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Approximation Methods for Navier-Stokes Problems: by R. Rautmann

By R. Rautmann

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This type of correction to the ideal gas EOS is said to have first been suggested by Clausius. 49). Sometimes, this equation is also called the Noble-Abel EOS. In the study of propulsion systems, gaseous combustion products at very high densities are reasonably well 14 1. The Equations of Fluid Dynamics described by the covolume EOS. In its simplest version the covolume b is a constant and is determined experimentally or from equilibrium thermochemical calculations. 1 X w- 3 . 49) with b constant.

The shallow water equations are an approximation to the full free~ surface problem and result from the assumption that the vertical component of the acceleration Dv I Dt can be neglected. 138) gives p=pg(l)-y). 141) This is called the hydrostatic pressure relation. e. 138) are independent of y. Thus, the x and z components of velocity are also independent of y for all t if they were at a given time, t = 0, say. 145) Wt +uwx +wwz = -g'f/z. 137) with respect toy (vertical direction) to obtain + 1'1/ '1/ Ux dy 1 -h Wz dy -h + v\~h = 0.

5) can be expressed as conservation laws, namely au at af(u) +a;;-= 0' f(u) =au' au af(u) = 0 f( ) = ~ 2 at+ ax ' u 2u . 3 (Eigenvalues). 11) where I is the identity matrix. 2} are called the eigenvalues of the system. 44 2. Notions on Hyperbolic Partial Differential Equations Physically, eigenvalues represent speeds of propagation of information. Speeds will be measured positive in the direction of increasing x and negative otherwise. 4 (Eigenvectors). A right eigenvector of a matrix A corresponding to an eigenvalue Ai of A is a vector K(i) = [k~i), k~i), ...

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