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

Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods

Boundary-conforming coordinate transformations are used widely to map a flow region onto a computational space in which a finite-difference solution to the differential flow conservation laws is carried out. This method entails difficulties with maintenance of global conservation and with computation of the local volume element under time-dependent mappings that result from boundary motion. To improve the method, a differential ''geometric conservation law" (GCL) is formulated that governs the spatial volume element under an arbitrary mapping. The GCL is solved numerically along with the flow conservation laws using conservative difference operators. Numerical results are presented for implicit solutions of the unsteady Navier-Stokes equations and for explicit solutions of the steady supersonic flow equations.

Geometric Conservation Law and Its Application to Flow Computations on Moving Grids

The identification issue of coherent vortices is investigated on the basis of direct numerical simulation (DNS) and large-eddy simulations (LES) of turbulent flows. It is first shown that the pressure Laplacian is positive within a low-pressure tube of small cross section enclosed by convex isobaric surfaces in a uniform-density flow. Since this quantity is related to the second invariant Q of ∇ u , the Q criterion (region where Q is positive) is a necessary condition for the existence of such tubes. This eduction scheme is compared to other classical methods in incompressible simulations of isotropic turbulence: a mixing layer, a channel flow and a backward-facing step. Q-isosurfaces turn out to display very nice coherent vortices. This criterion is also used in combination with a conditional sampling method to discuss the characteristics of quasi-longitudinal vortices in a manipulated channel flow. The contribution of near-wall vortical structures to velocity and vorticity fluctuations is clearly isolat...

On coherent-vortex identification in turbulence

A class of high-resolution algorithms is developed for advection of a scalar quantity in a given incompressible flow field in one, two, or three space dimensions. Multidimensional transport is modeled using a wave-propagation approach in which the flux at each cell interface is built up on the basis of information propagating in the direction of this interface from neighboring cells. A high-resolution second-order method using slope limiters is quite easy to implement. For constant flow, a minor modification gives a third-order accurate method. These methods are stable for Courant numbers up to 1. Fortran implementations are available by anonymous ftp.

High-resolution conservative algorithms for advection in incompressible flow

Numerical method for time dependent compressible Navier-Stokes equations applied to axisymmetric flow field produced by hypervelocity impact, examining viscous effects

The Effect of Viscosity in Hypervelocity Impact Cratering

Shock wave interaction with laminar boundary layer on flat plate using modified Lax-Wendroff difference technique

Numerical solution of the interaction of a shock wave with a laminar boundary layer

The current paper introduces a generalized, nonorthogonal coordinate system for application to supersonic flows with such large angles between streamlines of the initial Cauchy data and the marching direction that the velocity component in the marching direction is actually subsonic. Finite-difference operators are developed to solve the three-dimensional, steady, inviscid conservation equations of fluid flow referenced to this general frame. Moreover, a new method of aligning the difference mesh to the bow shock wave is presented which eliminates both the necessity for differencing the free-stream flow properties and the spurious fluctuations that often arise in conservative variables when they are differenced across the discontinuity. Results are given for the case of flow past a conical surface.

Robert W. MacCormack

Papers

The Effect of Viscosity in Hypervelocity Impact Cratering

Numerical solution of the interaction of a shock wave with a laminar boundary layer

A generalized hyperbolic marching technique for three-dimensional supersonic flow with shocks