On subsonic, supersonic and hypersonic inflectional-wave/vortex interaction

An unstable inflection point developing in an oncoming two-dimensional boundary layer can give rise to nonlinear three-dimensional inflectional-wave/vortex interaction as described in recent papers by Hall and Smith [1], Brown et al. [2], and Smith et al. [3]. In the current study on the compressible range the flow is examined theoretically just downstream of the linear neutral position, in order to understand how the interaction may be initiated. The research addresses both moderately and strongly compressible regimes. In the latter regime the vorticity mode, the most dangerous one, is taken as the wave part, causing the hypersonic interaction to become concentrated in a thin temperature-adjustment layer lying at the outer edge of the boundary layer, just below the free stream. In both regimes, the result is a nonlinear integro-differential equation for the wave-pressure which implies four different types of downstream behaviour for the interaction-a far-downstream saturation, a finite-distance singularity, exponentially decaying waves (leaving pure vortex motion) or periodicity. In a principal finding of the study, the coefficients of the equation are worked out explicitly for hypersonic flow, and in particular for the case of unit Prandtl number and a Chapman fluid, where it is shown that for sufficiently high wall temperatures the wave angle of propagation must lie between 45° and 90° relative to the free-stream direction and also no periodic solutions may occur then. The theory applies also to wake flows and others. Connections with experimental findings are noted.     

Free-surface flow of a fluid body with an inner circular cylinder in impulsive motion

An analytical theory and numerical computations are developed for the two-dimensional free-surface flow of an initially circular layer of inviscid fluid surrounding a rigid circular cylinder. The two cylinders are initially concentric. The fluid packet is released from rest and the flow suddenly starts forced by gravity and by the simultaneous impulsive motion of the inner body. A small-time expansion of the fully nonlinear free-surface problem is developed and a closed-form solution is found up to third order for an arbitrary radius of the rigid cylinder. For the gravitational flow around the body at rest, the solution is extended up to fourth order. Free-surface profiles and hydrodynamic forces on the cylinder are calculated and discussed against numerical solutions of the exact unsteady nonlinear problem. Some basic features, such as the formation of an almost uniform layer surrounding the upstream side of the body, are captured by the theory quite well and only later on in time significant quantitative differences appear. Similarly, the behaviour of hydrodynamic loads is rather well predicted during initial stages preceding larger fluctuations observed on a longer time-scale.     

An inverse integral 3D compressible boundary layer method and coupling with transonic inviscid solution

Summary A new velocity profile, which has a simple expression and agrees well with experimental data in a wide range, is proposed in the present paper. Based on this profile, the governing equations of the 3D compressible inverse boundary layer method are deduced. The steady transonic viscous flow around a 3D wing can be calculated as follows: the inviscid flow is calculated by using nonisentropic full potential equation; the viscous flow is calculated by using present boundary layer method; the viscous and inviscid solutions are coupled by using semi-inverse method. Numerical results agree well with the experimental data and required computer resources are less, so that it has broad prospects in the engineering application.     

Directed fluid sheets and gravity waves in compressible and incompressible fluids

This paper extends the range of applicability of the theory of directed fluid sheets [1] to the propagation of fairly long gravity waves in a compressible inviscid fluid and in a uniform shear flow of an incompressible inviscid fluid over a stream of constant initial depth. Applications are also made to nonhomogeneous immiscible fluid layers and the effect of compressibility on wave propagation over a stream with a level bottom is examined.     

Steady two-layer flows over an obstacle

Nonlinear waves in a forced channel flow of two contiguous homogeneous fluids of different densities are considered. Each fluid layer is of finite depth. The forcing is due to an obstruction lying on the bottom. The study is restricted to steady flows. First a weakly nonlinear analysis is performed. At leading order the problem reduces to a forced Korteweg-de Vries equation, except near a critical value of the ratio of layer depths which leads to the vanishing of the nonlinear term. The weakly nonlinear results obtained by integrating the forced Korteweg-de Vries equation are validated by comparison with numerical results obtained by solving the full governing equations. The numerical method is based on boundary integral equation techniques. Although the problem of two-layer flows over an obstacle is a classical problem, several branches of solutions which have never been computed before are obtained.     

The stability of low-amplitude three-dimensional disturbances in a nonequilibrium compressible boundary layer

The stability of Tollmien-Schlichting waves propagating at an angle to the main flow in a nonequilibrium compressible supersonic boundary layer is investigated within the linear theory of hydrodynamic stability. The dependences of the critical Reynolds number on the degree of disequilibrium and on the Mach number of undisturbed flow are found at different angles of wave propagation. It is demonstrated that the critical Reynolds number in a nonequilibrium medium may decrease appreciably with increasing degree of disequilibrium, which results in the reduction of the characteristic length of the linear region of transition to turbulence.     

Propagation of waves in a fluid-saturated porous elastic solid

A theory is developed for the propagation of waves in a porous elastic solid containing a compressible viscous fluid using a homogenization process. The matrix is a lattice of periodically distributed gaps of arbitrary shape, the period of the lattice being small compared with the wave length. The present treatment is concerned with materials where fluid and solid are of comparable densities. Two cases are considered: the situation in which the pores are connected and that in which they are not. When pores are closed, the bulk medium behaves like an elastic medium; when they are connected, the fluid filtration and the bulk deformation are coupled. Boundary conditions, for macroscopic variables, at the interface between such a porous medium and the adjacent free flow are given.     

Compressibility effects on outflows in a two-fluid system. 1. Line source in cylindrical geometry

A two-dimensional line source outflow is considered, in which the evolution of a sharp interface separating an incompressible fluid from a bounding weakly compressible gas is analysed. Linear theory is applied, assuming that anisotropies in the source outflow are small, to develop an approximate solution for the interfacial evolution. The simplest solutions to the governing linearised equations require the presence of a high-order velocity singularity at the location of the line source. A spectral method is also developed to capture the nonlinear behaviour of the flow; after some finite time, curvature singularities are found to develop on the interface. Comparisons are made between the stability of the interface and its analogue which separates two incompressible fluids. It is found that when the bounding fluid is weakly compressible rather than incompressible, the stability of the interface is significantly increased.     

Effects of a liquid layer on thickness-shear vibrations of rectangular AT-cut quartz plates

Thickness-shear vibrations of rectangular AT-cut quartz with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The two-dimensional (2D) governing equations for vibrations of piezoelectric crystal plates given previously are used in the present study. The solutions for 1D shear wave and compressional wave in a liquid layer are obtained, and the stresses at the bottom of the liquid layer are used as approximations to the stresses exerted on the crystal surface in the plate equations. Closed form solutions are obtained for both free and piezoelectrically forced thickness-shear vibrations of a finite, rectangular AT-cut quartz plate in contact with a liquid layer of finite thickness. From the present solutions, a simple and explicit formula is deduced for the resonance frequency of the fundamental thickness-shear mode, which includes the effects of both shear and compressional waves in the liquid layer and the effect of the thickness-to-length ratio of the crystal plate. The formula reduces to the widely used frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz, computed as a function of the thickness of the adjacent liquid layer, agrees closely with the experimental data measured by Schneider and Martin (Anal. Chem., vol. 67, pp. 3324-3335, 1995)     

Flow feature aligned grid adaptation

A flow feature aligned grid adaptation method is proposed for the solution of Euler and Navier–Stokes equations for compressible flows, motivated by the desire for an efficient grid system for an accurate and robust solution method to best resolve flow features of interest. The method includes extraction of the flow features; generation of the embedded flow feature aligned structured blocks combined with unstructured grid generation for the rest of the flowfield; and adaptation of the hybrid grid for high flow feature resolution. The feature alignment makes it possible to maintain the high resolution property for both shock waves and shear layers of the approximate Riemann solvers and the higher order reconstruction schemes based on one‐dimensional derivation and dimensional splitting. High grid efficiency is obtained with highly anisotropic directional grid corresponding to the feature directions. The computational procedure is described in details in the paper and its application to flow solutions involving shock waves, boundary layers, wakes and shock boundary layer interaction are demonstrated. Its accuracy, efficiency and robustness are discussed in comparison with an anisotropic unstructured grid adaptations for the shock boundary layer interaction case. Copyright © 2006 John Wiley & Sons, Ltd.     

Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil with compressible constituents

In this paper, transfer matrix solutions for three-dimensional consolidation of a multi-layered soil considering the compressibility of pore fluid are presented. The derivation of the solutions starts with the fundamental differential equations of Biot’s three-dimensional consolidation theory, takes into account the compressibility of pore fluid in the Cartesian coordinate system, and introduces the extended displacement functions. The relationship of displacements, stresses, excess pore water pressure, and flux between the ground surface (z = 0) and an arbitrary depth z is established for Biot’s three-dimensional consolidation problem of a finite soil layer with compressible pore fluid by taking the Laplace transform with respect to t and the double Fourier transform with respect to x and y, respectively. Based on this relationship of the transfer matrix, the continuity between layers, and the boundary conditions, the solutions for Biot’s three-dimensional consolidation problem of a multi-layered soil with compressible constituents in a Laplace-Fourier transform domain is obtained. The final solutions in the physical domain are obtained by inverting the Laplace-Fourier transforms. Numerical analysis is carried out by using a corresponding program based on the solutions developed in this study. This analysis demonstrates that the compressibility of pore fluid has a remarkable effect on the process of consolidation.     

The geometry of the cartan space associated with general compressible fluid flows

In this paper, an attempt is made to develop a general theory of the motion of a compressible fluid by means of the geometry of the general space propounded by Cartan based on the concept of area. Assuming that the flow is irrotational and isentropic, it is shown in Section I, that the equation of motion of an inviscid compressible fluid is regarded as that of hypersurfaces in the Cartan space. Various hydrodynamical features of the compressible fluid motion including the “shock” phenomena are clarified by the metric properties of the space. The fluid motion in the general case, in which the above assumptions are not valid, is treated in Section 2. The physical quantities such as vorticity, entropy and the Croccian vector are expressed by means of the geometrical terminology of the Cartan space along lines parallel to the Section 1.     

Propagation and decay of acceleration waves in thermoconsolidating media

Summary The propagation of acceleration waves in a fluid-saturated porous medium is considered. The two-phase medium is the system consisting of a porous elastic solid skeleton, filled with a viscous compressible fluid. Two types of the media are taken into account: the medium composed of definite conductors and the medium composed of non-conductors. The method of singular surfaces has been used in these considerations. The acceleration waves in the medium consisting of non-conductors are not homentropic, in general. In this paper the conditions are determined which must be fulfilled to satisfy the acceleration waves to be homentropic.The propagation conditions of the waves are formulated and analysed. As usual in such a two-phase medium two longitudinal waves and one transverse wave are propagated. The growth equations of homothermal and homentropic waves are derived, and their solutions are analysed.     

Acoustic scattering of spherical waves incident on a long fluid-saturated poroelastic cylinder

Acoustic scattering of spherical waves generated by a monopole point source in a perfect (inviscid and ideal) compressible fluid by a fluid-saturated porous cylinder of infinite length is studied theoretically in the present study. The formulation utilizes the Biot theory of dynamic poroelasticity along with the appropriate wave-field expansions, the translational addition theorem for spherical wave functions, and the pertinent boundary conditions to obtain a closed-form solution in the form of infinite series. The analytical results are illustrated with a numerical example in which a monopole point source within water is located near a porous cylinder with a water-saturated Ridgefield sandstone formation. The numerical results reveal the effects of source excitation frequency, the cylinder interface permeability condition, and the location of the point source and the field point on the backscattered pressure magnitudes. Limiting cases are considered, and the obtained numerical results are validated by already well-known solutions.     

Note on axis-symmetric jet mixing of compressible dusty fluid

Summary Laminar jet mixing of a compressible dusty fluid issuing from a circular opening has been considered. Assuming that the jet mixing is under full expansion, the governing equations have been linearised and solved by successively using Hankel and Laplace transform techniques. Numerical computations of the integrals giving velocity, temperature and density of both the fluid and particle phase have been made to discuss results. The increase in the concentration of dust particles results in width of the jet is greater for compressible flow. The compressible dusty jet does not cool as fast as a compressible jet of clear fluid. The particle concentration decreases along the axial direction but increases towards the free jet boundary.     

A spectral method for Faraday waves in rectangular tanks

A theoretical study of Faraday waves in an ideal fluid is presented. A novel spectral technique is used to solve the nonlinear boundary conditions, reducing the system to a set of nonlinear ordinary differential equations for a set of Fourier coefficients. A simple weakly nonlinear theory is derived from this solution and found to capture adequately the behaviour of the system. Results for resonance in the full nonlinear system are explored in various depth regimes. Time-periodic solutions about the main (subharmonic) resonance are also studied in both the full and weakly nonlinear theories, and their stability calculated using Floquet theory. These are found to undergo several bifurcations which give rise to chaos for appropriate parameter values. The system is also considered with an additional damping term in order to emulate some effects of viscosity. This is found to combine the two branches of the periodic solutions of a particular mode.     

On thin film flow in hydrodynamic bearings with a radial step at finite Reynolds number

A classical engineering approach to thin film flow problems with localised geometric step features is to use the Reynolds equation. For applications to new generation hydrodynamic bearings with very small gap clearances and lift-generating features, any abrupt changes in the thickness of a film will break the validity of the Reynolds equation, which is based on lubrication theory. In this work, formal asymptotic expansions are used to match a numerical solution of a local formulation of the full Navier–Stokes equations near a step feature to a Reynolds equation model which is valid sufficiently far from the step, i.e. with smooth film thickness variation. The approach is used to model a pressurised bearing with an axisymmetric Rayleigh step feature. An efficient and accurate mathematical model is presented using matched asymptotic expansions for both incompressible and compressible fluid flows. This work quantifies the effect of inertia at the step and considers the validation of the classical approach of patching lubrication solutions across the step with specified compatibility conditions. A parametric study is undertaken to evaluate cases where the classical engineering approach is justified.     

A turbulent boundary layer with injection into a compressible fluid with a pressure gradient

We use the semiempirical theory of turbulence to study the effect exerted by the input of a homogeneous material through the main flow on the friction and on the heat transfer in the turbulent boundary layer, in a compressible fluid with a pressure gradient.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 16, No. 6, pp. 989–1001, June, 1969.     

Internal waves in an elliptical channel

Abstract

Internal waves in open channels of various depths are studied, in this paper, and a wetted cross section with an elliptical bottom is considered. The frequencies of the first three sloshing modes of internal waves in two superposed fluids contained in an elliptical channel are calculated for various ratios of the depths of the two layers. Numerical solutions converge to analytical solutions based on the shallow water theory as the depth of the thin lower layer approaches zero. Also, solutions for the frequencies of the longitudinal modes of progressive internal waves in two superposed fluid layers contained in an elliptical channel are calculated for various ratios of the depths of the two layers and for two different wave numbers k.     

可压缩欧拉方程在不变子空间中的精确解

欧拉方程是流体力学中非常重要的模型,被广泛应用于许多领域.构造它的精确解是数学物理中非常有意义的工作.精确解可以为理解它的非线性现象和物理意义提供具体的例子.本文旨在通过不变子空间方法构造可压缩欧拉方程的精确解.在变量变换意义下,由不变条件给出与可压缩方程相关的不变子空间;在这些不变子空间中,它被约化为一阶常微分方程组;通过求解这些常微分方程组,最终得到可压缩欧拉方程的一些精确解.