A note on the Kutta condition in Glauert's solution of the thin aerofoil problem

Glauert's classical solution of the thin aerofoil problem (a coordinate transformation, and splitting the solution into a sum of a singular part and an assumed regular part written as a Fourier sine series) is usually presented in textbooks on aerodynamics without a great deal of attention being paid to the rôle of the Kutta condition. Sometimes the solution is merely stated, apparently satisfying the Kutta condition automatically. Quite often, however, it is misleadingly suggested that it is by the choice of a sine series that the Kutta condition is satisfied.It is shown here that if Glauert's approach is interpreted in the context of generalised functions, (1) the whole solution, i.e. both the singular part and any non-Kutta condition solution, can be written as a sine-series, and (2) it is really the coordinate transformation which compels the Kutta condition to be satisfied, as it enhances the edge singularities from integrable to non-integrable, and so sifts out solutions not normally representable by a Fourier series.Furthermore, the present method provides a very direct way to construct other, more singular solutions.A practical consequence is that (at least, in principle) in numerical solutions based on Glauert's method, more is needed for the Kutta condition than a sine series expansion.     

A note on the solution of a time-dependent crack problem in an elastic composite

In an earlier paper[1] the problem of the interaction of a forced transient torsional wave and a penny-shaped crack terminating at a bi-material interface of an elastic composite was considered. The method used and results obtained therein assumed that the square of the shear wave speed of the outer member of the bi-material composite was less than twice of square of the shear wave speed of its inner member, i.e. $\text{¦?¦< 1}$. In this note it is shown how the problem can be solved if this restriction on ? is removed in order to cover all physically admissible elastic bi-materials that can be considered under the present model.     

A note on the porous-wavemaker problem

Summary A complete analytical solution is obtained, by using an integral transform method, for the porous-wavemaker problem, when the effect of surface tension is taken into account on the free surface of water of finite-depth in which surface waves are produced by small horizontal oscillations of a porous vertical plate. The final results are expressed in the form of convergent integrals as well as series and known results are reproduced when surface tension is neglected.     

A note on the wall-jet problem

Summary The wall-jet problem is considered with, in turn, the effects of blowing and suction through the wall and the wall moving. In both cases it is assumed that the effect has the appropriate power-law variation so as to maintain the similarity form. It is shown that the resulting ordinary differential equation has no acceptable solution satisfying the required boundary conditions. The wall jet problem in which the original momentum condition is retained, but allowing for both transpiration velocity and the wall velocity is derived and it is shown that this implies that a solution is possible only for suction. A solution to this problem is then obtained for the appropriate power-law variations to keep the similarity form.     

Singular solution of a water wave problem

The Green's function for water wave problem in an unbounded region (half space) bounded internally by a circular cylinder has been obtained by the use of an appropriate fourier transform in the vertical direction. The technique used in this investigation is specially useful when the liquid is internally bounded by cylindrical regions of the form D × I where D is any two dimensional region in the undisturbed free surface and I is the interval [0, ∞).     

A note on the wall-jet problem II

The wall-jet problem described in a previous paper in which the effects of suction through the wall and the wall moving were allowed so as to still satisfy the momentum condition for the wall jet is considered further. It is shown that, as well as the upper branch of solutions which were obtained previously, there is a lower branch of solutions, all of which have a region of reversed flow next to the wall.     

A note on the acoustic wave of an explosion

Abstract

The acoustic wave of an explosion in an unbounded fluid is solved. There is not only an implusive pressure but also an implusive dipole caused by the propogation of the explosion. And at the far field the major cause of damage is the implusive dipole.     

A formal expansion procedure for the internal solitary wave problem in a two-fluid system of constant topography

Summary Equations are derived for a two-dimensional internal solitary wave between two immiscible superimposed, inviscid and incompressible liquids bounded by two rigid planes in a channel of infinite length analogous to those derived by Epstein (Q. Appl. Math. 1974) for the case of two-dimensional free surface solitary water waves. The effect of surface tension at the surface of separation has been neglected. A pair of ordinary differential equations, of an infinite order, as well as an algebraic equation has been obtained and solved by a power series expansion in terms of a parameter a which depends mainly on the reciprocal of Froude number. Comparing similar powers of a leads to a set of non-linear, second-order differential equations. The analytical form of the wave profile has been found, up to a fourth order of the parameter a, in terms of the Froude number, density ratio between the two liquids, and the distance between the two planes bounding the two fluids. The effect of the Froude number, density ratio and the distance between the two planes, on the wave profile, maximum amplitude of the wave and the speed of a particle just above the interface, has been studied and illustrated.     

A note on discontinuous numerical solutions of the kinematic wave motion

This note discusses the numerical solution of the kinematic wave equation under those conditions when the solution contains a discontinuous shock. A finite element solution is described in which shocks are represented by discrete nodal discontinuities. The implementation of the method follows conventional finite element practice over the shockless regions of the solution domain which are coupled by frontal constraints. The basis of the method and examples of its application to the solution of the kinematic wave equation in one and two dimensions are presented.     

A variational numerical solution of the one‐dimensional wave propagation problem for an inhomogeneous medium

Abstract

Some basic properties of driving point impedance function Z(s) representing a new class of five‐element‐kind RLC‐FDNR networks are derived based on Tellegen's theorem. Any function which possesses these properties is then called a PR2 function (second order positive real function).

The realizability conditions for a PR2 function to be realizable as RLC‐FDNR networks have been described and given in theorems. Analytic tests of PR2 functions with examples using computer aids have been discussed. Furthermore, a modified Brune cycle has been proved successful for realizing this kind of networks. We have solved many typical examples using computer aids and concluded with a conjecture that any PR2 function is realizable with finite Brune cycles.     

A note on the line spring solution for a part-through crack

The line spring solutions used to solve part-through crack problems in plates are investigated by comparing with the results obtained from an accurate three-dimensional analysis using the body force method. Comparisons are made for the mode I stress intensity of a semi-elliptical crack with various aspect ratios a/b, i.e. the ratio of the width to the depth of the crack. The results indicate that the two methods agree well for cracks that are wider than they are deep (an aspect ratio greater than one).     

A note on one of the methods of solution of the diffusion equation

An account is given of one of the methods of solution of the equation which describes the variation of the radius of a water droplet whose temperature has become established.     

Possibility of useful mechanical energy from noise: the solitary wave train problem in the granular chain revisited

A momentary velocity perturbation at an edge of a granular chain with the grains barely touching one another and held between fixed walls propagates as a solitary wave whereas a long lived perturbation, even if it is noisy, ends up as a solitary wave train. Here, we extend our earlier work but with a force instead of a velocity perturbation. Such a perturbation can propagate an extended compression front into the system. We find that a snapshot of the distribution of grain compressions in the solitary wave train shows parabolic as opposed to an approximate exponential decay with the leading edge at the front of the traveling pulse and the trailing edge following it. The system’s time evolution depends on three independent parameters-the material properties, duration of perturbation and the characteristic amplitude of the perturbation. Hence, the coefficients used to describe the parabolic decay of the grain compressions in the solitary wave train depend on these three parameters. When a random finite duration force perturbation is applied we find that the randomness is smoothed out by the system, which in turn suggests that long granular chains (or equivalent systems, such as circuits) can be potentially useful in converting random noisy signals to organized solitary wave trains and hence to potentially usable energy.     

A note on the can-order policy for the two-item stochastic joint-replenishment problem

In this paper, a two-item continuous-review inventory system is studied. Demands for item 1 and item 2 occur at epochs generated by independent Poisson processes. In addition to the standard cost structure, there is economy of scale in joint replenishment. For the continuous joint replenishment problem, the literature proposes the can-order policy. Under this policy, an order is triggered by item 1 at its demand epoch, when its inventory position falls to its reorder level. In this situation, if the inventory position of item 2 is at or below its “can-order” level, item 2 is also included in this order and a discounted fixed ordering cost is charged for it. As a result, the inventory positions of both items are raised to their respective order-up-to levels. Reciprocally, the same procedure is valid at the demand epoch of item 2. In this study, this two-item inventory system is modeled as a semi-Markov decision process and a simple enumeration algorithm is proposed for its solution. We show that previous formulations of the problem do not necessarily converge to the best can-order policy by providing numerical examples.     

A note on the problem of edge crack in a semi-circular plate

International Journal of Fracture -     

A solution to the two-dimensional findpath problem

The 'findpath problem' a well-known problem in robotics, is the problem of finding a path for a moving solid among other solid obstacles. In this paper, a solution is proposed for the two-dimensional case where two point masses are required to move to designated areas or targets located in the horizontal plane while avoiding moving or stationary planar objects. The main tool used to solve the problem is the 'second or direct method of Liapunov', a powerful mathematical tool usually associated with the stability analysis of nonlinear systems. The theory developed from solving the two-dimensional findpath problem is then applied to the problem of cooperation between two planar robot arms. Computer simulations show the effectiveness of the proposed method     

A solution to Kronecker's problem

Consider [x ₁,...,x _n], the multivariate polynomial ring over integers involvingn variables. For a fixedn, we show that the ideal membership problem as well as the associated representation problem for [x ₁,...,x _n] are primitive recursive. The precise complexity bounds are easily expressible by functions in the Wainer hierarchy.Thus, we solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers. As a direct consequence, we also obtain a solution to certain foundational problem intrinsic to Kronecker's programme for constructive mathematics and provide an effective version of Hilbert's basis theorem. Our original interest in this area was aroused by Edwards' historical account of theKronecker's problem in the context of Kronecker's version of constructive mathematics.Supported by an Italian grant Italian MURST 40% Calcolo Algebraico e Simbolico 1993 and an NSF grant: #CCR-9002819.