Context: I am trying to derive an equation given in a Journal of Fluid Mechanics paper (2.2). It deals with the analysis of an axisymmetric turbulent wake where cylindrical coordinate system has been used (which to me is a little hard to understand as I typically deals in Cartesian system).
We start with what is called as a momentum equation (simplified version after certain assumptions):
$$U_\infty \frac{\partial}{\partial x} (U - U_\infty) = - \frac{1}{r} \frac{\partial}{\partial r} (r \ \overline{uv})$$
Here, $r$ is the radial direction. The axial ($x$) velocity is defined as $U$ and $U_\infty$ is a constant freestream velocity. The term $\overline{uv}$ is called as a turbulent stress which tends to zero if $r$ tends to infinity (i.e. restricted within a finite radial distance).
The authors integrate this equation over a cross-section to yield: $$U_\infty \int_0^\infty (U_\infty - U) r \ dr \approx \theta^2U_\infty^2$$ where $\theta$ is the momentum thickness. I think I can tweak the variables to get the $\theta$ in the equation, but I am not able to understand how exactly would we take a cross-section and then integrate over it? Any leads would be appreciated.
PS: Another equation that hasn't been mentioned is a continuity equation that is written as: $$\frac{\partial U}{\partial x} + \frac{1}{r} \frac{\partial}{\partial x} (r V) = 0$$
HINT
We have deal with the boundary layer in the axisymmetrical stream, which behavior is invariant to the Reinnolds number $$\text{Re} = \dfrac{U_\infty l}{\nu},$$ where $\,\nu\,$ is the fluid viscosity and $\,l\,$ is the physical layer thickness.
The stress in the turbulent case is quadratic function of the Reinolds number.
We have the radial coordinate $\,r,\,$ which is scaled to the parameter $\,l\,$ and has the lower limit $\,0\,$ (at the surface) and the upper limit $\,\infty\,$ (freestream, or "local infinity"),
The linear scaling of the axial coordinate $\,x\,$ requires the factor $\,R\,$ (physical radial size), and the integration by this coordiate leads to the average value of the square of the "axial" Reinolds number, wherein the factor $\,R^2\,$ is hidden in the momentum thickness.