Page 65 - Computational Fluid Dynamics for Engineers

50 2. Conservation Equations

n
E
§
+
I /// > « +ljw- " = ISM «r-*<°
Q S
ff _ ^ - (2.2.28)
— / / (~Q C -n — pn-V + cr nn • V + a sfa • V) dS
S
Here the vector q denotes the volumetric rate of heat addition per unit mass
and q c denotes the heat conduction vector.
The above equation is for compressible flow and, as such, it has more terms
than the incompressible energy equation given by Eq. (2.2.11). It is, however, the
same as the differential form of the energy equation given by Eq. (2.2.15a) for
compressible flows. The first term on the left-hand side of Eq. (2.2.28) represents
the time rate of change of total energy Et inside the control volume due to
transient variations of flow-field variables, and the second term represents the
net rate of change of total energy across the control surface. The terms on the
right-hand side of Eq. (2.2.28) correspond to the terms on the right-hand side
of Eq. (2.2.24).

2.2.3 Navier-Stokes Equations: Vector-Variable Form

As we shall see in Chapters 4, 5 and 10 to 12, before the application of the
numerical methods to the conservation equations, it is convenient to combine
the continuity, momentum and energy equations into a compact vector-variable
form. With / denoting a length scale, the speed of sound, a, denoting the velocity
scale, the parameters g, u, v, p, a xx, a xy, o- yy, Et, \±,t and x, y, with oo referring
to freestream quantities, can be expressed in nondimensional form as

Q — U V p
» = , u v = a , P — a 2 '
Qoo ^oo oo Poo oo
0~xxL O'xyL Cyyl
:x — 2~' 0~xy - °yy ~ V a2 ' (2.2.29)

Et x y
E t 2~> M = i

Then the compressible Navier-Stokes equations in a Cartesian coordinate sys-
tem, given by Eqs. (2.2.12b), (2.2.18), (2.2.19) and (2.2.15a) can be written
in dimensionless form, without body forces or external heat addition for two-
dimensional flows, as

dt dx dy Re \ dx dy J

For simplicity, the ~ will be dropped in dimensionless quantities, and the
Reynolds number Re is defined by