### Continuum approach

#### Physical model and governing equations

In many situations of interest, compositional gravity currents and turbidity currents are driven by small density differences not exceeding *O*(1*%*). Under such conditions, the Boussinesq approximation can be employed, which treats the density as constant in the momentum equation with the exception of the body force terms. When dealing with turbidity currents, we account for the dispersed particle phase by means of a Eulerian-Eulerian formulation, which means that we employ a continuum equation for the particle concentration field, rather than tracking particles individually in a Lagrangian fashion.

In the following, it will be important to carefully distinguish between dimensional and dimensionless variables. Towards this end, we will employ the tilde symbol to indicate a dimensional variable, whereas variables without the tilde symbol are dimensionless. Under the Boussinesq approximation, the dimensional governing equations for compositional gravity currents driven by salinity and/or temperature gradients can be written as

$$\begin{array}{@{}rcl@{}} \frac {\partial \widetilde{u}_{j}}{\partial \widetilde{x}_{j}} &=& 0, \end{array} $$

(1)

$$\begin{array}{@{}rcl@{}} \frac{\partial \widetilde{u}_{i} }{\partial \widetilde{t}} +\frac{\partial\left(\widetilde{u}_{i}\widetilde{u}_{j}\right)}{\partial \widetilde{x}_{j}} &=& -\frac{1}{\widetilde{\rho}_{1}} \frac{\partial {\widetilde{p}} }{\partial \widetilde{x}_{i}} + \widetilde{\nu}\frac{\partial^{2} \widetilde{u}_{i}}{\partial \widetilde{x}_{j} \partial \widetilde{x}_{j}} +\frac{\widetilde{\rho}\widetilde{g}}{\widetilde{\rho}_{1}}e_{i}^{g}, \end{array} $$

(2)

$$\begin{array}{@{}rcl@{}} \frac{\partial {\widetilde{\rho}} }{\partial \widetilde{t}} +\frac{\partial\left({\widetilde{\rho}}{\widetilde{u}}_{j}\right)}{\partial \widetilde{x}_{j}} &=& \widetilde{\alpha}\frac{\partial^{2} {\widetilde{\rho}}}{\partial \widetilde{x}_{j} \partial \widetilde{x}_{j}}\ . \end{array} $$

(3)

Here, \(\widetilde {u}_{i}\) denotes the velocity vector, \(\widetilde {p}\) the pressure, \(\widetilde {\rho }\) the density, \(\widetilde {g}\) the gravitational acceleration, \(e_{i}^{g}\) the unit vector pointing in the direction of gravity, \(\widetilde {\nu }\) the kinematic viscosity, and \(\widetilde {\alpha }\) the molecular diffusivity of the density field. We nondimensionalize the above Eqs. (1)–(3) by a reference length scale, such as the domain half height \(\widetilde {H}/2\) of a lock-exchange flow (Meiburg et al. 2015; Nasr-Azadani and Meiburg 2014; Necker et al. 2002, 2005), the current density \(\widetilde {\rho }_{1}\), and the buoyancy velocity \(\widetilde {u}_{b}\)

$$ \widetilde{u}_{b} = \sqrt{\widetilde{g'}\;\widetilde{H}/2} \ . $$

(4)

Here, \(\widetilde {g'}\) indicates the reduced gravity

$$ \widetilde{g'}=\widetilde{g} \;\frac{\widetilde{\rho}_{1}\; - \widetilde{\rho}_{2}}{\widetilde{\rho}_{1}} \ . $$

(5)

where \(\widetilde {\rho }_{2}\) represents the ambient density. After nondimensionalization, we obtain

$$\begin{array}{@{}rcl@{}} \frac {\partial u_{j}}{\partial x_{j}} &=& 0, \end{array} $$

(6)

$$\begin{array}{@{}rcl@{}} \frac{\partial u_{i} }{\partial t} +\frac{\partial\left(u_{i}u_{j}\right)}{\partial x_{j}} &=&-\frac{\partial {p} }{\partial x_{i}} + \frac{1}{Re}\frac{\partial^{2} {u}_{i}}{\partial x_{j} \partial x_{j}} +\rho e_{i}^{g}, \end{array} $$

(7)

$$\begin{array}{@{}rcl@{}} \frac{\partial {\rho} }{\partial t} +\frac{\partial\left({\rho}{u}_{j}\right)}{\partial x_{j}} &=&\frac{1}{ReSc}\frac{\partial^{2} {\rho}}{\partial x_{j} \partial x_{j}} \ . \end{array} $$

(8)

Here, the nondimensional pressure *p* and density *ρ* are given by

$$ p = \frac{\widetilde{p}}{\widetilde{\rho}_{1}\widetilde{u}_{b}^{2}} \, \ \ \ \rho = \frac{\widetilde{\rho}-\widetilde{\rho}_{2}}{\widetilde{\rho}_{1}\; - \widetilde{\rho}_{2}} \ . $$

(9)

The nondimensionlization of the governing equations gives rise to two dimensionless parameters in the form of the Reynolds number *Re* and the Schmidt number *Sc*

$$ Re = \frac{\widetilde{u}_{b} \widetilde{H}}{2\widetilde{\nu}} \, \ \ \ Sc=\frac{\widetilde{\nu}}{\widetilde{\alpha}} \ . $$

(10)

While the Reynolds number indicates the ratio of inertial to viscous forces, the Schmidt number represents the ratio of kinematic fluid viscosity to molecular diffusivity of the density field.

When the driving density difference is due to gradients in particle loading, rather than salinity or temperature gradients, the above set of equations no longer provides a full description of the flow. Particles settle within the fluid, so that the scalar concentration field no longer moves with the fluid velocity. In addition, particle-particle interactions can result in such effects as hindered settling (Ham and Homsy 1988), increased effective viscosity, and non-Newtonian dynamics (Guazzelli and Morris 2011), thereby further complicating the picture. However, away from the sediment bed, turbidity currents are often quite dilute, with the volume fraction of the suspended sediment phase being well below *O*(1*%*). Under such conditions, particle-particle interactions can usually be neglected, so that the particle settling velocity remains the key difference (along with erosion) that distinguishes turbidity currents from compositional gravity currents.

Due to the small particle volume fraction of dilute turbidity currents, the volumetric displacement of fluid by the particulate phase can usually be neglected, allowing us to consider the fluid velocity field to be divergence-free. Rather, the particle-fluid interaction occurs primarily through the exchange of momentum, so that it suffices to account for the presence of the particles in the fluid momentum equation. In the following, we assume that the particle diameter \(\widetilde {d}_{p}\) is smaller than the smallest length scale of the flow, such as the Kolmogorov scale in turbulent flow. In addition, we consider only particles whose aerodynamic response time \(\widetilde {t}_{p}\) is significantly smaller than the smallest time scale of the flow \(\widetilde {t}_{f}\), so that the particle Stokes number \(St = \widetilde {t}_{p} / \widetilde {t}_{f} \ll O(1)\) (Raju and Meiburg 1995). Here, the aerodynamic response time is defined as

$$ \widetilde{t}_{p} =\frac{\widetilde{\rho}_{p} \widetilde{d}_{p}^{2}}{18 \widetilde{\mu}} \, $$

(11)

with \(\widetilde {\rho }_{p}\) indicating the particle material density and \(\widetilde {\mu }\) denoting the dynamic viscosity of the fluid. Such particles can then be assumed to move with a velocity \(\widetilde {u}_{p,i}\) that is obtained by superimposing the local fluid velocity \(\widetilde {u}_{i}\) and the particle settling velocity \(\widetilde {u}_{s} e_{i}^{g}\)

$$ \widetilde{u}_{p,i} = \widetilde{u}_{i} + \widetilde{u}_{s} e_{i}^{g} \, $$

(12)

where \(\widetilde {u}_{s}\) follows from balancing the gravitational force with the Stokes drag force

$$ \widetilde{F}_{i} = 3\pi\widetilde{\mu}\widetilde{d}_{p}(\widetilde{u}_{i}-\widetilde{u}_{p,i}) $$

(13)

as

$$ \widetilde{u}_{s} = \frac{\widetilde{d}_{p}^{2}(\widetilde{\rho_{p}}-\widetilde{\rho})\widetilde{g}}{18\widetilde{\mu}} \ . $$

(14)

Note that this implies that the particle velocity field is single-valued and divergence-free, so that monodisperse particles do not, for example, accumulate near stagnation points or get ejected from vortex centers. Hence, we can describe the spatio-temporal evolution of the particle number concentration field \(\widetilde {c}\) in a Eulerian fashion by the transport equation

$$ \frac{\partial {\widetilde{c}} }{\partial \widetilde{t}} +\frac{\partial\left(\widetilde{c}\left(\widetilde{u}_{j}+\widetilde{u}_{s}e_{j}^{g}\right)\right)}{\partial \widetilde{x}_{j}} =\widetilde{\alpha}\frac{\partial^{2} \widetilde{c}}{\partial \widetilde{x}_{j} \partial \widetilde{x}_{j}} \ . $$

(15)

The diffusion term in Eq. (15) represents a model for the decay of concentration gradients due to the hydrodynamic diffusion of particles and/or slight variations in particle size and shape (Davis and Hassen 1988; Ham and Homsy 1988).

The motion of the fluid phase is described by the incompressible continuity equation and the Navier-Stokes equation augmented by the force exerted on the fluid by the particles, which is equal and opposite to the Stokes drag force acting on the particles. In a dimensional form, these equations read

$$\begin{array}{@{}rcl@{}} \frac {\partial \widetilde{u}_{j}}{\partial \widetilde{x}_{j}} &=& 0, \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} \frac{\partial \widetilde{u}_{i} }{\partial \widetilde{t}} +\frac{\partial\left(\widetilde{u}_{i}\widetilde{u}_{j}\right)}{\partial \widetilde{x}_{j}} &=&-\frac{1}{\widetilde{\rho}} \frac{\partial {\widetilde{p}} }{\partial \widetilde{x}_{i}} + \widetilde{\nu}\frac{\partial^{2} \widetilde{u}_{i}}{\partial \widetilde{x}_{j} \partial \widetilde{x}_{j}} +\frac{\widetilde{c}}{\widetilde{\rho}} \widetilde{F}_{i}, \end{array} $$

(17)

As we had done for compositional gravity currents, we use the domain half height \(\widetilde {H}/2\) and buoyancy velocity \(\widetilde {u}_{b}\) for nondimensionalization. The reduced gravity \(\widetilde {g}'\) appearing in the calculation of \(\widetilde {u}_{b}\) can now be calculated as

$$ \widetilde{g}' = \frac{\pi(\widetilde{\rho}_{p}-\widetilde{\rho})\widetilde{c}_{0}\widetilde{d}_{p}^{3}} {6\widetilde{\rho}} \ \widetilde{g} \, $$

(18)

where \(\widetilde {c}_{0}\) indicates a reference number concentration of particles in the suspension. After nondimensionalization, we obtain

$$\begin{array}{@{}rcl@{}} \frac {\partial u_{j}}{\partial x_{j}} &\,=\,& 0 \, \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} \frac{\partial u_{i} }{\partial t} +\frac{\partial\left(u_{i}u_{j}\right)}{\partial x_{j}} &\,=\,& \,-\,\frac{\partial {p} }{\partial x_{i}} \,+\, \frac{1}{Re}\frac{\partial^{2} {u}_{i}}{\partial x_{j} \partial x_{j}} \,+\,c e_{i}^{g}, \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} \frac{\partial {c} }{\partial t} +\frac{\partial\left({c}\left({u}_{j}+u_{s}e_{j}^{g}\right)\right)}{\partial x_{j}} &\,=\,& \frac{1}{ReSc}\frac{\partial^{2} {c}}{\partial x_{j} \partial x_{j}} \ . \end{array} $$

(21)

For polydisperse suspensions containing particles of different sizes, the above approach can easily be extended by solving one concentration equation for each particle size and corresponding settling velocity (Nasr-Azadani and Meiburg 2014). Note that the set of governing equations for turbidity currents (19)–(21) differs from the corresponding set for compositional gravity currents (6)–(8) only by the additional settling velocity term in the concentration equation. In the following, we employ Eqs. (19)–(21) for both types of currents, with the tacit assumption that the settling velocity vanishes for compositional gravity currents.

Direct numerical simulations (DNS) represent the most accurate computational approach for studying gravity currents. In DNS, all scales of motion, from the integral scales dictated by the boundary conditions down to the dissipative Kolmogorov scale determined by viscosity, are explicitly resolved. However, for the case of turbidity currents, when the particle diameter is smaller than the Kolmogorov scale, the fluid motion around each particle is usually not resolved, due to the prohibitive computational cost. Nevertheless, the drag law accurately captures the exchange of momentum between the two phases at scales smaller than the Kolmogorov scale, so that the approach described above is still referred to as DNS.

Consistent with the above arguments, the grid spacing required for DNS is of the order of the Kolmogorov scale, while the time step needs to be of the same order as the time scales of the smallest eddies. Due to the large disparity between integral and Kolmogorov scales at high Reynolds numbers, the computational cost of DNS scales as *Re*
^{3}, so that the DNS approach is effectively limited to laboratory scale Reynolds numbers. The first DNS simulations of gravity currents in a lock-exchange configuration were reported by Härtel et al. (2000) for *Re* = 1225. Necker et al. (2002) extended this work to turbidity currents at *Re* = 2240. More recent simulations of lock-exchange gravity currents by Cantero et al. (2008) were able to reach *Re* = 15,000, which corresponds to a laboratory scale current of height 0.5 m with a front velocity of 3 cm/s.

DNS simulations can provide detailed information on the structure and statistics of the flow, on the various components of its energy budget, on the mixing behavior, and many additional aspects. As a case in point, the simulations by Härtel et al. (2000) explored the detailed flow topology near the current front and demonstrated that the stagnation point is located a significant distance behind the nose of the current. DNS results are furthermore very useful for testing the accuracy and identifying any deficiencies in larger-scale LES and RANS models (Yeh et al. 2013). Thus, while they are currently limited to laboratory scale currents, DNS simulations represent an excellent research tool for exploring the detailed physics of moderate Reynolds number gravity currents and for constructing larger-scale models for higher Reynolds number applications.