| Abstract: | We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel with general cross-section whose width is (f(z), 0 le zle H), and the gravity acceleration g acts in (-z) direction. (The classical rectangular cross-section is covered by the particular case (f(z) =) const.) We assume a two-layer system of homogeneous fluids of constant densities (rho _{c}) (current, of height (h < H )) and smaller (rho _{a}) (ambient, filling the remaining part of the channel). We focus attention on the calculation and assessment of the nose Froude-number condition (Fr = U/(g' h)^{1/2}); here U is the speed of propagation of the current and (g' = (rho _{c}/rho _{a}-1) g) is the reduced gravity. We first revisit the steady-state current, and derive compact insightful expressions of Fr and energy dissipation as a function of (varphi ) ((=) area fraction occupied by the current in the cross-section). We show that the head loss (delta _0) on the stagnation line is formally a degree of freedom in the determination of (Fr(varphi )), and we clarify the strong connections with the head loss (delta ) in the ambient fluid, and with the overall rate of dissipation (dot{{mathcal{D}}}). We demonstrate that the closure (delta _0 = 0) [suggested by Benjamin (J Fluid Mech 31, 209–248, 1968) for the rectangular cross-section] produces in general the smallest Fr for a given (varphi ); the results are valid for a significant range ([0, varphi _{max }]), in which the current is dissipative, except for the point (varphi _{max }) where (delta = dot{{mathcal{D}}} = 0). We show that imposing the closure (delta = dot{{mathcal{D}}} = 0), which corresponds to an energy-conserving or non-dissipative current, produces in general unacceptable restrictions of the range of validity, and large values of Fr; in particular, deep currents ((varphi < 0.3) say) must be excluded because they are inherently dissipative. On the other hand, the compromise closure (delta (varphi ) =delta _0(varphi )) produces the simple (Fr(varphi ) = sqrt{2}(1 - varphi )) formula whose values and dissipation properties are very close, and the range of validity is identical, to these obtained with Benjamin’s closure (moreover, we show that this corresponds to circulation-conservation solutions). The results are illustrated for practical cross-section geometries (rectangle, (Delta ) and (nabla ) triangle, circle, and the general power-law (f(z) = b z ^alpha ) ((b>0, alpha ge 0, 0< z le H)). Next, we investigate the connection of the steady-state results with the time-dependent current, and show that in a lock-released current the rate of dissipation of the system is equal to, or larger than, that obtained for Fr corresponding to the conditions at the nose of the current. The results and insights of this study cover a wide range of cross-section geometry and apply to both Boussinesq and non-Boussinesq systems; they reveal a remarkable robustness of Fr as a function of (varphi ). |
