By Eleuterio F. Toro
Offers a accomplished, coherent and functional presentation of Riemann Solvers and Numerical equipment. Designed to supply an figuring out of the elemental ideas, the underlying concept, and the mandatory details of the sensible implementation of those strategies.
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Additional info for Riemann solvers and numerical methods for fluid dynamics: a practical introduction
Sample text
5 (Hyperbolic System). 2) is said t o be hyperbolic at a point ( x ,t ) if A has m real eigenvalues X I , . . , A, and a corresponding set of m linearly independent right eigenvectors K ( l ) ,. . , K(,). T h e system i s said t o be strictly hyperbolic if the eigenvalues X i are all distinct. Kote that strict hyperbolicity implies hyperbolicity, because real and distinct eigenvalues ensure the existence of a set of linearly independent eigenvectors. 2) is said to be elliptic at a point (x,t ) if none of the eigenvalues A, of A are real.
In this book we are interested in numerical methods to solve the governing equations. 1 . 7) by adding the physical effects of viscosity. 7) that are modified. The stresses in a fluid, given by a tensor S, are due to the effects of the thermodynamic pressure p and the viscous stresses. 10) and 17 is the viscous stress tensor. It is desirable to express S in terms of flow variables already defined. For the pressure contribution this has already been achieved by defining p in terms of other thermodynamic variables via an equation of state.
In this case 9 ( t ) in (1 66) becomes the total mass in the volume I - . By assuming t h a t n o mass as generated or annahrlated wrthzn V and recalling that the surface A moves with the fluid velocity. which means that no mass flows across t h e surface. This integral conservation law may be generalised t o include sources of mass, which will then appear as additional integral terms. 5 Integral Form of the Equations 21 and thus is the time-rate of change of the mass enclosed by the volume V. 71) is the net mass inflow, per unit time, over the mass outflow.