Available online at ScienceDirect. Procedia IUTAM 15 (2015 ) 26 33

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1 Available online at ScienceDirect Procedia IUTAM 15 (2015 ) IUTAM Symposium on Multiphase flows with phase change: challenges and opportunities, Hyderabad, India (December 08 December 11, 2014) Sediment wave formation caused by erosional and depositional turbidity currents: A numerical investigation Gary Hoffmann a,b, Mohamad M. Nasr-Azadani b, Eckart Meiburg b, a Department of Hydrographic Lidar, Fugro Pelagos, Inc., 3574 Ruffin Rd., San Diego, CA 92123, USA b Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA Abstract Results are presented from two-dimensional direct numerical simulations of sediment wave formation via a succession of erosional and depositional turbidity currents. For currents propagating down a slope, we observe the formation of upstream migrating periodic sediment waves. The strength of the wave formation process is discussed as function of the governing dimensionless parameters. c The Authors. Published by by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of Indian Institute of Technology, Hyderabad. Peer-review under responsibility of Indian Institute of Technology, Hyderabad. Keywords: turbidity currents, sediment waves, numerical simulation 1. Introduction Turbidity currents represent a special class of gravity currents in lakes or the ocean in which the density difference between the current and the ambient fluid is caused by suspended particles ( 10 ). As a result of their depositional and/or erosional nature, various topographical features can form in the submarine environment ( 5, 4 ), such as meandering channels, levees, gullies, and sediment waves ( 19, 11, 12, 18 ). In the present study our focus will be on sediment waves, which are large, dune-like structures that form over time as a result of differential deposition/erosion from successive turbidity currents. Numerous authors have studied turbidity-current generated sediment waves in the field ( 17, 7, 8, 6 ). In spite of recent linear stability work ( 9 ), detailed information about their formation and maintenance remains largely unavailable, for example on the influence of the flow parameters on the wavelength of the waves. While most previous modeling efforts have considered depth-averaged approaches for modeling turbidity currents ( 7 ), more recently depth-resolved numerical simulations of turbidity currents have become feasible as a result of increased computational power ( 1 and 16 ). Specifically, 16 simulated successive turbidity currents that resulted in the development of sediment waves on the lee side of an initial obstacle. Here we will explore the possibility of sediment wave generation without the presence of such an obstacle. Corresponding author. Tel.: ; fax: address: meiburg@engineering.ucsb.edu The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of Indian Institute of Technology, Hyderabad. doi: /j.piutam

2 Gary Hoffmann et al. / Procedia IUTAM 15 ( 2015 ) Fig. 1. Schematic of the lock-exchange configuration employed to produce turbidity currents flowing down a slope. The heavier suspension of density ˆρ 1 is initially at rest and separated from the lighter ambient fluid of density ˆρ 0. Upon release, a bottom hugging turbidity current forms and travels down the slope. The solid boundary (Γ) is assumed to be erodible. Hence, as the current propagates along the bottom surface, particles can be resuspended back into the fluid due to shear stress acting on the erodible bed. Towards this end, we will simulate two-dimensional turbidity currents using direct numerical simulations (DNS), in order to capture any important depth-varying flow structures that may contribute to wave formation on an initially flat surface. The simulations to be discussed in the following will consider successive turbidity flows produced in a lock-exchange configuration flowing down an inclined slope (see figure 1). 2. Model description and governing equations Repeated occurance of turbidity currents in submarine environments has a significant role in the formation of various underwater shapes. In the present investigation, we study the formation of sediment waves, as their formation caused by turbidity currents is of particular interest in geological and environmental studies. Figure 1 shows the schematic of the current problem setup. At time zero, the suspension is at rest and includes particle of uniform size. Then, the membrane is removed and a turbidity current forms due to density difference caused by the suspended particulate matter. As the current evolves and travels down the inclined surface, we allow for particles to settle on the bottom topography. Turbidity currents in natural situations can give rise to both bedload transport and suspended loads. We focus on the influence of suspended sediment loads, based on the model proposed by 2 to account for erosion from the sediment bed into the flow. We conduct DNS simulations by means of our in-house code TURBINS ( 14 ). TURBINS employs a finite-difference discretization of the Navier-Stokes equations, along with a fractional projection method and third order TVD-RK3 time integration. Numerical details, validation data, and comparisons with experiments are provided in 15, 14,and 13. We present a brief summary of the governing equations in the following. Evolution of the fluid motion is governed by the incompressible Navier-Stokes equations in the Boussinesq approximation u=0, (1) u t + u u= p+ 1 Re 2 u+ce g, (2) where Re and e g denote the Reynolds number and the unit vector acting in the direction of gravity, respectively. The Reynolds number in equation (2) is given as Re= ûbĥ/2 ˆν, (3)

3 28 Gary Hoffmann et al. / Procedia IUTAM 15 ( 2015 ) where û b, Ĥ and ˆν indicate, respectively, the buoyancy velocity, lock height, and the kinematic viscosity of the fluid. In the above set of equations, we employ the buoyancy velocity Ĥ (ˆρ p ˆρ 0 )C 0 û b = ĝ, (4) 2 ˆρ 0 and half of the lock height Ĥ/2 (see figure 1) to make flow quantities dimensionless. In equation (4), ˆρ p,ˆρ 0,andC 0 denote particle material density, density of the ambient fluid, and the initial volume fraction of particles in the lock, respectively. We remark that aˆ-sign refers to a dimensional quantity. Assuming a dilute particulate phase, we employ a continuum description of the sediment concentration field c(x, t) and evolve it in an Eulerian manner by c t + (u+u se g ) c= 1 ReS c 2 c. Here, Sc represents the Schmidt number associated with the diffusion of the sediment concentration field c. We generally employ Sc= 1 and refer to 3 for a discussion of the influence of the Schmidt number on the dynamics of gravity currents. As for the dimensionless settling velocity u s, for small particle sizes and in the low Reynolds number regime, we employ the Stokes settling velocity for spherical particles Erosion and deposition The detailed grain-scale dynamics of sediment erosion are very complex and not yet well understood. To capture the essential aspects of this phenomenon, we employ the simplified model of 2. While this empirical model does not resolve the turbulent motions near the bottom boundary, it does provide the erosional inflow of particles into the current as a diffusive flux ( 1 ), which may capture the true dynamics more closely thanalternate approaches, such as the somewhat arbitrary distribution of the eroded particles over several computational grid spacings adjacent to the bottom boundary employed by other authors. Everywhere along the bottom surfaceγ(see figure 1), we enforce a Neumann boundary condition with a non-zero flux, i.e. 1 c ScRe η = u se s. Here,ηand E s denote the wall-normal distance and the erosion flux, respectively. The resuspension flux E s is defined as E s = 1 C 0 az 5 1+ a 0.3 Z5, with a= and Z as the erosion parameter. 2 set the maximum value of E s = 0.3/C 0, thereby providing a saturation mechanism for particle erosion from the bed. Following the erosion model proposed by 2, Z in equation (7) is given as Z= u u s Re 1.23 p if Re p 2.36, u u s Re 0.6 p if Re p > In equation (8), u and Re p denote the dimensionless shear velocity computed at the bottom wall u 1 u t = Re η, η=0 (5) (6) (7) (8) (9)

4 Gary Hoffmann et al. / Procedia IUTAM 15 ( 2015 ) and the particle Reynolds number Re p = ˆd p ĝ ˆd p (ˆρ p ˆρ 0 )/ˆρ 0. ˆν In the above equations, u t and ˆd p represent, respectively, the tangential velocity at the bottom boundary and the particle diameter. The computational setup employs a channel of size L x L y = The Cartesian grid in the x-andy-directions is uniformly spaced with x = and y = We enforce no-slip conditions everywhere on the boundaries except for the right wall (at x=l x ) where a non-reflective convective outflow condition of the form q t + Ū q x = 0, is employed. Here, Ū represents the maximum u-velocity value in the domain, while q can refer to any transported flow variable. For the density field c, we impose no-flux conditions at the top wall. The lock has dimesions of L s H=5 2 (see figure 1). The suspension in the lock region contains spherical particles of material density ˆρ p = 3, 200 kg/m 3 and diameter ˆd p = m (silt). We assume C 0 = 0.01 which results in a dimensionless settling velocity equal to u s = Water with density ˆρ 0 = 1, 000 kg/m 3 and kinematic viscosity equal to ˆν=10 6 m 2 /s is assumed for the ambient fluid. Thus, the particle Reynolds number has a value of Re p = 0.58, cf. equation (10). (10) (11) 3. Results and discussions In order to examine erosion on a macroscale, we consider the formation of sediment waves under the influence of ten succuessive turbidity currents Setup For these simulations, the characteristic length scale was chosen to be half the lock height. The lock was set atop an initially flat ramp, and filled with 1% concentration monodisperse sediment. Thanks to the immersed boundary method employed in TURBINS, shear stress at the fluid-sediment boundary could be accurately modeled, and even sub-grid scale changes in the bottom geometry resulted in changes to the behavior of the flow. However, because TURBINS does not dynamically update the bottom geometry during the simulation, we could only update it in between successive flows. Additionally, in order to ensure that the total suspension mass released in each each flow was the same, we cleared the lock of any sediment that may have deposited during the previous simulation. For each combination of input parameters (settling velocity, flow height, Reynolds number, and slope) we modeled 10 successive flows in hope of capturing the characteristic upslope migration that has been observed in such features. In the simulations, bottom shear stress (and thus entrainment, which is modeled as being proportional to shear stress to the 5th power) is high at the flow front and drops off rapidly. However, shear stress then increases again several times behind the flow front. This behavior appears to be related to the formation of Kelvin-Helmholtz instabilities at the upper interface of the flow (see figure 2). Our baseline simulations were performed with Re=2, 000, u s = 0.01, and slope=0.04 (see figure 3). In this series of flows, several upslope migrating waveforms developed on the initially flat ramp. Two locations were significantly net erosive: the top of the ramp and about halfway down the ramp. We now compare this baseline case against several others, in each of which one parameter was changed while keeping the others constant Effects of Re with constant u s and slope Increasing the Reynolds number from 2,000 to 3,000 greatly dampened the development of sediment waves on the ramp (see figure 4). We infer that the main reason for this is that an increased Reynolds number implies a larger effective characteristic length scale (if viscosity, gravity etc. are kept constant), and thus a larger effective particle diameter. As the Reynolds number increases, with all other parameters remaining the same, erosion decreases.

5 30 Gary Hoffmann et al. / Procedia IUTAM 15 ( 2015 ) Fig. 2. Temporal evolution of our baseline flow with Re=2, 000, u s = 0.01, and slope=0.04. Dark and white correspond to particle concetration of c=1andc=0. Adjacent to the bottom surface on the ramp, concentration can exceed the value of one due to resuspenion of particles. Fig. 3. Deposit profiles of 10 successive flows with Re=2, 000, u s = 0.01, and slope=0.04.

6 Gary Hoffmann et al. / Procedia IUTAM 15 ( 2015 ) Fig. 4. Deposit profiles of 10 successive flows with Re=3, 000, u s = 0.01, and slope=0.04. Fig. 5. Deposit profiles of 10 successive flows with Re=2, 000, u s = 0.02, and slope= Effects of u s with constant Re and slope Similarly, increasing the settling velocity of the sediment has a strong dampening effect on the development of sediment waves on the ramp (see figure 5). In this case, we doubled the settling velocity from u s = 0.01 to u s = 0.02, which also represents an increase in particle diameter and, thus, a decrease in the ability of the flow to entrain sediment into the water column. Notably, both these simulations and the simulations at Re = 3, 000 resulted in the development of Kelvin-Helmholtz instabilities at the upper interface of the flows Effects of slope with constant Re and u s Lastly, halving the slope from 0.04 to 0.02 has a remarkably small effect on the development of sediment waves on the ramp (see figure 6). Several sets of upslope-migrating waveforms still develop. The primary difference between this set of simulations and our baseline case is that here the flows are no longer net-erosive anywhere on the ramp. So the decrease in potential energy and gravitational acceleration of the sediment-laden fluid does result in a small decrease in erosive potential, but not enough to significantly dampen the development of waveforms. Overall, it appears that a better understanding of erosion on the microscale is critically important to any real understanding of

7 32 Gary Hoffmann et al. / Procedia IUTAM 15 ( 2015 ) Fig. 6. Deposit profiles of 10 successive flows with Re=2, 000, u s = 0.01, and slope=0.02. erosion on the macroscale, and thus to an understanding of sedimentary phenomena that rely heavily upon erosion, such as turbidity-current-generated sediment waves. Acknowledgment The authors gratefully acknowledge support under NSF grant OCE References 1. F. Blanchette, M. Strauss, E. Meiburg, B. Kneller, and M. E. Glinsky. High-resolution numerical simulations of resuspending gravity currents: Conditions for self-sustainment. Journal of Geophysical Research, 110(C12):C120 C122, M. Garcia and G. Parker. Experiments on the entrainment of sediment into suspension by a dense bottom current. Journal of Geophysical Research, 98(3): , C. Härtel, F. Carlsson, and M. Thunblom. Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability. Journal of Fluid Mechanics, 418: , B. Kneller and C. Buckee. The structure and fluid mechanics of turbidity currents: a review of some recent studies and their geological implications. Sedimentology, 47(s1):62 94, B. Kneller and W. D. McCaffrey. Depositional effects of flow nonuniformity and stratification within turbidity currents approaching a bounding slope; deflection, reflection, and facies variation. Journal of Sedimentary Research, 69(5): , S. Kostic. Modeling of submarine cyclic steps: Controls on their formation, migration, and architecture. Geosphere, 7(2): , Y. Kubo and T. Nakajima. Laboratory experiments and numerical simulation of sediment-wave formation by turbidity currents. Marine Geology, 192(1): , H. J. Lee, J. P. Syvitski, G. Parker, D. Orange, J. Locat, E. Hutton, and J. Imran. Distinguishing sediment waves from slope failure deposits: field examples, including the humboldt slide, and modelling results. Marine Geology, 192(1):79 104, L. Lesshafft, B. Hall, E. Meiburg, and B. Kneller. Deep-water sediment wave formation: linear stability analysis of coupled flow/bed interaction. Journal of Fluid Mechanics, 680(1): , E. Meiburg and B. Kneller. Turbidity currents and their deposits. Annual Review of Fluid Mechanics, 42(1): , S. Migeon, B. Savoye, E. Zanella, T. Mulder, J. C. Faugères, and O. Weber. Detailed seismic-reflection and sedimentary study of turbidite sediment waves on the Var Sedimentary Ridge (SE France): significance for sediment transport and deposition and for the mechanisms of sediment-wave construction. Marine and Petroleum Geology, 18(2): , T. Nakajima and M. Satoh. The formation of large mudwaves by turbidity currents on the levees of the Toyama deep-sea channel, Japan Sea. Sedimentology, 48(2): , M. M. Nasr-Azadani, B. Hall, and E. Meiburg. Polydisperse turbidity currents propagating over complex topography: Comparison of experimental and depth-resolved simulation results. Computers&Geosciences, 53(0): , M. M. Nasr-Azadani and E. Meiburg. TURBINS: An immersed boundary, Navier-Stokes code for the simulation of gravity and turbidity currents interacting with complex topographies. Computers&Fluids, 45(1):14 28, F. Necker, C. Härtel, L. Kleiser, and E. Meiburg. High-resolution simulations of particle-driven gravity currents. International Journal of Multiphase Flow, 28(2): , M. Strauss and M. E. Glinsky. Turbidity current flow over an erodible obstacle and phases of sediment wave generation. Journal of Geophysical Research, 117(C6):C06007, 2012.

8 Gary Hoffmann et al. / Procedia IUTAM 15 ( 2015 ) R. B. Wynn, D. G. Masson, D. A. V. Stow, and P. P. E. Weaver. Turbidity current sediment waves on the submarine slopes of the western Canary Islands. Marine Geology, 163(1-4): , R. B. Wynn and D. A. V. Stow. Classification and characterisation of deep-water sediment waves. Marine Geology, 192(1-3):7 22, R. B. Wynn, G. Weaver, P. P. E. Ercilla, D. A. V. Stow, and D. G. Masson. Sedimentary processes in the Selvage sediment-wave field, NE Atlantic: new insights into the formation of sediment waves by turbidity currents. Sedimentology, 47(6): , 2000.