High-resolution simulations of turbidity currents
- Edward Biegert^{1},
- Bernhard Vowinckel^{1},
- Raphael Ouillon^{1} and
- Eckart Meiburg^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s40645-017-0147-4
© The Author(s) 2017
Received: 24 June 2017
Accepted: 9 October 2017
Published: 16 November 2017
Abstract
Keywords
Introduction
Turbidity currents are particle-laden flows in the ocean that are driven by gravity (Meiburg and Kneller 2010). Particle concentrations are usually sufficiently low far away from the sediment bed so that particle-particle interactions play a small or negligible role throughout most of the body of the current. In this region, the Boussinesq approximation of the Navier-Stokes equations, in conjunction with a continuum formulation for the sediment concentration, is well-suited to capture the dynamics of the flow. However, near the sediment bed particle concentrations can be very high, which can potentially result in complex non-Newtonian behavior, hindered settling, and other effects. Here, we describe the above two different simulation approaches, along with representative results, which open up a path towards multiscale flow simulations via the μ(I) rheology (Cassar et al. 2005; Boyer et al. 2011; Aussillous et al. 2013).
Methods
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.
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.
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).
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.
Results and discussion
Continuum approach results
where \(\tilde {\rho }_{0}\) is a reference density.
A finite difference code is used to solve the equations, and the MPI library is used for parallelization. A third-order Runge-Kutta scheme with a three substep method is used to discretize the equations in time. The wall-normal viscous and diffusive terms are solved implicitly while the convective terms and the remaining viscous and diffusive terms are treated explicitly. To impose incompressibility, a projection method is used (Spalart et al. 1991) and a direct solver is used for the resulting Poisson equation. Slip-wall boundary conditions are used at the top and right walls and z-periodic boundary conditions are used at the lateral walls. The domain is assumed to be sufficiently long to neglect boundary effects in x, and the width of the domain is chosen so that the periodic boundary condition does not impact the flow development. Finally, an immersed boundary method is used to impose the no-slip condition on the slope (Nasr-Azadani and Meiburg 2011).
The direct impact of stratification is seen at later times (t≈15) when the current intrudes into the ambient, i.e. separates from the surface of the slope. The effects of stratification on intrusion depth are key in understanding the evolution of the suspended mass, deposition profiles and energy budgets of turbidity currents. Intrusion only occurs when the density of the current reaches the density of the ambient.
Comparison of numerically and experimentally measured front velocity
Exp 1 | Exp 2 | Exp 3 | |
---|---|---|---|
Re | 16,850 | 15,000 | 35,000 |
N | 3.66 | 2.39 | 2.77 |
v _{ s } | 0.001 | 0.0046 | 0.00046 |
(U _{ e } − U _{ s })/U _{ e } | 14% | − 11% | 16% |
The depth at which the current intrudes into the ambient also reveals a good agreement between the two approaches and validates the ability of numerical simulations to reproduce the dynamical features of turbidity currents moving into a stratified ambient. While it is extremely challenging to experimentally measure the velocity, particle concentration, and salinity fields of such 3D turbulent flows, direct numerical simulations give access to an entirely new set of data and opens the door to more accurate prediction tools and a deeper understanding of the underlying physics of gravity and turbidity currents in realistic environments at the scale of the lab.
Methods
Grain-resolving approach
Physical model and governing equations
When the concentration of particles grows large, particle-particle interactions become important and the aforementioned continuum approach is no longer applicable. For such cases, we have to account for the rheology of dense suspensions. A key element of progress with regard to the rheology of dense suspensions over the last decade has been the development of the so-called μ(I) approach, cf. (Guazzelli and Morris 2011; Boyer et al. 2011). Grain-resolving simulations of the type to be discussed in the following are expected to provide a tool for further investigating the validity of the assumptions underlying the derivation of the μ(I) rheology. One way to approach the simulation of dense suspensions is to fully resolve the particles interacting with the fluid by tracking each individual particle, evaluating the fluid no-slip condition at the particle surface and accounting for all the forces acting on the particle. Such simulations typically use grid resolutions of 10–25 grid cells per particle diameter to resolve the flow and are thus limited in scope to domains whose dimensions measure only tens to hundreds of diameters in length (Vowinckel et al. 2017; Kidanemariam and Uhlmann 2017). Thus, if simulating sand grains 100 μm in diameter, the domain dimensions would range in length from 1 mm to 1 cm. However, the idea is to use grain-resolved simulations to develop better models of sediment transport to be used in larger-scale simulations.
where ρ _{ f } is the fluid density and f _{ IBM } is the IBM force, which acts as a source term to enforce the no-slip condition at the particle surfaces. This force effectively couples the particle and fluid momentum equations. Though there are many ways to carry out this coupling, the method we employ for the particles uses regularized Dirac delta functions, which interpolate fluid velocities onto the particle surface and spread f _{ IBM } onto the fluid (Roma et al. 1999). Note that this implementation is different from that used to create the sloped lower wall in the turbidity current simulations of the previous section.
Here, m _{ p } is the particle mass, Γ _{ p } the fluid-particle interface, τ the hydrodynamic stress tensor, ρ _{ p } the particle density, V _{ p } the particle volume, g the gravitational acceleration, \(I_{p}~=~8\pi \rho _{p} R_{p}^{5}/15\) the moment of inertia, and R _{ p } the particle radius. Furthermore, the vector n is the outward-pointing normal on the interface Γ _{ p }, r = x − x _{ p } is the position vector of the surface point with respect to the center of mass x _{ p } of a particle, F _{ l } and T _{ l } are the force and torque due to lubrication forces, and F _{ c } and T _{ c } are the force and torque due to particle collisions. We evaluate the IBM force f _{ IBM } as well as the hydrodynamic force, F _{ h }, and torque, T _{ h }, using the approach of Kempe and Fröhlich (2012), fully resolving the hydrodynamic effects of the fluid on the particles as well as the particles on the fluid. The lubrication force, F _{ l }, and contact force, F _{ c }, model close-range particle-particle interactions. With the exception of the tangential lubrication force, the methods used to evaluate these forces are described and validated in detail by Biegert et al. (2017), but here, we present them briefly.
where ζ _{ t } is the tangential displacement vector representing accumulated slip between the two surfaces, μ is the coefficient of friction between the surfaces, and t is the unit normal vector in the tangential direction. The Coulomb friction criterion, represented by ||μ F _{ n }||, allows the two surfaces to slip past one another when large stresses are present. Similar to the normal coefficients, the tangential coefficients of stiffness and damping, k _{ t } and d _{ t }, are also adaptively calibrated.
Results and discussion
Grain-resolving results
Pressure-driven flow over dense sediment
Simulation parameters of the pressure-driven flow scenario, where \(u_{\tau }=\sqrt {\tau _{w}/\rho _{f}}\) is the friction velocity at the fluid/particle interface, U _{ b } is the bulk (average) velocity of the fluid, τ _{ w } is the shear stress at the fluid/particle interface, ν _{ f } the kinematic viscosity, L _{ x }, L _{ y }, and L _{ z } are the spatial extents of the computational domain in the Cartesian space, and h is the grid cell size
Re = U _{ b } L _{ y }/ν _{ f } | 9.9 |
D ^{+} = u _{ τ } D _{ p }/ν _{ f } | 0.39 |
Sh = τ _{ w }/[(ρ _{ p }−ρ _{ f })gD] | 0.97 |
ρ _{ p }/ρ _{ f } | 2.1 |
L _{ x } × L _{ y } × L _{ z } | 11.26D _{ p } × 22.52D _{ p } × 11.26D _{ p } |
h _{ f }/D _{ p } | 8.7 |
D _{ p }/h | 22.7 |
Some qualitative inferences can already be drawn from Fig. 4, where we have nondimensionalized velocities by the bulk fluid velocity \(U_{b}~=~\frac {1}{L_{y}}\int _{0}^{L_{y}} u \, \mathrm {d}y\). In the clear-water layer, a parabolic profile obeying the analytical solution of the classical Poiseuille flow can be observed. The lower end of this parabolic region, however, is not a no-slip wall, but a moving granular bed, which causes the symmetry axis of the flow profile to shift from h _{ f }/2 to a lower position. Inside the granular bed, a linear shear flow profile develops and since all particles are moving, this profile continues all the way to the bottom wall of the domain. This interesting behavior and the wealth of data obtained from the grain-resolving simulations opens up a wide range of analytical tools in terms of statistical description as well as physical modeling, which will be our focus in the future.
Shearing of dense suspensions
Physical and numerical simulation parameters for simulations of shear flows with dense suspensions
e _{dry} | μ _{ k } | μ _{ s } | ν | ζ _{min} | ρ _{ p }/ρ _{ f } | H/D _{ p } | D _{ p }/h | Re |
---|---|---|---|---|---|---|---|---|
0.97 | 0.15 | 0.8 | 0.22 | 3·10^{−3} R _{ p } | 1.011 | 10 | 25.6 | 10 |
Simulation scenarios
Scenario | L _{ x } × L _{ y } × L _{ z } | Re | ϕ _{ v } | \(t_{s} \dot {I}\) | \(t_{a} \dot {I}\) |
---|---|---|---|---|---|
Re10p42 | 2H × 1H × 1H | 10 | 0.42 | 85 | 35 |
Re10p54 | 2H × 1H × 1H | 10 | 0.54 | 150 | 30 |
Re40p54 | 1H × 2H × 1H | 40 | 0.54 | 20 | 20 |
The present study of a Couette-type flow supplements our simulations of pressure-driven flow described in the previous section to fully understand the rheologic behavior of dense suspensions of particles with different inertia in flows with different momentum supply.
Internal waves propagating over fully resolved sediment beds
We can track the front of the current using the location in the horizontal profile where the interface between light and heavy fluid starts to increase in height. Alternatively, we can track the front using the location where particles start to experience an enhanced lift force. A comparison of these two methods is shown in Fig. 9 c for the first few time units simulated. It can be seen that, indeed, during the initial stage of the simulation, where we can see a steep front of the internal wave, the force signal propagates quicker through the sediment bed than the actual propagation speed of the current would suggest. This effect, however, levels off over time as the wave continues to travel over the rough bed, constantly losing energy due to viscous dissipation.
Conclusions
The modeling of dilute, non-eroding turbidity currents has reached a mature level, as evidenced by the fact that high-resolution simulations have been able to reproduce many of the observations made in laboratory experiments (e.g., Nasr-Azadani et al. 2013). We are now able to account for some topographical complexity via the immersed boundary method. Some of the remaining challenges concern the extension to the very large Reynolds number values of field-scale flows and the frequent interaction with ambient phenomena in the ocean such as internal waves and tides, as well as the accurate modeling of erosion and resuspension in such high Reynolds number flows. However, a similar level of maturity has not yet been achieved with regard to the modeling of highly concentrated turbidity currents with significant erosion, resuspension, and bedload transport. Especially the dynamics of the near-bed region of such high-concentration currents in the form of dense suspensions is still poorly understood, as it is governed by intense particle-fluid and particle-particle interactions that give rise to strongly non-Newtonian dynamics and to mass and momentum exchanges between the current and the sediment bed. As a result, insight into the erosional and depositional behavior of such currents and the coupling between the motion of the current above the sediment bed and the fluid flow inside the bed is just beginning to emerge. Key progress has been accomplished with regard to understanding the rheology of dense suspensions over the last decade, through the development of the so-called μ(I) approach, cf. (Guazzelli and Morris 2011; Boyer et al. 2011). Grain-resolving simulations based on the approach outlined here will provide a tool for further investigating the validity of the assumptions underlying the derivation of the μ(I) rheology. The computational approach outlined and tested in the present paper holds great promise, as it is able to capture the grain-resolved dynamics of thick, mobile sediment beds and their coupled dynamics with the flow above. Simulations on this basis provide the opportunity to understand erosion and dense suspension rheology from a fundamental perspective, which can lead to better models for use at larger scales. This multiscale approach would thus further enrich our understanding of turbidity currents.
Declarations
Acknowledgements
Computational resources for this work were provided by the Extreme Science and Engineering Discovery Environment (XSEDE), supported by the National Science Foundation, USA, Grant No. TG-CTS150053.
Funding
This research is supported in part by the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under the contract no. DE-AC05-06OR23100. BV gratefully acknowledges the Feodor-Lynen scholarship provided by the Alexander von Humboldt Foundation, Germany.
Authors’ contributions
EB and BV developed the grain-resolving simulation approach. RO carried out the continuum formulation simulations. EM proposed the topic and conceived the study. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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