Page 97 - Aerodynamics for Engineering Students

80 Aerodynamics for Engineering Students

When the momentum equation is applied to an infinitesimal control volume (c.v.),
it can be written in the form:

Rate of increase of momentum within the C.V.

---
+ Net rate at which momentum leaves the C.V.
-,
(ii)

= Body force + pressure force + viscous force (2.59)
(iii) (iv) (4
We will consider now the evaluation of each of terms (i) to (v) in turn for the case
of two-dimensional incompressible flow.
Term (i) is dealt with in a similar way to Eqn (2.43), once it is recalled that
momentum is (mass) x (velocity), so Term (i) is given by
a aP;
-((pxvolume x i') =-6xSy x 1 = (2.60)
at at
To evaluate Term (ii) we will make use of Fig. 2.21 (c.f. Fig. 2.12). Note that the
rate at which momentum crosses any face of the control volume is (rate at which
mass crosses the face) x velocity. So if we denote the rate at which mass crosses a face
by h, Term (ii) is given by

h3 x $3 -hl x $1 +hZq x ?4-h2 x $2 (2.61)
But rj23 and ml are given by Eqns (2.38) and (2.39) respectively, and mz and h4 by
similar expressions. In a similar way it can be seen that, recalling ?= (u, v)
$1 = (u,v) - ($E)
;,
(;;,;;) :
(;;,;;):
v2= (u,v) - - - -, $4 = (u,v)+ -.- -
+

rillXrf1 m3xi73

Fig. 2.21