Page 87 - Aerodynamics for Engineering Students

70 Aerodynamics for Engineering Students

Yt

I P(x, y)

Fig. 2.11

Thus
4' = q: + 4:
and the direction of q relative to the radius vector is given by

p = tan- 1%
4n

Fluid acceleration
The equation of acceleration of a fluid mass is rather different from that of a vehicle,
for example, and a note on fluid acceleration follows. Let a fluid particle move from
P to Q in time St in a two-dimensional flow (Fig. 2.11). At the point P(x, y) the
velocity components are u and v. At the adjacent point Q(x+ Sx, y+ by) the
velocity components are u + 61.4 and v + Sv, i.e. in general the velocity component
has changed in each direction by an increment Su or Sv. This incremental change is the
result of a spatial displacement, and as u and v are functions of x and y the velocity
components at Q are
au au aV av
Sx
u + Su = u + - + -by and v + Sv = v + - SX + - Sy (2.34)
ax ay ax ay
The component of acceleration in the On direction is thus
d(u+Su) au dudx audy
--+--+--
-
dt at dxdt aydt
au au au
=-+u-+v- (2.35)
at ax ay
and in the Oy direction
d(v+ Sv) au dv dv (2.36)
-_ +u-+v-
dt - at ax ay
The change in other flow variables, such as pressure, between points P and Q may be dealt
with in a similar way. Thus, if the pressure takes the value p at P, at Q it takes the value

aP aP
p + sp = p + -6x + -6y (2.37)
ax ay