non-Boussinesq Gravity Currents Research Articles

Capturing the baroclinic effect in non-Boussinesq gravity currents

Direct numerical simulations of two-dimensional gravity currents with small and medium density variations are performed using different non-Boussinesq buoyancy approximations. Taking the full low-Mach-number approximation as the reference, the accuracy of several buoyancy terms are examined. It is found that all considered buoyancy terms performed well in the cases with small density variation. In the cases with medium density variation, the classical gravitational Boussinesq’s buoyancy term showed the lack of accuracy, and a simple correction did not make any improvement. In contrast, the recently introduced second-order buoyancy term showed a significantly higher accuracy. The present results and our previous derivations indicate that simple algebraic buoyancy approximations extended from the Boussinesq’s gravitational buoyancy are unlikely to achieve an accuracy beyond first order. Instead, it seems necessary to solve at least one extra Poisson equation for buoyancy terms to capture the higher-order baroclinic effect. An approximate analysis is also provided to show the leading term of the non-Boussinesq effect corresponding to gravity.

Boussinesq and non-Boussinesq gravity currents propagating on unbounded uniform slopes in the deceleration phase

Abstract

Large eddy simulation of non-Boussinesq gravity currents with a DG method

We present results of three-dimensional direct numerical simulations (DNS) and large eddy simulations (LES) of turbulent gravity currents with a discontinuous Galerkin finite elements method. In particular, we consider the lock-exchange test case as a benchmark for gravity currents. Since, to the best of our knowledge, non-Boussinesq three-dimensional reference DNS are not available in the literature for this test case, we first perform a DNS experiment. The DNS provides an accurate description of the turbulent phenomena and highlights some differences with respect to the Boussinesq regime, like the non-symmetric pattern in the evolution of instabilities at the interfacial region and the fact that less turbulent structures are present due to greater stratification. A periodic pattern is also evident in the time evolution of turbulent mixing. The DNS is then employed to assess the performance of different LES models. In particular, we have considered the isotropic dynamic model and an anisotropic dynamic model. The LES results provide a first indication about the superiority of dynamic models with respect to no-model LES. However, the considered Reynolds numbers in the non-Boussinesq context are still too low to draw firm conclusions about the superiority of the present explicit LES approach with respect to an implicit LES approach.

Non-Boussinesq gravity currents and surface waves generated by lock release in a circular-section channel: theoretical and experimental investigation

We present a combined theoretical and experimental study of lock-release inertial gravity currents (GCs) propagating in a horizontal channel of circular cross-section with open-top surface in the non-Boussinesq regime. A two-layer shallow-water (SW) model is developed for a generic shape of the cross-section with open top, and then implemented in a finite difference numerical code for the solution in a circular-cross-section channel of the type used in the experiments. The model predicts propagation with (almost) constant speed for a fairly long distance, accompanied by a depression of the ambient free open-top surface behind the front of the current. Sixteen experiments were conducted with a density ratio $r=0.587{-}0.939$ in full-depth and part-depth release conditions, measuring the front speed and the free-surface time series at four cross-sections. The channel was a circular tube 409 cm long, with a radius of 9.5 cm; the lengths of the locks were 52 and 103.5 cm. Density contrast was obtained by adding sodium chloride and dipotassium phosphate to fresh water. The theoretical values of the front speed and of the depression overestimate the experimental values, but they predict correctly their trend for varying parameters and provide reliable insights into the underlying mechanisms. In particular, we demonstrate that the circular cross-section increases the speed of propagation as compared to the standard rectangular cross-section case (for the same initial height and density ratio). The discrepancies between the SW predictions and the present experiments are of the same order of magnitude as those of previously published results for simpler systems (Boussinesq, rectangular). In addition to the depression, which is a wave bound to, and following the front of, the GC, the system also displays two kinds of free-surface waves, namely the initial bump (its amplitude is of the same order as the depression) and some short-length and low-amplitude waves in the tail of the bump. These free waves propagate with a celerity well predicted by the ‘fast’ eigenvalues of the mathematical model. Comparison is provided with the celerity of a solitary wave. It is expected that discrepancies between theory and experiments can be partly attributed to the presence of these waves. The reported insights and SW prediction method can be applied to a variety of cross-sections of practical interest (triangles, trapezoids, etc.).

A new highly scalable, high-order accurate framework for variable-density flows: Application to non-Boussinesq gravity currents

This paper introduces a new code “QuasIncompact3D” for solving the variable-density Navier–Stokes equations in the low-Mach number limit. It is derived from the Incompact3D framework which is designed for incompressible flows (Laizet and Lamballais, 2009). QuasIncompact3D is based on high-order accurate compact finite-differences (Lele, 1992), an efficient 2D domain decomposition (Laizet and Li, 2011) and a spectral Poisson solver. The first half of the paper focuses on introducing the low-Mach number governing equations, the numerical methods and the algorithm employed by QuasIncompact3D to solve them. Two approaches to forming the pressure-Poisson equation are presented: one based on an extrapolation that is efficient but limited to low density ratios and another one using an iterative approach suitable for higher density ratios. The scalability of QuasIncompact3D is demonstrated on several TIER-1/0 supercomputers using both approaches, showing good scaling up to 65k cores. Validations for incompressible and variable-density low-Mach number flows using the Taylor–Green vortex and a non-isothermal mixing layer, respectively, as test cases are then presented, followed by simulations of non-Boussinesq gravity currents in two- and three-dimensions. To the authors’ knowledge this is the first investigation of 3D non-Boussinesq gravity currents by means of Direct Numerical Simulation over a relatively long time evolution. It is found that 2D and 3D simulations of gravity currents show differences in the locations of the fronts, specifically that the fronts travel faster in three dimensions, but that it only becomes apparent after the initial stages. Our results also show that the difference in terms of front location decreases the further the flow is from Boussinesq conditions.

S.03.02 Molecular mechanisms of choosing alcohol over natural reward

This paper introduces a new code “QuasIncompact3D” for solving the variable-density Navier–Stokes equations in the low-Mach number limit. It is derived from the Incompact3D framework which is designed for incompressible flows (Laizet and Lamballais, 2009). QuasIncompact3D is based on high-order accurate compact finite-differences (Lele, 1992), an efficient 2D domain decomposition (Laizet and Li, 2011) and a spectral Poisson solver. The first half of the paper focuses on introducing the low-Mach number governing equations, the numerical methods and the algorithm employed by QuasIncompact3D to solve them. Two approaches to forming the pressure-Poisson equation are presented: one based on an extrapolation that is efficient but limited to low density ratios and another one using an iterative approach suitable for higher density ratios. The scalability of QuasIncompact3D is demonstrated on several TIER-1/0 supercomputers using both approaches, showing good scaling up to 65k cores. Validations for incompressible and variable-density low-Mach number flows using the Taylor–Green vortex and a non-isothermal mixing layer, respectively, as test cases are then presented, followed by simulations of non-Boussinesq gravity currents in two- and three-dimensions. To the authors’ knowledge this is the first investigation of 3D non-Boussinesq gravity currents by means of Direct Numerical Simulation over a relatively long time evolution. It is found that 2D and 3D simulations of gravity currents show differences in the locations of the fronts, specifically that the fronts travel faster in three dimensions, but that it only becomes apparent after the initial stages. Our results also show that the difference in terms of front location decreases the further the flow is from Boussinesq conditions.

Experimental study of smoke propagation in a tilted tunnel

In the context of an underground repository, one of the main concerns lies within the management of the smoke produced by a fire. The propagation front of smoke is the first indicator received by the detectors and this front can be represented by a gravity current. In this study, air/helium experiments on the non-Boussinesq gravity currents generated from an instantaneous buoyancy source in horizontal and tilted tunnels have been carried out. A large range of density ratios γ = ρc/ρ0 (0.29 ≤ γ ≤ 0.88), between the density of the release (ρc) and the ambient (ρ0) is investigated. First, similarly to the open ambient case, we note that the front of the gravity current propagates in three phases: the slumping, inertial and viscous phase. In each phase, a power-law exists between the front location of the gravity current (x) and the time (t). In the early inertial deceleration phase, x ∝ t2/3 is valid for both Boussinesq and non-Boussinesq gravity currents. Moreover, in the late deceleration phase, the viscous effects become more important and a power-law x ∝ t1/2 has been identified. Then, experiments on tilted tunnels with angles varying from 1% to 17 % have been performed to consider the influence of the slope. It is found that the transition between inertial and viscous propagation decreases when the slope increases. One possible explanation is that the mixing layer increases in the higher angle percentages which causes the buoyancy contained within the head of the density currents to decrease more rapidly. A high angle of slope can be seen potentially as a disadvantage for the propagation of the smoke.

A new front condition for non-Boussinesq gravity currents

ABSTRACTA new front condition for gravity currents in a two-layer shallow water framework with a free surface is developed by applying the theory of the weak formulation to a first order system of four partial differential equations written in conservation form. The Rankine–Hugoniot jump conditions are applied and a relation between the velocity and the height of the shock is obtained. Laboratory experiments on lock-release non-Boussinesq gravity currents are performed and the front propagation and the free surface displacement are evaluated. Front velocities, predicted by the present mathematical model, are compared to both laboratory measurements and the predictions of the Benjamin formula, which is based on the rigid lid approximation. The new jump condition is in a good agreement with the laboratory measurements, while the Benjamin formula exhibits an overestimation of the shock velocity in the non-Boussinesq cases. In addition, the free surface displacement near the shock predicted by the new model is in agreement with the laboratory measurements.

Modelling gravity currents without an energy closure

We extend the vorticity-based modelling approach of Borden & Meiburg (Phys. Fluids, vol. 25 (10), 2013, 101301) to non-Boussinesq gravity currents and derive an analytical expression for the Froude number without the need for an energy closure or any assumptions about the pressure. The Froude-number expression we obtain reduces to the correct form in the Boussinesq limit and agrees closely with simulation data. Via detailed comparisons with simulation results, we furthermore assess the validity of three key assumptions underlying both our as well as earlier models: (i) steady-state flow in the moving reference frame; (ii) inviscid flow; and (iii) horizontal flow sufficiently far in front of and behind the current. The current approach does not require an assumption of zero velocity in the current.

Thermal Theory for Non-Boussinesq Gravity Currents Propagating on Inclined Boundaries

In this study the author derived the thermal theory for non-Boussinesq gravity currents produced from an instantaneous buoyancy source propagating on an inclined boundary. For Boussinesq gravity currents on a slope, it is known that the gravity current front location history follows an asymptotic relationship, xf3/2∼t, where xf is the front location, and t is the time, when the gravity current is sufficiently far into the deceleration phase. For non-Boussinesq gravity currents, the distance for the acceleration phase is extended attributable to the non-Boussinesq effects. When the gravity current is sufficiently far into the deceleration phase, this paper shows that the non-Boussinesq gravity currents tend to approach similar asymptote in the deceleration phase as Boussinesq gravity currents, but the approach is less rapid in non-Boussinesq gravity currents.

Experimental and Numerical Investigation on the Motion of Discrete Air Pockets in Pressurized Water Flows

Entrapped air pockets are linked to various operational issues in closed pipe flows. Tracking of entrapped air pockets is a relevant task in numerical simulation of hydraulic systems subjected to air pocket entrapment. However, few studies have focused on the kinematics of discrete air pockets and the motion of these at various combinations of pipe flow velocities, air pocket volumes, and pipe slopes. Such information is relevant for the improvement of current numerical models for the simulation of transition between pressurized and free surface flows, and related air pocket entrapment. This work explores the link between pipe flow velocity, pipe slope, and entrapped air pocket celerities. Systematic experimental studies were performed and provided insights on the nature of entrapped air pocket motion in cases when buoyancy forces are either summed or opposed to flow drag. Results indicated that when flow drag and buoyancy forces are summed, the observed air pocket celerity can be approximated by the celerity in quiescent conditions plus the pipe velocity. This approximation fails in cases when drag force opposed buoyancy. This work also proposes a novel approach for the simulation of the air pocket motion as non-Boussinesq gravity currents. Modeling results compared favorably with experimental measurements in horizontal slope and pipe flow velocity cases. Such results are an indication of the feasibility of integrating this approach into more complex hydraulic models that are able to simulate flow regime transitions.

Non-Boussinesq gravity currents propagating on different bottom slopes

AbstractExperiments on the non-Boussinesq gravity currents generated from an instantaneous buoyancy source propagating on an inclined boundary in the slope angle range $0^{\circ } \le \theta \le 9^{\circ }$ with relative density difference in the range of $0.05 \le \epsilon \le 0.17$ are reported, where $\epsilon = (\rho _1-\rho _0)/\rho _0$, with $\rho _1$ and $\rho _0$ the densities of the heavy and light ambient fluids, respectively. We showed that a $3/2$ power-law, ${(x_f+x_0)}^{3/2}= K_M^{3/2} {B_0'}^{1/2} (t+t_{I0})$, exists between the front location measured from the virtual origin, $(x_f+x_0)$, and time, $t$, in the early deceleration phase for both the Boussinesq and non-Boussinesq cases, where $K_M$ is a measured empirical constant, $B_0'$ is the total released buoyancy, and $t_{I0}$ is the $t$-intercept. Our results show that $K_M$ not only increases as the relative density difference increases but also assumes its maximum value at $\theta \approx 6^{\circ }$ for sufficiently large relative density differences. In the late deceleration phase, the front location data deviate from the $3/2$ power-law and the flow patterns on $\theta =6^{\circ },9^{\circ }$ slopes are qualitatively different from those on $\theta =0^{\circ },2^{\circ }$. In the late deceleration phase, we showed that viscous effects could become more important and another power-law, ${(x_f+x_0)}^{2}= K_{V}^{2} {B_0'}^{2/3} {{A}^{1/3}_0} {\nu }^{-1/3} (t+t_{V0})$, applies for both the Boussinesq and non-Boussinesq cases, where $K_V$ is an empirical constant, $A_0$ is the initial volume of heavy fluid per unit width, $\nu $ is the kinematic viscosity of the fluids, and $t_{V0}$ is the $t$-intercept. Our results also show that $K_V$ increases as the relative density difference increases and $K_V$ assumes its maximum value at $\theta \approx 6^{\circ }$.

Two-layer shallow-water dam-break solutions for non-Boussinesq gravity currents in a wide range of fractional depth

We consider the dam-break initial stage of propagation of a gravity current released from a lock of length x0 and height h0 into an ambient fluid in a channel of height H*. The system contains heavy and light fluids, of densities ρH and ρL, respectively. When the Reynolds number is large, the resulting flow is governed by the parameters R = ρL/ρH and H = H*/h0. We focus attention on non-Boussinesq effects, when the parameter R is not close to 1; in this case significant differences appear between the ‘light’ (top surface) current and the ‘heavy’ (bottom) current. We use a shallow-water two-layer formulations. We show that ‘exact’ solutions of the thickness and speed of the current and ambient can be obtained by the method of characteristics. However, this requires a careful matching with the conditions at the front (ambient) and the back (reservoir). We show that a jump, instead of a rarefaction wave, propagates into the reservoir when H < Hcrit(R), and solutions for these jumps are presented and discussed. The theory is applied to the full-depth lock-exchange H = 1 problem, and the results are compared with previous hydraulic models. The application of the theory is also illustrated for more general cases H > 1, including comparisons with the one-layer model results of Ungarish (J. Fluid Mech., vol. 579, 2007, p. 373).Overall, the shallow-water two-layer theory yields consistent, self-contained, and physically acceptable analytical solutions for the dam-break problem over the full physical range of the R and H parameters, for both light-into-heavy and heavy-into-light gravity currents. The solution can be closed without adjustable constants or predetermined properties of the flow field. The thickness solution is formally valid until the jump, or rarefaction wave, hits the backwall; the speed of propagation prediction is valid until this reflected wave hits the nose, i.e. until the end of the slumping stage. This theory is a significant extension of the Boussinesq problem (recovered by the present solution for R = 1), which elucidates the non-Boussinesq effects during the first stage of propagation of lock-released gravity currents.

The propagation of high-Reynolds-number non-Boussinesq gravity currents in axisymmetric geometry

We consider the propagation of a non-Boussinesq gravity current in an axisymmetric configuration (full cylinder or wedge). The current of density ρc is released from rest from a lock of radius r0 and height h0 into an ambient fluid of density ρa in a container of height H. When the Reynolds number is large, the resulting flow is governed by the parameters ρc/ρa and H* = H/h0. We show that the one-layer shallow-water model, carefully combined with a Benjamin-type front condition, provides a versatile formulation for the thickness and speed of the current, without any adjustable constants. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1, and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio, H*. We obtain finite-difference solutions for the propagation and show that a self-similar behaviour develops for large times. This reveals the main features, in particular: (a) The heavy current propagates faster and its front is thinner than that for the light counterpart; (b) For large time, t, both the heavy and light currents spread like t1/2, but the thickness profiles display significant differences; (c) The energy-constrained propagation with the thickness of half-ambient-depth (when H* is close to 1) is a very limited occurrence, in contrast to the rectangular geometry counterpart in which this effect plays a major role. The predictions of the simple model are supported by some axisymmetric Navier–Stokes finite-difference simulations.

Existence and features of similarity solutions for non-Boussinesq gravity currents

In this paper, the flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions. The current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of much lower density. Unlike earlier investigators, we assume that the ambient fluid exerts no significant resisting action on the current, and therefore the flow depth is expected to drop to zero at the front in the absence of friction. In this context, the shallow-water equations are highly appropriate for computing the mean velocity and flow depth of the current. The boundary condition imposed at the front leads to technical mathematical difficulties. Indeed, unlike in the Boussinesq case, no regular solution to the shallow-water equations satisfies the downstream condition, but when the flow is supercritical at the channel inlet, it is possible to construct a piecewise solution by patching a regular solution to an exceptional solution, which represents the head behavior. To better understand this result and make sure that the result is physically relevant, we consider the Navier–Stokes equations within the high-Reynolds-number limit. Approximate similarity solutions can be worked out, which support our earlier analysis on the shallow-water equations. While the flow body is self-similar and weakly rotational, the head is not self-similar, but tends toward a self-similarity shape at long times. It is characterized by a strong vorticity, a straight free surface, and a nonuniform velocity profile, which becomes flatter and flatter with time.