Generation of water waves and bores by impulsive bottom flux

The inviscid free-surface flow due to an impulsive bottom flux on constant depth is investigated analytically and numerically. The following classes of two-dimensional flow are considered: an upwelling flow which is uniform over a half-plane, a line source/sink, and a dipole aligned along the bottom. The bottom flux is turned on impulsively and may decay with time. The fully nonlinear problem is solved numerically. A small-time asymptotic expansion to third order is found for the nonlinear problem. An asymptotic large-time solution is found for the linearized problem. A steady source will generate a pair of symmetric bores, and their breaking is investigated. A steady sink generates a depression wave if it is weak, and dip instability if it is strong. Wave breaking will occur for intermediate sink strengths. A decaying source emits solitary waves.     

High-Reynolds-number flow about a submerged circular cylinder induced by free-surface travelling waves on finite depth

We consider the fluid flow induced when free-surface travelling waves, on fluid whose depth is finite and uniform in its undisturbed state, pass over a submerged circular cylinder. The wave amplitude is assumed to be small, and a suitably defined Reynolds number large. Thus, the inviscid flow may be pursued by perturbation methods, as may viscous effects that are confined to thin boundary layers on the cylinder and bed beneath it. Particular attention is focused on the steady streaming motion, which induces a circulation about the cylinder. The consequences of this on bed scouring beneath the cylinder, when the bed is erodible, are considered.     

Vortex motion in the early stages of unsteady flow around a circular cylinder

In this paper, processes in the early stages of vortex motion and the development of flow structure behind an impulsively-started circular cylinder at high Reynolds number are investigated by combining the discrete vortex model with boundary layer theory, considering the separation of incoming flow boundary layer and rear shear layer in the recirculating flow region. The development of flow structure and vortex motion, particularly the formation and development of secondary vortex and a pair of secondary vortices and their effect on the flow field are calculated. The results clearly show that the flow structure and vortices motion went through a series of complicated processes before the symmetric main vortices change into asymmetric: development of main vortices induces secondary vortices; growth of the secondary vortices causes the main vortex sheets to break off and causes the symmetric main vortices to become “free” vortices, while a pair of secondary vortices is formed; then the vortex sheets, after breaking off, gradually extend downstream and the structure of a pair of secondary vortices becomes relaxed. These features of vortex motion look very much like the observed features in some available flow field visualizations. The action of the secondary vortices causes the main vortex sheets to break off and converts the main vortices into free vortices. This should be the immediate cause leading to the instability of the motion of the symmetric main vortices. The flow field structure such as the separation position of boundary layer and rear shear layer, the unsteady pressure distributions and the drag coefficient are calculated. Comparison with other results or experiments is also made. This work was presented at the First Asian Congress of Fluid Mechanics, Bangalore in December 1980.     

Cusp flows due to an extended sink in two dimensions

The steady two-dimensioanl potential flow of a finite-depth fluid into an extended or distributed sink, in which the free surface dips to form a cusp above the centre of the sink, is examined. The extended sink is a region where the vertical outflow velocity V is constant and uniform. Numerical solutions for the free-surface profiles are obtained by use of a boundary-integral technique. Solutions are only found for the supercritical case where the Froude numbers are greater than one. In the limiting case where the extended sink width tends to zero, the problem reduces to that of a line sink beneath the free surface, and comparisons are made to existing results for this type of flow.     

Wave scattering by a circular cylinder half-immersed in water with an ice-cover

The problem of scattering of water waves obliquely incident on a fixed long circular cylinder half-immersed in deep water with an ice-cover is investigated here. The ice-cover is modelled as an elastic plate of very small thickness. The problem is formulated using the method of multipoles. This leads to an infinite system of linear equations which are solved numerically by truncation. The reflection and transmission coefficients are obtained and depicted graphically against the wave number for various values of the angle of incidence and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover on these quantities. The effect of ice-cover is seen to increase the reflection coefficient and to decrease the transmission coefficient.     

Rapid plunging of a body partly submerged in water

The water flow is studied when a thin body, part-submerged in originally still water, is plunged vertically downward. The rapid motion is at medium-to-large Reynolds number. The flow structure consists mainly of two viscous nonlinear layers, namely the unsteady boundary layer on the plunging body and unsteady wake along the trailing free surface, and one inviscid linear region in the bulk of the water. The former produce unsteady displacements which force the motion in the inviscid bulk, along with mixed boundary conditions, and control the evolutions of the wake-side and upper free surfaces. Other regions present are also considered. Analytical and numerical solutions are described, including the effects of the high Froude number and of contact slippage, and the time to bubble formation on the body.     

Viscous oscillations in a circular cylinder with an elastic cover on the free surface

The effects of an elastic membrane on the viscous oscillations of liquid filling a circular cylindrical container are studied by using the natural viscous complex eigenfunctions of the problem. The free surface of the liquid is assumed to be fully covered by the membrane. By projecting the governing equations onto an appropriate basis, a nonlinear eigenvalue problem for the complex frequencies is obtained. This is then solved to obtain the modal frequencies as a function of the Reynolds number Re, the tension parameter τ, the mass parameter ζ and the liquid depth h. The zero velocity conditions on both the side and bottom walls are satisfied unlike in earlier studies where either only the sidewall or only the bottom wall conditions were met. Results are presented for the four lowest non-axisymmetric modes as a function of Re, h, τ and ζ. The elastic cover increases the slosh frequencies but only in comparison with an uncovered free surface with a contact line that is free to move; the frequencies are lower when compared with those of a free surface with pinned contact line. There are ranges of Re, h, τ and ζ for which the oscillations are overdamped and the sloshing is aperiodic. Though the frequencies and damping rates decrease for an increasing mass of the elastic cover, there exist ranges of Re, h and τ for which the heavier cover produces higher slosh frequencies.     

Wave scattering by a horizontal circular cylinder in a two-layer fluid with an ice-cover

In a two-layer fluid wherein the upper layer is of finite depth and bounded above by a thin but uniform layer of ice-cover modelled as a thin elastic sheet and the lower layer is infinitely deep below the interface, time-harmonic waves with a given frequency can propagate with two different wavenumbers. The wave of smaller wavenumber propagates along the ice-cover while wave of higher wavenumber propagates along the interface. In this paper problems of wave scattering by a horizontal circular cylinder submerged in either the lower or in the upper layer due to obliquely as well as normally incident wave trains of both the wave numbers are investigated by using the method of multipole expansions. The effect of the presence of ice-cover on the various reflection and transmission coefficients due to incident waves at the ice-cover and the interface is depicted graphically in a number of figures.     

Static deformations of functionally graded polar-orthotropic cylinders with elliptical inner and circular outer surfaces

Functionally graded materials (FGMs) enable one to tailor the spatial variation of material properties so as to fully use the material everywhere. For example, in a hollow circular cylinder one can vary, in the radial direction, the material moduli to make the hoop stress constant. Whereas the problem for a hollow cylinder with the inner and the outer surfaces circular has been studied, that of a cylinder with a circular outer surface and a non-circular inner surface or vice versa has not been investigated. We study here such a plane-strain problem when the cylinder material is polar-orthotropic, material properties vary exponentially in the radial direction, and deformations are independent of the axial coordinate. The problem is challenging since the cylinder thickness varies with the angular position of a point, and the cylinder material is inhomogeneous. Equilibrium equations are solved by expanding the radial and the circumferential displacements in Fourier series in the angular coordinate. The method of Frobenius series is used to solve ordinary differential equations for coefficients of the Fourier series, and boundary conditions are satisfied in the sense of Fourier series. A parametric study has been conducted that delineates effects on stresses of the eccentricity of the ellipse, the material property gradation index and loads applied on boundaries of the cylinder. The analytical solutions presented here will serve as benchmarks for comparing solutions derived by numerical methods.     

Analysis of the stressed state of a circular thin-walled cylinder under complex loading and with nonlinear hardening of the material

The generalized differential equations of plastic flow for a material with nonlinear hardening are derived using the Prager kinematic model. An example of numerical analysis for stress variation under elastoplastic deformation of a thin-walled cylinder of a structural carbon steel is given for different elastoplastic material models. Translated from Problemy Prochnosti, No. 3, pp. 58–65, May–June, 2009.     

负压条件下爆炸罐内爆炸引起筒体振动及噪声特性

为了研究负压条件下容器内爆炸引起的振动及噪声特性,研制一套34.8 L抽真空爆炸罐装置.实验用单发8号工业雷管作为爆炸源,采用NuBox8016型爆破振动测试仪、SZ4A型噪声振动测试仪分别测试简体振动速度及爆炸噪声声压级,并使用快速傅里叶变换(fast Fourier transform,FFT)以及经验模态分解(e...     

A mathematical model for an upper convected Maxwell fluid with an elastic core: Study of a limiting case

In this paper we formulate a mathematical model for a continuum which behaves like an upper convected visco-elastic Maxwell fluid if the stress is above a certain threshold and like a neo-Hookean elastic solid if the stress is below that threshold. The constitutive equations for each phase are derived within the context of the theory of natural configurations and by means of the criterion of the maximization of the rate of dissipation [11]. We then focus on a limiting case in which the continuum becomes an elastic-rigid body. In this limiting case the constitutive relation of the material becomes implicit and, although there is no energy dissipation, it cannot be included in the class of hyperelastic (or Green) bodies. The stress indeed cannot be expressed as a function of the strain. This class of materials was first introduced by Rajagopal in [15] and is the subject of the forthcoming papers [3] and [4].     

On collocation schemes for integral equations arising in slender-body approximations of flow past particles with circular cross-section

A collocation method is presented for solving a singular integral equation of the second kind arsing in the slender-body approximation of viscous flow past slender bodies. When the spectral representation of the integral operator is explicitly known, the collocation method is shown to recover the spectrum of the continuous operator. The approximation error is estimated for two discretizations of the integral operator, and convergence is proved. The collocation scheme is validated for several test cases and extended to situations where the spectrum is not explicitly known.     

Identification of the vortex mechanism of front stabilization in modeling of asymmetric flow of an incompressible fluid along a cylinder with a protruding disk

Based on numerical solution of the Navier—Stokes equations by the finite-difference method and physical modeling in a wind tunnel of laminar flow along a cylinder with a protruding disk the vortex mechanism of front stabilization and reduction in the drag of blunt bodies is investigated.     

Flow and heat transfer of a micropolar fluid in an axisymmetric stagnation flow on a cylinder with variable properties and suction (numerical study)

Summary. In this paper, an analysis is presented to study the effects of variable properties, density, viscosity and thermal conductivity of a micropolar fluid flow and heat transfer in an axisymmetric stagnation flow on a horizontal cylinder with suction, numerically. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. However, the fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The similarity solution is used to transform the problem under consideration into a boundary value problem of nonlinear coupled ordinary differential equations which are solved numerically by using the Chebyshev finite difference method (ChFD). Numerical results are carried out for various values of the dimensionless parameters of the problem. The numerical results show variable density, variable viscosity, variable thermal conductivity and micropolar parameters, which have significant influences on the azimuthal and the angular velocities and temperature profiles, shear stress, couple stress and the Nusselt number. The numerical results have demonstrated that with increasing temperature ratio parameter the azimuthal velocity decreases. With increasing variable viscosity parameter the temperature increases, whereas the azimuthal and the angular velocities decrease. Also, the azimuthal and the angular velocities increase and the temperature decreases as the variable conductivity parameter increases. Finally, the pressure increases as the suction parameter increases.     

Heat transfer during the flow of an incompressible fluid in a circular tube,allowing for axial heat flow,with boundary conditions of the first and second kind at the tube surface

An examination is made of heat transfer in a hydraulically stabilized laminar stream and in a two-layer dynamic flow model.     

Experimental and finite element analyses on stress intensity factors of an elliptical surface crack in a circular shaft under tension and bending

Experimental backtracking technique and finite element analysis have been employed to evaluate the stress intensities along the front of an elliptical surface crack in a cylindrical rod. The finite element solution covers a wide range of crack shapes loaded under end-free and end-constrained axial tension and pure bending. Convenient closed form stress intensity expressions along the whole crack front for each of the loading cases have been given in terms of the crack aspect ratio, crack depth ratio and place ratio.The closed form solutions have been compared against a number of representative solutions collected from the literature. It has been found that different finite element results for the interior points are generally in good mutual agreement, while solutions derived from other methods may sometimes indicate different trends. At the surface interception point agreement is less good because of a complication in the interpretation of stress intensity there.Experimental backtracking results on the end-constrained axial tension case corroborate well with the closed form solution presented. It suggests that the current closed form solution is adequate in describing the stress intensities along the whole crack front of real surface cracks in cylindrical rods.