Page 63 - Computational Fluid Dynamics for Engineers

48 2. Conservation Equations

2.2.2 Navier-Stokes Equations: Integral Form

The Navier-Stokes equations can also be derived for a finite control volume fixed
in space or moving with the fluid. The Navier-Stokes equations derived for a
fixed control volume, in either integral or differential form, are expressed in con-
servation form, while the equations for a moving control volume are necessarily
in nonconservation form. As we shall see later, in the numerical solution of the
conservation equations, the conservation form is preferable to avoid numerical
difficulties that may arise in some flows such as those containing shock waves.
The conservation form is also convenient in that the continuity, momentum and
energy equations can all be expressed by the same generic equation. A detailed
derivation of the Navier-Stokes equations in integral form is given in several
references, see for example Arpaci [2] and Anderson [4, 5]. Here we adopt the
notation and description in Anderson. For a finite control volume fixed in space,
with df2 denoting the control volume and dS the control surface, we first write
the conservation integral form of the continuity equation for a three-dimensional
compressible flow as

-|- fff gdn+ ff gV-dS = 0 (2.2.23)
Q S
The first term of this equation denotes the time rate of increase of mass

gdQ
/ / /
Q
inside the control volume, while the second term, with the sign convention that
positive mass flow
gV -dS
corresponds to outflow and negative to inflow, denotes the net mass flow out of
the control volume. Equation (2.2.23) is a specific example of a generic form of
the conservation integral equations, which can be written for a general unknown
variable U as follows:

^ fffudn+ ff F-dS = fffQ vdtt+ ffQsdS (2.2.24)
Q S Q S
As we shall see shortly, when the equations are expressed in integral form, we
are concerned with the flux of mass, momentum and energy into and out of the
volume. Typical examples of fluxes are mass flux, gV', flux of x-component of
momentum, guV, flux of ^/-component of momentum, gvV, flux of ^-component
of momentum, gwV, flux of internal energy, geV, and flux of total energy,
2
g(e + V /2)V. In Eq. (2.2.24), U is a quantity which can "accumulate" inside
the control volume, F is the flux associated with U which serves to increase or