Constant-volume similarity in a V-shaped channel ($h=[3(X^2-x^2)/(7Ct){]}_{+}^{1/3},X\propto t^{2/7}$)

The symmetric similarity solution for a localized fixed volume $V$ in a V-shaped channel gravity current has $h(x,t)=[3(X^2-x^2)/(7Ct){]}_{+}^{1/3}$. Integrating the similarity conservation of mass equation once gives $C{h}^4{h}_x=-2x{h}^2/(7t)$, and vanishing thickness at the two noses gives the profile. Its volume fixes $X=[V/(I\tan \alpha ){]}^{3/7}(7Ct/3{)}^{2/7}$, where $I={\int }_{-1}^1(1-s^2{)}^{2/3}ds$. At any fixed nonzero position, the thickness first rises after arrival and then falls; its maximum occurs when $X^2=7x^2/3$.