Page 101 - Aerodynamics for Engineering Students

84 Aerodynamics for Engineering Students

n--i
I
Start

(b) Dilation

(d) Distortion Rotation Shear
Fig. 2.24 Transformation of a fluid element as it moves through the flow field

2.7.2 Rate of shear strain
Consider Fig. 2.25. This shows an elemental control volume ABCD that initially at
time t = ti is square. After an interval of time 6t has elapsed ABCD has moved and
distorted into A’B‘C’D’. The velocities at t = ti at A, B and C are given by
auSx ausy avsx avsy
ay2,
ax2
UA = ------ vA=v------ (2.69a)
ax 2 ay 2
UB = u----+-- ausx ausy VB = v---+-- avsx avsy (2.69b)
ax 2 ay 2’ ax 2 ay 2

ausx ausy avsx avsy
uc=u+ ----- vc=v+----- (2.69~)
ax 2 ay 2 ’ ax 2 ay 2