Towards a generic non-reflective characteristic boundary condition for aeroacoustic simulations

A blended zonal characteristic boundary condition is proposed following a quantitative investigation of the performance of several non-reflective boundary conditions. Two test cases are considered that investigate the effects of acoustic and vortical plane waves impinging on the domain outflow region. A third test case investigates the effects of broadband turbulent flow impinging on a non-reflective outflow boundary condition. From these studies, two non-reflective boundary conditions based on a zonal characteristic method are found to provide a minimal acoustic response for impinging acoustic and vortical disturbances, respectively. These methods both make use of the transverse characteristic termsto improve performance, although each method uses a different inclusion of these terms.A final boundary condition is proposed that blends the performance of the two zonal characteristic methods. A blending function is used that switches the characteristic boundary condition smoothly between regions dominated by acoustic, or vortical, disturbances. The feasibility of this novel method is demonstrated on a test case where broadband turbulence impinges on a small section of an outflow region

Regularized characteristic boundary condition for the Lattice Boltzmann methods at high Reynolds number flows

This paper reports the investigations done to adapt the Characteristic Boundary Conditions (CBC) to the Lattice-Boltzmann formalism for high Reynolds number applications. Three CBC formalisms are implemented and tested in an open source LBM code: the baseline one-dimension inviscid (BL-LODI) approach, its extension including the effects of the transverse terms (CBC-2D) and a local streamline approach in which the problem is reformulated in the incident wave framework (LS-LODI). Then all implementations of the CBC methods are tested for a variety of test cases, ranging from canonical problems (such as 2D plane and spherical waves and 2D vortices) to a 2D NACA profile at high Reynolds number (Re = 100,000), representative of aeronautic applications. The LS-LODI approach provides the best results for pure acoustics waves (plane and spherical waves). However, it is not well suited to the outflow of a convected vortex for which the CBC-2D associated with a relaxation on density and transverse waves provides the best results. As regards numerical stability, a regularized adaptation is necessary to increase the Reynolds number. The so-called regularized FD adaptation, a modified regularized approach where the off-equilibrium part of the stress tensor is computed thanks to a finite difference scheme, is the only tested adaptation that can handle the high Reynolds computation

Strong Approximations of BSDEs in a domain

We study the strong approximation of a Backward SDE with finite stopping time horizon, namely the first exit time of a forward SDE from a cylindrical domain. We use the Euler scheme approach of Bouchard and Touzi, Zhang 04}. When the domain is piecewise smooth and under a non-characteristic boundary condition, we show that the associated strong error is at most of order h^{\frac14-\eps} where $h$ denotes the time step and \eps is any positive parameter. This rate corresponds to the strong exit time approximation. It is improved to h^{\frac12-\eps} when the exit time can be exactly simulated or for a weaker form of the approximation error. Importantly, these results are obtained without uniform ellipticity condition.Comment: 35 page

Finite element formulation of general boundary conditions for incompressible flows

We study the finite element formulation of general boundary conditions for incompressible flow problems. Distinguishing between the contributions from the inviscid and viscid parts of the equations, we use Nitsche's method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter, including the limit case of the Euler equations. In order to control the discrete kinetic energy, additional consistent terms are introduced. We treat the limit case as a (degenerate) system of hyperbolic equations, using a balanced spectral decomposition of the flux Jacobian matrix, in analogy with compressible flows. Then, following the theory of Friedrich's systems, the natural characteristic boundary condition is generalized to the considered physical boundary conditions. Several numerical experiments, including standard benchmarks for viscous flows as well as inviscid flows are presented

Spectral simulation of unsteady compressible flow past a circular cylinder

An unsteady compressible viscous wake flow past a circular cylinder was successfully simulated using spectral methods. A new approach in using the Chebyshev collocation method for periodic problems is introduced. It was further proved that the eigenvalues associated with the differentiation matrix are purely imaginary, reflecting the periodicity of the problem. It was been shown that the solution of a model problem has exponential growth in time if improper boundary conditions are used. A characteristic boundary condition, which is based on the characteristics of the Euler equations of gas dynamics, was derived for the spectral code. The primary vortex shedding frequency computed agrees well with the results in the literature for Mach = 0.4, Re = 80. No secondary frequency is observed in the power spectrum analysis of the pressure data

Numerical studies of the fluid and optical fields associated with complex cavity flows

Numerical solutions for the flowfield about several cavity configurations have been computed using the Reynolds averaged Navier-Stokes equations. Comparisons between numerical and experimental results are made in two dimensions for free shear layers and a rectangular cavity, and in three dimensions for the transonic aero-window problem of the Stratospheric Observatory for Infrared Astronomy (SOFIA). Results show that dominant acoustic frequencies and magnitudes of the self excited resonant cavity flows compare well with the experiment. In addition, solution sensitivity to artificial dissipation and grid resolution levels are determined. Optical path distortion due to the flow field is modelled geometrically and is found to match the experiment. The fluid field was computed using a diagonalized scheme within an overset mesh framework. An existing code, OVERFLOW, was utilized with the additions of characteristic boundary condition and output routines required for reduction of the unsteady data. The newly developed code is directly applicable to a generalized three dimensional structured grid zone. Details are provided in a paper included in Appendix A

An Efficiently Parallelized High-Order Aeroacoustics Solver Using a Characteristic-Based Multi-Block Interface Treatment and Optimized Compact Finite Differencing

This paper presents the development of a fourth-order finite difference computational aeroacoustics solver. The solver works with a structured multi-block grid domain strategy, and it has been parallelized efficiently by using an interface treatment based on the method of characteristics. More importantly, it extends the characteristic boundary condition developments of previous researchers by introducing a characteristic-based treatment at the multi-block interfaces. In addition, most characteristic methods do not satisfy Pfaff’s condition, which is a requirement for any mathematical relation to be valid. A mathematically-consistent and valid method is used in this work to derive the characteristic interface conditions. Furthermore, a robust and efficient approach for the matching of turbulence quantities at the multi-block interfaces is developed. Finally, the implementation of grid metric relations to minimise grid-induced errors has been adopted. The code was validated against a number of benchmark cases, which demonstrated its accuracy and robustness across a range of problem types

The role of numerical boundary procedures in the stability of perfectly matched layers

In this paper we address the temporal energy growth associated with numerical approximations of the perfectly matched layer (PML) for Maxwell's equations in first order form. In the literature, several studies have shown that a numerical method which is stable in the absence of the PML can become unstable when the PML is introduced. We demonstrate in this paper that this instability can be directly related to numerical treatment of boundary conditions in the PML. First, at the continuous level, we establish the stability of the constant coefficient initial boundary value problem for the PML. To enable the construction of stable numerical boundary procedures, we derive energy estimates for the variable coefficient PML. Second, we develop a high order accurate and stable numerical approximation for the PML using summation--by--parts finite difference operators to approximate spatial derivatives and weak enforcement of boundary conditions using penalties. By constructing analogous discrete energy estimates we show discrete stability and convergence of the numerical method. Numerical experiments verify the theoretical result