Obtain a self$-$similar solution for the downstream evolution of the two dimensional wake. We represent the mean streamwise velocity by $(U{)}_0-(U)=(U{)}_{s}f(),$ and the Reynolds shear stress by

$-$rho$(uv)=$ rho$(U{)}_s^{2}g(),$ where $f$ and $g$ are functions of $\frac{y}{l}$ only $($with $l$ being the transverse length scale of the flow.$)$

$($a$)$ By substituting into the mean flow streamwise momentum equation, show that a self$-$ similar solution is possible only if the quantities

$U_0$

are both independent of downstream location $(x).$

$($b$)$ By making use also of the momentum integral, show that the self$-$similar solution has the

forms

$l=A{x}^{\frac{1}{2}}\frac{,}{b}ar(U{)}_s=B{x}^{-\frac{1}{2}},$

where A and $B$ are constants to be determined.