Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity

We consider nonlinear finite-amplitude progressive shear-flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal-density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ ? 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical-layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

Resonant Generation of Finite-Amplitude Waves by Flow Past Topography on a β-Plane

The forced Korteweg-de Vries (fKdV) equation is the generic equation for resonant flow past an obstacle. However, for flow past topography on a β-plane, the case when the upstream flow is uniform is anomalous in that there is no quadratic nonlinear term in the fKdV equation. Here we show that in this important case an alternative theory is required and obtain a new evolution equation, which has some similarities to the fKdV equation with two significant differences. These are that a small-amplitude topography now produces finite-amplitude waves and the flow response is limited by a wave breakdown characterized by an incipient flow reversal. Various numerical solutions are described.     

Rossby Waves on a Shear Flow with Recirculation Cores

Large-amplitude Rossby waves riding on a background flow with a weak shear can be calculated up to a critical amplitude for which the meridional velocity, in a frame traveling with the wave, approaches zero at some point. Here we consider waves with an amplitude slightly greater than the critical amplitude by incorporating a region of recirculating fluid (vortex core) near this critical point. The effect of the vortex core is to introduce an extra nonlinear term into the equation for the wave amplitude proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. The main effect due to the vortex core is a broadening of the wave profile. Furthermore, we show that the vortex core family has a limiting amplitude, with the limiting amplitude corresponding to a semi-infinite bore.     

Existence of Neutral Rayleigh Waves in the Hagen–Poiseuille Flow Through a Pipe of Circular Cross Section

The inviscid neutral stability of Hagen–Poiseuille flow through a circular pipe is studied using both analytical and numerical techniques. A zero phase shift is applied across the critical surface to represent the effects of strong nonlinearity. Using a form of Sturm's comparison theorem it is possible to prove that no neutral solutions exist if a combination of the axial and azimuthal wave numbers of the perturbation exceeds a critical value. As a consequence, the physical problem admits only neutral solutions for an azimuthal wave number of unity.     

Ground State of the Conformal Flow on 𝕊3

We consider the conformal flow model derived in Bizoń et al. (2017) as a normal form for the conformally invariant cubic wave equation on 𝕊³. We prove that the energy attains a global constrained maximum at a family of particular stationary solutions that we call the ground state family. Using this fact and spectral properties of the linearized flow (which are interesting in their own right due to a supersymmetric structure), we prove nonlinear orbital stability of the ground state family. The main difficulty in the proof is due to the degeneracy of the ground state family as a constrained maximizer of the energy. © 2019 Wiley Periodicals, Inc.     

基于不动点方法求解非线性Falkner-Skan流动方程

Falkner-Skan流动方程描述绕楔面的流动,该方程具有很强的非线性.首先通过引入变换式,将原半无限大区域上的流动问题转化为有限区间上的两点边值问题.接着基于泛函分析中的不动点理论,采用不动点方法求解两点边值问题从而得到Falkner Skan流动方程的解.最后将不动点方法给出的结果和文献中的数值结果相比较,发现不动点方法得到的结果具有很高的精度,并且解的精度很容易通过迭代而不断得到提高.表明不动点方法是一种求解非线性微分方程行之有效的方法.     

一神非定常振荡流的稳定性分析

本文分析了一种非定常振荡的不稳定性问题.其特点是.应用偏微分方程特征理论以及O-S方程特征值的展开,求解扰动波的相函数而不是预先给定扰动波的波动形式.本文研究平面Poiseuille流与其垂向振荡流的组合流动系统.对于连续振荡源导致的波包演化,该系统存在不稳定性.     

Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

柱状固粒两相边界层流场稳定性研究

从运动方程和本构方程出发,推导得到了含柱状粒子两相流场的修正Orr-Sommerfeld方程,然后在边界层流场中,采用数值计算方法,得到了含柱状粒子流场的稳定性中性曲线,给出了流场失稳的临界雷诺数.结果表明在所述情况下,柱状粒子对流场起着抑制失稳的作用,而且抑制的程度随着柱状粒子体积分数和长径比的增加而提高.     

Hydromagnetic stability of plane Couette flow of an upper convected Maxwell fluid

Hydrodynamic stability of plane Couette flow of an upper convectedMaxwell fluid is investigated in presence of a transverse magneticfield assuming that the magnetic Prandtl number is sufficientlysmall. The resulting equation is a modified Orr–Sommerfeldequation. The equations of stability are solved numericallyusing Chebyshev collocation method with QZ algorithm. The criticalvalues of Reynolds number, wave number and wave speed are computedand the results are shown through the neutral curves. By increasingthe amount of elasticity to a certain value, it is shown that,as the Hartmann number increases, the minimum critical Reynoldsnumber decreases and it does not increase again in contrastto the Newtonian case.     

非牛顿幂律流体球向不定常渗流

本文研究了弱压缩非牛顿幂律流体球向不定常渗流,导出了抛物型偏微分非线性方程.球向扩散方程是其特殊情况.用Laplace变换的方法,找到了线性化后方程的解析解和渐近解.用影响半径的概念和平均值方法求得了近似解.渐近解和近似解的结构是相似的,从而丰富了非牛顿流体一维不定常渗流的理论.     

Numerical Simulation of Two-Phase Flow through Heterogeneous Porous Media

A mixed finite element method is combined to finite volume schemes on structured and unstructured grids for the approximation of the solution of incompressible flow in heterogeneous porous media. A series of numerical examples demonstrates the effectiveness of the methodology for a coupled system which includes an elliptic equation and a nonlinear degenerate diffusion–convection equation arising in modeling of flow and transport in porous media.     

旋转流体中圆球沿旋转轴运动产生的扰动

本文用线化理论分析了整体旋转的理想流体中有一个圆球沿旋转轴作匀速运动时流体的扰动,基于旋轴对称流动的假设导出了决定运动稳定性的扰动压力方程和扰动流函数方程.用简正模法分析了扰动流函数方程,得出了非平凡中性扰动的波数与波速必须满足的约束条件,并求出了扰动的精确表达式.文中得出结论,中性扰动共有三种可能的形式.     

Transcritical Flow Over a Hole

Transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. In the weakly nonlinear long-wave regime, this flow has been modeled with the forced Korteweg-de Vries (fKdV) equation, where numerical simulations and asymptotic solutions have demonstrated that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle. Further, it has been shown that when the obstacle is replaced by a step of semi-infinite length, it is found that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore. This result suggests that for flow over a hole, that is a step down followed by a step up, the two wavetrains generated will interact over the hole. In this paper, this situation is explored by numerical simulations of the fKdV equation.     

轴对称模型的Hagen-Poiseuille流的运动不稳定性

本文讨论流体通过圆管的运动不稳定性问题。作为流体运动所受的干扰波,我们考虑了一个非线性轴对称模型。它对应的相关振幅函数满足扩散方程,且由于复杂的分子运动和流体粘性的相互作用,当流体的雷诺数增大时其扩散系数会出现负值。如负扩散现象出现,在流体运动中出现的湍流段内会引起流体的能量集中,并扮演减少阻尼的角色。     

Solutions of the Ginzburg-Landau Equation of Interest in Shear Flow Transition

The Ginzburg-Landau equation may be used to describe the weakly nonlinear 2-dimensional evolution of a disturbance in plane Poiseuille flow at Reynolds number near critical. We consider a class of quasisteady solutions of this equation whose spatial variation may be periodic, quasiperiodic, or solitarywave- like. Of particular interest are solutions describing a transition from the laminar solution to finite amplitude states. The existence of these solutions suggests the existence of a similar class of solutions in the Navier-Stokes equations, describing pulses and fronts of instability in the flow.     

On the Modulations and Nonlinear Interactions of Waves and the Nonlinear Stability of a Near-Critical System

The nonlinear interactions and modulations of an n-dimensional wave and of a disturbance to a near-critical system governed by a general (n + 1)-dimensional system of equations are studied by perturbation methods. It is found that these modulations are governed by an evolution equation which is either by itself or coupled to a second equation, depending on the nature of the long wave solutions of the corresponding linearized system. When a single evolution equation exists, its leading terms are shown to give the nonlinear Schrödinger equation. Water waves and near-critical plane Poiseuille flow are discussed as examples.     

Interacting Wave Fronts and Rarefaction Waves in a Second Order Model of Nonlinear Thermoviscous Fluids

A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation.     

A Mean Flow First Harmonic Theory for Hydrodynamic Instabilities

A general theory is presented for nonlinear instabilities arising in steady hydrodynamic motions. For quasiparallel flows at high values of the Reynolds number it is found that for relatively small disturbance levels the usual ideas concerning the generation of higher harmonics and the subsequent modification of the fundamental may be overwhelmed by three dimensional interactions between the evolving mean flow and the first harmonic wave. The differences from and similarities to existing asymptotic and numerical studies are discussed. The theory developed applies to a variety of flow configurations. Numerical results are given for Poiseuille flow and the Blasius boundary layer. In addition the theory developed here is applied to simulate the instabilities produced in a boundary layer due to the presence of free stream disturbances.