A one-equation turbulence model for aerodynamic flows

Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to determine the steady transonic flow past an airfoil using an O mesh. Convergence to a steady state is accelerated by the use of a variable time step determined by the local Courant member, and the introduction of a forcing term proportional to the difference between the local total enthalpy and its free stream value.

Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes

This Key Note presents a summary of the development of the Finite Element Method in the field of Electromagnet ic Engineering, together with a description of several contributions of the authors to the Finite Element Method and its application to the solution of electromagnetic problems. First, a self-adaptive mesh scheme is presented in the context of the quasi-static and full-wave analysis of general anisotropic multiconductor arbitrary shaped waveguiding structures. A comparison between two a posteriori error estimates is done. The first one is based on the complete residual of the differential equations defining the problem. The second one is based on a recovery or smoothing technique of the electromagnetic field. Next, an implementation of the first family of Nedelec's curl-conforming elements done by the authors is outlined. Its features are highlighted and compared with other curl-conforming elements. A presentation of an iterative procedure using a numerically exact radiation condition for the analysis of open (scattering and radiation) problems follows. Other contributions of the authors, like the use of wavelet like basis functions and an implementation of a Time Domain Finite Element Method in the context of two-dimensional scattering problems are only mentioned due to the lack of space.

The finite element method in electromagnetics

A new set of benchmarks has been developed for the performance evaluation of highly parallel supercom puters. These consist of five "parallel kernel" bench marks and three "simulated application" benchmarks. Together they mimic the computation and data move ment characteristics of large-scale computational fluid dynamics applications. The principal distinguishing feature of these benchmarks is their "pencil and paper" specification-all details of these benchmarks are specified only algorithmically. In this way many of the difficulties associated with conventional bench- marking approaches on highly parallel systems are avoided.

/pdf/the-nas-parallel-benchmarks-1v48ldcocu.pdf

The Nas Parallel Benchmarks

https://ntrs.nasa.gov/api/citations/19860020304/downloads/19860020304.pdf

Pseudo-time algorithms for the Navier-Stokes equations

For low speed flows the use of the compressible fluid dynamic equations is inefficient. The use of an explicit scheme requires delta t to be bounded by 1/c. However, the physical parameters change over time scales of order 1/u which is much larger. Hence, it is not appropriate to use explicit schemes for very subsonic flows. Implicit schemes are hard to vectorize and frequently do not converge quickly for very subsonic flows. If one is only interested in the steady state then a minor change to an existing code can greatly increase the efficiency of an explicit method. Even when using an implicit method the proposed changes increase the efficiency of the scheme. The Euler equations for low speed flows will be considered first and then incompressible flows. The method is generalized to include viscous effects. Supersonic flow is accelerated by essentially decoupling the equations.

/pdf/fast-solutions-to-the-steady-state-compressible-and-1h7b6g2xgm.pdf

Fast solutions to the steady state compressible and incompressible fluid dynamic equations

A model-based non destructive testing (NDT) method is proposed for damage identification in elastic structures, incorporating computational time reversal (TR) analysis. Identification is performed by advancing elastic wave signals, measured at discrete sensor locations, backward in time. In contrast to a previous study, which was purely numerical and employed only synthesized data, here an experimental system with displacement sensors is used to provide physical measurements at the sensor locations. The performance of the system is demonstrated by considering two problems of a thin metal plate in a plane stress state. The first problem, which represents passive damage identification, consists in finding the location of a small impact region from remote measurements. The second problem is the identification of the location of a square hole in the plate. The difficulties one encounters in applying this identification method and ways to overcome them are described. It is concluded that while this is a viable model-based identification method, which may lead, after further development, to a practical NDT procedure, one must be careful when drawing conclusions about its performance based solely on numerical experiments with synthesized data.

/pdf/computational-time-reversal-for-ndt-applications-using-54une89w20.pdf

Computational Time Reversal for NDT Applications Using Experimental Data

A composite scheme is presented which combines the properties of the Lax–Wendroff and leapfrog algorithms. The stability properties in one dimension are analyzed for both the pure initial value and for the initial boundary value problem. In two space dimensions one must be careful which generalization of Lax–Wendroff and which generalization of leapfrog one uses. It is shown that some combinations are stable while others are unconditionally unstable. Numerical results are presented confirming the improved behavior of the composite scheme.

Composite methods for hyperbolic equations

Publisher Summary
This chapter discusses iterative methods for solving the two-dimensional Helmholtz equation Δu + k2n (x, y) u = 0. The function n (x, y) is a piecewise smooth function. It presents the case where n (x,y) is positive over a significant region. This equation arises in scattering theory and underwater acoustics. The chapter focuses on the numerical difficulties, and methods for overcoming them, associated with solving the discretized approximation to the equation with appropriate boundary conditions. The assumption n < 0 implies that the eigenvalues of Δ + k2n have both positive and negative real parts. The discretization of the equation results in a linear system of equations Ax = b. In applications, it is usually of interest to solve the two-dimensional Helmholtz equation in a region that is unbounded in at least one direction. To solve such a problem on a computer, the region is truncated and some approximation to a radiation condition is specified on an artificial boundary. This boundary condition can be either a global condition represented by an integral operator along the boundary or a local differential operator.

Eli Turkel

Papers

Pseudo-time algorithms for the Navier-Stokes equations

Fast solutions to the steady state compressible and incompressible fluid dynamic equations

Computational Time Reversal for NDT Applications Using Experimental Data

Composite methods for hyperbolic equations

Preconditioned conjugate gradient methods for the helmholtz equation