William G. Szymczak

A Numerical Method for the Incompressible Navier--Stokes Equations Based on an Approximate Projection

In this method we present a fractional step discretization of the time-dependent incompressible Navier--Stokes equations. The method is based on a projection formulation in which we first solve diffusion--convection equations to predict intermediate velocities, which are then projected onto the space of divergence-free vector fields. Our treatment of the diffusion--convection step uses a specialized second-order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high Reynolds number. In contrast to conventional projection-type discretizations that impose a discrete form of the divergence-free constraint, we only approximately impose the constraint; i.e., the velocity field we compute is not exactly divergence-free. The approximate projection is computed using a conventional discretization of the Laplacian and the resulting linear system is solved using conventional multigrid methods. Numerical examples are presented to validate the second-order convergence of the method for Euler, finite Reynolds number, and Stokes flow. A second example illustrating the behavior of the algorithm on an unstable shear layer is also presented.

A numerical algorithm for hydrodynamic free boundary problems

A generalized formulation of inviscid incompressible hydrodynamics as a system of conservation laws subject to a one-sided density constraint is used as the basis of a numerical algorithm for a variety of hydrodynamic free surface problems. Benchmark calculations for colliding masses of fluid and for the motion of a spherically symmetric bubble are compared with theoretical predictions. Also shown are profiles calculated for an evolving underwater bubble near a wall. Energy dissipation is introduced as a measure of turbulence and is used in analyzing the numerical results. Convergence behavior of the numerical algorithm is discussed.

A Second-Order Projection Method for the Incompressible Navier Stokes Equations on Quadrilateral Grids

This paper describes a second-order projection method for the incompressible Navier-Stokes equations on a logically-rectangular quadrilateral grid. The method uses a second-order fractional step scheme in which one first solves diffusion-convection equations to predict intermediate velocities which are then projected onto the space of divergence-free vector fields. The spatial discretization of the diffusion-convection equations is accomplished by formally transforming the equations to a uniform computational space. The diffusion terms are then discretized using standard finite-difference approximations. The convection terms are discretized using a second-order Godunov method that provides a robust discretization of these terms at high Reynolds number. The projection is approximated using a Galerkin procedure that uses a local basis for discretely divergence-free vector fields. Numerical results are presented illustrating the performance of the method.

Numerical Solution of Hydrodynamic Free Boundary Problems

A generalized formulation of hydrodynamics as a system of conservation laws subject to a one-sided density constraint is used to solve a variety of hydrodynamic free surface problems. Consideration of the time derivative of the density constraint gives an alternative formulation in which an additional one-sided constraint on the divergence of the velocity appears. A corresponding numerical algorithm is presented, and benchmark calculations are compared with theoretical predictions. Also shown are profiles calculated for an evolving underwater bubble. Energy dissipation is introduced as a measure of turbulence and is used in analyzing the numerical results.

Multiple discrete solutions of the incompressible steady-state Navier-Stokes equations

SummaryThis paper discusses the computation of multiple solutions of various discretizations of the steady state incompressible Navier-Stokes equations. Solution paths (α,R) satisfying the discrete system of equationsH(α,R)=0, where α represents the discrete flow field andR is the Reynolds number, are computed using a pseudo arc-length continuation procedure. For flows over the back end of an axially symmetric body with a cusped tail in a coaxial circular cylinder, the solution paths often exhibit “hairpin” turning points. Dependence of the paths on the mesh spacing and various selections for the discretizations of the convective and diffusive terms are presented.

Modeling three-dimensional acoustic scattering from targets near an elastic bottom using an interior-transmission formulation

For targets near the sediment–fluid interface, the scattering response is fundamentally influenced by the characterization of the sediment in the model. We show that if the model consists of a three-dimensional elastic sediment with acoustic fluid on top, then the use of perfectly matched-layer (PML) approximation for the truncation of the infinite exterior domain for scattering applications has fundamental problems and gives erroneous results. We present a novel formulation using the an interior-transmission representation of the scattering problem where the exterior truncation with PML does not induce errors in the result. Numerical examples will be presented to verify the application of this formulation to scattering from elastic targets near a fluid–sediment interface.

Supersonic flows and their interaction with propagating acoustic signals: Acoustic black holes in the laboratory.

Work in particle physics and general relativity (GR) has established that deep connections exist between acoustics and GR. Most remarkable is the fact that acoustic wave propagation in fluids is governed by an effective Lorentzian spacetime geometry: Acoustic waves follow the geodesics of a (curved) acoustic metric. This provides an entirely new way of looking at acoustic propagation, and in principle provides valuable theoretical tools since much of the machinery developed by the GR community over the past several decades can be directly applied to acoustic systems expressed in this framework. Notably, supersonic liquid flows are predicted to have completely analogous properties to spacetime regions near (gravitational) black holes. We present the status of a developing research program at NRL, designed to begin exploring these connections via laboratory experiments, numerical simulations, and theoretical development. [Work supported by the Office of Naval Research.]

AN ALGORITHM FOR DIRECT SIMULATION OF LINEAR WAVE PROPAGATION IN IRREGULAR REGIONS

A numerical algorithm has been developed to simulate linear wave propagation in media containing irregular inhomogeneities, especially irregular voids in fluids. The computational domain is extended to include the regions occupied by the inhomogeneities through replacing the boundaries with properly chosen sources. The solution corresponding to Dirichlet boundary conditions on the inhomogeneities is presented. This algorithm can be used to calculate linear wave propagation in a fluid medium with multiple bubbles.

A hydro‐acoustic source model in calculating noise field generated by breaking waves

To develop a complete model for the breaking wave noise, it is necessary to relate the source quantities to the physical parameters of the wave‐breaking and noise‐generation processes. In this paper, the source structure of an individual breaking wave is simulated using a coupled hydro‐acoustic model, which incorporates the physical processes underlying the mechanisms of the generation of the noise. The physical processes of wave formation and breaking are modeled using a generalized hydrodynamics formulation, providing the hydrodynamic parameters, such as pressure variations and air cavity shapes, etc., for the acoustic calculation. In the acoustic simulation, an algorithm has been developed in handling wave propagation in irregular regions, such as the bubbly liquid generated by wave‐breaking. For the noise field modeling, the locations and occurrence times for the individual breaking waves are specified as stochastic quantities using Poisson simulations and the total noise field is calculated as the su...