Page 64 - Computational Fluid Dynamics for Engineers

2.2 Navier-Stokes Equations 49

decrease U due to the flow into or out of the control volume across the control
surface, Q v represents any possible sources of U locally inside the volume, and
denotes any possible sources of U on the control surface. For the continuity
Q s
equation, U is g, F is gV and Q v = Q s = 0.
The components of the momentum equation, consistent with the generic
integral form of Eq. (2.2.24), are derived and discussed by Anderson [5]. They
may be summarized by the following equations:
^-component of the momentum equation
d
- jjj(Qu) dQ + jj{QuV) • dS
Q S (2.2.25)
=
(~pn • i + cr nn • i + a sm • i) dS + / / / {gf x) dQ
S Q
^-component of the momentum equation
d

Q S (2.2.26)
= / / {-pn • j + a nn • j + a srh • j) dS + / / / (gf y) dQ
s n
z-component of the momentum equation
{ew d§
M IJJ {0W) dn + II ^'

f f -> -> -* -> fff (2.2.27)
— / / (~pn ' k + cr nn - k + <j sm • k)dS + / / / (gf z) dQ
S Q
Here ft is a unit vector perpendicular to the infinitesimal control surface dS and
m is a unit vector tangent to the surface and pointing in the direction of the
viscous shear stress that acts on the surface.
As with the differential form of the momentum equations, Eqs. (2.2.2) to
(2.2.4), the left-hand sides of Eqs. (2.2.25) to (2.2.27) represent momentum flux
rates with the first term representing the time rate of change of momentum due
to unsteady fluctuations of flow properties inside the control volume and the
second term representing the net flow of momentum out of the control volume
across the surface S. The right-hand sides of Eqs. (2.2.25) to (2.2.27) represent
the sum of the net forces acting on the fluid as it flows through the control
volume. The first term represents the components of the surface forces which
are composed of pressure and viscous forces, while the second term represents
the body force.
The conservation integral form of the energy equation in the form of Eq.
(2.2.24) is