2.1. Experimental set-up

The experimental set-up is shown in figure 1. The experiments are carried out in a glass square tank ( $120~\text{cm}\times 120~\text{cm}\times 40~\text{cm}$ ) at the centre of which we place a hollow cylinder of equivalent radius $R_{0}$ , filled up to a height $h_{0}$ with a fluid of different density than the ambient fluid of height $H$ . Two different cross-sectional shapes are considered, namely a circular section (CS) for verification and comparison with previous results and a rounded rectangular section (RR), i.e. a rectangle where the two shorter edges are replaced by semicircles. The initial horizontal aspect ratio ${\it\chi}_{0}$ of the rounded rectangular cylinder, here defined as the ratio of the longest to the shortest side, is ${\it\chi}_{0}=3.8$ . Fluorescent dye is added to the fluid inside the cylinder. Black light tubes mounted on two sides of the tank illuminate the fluorescent dye inside a dark room, allowing the current to be solely visible.

Figure 1.

Experimental set-up: $\unicode[STIX]{x2460}$  tank support, $\unicode[STIX]{x2461}$  pulley support, $\unicode[STIX]{x2462}$  hollow cylinder, $\unicode[STIX]{x2463}$  pulley weight, $\unicode[STIX]{x2464}$  tank,  $\unicode[STIX]{x2465}$  mirror,  $\unicode[STIX]{x2466}$  camera. The experiments are carried out with two different cross-sectional geometries, namely a circular section (CS) and a rounded rectangle (RR). Both geometries have roughly the same cross-sectional area, leading to an equivalent radius $R_{0}$ of 4.6 and 4.7 cm for the CS and RR geometry, respectively, $R_{0}$ being calculated from the surface area $A$ as $R_{0}=\sqrt{A/{\rm\pi}}$ . An inclined mirror is placed underneath the tank, so that the current’s evolution is recorded from both the side and below. The square tank dimensions are $120~\text{cm}\times 120~\text{cm}\times 40~\text{cm}$ , corresponding to approximately $26R_{0}\times 26R_{0}\times 9R_{0}$ . Gravity is pointing towards the $z$ direction.

Unless stated otherwise, the ambient fluid is tap water of density ${\it\rho}_{a}=1000~\text{kg}~\text{m}^{-3}$ while the current consists of salty water of density ${\it\rho}_{c}=1100~\text{kg}~\text{m}^{-3}$ . The depth ratio $h_{0}/H$ (ratio of lock height to that of the ambient) for all the experiments was held at unity. The initial vertical aspect ratio ${\it\lambda}=h_{0}/R_{0}$ (height/radius) was varied between 0.25 and 7. The radius $R_{0}$ for the non-axisymmetric cross-sections is calculated from the surface area $A$ via $R_{0}=\sqrt{A/{\rm\pi}}\approx 4.6$  cm. The two geometries were chosen to have roughly the same cross-sectional area, so that, for a fixed initial height, the volume of release is constant whatever the initial cross-sectional shape. The tank and the cylinder are simultaneously filled. When the desired vertical aspect ratio is reached, the water in the tank is given sufficient time to reach a stagnant state. The hollow cylinder is then raised rapidly via a pulley system connected to a weight.

The front location and the current’s height are measured using a mirror placed beneath the tank, which allows for a plan bottom view of the front evolution (e.g. figure 2), while the side view of the current provides information about the evolution of the current height (e.g. figure 5). The pixel resolution was approximately $R_{0}/82$ (0.5 mm) and $h_{0}/44$ (2 mm) for a ${\it\lambda}=2$ release in the horizontal and vertical directions, respectively. The front location is obtained from the plan view images using the MATLAB^® Graphics in-built function Imread^®, where each pixel is assigned a value in the intensity range $[0,255]$ . All values between 1 and 254 can be considered as different shades of grey (0 corresponding to the black colour). The front is easily determined since there is a significant jump (within a few pixels) in the intensity levels at the current–ambient interface. Note that the location of the front was found to be insensitive to the chosen cut-off value. As can be seen from the example in figure 2, the location of the front is well extracted.

Figure 2.

Example of front location detection from experiments. A plan view of the current (Exp 2 at $t=5.9$ ) is shown in (a) along with the white star symbols corresponding to the extracted front location. The close-up views in (b,c) show good agreement between the extracted and the actual front location. (See supplementary movie available at http://dx.doi.org/10.1017/jfm.2015.580.)

The local front location is then computed as follows. The local radial location of the front $\overline{r_{N}}({\it\theta}t)$ , along the azimuthal ${\it\theta}$ direction is first calculated by averaging along a sector of angle $2{\it\alpha}_{0}={\rm\pi}/36$ around the front location as

where the ${\it\theta}$ coordinate’s origin is taken along the $x$ direction and $r_{N}({\it\theta}t)$ is the radial distance at time $t$ between the initial centre of mass of the current and a point at the front of the current. We further use the symmetry of the flow, when applicable, by taking the average value of $\overline{r_{N}}({\it\theta},t)$ along the symmetry directions. For instance, the ‘fast’ front position in figure 3 is computed as $\overline{r_{F}}(t)=[\overline{r_{N}}({\it\theta}=0,t)+\overline{r_{N}}({\it\theta}={\rm\pi},t)]/2$ and similarly for the ‘slow’ front position, i.e. $\overline{r_{S}}(t)=[\overline{r_{N}}({\it\theta}={\rm\pi}/2,t)+\overline{r_{N}}({\it\theta}=3{\rm\pi}/2,t)]/2$ . For all the non-axisymmetric cases, we selected the fastest and slowest sections of the front, all other directions hence having velocity magnitudes bounded between these values.

Figure 3.

Time evolution of the front position of the ‘slow’ and ‘fast’ fronts of RR currents for various initial orientations of the rounded rectangular cylinder relative to the tank walls. The angle between the initial longest side of the RR cylinder and the tank wall is $0^{\circ }$ (Exp 2), $45^{\circ }$ (Exp 3) and $90^{\circ }$ (Exp 4), respectively. The front location is averaged over a small sector of width $2{\it\alpha}={\rm\pi}/36$ . Note that the tank walls are located at a minimum distance of $13R_{0}$ from the centre of mass of the current, which corresponds to $r_{N}-R_{0}=6$ here (recall that the front position is scaled by $h_{0}$ and that ${\it\lambda}=h_{0}/R_{0}=2$ here). For comparison, the results obtained with Sim 1, for which the ‘outer walls’ are at a minimum distance of $16R_{0}$ , is also plotted.

2.2. Preliminary verifications

As a preliminary verification, we performed two sets of experiments, in order to check that the outer vertical walls of the tank did not affect the dynamics of the non-circular gravity currents. In the first set of experiments, we consider three rounded rectangular releases under nominally identical conditions except that the initial orientation of the RR cylinder relative to the tank walls is varied; in particular, the angle between the initial longest side of the RR cylinder and the tank wall is $0^{\circ }$ , $45^{\circ }$ and $90^{\circ }$ in the experiments denoted as Exp 2, 3 and 4 in table 1, respectively. The temporal evolution of the slow and fast fronts is displayed in figure 3. The dynamics of the fronts is observed to be similar in all cases. The slight difference between the three realizations is indicative of experimental measurement uncertainty, which is much smaller than the observed difference between the fast and slow fronts.

Table 1.

Velocity ratio $u_{F}/u_{S}$ and height ratio $h_{F}/h_{S}$ between the fastest and slowest points of some gravity currents of initially arbitrary shape. The mean quantities are computed for $0\leqslant t\leqslant 22$ ( ${\it\lambda}=1.4$ ),  $0\leqslant t\leqslant 12.6$ ( ${\it\lambda}=2$ ) and $0\leqslant t\leqslant 6.5$ ( ${\it\lambda}=4$ ). CS, RR and TR refer to the circle, rounded rectangle and true rectangle, respectively (see figure 1). The initial vertical aspect ratio is ${\it\lambda}=h_{0}/R_{0}$ . (See supplementary movies.)

a Mean quantities are computed for $0\leqslant t\leqslant t_{f}$ , with $t_{f}$ corresponding to the maximum time shown in figure 18.

b Mean velocity ratio computed for the post-acceleration stage $2200\leqslant t\leqslant 33\,500$ .

The second set of experiments consists of placing two vertical panels, each at an opposite end of the tank and having the same width and height as the tank, at a distance of 10 cm (that is, roughly $2R_{0}$ ) from the tank walls normal to the $x$ axis. In that case, the size of the tank is shorter in one direction by approximately $4R_{0}$ . We repeated the previous experiments of rounded rectangular releases in this smaller tank and compared the temporal evolution of the front position and velocity. It was found that the dynamics of the current were not affected by the presence of the wall as long as the current’s front was at a distance larger than $2R_{0}$ from the lateral walls (not shown). The above tests allow us to conclude that the dynamics of the non-circular gravity currents shown in the present experiments is not influenced by the presence, shape or orientation of the walls of the tank. This point will be further strengthened in § 7.5 by showing that the velocity of the ambient fluid beyond the front is much smaller than that of the current.

As a final verification, we use simple estimates to show that both the slumping and the inertial self-similar regimes of propagation are covered in the present experiments and simulations. By matching the (nearly) constant velocity during the slumping phase with the inertial phase scaling of a circular current, the transition time from the slumping to the inertial phase can be estimated as (Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007a )

Hoult (Reference Hoult1972) and Huppert & Simpson (Reference Huppert and Simpson1980) have proposed the values ${\it\xi}_{c}=1.3$ and ${\it\xi}_{c}=1.16$ , respectively. The constant $F_{c,sl}\approx 0.3$ represents the mean front velocity during the slumping phase. For our axisymmetric release (Exp 1 and Sim 2), the transition time computed from (2.2) is $t_{SI}\approx 4.2$ (respectively 3.3), for ${\it\xi}_{c}=1.3$ (respectively 1.16). These values are well below the characteristic duration of the experiments and simulations, which is approximately 14. As will be confirmed later, the gravity currents presented here undergo the acceleration phase, the slumping phase and eventually the inertial self-similar phase.

2.3. Numerical procedure

In this paper, all the variables are dimensionless, choosing $h_{0}$ as length scale, $U=\sqrt{g^{\prime }h_{0}}$ as velocity scale, $h_{0}/U$ as time scale and ${\it\rho}_{a}$ as density scale ( $g^{\prime }$ is the reduced gravity defined as $g^{\prime }=g({\it\rho}_{c}-{\it\rho}_{a})/{\it\rho}_{a}$ ). We define $x$ ( $y$ ) as the direction of the major (minor) axis when applicable, and $z$ as the direction parallel to gravity.

The physical configuration of the simulations is identical to the experimental set-up. We solve the concentration equation along with the incompressible Navier–Stokes equations. The system reads in dimensionless form

where $\boldsymbol{u}$ , $p$ and ${\it\rho}$ are the local velocity, total pressure and density in the flow, respectively, and $\boldsymbol{e}^{g}$ is a unit vector pointing in the direction of gravity. Two dimensionless parameters have been introduced in (2.4) and (2.5), namely the Reynolds number and the Schmidt number defined as

Here ${\it\kappa}$ is the molecular diffusivity and ${\it\nu}$ is the kinematic viscosity of the fluid. Equations (2.3)–(2.5) are solved inside a rectangular box of size $15\times 15\times 1$ with a spectral code (Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007a ,Reference Cantero, Balachandar and Garcia b ). Note that the experimental tank size was approximately $12\times 12$ in the $x$ – $y$ plane for ${\it\lambda}=2$ . No-slip and free-slip boundary conditions are imposed for the velocity at the bottom and top walls, respectively, while periodic boundary conditions are imposed at the sidewalls. Zero normal gradient is imposed for the density field at the bottom and top walls. Fourier expansions are used along the two horizontal periodic directions, and a Chebyshev expansion with Gauss–Lobatto quadrature points (Canuto et al. Reference Canuto, Hussaini, Quarteroni and Zang1988) is used along the vertical non-periodic direction. The reader is referred to Cortese & Balachandar (Reference Cortese and Balachandar1995) and Cantero et al. (Reference Cantero, Balachandar and Garcia2007b ) for a detailed description of the numerical approach and for results obtained with the same code in axisymmetric configurations, respectively.

In the present work, we simulate the collapse of a non-axisymmetric patch of heavy fluid at $\mathit{Re}=8950$ with a grid resolution of $880\times 880\times 179$ corresponding to approximately 140 million grid points. The numerical resolution was selected to have between four and six decades of decay in the energy spectrum for all the variables, and the time step was selected to produce a Courant number smaller than 0.5. In the simulations, the Schmidt number is set to unity. Note that this value is smaller than that of a saline gravity current, for which $\mathit{Sc}\approx 700$ , but it has been shown that the dynamics of gravity currents is independent of the Schmidt number as long as the Reynolds number is large, which is the case here (Bonometti & Balachandar Reference Bonometti and Balachandar2008).

3.1. Self-similarity of the front shape in non-axisymmetric releases

The temporal evolution of the spreading of gravity currents with two different initial cross-sectional shapes is presented in figure 4(a,b). The figure shows a plan view of a composite image of the front evolution for each experiment at various instants in time. The circular release is shown for comparison in panel (a). While the case of the circular release shows small undulations at the front due to the lobe and cleft instability (Simpson Reference Simpson1972; Härtel, Fredrik & Mattias Reference Härtel, Fredrik and Mattias2000), the current retains its overall symmetry as it propagates outwards. Conversely, for the RR current, as shown in Zgheib et al. (Reference Zgheib, Bonometti and Balacandar2014), the long-time circumferential shape is approximately an ellipse, but with switched major and minor axes as compared to the initial shape. Lobes and clefts are observed at the front even in this non-axisymmetric release. Note that the characteristic sizes of these lobes and clefts are an order of magnitude smaller than the length scale of the larger-scale flow pattern.

Figure 4.

Evolution of the front contours for (a) a circular (Exp 1) and (b) a rounded rectangular release (Exp 2). Panel (c) is similar to panel (b) using the self-similar variables ${\it\eta}=xt^{-1/2}$ and ${\it\zeta}=yt^{-1/2}$ , respectively. Note that here the variables are dimensionless, so that in dimensional form the self-similar variables would become ${\it\eta}=(g^{\prime }h_{0}^{3})^{-1/4}\tilde{x}\tilde{t}^{-1/2}$ and ${\it\zeta}=(g^{\prime }h_{0}^{3})^{-1/4}{\tilde{y}}\tilde{t}^{-1/2}$ , respectively (where the tilde denotes a dimensional variable). The contours are plotted from $t_{0}=2.1$ (red curve) to $t_{f}=13.8$ (blue curve) in steps of ${\rm\Delta}t=1.3$ . Plotted in green is the curve corresponding to a time of $t=8.6$ for which the shape becomes roughly self-similar. It is noteworthy that, at the time $t_{f}$ , the currents have crossed a minimum distance of seven or eight initial equivalent radii, while the tank walls are located at $x=y\approx \pm 6$ (axes are scaled by the initial height of the ambient $H$ ). In panels (a,b), the dashed line represents the initial location of the hollow cylinder.

It is noteworthy that the phrase ‘switching of the major and minor axes’ is one that has been consistently used to describe the evolution of elliptic free jets (Quinn Reference Quinn1989; Gutmark & Grinstein Reference Gutmark and Grinstein1999). As an elliptic jet propagates downstream, its shear layer along the minor axis plane grows at a faster rate compared to the shear layer along the major axis plane. This unequal growth rate results in a crossover point at a downstream location from the nozzle, where the jet temporarily attains a circular-like cross-section before its major and minor axes switch. Throughout this study, we will employ ‘switching of axes’ for the RR case to denote that, owing to a relatively faster propagation of the front along the minor axis ( $y$ axis) compared to a relatively slower spreading along its major axis ( $x$ axis) as shown in figure 4, the major and minor axes will eventually switch, making the initial minor axis of the RR geometry the major axis of the current at later stages of spreading.

In the case of planar and axisymmetric releases, it has been shown that, provided the Reynolds number is large enough, the current enters a self-similar inertial regime in which the evolution of the front position scales as $t^{2/3}$ and $t^{1/2}$ , respectively (Hoult Reference Hoult1972; Huppert & Simpson Reference Huppert and Simpson1980; Ungarish Reference Ungarish2009). The corresponding front velocities in the inertial regime scale as $t^{-1/3}$ and $t^{-1/2}$ for the planar and the axisymmetric currents, respectively. Regarding non-axisymmetric releases, Zgheib et al. (Reference Zgheib, Bonometti and Balacandar2014) plotted in their figure 4(b) the time evolution of the velocity of the fast and slow fronts of an initially rounded rectangular release and observed that, at the later times, it roughly follows $t^{-1/2}$ . Here we verify that the self-similar behaviour is valid for the entire propagating front. Self-similar evolution of an axisymmetric current can be expressed as (provided the front remains convex in shape)

where $f({\it\theta})$ is the self-similar shape of the front. It follows that the self-similar front velocity reads

and, provided a constant Froude number applies (as will be the case for a current spreading in a deep ambient), the self-similar front height around the circumference of the current can be expressed as

The function $f({\it\theta})$ is unknown a priori, but we shall propose in § 4.3 a simple empirical expression for this function, which is able to describe reasonably well the azimuthal variation of $r_{N}$ , $u_{N}$ and $h_{N}$ in the self-similar regime. Furthermore, in the inertial and viscous self-similar regimes, the front velocity has been shown to follow a power-law behaviour of the form (Fay Reference Fay and Hoult1969; Fannelop & Waldman Reference Fannelop and Waldman1971; Hoult Reference Hoult1972; Huppert & Simpson Reference Huppert and Simpson1980; Rottman & Simpson Reference Rottman and Simpson1983; Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007a )

where the power-law exponent takes the value $-1/2$ in the inertial regime of axisymmetric spreading and $-4/5$ or $-7/8$ in the viscous regime (Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007a ). The corresponding power-law evolutions of the mean radius and height are given by

It is noteworthy that the power law $H_{N}(t)\propto t^{2{\it\alpha}}$ is derived using the constant Huppert & Simpson (Reference Huppert and Simpson1980) Froude-number relation $\mathit{Fr}=U_{N}/\sqrt{H_{N}}=1.19$ , which is valid only in the inertial regime. Therefore it follows that the power law for $H_{N}$ is also valid only in the inertial regime. Based on an estimate of inertial-to-viscous transition time (Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007a ) we expect the dominant spreading of the rounded rectangle shown in figure 4(b) to be in the inertial regime. To test whether the non-axisymmetric spreading of the RR release is indeed self-similar and predominantly in the inertial regime, we replot in figure 4(c) the contours of the front in the scaled coordinates ${\it\eta}=xt^{-1/2}$ and ${\it\zeta}=yt^{-1/2}$ . It can be observed that the rounded rectangle reaches a self-similar shape resembling an ellipse, with its major and minor axes different from those of the initial release.

In § 2.1, we defined the initial aspect ratio ${\it\chi}_{0}$ of the release as the ratio of the longest side to the shortest side of the initial cross-section. Similarly, we define the self-similar aspect ratio ${\it\chi}_{\infty }$ as the ratio of the longest to the shortest sides. In the present case of the rounded rectangular release displayed in figure 4(c), we have ${\it\chi}_{0}=3.8$ and ${\it\chi}_{\infty }\approx 1.39$ , respectively. Note that in figure 4(c) we have varied the value ${\it\alpha}$ in the range $-2/3\leqslant {\it\alpha}\leqslant -1/2$ and verified that the value ${\it\alpha}=-1/2$ gives the best collapse in terms of self-similar shape. This was also confirmed by a best fit for the slope in the log–log plots of time versus local fast and slow front positions. This analysis was done both for the experimental data and for the simulation results to be discussed below. Overall, figure 4 confirms that the initially non-circular gravity current eventually reaches a self-similar shape (in the inertial regime), which is non-axisymmetric. We shall see later that the present finding is supported by both direct numerical simulations and results from the EBM.

3.2. Local front Froude number of non-axisymmetric gravity current

Table 1 presents the velocity and height ratios for various releases. The local fast and slow front velocities $u_{F}$ and $u_{S}$ are computed from $\overline{r_{F}}(t)$ and $\overline{r_{S}}(t)$ by differentiating in time. The maximum velocity ratio between the fastest front and the slowest front for the RR case is in the range 2.2–2.6 for a vertical aspect ratio of  $2$ . This indicates that the local instantaneous fast front can be up to 2–3 times faster than the slowest portion of the front. This strong variation of local front velocity is confirmed by the measured mean velocity ratio, which is approximately 1.9–2 for the RR current ( ${\it\lambda}=2$ ). The mean velocity ratio in table 1 is computed as

where, as indicated in table 1, $t_{max}=22,~12.6$ and 6.5 for ${\it\lambda}=1.4,~2$ and 4, respectively.

In figure 5(a) we present a side view of the evolution of the current for the RR and CS cases. For the circular release, the height is observed to be roughly uniform at all stages of propagation. The RR current shows clear variations in the current’s thickness, in particular between the central region (corresponding to a spreading along the minor $y$ axis) and the edges (corresponding to a spreading along the major $x$ axis). For instance, at $t\geqslant 2.6$ , the current is already thicker at the midplane than at the edges (note that the height is almost uniform for the CS current at the same dimensionless time). At $t=5.2$ , the height at the midplane is 3–4 times larger than at the edge. The height contrast along the front circumference decreases with time, as the absolute height is decreasing. The height ratio, however, is still larger than unity (approximately 2) at time $t=13$ as the current has crossed a distance of approximately $10R_{0}$ . Moreover, as can be seen from the $x$ – $y$ plan-view images (see figure 2), the maximum height of the current is located close to the front, in the ‘head’ of the gravity current. Note that this is in line with the recent laser-induced fluorescence measurements of Sahuri et al. (Reference Sahuri, Kaminski, Flynn and Ungarish2015) (see figure 4 in their paper). Therefore, it is reasonable to consider that the observed thickness in figure 5 (especially along the minor axis) corresponds to the front height of the current as opposed to that of the interior body of the current, which is significantly smaller, and hence hidden in the snapshots of figure 5. The same RR configuration was simulated, and in figure 6(a) we present the current’s height evolution via isosurfaces of density. In figure 6(b), we display the corresponding contours of the current height. Similar to the experimental findings, the heavy fluid is observed to aggregate along its periphery, with a clear distinction in thickness of approximately a factor of two between the minor and major axes. It is in fact this height inhomogeneity that leads to local velocity variations.

Figure 5.

Side views of the (a) rounded rectangular and (b) circular gravity currents: (a) Exp 2 initial shape; (b) Exp 1 initial shape. Contrary to the RR release, the current’s thickness is roughly uniform for the circular release during spreading.

Figure 6.

Height of the RR gravity current (Sim 1). (a) Isosurfaces ${\it\rho}=10^{-2}$ . (b) Distribution of the height along the horizontal plane. Note that the local height strongly varies along the circumference of the current, being maximum along the $y$ direction.

The present simulation results enable us to compare the value of the local Froude number along the front and specifically at the slow and fast sections of the fronts. One may assess the relevance of the various Froude functions reported in the literature with respect to the propagation of non-circular gravity currents. The simulations give access to local instantaneous front height $h_{N}$ and velocity $u_{N}$ information, and hence allow us to compute the Froude number as $\mathit{Fr}=u_{N}/\sqrt{h_{N}}$ . In order to evaluate the height $h_{N}$ for the slow and fast fronts, first the local height of the current, $h$ , is defined as

This local current height is then averaged over a wedge of $5^{\circ }$ aligned along the $x$ (slow front) and $y$ (fast front) axes, the averaging being performed over a distance extending between the front of the current and the location of the maximum height in the head.

The instantaneous local Froude number $\mathit{Fr}=u_{N}/\sqrt{h_{N}}$ of the slow and fast sections of the front of an initially rounded rectangular release is plotted in figure 7. The fast front Fr fluctuates in the range 0.9–1.1 for $2\leqslant t\leqslant 10$ and decreases to approximately 0.7 at later times (in the self-similar inertial regime). On the contrary, the slow front Fr is significantly lower at early times (in the range 0.6–0.8 for $2\leqslant t\leqslant 10$ ) but seems to catch up with the fast front at later times, i.e. $\mathit{Fr}\approx 0.7$ for $t>10$ . The larger value of the fast front Fr as compared to that of the slow front during the early stage of spreading suggests that the increase in front velocity due to mass redistribution inside the current is larger than the increase of height. Conversely, at late times (here $t>10$ ) the evolutions of the front velocity and height are similar for both the fast and slow fronts, as the values of the local Froude number are roughly similar. This is in line with the fact that the current has entered the self-similar inertial phase.

Figure 7.

Time evolution of the local Froude number at (a) the fast front and (b) the slow front: $\mathit{Fr}=u_{N}/\sqrt{h_{N}}$ , Sim 1 (——); $\mathit{Fr}^{HS}(a)=\min (a^{-1/3},1.19)$ (-  $\cdot$  -  $\cdot$  -);  $\mathit{Fr}^{B}(a)=\sqrt{a(1-a)(2-a)/(1+a)}$ (- - - -); $\mathit{Fr}^{BM}(a)=\sqrt{2a}(1-a)$ (– – –). Here, $a$ is defined as $a=h_{N}/H$ .

Numerical simulations can also be used to evaluate the other models of Froude functions. These models generally depend on the ratio $a=h_{N}/H$ of the nose height of the current to that of the ambient. We consider in the following three models of Froude functions, namely Benjamin’s (Reference Benjamin1968) front condition

which is valid for $a\leqslant 1/2$ , Huppert & Simpson’s (Reference Huppert and Simpson1980) relation

and Borden & Meiburg’s (Reference Borden and Meiburg2013) circulation-based model, which reads

The Benjamin and BM models give almost identical results and consistently underpredict the local Froude number by approximately 50 %. Alternatively, the HS correlation is in reasonable agreement with the simulated fast front Froude number, but overestimates the slow front Froude number by approximately 30 % for the whole duration of spreading. It should be noted that good agreement cannot be expected for this comparison as the above Froude models were derived for quasi-steady two-dimensional currents, which is not the case for the present set of experiments and simulations. In addition, the models assume a hydrostatic flow (i.e. a gravity current that is long and thin), which is not necessarily the case for early times in the present experiments and simulations.

4.1. Equations and assumptions

The classical box model generally used for predicting the evolution of gravity currents (Huppert & Simpson Reference Huppert and Simpson1980; Dade & Huppert Reference Dade and Huppert1995) has been shown to reproduce admirably the dynamics of axisymmetric and planar releases (see e.g. Ungarish & Zemach Reference Ungarish and Zemach2005). In the case of finite releases, the box model assumes the fluids to be immiscible, with negligible entrainment with the ambient, so that the mass and volume of the current are conserved throughout the duration of spreading. Additionally, the height is taken to be uniform along the body of the current and the current is advanced outwards normal to the front with a velocity proportional to the square root of the height. According to this model, the height intrinsically remains uniform along the circumference of the patch, so the speed of propagation is uniform along the current’s front during all phases of spreading. Therefore, using the classical box model, an initially non-axisymmetric current inevitably becomes axisymmetric.

The EBM proposed by Zgheib et al. (Reference Zgheib, Bonometti and Balacandar2014) is based on the partitioning of the initial release using inward rays normal to the front. An example of such partitioning is given in figure 8(b). Here, each segment of the front is now associated with a sub-volume of initial release. Once the various sub-volumes are obtained, the same procedure as in the classical box model is applied locally for each sub-volume, where the front is advanced outwards normal to itself. More particularly in the EBM, the current is defined by the front position $\{x_{N}(s,t),y_{N}(s,t)\}$ , the height $h_{N}(s,t)$ and the outward normal front velocity $u_{N}(s,t)$ , where $s$ is the distance measured along the circumference of the front. An additional variable, namely the area per arclength ${\it\sigma}(s,t)$ , is also used in the model (figure 8 a). An integration of ${\it\sigma}(s,t)$ over the entire arclength of the advancing front yields the total area covered by the planform of the advancing current. The EBM can be summarized by the following set of coupled equations (Zgheib et al. Reference Zgheib, Bonometti and Balacandar2014):

where $\mathit{Fr}$ is the Froude number, which is here calculated from the Huppert & Simpson (Reference Huppert and Simpson1980) empirical relation. Note that any other model of Froude-number function could be used without loss of generality, provided this function is applicable for the whole range of height ratio of nose to ambient, $h_{N}/H$ , considered here. All variables are dimensionless. Equations (4.1)–(4.3) refer to the Froude front condition, kinematic relations and mass conservation, respectively. In (4.2a ), the current is restricted to normal outward spreading with velocity $u_{N}$ . It will be shown below that this is a good approximation despite the non-uniform height distribution along the front, which might induce a tangential velocity component. The increase in the current’s surface area is captured in (4.2b ). This step is non-existent in the classical box model, as the area increase can be directly inferred from the radius of the current.

Analytical solutions of (4.1)–(4.3) are not feasible in the case of arbitrary initial patches. However, the system may be solved numerically. Details of the numerical procedure used for solving (4.1)–(4.3) and verification of spatial and temporal convergence are given in appendix A.

4.2. Examination of the extended box model

The EBM involves various approximations that can be summarized as follows. (H1) The volume of initial release is partitioned with the help of inward-propagating (normal to the front) geometric rays, and accordingly different sub-volumes are assigned to the different portions of the front. (H2) As the current propagates, the height of the current is not taken to be constant over the entire release. It varies along the front depending on the local speed of propagation. (H3) The velocity of propagation is taken to be normal to the front. Since there is variation in the height of the current along the front, it can be expected that there is some cross-flow (tangential velocity) induced by this variation in the current height. However, since the pressure gradient normal to the front is expected to greatly exceed the tangential gradient at the front, the current velocity is likely to be predominantly normal to the front. (H4) Finally, we assume that, even in the present case of non-axisymmetric propagation, the Huppert–Simpson front relation can be used to express the front velocity in terms of local front height. Here we examine these assumptions relative to the results of fully resolved simulations.

Figure 9.

(a,b) Azimuthal evolution of the normal-to-the-front $u_{n}$ and tangential component $u_{{\it\theta}}$ of the front velocity of a rounded rectangular gravity current at two different time instants (Sim 1). Also plotted is the velocity obtained via the Huppert & Simpson (Reference Huppert and Simpson1980) Froude function (see (3.9)) using the mean height (dash-dotted red line) and the maximum height (solid black line) in the head of the current extracted from the fully resolved simulation. (c,d) Height distribution in the RR current before and after the switch of axes at $t=1.5$ and $t=7$ , respectively. The streamlines of the vertically averaged velocity field in the current are also plotted in the upper right corner.

Let us first examine the direction of fluid velocity at the front of the current. To focus on the velocity of the outward-propagating current and eliminate the contribution from the inward-propagating ambient, we define the depth-averaged velocity of the current as follows:

Recall that ${\it\rho}=1$ in the current and ${\it\rho}=0$ in the ambient. From (4.4), one can extract the velocity along the front and compute the normal-to-the-front and tangential components of the front velocity $u_{n}$ and $u_{{\it\theta}}$ as plotted in figure 9. The simulation results indicate that the normal velocity is an order of magnitude larger than the tangential component of velocity over the entire front of the current. Furthermore, the faster propagation of the current along the $y$ axis ( ${\it\theta}={\rm\pi}/2$ ) is clear. Integrating over the entire front, we find the average normal-to-the-front and tangential front velocities to be approximately 0.37 (0.19) and 0.03 (0.03) at $t=1.5$ (7), respectively. Interestingly, the tangential velocity is mostly positive at $t=1.5$ (when the height is nearly uniform), indicating a slight cross-flow towards the fast front and initial accumulation of excess material at the fast front. Alternatively, $u_{{\it\theta}}$ is mostly negative at $t=7$ (when the height is much larger at the fast front), in line with the expectation that there may be some cross-flow induced by the hydrostatic pressure gradient stemming from the variation in the current height. Overall, this corroborates approximation H3. Secondly, approximation H2 can be readily verified thanks to figures 5(a) and 6(b), which show that the height of the current is not homogeneous along the front during spreading.

We also present in figure 9 the normal-to-the-front velocity estimated by Huppert & Simpson’s (Reference Huppert and Simpson1980) front Froude-number relation using both the head of the current’s mean height and the maximum height taken from the simulation. At the early time $t=1.5$ , reasonable agreement is observed between the simulation results and the prediction. However at $t=7$ , the HS prediction is significantly larger by 45 % than the simulation results. It is noteworthy that, even though the simulated front velocity in the present case is consistently lower than the HS prediction, the EBM with the HS front velocity is capable of predicting the front motion reasonably well (Zgheib et al. Reference Zgheib, Bonometti and Balacandar2014). This is not a contradiction: in the box model, the height of the current is underpredicted since the current is taken to be of uniform height. This underprediction of the front height somewhat compensates the presently observed overestimation of the front velocity given by the Huppert and Simpson Froude-number relation.

Two snapshots of the height distribution of the rounded rectangular release are presented in figure 9(b) together with streamlines (evaluated from the vertically averaged velocity defined in (4.4)). At early times, the streamlines resemble the inward-propagating geometric rays (which are normal to the front) shown in figure 8(b). At later times, the streamlines remain preferentially normal to the front. This is consistent with the earlier observation that the velocity of the current is dominantly oriented along the normal direction. This also provides some support for approximation H1 that the initial partitioning of the release volume is dictated by the inward-propagating geometric rays normal to the front.

4.3. A scaling law for the final shape of non-circular gravity currents

Scaling laws for flows such as turbulent jets, plumes and gravity currents under complex conditions such as cross-flow ambient are important for evaluating particular flow configurations (List Reference List1982). In this section we use the EBM to analyse the characteristic self-similar development of a non-axisymmetric gravity current. As shown earlier by experiments, an initially non-circular gravity current eventually reaches an inertial self-similar shape that is non-axisymmetric. Figure 10(a) presents the evolution of the front obtained from an EBM calculation for an initially elliptical release. Here, the initial length of the major and minor axes of the ellipse are 0.90 and 0.24, respectively. These dimensions correspond to an initial horizontal aspect ratio of ${\it\chi}_{0}\approx 3.8$ and a vertical aspect ratio of ${\it\lambda}\approx 2$ . We show 11 contours of the front in the ( ${\it\eta}=xt^{-1/2}$ ,  ${\it\zeta}=yt^{-1/2}$ ) plane from an initial non-dimensional time of $t=100$ to a final time of $t=200$ with a time increment of ${\rm\Delta}t=10$ . Clearly, the current has reached a near self-similar shape.

Figure 10.

(a) Evolution of the front contour of an initially elliptical release of horizontal initial aspect ratio ${\it\chi}_{0}=3.8$ in the ( ${\it\eta}=xt^{-1/2}$ ,  ${\it\zeta}=yt^{-1/2}$ ) plane obtained from an EBM simulation. Here, the contours are plotted from $t_{0}=100$ to $t_{f}=200$ in steps of ${\rm\Delta}t=10$ . (b) Self-similar horizontal aspect ratio ${\it\chi}_{\infty }$ of the front contour of non-axisymmetric gravity currents as a function of the horizontal initial aspect ratio ${\it\chi}_{0}$ : ▾, ◂, ▸, experiments 2, 3, 4, respectively; $\ast$ , ○,  $\#,+$ , fully resolved simulations 1, 2, 3, 4 respectively; ▫, ♢, EBM simulations with releases of initially elliptical and rounded rectangular shape, respectively; ——, correlation ${\it\chi}_{\infty }=1+(\ln {\it\chi}_{0})/3$ .

We performed a series of calculations with the EBM where the initial horizontal aspect ratio ${\it\chi}_{0}$ of the non-axisymmetric gravity current was varied in the range $1\leqslant {\it\chi}_{0}\leqslant 20$ . For each ${\it\chi}_{0}$ case, the self-similar aspect ratio ${\it\chi}_{\infty }$ was measured. To be specific, we take the value of ${\it\chi}_{\infty }$ at a sufficiently large time, here $t=200$ , so the self-similar regime was reached. EBM calculations were performed for two initial non-circular shapes, namely elliptical and rounded rectangular shapes. The EBM results are summarized in figure 10(b). For comparison, we also plotted the results obtained by the experiments and simulations of table 1. All the data roughly follow a similar trend, which can be modelled by the following simple empirical relation:

In the case of a circular release, it can be shown that the temporal evolutions of the front height $h_{N}$ , radius $r_{N}$ and normal-to-the-front velocity $u_{N}$ in the inertial self-similar regime scale as $t^{2{\it\alpha}}$ , $t^{-{\it\alpha}}$ and $t^{{\it\alpha}}$ with ${\it\alpha}=-1/2$ (see e.g. Ungarish Reference Ungarish2009, p. 122). Figure 11 displays the azimuthal variation of the front height, radius and normal-to-the-front velocity using the aforementioned scaling from Sim 1 ( ${\it\chi}_{0}=3.8$ ) and Sim 3 ( ${\it\chi}_{0}=8$ ). Here ${\it\theta}$ is the angle measured anticlockwise from the $x$ axis. We observe that the normalized $r_{N}$ , $u_{N}$ and $h_{N}$ for both simulations reach a self-similar profile that resembles a sinusoidal curve. This suggests that the function $f({\it\theta})$ introduced in (3.1)–(3.3) may be written as $f({\it\theta})=1+g({\it\theta})$ , with $g({\it\theta})$ being a sinusoidal function to be determined. For Sim 1 ( ${\it\chi}_{0}=3.8$ ), we plot the azimuthal dependence of radius, speed and height from $t=3.15$ (red curve) to $t=17.15$ (blue curve) with a constant time increment of ${\rm\Delta}t=1.75$ . The green curve at $t=8.4$ represents the time at which these quantities become roughly self-similar. Similarly for Sim 3 ( ${\it\chi}_{0}=8$ ), we plot the azimuthal evolution from $t=4$ (red curve) to $t=22$ (blue curve) with a constant time increment of ${\rm\Delta}t=2$ . The green curve at $t=16$ represents the time beyond which these quantities become roughly self-similar. Beyond the self-similar phase, the height, speed and radius are observed to attain a minimum value at ${\it\theta}=0,{\rm\pi}$ and $2{\rm\pi}$ , and a maximum value at ${\it\theta}={\rm\pi}/2$ and $3{\rm\pi}/2$ . This self-similar shape is indicative of an elliptical like shape whose minor axis coincides with the ${\it\theta}=0$ line, which corresponds to the $x$ axis in the $x$ – $y$ plane.

Figure 11.

Evolution for ${\it\chi}_{0}=3.8$ (Sim 1) of the front contours (a,d), front speed (b,e) and mean front height (c,f). The contours are plotted from $t_{0}=3.15$ (red curve) to $t_{f}=17.15$ (blue curve) in steps of ${\rm\Delta}t=1.75$ . Plotted in green is the curve corresponding the time $t=8.4$ for which these quantities become roughly self-similar. Evolution for ${\it\chi}_{0}=8$ (Sim 3) of the front contours (g,j), front speed (h,k) and mean front height (i,l). The contours are plotted from $t_{0}=4$ (red curve) to $t_{f}=22$ (blue curve) in steps of ${\rm\Delta}t=2$ . Plotted in green is the curve corresponding the time $t=16$ for which these quantities become roughly self-similar.

Subtracting the mean value from each curve, the scaled radius ( $r_{N}t_{f}^{-1/2}$ ), front speed ( $u_{N}t_{f}^{1/2}$ ) and front height ( $h_{N}t_{f}$ ) may be approximately described by a single sinusoidal function of the form

where $A$ and ${\it\theta}_{0}$ represent the amplitude and phase angle, respectively. The phase angle ${\it\theta}_{0}$ is the angle the $x$ axis makes with the major axis of the rounded rectangle and, in the present case, ${\it\theta}_{0}=0$ . The amplitude $A$ is obtained from the average root mean square (r.m.s.) value of the three curves (radius, speed and height) as $A\approx 0.22$ for Sim 1 and $A\approx 0.32$ for Sim 3, for which ${\it\chi}_{0}=3.8$ and 8, respectively. Since $A$ is obviously 0 for ${\it\chi}_{0}=1$ , the present results indicate that $A$ increases as ${\it\chi}_{0}$ is increased, i.e. the larger the non-axisymmetry of the initial release, the higher the contrast of the front height and speed along the front contour. This is in line with (4.5), which predicts a larger self-similar aspect ratio for larger initial horizontal aspect ratios. In fact, we found that the amplitude $A$ of $g({\it\theta})$ can be nicely described by the empirical relation $A\approx (\ln {\it\chi}_{0})/6$ , at least for the three values of ${\it\chi}_{0}$ used in the simulations (namely ${\it\chi}_{0}=1,3.8$ and 8). As for the EBM data, an empirical relation of the form $A\approx (2/9)\ln {\it\chi}_{0}$ provides a better fit in the range ${\it\chi}_{0}=[1,20]$ . Note that, in computing $A$ from the EBM, the values from the height were discarded since the height is consistently underpredicted (recall that the height of the current in the EBM is averaged over the entire sector, which is not the case in reality, as seen in figures 2 and 6).

In figure 12, we plot the azimuthal evolution at the end of each simulation for the scaled front location ( $r_{N}t_{f}^{-1/2}$ ), speed ( $u_{N}t_{f}^{1/2}$ ) and height ( $h_{N}t_{f}$ ), for which the mean value of each curve has been subtracted. We plot the results from Sim 1 and Sim 3 and observe good agreement between the three curves and the sinusoidal function $g({\it\theta})$ defined in (4.6). It follows that, with the knowledge of the self-similar shape of the front, one could roughly predict the front height (or front speed) of the current provided the front height (or front speed) is known at some azimuthal orientation.

Figure 12.

Azimuthal evolution at the final time of the scaled front location (red solid), speed (green dashed) and height (blue dash-dotted) from (a) Sim 1 ( ${\it\chi}_{0}=3.8$ ) and (b) Sim 3 ( ${\it\chi}_{0}=8$ ). The thick solid black line corresponds to $f({\it\theta})$ from (4.6). The mean value is subtracted from each curve.