Small Perturbation Solution Research Articles

Mass transport in water waves over an elastic bed

The mass transport velocity in water waves propagating over an elastic bed is investigated. Water is assumed to be incompressible and slightly viscous. The elastic bed is also incompressible and satisfies the Hooke’s law. For a small amplitude progressive wave perturbation solutions via a boundary-layer approach are obtained. Because the wave amplitude is usually larger than the viscous boundary layer thickness and because the free surface and the interface between water and the elastic bed are moving, an orthogonal curvilinear coordinate system (Longuet-Higgins 1953) is used in the analysis of free surface and interfacial boundary layers so that boundary conditions can be applied on the actual moving surfaces. Analytical solutions for the mass transport velocity inside the boundary layer adjacent to the elastic seabed and in the core region of the water column are obtained. The mass transport velocity above a soft elastic bed could be twice of that over a rigid bed in the shallow water.

Forecasting scale-dependent dispersion from spills in heterogeneous aquifers

Existing theories of flow and contaminant transport in aquifers are either based on Monte Carlo simulations or small perturbation solutions of the governing stochastic partial differential equations, which limit the applications to cases of small variances in the physical parameters. In this article scale-dependent models to predict mean contaminant concentration from point sources (well injection), and non-point sources (ground surface spills) in heterogeneous aquifers are developed and tested. A general analytic technique, the Neumann expansion, is used in the solution of the equations. This method does not require the assumptions of small perturbation, logarithmic transformations, or a specific probability law in the random quantities. In addition, aquifer parameters such as the mean pore velocity are functionally defined in terms of the underlying groundwater flow problem, and field-measurable aquifer hydrogeologic properties such as mean transmissivity, mean hydraulic gradient, mean porosity and aquifer thickness. Comparison with theoretical models, and verification with field tracer tests at the Borden aquifer indicated that the proposed models reproduced the enhanced longitudinal dispersion as a function of distance reported in the literature.

Like‐ and cross‐polarized transmission scatter cross sections for random rough surfaces: Full wave solutions

In this work full wave expressions are derived for the incoherent diffuse scatter cross sections (per unit area) for waves transmitted across rough surfaces. Both the like‐ and cross‐polarized cross sections are considered since the surfaces are rough in two dimensions. The full wave solutions are compared with the small perturbation solutions as well as with the physical/geometrical solutions. It is shown that when the heights and slopes are of the same order of smallness, the full wave solutions are in agreement with the small perturbation solution, while in the high‐frequency limit the full wave solution is in agreement with physical/geometrical optics solutions, provided that specular (stationary phase) points exist on the surface. Several illustrative examples are considered and the refractive index is assumed to be larger or smaller than unity. The critical angle for total internal reflection has a significant impact on diffuse scattering in the nonspecular direction. Since the full wave solutions account for specular point scattering as well as Bragg type scattering in a uniform, self‐consistent manner, it is not necessary to adopt the two‐scale model for the rough surfaces.

Electromagnetic scattering and depolarization across rough surfaces: Full wave analysis

Full wave solutions are derived for vertically and horizontally polarized waves diffusely scattered across an interface that is two‐dimensionally rough separating two different propagating media. Since the normal to the rough surface is not restricted to the reference plane of incidence, the waves are depolarized upon scattering; and the single scattered radiation fields are expressed as integrals of a surface element transmission scattering matrix that also accounts for coupling between the vertically and horizontally polarized waves. The integrations are over the rough surface area as well as the complete two‐dimensional wave spectra of the radiation fields. The full wave solutions satisfy the duality and reciprocity relationships in electromagnetic theory, and the surface element scattering matrix is invariant to coordinate transformations. It is shown that in the high‐frequency limit the full wave solutions reduce to the physical optics solutions, while in the low‐frequency limit (for small mean square heights and slopes) the full wave solutions reduce to Rice's (1951) small perturbation solutions. Thus, the full wave solution accounts for specular point scattering as well as diffuse, Bragg‐type scattering in a unified, self‐consistent manner. It is therefore not necessary to use hybrid, perturbation and physical optics approaches (based on two‐scale models of composite surfaces with large and small roughness scales) to determine the like‐ and cross‐polarized fields scattered across the rough surface.

Green’s functions for the full-wave acoustic radiation fields diffusely scattered by one- and two-dimensionally rough surfaces

Green’s functions are derived for the acoustic radiation fields diffusely scattered by rough surfaces. The media on both sides of the rough interface are characterized by their bulk modulus and equilibrium density. Absorption is accounted for through the relaxation time parameter. The Green’s functions are expressed as integrals over the rough surface variables, as well as the incident and scatter wave vector variables. Special forms of the Green’s functions are obtained when the distances from the rough surface to the observation point and the source point are large compared to the acoustic wavelength and when these distances are also much larger than the lateral dimensions of the excited rough surface. In the appropriate limits, the full-wave solution is shown to be in good agreement with the small perturbation solution and the physical acoustics solution for the acoustic pressure scattered by random rough surfaces. Thus, it accounts for specular point scatter as well as Bragg-type scatter in a unified self-consistent manner. An approximate small-slope approximation, which is suitable for the sea surface, is also presented.

Flow between eccentric rotating cylinders: Bifurcation and stability

The effect of cylinder eccentricity on Couette-Taylor transition is investigated here for flow between infinite rotating cylinders. The method of analysis is Fourier expansion of the conservation equations in the axial direction, followed by projection onto a polynomial subspace. Critical points of the solution, which are characterized by singularity of the Jacobian matrix, are located via parametric continuation. This computational scheme permits an extension of the DiPrima and Stuart results to higher values of the eccentricity ratio; it also makes it possible to move far from the critical point into the supercritical Reynolds number regime. The first bifurcation from Couette flow is found to be supercritical, while supercritical flow is shown to consist of regions of plane motion with recirculation, separating toroidal vortex regions. The domain of recirculating flow is asymmetric with respect to the line of centers, and on increasing the supercritical Reynolds number its axial dimension decreases while its radial dimension, in the plane separating the vortex cells, increases. The results established here for critical Reynolds number agree well with those of DiPrima and Stuart at eccentricities where their small perturbation solution is applicable, and there is good agreement with the experimental data of Vohr. The torque calculations compare favorably with the experimental data of Donnelly and Simon for the concentric case and with the data of Castle and Mobbs for non-zero eccentricity.

Pseudospectral calculations of viscoelastic flow in a periodically constricted tube

The numerical implementation and results of a full pseudospectral method applied to the solution of an axisymmetric steady-state viscoelastic flow problem — the flow through a periodically constricted tube — are discussed. The method involves a double series expansion of the variables in terms of Fourier functions (sin and cos) in the periodic (axial) direction and Chebychev polynomials in the radial one. Results obtained with the Oldroyd-B viscoelastic fluid model are compared against those calculated using mixed pseudospectral / finite difference formulations. When viscoelasticity is absent (Newtonian viscous flow) the full pseudospectral calculations are extremely accurate; the small amplitude perturbation solution can be reproduced up to six digits of accuracy. The computed results in the Newtonian limit converge exponentially fast with the number of modes involved in either the radial or the axial directions. Furthermore, the results are in excellent agreement with the predictions obtained using independent numerical techniques. However, the accuracy of the calculations decreases with increasing elasticity in the flow, to the point where it becomes inferior to the accuracy obtained by the mixed techniques involving similar or less computational effort. The latter methods are shown to be much more robust, their accuracy being only slightly affected with increasing elasticity in the flow.

Acoustic backscatter cross sections for rough sea surface excited from above and below the air–water interface

Acoustic backscatter cross sections (normalized scattered pressure intensities) are evaluated for rough seas using a unified full-wave approach that accounts for specular point scatter and Bragg scatter in a self-consistent manner. Thus the two-scale model of the sea surface is not used in these investigations. Different surface wind speeds are considered. The full-wave cross sections are shown to be significantly different for excitations from above or below the air–water interface. The full-wave solutions are compared with the corresponding physical-acoustic and small-perturbation solutions.

Full wave analysis for rough surface diffuse, incoherent radar cross sections with height-slope correlations included

The bistatic scattering cross sections are derived for rough one-dimensional perfectly conducting surfaces using the full wave approach. The surfaces are characterized by four-dimensional Gaussian joint probability density functions for heights and slopes. Thus, correlations between the rough surface heights and slopes are accounted for in the analysis. Convergence of the formal series solution is considered. Self-shadowing effects are included. The full-wave solutions are compared with the small perturbation solutions, which are polarization dependent, and the specular point (physical optics) solutions, which are independent of polarization. Both the physical optics and the small perturbation solutions can be obtained from the full-wave solution.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Acoustic scattering by two-dimensionally rough interfaces between dissipative acoustic media—Full wave, physical acoustics, and perturbation solutions

Explicit expressions for the acoustic pressure and velocity scattered by two-dimensionally rough surfaces are derived using a full-wave approach. The conditions under which these solutions merge with the physical acoustic and small perturbation solutions in the high- and low-frequency limits are given. The acoustic media on both sides of the rough interface are characterized by their bulk modulus, equilibrium density, and relaxation time (to account for dissipation); the Dirichlet and Neumann boundary conditions are treated as special cases. The closed-form full-wave expressions for the surface element scattering coefficients are significantly different for these special cases. However, the corresponding physical optics solutions differ only in the sign of the acoustic reflection coefficient. The full-wave solution can be applied to composite surfaces with a broad range of roughness scales. Since it accounts for specular point and diffuse scattering in a unified self-consistent manner, there is no need to adopt a two-scale model of the rough surface. Thus the full-wave expressions for the rough surface scattered fields are also more suitable for application to broadband (transient) excitation problems and for the solution of inverse problems.

Prediction by the method of moments of transport in a heterogeneous formation

Natural geologic formations are highly heterogeneous. It is impossible and often unnecessary to describe in deterministic terms the spatial variability of their properties. However, the hydrogeologic parameters may be represented in probabilistic terms. Prediction of solute transport may then be defined as the derivation of the probabilistic properties of concentration. This work deals with the first two integral (or spatial) moments of solute concentration in a heterogeneous formation of infinite extent. The first moment is the vector of the mean position of the centroid of the plume and, in a generalized sense, represents advection. The second moment is the matrix of dyadics of the mean squared displacement about the average position of the centroid of the plume and, again in a generalized sense, represents dispersion. Assuming that the mixing at the laboratory scale is Fickian, with random but time-invariant velocities and lab-scale dispersion matrices, the differential equations satisfied by the first two moments are derived. An analytical first-order (or small perturbation) solution is obtained for stationary velocity, and compared with a numerical solution.

Numerical simulation of scattering from simple and composite random surfaces

Numerical simulation studies of backscattering from one-dimensional, statistically known, perfectly conducting random surfaces are carried out to examine (1) the scattering characteristics in the intermediate-frequency region, where neither the low- nor the high-frequency approximation is applicable, (2) the validity of the two-scale concept in rough-surface scattering, and (3) applicability of the wavelength filtering concept to simple and composite surfaces. It is found that in the intermediate-frequency region the angular scattering curves for both the horizontal and vertical polarizations decrease at a slower rate than that predicted by a first-order small perturbation solution and that the Kirchhoff solution always lies between the two polarizations. In addition, the level difference between the horizontal and vertical polarization scattering coefficients is smaller than that of the perturbation solution. The two-scale concept has no general applicability. Its application is restricted to two-scale surfaces where one scale satisfies the Kirchhoff approximation and the other the small perturbation assumptions such that the two scales dominate scattering in different angular regions. The concept of wavelength filtering is applicable to simple surfaces or to each component of a composite surface in the low-frequency region where scattering is dominated by a narrow band of surface frequency components but loses its validity gradually as we approach the high-frequency limit where scattering is determined by the surface slope distribution. It is not the explanation of why the surface components of a composite surface may dominate scattering over different angular regions.

Friction coefficients of rippled beds in oscillatory flow

Dissipation and friction factors have been computed for rippled beds in oscillatory flow using a finite-difference solution of the two-dimensional equation of vorticity. The computations show good agreement with available experimental results and also, at very low Reynolds numbers, with the results of a small-perturbation solution of the vorticity equation.

Connection between various small-waveheight solutions of the problem of scattering from the ocean surface

Careful developments of a number of approximate solutions of the problem of point source acoustic scattering from the ocean surface are presented. The regime of interest is the small-waveheight regime, where the rms waveheight of the ocean surface is small compared to the acoustic wavelength and the correlation length of the ocean surface. The approximate solutions fall into two categories: those derived by utilizing the Kirchhoff approximation and those based on a small-waveheight perturbation expansion. It is shown that within the small-waveheight regime the small-perturbation solution is superior to any solution based on the Kirchhoff approximation, the former being uniformly valid in the ratio of acoustic wavelength to surface-wave wavelength. In that portion of the small-waveheight regime where the ratio of acoustic wavelength to surface-wave wavelength is small, all approximate solutions are shown to be in agreement. Pitfalls which can and have occurred in the misapplication of the Kirchhoff approximation are pointed out.

Internal waves in a viscous atmosphere

The way in which internal waves change in amplitude as they propagate through an incompressible fluid or an isothermal atmosphere is considered. A similarity solution for the small amplitude isolated viscous internal wave which is generated by a localized two-dimensional disturbance or energy source was given by Thomas &amp; Stevenson (1972). It will be shown how summations or superpositions of this solution may be used to examine the behaviour of groups of internal waves. In particular the paper considers the waves produced by an infinite number of sources distributed in a horizontal plane such that they produce a sinusoidal velocity distribution. The results of this analysis lead to a new small perturbation solution of the linearized equations.

Image system solution for store aerodynamics with interference. I

The mutual aerodynamic interference problem for external aircraft-stores has been analyzed using the image system technique. To facilitate this analysis it has been assumed that small perturbation solutions are valid. It is further assumed that the external stores are slender, axisymmetric bodies and that the mutual interference can be analyzed by first assuming a cross-flow solution. Both body-body and wing-body interference solutions have been obtained. The first were considered in Part I of this paper. For the wing-body solution, the image systems in the cross-flow plane consist of vortex pairs appropriately located by using the Milne-Thomson circle theorem. The strengths of the image vortex system are then determined by satisfying the body boundary conditions at designated control points on the external stores. Good agreement has been found between the theoretical and experimental results.

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

A fast direct (noniterative) “Cauchy-Riemann Solver” is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver (Buneman's method, but with the cyclic reduction of simple tridiagonal matrices replaced by the Thomas algorithm). The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Asymptotic solution to the problem of optimal low-thrust energy increase.

HE problem to be considered is that of optimal ascent from an initial circular planetary orbit to some specified final energy level by a spacecraft equipped with a low-thrust engine. Optimal will be defined as minimum time; and since it will be assumed that the engine produces continuous thrust with constant thrust-acceleration, fuel expenditure is minimized. Analytical solutions to this problem have previously been found by Lawden, 1 and by Breakwell and Rauch.2 Until now, Lawden's was the only analysis which used a small parameter perturbation approach; and his results failed to predict the oscillatory nature of the optimal control program. Breakwell and Rauch's work was directed primarily toward the analytical representation of a nominal trajectory and guidance coefficients for a neighboring optimal guidance scheme. Their solution is basically a series solution in the radial distance, but also contains some additional correction terms which were found by developing a set of defining differential equations, assuming a periodic solution with variable coefficients, and employing the method of averaging to determine those coefficients. The solution correctly, represents the characteristics of the control program and trajectory and, for at least eight revolutions, matches a numerically generated solution to within 1%. The use of the radial distance as independent variable and the series form of the solution, however, make an analysis of the motion and a comparison with other spiral trajectories difficult. The purpose of this Note is to present an accurate small parameter perturbation solution to the problem. In addition, the optimal trajectory is analyzed and compared with a tangential thrust trajectory. Analysis The problem is formulated using the equations of motion

Focusing of Finite-Amplitude Cylindrical and Spherical Sound Waves in a Viscous and Heat-Conducting Medium

The focusing of cylindrical and spherical pulses of finite amplitude in a medium of small viscosity and heat conductivity has been studied by dividing the region of interest into three parts: converging, interaction, and diverging regions. In the converging region, the flow field is governed by the radial Burgers equation with a small parameter multiplying the term with second derivatives. The method of matched asymptotic expansion is found applicable to this equation with an N wave as an initial condition; a composite solution obtained describes a converging N wave with increasing amplitude and wavelength. The front and rear shocks of the N wave are locally described by Taylor's shock structure. In the interaction region, no small perturbation solution exists for the shocks. However, the flow field between the front and rear shocks satisfies to first order the inviscid linear wave equation, which is solved by the Fourier transform technique. The treatment of the diverging region is similar to that of the converging region, except for trivial changes in the analysis. It is shown that the effect of entropy increase on the failure of the solution at the focus is inconsequential if certain restrictions on the initial strength and wavelength are satisfied.

Unsteady Magnetohydrodynamic Flow Past Thin Bodies with Inwardly Diffusing Magnetic Field

A rational small-perturbation theory is proposed for treating an initial- and boundary-value problem in magnetohydrodynamic flow past thin symmetrical bodies with the thickness parameter ε as the perturbation parameter. An irrotational flow over the body is established initially in the total absence of electromagnetic interactions. The fluid is assumed to be incompressible, inviscid, and electrically conducting. Moreover, it is bounded by two infinite parallel planes outside which vacuum prevails. The external magnetic field is then imposed by turning on current sheets of constant intensity at the fluid-vacuum boundaries. The zeroth- and first-order solutions are obtained analytically for an arbitrary body shape. The establishment of Lary's steady flow past a thin body at super- or sub-Alfvénic conditions, the existence of small-perturbation solutions at exactly Alfvénic flows, and the appearance of an accumulation point of poles to characterize the transient behavior of a physical system are the interesting features which emerge in the course of the present work.