Page 61 - Computational Fluid Dynamics for Engineers

46 2. Conservation Equations

-j£ + V • EtV = q h - V • q + V • {Ilij • V) + gf • V (2.2.15a)

or in terms of internal energy per unit mass, e, as

DP C)U '
Q^+P{V-Y) = q h-V-q + Oij^ (2.2.15b)

In the above equations, IIij represents the stress tensor given by

Ilij = —pSij + (Tij (2.2.16a)
and (T{j (dui/dxj) represents the dissipation function ^ given by

$ = fj, \(—\ 2 l(—\ 2 2^—V (— — V (— —V
Vdx) \dyj \dz J \dx dy) \dy dzj

2
dtvV
H du 9w\ _2/du 3 \dx dv_ dz J (2.2.16b)
J
dy
dz
dx
Finally, we need an equation of state for the fluid to relate p, g and e. The
commonest example is the perfect gas law
p = (7 - l)c vgT

where
e = c vT, 7 = c p/c v

with c v and c p representing the specific heats in constant volume and pressure,
respectively.
The terms on the left-hand side of the energy equation given by Eq. (2.2.15a)
represent the rate of increase of total energy in the control volume (per unit
volume) and the rate of total energy lost by convection through the control
volume (per unit volume), respectively. The first term on the right-hand side of
the equation represents the heat produced per unit volume by external agencies,
the second term represents the rate of heat lost by conduction through the
control volume (per unit volume), and the third and fourth terms represent
the work done on the control volume by the surface forces and body forces,
respectively (per unit volume).
In the numerical solution of the conservation equations, it is often preferable
to express them in "divergence form" or "conservation form" to avoid numerical
difficulties that may arise in some flows, such as flows containing shock waves,
when the nondivergence form or the nonconservation form of the equations is
used. In conservation form the coefficients of the derivative terms can be con-
stant or variable; if variable, their derivatives do not appear in the equation. For
example, the continuity equation (2.2.12a) is in the conservation form. However,
if it is written as