Short-wave asymptotics for weak shock waves and solitons in mechanics

Some recent applications of the theory of non-linear waves in smoothly inhomogeneous and weakly dissipative media are discussed in the paper. The possibilities of “linear-ray” approximation when the non-linear self-refraction effects may be neglected in comparison with the non-linear wave distortion along the rays are demonstrated for weak acoustic shocks in stratified atmospheres and ocean, the solitary waves in shallow water of variable depth and the solitons in elastic rods.     

Experimental investigation of cylindrically diverging solitons in an electric lattice

The results of an experimental investigation of cylindrical solitons in a two-dimensional electric LC-lattice are given. It is shown that in the continuum limit, propagation of cylindrical waves far from the center of symmetry in such a lattice may be described for each ray tube by a known modification of the Korteweg-de Vries equation which takes account of the cylindrical divergence. The dispersion term in this equation depends on the direction of wave propagation relative to the direction of the main axes of the lattice. Formation of solitons from non-soliton-shaped pulses was observed. The variations of soliton amplitude and duration with distance have been determined. They agree well with the numerical calculations by Maxon & Viecelli [2] and Dorfman [9]. Comparison of the obtained experimental data with the known theoretical laws of amplitude attenuation for diverging solitons [2, 12, 14] seems to favor the validity of the law A r^−2/3 rather than A r^−1/2.     

A study of main and secondary resonances in non-linear multi-degree-of-freedom vibrating systems

A uniform study of all types of resonances that can occur in non-linear, dissipative multi-degree-of-freedom systems subject to sinusoidal excitation is presented. The theoretical investigation is based on a harmonic or multi-harmonic solution and the Ritz method. The new approach suggests that non-linear normal mode shape or the so-called “coupled” non-linear mode shapes are those which are retained in resonance conditions, no matter what type of resonancemain, or secondary, periodic or almost-periodic.

By introducing the concept of non-linear normal coordinates the response of an n-degree-of-freedom system is described, to a satisfactory degree of accuracy, by a single coordinate in the case of main or secondary-periodic resonance, or by p coordinates in the case of almost-periodic (combination) resonance with p harmonic components.

Numerical examples indicate good agreement of theoretical and analog computer results and illustrate considerable discrepancies between resonance curves calculated by the commonly used “single linear mode approach” and the suggested “single non-linear mode approach”.     

Internal solitary waves moving over the low slopes of topographies

Internal solitary waves moving over uneven bottoms are analyzed based on the reductive perturbation method, in which the amplitude, slope and horizontal lengthscale of a topography on the bottom are of the orders of , ^5/2 and ^−3/2, respectively, where the small parameter is also a measure of the wave amplitude. A free surface condition is adopted at the top of the fluid layer. That condition contains two parameters, δ and Δ, the first of which concerns the discontinuity of the basic density between the outer layer and the inner one; the second concerns the discontinuity of the mean density between them. An amplitude equation for the disturbance of order decomposes into a Korteweg-de Vries (KdV) equation and a system of algebraic equations for a stationary disturbance around a topography on the bottom. Solitary waves moving over a localized hill are studied in a simple case where both the basic flow speed and the Brunt-Vaisalla frequency are constant over the fluid layer. For this case, the expression for the amplitude of the stationary disturbance contains singular points with respect to basic flow speed. These singularities correspond to the resonant conditions modified by the free surface condition. The advancing speeds of solitary waves are changed by the influence of bottom topography, in a case where the long internal waves propagate in the direction opposite to the basic flow, but their waveforms remain almost unchanged.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

Solutions of some classes of second order non-linear ordinary differential equations

A second order non-linear ordinary differential equation satisfied by a homogeneous function of u and v where u is a solution of the linear equation ÿ + p(t)ÿ + r(t)y = 0 and v = ωu, ω being an arbitrary function of t, is obtained. Defining ω suitably in two specific cases, solutions are obtained for a non-linear equation of the form ÿ + p(t)ÿ + q(t)y = μÿ²y^{−1 +} f(t)yⁿ where μ ≠ 1, n≠ 1. Applying our results, some classes of equations of the above type possessing solutions involving two or one or no arbitrary constants are derived. Some illustrative examples are also discussed.     

Formation of solitary waves on gas-sheared liquid layers

The origin of solitary waves on gas-liquid sheared layers is studied by comparing the behavior of the wave field at sufficiently low liquid Reynolds number, R_L, where solitary waves are observed to form, to measurements at higher R_L where solitary waves do not occur. Observations of the wave field with high-speed video imaging suggest that solitary waves, which appear as a secondary transition of the stratified gas-liquid interface, emanate from existing dominant waves, but that not all dominant waves are transformed. From measurements of interface tracings it is found that for low R_L, waves which have amplitude/substrate depth (a/h) ratios of 0.5–1 occur while for higher R_L, no such waves are observed. A comparison of amplitude/wavelength ratios shows no distinction for different R_L. Consequently, it is conjectured that solitary waves originate from waves with sufficiently large a/h ratios; this change of form being similar to wave breaking. The dimensionless wavenumber is found to be smaller at low R_L, where solitary waves are observed. This suggests that perhaps, larger precursor (to solitary wave) waves are possible because the degree of dispersion, which acts to break waves into separate modes, is lower.     

Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation

The exact equations for surface waves over an uneven bottom can be formulated as a Hamiltonian system, with the total energy of the fluid as Hamiltonian. If the bottom variations are over a length scale L that is longer than the characteristic wavelength ℓ, approximating the kinetic energy for the case of “rather long and rather low” waves gives Boussinesq type of equations. If in the case of an even bottom one restricts further to uni-directional waves, the Korteweg-de Vries (KdV) is obtained. For slowly varying bottom this uni-directionalization will be studied in detail in this part I, in a very direct way which is simpler than other derivations found in the literature. The surface elevation is shown to be described by a forced KdV-type of equation. The modification of the obtained KdV-equation shares the property of the standard KdV-equation that it has a Hamiltonian structure, but now the structure map depends explicitly on the spatial variable through the bottom topography. The forcing is derived explicitly, and the order of the forcing, compared to the first order contributions of dispersion and nonlinearity in KdV, is shown to depend on the ratio between ℓ and L; for very mild bottom variations, the forcing is negligible. For localized topography the effect of this forcing is investigated. In part II the distortion of solitary waves will be studied.     

An application of Nagumo''s lemma to some singularly perturbed systems

The existence and asymptotic behavior as ε → 0⁺ of periodic, almost periodic, and bounded solutions of the differential system x = f(t, x, y, ε), Ωy′ = g(t, x, y, ε), are considered where x, f; are n-vectors, y, g are m-vectors and Ω = diag{ε^h₁}…, ε^h_m for integral h_i, h₁ h₂ …, h_m. The principal tools are a lemma of Nagumo which allows the construction of appropriate upper and lower solutions and the asymptotic theory of singularly perturbed linear differential systems.     

Impingement heat transfer at a circular cylinder due to a submerged slot jet of water

An experimental investigation was carried out on the heat transfer due to a submerged slot jet of water impinging on a circular cylinder in crossflow. The cylinder diameter and the slot width are of the same order of magnitude, specifically D_s = 2.0 and 3.0 mm and D_c = 2.5 and 3.0 mm. The experimental apparatus allowed variation of the slot width, the cylinder diameter, and the distance from nozxle exit to heater. Conditions of impingement from the bottom (ascending flow) were taken into consideration as well as impingement from above (descending flow). The Nusselt number was determined as a function of Reynolds and Prandtl numbers in the range 1.5 × 10³ < Re < 2.0 × 10⁴, 2.7 < Pr < 7.0, and 1.5 ≤ z/D_s ≤ 10. The experimental data were correlated with a simple equation that fits 90% of the data with a precision of 20%.     

A time-reversal method for an acoustical pulse propagating in randomly layered media

In the recent years a considerable amount of mathematical work has been devoted to the study of reflected signals obtained by the propagation of pulses in randomly layered media. We refer to [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33 (4), 519–625 (1991)] for an extensive survey and applications to inverse problems. The analysis is based on separation of scales between the correlation scale of the inhomogeneities present in the medium, the typical wavelengths of the pulse and the macroscopic variations of the medium. On the other hand, in the context of ultrasounds, time-reversal mirrors have been developed and their effects have been studied experimentally by Mathias Fink and his team at the Laboratoire Ondes et Acoustique (ESPCI-Paris). We refer to: [M. Fink, “Time reversal mirrors”, J. Phys. D: Appl. Phys. 26, 1333–1350 (1993)]. Our goal is to present a mathematical analysis of a time-reversal method for analyzing reflected signals in the model described in [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33(4), 519–625 (1991)]. We restrict our analysis to the one-dimensional case, the three-dimensional layered case being the content of a forthcoming paper. It is noticeable that we do not introduce new mathematics in the problem but simply put together an already existing mathematical theory and a new device, the time-reversal mirror.     

Rarefactive solitary waves in two-phase fluid flow of compacting media

Rarefactive solitary wave solutions of a third order nonlinear partial differential equation derived by Scott and Stevenson (Geophys. Res. Lett. 11, 1161–1164 (1984)) to describe the one-dimensional migration of melt under the action of gravity through the Earth's mantle are investigated. The partial differential equation contains two parameters, n and m, which are the exponents in power laws relating, respectively, the permeability of the medium and the bulk and shear viscosities of the solid matrix to the voidage. It is proved that, for any value of m, rarefactive solitary wave solutions satisfying certain physically reasonable boundary conditions always exist ifn>1 but do not exist if 0n1. It is also proved that the speed of the solitary wave is an increasing function of the amplitude of the wave. Six new exact rarefactive solitary wave solutions, four of which are expressed in terms of elementary functions and two in terms of elliptic integrals, are derived for six sets of values of n and m. The large amplitude approximation is considered and the results of Scott and Stevenson for n>2, m=0 and n>1, m=1 are extended to n>1 and all m0. It is shown that, for sufficiently large amplitude, larger amplitude solitary waves are broader in width if 0m1 and are narrower in width if m>1.     

Steady laminar flow through thin curved pipes

Let R, τ denote, respectively, the radius of curvature and radius of torsion of the pipe (centre-line) and let a be a typical cross-sectional diameter.

The major part of the present paper addresses the case of flows through pipes of constant cross-section; (Re)²(a/R), Re(a/τ), (a/R) and (a/τ) all being small. Re is the Reynolds number for the flow. It is found that, even without further specifications of the details of the pipe, many important results can be obtained about the secondary flow which occurs and the pressure losses resulting from it. For example, it is shown that an important feature of such flows is valid for any corss-sectional shape; this was not obvious from previous works which treated only special cases having significant symmetries. Also, a new method for calculating the modified axial pressure gradient is presented which reduces dramatically the amount of work required therefor.

The remainder of the paper presents some results for similar flows through pipes of varying cross-section.     

Internal waves in a two-layer electrically conducting fluid in the presence of a magnetic field

The propagation of linear and nonlinear internal waves along the interface between two weakly conducting media differing in density and electrical conductivity is investigated and the influence of MHD interaction effects on their characteristics is analyzed. It is shown that in this system the waves propagate with dispersion and dissipation, and for harmonic waves of infinitesimal amplitude there exists a range of wave numbers on which propagating modes do not exist. For waves of finite amplitude a nonlinear Schrödinger equation with a dissipative perturbation is obtained and its asymptotic solution is found. It is established that the presence of electrical conductivity and an applied magnetic field leads to a decrease in the amplitude and the frequency of the envelope of the wave train.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–108, September–October, 1990.     

Interaction and “apparent” reconnection of 3D vortex tubes via direct numerical simulations

We present evidence of: “binding” of anti-parallel vortex tube segments; strong noncircular core development; evolution of new secondary finger-like vortex structures: and finally “apparent” vortex reconnections due to entanglement. The latter three processes are not present in Biot-Savart filament simulations.     

The turbulent character and pressure loss produced by periodic symmetric ribs in a circular duct

The structural character and steady-state statistics of the turbulence inside a rib-wall circular duct is investigated by the large-eddy simulation (LES) methodology. The impetus of this study is to gain an understanding of the principle physics attributing to minimizing the pressure recovered (or maximizing the pressure loss) within the core flow. For a rib periodicity with height (h) to pitch (p) ratio p/h=5, the computational results show that the majority of turbulence produced due to the rib’s presence is concentrated near the rib crest leading edge. Pairs of counter-rotating streamwise vortices form soon after the leading edge that are quickly convected radially toward the core flow. The turbulent activity within the duct trough region is negligible compared to the turbulence levels of the core flow. At this rib periodicity, the separated shear layers from the trailing edge of each rib nearly reattach to the trough floor before reaching the next rib. The resultant irrecoverable pressure loss in the form a centerline frictional coefficient is verified by an ‘at-sea’ test on board a US Navy submarine. Based on the duct diameter, their Reynolds numbers are Re_D^LES=8×10³ and (Re_D^exp)_avg=4×10⁶, respectively.     

Experimental study of turbulence in isothermal and stably stratified homogeneous shear flows: Mean flow measurements

The evolution of freestream turbulence under the combined action of linear shear and stable linear temperature profile is investigated. The experiment is carried out in a small, open circuit, low-speed test cell that uses air as working fluid. The temperature gradient formed at the entrance to the test section by means of an array of 24 horizontal, differentially heated elements is varied to get a maximum Brunt-Vaisala frequency N_o[=({g/T_m}{∂T/∂y})^1/2] of 3.1⁻¹. Linear velocity profiles are produced using screens of variable mesh size. The Reynolds number Re_M based on centre-line velocity and mesh size is varied from 80 to 175. Isothermal studies are carried out in four different experiments with varying velocity gradients. The effect of inlet turbulence level on growth of turbulence is studied in these flows by keeping the shear parameter Sh (=(x/u)(∂u/∂y)) constant. The range of shear parameters considered is 2.5–7.0. Shear and stratification combined produce a maximum gradient Richardson number Ri_g (= N_o²/(∂u/∂y)²) of 0.0145. Results have been presented in terms of evolution of variance of velocity fluctuations, Reynolds shear stress and temperature fluctuations. Measurements show the following: In isothermal flows the growth rate of turbulence quantities depends on both shear parameter and inlet turbulence level. There are distinct stages in the evolution of the flow and that can be identified by the power-law exponent of growth of turbulence. Shear is seen to promote the growth of turbulence and accelerate it towards a fully developed equilibrium state. Stratification initially suppresses the growth of turbulence and, hence, enhances the degree of underdevelopment. Under these conditions shear becomes active and subsequently enhances the growth rate of turbulence quantities.     

Waves on vortex cores

Here we discuss the types of waves which can be supported on compact regions of vorticity. This is a subject first studied by Lord Kelvin for waves propagating along the vortex lines of the Rankine and hollow-core vorticity distributions. Kelvin's major interest was in the stability of vortex rings and the numerous resurgencies in interest have usually been driven by practically important phenomena, e.g., the observations of vortex breakdown and waves on tornado and aircraft vortices in the early 1960's and more recently in technologically and geophysically significant flows and on the quantised vortices of super fluid HeII.

The major wave-types of interest are of varicose, helicoidal and fluted form and represent a periodic swelling and contraction, a bending and a “krinkling” of the core, respectively. The first two propagate along and the third around the vortex lines. All have been studied theoretically, experimentally and numerically in the limit of small wave amplitude and their major characteristics are now clear. Of particular interest is the extension of these results to the non-linear regime in which case the two first types are known to exhibit solitary wave or soliton characteristics in certain parameter ranges. It is these non-linear waves which often dominate observations of vortex flows both in nature and in technological applications and which have caused much controversy in the interpretation of results found under complex circumstances of flow and apparatus geometry.     

Fluid flow and heat transfer around circular cylinder – flat and curved plates combinations

Experimental studies were carried out to investigate the fluid flow and heat transfer around a heated circular cylinder which was placed at various distances of a wall boundary with different geometries (flat or curved plate) with subcritical Reynolds number ranging from 3.5×10³ to 10⁴. The effects of plate geometry (aspect ratio: W|H=1.0,1.5 and 2.0, and rim angle, φ=0°,60°,90°, and 120°) and gap ratio, (G|D=0.0,0.86,2.0,7.0,10.0) on the fluid flow and heat transfer characteristics (static pressure around cylinder surface, wake width, base pressure, pressure drag coefficients, velocity distribution, and both local and mean Nusselt numbers) were presented. Also flow visualization was carried out to illustrate the flow patterns around the cylinder at various gap ratios (G|D). It was found that the heat transfer and fluid flow characteristics are dependent on the plate geometry at all tested gap ratios, except for G|D=7.0 and 10.0, they are independent of the plate geometry.