Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.     

Spatial Simple Waves on a Shear Flow

The paper studies simple waves of the shallowwater equations describing threedimensional wave motions of a rotational liquid in a freeboundary layer. Simple wave equations are derived for the general case. The existence of unsteady or steady simple waves adjacent continuously to a given steady shear flow along a characteristic surface is proved. Exact solutions of the equations describing steady simple waves were found. These solutions can be treated as extension of Prandtl–Mayer waves for sheared flows. For shearless flows, a general solution of the system of equations describing unsteady spatial simple waves was found.     

Solitary wave trains in a cold plasma

The existence of traveling solitary waves, the products of modulation instability in a cold quasi-neutral plasma, is considered. Solitary waves of this type (solitary wave trains) are formed as a result of bifurcation from a nonzero wave number of the linear wave spectrum. It is shown that the complete system of equations describing the wave process in a cold plasma has solutions of the solitary wave train type, at least when the undisturbed magnetic field is perpendicular to the wave front. Sufficient conditions of existence of solitary wave trains in weakly dispersive media are also formulated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 154–161, September–October, 1996.     

Unsteady wave motions of a fluid above an inclined floor

The propagation of unsteady waves above a flat inclined floor within the framework of a linear dispersion model was first studied in [1]. This paper shows how to solve the three-dimensional problem for the case = /4, where is the angle of inclination of the floor plane to the free surface. The two-dimensional problem was studied in [2–4]. In articles [2, 3] asymptotic solutions were found for the Cauchy-Poisson problem for certain values of . In [4], a method is proposed for solving the problem of the wave motion of a fluid due to the displacement of a section of the floor of the basin. However, the complicated structure of the expression obtained by reducing the problem to an inhomogeneous functional equation makes it impossible to study the solution. The aim of the present work is to obtain some exact solutions for the two- and three-dimensional problems of unsteady waves above an inclined floor, which are suitable for calculations and asymptotic estimates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 65–70, November–December, 1984.     

Wave angle for oblique detonation waves

The flow field associated with a steady, planar, oblique detonation wave is discussed. A revision is provided for- diagrams, where is the wave angle and is the ramp angle. A new solution is proposed for weak underdriven detonation waves that does not violate the second law. A Taylor wave, encountered in unsteady detonation waves, is required. Uniqueness and hysteresis effects are also discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Internal waves generated by the turbulent wake of a sphere

Internal waves generated by the turbulent wake of a sphere travelling horizontally through a linearly stratified fluid were studied using shadowgraph and particle-streak photography. The Reynolds and internal Froude number ranges considered were 2,000 Re 12,900 and 2.0 Fi 28.0, respectively. Two quite distinct flow regimes based on the structure of the turbulent wake were identified. In one, the wake is characterized by large-scale coherent structures. In the other, the wake, as viewed on a side-view shadowgraph, grows in a roughly symmetric fashion to a maximum height and then collapses slowly; such flows are termed the smallscale structures regime.Wave lengths and maximum wave heights of the internal waves were measured as functions of Nt and Fi, where N is the Brunt-Väisälä frequency and t the time. It was found that the wave lengths scale well with the streamwise dimension of the spiralling coherent structures. The maximum amplitude of the internal waves were found to scale with the vertical dimension of the turbulent wake, upon varying the internal Froude number.     

Investigation of a “hanging” shock wave near a generating point

The hodograph method is used to plot a hanging shock wave in the plane nonequilibrium supersonic flow of an ideal gas. This paper considers the general case of an analytical solution in the plane of the hodograph at the point of generation of the shock wave. A type of limiting line is established which makes it possible to plot a shock wave (it is found that the shock wave may not extend over the whole flow, with a convolution in the physical plane).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–37, November–December, 1971.     

Protective properties of hot‐air screens

An analytical and numerical study of protective properties of hotair screens is performed for a plane wave with a triangular profile. The reduction of the shockwave pressure behind the screen is only observed at temperatures higher a certain critical value; at lower temperatures, the pressure is higher than that in a shock wave traveling in an isothermal gas. With allowance for real properties of air, an analytical relation between the critical temperature of the hot screen and the incidentwave intensity is obtained.     

Doubly periodic and quasiperiodic wave regimes in a liquid film flowing down an inclined plane,their stability and bifurcations

A nonlinear evolution equation frequently encountered in modeling the behavior of disturbances in various nonconservative media, for example, in problems of the hydrodynamics of liquid film flow, is considered. Wave solutions of this equation, regular in space and both periodic and quasiperiodic in time, branching off from steady and steady-state traveling waves are found numerically. The stability and bifurcations are analyzed for some of the solutions obtained. As a result, a bifurcation chain is found for solutions stable with respect to disturbances of the same spatial period. It is shown that the bifurcations are related to the loss of certain symmetries of the initial solution. It is demonstrated that as the bifurcation parameter increases it is possible to distinguish in the structure of the solutions intervals of quiet behavior and intervals of intense outbursts.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 98–107, July–August, 1992.     

Inertial phase separation in rotating self-gravitating media

The centrifugal separation of foreign inclusions (particles) in a rotating spherical volume of a self-gravitating medium is considered in the hydrodynamic approximation. Using the full Lagrangian approach, the particle trajectories and radial concentration profiles are studied for a rigid-body velocity distribution in the carrier phase. The regimes of continuum and free-molecular flow around the particles are considered. The cases of a heavy (with density greater than that of the carrier phase and traveling toward the center) and a light-weight (traveling toward the periphery) admixture are investigated. Analytical and numerical solutions corresponding to steady-state spherically symmetric boundary conditions for the dispersed phase are found. It is shown that the presence of rotation may result in a significant angular anisotropy of the radial particle concentration distributions and, in particular, in the formation of ring-shaped accumulation zones of heavy inclusions in the equatorial plane. The solutions obtained can be used to explain the mechanisms of onset of density nonuniformities in planet cores, the formation of planetary systems from gas-particle clouds, and the behavior of aerosol particles in atmospheric vortices.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 86–100.Original Russian Text Copyright © 2004 by Ahuja, Belonoshko, Johansson, and Osiptsov.     

Propagation of nonlinear waves in gas-liquid media with a gas content variable in space

A study is made of the propagation of nonlinear waves in gas-liquid systems of bubble structure with a gas content which is variable in the direction of propagation of the wave. It is shown that it is possible in a number of cases to amplify the pressure waves. Restrictions on the degree of homogeneity of the gas content are obtained such that, when they are fulfilled, this amplification is possible. The study also covers the effect of inhomogeneity in the gas content on the structure of nonlinear steady waves of the soliton and shock wave types.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 180–183, January–February, 1986.     

Unsteady effects in a thin viscous shock layer near a three-dimensional stagnation point of a body moving with given acceleration and deceleration

An unsteady viscous shock layer near a stagnation point is studied. The Navier-Stokes equations are analyzed in the limit 1, Re₀ , df/dt = ^n-mF(t/^m). The Reynolds number Re₀ is defined in the paper by Eq. (1.3) (df/dt is the velocity of the body with respect to an inertial frame of reference moving with the original steady velocity –V'_t8, ₂ = ( – 1)/( + 1)). Various flow regimes in the case 1, l, n max(2m, m + 1), m 0, where ² = 1/Re₀ are analyzed. Equations are derived that generalize the asymptotic analysis to the case of a viscous unsteady flow of gas in a thin three-dimensional shock layer. The problem of a thin unsteady viscous shock layer near the stagnation point of a body with two curvatures is formulated. Examples of numerical solution are given for different ratios of the principal curvatures of the body, the wall temperature, the parameters of the original steady flow, and the acceleration and deceleration regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–111, March–April, 1981.I thank Yu. D. Shevelev for a fruitful discussion of the work.     

The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c0. Convergence results for solutions are obtained at the singular perturbation limit c 0.     

Nearly Spherically Symmetric Expanding Fronts in a Bistable Reaction-Diffusion Equation

We consider nearly spherically symmetric expanding fronts in the scalar bistable reaction-diffusion equation on R^N. As t, the front is known to look more and more like a sphere under the rescaling of the radius to unity. In this paper we prove that, if the initial state is spherically symmetric and approximated by a one-dimensional traveling wave with a sufficiently large radius, then the solution is approximated uniformly for all t0 without the rescaling of the radius by the one-dimensional traveling wave with the speed of V=c–(N–1), where c>0 is the speed of the one-dimensional traveling wave solution and the mean curvature of the sphere. We further show that, if the initial state is a slightly perturbed one from the spherical front, the difference between the actual front and the expanding sphere hardly grows or decays for all t0, although the relative magnitude of the perturbation to the radius of the sphere decreases to zero.     

Performance prediction of dual cylindrical wave laser devices for flow measurements

The dual cylindrical wave system is a variant of laser Doppler velocimetry, in which two cylindrical waves of laser light are used to illuminate a moving particle. This instrument is being used for local measurement of the unsteady skin friction in turbulent boundary layers, as well as droplet sizing in spray flows. In the present work, performance of these new devices is examined using the electromagnetic theory of light. Various requirements for the design and operation of these instruments have been further elaborated and extended. The accuracy of the previous experimental results has also been considered. The optics-related errors are shown to be negligible in the measurements of streamwise as well as spanwise wall velocity gradients. However, rigorous simulations appear to be essential for proper calibration of the particle sizing device.List of symbols A, B, C three particle positions - a half-width of an optical slit - a _gm amplitude of a plane wave in the spectrum of a cylindrical wave - d _f fringe spacing - d _p particle diameter - E amplitude of the electric oscillation in the optical field - E _c combined electric field of two cylindrical waves - E _o maximum strength of the electric field at the source of a cylindrical wave - E _s electric field of a scattered wave - E _y time-dependent electric field in the case of electric polarization - f characteristic length for the phase of the scattering amplitude - f _a anisotropic frequency - f _D Doppler frequency - F _DCW transfer function of DCW system for particle sizing - F _pDA Phase Doppler transfer function - g wall velocity gradient - g _m measured wall velocity gradient - I ₀, I₂ integrals in the asymptotic expansion of the scattering amplitude - I _s intensity of the scattered light - k wave number of laser light in the fluid medium - m refractive index of the particle relative to the surrounding medium - N ₀ nominal number of fringes resulting from interference of two cylindrical waves - P phase of a plane wave - P ₁, P₂ phases of plane waves from downstream and upstream cylindrical waves respectively - P _s scattered light power at a receiving aperture - r unit vector in the direction of light scattering - r _D distance of the signal detector from the particle center - S scattering amplitude of a cylindrical wave - S ₁, S₂ Scattering amplitudes of the cylindrical waves emanating from S₁ and S₂ respectively - magnitude of the scattering amplitude for a plane wave - S _c combined scattering amplitude of two cylindrical waves - S₁, S₂ downstream and upstream sources of cylindrical waves, respectively - S scattering amplitude of a plane wave - s half-spacing between sources of the cylindrical waves - t time - u velocity along x-axis - w ₀ 1/e half-width of the field distribution at the waist of a laser sheet - X ₀ nominal width of the fringe volume along the particle path - X particle position in the measuring volume - x, y, z Cartesian coordinates Greek symbols angle of the direction of wave propagation from x-axis - coefficient of the second-order term in the phase function of a cylindrical wave - angular size of the signal receiving aperture - incremental used for numerical differentiation the ratio of to _p - — ₀ integration parameter for I₀ and I₂ - half-angle between the directions of propagation of two waves - wavelength of laser in the fluid medium - ₀ wavelength of laser in vacuum - parameter defining the direction of propagation of a plane wave - 1/e 1/e half-width of the function A - ₀ direction of propagation of the dominant plane wave in the spectrum - _0s the direction of propagation of the plane wave that contributes predominantly to scattering in a particular direction - _p the value of . corresponding to one cycle of P - _s change in corresponding to a lobe of the scattering amplitude - a dimensional form of that determines lobes in the scattered field - signal phase, ₀ + _a - _a anisotropic phase shift - ₀ phase difference between two indicent waves - off-axis angle - elevation angle - circular frequency of laser light     

Conical flows in the neighborhood of breaks in the edges of surfaces separating cocurrent supersonic streams of perfect gas

An investigation is made into the conical flows which occur when a perfect (inviscid and nonheat conducting) gas flows over the terminal edges of surfaces with breaks separating an external and an internal flow with velocity vectors parallel to the line of intersection of the surfaces. Such flows are observed, in particular, in the neighborhood of breaks in the outlet edge of a nozzle of rectangular cross section with a straight or skewed exit plane under conditions of underexpanded flow of a supersonic jet into a cocurrent supersonic stream. By means of a linear analysis flow patterns corresponding to various flow interaction regimes and edge geometries are constructed and a law of similarity is formulated. The validity of the results thus obtained is confirmed by examples of the numerical solution of the complete nonlinear system of Euler equations. In this connection, within the framework of the approach outlined in [1], as a rule, together with the shocks and characteristic surfaces bounding the conical flow in question, the shear discontinuity separating the external and internal streams is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 119–127, January–February, 1935.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.     

Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers

The spatial development of disturbances with small and moderate amplitudes in a two-dimensional (2-D) supersonic flat-plate boundary layer at Mach 4.8 is investigated using direct numerical simulations based on the compressible 3-D Navier-Stokes equations. Disturbances are introduced into the boundary layer by blowing and suction within a narrow disturbance strip at the wall. In response to the timewise periodic forcing, two types of disturbance waves are generated, a first-mode wave and a multiple-viscous-solution. The multiple-viscous-solution was described by Mack (1969, 1984) but was not seen before in a direct numerical simulation. The results of the simulations are compared with results of linear stability theory, and the agreement is very good. In simulations for larger amplitudes, fundamental resonance is observed, where both types of 3-D waves are nonlinearly amplified and synchronize their phase velocities with the 2-D disturbance waves. Subharmonic resonance is found for 3-D waves with large wave numbers, where the phase velocities of the linear 2-D and 3-D waves are nearly the same.This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn-Bad Godesberg, as part of SFB 259.