On the front conditions for gravity currents in channels of general cross-section

We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel with general cross-section whose width is \(f(z), 0 \le z\le H\), 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 \(\varphi \).     

Drag coefficient for rigid vegetation in subcritical open-channel flow

Drag coefficient has been commonly used as a quantifying parameter to represent the vegetative drag, i.e., resistance to the flow by vegetation. In this study, the measured data on the drag coefficient for rigid vegetation in subcritical open-channel flow reported in previous studies are collected and preprocessed for multi-parameter analysis. The effect of Froude number (Fr) on the drag coefficient for rigid vegetation in subcritical flow cannot be ignored, especially when \(Fr < 0.12\). The drag coefficient is observed to exponentially decrease with the stem Reynolds number (R _d ) and logarithmically decreased with the vegetation density (λ) when \(0.012 < \lambda < 0.12\). The relative submergence (h ^* ) has a significant effect on the drag coefficient, and a positive logarithmic relationship is summarized. A simplified three-stage empirical formula is obtained based on the divisions of Fr. Laboratory tests (with \(Fr < 0.02\)) prove that the present empirical model has higher precision compared with existing models.     

Thin-layer models for gravity currents in channels of general cross-section area,a review

We present a brief review of the recent investigations on gravity currents in horizontal channels with non-rectangular cross-section area (such as triangle, \(\bigvee \)-valley, circle/semi-circle, trapezoid) which occur in nature (e.g., rivers) and constructed environment (tunnels, reservoirs, canals). To be specific, we discuss the propagation of a gravity current (GC) in a horizontal channel along the horizontal coordinate x, with gravity g acting in the \(-z\) direction, and y the horizontal–lateral coordinate. The bottom and top of the channel are at \(z=0,H\). The “standard” problem is concerned with 2D flow in a channel with rectangular (or laterally unbounded) cross-section area (CSA). Recent investigations have successfully extended the standard knowledge to the channels of CSA given by the quite general \(-f_1(z)\le y \le f_2(z)\) for \(0 \le z \le H\). This includes the practical \(\bigvee \)-valley, triangle, circle/semi-circle and trapezoid; these geometries may be in “up” or “down” setting with respect to gravity, e.g., \(\bigtriangleup \) and \(\bigtriangledown \). The major objective of the extended theory is to predict the height of the interface \(z=h(x,t)\) and the velocity (averaged over the CSA) u(x, t), where t is time; the prediction includes the speed and position of the nose \(u_N(t), x_N(t)\). We show that the motion is governed by a set of simplified equations, called “model,” that provides versatile and insightful solutions and trends. The emphasis in on a high-Reynolds-number current whose motion is dominated by buoyancy–inertia balance; in particular a GC released from a lock, which also contains general effects such as front and internal jumps (shocks), and reflected bore. We discuss two-layer, one-layer, and box models; Boussinesq and non-Boussinesq systems; compositional and particle-driven cases; and the effect of stratification of the ambient fluid. The models are self-contained, and admit realistic initial and boundary conditions. The governing equations are amenable to analytical solutions in some special circumstances. Some salient features of the buoyancy-viscous regime, and the estimate for the length at which transition to this regime takes place, are also presented. Some experimental support to the theory, and open questions for further investigations, are also mentioned. The major conclusions are (1) The CSA geometry has significant influence on the motion of the GC; and (2) The new theory is a useful, very significant, extension of the standard two-dimensional GC problem. The standard current is just a particular case, \(f_{1,2} =\) constants, among many other covered by the new theory.     

Penetrative convection in slender containers

An experimental study was conducted to investigate the penetration of a convective mixed layer into an overlying stably (solutally) stratified layer contained in a narrow, tall vessel when the fluid is subjected to a destabilizing heat flux from below. The interest was the evolution of the bottom mixed-layer height (\(h\)) with time (\(t\)) in the presence of side-wall effects, but without the formation of conventional double-diffusive layers. The side-wall effects are expected at small mixed-layer aspect ratios, \(\varGamma_{h} = (W/h)\), where \(W\) is the container width. This case has not been studied hitherto, although there are important environmental and industrial applications. The mixed-layer growth laws for low aspect ratio convection were formulated by assuming a balance between the vertical kinetic energy flux at the interface and the rate of change of potential energy of the fluid system due to turbulent entrainment. The effects of sidewalls were considered using similarity arguments, by taking characteristic rms velocities to be a function of \(\varGamma_{h}\), in addition to buoyancy flux (\(q_{0}\)) and \(h\). In all stages of evolution, the similarity variables \(\xi = h/W\) and \(t^{\prime } = Nt/A\), where \(A = N^{3} W^{2} /4q_{0}\) and \(N\) is the buoyancy frequency, scaled the mixed-layer evolution data remarkably well. Significant wall effects were noted when \(\varGamma_{h} < 1\), and for this case the interfacial vertical turbulent velocity and length scales were identified via scaling arguments and experimental data.     

Experiments on gravity currents propagating on unbounded uniform slopes

Gravity currents propagating on \(12^\circ \), \(9^\circ \), \(6^\circ \), \(3^\circ \) unbounded uniform slopes and on an unbounded horizontal boundary are reported. Results show that there are two stages of the deceleration phase. In the early stage of the deceleration phase, the front location history follows \({(x_f+x_0)}^2 = {(K_I B)}^{1/2} (t+t_{I})\), where \((x_f+x_0)\) is the front location measured from the virtual origin, \(K_I\) an experimental constant, B the total buoyancy, t time and \(t_I\) the t-intercept. In the late stage of the deceleration phase for the gravity currents on \(12^\circ \), \(9^\circ \), \(6^\circ \) unbounded uniform slopes, the front location history follows \({(x_f+x_0)}^{8/3} = K_{VS} {{B}^{2/3} V^{2/9}_0 }{\nu }^{-1/3} ({t+t_{VS}})\), where \(K_{VS}\) is an experimental constant, \(V_0\) the initial volume of heavy fluid, \(\nu \) the kinematic viscosity and \(t_{VS}\) the t-intercept. In the late stage of the deceleration phase for the gravity currents on a \(3^\circ \) unbounded uniform slope and on an unbounded horizontal boundary, the front location history follows \({(x_f+x_0)}^{4} = K_{VM} {{B}^{2/3} V^{2/3}_0 }{\nu }^{-1/3} ({t+t_{VM}})\), where \(K_{VM}\) is an experimental constant and \(t_{VM}\) the t-intercept. Two qualitatively different flow morphologies are identified in the late stage of the deceleration phase. For the gravity currents on \(12^\circ \), \(9^\circ \), \(6^\circ \) unbounded uniform slopes, an ‘active’ head separates from the body of the current. For the gravity currents on a \(3^\circ \) unbounded uniform slope and on an unbounded horizontal boundary, the gravity currents maintain an integrated shape throughout the motion. Results indicate two possible routes to the final stage of the gravity currents on unbounded uniform slopes.