Gravity current in a channel of general cross-section with open top surface

Consider the idealized steady-state gravity current of height h and density ρ₁ that propagates into an ambient motionless fluid of height H and density ρ₂ (< ρ₁) in a channel of general cross-section (rectangle, triangle, semi-circle, trapezoid, etc.) with an upper surface open to the atmosphere (open channel) at high Reynolds number. The current propagates with speed U and causes a depth decrease χ of the top surface. This is a significant extension of the solution for the fixed-top channel χ = 0. Here the determination of χ is a part of the problem. For a given cross-section geometry, the dimensionless parameters of the problem are a=h/H and r = ρ₂/ρ₁. We show that a control-volume analysis determines χ/H and Fr = U/(g' h)^½ as functions of a and r, where g' = (1/r -1)g is the reduced gravity. The system satisfies balance of volume and momentum (explicitly), and vorticity (implicitly). We present solutions and insights for various cross-section geometries. For a Boussinesq system with r ≈ 1, we obtain χ/H ≈ 0 , and the present Fr and dissipation results differ only slightly from the fixed-top predictions, as expected.