Hydrodynamic Stability Problems Research Articles

Bounds on the eigenvalues for the circular Rayleigh problem of hydrodynamic stability

Bounds on the eigenvalues for the circular Rayleigh problem of hydrodynamic stability

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. III. Asymptotics of branching

The previous paper of this series presented the results of a numerical investigation of the dependence of the dominant growth rates of Bloch eigenmodes on the diffusivity parameters (the molecular viscosity ν and molecular magnetic diffusivity η) in three linear stability problems: the kinematic dynamo problem, and the hydrodynamic and MHD stability problems for steady spaceperiodic flows and MHD states. The dominant eigenmodes (i.e., the stability modes, whose growth rates are maximum over the wave vector q of the planar wave involved in the Bloch modes) comprise branches. In some branches, the dominant growth rates are attained for constant half-integer q. In all the three stability problems for parity-invariant steady states, offshoot branches, stemming from the branches of this type, were found, in which the dominant growth rates are attained for q depending on ν and/or η. We consider now such a branching of the dominant magnetic modes in the kinematic dynamo problem, where an offshoot stems from a branch of neutral eigenmodes for q = 0, and construct power series expansions for the offshoots and the associated eigenvalues of the magnetic induction operator near the point of bifurcation. We show that the branching occurs for the molecular magnetic diffusivities, for which the two eigenvalues of the eddy diffusivity operator become imaginary, and magnetic field generation by the mechanism of the negative eddy diffusivity ceases. The details of branching in the other linear stability problems under consideration are distinct.

Exact solutions for the description of nonuniform unidirectional flows of magnetic fluids in the Lin–Sidorov–Aristov class

The paper considers the exact integration of magnetic hydrodynamic equations for describing nonuniform unidirectional flows of viscous incompressible fluids. The construction of an exact solution is based on the well-known representation of hydrodynamic fields as the Lin–Sidorov–Aristov class. The 3d magnetic field is described by linear forms with respect to two spatial coordinates (longitudinal, or horizontal). The coefficients of the linear forms depend on the third coordinate and time. In view of the incompressibility condition, the 1D velocity field depends on two coordinates and time. The pressure is shown to be determined by a quadratic form with constant coefficients. These coefficients are determined by pressure distribution on the known (free) boundary. The exact solution is illustrated by the integration of non-1D hydrodynamic fields in the case of the steady motion of a conducting viscous incompressible fluid. This solution is polynomial, and it will be useful for the formulation of new problems of hydrodynamic stability.

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

We consider Bloch eigenmodes of three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states comprised of randomly generated Fourier coefficients and having energy spectra of three types: exponentially decaying, Kolmogorov with a cut off, or involving a small number of harmonics (“big eddies”). A Bloch mode is a product of a field of the same periodicity as the perturbed state and a planar harmonic wave, exp(iq · x). Such a mode is characterized by the ratio of spatial scales which, for simplicity, we identify with the length |q| < 1 of the Bloch wave vector q. Computations have revealed that the Bloch modes, whose growth rates are maximum over q, feature the scale ratio that decreases on increasing the nondimensionalized molecular diffusivity and/or viscosity from 0.03 to 0.3, and the scale separation is high (i.e., |q| is small) only for large molecular diffusivities. Largely this conclusion holds for all the three stability problems and all the three energy spectra types under consideration. Thus, in a natural MHD system not affected by strong diffusion, a given scale range gives rise to perturbations involving only moderately larger spatial scales (i.e., |q| only moderately small), and the MHD evolution consists of a cascade of processes, each generating a slightly larger spatial scale; flows or magnetic fields characterized by a high scale separation are not produced. This cascade is unlikely to be amenable to a linear description. Consequently, our results question the allegedly high role of the α-effect and eddy diffusivity that are based on spatial scale separation, as the primary instability or magnetic field generating mechanisms in astrophysical applications. The Braginskii magnetic α-effect in a weakly non-axisymmetric flow, often used for explanation of the solar and geodynamo, is advantageous not being upset by a similar deficiency.

Линейные возмущения Блоховского типа пространственно-периодических магнитогидодинамических стационарных состояний. I. Математические вопросы

We consider Bloch eigenmodes in three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states. A Bloch mode is a product of a field of the same periodicity, as the state subjected to perturbation, and a planar harmonic wave, exp(iqx). The complex exponential cancels out from the equations of the respective eigenvalue problem, and the wave vector q remains in the equations as a numeric parameter. The resultant problem has a significant advantage from the numerical viewpoint: while the Bloch mode involves two independent spatial scales, its growth rate can be computed in the periodicity box of the perturbed state. The three-dimensional space, where q resides, splits into a number of regions, inside which the growth rate is a smooth function of q. In preparation for a numerical study of the dominant (i.e., the largest over q) growth rates, we have derived expressions for the gradient of the growth rate in q and proven that, for parity-invariant flows and MHD steady states or when the respective eigenvalue of the stability operator is real, half-integer q (whose all components are integer or half-integer) are stationary points of the growth rate. In prior works it was established by asymptotic methods that high spatial scale separation (small q) gives rise to the phenomena of the α-effect or, for parity-invariant steady states, of the eddy diffusivity. We review these findings tailoring them to the prospective numerical applications.

On the Short Wave Instability of the Liquid/Gas Contact Surface in Porous Media

We consider a problem of hydrodynamic stability of the liquid displacement by gas in a porous medium in the case when a light gas is located above the liquid. The onset of instability and the evolution of the small shortwave perturbations are investigated. We show that when using the Darcy filtration law, the onset of instability may take place at an infinitely large wavenumber when the normal modes method is inapplicable. The results of numerical simulation of the nonlinear problem indicate that the anomalous growth of the amplitude of shortwave small perturbations persists, but the growth rate of amplitude decreases significantly compared to the results of linear analysis. An analysis of the stability of the gas/liquid interface is also carried out using a network model of a porous medium. It is shown that the results of surface evolution calculations obtained using the network model are in qualitative agreement with the results of the continual approach, but the continual model predicts a higher velocity of the interfacial surfaces in the capillaries. The growth rate of perturbations in the network model also increases with decreasing perturbation wavelength at a constant amplitude.

Eigenvalue bounds for the circular orr-sommerfeld problem of hydrodynamic stability

Eigenvalue bounds for the circular orr-sommerfeld problem of hydrodynamic stability

On compressible circular Rayleigh problem of hydrodynamic stability

On compressible circular Rayleigh problem of hydrodynamic stability

A phase‐field lattice Boltzmann method for the solution of water‐entry and water‐exit problems

AbstractIn this paper, an original phase‐field lattice Boltzmann scheme for a system composed of two immiscible and incompressible fluids interacting with moving solids is developed, presented, and tested for the first time. The proposed approach is benchmarked against experimental data and smoothed particle hydrodynamics simulations of four well‐established two‐dimensional problems where a rigid body interacts with the interface between the two fluids. The outlined methodology represents a very good candidate to perform fluid‐structure interaction simulations of hydrodynamic stability problems as well as water‐entry and water‐exit problems.

Comparative Study on Sixth Order Boundary Value Problems with Application to Linear Hydrodynamic Stability Problem and Benard Layer Eigenvalue Problem

Numerical estimation for higher order eigenvalue problems are promising and has accomplished significant importance, mainly due to existence of higher order derivatives and boundary conditions relating to higher order derivatives of the unknown functions. In this article, we perform a numerical study of linear hydrodynamic stability of a fluid motion caused by an erratic gravity field. We employ two methods, collocation and spectral collocation based on Bernstein and Legendre polynomials to solve the linear hydrodynamic stability problems and Benard type convection problems. In order to handle boundary conditions, our techniques state all the unknown coefficients of boundary conditions derivatives in terms of known co-efficient. The schemes have been carried out to several test problems to establish the efficiency of the two methods.

Note on Instability Regions in the Circular Rayleigh Problem of Hydrodynamic Stability

Batchelor and Gill’s semicircular bound on the range of the complex wave velocity of an arbitrary unstable mode in the stability problem of inviscid incompressible homogeneous axial flows with respect to axisymmetric disturbances is further reduced. Significance In the circular Rayleigh problem of hydrodynamic stability, we have obtained a parabolic instability region. The significance of this result is that it depends on the discriminant $$\Psi (r)$$ which plays a vital role in deciding whether a basic axial flow is stable or not to axisymmetric disturbances. However, our parabolic instability region is an unbounded one unlike the semicircular instability region of Batchelor and Gill, which has the weakness that it does not depend on $$\Psi (r)$$ . So we have demonstrated that our parabola intersects the semicircle of Batchelor and Gill for two classes of basic flows, and consequently the semicircular instability region of Batchelor and Gill is further reduced. Moreover, we have stated the stability problem consisting of a second-order ODE and associated boundary conditions in three different but related variables. The significance of this is that further analytical results can be obtained for this problem, and some of these are presented in our subsequent paper.

A New Scheme Using Cubic B-Spline to Solve Non-Linear Differential Equations Arising in Visco-Elastic Flows and Hydrodynamic Stability Problems

This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) to produce a new system of linear equations. The algorithm developed here is not only for the approximation solutions of the 10th order BVPs but also estimate from 1st derivative to 10th derivative of the exact solution as well. Some examples are illustrated to show the feasibility and competence of the proposed scheme.

On the problem of hydrodynamic stability

Mathematical models of hydrodynamic stability for single-phase flows are obtained in papers using the small perturbation method. The spectral-grid method is used to approximate the stability equations. It combines the high accuracy of the spectral method of non-uniform grids and allows you to immediately determine all the eigenvalues of the problem under consideration. Figs.: 1. Tabl.: 1. Refs.: 11 titles. Keywords: hydrodynamic stability; small perturbation method; approximate the stability equations; spectral-grid method; non-uniform grid method.

Comparative study of different non-reflecting boundary conditions for compressible flows

The numerical simulation of hydrodynamic stability and aeroacoustic problems requires the use of high-order, low-dispersion and low-dissipation numerical methods. It also requires appropriate boundary conditions to avoid reflections of outgoing waves at the boundaries of the computational domain. There are many different methods to avoid wave reflection at the boundaries such as the buffer zone and boundary conditions based on characteristic equations. This paper considers the use of a methodology called perfectly matched layer (PML). The PML is evaluated for the simulation of an acoustic pulse in a uniform flow and the Kelvin–Helmholtz instability in a mixing layer using the linear and nonlinear form of the Euler equation. PML results are compared with other non-reflecting boundary condition methods in terms of effectiveness and computational cost. The other non-reflecting boundary conditions implemented were the buffer zone (BZ), widely used in aeroacoustic and hydrodynamic problems, and the energy transfer and annihilation (ETA), a very simple boundary condition to be implemented. The results show that the PML is an effective boundary condition method, but can be computationally expensive. The PML is also more complex to implement and requires careful stability analysis. The other boundary conditions, the BZ and the ETA, are also effective and may perform better than the PML depending on the flow conditions. These two methods have an advantage in terms of robustness and are much simpler to implement than the PML.

An improved instability region for the extended Rayleigh problem of hydrodynamic stability

For the extended Rayleigh problem of hydrodynamics stability dealing with homogeneous shear flows with variable cross section, we have obtained a parabolic instability region. This improved parabolic instability region intersects with the semicircular instability region under certain condition. The validity of the result is illustrated with an example of basic flows. Furthermore, we have obtained a bound for the complex part of the phase velocity.

On sufficient conditions for stability in the circular Rayleigh problem of hydrodynamic stability

Two sufficient conditions for stability are presented for the circular Rayleigh problem of hydrodynamic stability.

On the Long Wave Stability of Shear Flows in Sea Straits of Arbitrary Cross Section

We consider an eigen value problem of hydrodynamic stability which deals with inviscid, incompressible, density stratified shear flows of variable topography. In this paper, we obtained a condition for long wave stability i.e., if the wave number k is less than or equal to critical wave number $$k_{c}$$ which is greater than zero then the flow is stable. This result is illustrated with examples. An instability region which depends on various parameters is derived for an unstable mode and an estimate for the growth rate of an unstable mode is also obtained.

A Unified Instability Region for the Extended Taylor–Goldstein Problem of Hydrodynamic Stability

AbstractWe consider inviscid, incompressible shear flows with variable density and variable cross section. For this problem, we derived a new estimate for the growth rate of an unstable mode and a parabolic instability region which intersects semiellipse instability region under some condition.

A unified solution of several classical hydrodynamic stability problems

A unified solution of several classical hydrodynamic stability problems

A Newton-type method for non-linear eigenproblems

Abstract This work is devoted to computations of invariant pairs associated with separated groups of finite eigenvalues of large regular non-linear matrix pencils. It is proposed to combine the method of successive linear problems with a Newton-type method designed for partial linear eigenproblems and a deflation procedure. This combination is illustrated with a typical hydrodynamic spatial stability problem.