Interdisciplinary Applied Mathematics

Скачать в pdf «Interdisciplinary Applied Mathematics»

(2.5a)


De


p— = -pV-v — V-q + Ф,


(2.5b)

where Ф = т • Vv is the dissipation function and D/Dt = d/dt + v • V is the material derivative.


In addition    to    the    governing    conservation    laws,    an equation    of    state    is


required. For ideal gases, it has the simple form


p = pRT,    (2.6)


where R is the ideal gas constant defined as the difference of the constant specific heats; that is, R = Cp — Cv, where Cv = ^p and Cp = jCv with Y the adiabatic index. For ideal gases, the energy equation can be rewritten in terms of the temperature, since e = p/(p(Y — 1)) = CvT, and so equation (2.5b) becomes


pCv ^ = -pV-v + V- [kVT] + Ф.    (2.7)


The system of equations (2.4a; 2.5a), (2.6), and (2.7) is called compressible Navier-Stokes equations, contains six unknown variables (p, v,p,T) with six scalar equations. Mathematically, it is an incomplete parabolic system, since there are no second-order derivative terms in the continuity equation.


A hyperbolic system arises in the case of inviscid flow, that is, p = 0 (assuming that    we    also    neglect    heat    losses    by    thermal diffusion,    that    is,


к = 0). In that case we obtain the Euler equations, which in the absence of external forces or heat sources have the form


dp


dt



+ V-(pv)



0,



d(pv)


dt



+ V-(pvv)



—Vp,



dE


~dt



+ V-[(E + p)v]



0.



(2.8a)


(2.8b)


(2.8c)


This system admits discontinuous solutions, and it can also describe the transition from a supersonic flow (where |v| > cs) to subsonic flow (where |v| < cs), where cs = (pRT)1/2 is the speed of sound. Typically, the transition is obtained through a shock wave, which represents a discontinuity in flow variables. In such a region the integral form of the equations should be used by analogy with equations (2.1a)-(2.1c).

Скачать в pdf «Interdisciplinary Applied Mathematics»

Метки