Radiation and Wall Boundary Conditions for Computational Aeroacoustics: A Review

A review of unsteady computational boundary conditions for computational aeroacoustics (CAA) problems is presented. This review is meant to serve as a general overview of previous work on solid wall, radiation and outflow boundary conditions that have been proposed and used in CAA calculations. Both the physical nature of the boundary condition problem as well as the numerical considerations affecting their implementation are discussed.     

Nonreflecting boundary conditions based on nonlinear multidimensional characteristics

Nonlinear characteristic boundary conditions based on nonlinear multidimensional characteristics are proposed for 2‐ and 3‐D compressible Navier–Stokes equations with/without scalar transport equations. This approach is consistent with the flow physics and transport properties. Based on the theory of characteristics, which is a rigorous mathematical technique, multidimensional flows can be decomposed into acoustic, entropy, and vorticity waves. Nonreflecting boundary conditions are derived by setting corresponding characteristic variables of incoming waves to zero and by partially damping the source terms of the incoming acoustic waves. In order to obtain the resulting optimal damping coefficient, analysis is performed for problems of pure acoustic plane wave propagation and arbitrary flows. The proposed boundary conditions are tested on two benchmark problems: cylindrical acoustic wave propagation and the wake flow behind a cylinder with strong periodic vortex convected out of the computational domain. This new approach substantially minimizes the spurious wave reflections of pressure, density, temperature, and velocity as well as vorticity from the artificial boundaries, where strong multidimensional flow effects exist. The numerical simulations yield accurate results, confirm the optimal damping coefficient obtained from analysis, and verify that the method substantially improves the 1‐D characteristics‐based nonreflecting boundary conditions for complex multidimensional flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

Towards a transparent boundary condition for compressible Navier–Stokes equations

A new artificial boundary condition for two‐dimensional subsonic flows governed by the compressible Navier–Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow, as well as a convection velocity, is used to properly discretize the terms corresponding to the entering waves. Numerical tests on various classical model problems, whose solutions are known, and comparisons with other boundary conditions (BCs), show the efficiency of the BC. Direct numerical simulations of more complex flows over a dihedral plate are simulated, without creation of acoustic waves going back in the flow. Copyright © 2001 John Wiley & Sons, Ltd.     

Absorbing Boundary Conditions

Absorbing boundary conditions for computational aeroacoustics (CAA) are reviewed. Commonly used absorbing zonal techniques, such as sponge layers and buffer zones, as well as perfectly matched layers (PML) are discussed. The basic ideas and central results of these methods are surveyed and summarized. Special attention will be given to the recently emerged PML technique and its application to CAA. Numerical examples are presented for PML in duct acoustics. A comparison of PML and non-PML absorbing boundary conditions will also be given.     

On the shoreline boundary conditions for Boussinesq‐type models

We propose and illustrate a novel type of shoreline boundary conditions for Boussinesq‐type models. On the basis of characteristic equations of the non‐linear shallow water equations, boundary conditions are developed equations that can suitably model the motion of the instantaneous shoreline. Such boundary conditions are then implemented in a numerical solver for a specific set of Boussinesq‐type equations, which have been proved very effective for near‐shore modelling. Finally, a number of tests are performed to validate and illustrate the behaviour of the new conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

A PSEUDOSPECTRAL CODE FOR CONVECTION WITH AN ANALYTICAL/NUMERICAL IMPLEMENTATION OF HORIZONTAL BOUNDARY CONDITIONS

A new code for simulating convection in a horizontal layer of fluid is described. The code can be used to study the usual Rayleigh –Bénard convection problem but can also incorporate internal heating, rotation and the vortex force responsible for Langmuir circulation. Boundary conditions in the horizontal directions are periodic, but a wide range of conditions may be imposed on the upper and lower boundaries. A novel feature of the method is the way in which these boundary conditions are implemented through the following analytical/numerical technique. The governing partial differential equations are reduced to a number of inhomogeneous second-order ODEs for the horizontal Fourier modes. The solutions to these are then written as the sum of a particular integral and a complementary function. The former is easily computed (numerically) without regard to the boundary conditions and the latter is then selected (analytically/numerically) to ensure that the boundary conditions are met. We apply our code to the problem of highly supercritical thermal convection in a shear flow. We compare our results with simulations in the literature and, by integrating over a longer time interval, find flow features not observed in the previous simulations, including stable time-dependent states, multiple stable equilibria and chaos. © 1997 John Wiley & Sons, Ltd.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

Vortex‐in‐cell method combined with a boundary element method for incompressible viscous flow analysis

In this study, an immersed boundary vortex‐in‐cell (VIC) method for simulating the incompressible flow external to two‐dimensional and three‐dimensional bodies is presented. The vorticity transport equation, which is the governing equation of the VIC method, is represented in a Lagrangian form and solved by the vortex blob representation of the flow field. In the present scheme, the treatment of convection and diffusion is based on the classical fractional step algorithm. The rotational component of the velocity is obtained by solving Poisson's equation using an FFT method on a regular Cartesian grid, and the solenoidal component is determined from solving an integral equation using the panel method for the convection term, and the diffusion term is implemented by a particle strength exchange scheme. Both the no‐slip and no‐through flow conditions associated with the surface boundary condition are satisfied by diffusing vortex sheet and distributing singularities on the body, respectively. The present method is distinguished from other methods by the use of the panel method for the enforcement of the no‐through flow condition. The panel method completes making use of the immersed boundary nature inherent in the VIC method and can be also adopted for the calculation of the pressure field. The overall process is parallelized using message passing interface to manage the extensive computational load in the three‐dimensional flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Pseudo‐characteristic formulation and dynamic boundary conditions for computational aeroacoustics

The present paper deals with the use of the pseudo‐characteristic formulation of the Navier–Stokes and Euler equations recently introduced by Sesterhenn (Comput. Fluid. 2001; 30 :37–67) for the simulation of acoustic wave propagation. The emphasis is put on the formulation of an efficient method on structured curvilinear grids, along with the definition and implementation of efficient boundary conditions. The cases of inflow, outflow, rigid/compliant walls and walls with prescribed impedance are addressed. The proposed boundary conditions are assessed on generic cases. The pseudo‐characteristic formulation enables a straightforward and optimal use of high‐order upwind dispersion‐relation‐preserving schemes, yielding an efficient method. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-homogeneous radiation and outflow boundary conditions simulating incoming acoustic and vorticity waves for exterior computational aeroacoustics problems

A set of non-homogeneous radiation and outflow boundary conditions which automatically generate prescribed incoming acoustic or vorticity waves and, at the same time, are almost transparent to outgoing sound waves produced internally in a finite computation domain is proposed. This type of boundary condition is needed for the numerical solution of many exterior aeroacoustics problems. In computational aeroacoustics, the computation scheme must be as non-dispersive and non-dissipative as possible. It must also support waves with wave speeds which are nearly the same as those of the original linearized Euler equations. To meet these requirements, a high-order/large-stencil scheme is often necessary. The proposed non-homogeneous radiation and outflow boundary conditions are designed primarily for use in conjunction with such high-order/large-stencil finite difference schemes. © 1998 John Wiley & Sons, Ltd.     

Toward low‐noise synthetic turbulent inflow conditions for aeroacoustic calculations

The accuracy of boundary conditions for computational aeroacoustics is a well‐known challenge, due in part to the necessity of truncating the flow domain and replacing the analytical boundary conditions at infinity with numerical boundary conditions. In particular, the inflow boundary condition involving turbulent velocity or scalar fields is likely to introduce spurious waves into the domain, therefore degrading the flow behavior and deteriorating the physical acoustic waves. In this work, a method to generate low‐noise, divergence‐free, synthetic turbulence for inflow boundary conditions is proposed. It relies on the classical view of turbulence as a superposition of random eddies convected with the mean flow. Within the proposed model, the vector potential and the requirement that the individual eddies must satisfy the linearized momentum equations about the mean flow are used. The model is tested using isolated eddies convected through the inflow boundary and an experimental benchmark data for spatially decaying isotropic turbulence. Copyright © 2013 John Wiley & Sons, Ltd.     

Fourth‐order accurate compact‐difference discretization method for Helmholtz and incompressible Navier–Stokes equations

A fourth‐order accurate solution method for the three‐dimensional Helmholtz equations is described that is based on a compact finite‐difference stencil for the Laplace operator. Similar discretization methods for the Poisson equation have been presented by various researchers for Dirichlet boundary conditions. Here, the complicated issue of imposing Neumann boundary conditions is described in detail. The method is then applied to model Helmholtz problems to verify the accuracy of the discretization method. The implementation of the solution method is also described. The Helmholtz solver is used as the basis for a fourth‐order accurate solver for the incompressible Navier–Stokes equations. Numerical results obtained with this Navier–Stokes solver for the temporal evolution of a three‐dimensional instability in a counter‐rotating vortex pair are discussed. The time‐accurate Navier–Stokes simulations show the resolving properties of the developed discretization method and the correct prediction of the initial growth rate of the three‐dimensional instability in the vortex pair. Copyright © 2004 John Wiley & Sons, Ltd.     

Recent advances of computational aeroacoustics

Computational aeroacoustics (CAA) is an interdiscipline of aeroacoustics and computational fluid dynamics (CFD) for the investigation of sound generation and propagation from various aeroacoustics problems. In this review, the foundation and research scope of CAA are introduced firstly. A review of the early advances and applications of CAA is then briefly surveyed, focusing on two key issues, namely, high order finite difference scheme and non-reflecting boundary condition. Furthermore, the advances of CAA during the past five years are highlighted. Finally, the future prospective of CAA is briefly discussed.     

关于气动声学数值计算的方法与进展

气动声学数值计算是近年才出现的研究领域。本文介绍了气动声学数值计算的方法和有关的问题、边界条件的处理以及计算非线性声波的数值方法和进展。讨论了计算气动声学(CAA)的特性及其与计算流体力学(CFD)的差异，指出气动声学数值方法的关键是建立能保持色散关系的差分方程和正确处理无反射边界条件。对于非线性声波传播的问题，为了得到正确的解，应注意提高差分格式对短波的分辨能力，同时发展能抑制“伪”振荡(短波)而对长波基本不起作用的数值方法。     

An improved immersed boundary‐lattice Boltzmann method based on force correction technique

In this paper, an improved immersed boundary‐lattice Boltzmann method based on the force correction technique is presented for fluid‐structure interaction problems including the moving boundary interfaces. By introducing a force correction coefficient, the non‐slip boundary conditions are much better enforced compared with the conventional immersed boundary‐lattice Boltzmann methods. In addition, the implicit and iterative calculations are avoided; thus, the computational cost is reduced dramatically. Several numerical experiments are carried out to test the efficiency of the method. It is found that the method has the second‐order accuracy, and the non‐slip boundary conditions are enforced indeed. The numerical results also show that the present method is a suitable tool for fluid‐structure interaction problems involving complex moving boundaries.     

On the stability and dissipation of wall boundary conditions for compressible flows

Characteristic formulations for boundary conditions have demonstrated their effectiveness to handle inlets and outlets, especially to avoid acoustic wave reflections. At walls, however, most authors use simple Dirichlet or Neumann boundary conditions, where the normal velocity (or pressure gradient) is set to zero. This paper demonstrates that there are significant differences between characteristic and Dirichlet methods at a wall and that simulations are more stable when using walls modelled with a characteristic wave decomposition. The derivation of characteristic methods yields an additional boundary term in the continuity equation, which explains their increased stability. This term also allows to handle the two acoustic waves going towards and away from the wall in a consistent manner. Those observations are confirmed by stability matrix analysis and one‐ and two‐dimensional simulations of acoustic modes in cavities. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional vortex simulation of unsteady flow in hydraulic turbines

On the basis of the Helmholtz decomposition, a grid‐free numerical scheme is provided for the solution of unsteady flow in hydraulic turbines. The Lagrangian vortex method is utilized to evaluate the convection and stretch of the vorticity, and the BEM is used to solve the Neumann problem to define the potential flow. The no‐slip boundary condition is satisfied by generating vortex sticks at the solid surface. A semi‐analytical regularization technique is applied to evaluate the singular boundary surface integrals of the potential velocity and its gradients accurately. The fast multipole method was extended to evaluate the velocity and velocity gradients induced by the discretized vortex blobs in the Lagrangian vortex method. The successful simulation for the unsteady flow through a hydraulic turbine's runner has manifested the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Simulating hydrostatic and non‐hydrostatic oceanic flows

The thin aspect ratio of oceanic basins is simultaneously a complication to contend with when developing ocean models and an opportunity to simplify the equations of motion. Here we discuss these two aspects of this geometric feature in the context of hydrostatic and non‐hydrostatic ocean models. A simple analysis shows that the horizontal viscous operator in the hydrostatic primitive equations plays a central role in the specification of boundary conditions on the lateral vertical surfaces bounding the domain. The asymptotic analysis shows that for very thin aspect ratios the leading‐order flow cannot be closed unless additional terms in the equations are considered, namely either the horizontal viscous forces or the non‐hydrostatic pressure forces. In either case, narrow boundary layers must be resolved in order to close the circulation properly. The computational cost increases substantially when non‐hydrostatic effects are taken into account. Copyright © 2008 John Wiley & Sons, Ltd.     

THE APPLICATION OF A FINITE DIFFERENCE/FINITE VOLUME ALGORITHM TO SOLVE MAXWELL'S EQUATIONS IN THE TIME DOMAIN

Abstract

A finite volume/finite difference method based on Ni's multigrid formulation is introduced for the solution of Maxwell's equations. The scheme is presented for the cases of transverse magnetic scattering from two-dimensional circular and square cylinders, as well as from NACA 0012 airfoil. The codes are validated against the traditional Method of Moments, which is analogous to a panel method in CFD. The circular cylinder scattering is compared to the analytical series solution for better understanding how the roles of numerical dispersion and dissipation errors affect the solution. The reflecting boundary conditions are modeled by the idea of inducing fields inside the conductor and a method of modeling the singularities that arise at a sharp corner is presented. Absorbing boundary conditions are modeled by integrating along the characteristic compatibility equations in the direction of the outgoing wave.