Question

Derivation of the equation of change for the Reynolds stresses. At the end of S5.2 it was pointed out that there is an equation of change for the Reynolds stresses. This can be derived by 

(a) Multiplying the ith component of the vector form of Eq. 5.2-5 by vj and time smoothing, 

(b) Multiplying the jth component of the vector form of Eq. 5.2-5 by v’ and time smoothing, and 

(c) Adding the results of (a) and (b) Show that one finally gets Equations 5.2-10 and 11 will be needed in this development.   

Answer 1

erivation of the equation of change for the Reynolds stresses 

Multiplication of the ith component of Eq. 5.25 by vi’ and time-smoothing gives for constant p gives (in the Cartesian tensor notation of SA.9, with the Einstein summation convention and with the shorthand notation ∂t ≡ ∂/∂t):

Then we write the same equation with I and j interchanged. When the two equations are added we get, term by term:

 

Combining the above gives:

Or

 

The two terms on the left side are the substantial derivative term in Eq. 5D.1-1. The remainder of the terms is set out in the same order as in Eq. 5D.1-1.