The viscous Boussinesq equations are used to simulate the unsteady flow from an elevated source of a plume of heavier fluid into a lighter fluid. The solution is obtained with a spectral method. The source is located at some height above the bottom of a vertically confined layer and the flow is two dimensional. Results indicate that the small, initially circular bubble of heavier fluid expands until the top reaches some height above the source, after which it levels off and starts to flow downwards in a vertical plume before spreading horizontally. We discuss the results for different values of Reynolds number, flow rate, and density differential. It is found that there are several different plume types, the behaviours of which depend on the flow rate. At low flows, the plume forms an inverted mushroom shape, some parts of which may separate as it falls downward. At moderate flow rates, the inverted mushroom plume remains connected until it reaches the base, after which it spreads horizontally as a gravity current. Finally, at large flow rates, the central blob expands outward until it hits the bottom, after which it spreads horizontally. When the Reynolds number is small and viscosity is relatively high, the interface between the expanding region and the ambient fluid is very stable and exhibits few deviations and very little mixing, but as the Reynolds number increases, spirals begin to form around the edges, thus, enhancing the mixing.
Hinweise
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1 Introduction
Studies of situations in which water is pumped into a pond, reservoir or the ocean, or those in which gas is released into the atmosphere, are important in better understanding and controlling water and air quality and dispersal of pollutants. In most situations, this will involve fluid of one density entering a region of a different density.
The most obvious example of this is a simple fountain, in which water is pumped into the air. In that case, the density difference is such that the flow is essentially a high Reynolds number flow in which viscosity plays almost no role and the fluid can be considered inviscid with a free boundary between the water and the air. Classical solutions exist for these problems, and these are often used as a basis for more complicated situations. In cases where it is water entering water or a gas entering air, there will be viscous effects at the interface between the inflowing fluid and the ambient fluid. Furthermore, the ambient fluid may be of approximately constant density or it may be stratified. These situations present different and more complicated challenges for the modeller, requiring a simulation that requires a full representation of the Navier-Stokes equations for example. A very detailed discussion of earlier models is considered by Hunt and Burridge [1].
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One important application of this work that has recently arisen is the process by which water from desalination plants is returned to the ocean [2]. Highly concentrated brine is pumped into the ocean where it is dispersed from elevated outlets. It is to be hoped that this process will minimise the environmental impact of this highly saline water on the sea bed [3] and so it is important to generate as much mixing with the ambient fluid as possible.
Here, we implement a spectral method to simulate this situation using the viscous Boussinesq equations. A recent paper considered the similar problem of a heavier fluid pumped into a region with a less dense fluid from “ground level”, i.e. at the bottom of the region [4]. The spectral approach was developed by Forbes, Russell and others to consider Rayleigh-Taylor instability, [5], the Kelvin-Helmholtz instability [6] and a plume of light fluid emanating from the base of a channel containing heavier fluid [7].
Flows such as these are characterised by the Froude number, defined as
where \(\nu \) is kinematic viscosity. The quantities for speed and length need to be chosen appropriately, and so although there are obvious scales in this problem, the choice here is dependent on only the flow from the source and not the surrounding geometry. These scales will be explained later to represent the outflow from the source. The Froude number measures the speed of the outgoing jet while the Reynolds number indicates the relative importance of viscosity.
Fig. 1
A schematic of the situation under consideration. The source is located at some height above a solid boundary, and there are horizontal boundaries at \(y=0,h\). The point \(h_{mid}\) is the height of the centre of the outflowing region directly above the source. A heavier fluid flows from the source into a lighter fluid, with the two being separated by an interface
×
A “bubble” of fluid that is denser than the ambient fluid is tracked as it emanates from the source and grows in size see Fig. 1. At some point, this bubble reaches a size where gravitational effects start to become important and the plume travels downward until it hits the base and starts to spread horizontally (Fig. 1) in the form of a gravity current [8‐10]. However, our major interest is not in this outflowing gravity current, but on earlier stages of the flow, and how the two fluids interact as the flow parameters change.
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There are many earlier investigations of such flows from an elevated source. For example, numerical simulations by Williamson et al. [11] for the flow of a turbulent, weak fountain showed the behaviour over a range of low Froude numbers and Reynolds number \(Re=20-3500\). For slow flows in the range \(F<0.4\), the Reynolds number has little effect on the character of the flow, although the resulting gravity current intrusion does change with the Reynolds number. At moderate values of Froude number, say \(0.4< F < 2.1\) the impact of the source momentum flux grows and the structure of the flow changes to one where there is an upward flow and a cap region where the flow stagnates and then inverts. Again, they found that the impact of the source velocity profile is more important than the Reynolds number and the effect of turbulence is very small. Hunt and Coffey [12] presented analytical solutions for flow from a horizontal source of width \(2b_0\) from on the bottom of the region into an initially stagnant environment of uniform density and found the initial rise height of the plume, \(z_m\). They also found different flow regimes for \(F \gg 1\), \(F=\mathcal {O}(1)\), \(F \ll 1\). Lin and Armfield [13] considered similar fountain flows at medium Froude and Reynolds numbers. Kaye and Hunt [14] incorporated turbulence into the fountain and for large Froude number they found that \(\frac{z_m}{r_0} \sim F\), where \(r_0\) is the source radius, while for moderate values, the relationship \(\frac{z_m}{r_0} \sim F^2\) was found and the rise height was found to be independent of the entrainment. At small flow rates the flow is hydraulically controlled and \(\frac{z_m}{r_0} \sim F^{2/3}\). Van den Bremer and Hunt [15] presented closed-form solutions for a two-dimensional planar turbulent fountain or rising plume in an ambient fluid of uniform density. They showed the results to be valid for both Boussinesq and non-Boussinesq assumptions. These simulations all indicate that the Froude number is a major factor in determining the general behaviour of the flow. Here we will investigate this further.
Here we are particularly interested in times after the initiation of the flow from the source, but before it evolves into the outflowing gravity current. Shraida et al. [4] used such a spectral method to solve the viscous Boussinesq equations and study the unsteady flow of a plume of heavy fluid into a lighter fluid when the source is located on the bottom of a channel, and this paper extends that work. We consider the extra features of the flow caused by the elevation of the source of fluid. The elevation of the sink allows the initial outflow to develop more fully before the more dense fluid starts to flow downward, and it is found there are different types of behaviour that depend on the Froude number, elevation and Reynolds number. At low flows, the central resulting plume resembles an inverted mushroom and separates as it falls downward, while at medium flow rates, a similarly shaped plume forms and stays connected with spiral flows developing on the interface. At high flow rates, a large circular blob expands rapidly outward and down, going directly into an outflowing gravity current with spiral flows at its head.
The general shape of the resulting plume depends almost exclusively on the Froude number, but variations in the behaviour of the interface between the two regions of different density are determined by the effects of viscosity (Reynolds number). At large values of Reynolds number, there is a significant amount of mixing at the interface.
2 Problem formulation and solution
The spectral method used by Russell et al. [7] and Shraida et al. [4] is to be used to study the flow emanating from a two-dimensional source into a lighter fluid. The method uses a background flow and then uses a Fourier series to compute the correction to this flow.
2.1 Background flow
The background flow is computed by finding the potential flow in the region due to a source at the specified location. In other words, we use the irrotational flow of an inviscid, incompressible fluid from a source. We denote the variables for this background flow with a subscript 0. We define a velocity potential \(\phi _0(x,y)\) such that the velocity field, \(\textbf{q}_0\) is given by \(\textbf{q}_0=\nabla \phi _0\), and under these assumptions \(\phi _0(x,y)\) satisfies Laplace’s equation:
and since there can be no flow through the solid boundaries,
$$\begin{aligned} \begin{aligned} \phi _{0x}&=0 \text{ on } x=0,\ 0<y<h, y\ne h_s, \\ \phi _{0y}&=0 \text{ on } y=0,h. \end{aligned} \end{aligned}$$
(5)
where h is the height of the duct and \(h_s\) is the elevation of the source and F is the Froude number.
These equations are couched in terms of the non-dimensional variables. To derive these, we note that there are several length scales in the problem, such as the height of the source and the height of the region, but neither of these are related to the flow from the source, which occurs in isolation from the boundaries for quite a long time. For this source flow, there is no obvious length scale and so we non-dimensionalised with respect to the length scale as a combination of the other variables, i.e. \(H=(m^2/g')^{1/3}\) and similarly for the velocity \(U=(mg')^{1/3}\), where \(g'=\varDelta \rho g / \rho _0\) is known as the reduced gravity, with \(\rho _0\) the density of the fluid emanating from the source and \(\varDelta \rho \) being the difference between the source density and the ambient fluid. The quantity m is the flux from the source, and g is gravitational acceleration. Therefore, the densimetric Froude number, F, is defined as \(F=m/(g'H^3)^{1/2}\), which is the usual definition if we note that \(U=m/H\). Thus all of these quantities are determined from the outflow source rather than the full domain.
The solution for the background flow can be obtained using conformal mapping. A complex potential, \(f_0(z)=\phi _0+i\psi _0\), where \(\psi _0\) is the streamfunction, can be defined where \(f_0'(z)=u_0-iv_0\) provides the velocity field and \(z=x+iy\) is the complex variable in this coordinate system. The appropriate solution for flow from a line source situated off the base at \(z=ih_s\) is
The velocities computed in this way will be used as a base flow in all subsequent calculations and will be perturbed to find the actual velocity everywhere using the spectral method.
2.2 Boussinesq solution
Following the method of Forbes et al. [5], Forbes et al. [6], and Russell et al. [7], we now formulate a spectral solution method for the vorticity equations. Complete details are given in Russell et al. [7], but for completeness, we summarise them here. The Boussinesq approximation assumes effects of the density variation in a fluid flow are confined to the buoyancy term, i.e the term containing gravitational effects, and that it can be neglected in the rest of the equations as it is relatively small. In all other terms, the density is assumed to be constant. In the work presented here, this is an accurate representation except perhaps in the region close to the interface between the two fluids, but even here it is expected the effect would be secondary. Under the Boussinesq approximation, the continuity equation reduces to the incompressible form, as in Sect. 2.1.
Thus to represent density \(\rho \) in Boussinesq theory, we need to split the continuity equation
where \(\textbf{q}=(u,v)\) is the velocity vector and \(\sigma \) is a small dimensionless diffusion coefficient that may be related to a Prandtl number [7], to give the two conditions:
$$\begin{aligned} & \text{ div }(\textbf{q})=0 \end{aligned}$$
where \(\hat{\rho }\) is a perturbation from the constant density of the inflowing fluid. Implementing this approximation gives the Navier-Stokes-Boussinesq equation [16]:
can be used to satisfy the incompressibility conditions, and taking the vector curl of the Navier-Stokes-Boussinesq equation (12) gives the 2-D Boussinesq vorticity equation:
where \(\varPsi _0\) is determined from the earlier conformal mapping, (6). The choice of eigenvalues in the x direction, at \(x=\pm L\), is to allow the far field solution to be the inviscid flow solution. So long as the horizontal extent of the domain \(x \in (-L,L)\) is greater than the width of the plume throughout the simulation, this truncation has very little impact on the flow.
The velocity components u and v can be obtained by differentiation of (15), so that
where \(u_0\) and \(v_0\) are given by (7). Observe that \(u_0\) and \(v_0\) satisfy the boundary conditions (5). Note also that the horizontal velocity u is anti-symmetric in x, and the vertical velocity v is symmetric in x. The vorticity \(\omega \) can, therefore, be written as follows:
Comparison for the case of \(F=0.75\), \(Re=500\), \(h_s=2\), \(H=6\) at two different times, \(t=6\) (the inner rings) and \(t=10\) (the outer rings). The contours are for density perturbations \(\hat{\rho }= -0.04, -.03, -0.02, -0.01\), with \(D=0.05\), and the dashed lines are the integration over time following the location of the particles initially coincident with the starting bubble, integrated along the streamlines of the inviscid flow field given in (7)
×
At the location of the source, \((x,y)=(0,h_s)\), the density perturbation \(\hat{\rho }\) needs to be zero. In order to retain an appropriate form, we define the representation
The height of the mid point (\(h_{mid}\)) as a function of time for different values of source height, \(h_s\), F=0.5, and Re=100. \(h_s\)=0 (the dashed line) and for \(h_s\)=0.5,1,1.5,2,3,4 corresponds to the lines from the bottom to the top. The top of the plume levels off quite quickly and the flow becomes locally steady
×
The function \(G_{x,y}\) ensures that \(\hat{\rho }=0\) at \((0,h_s)\). Russell et al. [7] used \(G_{x,y}=x^2+(y-h_s)^2\) but in this work, we used
which in the limit \((x,y) \rightarrow \ (0,h_s)\) is equivalent, but was shown to give better convergence near the interface of the two regions of different density [4]. Using equations (31)-(44) (see Appendix A), it is possible to derive a matrix equation for the time derivatives \(B_{00}^{'}, B_{0n}^{'} \ \text{ and } \ B_{mn}^{'}\). These equations can be solved numerically at each new time step. In matrix form, \(PB'(t)=R\), this is
and the initial density profile provides a new right-hand-side vector S, such that \(PB(0)=S\), where
$$\begin{aligned} & S_{00}=\int _{0}^{h} \int _{-L}^{L}\hat{\rho }(x,y,0)\ \text{ d } x \ \text{ d } y, \end{aligned}$$
(23)
$$\begin{aligned} & S_{0l}=\int _{0}^{h} \int _{-L}^{L}\hat{\rho }(x,y,0) \cos \left( \frac{l \pi y}{h}\right) \text{ d } x \ \text{ d } y, \text{ for } l=1,\dots ,N, \end{aligned}$$
(24)
$$\begin{aligned} & S_{kl}=\int _{0}^{h} \int _{-L}^{L}\hat{\rho }(x,y,0)\cos \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right) \text{ d } x \ \text{ d } y, \end{aligned}$$
(25)
for \(l=1,\dots ,N,\) and \(k=1,\dots ,M\). The initial density perturbation is \(D-1\), where \(D=\rho _2/\rho _1\), everywhere except for a small region of radius \(\alpha \) about the source within which the density perturbation is zero, i.e.
Definitions of the coefficients and quantities in these equations can be seen in the Appendix. The quantities in matrix P do not change, and hence, the inverse \(P^{-1}\) needs only be computed once. Using the initial conditions determined from \({B(0)}=P^{-1}S\), we can use the fact that \(B=P^{-1}R\) to step through time, updating the Fourier coefficients using a fourth-order Runge-Kutta scheme. Thus, the evolution of the flow field and density perturbation can be computed.
The method was tested with different numbers of coefficients, and it was found that for smaller values of flow rate and Reynolds number, e.g. \(Re=100-1000\), a smaller number of coefficients, say \(N=48\), was sufficient to produce converged solutions when compared with say \(N=64\). However, as the Reynolds number increased to 5000, \(N=72\) was used to produce the simulation results. This is not surprising due to the intricate patterns developed in the simulation. Even with this number of coefficients, for the case with \(Re=5000\), there is some breakup of the solutions that may be of numerical origin. Simulations with \(N=72\) took several hours (up to 20 hours) to compute up to time \(t=25\) with \(Re=5000\). The time for simulations with higher numbers of coefficients would be prohibitive with a desktop computer.
3 Results
The problem under consideration is the release of fluid from an elevated source into an ambient fluid of lower density. The force of gravity ensures that the plume which is generated flows downward, hits the base and then spreads like those described in [4]. The spreading takes the form of a gravity current as described in the work of Benjamin [8]. Simulations in [4] and [10] show some of the characteristics of these gravity currents, including the development of the head and breaking at the front of the current. However, this is not the goal of this work and in what follows we consider the initiation and downward flow of the plume itself.
To begin we do a simple test on whether the method is working correctly. Figure 2 shows the early stages of a flow with \(F=0.75\), \(Re=500\), \(h_s=2\), and \(H=6\) at \(t=6\) (the inner rings) and \(t=10\) (the outer rings). The dashed lines are the location of the particles that were initially on the surface of the starting interface in the full simulation, if one tracks them by integrating along the streamlines of the inviscid flow (as if there were no interface) given in (7). The similarity at \(t=6\) indicates that the flow is almost totally independent of both gravity and viscosity for a significant time. Even at \(t=10\), it appears to be the effect of gravity that causes the differences rather than any viscous effects.
An important factor in this problem is the upward movement of the heavier fluid as it is pushed into the surrounding, lighter, fluid. The central point on the interface, \(h_{mid}\), directly above the source, rises up, and then remains steady as the plume grows and travels downward. This is shown in Fig. 3, which shows the height of this point as a function of time for different source heights at \(F=0.5\), \(Re=100\). Figure 4 shows \(h_{mid}\) plotted against \(h_s\) at different values of Froude number. When the source is close to the bottom this elevation is almost constant, but once the source gets more than a height of \(h_s\approx 1.5\) the value starts to rise. This is a clear effect of the proximity of the base directing more of the flow upward.
Fig. 4
The final (quasi-steady) height of the plume \(h_{mid}\) for different values of source height \(h_s\) and Froude number F and Re=100. For small values of \(h_s\), \(h_{mid}\) remains approximately constant for each value of F, but once \(h_s\) gets above around F approx 1.5, the distance from \(h_s\) to \(h_{mid}\) remains approximately constant
×
This result is reinforced in Fig. 5, which shows the final (quasi-steady) distance from the source to the top of the plume, i.e. \(h_{mid}-h_s\) for different values of source height \(h_s\) and Froude number F. At small values of \(h_s\), the separation decreases rapidly with \(h_s\) until it becomes approximately constant for each value of F once \(h_s\) gets above \(h_s \approx 1.5\). This suggests that there is a source height beneath which upflow will stop at a height determined purely by the Froude number and the location of the base, while if the source is above this height the upflow stops a certain distance above the source that is determined by the Froude number, i.e. the dynamical interaction between the upward flow and the ambient, stagnant fluid.
Fig. 5
The height of top of the plume above the source for different values of Froude number F and Re=100. At moderate F, for values above \(h_s \approx 1.5\), the height difference stabilises for each Froude number
Fig. 6
The midpoint height for different values of source height and different values of F and Re, for Re=100 (dashed line) and for Re=1000 (solid line)
×
×
It is of interest to consider the effect of viscosity on this height, \(h_{mid}\) and Fig. 6 shows the value for different values of source height and Froude number, F, for Reynolds numbers, Re=100,1000. It is clear that the Reynolds number has almost no effect on the height and hence that this elevation, \(h_{mid}\) is determined mainly by the Froude number.
Fig. 7
Plume shapes for a small value of \(F=0.01\), \(h_s=4\) and \(Re=1000\). The number of coefficients is \(N=72\). At \(t=25\) the central plume begins to segment and the outer edge of the mushroom begins to break away as a small spiral forms
Fig. 8
Plume shapes for \(F=0.1\), \(h_s=4\) and \(Re=1000\). The number of coefficients is \(N=72\). At \(t=15\) spirals form at the outer edges of the plume. By time \(t=25\), these spirals have grown and a second one is beginning to form. Contours shown are \(D-1= -.04, -0.025, -0.01\), which correspond to the edges and middle of the interface
Fig. 9
Plume shapes for \(F=1\), \(h_s=4\) at different times \(t=15, 20, 25\) of \(Re=1000\), \(N=64\). Contours shown are \(D-1= -.04, -0.025, -0.01\), which correspond to the interface. The outer fluid has value \(D-1=-0.05\) and the inner fluid has \(D-1=0\). The flow expands until it approaches the base, then begins to move outward as a gravity current
×
×
×
Depending on the strength of the flow, the plumes formed fall into several different categories. At very low flow rates, e.g. \(F=0.01\) the plume falls downward and forms a kind of inverted mushroom shape. Figure 7 shows such a flow at \(t=20\) and \(t=25\), with \(F=0.01\) and 72 coefficients in the series, all at \(Re=1000\). The plume is very small and very quickly forms into the inverted mushroom shape. At \(t=25\) the long central plume down to the mushroom begins to separate into small segments, and the outer edges of the mushroom begin to separate as a small spiral forms. The flow from the source is not sufficient to compensate for the downward pull of gravity, causing this separation.
At slightly larger flow rates, e.g. \(F=0.1\), a similar plume forms and progresses downward, but it is much “thicker” and sustains itself much longer, as shown in Fig. 8 longer. At \(t=15\), the mushroom is much stronger, and spirals are forming at the outer edges. At \(t=25\), the initial spirals have grown and a secondary spiral is beginning just below. The central plume has narrowed near the bottom as it accelerates downward.
Figure 9 is a very high flow case with \(F=1\), \(Re=1000\), at different times, \(t=15, 20, 25\). The outward flow is so strong that gravity plays only a minor role until the time when the expanding outflow reaches the base, at which time it spreads horizontally as a gravity current. Interestingly, the spirals still form on the outer edges and these are ultimately absorbed into the head of the current. The region above the source still exhibits the levelling off of the central point, but there is no mushroom shape. Flows such as this, without the mushroom shape, form once the Froude number becomes greater than \(F\approx 0.5\) as \(h_s\) increases.
Fig. 10
Plume shapes for \(F=0.25\), \(h_s=4\) and different values of \(Re=100, 1000\) and 5000 at \(t=20\). Contours shown are \(D-1= -.04, -0.025, -0.01\), which correspond to the interface. The outer fluid has value \(D-1=-0.05\) and the inner fluid has \(D-1=0\). As the viscosity decreases (Re increases), the billowing at the outer edge of the plume increases dramatically. The case \(Re=5000\) has multiple spirals formed around the edges
×
It is clear that the middle values of F potentially generate the largest amount of interfacial mixing, and it is of interest to see how the flows vary with Reynolds number. Figure 10 shows three examples at time \(t=25\) with \(R=100, 1000\) and 5000. At low Reynolds number the flow is quite stable and the interface with the surrounding fluid remains quite “smooth”. The viscosity acts to maintain the smooth surface. At \(Re=1000\) a few bumps appear on the interface and there are some spirals beginning to form on the edge of the mushroom and along the base of the plume. However, at \(Re=5000\) it is clear that some further instabilities have occurred on the interface leading to multiple Kelvin-Helmholtz spirals beginning to form at regions where the vortex sheet between the expanding fluid meets the relatively stagnant outer fluid. The spirals form close to the front of the downward travelling plume, then move up around the interface and then another pair begins to form below. In order to obtain these pictures, \(N=72\) coefficients were required, taking about 12 hours on an i7 Intel chip. The cases of lower Reynolds number only required \(N=48\) coefficients to obtain converged plots. At this time, \(t=25\), the spirals are beginning to lose their integrity in the figure, especially for the case \(Re=5000\). It is likely that a larger value for the number of coefficients and a smaller time step would be required to obtain converged images over longer times.
4 Concluding remarks
This paper implements a spectral method to solve the viscous Boussinesq equations, and considers the unsteady flow of a plume generated by a source located off the bottom of a horizontal duct of fixed height. The fluid emanating from the source is heavier than the ambient fluid into which it flows. This investigation considered the effects of Froude number, Reynolds number and source location.
The numerical method proved very effective and is very efficient due to the pre-calculation of the matrix quantities in the time-stepping. Simulations can be performed in a couple of hours using \(\texttt {Matlab}^{\texttt {TM}}\) or Octave out to times of \(t=30\) with \(N=64\) coefficients on a standard laptop computer. The development of the Kelvin-Helmholtz spirals on the interface and the success for large \(Re=5000\) are a nice indicator of the effectiveness of the simulations.
The early part of the flow follows quite closely the streamlines of the inviscid solution. Next, gravity begins to play a role. The initial rise height of the plume is linear in Froude number except when the source is close to the base, in which case upward flow is a little stronger. The general shape and extent of the plume were shown to depend on the inertial effects (Froude number) with some influence from the lower horizontal boundary. Reynolds number (viscous effects) had an impact on the stability of the interface between the inflowing water and the ambient fluid, with higher Reynolds number flows developing a convoluted surface shape involving spirals and oscillations and thus more mixing. Viscosity therefore had a stabilising effect on the interface between the two fluid regions.
In the context of desalination outfalls, it is of interest to consider the amount of mixing. It is undesirable for a highly saline gravity current to travel outward from the outlets, and so the more mixing that can be induced, the better it is for the seabed environment. The results of this work indicate that this mixing is most pronouced in the range of Froude numbers from \(F=0.1-0.4\), since, for realistic values of Reynolds number (\(Re>1000\)) this flow induces the most mixing before the gravity current forms. Clearly for small flows \(F\approx 0.01\) the saline output would be more diluted, but in general flows from desalination plants are dictated by plant operations. In practice this suggests having a large number of outlets with lower flows is more environmentally friendly than a few high flow outlets which would generate significant hyper-saline gravity currents. It also suggests it might be wise to have the sources situated higher off the seabed to maximise the time available for the plume to mix.
In general, the results of this work show that the most important factor determining the major features of the flow is the Froude number. As the Reynolds number increases, the interface between the central and outer fluids develops increasing amounts of instability resulting in the formation of multiple spirals. These spirals greatly enhance the mixing.
Declarations
Competing interests
The authors declare no competing interests.
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the constant \(\delta _{mn}\) is defined in equation (19). We Fourier analyse this expression by multiplying by \(\sin \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right) \) and integrating over the two-dimensional domain \(-L \le x \le L\), \(0 \le y \le h\), the orthogonality relations then yield
$$\begin{aligned} \begin{aligned} A_{kl}{'}(t)&=-\frac{1}{Re} \delta _{kl}^{4}A_{kl}(t) \\&\quad -\frac{2}{\delta _{kl}^{2}Lh}\int _{0}^{h} \int _{-L}^{L}\left[ u \frac{\partial \omega }{\partial x} + v \frac{\partial \omega }{\partial y}+ \frac{\partial \hat{\rho }}{\partial x} \right] \sin \left( \frac{k \pi x}{L}\right) \sin \left( \frac{l \pi y}{h}\right) \text{ d } x \text{ d } y , \end{aligned} \end{aligned}$$
(28)
for \(k={1,...,M}\) and \(l={1,...,N}\). The second system of differential equations is obtained by considering the density perturbation function \(\hat{\rho }\). However, we observe that singular behaviour near the source requires the extra factor \(G_{x,y}\) in the representation (20) for \(\hat{\rho }\), and as a result, the orthogonality of the trigonometric functions no longer produces a decoupled system. Substituting our expression for \(\hat{\rho }\) into equation (11) gives
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