Three-dimensional Parabolic Equation Research Articles

A three-dimensional parabolic equation model that includes the effects of rough boundaries

A three-dimensional parabolic equation (3DPE) model that handles wide propagation angles in depth, narrow propagation angles in azimuth, and rough boundaries is derived and solved numerically with the method of alternating directions. A homogeneous boundary condition, which is easily incorporated into the numerical solution of the 3DPE, is applied at the ocean surface to approximate the effects of rough boundaries in terms of a reflection coefficient that depends on grazing angle. Calculations are presented to demonstrate the accuracy of the 3DPE and to demonstrate that horizontal coupling, the effects due to the term involving azimuth derivatives, can be important in shallow water (previous work has focused on deep-water applications). The model is applied to problems involving rough boundaries and range dependence.

A parabolic equation model for scattering in the ocean

The small-angle-of-propagation limit and the method of matched asymptotics are applied to derive an efficient model for solving realistic underwater acoustics problems involving both propagation and scattering from a submerged object. The propagation and scattering aspects of the waveguide scattering problem are decoupled by approximating the waveguide Green’s function on the surface of the scatterer. For low frequencies, the small-angle limit also allows one to approximate the incident field with a horizontally propagating plane wave and the scattered field with an azimuthally specular point-source field. With these approximations, scattering calculations can be performed efficiently in the time domain. Calculations involving the three-dimensional parabolic equation and the time-domain parabolic equation are presented.

Numerical computation of convective heat transfer in the bend tube flow by time-dependent three-dimensional parabolic equations.

Time-dependent three-dimensional N-S and energy equations are solved numerically for the laminar flow and heat transfer in a curved tube with straight inlet and outlet sections. The results clearly explain the effects of curvature in the upstream of the straight section and the existence reverse flow in the main flow direction of the curved section. The heated section adopted only in the curved tube does not show much effect on the upstream straight section. In the reverse flow field which appeared in the curvature, profound heat flow was found to wards the upstream. To explain the reverse flow effects, the dissipation term in the main flow direction becomes important and can be treated numerically.

A mathematical model for the three-dimentional ocean sound propagation

A model is developed mathematically to represent sound propagation in a three-dimensional ocean. The complete development is based on characteristics of the physical environment, mathematical theory, and computational accuracy. While the two-dimentional underwater acoustic wave propagation problem is not yet solved completely for range-dependent environments,three-dimentional environmental effects, such as fronts and eddies, often cannot be neglected. To predict underwater sound propagation, one usually deals with the solution of the Helmholtz (reduced wave) equation. This elliptical equation, along with a set of boundary conditions including a wall condition at the maximum range, forms a well-posed problem, which is pure boundary-value problem. An existing approach to economically solve this three-dimensional range-dependent problem is by means of a two-dimensional parabolic partial differential equation. This parabolic approximation approach, within the limitation of mathematical and acoustical approximations, offers efficient solutions to a class of long-range propagation problems. The parabolic wave equation is much easier to solve than the elliptic equation; one major saving is the removal of the wall boundary condition at the maximum range. The application of the two-dimensional parabolic wave equation to a number of realistic problems has been successful. We discuss the extension of the parabolic equation approach to three-dimensional problems. This paper begins with general considerations of the three-dimensional elliptic wave equation and shows how to transform this equation into parabolic equations which are easier to solve. The development of this paper focuses on wide angle three-dimensional underwater acoustic propagation and accommodates as a special case prevoius developments by other authors. In the course of our development, the physical properties, mathematical validity, and computational accuracy are the primary factors considered. We describe how parabolic wave equations are derived and how wide angle propagation is taken into consideration. Then, a discussion of the limitations and the advantages of the parabolic equation approximation is highlighted. These provide the background for the mathematical formulation of three-dimensional underwater acoustic wave propagation models. Modelling the mathematical solution to three-dimensional underwater acoustic wave propagation involves difficulties both in describing the theoretical acoustics and in performing the large scale computations. We have used the mathematical and physical properties of the problem to simplify considerably. Simplications allow us to introduce a three-dimensional mathematical model for underwater acoustic propagation predictions. Our wide angle three-dimensional parabolic equation model is theoretically justifiable and computationally accurate. This model offers a variety of capabilities to handle a class of long-range propagation problems under acoustical environments with three-dimensional variations.

Prediction of secondary vortex flowfields generated by an interacting multiple free-jet configuration

APENALTY finite element algorithm for solution of the three-dimensional parabolic Navier-Stokes equations for subsonic turbulent flows is applied to the prediction of secondary vortex flowfields induced by a multiple free-jet configuration. The combined action of rapid decay of the initial high-speed parallel jets, turbulence level, induced entrainment from the far field, and geometric discreteness of a four-jet configuration, is predicted to produce a persistent system of eight counterrotating vortex pairs in the plane normal to the initial jet axis. Numerical predictions are interpreted and compared with available experimental data. Contents As detailed in the complete paper,1 a penalty finite element numerical algorithm is well suited to the solution of subsonic, three-dimensional multiple free-jet problem definitions. The abiding predominance of a preferred (axial) flow direction permits an order-of-magn itude analysis confirming that associated diffusion effects are insignificant. Discarding these terms yields the thin-layer Navier-Stokes (TLNS) approximation; provided a suitable pressure interaction procedure exists, solution of the TLNS system can be cast into the efficient space-marching, initial-value, parabolic NavierStokes (PNS) equation set. The ordering analysis also confirms that the transverse plane momentum equations govern pressure modifications to first order, while the continuity equation is the principal (divergence-free) momentum constraint statement. The key features of the PNS penalty finite element algorithm, as applied to the free-jet problem class, directly address these fundamental issues by 1) establishment of a quasilinear pressure Poisson equation, with complementary and particular solutions that readily admit interaction and far-field boundary condition requirements; and 2) a specific continuity equation constraint yielding efficient and accurate global communication of the local momentum defect relaxa

Difference schemes for the parabolic wave equation in ocean acoustics

We introduce and analyze a collection of difference schemes for the numerical solution of a model multidimensional equation of Schrödinger type with applications to the three-dimensional parabolic wave equation arising from the sound propagation in the ocean. This collection of methods includes explicit and implicit schemes, two-level and three-level schemes and real and complex schemes. Many of these are analogous to classical schemes for the heat equation and the wave equation, but some schemes are unique to the Schrödinger equation. Von Neumann type stability results are given for all schemes. Numerical results arising from the application to an ocean acoustic problem are presented.

Aspects of three-dimensional parabolic equation computations

In some problems of interest, sound propagation in the ocean involves significant variations in all three space dimensions. A review is provided of physical situations where three-dimensionality occurs and of physical mechanisms which can cause it. Parabolic approximations to the Helmholtz equation are described which are appropriate for three-dimensional propagation problems. Particular attention is given to estimates of limits of validity for the parabolic equations along with algorithms for their numerical calculation. Some computational examples are discussed for finite-difference approximations. Simple analytical solutions are suggested which are useful for three-dimensional test computations. These solutions illustrate comparisons between the Helmholtz equation and the parabolic approximation. In addition, they can show transition between two- and three-dimensional propagation characteristics and can provide accuracy tests for numerical computations.

A wide-angle three-dimensional parabolic wave equation

A third-order partial differential equation with wide-angle capability is formulated to predict three-dimensional underwater sound propagation. The development is based on physical acoustic characteristics and mathematical theory. Both operator and asymptotic analyses are given to thoroughly discuss the validity of the formulation. Physical conditions are indicated when a three-dimensional approach is needed.

An approximation to the three-dimensional parabolic-equation method for acoustic propagation

We have reported previously [Ralph N. Baer, J. Acoust. Soc. Am. 69, 70–75 (1981)] on an algorithm, based on the parabolic approximation to the reduced wave equation, for the propagation of sound in three dimensions in the ocean. We present here a simpler algorithm: solve N two-dimensional problems and combine the results to form an approximate three-dimensional solution. Analytic and numerical results show that this N×2D approach is an excellent approximation to the original algorithm for realistic ocean environments, even those where fronts and eddies are present, provided redirection of energy in azimuth due to boundary interaction is not important. We compare the two computer models based on these algorithms for several test cases by considering how they distribute energy spatially, and by simulating the performance of a hypothetical horizontal array of hydrophones placed in the calculated complex-valued acoustic fields. In an example of propagation through a large and strong ocean eddy, the N×2D method predicts intensities and array performance parameters (such as array signal gain) which are typically within 8% of the values from the full 3D algorithm. In another example, with a large linear cross-range gradient on the index of refraction, the N×2D and 3D solutions are essentially identical out to a range of 120 km. The new method has made it practical to consider higher source frequencies and has reduced the substantial computer time of the three-dimensional program.