Page 108 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 91

2.8.2 The derivation of the Navier-Stokes equations
Restricting our derivation to two-dimensional flow, Eqn (2.87) with (2.72a) and
(2.73) gives
dU dV au av
a,,=2p--, orV=2p--, CTxy=uyx=p (ay -+-) ax (2.90)
ax aY
So the right-hand side of the momentum Eqns (2.66a) becomes
- +p- -+-
aP
g,--+2p- ax ax :(;; 3
ax
(au)
-gx--+p ap -+- a2U a2u) +p- a (au -+- av) (2.91)
-
ax (ax2 ay2 ax ax ay
v
=0, Eqn (2.46)
The right-hand side of (2.66b) can be dealt with in a similar way. Thus the momen-
tum equations (2.66a,b) can be written in the form
-+-
p -+u-+v- ay =pgx--+p (8x2 9 (2.92a)
<g ax
ap
ax
au
ay2
a2U
av ap a2v $7
p -+u-+v- "1 =pg --+p -+- (2.92b)
(E ax ay y ay (ax2 ay2
This form of the momentum equations is known as the Navier-Stokes equations for
two-dimensional flow. With the inclusion of the continuity equation
au av
-+-=O (2.93)
ax ay
we now have three governing equations for three unknown flow variables u, v, p.
The Navier-Stokes equations for three-dimensional incompressible flows are given
below:
au av aw
-+-+-=O (2.94)
ax ay az

au au au au ap azU 8% aZU
ax ay az
ax + (32 + ayz + 32)
p(at + u- + v- + w -) = pg, - - (2.95a)
aZv
av av =pgy--+p -++++) azv
ap
p -+u-+v-+w- a2 (2.95b)
(E ax ay aZ ay (ax2 ay2 az2
aw aw ap a2W a2W a2W
p =pgz---+p -+-+-) (2.95~)
az 6x2 ay2 az2

2.9 Properties of the Navier-Stokes equations

At first sight the Navier-Stokes equations, especially the three-dimensional version,
Eqns (2.95), may appear rather formidable. It is important to recall that they are
nothing more than the application of Newton's second law of motion to fluid flow.