nonlinear stability of a rectangular porous channel saturated by a fluid is here investigated. The aspect ratio of the channel is assumed to be variable. The channel walls are considered impermeable and adiabatic except for the horizontal top which is assumed to be isothermal. The viscous dissipation is acting inside the channel as internal heat generator. A basic throughflow is imposed, and the nonlinear convective stability is investigated by means of the generalized integral transform technique. The neutral stability curve is compared with the one obtained by the linear stability analysis already present in the literature. The growth rate analysis of different unstable modes is performed. The Nusselt number is investigated for several supercritical configurations in order to better understand how the system behaves when conditions far away from neutral stability are considered. The patterns of the neutrally stable convective cells are also reported. Nonlinear simulations support the results obtained by means of the linear stability analysis, confirming that viscous dissipation alone is indeed capable of inducing mixed convection. Low Gebhart or high Péclet numbers lead to a transient overheating of the originally motionless fluid before it settles in its convective steady state.

Stability Analysis of a Thin Pseudoplastic Fluid with Condensation Effects Flowing on a Rotating Circular Disk

2013 ◽  
Vol 699 ◽  
pp. 413-421
Author(s):  
Ming Che Lin
Keyword(s):  
Linear Stability ◽  
Kinematic Model ◽  
Circular Disk ◽  
Vital Role ◽  
Linear Growth Rate ◽  
Flow Index ◽  
Neutral Stability ◽  
Pseudoplastic Fluid ◽  
Linear Solution ◽  
By Products

This paper investigates the linear stability of a thin axisymmetric pseudoplastic fluid with condensation effects flowing on a rotating circular disk. Long-wave perturbation analysis is proposed to derive a generalized kinematic model of the physical system with a small Reynolds number. The method of normal mode is applied to study the linear stability. The neutral stability curve and the linear growth rate are obtained subsequently as the by-products of linear solution. The study reveals that the rotation number generates a destabilizing effect in pseudoplastic fluid. The degree of the flow index n plays a vital role in stabilizing the film flow.

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).

A two-lane lattice model considering taillight effect and man–machine hybrid driving

2020 ◽  
Vol 34 (32) ◽  
pp. 2050365
Author(s):  
Siyuan Chen ◽  
Changxi Ma ◽  
Jinchou Gong
Keyword(s):  
Traffic Flow ◽  
Lattice Model ◽  
Critical Density ◽  
Mkdv Equation ◽  
Flow Stability ◽  
Stability Conditions ◽  
Neutral Stability ◽  
Nonlinear Stability Analysis ◽  
Traffic Flow Stability ◽  
The Stability

At present, drivers can rely on road communication technology to obtain the current traffic status information, and the development of intelligent transportation makes self-driving possible. In this paper, considering the mixed traffic flow with self-driving vehicles and the taillight effect, a new macro-two-lane lattice model is established. Combined with the concept of critical density, the judgment conditions for vehicles to take braking measures are given. Based on the linear analysis, the stability conditions of the new model are obtained, and the mKdV equation describing the evolution mechanism of density waves is derived through the nonlinear stability analysis. Finally, with the help of numerical simulation, the phase diagram and kink–anti-kink waveform of neutral stability conditions are obtained, and the effects of different parameters of the model on traffic flow stability are analyzed. The results show that the braking probability, the proportion of self-driving vehicles and the critical density have significant effects on the traffic flow stability. Considering taillight effect and increasing the mixing ratio of self-driving vehicles can effectively enhance the stability of traffic flow, but a larger critical density will destroy the stability of traffic flow.

Linear stability of rotating Hagen-Poiseuille flow

1981 ◽  
Vol 108 ◽  
pp. 101-125 ◽  
Author(s):  
Fredrick W. Cotton ◽  
Harold Salwen
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Neutral Stability ◽  
Unstable Region ◽  
Azimuthal Index ◽  
Expansion Technique ◽  
Simple Scaling

Linear stability of rotating Hagen-Poiseuille flow has been investigated by an orthonormal expansion technique, confirming results by Pedley and Mackrodt and extending those results to higher values of the wavenumber |α|, the Reynolds number R, and the azimuthal index n. For |α| [gsim ] 2, the unstable region is pushed to considerably higher values of R and the angular velocity, Ω. In this region, the neutral stability curves obey a simple scaling, consistent with the unstable modes being centre modes. For n = 1, individual neutral stability curves have been calculated for several of the low-lying eigenmodes, revealing a complicated coupling between modes which manifests itself in kinks, cusps and loops in the neutral stability curves; points of degeneracy in the R, Ω plane; and branching behaviour on curves which circle a point of degeneracy.

Linear Stability of Momentum Boundary Layer Flow and Heat Transfer Over a Moving Wedge

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
Noor E. Misbah ◽  
M. C. Bharathi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Linear Stability ◽  
Boundary Layer Flow ◽  
Shooting Methods ◽  
Flow And Heat Transfer ◽  
Moving Wedge ◽  
Layer Flow ◽  
Steady Solutions ◽  
Similar Solutions

Abstract This paper studies the linear stability of the unsteady boundary-layer flow and heat transfer over a moving wedge. Both mainstream flow outside the boundary layer and the wedge velocities are approximated by the power of the distance along the wedge wall. In a similar manner, the temperature of the wedge is approximated by the power of the distance that leads to a wall exponent temperature parameter. The governing boundary layer equations admit a class of self-similar solutions under these approximations. The Chebyshev collocation and shooting methods are utilized to predict the upper and lower branch solutions for various parameters. For these two solutions, the velocity, temperature profiles, wall shear-stress, and temperature gradient are entirely different and need to be assessed for their stability as to which of these solutions is practically realizable. It is shown that algebraically growing steady solutions do exist and their effects are significant in the unsteady context. The resulting eigenvalue problem determines whether or not the steady solutions are stable. There are interesting results that are linked to bypass an important class of boundary layer flow and heat transfer. The hydrodynamics behind these results are discussed in some detail.

On the generation of waves by wind

1980 ◽  
Vol 298 (1441) ◽  
pp. 451-494 ◽  
Keyword(s):  
Surface Waves ◽  
Linear Stability ◽  
Small Amplitude ◽  
Nonlinear Effects ◽  
Amplitude Equation ◽  
Basic Flow ◽  
Parallel Plates ◽  
Cubic Nonlinearity ◽  
Linear Effects ◽  
Layer Flow

The fully developed laminar flow of air over water confined between two infinite parallel plates was used to study nonlinear effects in the generation of surface waves. A linear stability analysis of the basic flow was made and the conditions at which small amplitude surface waves first begin to grow were determined. Then, following Stewartson & Stuart (1971), the nonlinear stability of the flow was examined and the usual parabolic equation with cubic nonlinearity obtained for the amplitude of the disturbances. The calculation of the linear stability characteristics and the coefficients appearing in the amplitude equation was a lengthy computational task, with most interest centred on the coefficient of the nonlinear terms in the amplitude equation. In two profiles, used as crude models of a boundary layer flow of air over water, the calculations indicated that, over a range of parameters, the non-linear effects would reduce the growth rate of the surface waves and hence lead to equilibrium amplitude waves.

Convective Instability of the Darcy Flow in a Horizontal Layer With Symmetric Wall Heat Fluxes and Local Thermal Nonequilibrium

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
A. Barletta ◽  
M. Celli ◽  
A. V. Kuznetsov
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Solid Phase ◽  
Fluid Phase ◽  
Heat Fluxes ◽  
Local Thermal Equilibrium ◽  
Neutral Stability ◽  
Thermal Nonequilibrium ◽  
Local Thermal Nonequilibrium ◽  
Darcy Rayleigh Number

The linear stability of the parallel Darcy throughflow in a horizontal plane porous layer with impermeable boundaries subject to a symmetric net heating or cooling is investigated. The onset conditions for the secondary thermoconvective flow are expressed through a neutral stability bound for the Darcy–Rayleigh number associated with the uniform heat flux supplied or removed from the walls. The study is performed by taking into account a condition of local thermal nonequilibrium between the solid phase and the fluid phase. The linear stability analysis is carried out according to the normal modes' decomposition of the perturbations to the basic state. The governing equations for the disturbances are solved numerically as an eigenvalue problem leading to the neutral stability condition. If compared with the asymptotic condition of local thermal equilibrium, the regime of local nonequilibrium manifests an enhanced instability. This behavior is displayed by lower critical values of the Darcy–Rayleigh number, eventually tending to zero when the thermal conductivity of the solid phase is much larger than the conductivity of the fluid phase. In this special limit, which can be invoked as an approximate model of a gas-saturated metallic foam, the basic throughflow is always unstable to external disturbances of arbitrarily small amplitude.

Separation effects in Ekman layer flow over ridges

1979 ◽  
Vol 105 (443) ◽  
pp. 129-146 ◽  
Author(s):  
P. J. Mason ◽  
R. I. Sykes
Keyword(s):  
Ekman Layer ◽  
Layer Flow
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