Triangular-profile line plume ($f(\xi )=(1-|\xi |{)}_{+}$)

A two-sided line plume whose axial velocity and density anomaly both have triangular transverse profiles. For half-width $b$, ${\int }_{-b}^{b}f(x/b{)}^{n}dx=2b/(n+1)$. These profile moments set the different coefficients of stored mass, momentum, and their fluxes. It differs from a wall plume because both edges entrain ambient fluid.

In the Boussinesq approximation, density-weighted fluxes are $Q={\rho }_{0}bW$, $M=(2/3){\rho }_{0}b{W}^2$ and $F=(2/3)gbW({\rho }_0-\rho )$. With two-sided Batchelor entrainment $e=\alpha W$, the integral balances areThe factors $4/3$ and $3$ depend on the triangular profile and the definition of the entrainment coefficient.

The unsteady Boussinesq triangular-profile line-plume balances admit a buoyant similarity solution $Q=\alpha {\rho }_0{z}^2/\tau$, $M=(2/3)\alpha {\rho }_0{z}^3/{\tau }^2$ and $F=(2/3)\alpha {\rho }_0{z}^3/{\tau }^3$, with $\tau =t+t_{\ast }>0$. It reconstructs $b=\alpha z$, $W=z/\tau$ and reduced gravity $z/{\tau }^2$. Origins can be shifted, but the zero-time unshifted solution is singular. Source and initial data are required to select the physical member and region of validity.

For a triangular-profile line plume with centreline density $\rho$, speed $W$, half-width $b$, ambient density ${\rho }_0$, and inward relative edge speed $e$, volume, mass and momentum conservation giveSubtracting mass conservation from ${\rho }_0$ times volume conservation also gives buoyancy conservation. Batchelor entrainment supplies a closure for $e$.