Vorticity Versus Strain Rate in 2-D

Consider a rectangular particle of initial size (Ax, Ay) near the wall y = 0. On the upper edge the velocity is u = Au. Equivalently, this corresponds to moving with a particle located at (0, 0) in a velocity field where v = 0. After a short time At the particle is sheared and takes the shape of a parallelogram. The strain rate is defined as є1і2 = limAy^0 Ay = &y. Similarly, considering a velocity field with u = 0 and upon superposition with the previous flow field, one finds the following strain rate, Fig.8.3

du dv

£1,2 = £2,1 = +

dy dx

(a) y

Vorticity Versus Strain Rate in 2-D

Fig. 8.3 Shear a in x-direction and b in ^-direction

 

(b) y

 

x

 

Further, we define the following strain rates

du. d v

£1,1 = 2— and £2,2 = 2— dx dy

 

(8.3)

 

Vorticity is defined as

 

du dv

d y + dx

 

(8.4)

 

and the angular velocity of the particle as

 

1 1 ( du dv

Q = w = — +

2 2 dy dx

 

(8.5)

 

A general deformation reads:

 

u(x + Ax, y + Ay, t) = u(x, y, t) + Au(x, y, t) +———–

v(x + Ax, y + Ay, t) = v(x, y, t) + Av(x, y, t) + •••

 

(8.6)

 

where

 

Au(x, y, t) = du Ax + Ц Ay + Av(x, y, t) = dvAx + dv Ay + ■

 

(8.7)

 

or, in matrix form

 

(8.8)

 

Vorticity Versus Strain Rate in 2-D

Note: any square matrix A can be decomposed uniquely into the sum of a sym­metric and antisymmetric matrices

Подпись: (8.9)A = 1 (A + A*) + 1 (A – A)

where A* is the transposed of A. Hence

Vorticity Versus Strain Rate in 2-D

1 / £1,1 ё 1,2

2 £1,2 £2,2

1 ( 0 – из

= 2 U 0

 

(8.10)

 

(8.11)

 

Vorticity Versus Strain Rate in 2-D

Vorticity Versus Strain Rate in 2-D

Therefore

u(x + Ax, y + Ay, t)
v(x + Ax, y + Ay, t)
general 2 – D motion

Vorticity Versus Strain Rate in 2-D

(8.12)

 

Vorticity Versus Strain Rate in 2-D

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