Page 60 - Computational Fluid Dynamics for Engineers

2.2 Navier-Stokes Equations 45

thermodynamics and states that the increase at energy in a system (i.e. control
volume) is equal to heat added to the system plus the work done on the system.
For incompressible flows, the work done on the system is negligible, and the
energy equation may be written as
DT (dq x dq v dq z\ . , n n , ^

where q is the conduction heat transfer rate per unit area in the three orthogonal
directions, qh is the heat "source" (i.e. radiation, chemical reactions) and c p is
the specific heat. The conduction heat-transfer terms may be written in the
form
2
2
2
(d T d T d T\
I dx 2 dy 2 dz 2 I
where k is the constant thermal conductivity.

Compressible Flows
For compressible flows, the Navier-Stokes equations are similar to those given
by Eqs. (2.2.1) to (2.2.4) for incompressible flows. Since the fluid properties
now also vary with temperature, the continuity and momentum equations are
coupled to the energy equation, and the solution of the energy equation provides
the temperature distribution in the flowfield. These equations are discussed in
some detail in several references, see for example [1,2], and are summarized
below for an unsteady compressible three-dimensional flow.
The continuity equation is

~+^-(QY) = 0 (2.2.12a)

For a Cartesian coordinate system, it becomes
+
-
i + K<««» !;<»») + |(«"» = ° (22 12b)
The momentum equations are identical to those given by Eqs. (2.2.2) to (2.2.4)
provided that, with Sij denoting the Kronecker delta function (6ij = 1, if i = j
and 6ij = 0 if i ^ j), the viscous stress tensor aij is written as
dui duj\ 2 du k
dij = /i lJ (i,j,k = 1,2,3) (2.2.13)
dxj ' dxi J 3 dxk
The energy equation can be written either in terms of total energy per unit
volume, Et,
2 2 14
Et = Q\e+yP\ ( - - )

as