I was recently reading an article about organized waves in turbulent flows. At one point in the theoretical part of the study, the author presents the equations for the kinetic energy of the organized waves. They then assume that certain terms, such as: $$\frac{\partial}{\partial x_j}\overline{\tilde{u}_j(\tilde{p}+\frac 12 \tilde{u}_i\tilde{u}_i)}$$
should vanish when integrated over large enough volumes, and that this means that this quantity represents an internal exchange of energy. Normally I would have given the argument the other way around, but that's not the case in the article (I may have misunderstood, or maybe they did mean it the other way around but there was a typo?). The tildes here represent the organized component of velocity ($\langle f\rangle=\overline f + \tilde f$ where angle brackets represent a phase average) and the overline is the time average. The domain is supposed to be infinite (no solid boundaries).
The obvious first steps would be:
$$\int_V\frac{\partial}{\partial x_j}\overline{\tilde{u}_j(\tilde{p}+\frac 12 \tilde{u}_i\tilde{u}_i)}dV=\overline{\int_V\frac{\partial}{\partial x_j}\tilde{u}_j(\tilde{p}+\frac 12 \tilde{u}_i\tilde{u}_i)dV}$$ $$=\overline{\int_{\partial V} (\tilde u_j\cdot n_j)(\tilde{p}+\frac 12 \tilde{u}_i\tilde{u}_i)dS}$$ where $n$ is the vector normal to a given infinitesimal surface $dS$. I feel the answer is staring me in the face, but I can't see it. Any hint would be appreciated.
PS: The full article by Reynolds and Hussain can be found here