1 Introduction

Gravity current shapes to form a head as the front of the current advances through a surrounding fluid of different density [1, 9, 10, 13, 18]. Sudden release of dense fluids, such as earthquake-triggered turbidities, can produce gravity currents with great destructive forces [8, 21]. As much as 20 m/s of current velocity on the continental slope has been reported as a consequence of submarine landslides [11]. The total buoyancy in the gravity-current head may increase if the velocity on the floor is sufficiently high to cause re-suspension of the particulate matter from the floor. The advancement of the particulates toward the front and their re-entrainment into the wake is a mechanism responsible for the transport of fine sediments over long distances on the ocean floor [24]. The thickness of the boundary layer can play a critical role in the advance of the gravity-current head. Britter and Simpson [2] have shown how the structure of the head can change depending on slip versus no-slip boundary conditions. At a high Reynolds number, the boundary-layer thickness can be quite small compared with the overall dimensions of the current head. The dynamics of the gravity current over wide ranges of parameter space has been studied in the laboratory by Rottman and Simpson [22], Thomas et al. [25], Marino et al. [15] and Nogueira et al. [18]. The experimental results were correlated mostly in terms of depth-averaged internal hydraulic models. The Direct Numerical Simulations (DNS) model and the Large Eddy Simulations (LES) model have been used by Hartel et al. [9, 10], Thomas et al. [25], Cantero et al. [3, 7] and Ooi et al. [19, 20] to study some aspect of the gravity current in detail. Well-resolved DNS can provide all turbulence-length scales down to the dissipative range. LES with correct modelling of the sub-grid scales can be an efficient alternative, covering a wider range of parameter space utilizing affordable computation resources. DNS were conducted by Cantero et al. [3, 7] up to a Reynolds number of 8950. High Reynolds number LES simulations were carried out by Ooi et al. [20] up to a Reynolds number of 12,600.

The present study is focused on the gravity-current head produced by the release of a relatively small volume of dense fluid behind a tall lock gate. The numerical simulation is based on a two-dimensional inviscid model. The challenge in the numerical simulation is to capture the turbulent interface in the head without the artifact of numerical diffusion. In the classical finite volume method, fluxes are estimated on the surface of the finite volume using a truncation series. Spurious numerical oscillations and false numerical diffusion are the consequences, particularly in regions across sharp discontinuities. Artificial diffusion often is introduced synthetically to gain computational stability. Occasional switching to a diffusive upwind scheme, for example, is one classic strategy to manage the numerical oscillations (see a review of the classical method by Karimpour Ghannadi and Chu [12]). The method of the Lagrangian block simulation (LBS) on the other hand is very different. Fluxes are not estimated so there are no numerical oscillation and diffusion errors from the truncation. Validation of the LBS method has been conducted in the calculation for the advance of wet water on dry bed by Tan and Chu [26, 27] and in the simulations for the laminar-and-turbulent interfaces by Chu and Altai [4–6]. In the absence of spurious oscillations and false diffusion, the best possible inviscid model of the gravity-current head is constructed by the LBS.

2 Lagrangian block simulation

The inviscid model is constructed by setting the real viscosity to zero in the LBS. The general method as delineated in Fig. 1 has a Lagrangian advection step and a diffusion step. The diffusion step however is ignored in the present simulation for the inviscid model. The block-simulation procedure consists of (1) Lagrangian advection, (2) division into portions, and (3) reassembly of the portions into new blocks by the second-moment method. The blocks are renewed in each time increment to prevent excessive distortion. The numerical error associated with the LBS is not cumulative, and false-diffusion error has been shown to be negligible by Chu and Altai [6]. A couple of simple advection experiments are given in the “Appendix” just to show how the LBS can capture discontinuities without the false diffusion and spurious oscillation errors.

Fig. 1
figure 1

Lagrangian block in the Eulerian cells a before and b after the advection and diffusion steps. The Lagrangian block simulation method and the notation details in this figure are given in Chu and Altai [6]

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3 Formulation

The LBS for the gravity-current head is based on the Boussinesq approximation, and the two-dimensional stream-function and vorticity formulation. The stream function ψ is introduced to define the x- and y-component of the velocity vector (u, v):

$$u = + \frac{\partial \psi }{\partial y},\quad v = - \frac{\partial \psi }{\partial x}$$
(1)

The vorticity \(\zeta\) is

$$\zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$$
(2)

This definition for the vorticity expressed in terms of stream function ψ is the Poisson equation:

$$\zeta = - \left[ {\frac{{\partial^{2} \psi }}{{\partial x^{2} }} + \frac{{\partial^{2} \psi }}{{\partial y^{2} }}} \right]$$
(3)

The production of the vorticity is determined by the x- and y-momentum equations as follows:

$$\frac{Du}{Dt} = - \frac{1}{\rho }\frac{\partial p}{\partial x}$$
(4)
$$\frac{Dv}{Dt} = - \frac{1}{\rho }\frac{\partial p}{\partial y} + g^{\prime }$$
(5)

Subtracting the x-differentiation of Eq. 5 from the y-differentiation of Eq. 4 gives the Lagrangian vorticity equation

$$\frac{D\zeta }{Dt} = \frac{{\partial g^{\prime } }}{\partial x}$$
(6)

The buoyancy force per unit mass of the fluid is \(g^{{\prime }} = g(\Delta \rho /\rho )\). The scalar transports of the buoyancy and the vorticity are carried out by Lagrangian block advection without the diffusion step. Two systems of blocks are employed in the simulation. One system of blocks is for the advection of the buoyancy \(g^{\prime }\). A separate system of blocks is for the advection of the vorticity \(\zeta\). The gradients in the Eqs. 4, 5 and 6 are evaluated directly on the Eulerian grid by central differencing. This evaluation for the gradients in the LBS is to be distinguished from the particle methods. In the Smooth Particle Hydrodynamic (SPH) method for example, the gradients are determined by the weighting of the influence by the nearby particles [16].

The numerical solution of the Poisson equation for the stream function and the velocity is obtained by the Gauss–Seidel iteration method. Convergence is declared when the velocity difference between the subsequent iterations is less than 10−5 time of the maximum velocity. The simulations are conducted in a tank of a length L equal to four times its height H tank using 384 blocks over the height of the tank and 1536 blocks along the length of the tank. The time-step sizes are such that the Courant number associated with the maximum velocity is 0.2. The dense fluid is released from a small lock of length L o equal to one-eighth of the tank’s height H tank, i.e., L o  = 1/8H tank. Three numerical experiments referred to as the full-height, half-height and quarter-height releases are conducted with the height of the lock H = H tank, ½H tank and ¼H tank, i.e., H = 8 L o , 4 L o and 2 L o , respectively. Figures 2 and 3 show the buoyancy profiles \(g^{\prime } /g^{\prime}_{o}\) of the gravity current as the heads advance toward the end of the tank. The height H and length L o of the locks are defined by the dashed lines in the figures. These releases from a tall lock gate of small volume have produced prominent heads with a weak trailing current behind the heads. The development depends on the height of the lock. For the full-height release with H = 8 L o as shown in Fig. 2, the vorticity in the turbulent mixing layer feeding into the wake is able to organize itself to produce one highly coherent vortex within the gravity-current head. In contrast, the current head produced by the half-height release with H = 4 L o is less organized as shown in Fig. 3. The structural difference between the coherent structure of the current head in Fig. 2 and the less coherent structure in Fig. 3 will be quantitatively analyzed and explained in the subsequent sections.

Fig. 2
figure 2

Development of the buoyancy profiles \(g^{\prime } /g^{\prime }_{o}\) of the gravity-current head produced by the full-height release of a source fluid of length L o and height H = 8 L o . The dashed line marks the initial position of the source fluid of reduced gravity \(g_{o}^{\prime }\) behind the dam

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Fig. 3
figure 3

Development of the buoyancy profiles \(g^{\prime } /g_{o}\) of the gravity-current head produced by the half-height release of source fluid of a length L o and a height H = 4 L o . The dashed line marks the initial position of the source fluid of reduced gravity \(g_{o}^{\prime }\)

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4 Front speed at the leading edge

The speed of the front at the leading edge of the gravity-current head is analyzed first. Figure 4 shows front speed u f normalized by \(\sqrt {g_{o}^{\prime } H}\) obtained by the LBS for the full-height release, half-height release and quarter-height release. The front speed and the return velocity external to the current head depend on the dam’s height H and the tank’s height H tank. On average, the front speed tends to decrease slightly due to the entrainment of ambient fluid which moves in the opposite direction to the gravity current. The decrease in the front speed is not monotonic. The variations from the monotonic are closely related to the dynamics of the Kelvin–Helmholtz instability in the mixing layer and the feeding of the vorticity from the mixing layer into the wake. This structural dynamics of the current head have been observed and explained in the laboratory experimental study by Nogueira et al. [18].

Fig. 4
figure 4

Front speed u f plotted versus the front distance x f from the upstream end of the tank. The triangular, square and circle symbols are the data obtained from full-height, half-height and quarter-height simulations, respectively. The red open circles denote the frontal speed at the end of the acceleration phase, which is estimated by tracing back to the initial position of the front at x f  = L o

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As pointed out in the laboratory experiment by Rottman and Simpson [22], Marino et al. [15] and Nogueira et al. [17, 18], the development of the gravity current consists of an initial phase when the development is dependent on the initial conditions, and an asymptotic state when the advance of the head is dependent on the total buoyancy at the source. The front speed is zero immediately after the removal of the dam. It accelerates rapidly to a maximum during the acceleration phase. The red open circles in Fig. 4 are the front speeds at the end of the acceleration phase that are obtained by tracing back to the initial positions at x f  = L o . Shin et al. [23] and Marino et al. [15] have examined the role of the ambient current on the front speed. Shin et al. [23] developed a hydraulic theory to correlate the front-speed Froude number to the dam-to-tank height ratio H/H tank by the formula:

$$\frac{{u_{f} }}{{\sqrt {g_{o}^{\prime } H} }} = \frac{1}{2}\sqrt {2 - \frac{H}{{H_{\text{tank}} }}}$$
(7)

This theory by Shin et al. [23] and the experimental data of Marino et al. [15] are included in Fig. 5 to compare with the front speeds at the end of the acceleration phase obtained from the inviscid model simulations. The hydraulic theory of Shin et al. [23] does not include the entrainment. It is therefore not surprising that applying their theory gives a slightly higher front speed than occurs in the results of the experimental data of Marino et al. [15] and the results of the present simulations. However, the agreement between the present inviscid model and the experimental observation of Marino et al. [15] was not entirely anticipated. We offer the following explanation. The data of Marino et al. in Fig. 5 was collected shortly after the acceleration phase in currents of very high Reynolds numbers. In this early stage of development, the inertia effect is still the dominant effect over the viscosity. The inviscid model therefore is an acceptable approximation.

Fig. 5
figure 5

Frontal speed of the 2D inviscid model compared with the laboratory experimental data by Marino et al. [15] and the hydraulic theory of Shin et al. [23]. The red open circles denote the front speed at the end of the acceleration phase obtained from tracing back the LBS data in Fig. 4

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5 Structural coherence of the head by full-height release (H/L o  = 8)

The full-height release with the aspect ratio of H/L o = 8 has produced the most coherent structure of the gravity-current head. Figure 2a shows the detachment of this gravity current from the tank at the dimensionless time \(t\sqrt {g_{o}^{\prime }/ H} = 2.29\). The head shown in Fig. 2b for \(t\sqrt {g_{o}^{\prime }/ H} = 4.90\) has multiple eddies in the wake. Finally at time \(t\sqrt {g_{o}^{\prime }/ H} = 7.50\) as shown in Fig. 2c, the wake and the head are combined to form one large coherent vortex. Figure 6 is a close-up view of the vorticity distribution and the velocity of this gravity-current head at the instant after the front has advanced a distance x f  = 14.9L o. The Kelvin–Helmholtz instability initiates the turbulent mixing layer from the leading edge. Vorticity is produced in the mixing layer and then transported along the mixing layer to the back of the gravity-current head to form a wake. A prominent circulation region is observed in the wake region. The maximum velocity occurs at the impingement of the wake vortex on the floor. This full-height release has produced a gravity-current head without much of the trailing current that otherwise would be feeding from behind the head. This is remarkable as the head is now charging forward without much of the current that initiates it. The relation between the head and the gravity current in this full-height release is akin to the formation of a smoke ring and the perfect impulse that blows the ring.

Fig. 6
figure 6

The vorticity pattern and the velocity vectors delineating the coherent structure of the gravity-current head produced by the full-height release, after the leading edge has advanced a distance x f  = 14.9 L o. The shape of the current head is defined by the current’s height h max and the distance (x m − x f ) from the leading edge to the location where the maximum floor velocity u max occurs

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The organization of the vorticity in the wake and the impingement of the coherent wake vortex on the floor have produced a maximum velocity u max in the wake on the floor. The distance from the leading edge to the floor velocity maximum is (x f  − x m). This length (x f  − x m) and the maximum height h m define the shape of the current head. The numerical simulations have determined the maximum floor velocity u max, the length of the head (x f  − x m) and the height of the head h m. Figure 7 shows the values of these key parameters: (a) u max/u f, (b) (x f  − x m)/h max (c) h max/H, and (d) KE/PE o, PE/PE o and (KE + PE)/PE o. The nominal value of the floor maximum velocity is u max/u f  ~ 3.1 as the current head advances toward the end of the tank. This is marked as a red dashed line Fig. 7a. There is however considerable variation from this nominal value as the shape of the current head is not entirely constant. The head length-to-height ratio that defines the shape of the gravity-current head has a nominal value of (x f  − x m)/h max ~ 1.3. This and the variation from the value are shown in Fig. 7b. The head-height to the lock-height ratio has a nominal value of h max/H ~ 0.28 as shown in Fig. 7c. The height of the gravity-current head h max is determined from the present simulation by the cut off at the 5% boundary where \(g^{\prime } /g_{o}^{\prime } = 0.05\).

Fig. 7
figure 7

The key parameters of the gravity-current head produced by the full-height release: a u max/u f , b (x f  − x m)/h max, c h max/H, and d KE/PE o denoted by the triangle symbol, PE/PE o by the square symbols and (KE + PE)/PE o by the diamond symbol. The nominal values for these key parameters are estimated and then marked by the red dashed lines in the figure despite the considerable variations from the nominal values

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In the inviscid model the energy is conserved. The potential energy PE and the kinetic energy KE of the gravity-current head are determined from the numerical simulation using the following formulae:

$$PE = \sum\limits_{i = 1,M} {\sum\limits_{j = 1,N} {g_{ij}^{{\prime }} y_{ij} } } \quad {\text{and}}\quad KE = \sum\limits_{i = 1,M} {\sum\limits_{j = 1,N} {\frac{1}{2}\left( {u_{ij}^{2} + v_{ij}^{2} } \right)} }$$
(8)

in which \(g_{ij}^{{\prime }}\) is the buoyancy, (u ij , v ij ) the velocity components and y ij the elevation at the i–j cell. The grid for the numerical simulations is i = 1 to M and j = 1 to N in which M = 1536 and N = 384. Figure 7d shows the variation of the kinetic and potential energy with distance of advance for this gravity-current head produced by the full-height release. The advection of 1536 × 384 blocks on the grid with 1536 × 384 cells conserves the energy accurately. Normalized by the initial potential energy PE o, the kinetic energy approaches the value of KE/PE o ≈ 0.80. The potential energy has a value of KE/PE o ≈ 0.20 as the head advances toward the end of the tank. There are some variations in the kinetic energy and the potential energy. The total energy KE + PE is perfectly constant throughout. In this simulation using 1536 × 384 blocks, the energy ratio (KE + PE)/PE o is closely equal to unity for all time with an error of no more than 0.27%. Since the energy is conserved in the inviscid model, the deviation from the unity becomes a measure of the accuracy of the numerical simulation.

6 Half-height and quarter-height releases (H/L o  = 4 and 2)

The less coherent structure of the gravity-current head is produced by the half-height and the quarter-height releases. The lock height-to-length aspect ratios of these releases are H/L o  = 4 and H/L o  = 2, respectively, and are comparable to the aspect ratio of H/L o  = 4 to 1 in the laboratory experiment of Marino et al. [15] and the H/L o  = 1.33 in the laboratory experiment of Nogueira et al. [18]. Figure 3 shows the buoyancy profiles \(g^{\prime } /g_{\text{o}}^{\prime }\) produced by the half-height release. The gravity-current head now has a wake and a weak current following the wake. The vorticity from the mixing layer organizes itself in the wake into large and small vortexes that interact to produce unsteady dynamic structures with considerable time and space variations. Such unsteady dynamics of the current head in a similar range of lock aspect ratios are well documented and explained by Nogueira et al. [18]. The nominal values of the key parameters of the current head produced by the half-height release and quarter-height release are presented in Figs. 8 and 9 using the same format as in Fig. 7 for the full-height release. There are considerable variations in and deviations from the values of these parameters. Rough estimates of their deviations from the nominal values nevertheless are made. The nominal values are marked by the red dashed lines shown in the figures. Table 1 provides the nominal values and also the rough estimates of the deviations for comparison among the three simulation experiments.

Fig. 8
figure 8

The key parameters of the gravity-current head produced by the half-height lock release: a u max/u f , b (x f  − x m)/h max, c h max/H, and d KE/PE o denoted by the square symbol, PE/PE o by the triangle symbols and (KE + PE)/PE o by the diamond symbol. For comparison with other simulations, the nominal values for the key parameters are estimated and then marked by the red dashed lines in the figure

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Fig. 9
figure 9

The key parameters of the gravity-current head produced by the quarter-height lock release: a u max/u f , b (x f  − x m)/h max, c h max/H, and d KE/PE o denoted by the square symbol, PE/PE o by the triangle symbols and (KE + PE)/PE o by the diamond symbol. For comparison with other simulations, the nominal values for the key parameters are estimated and then marked by the red dashed lines in the figure

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Table 1 Nominal values and deviations from the values of the key parameters for the gravity-current head produced by the full-height, half-height and quarter-height releases
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The current head initiated by the full-height release with the highest aspect ratio H/L o  = 8 is structurally the most coherent and has the fewest variations among the three simulation experiments. The current head of the full-height release also has the highest relative floor velocity u max/u f , the smallest length-to-height ratio (x f  − x m)/h max, and the lowest height h max/H compared with the less-coherent current heads produced by the half-height release and the quarter-height release.

7 Energy conservation and computational error

The numerical computation accuracy of the three simulation experiments is not exactly the same. The total energy (KE + PE) would be equal to its initial potential PE o if the energy were perfectly conserved. The maximum deviation of the ratio (KE + PE)/PE o from the unity is 0.27% for full-height release, 0.65% for the half-height release and 2.1% for the quarter-height release. These computational errors can be correlated with the number of blocks and cells over the height of the current head. For the full-height release, the height of the current head is h max ≈ 0.28 H tank. The number of computation cells is 108 (≈384 × 0.28) over the height of the head. For the half-height and quarter-height releases, the height of the head are h max ≈ 0.42 H = 0.21 H tank and h max ≈ 0.55 H = 0.14 H tank. The corresponding numbers are 81 (≈384 × 0.21) computation cells and 54 (≈384 × 0.14) computation cells over the height of the head. There are fewer computation cells over the height of the current head for the half-height and quarter-height releases. This problem of resolution is reflected in the energy conservation. It explains why the ratios (KE + PE)/PE o for the half-height release and quarter-release deviate from the unity by as much as 0.65 and 2.1%, respectively.

8 Discussion and conclusion

The gravity-current heads produced by the release of a relatively small volume of source fluid from behind a tall lock gate are studied for structural coherence of the current heads. Three LBS based on a two-dimensional inviscid model are conducted for the lock aspect ratios H/L o  = 8, 4 and 2. The current head produced by the highest lock aspect ratio H/L o  = 8 is most coherent and has the highest floor velocity and the least trailing current behind the head. The simulations based on the two-dimensional inviscid model cannot perfectly reproduce reality. They do however suggest the possibility of a coherent structure with high floor velocity. The association of structural coherence with the floor velocity is significant. The high floor velocity may explain the destructive force of the earthquake-triggered turbidities reported by Heezen and Ewing [8], Piper et al. [21] and Hsu et al. [11]. In a coordinated system moving with the front velocity, the dense fluid is pushed from the back toward the current head with a relative speed equal to (u max − u t ) and from the front with a relative speed equal to u t . The convergence of the external flow inward is balanced by the tendency of the dense fluid to spread outward along the floor. This is a necessary condition for a gravity-current head to maintain its structural integrity so that the fine sediments can travel with the head over long distances on the ocean floor without significantly changing the form of the head. The high velocity on the floor could also be responsible for the re-suspension of the fine sediment and possible increase of the total buoyancy within the gravity-current head.