One-dimensional nonstationary motions of a liquid

An effective method is developed for solving the problem of the nonstationary motion of a liquid with plane, cylindrical, and spherical symmetry [1]. It is based on the idea of dividing the region of disturbed motion into two parts and using matched asymptotic expansions. Solutions are presented to typical problems associated with the motion of a piston, and these make it possible to obtain the solution to problems of an explosion in a liquid, oscillations of a bubble, and so forth. It is also shown that the well-known solutions for such problems given, for example, in the book of Naugol'nykh and Roi on the basis of the acoustic approximation with allowance for nonlinear terms are incorrect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1980.     

Slow nonstationary vertical motions of a die on the surface of an elastic half-space

We consider the dynamic contact problem on vertical motions of an absolutely rigid body on an elastic half-space. We assume that the contact region does not vary during the motion and there is no friction under the die bottom. We construct an approximate solution of the problem under the assumption that the variation in the contact pressure under the die bottom on the time interval in which the Rayleigh wave runs the distance equal to the contact area diameter is small. Computational formulas are obtained for the cases of circular and elliptic dies.     

Self-similarity of a convective heat-transfer problem

We study the system of differential equations for the fluid velocity and fluid temperature in a two-dimensional channel and also in a circular tube in a region of stabilized heat transfer. On the tube walls we specify boundary conditions of the second kind; we assume that the viscosity depends exponentially on the temperature. We consider the conditions under which one-dimensional nonisothermal flows arise.     

Mechanism of three-dimensional boundary-layer receptivity

Boundary-layer receptivity is always a hot issue in laminar-turbulent transition. Most actual laminar-turbulent transitions belong to three-dimensional flows. An infinite back-swept flat-plate boundary layer is a typical three-dimensional flow. Study of its receptivity is important both in theory and applications. In this paper, a freestream turbulence model is established. A modified fourth-order Runge-Kutta scheme is used for time marching, and compact finite difference schemes are used for space discretization. On these bases, whether unsteady cross-flow vortices can be excited in the three-dimensional boundary layer(the infinite back-swept flat-plate boundary layer) by free-stream turbulence is studied numerically. If so, effects of the level and the direction of free-stream turbulence on the three-dimensional boundary-layer receptivity are further studied. Differences of the three-dimensional boundary-layer receptivity are then discussed by considering the non-parallel effect, influence of the leading-edge stagnation point of the flat plate, and variation of the back-swept angle separately. Intensive studies on the three-dimensional boundary-layer receptivity will benefit the development of the hydrodynamic stability theory, and provide a theoretical basis for prediction and control of laminar-turbulent transition.     

Characterization of hypersonic roughness-induced boundary-layer transition

The flow-field structure in the vicinity and in the wake of an isolated 3D roughness element has been studied. Different experimental techniques have been coupled and supported by CFD simulation for a good understanding of the flow-field topology. The results have shown strong flow-field similarities for different roughness elements. A model describing the flow structure and interaction mechanisms has been proposed. This model is in good agreement with experimental and CFD results as well as the literature.     

Generalized variables in boundary-layer theory

Independent variables are widely used in boundary-layer theory to construct efficient methods of solving problems. The Dorodnitsyn variables in Lees' form [1] are the most common and general. This form combines the transformations proposed by Dorodnitsyn [2], Blasius [3], and Mangler-Stepanov [4, 5]. As is well known, transformation of the boundary-layer equations to Dorodnitsyn variables in Lees' form leads to a generalized single system of equations describing plane and axisymmetric gas flows. An analogous generalization of the Mises [6] and Crocco [7] variables is carried out below.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 166–168, September–October, 1976.     

Asymptotic solutions of non-self-similar boundary-layer equations

In this paper asymptotic solutions near the outer boundary are found for the non-self-similar laminar boundary-layer equations in an incompressible fluid and a compressible gas. It is shown that the nature of the asymptotic solution is determined by the form of the initial velocity and enthalpy profiles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 29–34, November–December, 1976.In conclusion, the author is grateful to N. M. Belyanin and F. A. Slobodkina for attention to the research and discussion of the results.     

Calculation of nonstationary viscous flows

The present study will consider the problem of formation of a nonstationary flow of a viscous, thermally conductive gas around a semiinfinite plate, with the gas being set in motion instantaneously from a state of rest (problem I) and behind a shock wave incident upon the gas (problem H). A numerical method will be used, based on integration of the complete system of Navier-Stokes equations. Special attention is given in the solution to these times when the simplifying assumptions of boundary layer theory are invalid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–171, November–December, 1978.The authors thank Yu. A. Dem'yanov for evaluating the results of the study.     

Turbulent boundary-layer analysis using finite elements

The use of finite element methods for turbulent boundary-layer flow is relatively recent and of limited extent.¹ In the present study, we extend the group variable approach of Fletcher and Fleer^2,3 to treat turbulent boundary layer flows with heat transfer using a two-equation turbulence model. The main concepts in the formulations include a Dorodnitsyn-type transformation which uses a velocity component as the transverse variable, a ‘variational’ formulation for the transformed equations using special test functions and development of a two-equation turbulence model in terms of the turbulent kinetic energy and turbulence dissipation rate as additional field variables. Several numerical test cases have been examined comparing the results with finite difference calculations and comparing the two-equation turbulence model with an algebraic turbulence model.     

Wave-envelope steepening in the Blasius boundary-layer

Previous experiments into the evolution of small amplitude disturbances to the Blasius boundary layer have shown that modulated waves become nonlinear at lower amplitudes than unmodulated waves. In this paper we propose a mechanism that may account for this behaviour. It involves a wave-envelope steepening scenario analogous to water-wave overturning and shock formation. Larger amplitude parts of a modulated wave travel at a different speed to lower amplitude parts, due to the proposed nonlinear mechanism, leading to an asymmetry between the steepness of decaying and growing sections. These effects occur in the higher order Ginzburg–Landau equation, so this may be a useful model for the process. Results from a windtunnel experiment, and a direct numerical simulation, will be presented and analysed for this effect. Both show a clear progressive asymmetry developing as the amplitude, and hence nonlinearity, are increased. Comparison between the experiment and simulation highlights key differences between two- and three-dimensional nonlinear evolutions.     

Asymptotic theory of nonstationary separation

Nonstationary separation of a laminar boundary layer in a incompressible fluid is studied on the basis of asymptotic analysis of the Navier-Stokes equations at large Reynolds numbers. The case when the point of separation moves upstream is considered. It is shown that under certain restrictions on the acceleration with which the motion of the point of separation occurs the local solution depends on the time as on a parameter. As in the stationary case, the separation occurs spontaneously under the influence of the large local pressure gradient. An important difference is however that the mechanism of nonstationary separation is inviscid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 21–32, November–December, 1979.I thank A. I. Ruban for great interest and assistance in the work, and also O. S. Ryzhov for a number of helpful comments.     

The upper-branch stability of compressible boundary-layer flows

The upper-branch linear and nonlinear stability of compressible boundary-layer flows is studied using the approach of Smith and Bodonyi (1982) for a similar incompressible problem. Both pressure gradient boundary layers and Blasius flow are considered with and without heat transfer and the neutral eigenrelations incorporating compressibility effects are explicitly obtained. The compressible nonlinear viscous critical-layer equations are derived and solved numerically and the results indicate some solutions with positive phase shift across the critical layer. Various limiting cases are investigated including the case of much larger disturbance amplitudes and this indicates the structure for the strongly nonlinear critical layer of the Benney-Bergeron (1969) type. Finally, we also show how a match with the inviscid neutral inflexional modes arising from the generalized inflexion-point criterion is achieved.J. Cole is grateful to the Science and Engineering Research Council of Great Britain for financial support. J. Gajjar gratefully acknowledges the support of ICOMP, NASA Lewis Research Center, Cleveland, Ohio, where part of this work was done, and he is also grateful to the Computation Center at Iowa State University for a grant which enabled the numerical work in the paper to be completed. The permanent address of the authors is the Mathematics Department, Exeter University, Exeter EX4 4QE, England.     

Self-similarity in rock cracking and related complex critical exponents

The purpose of this paper is to further confirm some interpretations we made previously from materials or rock structure in term of the requirement of the introduction of complex critical exponents, where catastrophic brittle fracture is considered as a kind of second-order phase transition by analogy with percolation phenomenon. We propose here, using acoustic emission measurement data, a more complete experimental validation to support our previous conjecture that, “the higher the grain size and power supply, the longer range the interaction, and therefore the higher the imaginary part of complex critical exponents,” [Moura, A., Lei, X.L, Nishizawa, O., 2005. Prediction scheme for the catastrophic failure of highly loaded brittle materials or rocks. J. Mech. Phys. Solids 53(11), 2435-2455].