Eckart Meiburg Modeling Gravity and Turbidity …...from turbidity current deposits plays an...

Eckart Meiburg1

Department of Mechanical Engineering,

University of California,

Santa Barbara, CA 93106

e-mail: [email protected]

Senthil RadhakrishnanDepartment of Mechanical Engineering,




Mohamad Nasr-AzadaniDepartment of Mechanical Engineering,




Modeling Gravity and TurbidityCurrents: ComputationalApproaches and ChallengesIn this review article, we discuss recent progress with regard to modeling gravity-driven,high Reynolds number currents, with the emphasis on depth-resolving, high-resolutionsimulations. The initial sections describe new developments in the conceptual modelingof such currents for the purpose of identifying the Froude number–current height rela-tionship, in the spirit of the pioneering work by von K�arm�an and Benjamin. A brief intro-duction to depth-averaged approaches follows, including box models and shallow waterequations. Subsequently, we provide a detailed review of depth-resolving modeling strat-egies, including direct numerical simulations (DNS), large-eddy simulations (LES), andReynolds-averaged Navier–Stokes (RANS) simulations. The strengths and challengesassociated with these respective approaches are discussed by highlighting representativecomputational results obtained in recent years. [DOI: 10.1115/1.4031040]

1 Introduction

Gravity currents represent a ubiquitous phenomenon in natureand technology. They constitute primarily horizontal flows thatare driven by hydrostatic pressure gradients due to variations intemperature or chemical composition. Examples of atmosphericgravity currents include sea breezes and thunderstorm outflows,while the Mediterranean and Red Sea outflows represent impor-tant oceanic gravity currents. Within the realm of technical appli-cations, gravity currents may be encountered under a large varietyof circumstances, including the heating and cooling of buildings,during tunnel fires, within water treatment facilities, as oil slickson the ocean’s surface, or during CO2 sequestration in depletedoil reservoirs. A highly accessible introduction to the beautifulfield of gravity current research can be found in the book by Simp-son [1], while a more mathematically rigorous treatment of thetopic is provided by Ungarish [2]. For readers interested specifi-cally in environmental gravity-driven flows, the individual chap-ters in Ref. [3] summarize the state of the art and are well suitedas entry points for carrying out research in this field.

Frequent gravity currents are driven by pressure gradientsresulting from spatial variations in particle loading, such as in de-sert sand storms (“haboobs”) and powder snow avalanches [4].Turbidity currents—derived from “turbid,” meaning “muddy”—represent an important class of such particle-driven flows [5].They can be encountered in lakes as well as the ocean, where theyare driven by the density difference between turbid water contain-ing sand and/or clay, and clear ambient water. In freshwater reser-voirs, they contribute to the loss of storage capacity over time,and in the world’s oceans they represent a key mechanism fortransporting sediment from the continental shelves to the deepsea. Large turbidity currents can last for hours or even days, andthey can propagate over vast distances in excess of O (1000 km),such as along the North Atlantic Mid-Ocean Channel [6]. Theirinteraction with the seafloor via erosion and deposition is respon-sible for the formation of large-scale features such as submarinesediment waves, dunes, and canyons. Over geological time scales,the deposits from turbidity currents (turbidites) can reach enor-mous scales of up to O (106 km3), such as the Bengal Fan [7].Under certain ambient conditions, the organic matter contained in

the sediment may form hydrocarbons, so that sedimentary rockfrom turbidity current deposits plays an important role in oil andgas exploration [8]. From an engineering point of view, turbiditycurrents pose a significant hazard to submarine oil pipelines, wellheads, and telecommunication cables.

The gravity currents can form under such a wide variety of con-ditions and render them a particularly fascinating research topic.They can be associated with opposite ends of the Reynolds num-ber spectrum (magma flows versus atmospheric currents), so thatthey are governed by different balances between inertial, viscous,and gravitational forces. They can be nonconservative in that theirexcess density varies with time (eroding or depositing turbiditycurrents), they can be Boussinesq or non-Boussinesq in nature(seabreezes versus powder snow avalanches), they can give rise tonon-Newtonian dynamics (debris flows), and they can be linked tochemical reactions or to the preferential rejection of salt duringthe freezing of water. Especially, turbidity currents are multiscalein nature, as their large-scale evolution is closely tied to themicroscale mechanisms of erosion and resuspension. Gravity cur-rents can exist in ambient environments that exhibit velocityshear, such as in thunderstorm outflows [9], they can interact witha background density stratification, thereby triggering the forma-tion of internal gravity waves [10], and their dynamics can beaffected by complex topography [11]. Frequently, several of theabove effects conspire to render the flow particularly complex,such as in certain types of snow avalanches, which may be non-Boussinesq, non-Newtonian, nonconservative, multiscale, andhighly turbulent in nature. As a result, it is usually difficult to clas-sify naturally occurring gravity-driven flows into neatly separatedcategories according to which of the many potentially relevantphysical mechanisms may or may not be influential under a partic-ular set of circumstances.

Various aspects of gravity current research have been reviewedin earlier articles. Hopfinger [4] provides an overview of powdersnow avalanche research, while Rottman and Linden [12] summa-rize the basic scaling laws and force balances for idealized com-positional gravity currents. Huppert [13] surveys a wide range oftopics related to gravity-driven geophysical flows, while in Ref.[14] the same author focuses more exclusively on box models andshallow water equations for turbidity currents. In Ref. [15], he dis-cusses aspects of dilute as well as concentrated particle-laden cur-rents, along with dense granular flows. Kneller and Buckee [16]review theoretical approaches and experimental data for turbiditycurrents from a geological perspective. Parsons et al. [17] provide

1Corresponding author.Manuscript received January 13, 2015; final manuscript received July 8, 2015;

published online July 27, 2015. Assoc. Editor: Herman J. H. Clercx.

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an overview over sediment gravity flows in the ocean, while therecent article by Meiburg and Kneller [5] reviews specifically tur-bidity currents and their deposits.

Partly as a consequence of this multitude of relevant flowregimes, a wide variety of modeling approaches for gravity andturbidity currents have evolved, spanning the entire range fromdimensional analysis to high-resolution DNS. In the following, wewill review these modeling approaches, with an emphasis ondepth-resolving Navier–Stokes simulations.

Section 2 highlights several conceptual models of high Reyn-olds number gravity currents. While these models are frequentlybased on such simplifying assumptions as steady-state and invis-cid flow, they nevertheless provide fundamental insight into thebasic mechanisms driving such currents, and specifically into thescaling laws governing their front velocity. Section 3 briefly dis-cusses the fundamental concepts behind depth-averaged model-ing, as exemplified by box models and shallow water approaches.Section 4 reviews depth-resolving modeling approaches based onthe full Navier–Stokes equations, including LES and RANS simu-lations. It furthermore highlights some areas of research wheresuch simulations have contributed to our physical understandingin recent years, such as interactions between gravity currents andengineering installations or seafloor topography, non-Boussinesqgravity currents, and gravity currents propagating in stratifiedambients. Section 5 identifies some of the current researchchallenges in the field and provides a brief outlook.

2 Conceptual Models: The Froude Number

as a Function of the Current Height

Attempts to determine the front velocity of a gravity current asa function of its height and its excess density date back at leastthree quarters of a century, when von K�arm�an [18] introduced theidealized gravity current model shown in Fig. 1(a). He consideredthe flow in the reference frame moving with the current front, andinvoked three main simplifying assumptions: (i) the flow is steadyin this reference frame; (ii) the flow is inviscid; and (iii) the fluidinside the current is at rest. By applying Bernoulli’s law along thestreamlines C-O and O-A, i.e., by assuming that the mechanicalenergy is conserved along these streamlines, he obtained for theFroude number

Fh ¼Uffiffiffiffiffiffig0hp ¼

ffiffiffi2

r

r(1)

where U denotes the front velocity of the gravity current, h repre-sents its height, g0 ¼ gðq1 � q2Þ=q1 indicates the reduced gravity,and r¼q2/q1 refers to the density ratio.

Benjamin [19] objected to von K�arm�an’s analysis on thegrounds that Bernoulli’s equation should not be assumed to holdalong streamline O-A, due to the dissipation that occurs in thisinterfacial region as a result of the velocity shear between the cur-rent and the ambient. Benjamin instead considered a correspond-ing gravity current in a channel of finite depth H, as shown in

Fig. 1(b). By applying the same three simplifying assumptions asvon K�arm�an, and also considering the pressure distributions farup- and downstream of the current front to be hydrostatic,Benjamin was able to write the conservation laws for mass andhorizontal momentum flux as

UH ¼ U2ðH � hÞ (2)

pCH þ q2U2H ¼ pBH þ 1

2g q1 � q2ð Þh2 (3)

� g q1 � q2ð ÞHhþ q2U22 H � hð Þ (4)

For a given set of values for current thickness, channel height, anddensity ratio, the above relationships represent two equations forthe three unknowns, U, U2, and pB� pC, so that one additionalequation is required. To close the problem, Benjamin followedvon K�arm�an’s approach and applied Bernoulli’s law; however, heeffectively did so along the bottom wall C–B of the channel,rather than along the interface as von K�arm�an had done. For a cur-rent of fractional height a¼ h/H, Benjamin thus obtained theFroude number

FH;b ¼Uffiffiffiffiffiffiffiffig0Hp ¼ að1� aÞð2� aÞ

rð1þ aÞ

� �1=2

(5)

Note that the Froude number Fh based on the current height isrelated to the Froude number FH based on the channel height byFh¼FHa�1=2.

By then applying Bernoulli’s equation along the top of thechannel (D–E), Benjamin showed that an energy-conserving cur-rent is possible only for a¼ 1/2; currents with a> 1/2 require aninput of energy to be realized. Several studies attempted to derivethe Froude number by combining global energy considerationswith the solution of a lock-release current, as reported and dis-cussed by Yih [20], Shin et al. [21], and Ungarish [22,23]. Suchflows are not of the “steady-state” type considered here, and thedetails are beyond the scope of this brief outline.

In the context of Boussinesq gravity currents, Borden and Mei-burg [24] show that invoking an energy closure assumption �a lavon K�arm�an and Benjamin becomes unnecessary if the conserva-tion of vertical momentum is enforced, along with the conserva-tion of mass and horizontal momentum. This approach bypassesthe search for the “correct: energy closure entirely, as the conser-vation of energy or head loss arguments never enter the picturewhen calculating the front velocity. While there is no flow ofvertical momentum into or out of the control volume BCDE inFig. 1(b), the importance of vertical momentum conservationnevertheless is immediately obvious: inside the control volume,the ambient fluid is first accelerated and then decelerated in thevertical direction, which affects the pressure profiles along the topand bottom walls. In turn, these profiles determine the pressurejump pB� pC across the current front, for which the need of anadditional equation originally arose. Borden and Meiburg [24]show that the conservation of vertical momentum can easily be

Fig. 1 Idealized gravity current models introduced by (a) von K�arm�an [18] and (b) Benjamin[19]

040802-2 / Vol. 67, JULY 2015 Transactions of the ASME


accounted for by considering the linear combination of the differ-ential versions of the steady-state, inviscid, horizontal, and verti-cal momentum equations, in the form of the dimensionalBoussinesq vorticity equation

u � rx ¼ �g0@q@x

(6)

where x ¼ @v=@x� @u=@y denotes vorticity, q indicates the nor-malized density, and x, y, u, and v represent the horizontal andvertical directions and velocity components, respectively. By inte-grating Eq. (6) over the control volume, we obtain a relation gov-erning the total circulation around the control volumeþ

xu � ndS ¼ð�g0

@q@x

dA (7)

Equation (7) states that the flow of vorticity into and out of thecontrol volume is balanced by the baroclinic generation of vortic-ity inside the control volume. For a sharp interface, the area inte-gral over the baroclinic term becomes g0h. Furthermore, novorticity enters the control volume, and the flow of vorticity out ofthe control volume is confined to the vortex sheet between the cur-rent and the ambient. The vorticity flux carried by this sheetequals the vortex sheet strength, U2, multiplied by the sheet’sprincipal velocity, uPV¼U2/2 [25,26]. Equation (7) thusreduces to

1

2U2

2 ¼ g0h (8)

Combining the vorticity conservation relationship Eq. (8) with thecontinuity Eq. (2) immediately produces

FH;c ¼ffiffiffiffiffi2apð1� aÞ (9)

Borden and Meiburg [24] show that for Boussinesq currents thisrelationship between the Froude number and the current heightresults in better agreement with DNS simulation results regardingthe vorticity flux than Benjamin’s relationship (5). We note that inthe above analysis, the pressure jump pB� pC across the currentfront has become decoupled from the problem of determining Uand U2, which were evaluated from the conservation of mass andvorticity alone. Up to this point, we have used the conservation ofhorizontal momentum only in linear combination with the conser-vation of vertical momentum, i.e., as the vorticity equation. Con-sequently, if desired, the pressure jump pB� pC across the currentfront can now be determined from the horizontal momentumequation, as shown in Ref. [24]. The decoupling of the pressure inthe above analysis very much corresponds to employing thestreamfunction-vorticity formulation of the Navier–Stokes equa-tions, which allows for the numerical simulation of incompressi-ble flow fields without having to explicitly calculate the pressure.We note that, by accounting for the conservation of mass, hori-zontal, and vertical momentum, the above analysis does not haveto invoke any assumptions about energy conservation. Rather,individual terms in the energy equation can now be evaluated, sothat the overall loss of energy can be calculated a posteriori, ratherthan assumed a priori. This modeling concept employing the con-servation of vorticity holds considerable promise for gravity cur-rents propagating under more complex conditions, such as intosheared ambients [27].

Most recently, the vorticity-based approach has been extendedto the non-Boussinesq regime by Konopliv et al. [28], who showthat for a density ratio r ¼ q2=q1 one obtains a Froude number of

FH;c ¼ffiffiffiffiffi2ar

rð1� aÞ (10)

In the limit of small density contrasts r� 1, so that the Boussinesqresult is recovered. We note, however, that in the non-Boussinesqcase the pressure no longer fully decouples from the equations forthe velocity field. Other recent extensions of conceptual gravitycurrent models address such issues as two-layer and linearly strati-fied ambients, nonrectangular cross sections, in- and outflow, aswell as fluctuations about the average [29–33].

For both Boussinesq and non-Boussinesq currents, Fig. 2 com-pares the different model predictions for the Froude number as afunction of the fractional current height a. Note that Froudenumber in the figure is defined in terms of the current heightand it includes the density ratio factor, i.e., Fh;k ¼ Fh

ffiffiffirp

;Fh;b

¼ FH;ba�1=2ffiffiffirp

and Fh;c ¼ FH;ca�1=2ffiffiffirp

. For currents occupyinghalf the channel height, both Benjamin and circulation modelspredict a Froude number of 1=

ffiffiffi2p

. The predictions by Benjamin’smodel and the circulation model show small differences for0< a< 0.5.

3 Depth-Averaged Models

It is useful to distinguish between continuous inflow gravitycurrents and those that originate from the release of a finite vol-ume of dense fluid, such as lock-exchange flows. Under certainconditions, much insight into both classes of flows can be gainedfrom simplified modeling approaches in conjunction with dimen-sional scaling arguments. Especially successful in this regard havebeen so-called box models for finite volume currents, and shallowwater analysis for currents whose length greatly exceeds theirdepth. In the following, we will briefly discuss the basic conceptsunderlying these modeling approaches.

Under many conditions, gravity currents form due to the releaseof a finite volume of dense fluid, a process that can be modeled bythe prototypical case of lock-release currents, cf. Fig. 3. This con-figuration has served as the basis of numerous experimental, theo-retical, and computational investigations into the physicalmechanisms governing gravity currents. Consider a rectangularchannel of height H and length L. The channel is filled with twofluids of different densities that are initially separated by a barrier.While the “lock” of height d holds a fluid of density q1, the ambi-ent fluid has lower density q2. This initial configuration causes adiscontinuity of the hydrostatic pressure across the barrier, whichsets up a predominantly horizontal flow once the barrier isremoved. The case of d¼H is referred to as a “full depth current.”Here the denser fluid forms a negatively buoyant gravity currentpropagating rightward along the bottom of the channel, while thelighter fluid moves leftward along the top as a positively buoyant

Fig. 2 Model predictions for the Froude number as a functionof the fractional current height a for gravity currents: Benja-min’s model (solid line), circulation model (dashed-dotted line),and von K�arm�an model (solid circle)

Applied Mechanics Reviews JULY 2015, Vol. 67 / 040802-3


current. When d<H, we obtain a “partial depth current,” forwhich the left-moving buoyant current takes the form of a rarefac-tion wave or bore propagating along the horizontal interfacebetween the two fluids.

Huppert and Simpson [34] demonstrated that under high Reyn-olds number conditions full depth lock-release flows go through awell-defined sequence of stages. During the initial “slumpingphase,” the current front travels for about O (5� 10) lock lengthsat constant velocity. This phase frequently comes to an end whenthe reflected bore generated at the left wall catches up with thefront. The current subsequently enters a second stage governed bythe balance of gravity and inertia, during which the influence ofthe ambient flow becomes negligible and the front locationevolves as t2=3. At late times, the influence of inertial forces sub-sides and a new balance between gravitational and viscous forcesforms, resulting in the front location advancing as t1=5. These scal-ing laws have been confirmed experimentally by a number ofauthors, among them [34–36].

3.1 Box Models. Conceptually simple box models, which canbe formulated for both two-dimensional and axisymmetric geome-tries, usually do not consider ambient fluid entrainment. Theyassume that the gravity current evolves in the form of constantarea rectangles, and neglect any vertical or streamwise variationsinside the current. The box model concept was extended to turbid-ity currents by Dade and Huppert [37], and Gladstone and Woods[38], and it is reviewed in some detail by Ungarish [2].

3.2 Shallow Water Models. Gravity currents whose length issubstantially larger than their depth can frequently be approxi-mated by depth-averaged or shallow water models. This approachwas introduced for compositional gravity currents by Rottman andSimpson [36], and later extended to turbidity currents [39–41].Earlier reviews are provided in Refs. [14,15] and [17] and espe-cially in the recent book by Ungarish [2]. In contrast to the abovebox models, shallow water models allow for streamwise variationsin the current height and velocity. They consider the current to bewell mixed, so that there are no density or velocity variations inthe vertical direction. Furthermore, they assume that viscousforces are negligible and that vertical accelerations are small, sothat the pressure field is purely hydrostatic. Entrainment at the topof the current is typically neglected, so that ambient fluid is usu-ally neither entrained nor detrained, although Johnson and Hogg[42] recently introduced a shallow water model for gravity cur-rents with entrainment. Kowalski and McElwaine [43] develop ashallow water model that accounts for variations in the lateral

direction, which is frequently important in avalanche or granularflows. Other recent extensions model the effects of nonrectangularcross sections [44–46]. When employing shallow water models,the implications of the assumptions underlying these models needto be carefully considered on a case-by-case basis, since they maybe too restrictive for some applications. For example, under someconditions turbulent mixing may result in substantial entrainmentalong the current-ambient interface. Similarly, during the laterstages of turbidity currents, when the decaying turbulence is nolonger fully able to distribute the particles across the entire currentheight, significant detrainment may occur.

For deeply submerged gravity and turbidity currents, the motionof the overlying fluid can frequently be neglected to a goodapproximation, which results in the so-called single-layer shallowwater equations for the current height h(x, t) and velocity u(x, t) asa function of the streamwise coordinate x and time t

@u

@tþ u

@u

@x¼ �g0

@h

@x(11)

@h

@tþ @uh

@x¼ 0 (12)

These equations, which express the conservation of mass and mo-mentum for the current fluid, represent a hyperbolic system inwhich the current front appears as a shocklike discontinuity.Hence, they cannot capture the detailed dynamics in the vicinityof the current tip and require a closure in terms of a relationshipbetween the current velocity and its fractional height, which canbe provided by models such as those in Refs. [19] and [24] dis-cussed above. Under certain conditions, their relative simplicityallows for the derivation of similarity solutions that have beenconfirmed by numerous laboratory experiments, cf. the discussionby Linden [47]. Consequently, the shallow water equations havebeen successfully applied to a large variety of flow situations. Oneexample is given in the work of Gonzalez-Juez and Meiburg [48],who extended earlier models by Lane-Serff et al. [49] and othersin order to investigate the interaction of gravity currents withsubmarine pipelines. When compared to depth-resolving, high-resolution simulations, the shallow water estimates of themaximum drag were typically found to be accurate to within O(10%). On the other hand, these models do not give informationon such aspects as vortex shedding from the pipeline, along withthe associated excitation frequencies. In another example, Birmanet al. [50] employ a shallow water model for turbidity currentsoverflowing the levees of a submarine canyon. Their analysisdemonstrates that the entrainment of ambient fluid by the overflowcurrent governs the shape of the evolving levee. Negligibleentrainment rates are seen to lead to exponentially decaying leveeshapes, whereas power-law levee shapes emerge for constantentrainment rates.

For shallow gravity currents, i.e., when the depth of the overly-ing ambient fluid layer is of the same order as the current height,it is necessary to account for the dynamics of the ambient fluidand its coupling with the gravity current. This can be accom-plished by formulating the so-called two-layer shallow waterequations [11], which account for the conservation of mass andstreamwise momentum in both the current and the ambient. Suchtwo-layer models are reviewed in some detail in the recent bookby Ungarish [2].

4 Depth-Resolving Simulation Approaches

In the following, we will begin by formulating the set of gov-erning equations commonly employed for depth-resolved simula-tions. Subsequently, we will review the different simulationapproaches based on these equations, such as DNS, LES, andRANS simulations. A discussion of some representative resultsobtained via these simulation approaches will follow.

Fig. 3 Schematic of a lock-exchange configuration. The lockregion to the left initially contains a fluid of higher density q1,which is separated from the lighter fluid of density q2 to theright by a partition. The hydrostatic pressure difference acrossthe partition generates a gravity current along the bottom wallwhen the partition is removed. Gravity acts in the verticaldirection 2y.



4.1 Governing Equations. In many situations of interest,compositional gravity currents and turbidity currents are driven bysmall density differences not exceeding O (1%). Under such con-ditions, the Boussinesq approximation can be employed, whichtreats the density as constant in the momentum equation with theexception of the body force terms. When dealing with turbiditycurrents, we account for the dispersed particle phase by means ofan Eulerian–Eulerian formulation, which means that we employ acontinuum equation for the particle concentration field, ratherthan tracking particles individually in a Lagrangian fashion.

In the following, it will be important to carefully distinguishbetween dimensional and dimensionless variables. Toward thisend, we will employ the tilde symbol to indicate a dimensionalvariable, whereas variables without the tilde symbol are dimen-sionless. Under the Boussinesq approximation, the dimensionalgoverning equations for compositional gravity currents driven bysalinity and/or temperature gradients can be written as

@~uj

@~xj¼ 0 (13)

@~ui

@~tþ@ ~ui ~uj

� �@~xj

¼ � 1

~q1

@~p

@~xiþ ~�

@2~ui

@~xj@~xjþ ~q~g

~q1

egi (14)

@~q@~tþ@ ~q~uj

� �@~xj

¼ ~a@2 ~q@~xj@~xj

(15)

where ~ui denotes the velocity vector, ~p the pressure, ~q the density,~g the gravitational acceleration, eg

i the unit vector pointing in thedirection of gravity, ~� the kinematic viscosity, and ~a the moleculardiffusivity of the density field. We nondimensionalize Eqs.(13)–(15) by a reference length scale, such as the domain half-

height ~H=2 of a lock-exchange flow (Fig. 3), the current density~q1, and the buoyancy velocity ~ub

~ub ¼ffiffiffiffiffiffiffiffiffiffiffiffieg0 ~H=2

q(16)

where eg0 indicates the reduced gravity

eg0 ¼ ~g~q1 � ~q2

~q1

(17)

where ~q2 represents the ambient density. For partial depth lockreleases, it may be more appropriate to form the relevant buoy-ancy velocity with the lock height, rather than the channel depth.After nondimensionalization, we obtain

@uj

@xj¼ 0 (18)

@ui

@tþ@ uiuj

� �@xj

¼ � @p

@xiþ 1

Re

@2ui

@xj@xjþ qeg

i (19)

@q@tþ@ quj

� �@xj

¼ 1

ReSc

@2q@xj@xj

(20)

Here the nondimensional pressure p and density q are given by

p ¼ ~p

~q1~u2b

;q ¼ ~q� ~q2

~q1 � ~q2

(21)

The nondimensionalization of the governing equations gives riseto two dimensionless parameters in the form of the Reynolds num-ber Re and the Schmidt number Sc

Re ¼ ~ub~H

2~�; Sc ¼ ~�

~a(22)

While the Reynolds number indicates the ratio of inertial to vis-cous forces, the Schmidt number represents the ratio of kinematicfluid viscosity to molecular diffusivity of the density field. Weremark that the aspect ratio of the lock and, for partial depth lockreleases, the ratio of the lock height to the channel height enter asan additional dimensionless parameter.

When the driving density difference is due to gradients in parti-cle loading, rather than salinity or temperature gradients, theabove set of equations no longer provides a full description of theflow. Particles settle within the fluid, so that the scalar concentra-tion field no longer moves with the fluid velocity. In addition,particle–particle interactions can result in such effects as hinderedsettling [51], increased effective viscosity and non-Newtonian dy-namics [52], thereby further complicating the picture. However,away from the sediment bed turbidity currents are often quitedilute, with the volume fraction of the suspended sediment phasebeing well below O (1%). Under such conditions, particle–particleinteractions can usually be neglected, so that the particle settlingvelocity remains as the key difference (along with erosion)that distinguishes turbidity currents from compositional gravitycurrents.

Due to the small particle volume fraction of dilute turbidity cur-rents, the volumetric displacement of fluid by the particulate phasecan usually be neglected, so that we can consider the fluid velocityfield as divergence free. Rather, the particle–fluid interactionoccurs primarily through the exchange of momentum, so that itsuffices to account for the presence of the particles in the fluidmomentum equation. In the following, we assume that the particle

diameter ~dp is smaller than the smallest length scale of the flow,such as the Kolmogorov scale in turbulent flow. In addition, weconsider only particles whose aerodynamic response time ~tp is sig-

nificantly smaller than the smallest time scale of the flow ~tf , so

that the particle Stokes number St ¼ ~tp=~tf � Oð1Þ [53]. Here theaerodynamic response time is defined as

~tp ¼~qped2p

18~l(23)

with ~qp indicating the particle material density and ~l denoting thedynamic viscosity of the fluid. Such particles can then be assumedto move with a velocity ~up;i that is obtained by superimposing thelocal fluid velocity ~ui and the particle settling velocity ~use

gi

~up;i ¼ ~ui þ ~usegi (24)

where ~us follows from balancing the gravitational force with theStokes drag force

~Fi ¼ 3p~l~dpð~ui � ~up;iÞ (25)

as

~us ¼~d2

pð eqp � ~qÞ~g18~l

(26)

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 ejectedfrom vortex centers. Hence, we can describe the spatio-temporalevolution of the particle number concentration field ~c in an Euler-ian fashion by the transport equation

@~c

@~tþ@ ~cð~uj þ ~use

gj Þ

� �@~xj

¼ ~a@2~c

@~xj@~xj(27)

The diffusion term in Eq. (27) represents a model for thedecay of concentration gradients due to the hydrodynamic



diffusion of particles and/or slight variations in particle size andshape [51,54].

The motion of the fluid phase is described by the incompressi-ble continuity equation and the Navier–Stokes equation aug-mented by the force exerted on the fluid by the particles, which isequal and opposite to the Stokes drag force acting on the particles.In dimensional form, these equations read

@~uj

@~xj¼ 0 (28)

@~ui

@~tþ@ ~ui ~uj

� �@~xj

¼ � 1

~q@~p

@~xiþ ~�

@2 ~ui

@~xj@~xjþ ~c

~q~Fi (29)

As we had done for compositional gravity currents, we use thedomain half height ~H=2 and buoyancy velocity ~ub for nondimen-sionalization. The reduced gravity ~g0 appearing in the calculationof ~ub can now be calculated as

~g0 ¼pð~qp � ~qÞ~c0

~d3p

6~q~g (30)

where ~c0 indicates a reference number concentration of particlesin the suspension. After nondimensionalization, we obtain

@uj

@xj¼ 0 (31)

@ui

@tþ@ uiuj

� �@xj

¼ � @p

@xiþ 1

Re

@2ui

@xj@xjþ ceg

i (32)

@c

@tþ@ cðuj þ use

gj Þ

� �@xj

¼ 1

ReSc

@2c

@xj@xj(33)

For polydisperse suspensions containing particles of differentsizes, the above approach can easily be extended by solving oneconcentration equation for each particle size and correspondingsettling velocity [55]. Note that the set of governing equations forturbidity currents (31)–(33) differs from the corresponding set forcompositional gravity currents (18)–(20) only by the additionalsettling velocity term in the concentration equation. In the follow-ing, we employ Eqs. (31)–(33) for both types of currents, with thetacit assumption that the settling velocity vanishes for composi-tional gravity currents.

4.2 DNS. DNS represent the most accurate computationalapproach for studying gravity currents. In DNS all scales ofmotion, from the integral scales dictated by the boundary condi-tions down to the dissipative Kolmogorov scale determined byviscosity, are explicitly resolved. However, for the case of turbid-ity currents, when the particle diameter is smaller than the Kolmo-gorov scale, the fluid motion around each particle is usually notresolved, due to the prohibitive computational cost. Nevertheless,the drag law accurately captures the exchange of momentumbetween the two phases at scales smaller than the Kolmogorovscale, so that the approach described above is still referred to asDNS.

Consistent with the above arguments, the grid spacing requiredfor DNS is of the order of the Kolmogorov scale, while the timestep needs to be of the same order as the time scales of the small-est eddies. Because of the large disparity between integral andKolmogorov scales at high Reynolds numbers, the computationalcost of DNS scales as Re3, so that the DNS approach is effectivelylimited to laboratory scale Reynolds numbers. The first DNS sim-ulations of gravity currents in a lock-exchange configuration, suchas the one shown in Fig. 3, were reported by H€artel et al. [56] forRe¼ 1225. Necker et al. [57] extended this work to turbidity cur-rents at Re¼ 2240. More recent simulations of lock-exchange

gravity currents by Cantero et al. [58] were able to reachRe¼ 15,000, which corresponds to a laboratory scale current ofheight 0.5 m with a front velocity of 3 cm/s.

DNS simulations can provide detailed information on the struc-ture and statistics of the flow, on the various components of itsenergy budget, on the mixing behavior and many additionalaspects. As a case in point, the simulations by H€artel et al. [56]explored the detailed flow topology near the current front anddemonstrated that the stagnation point is located at a significantdistance behind the nose of the current. DNS results are further-more very useful for testing the accuracy and identifying any defi-ciencies in larger scale LES and RANS models [59]. Thus, whilethey are currently limited to laboratory scale currents, DNS simu-lations represent an excellent research tool for exploring thedetailed physics of moderate Reynolds number gravity currents,and for constructing larger scale models for higher Reynoldsnumber applications.

4.3 Large-Eddy Simulation. LES require fewer computa-tional resources than DNS, as they resolve only the energy con-taining large eddies, while modeling all of the scales of motionbelow a cutoff. This approach often results in the majority of thedissipative scales being modeled [60,61]. The cutoff is determinedby a filter width that depends on the grid spacing employed in thesimulation. In the LES technique, the governing equations areobtained by filtering the Navier–Stokes and the concentrationequations. The filtered governing equations read

@�uj

@�xj¼ 0 (34)

@�ui

@tþ@ �ui �uj

� �@xj

¼ � @�p

@xiþ 1

Re

@2 �ui

@xj@xj� @sij

@xjþ �ceg

i (35)

@�c

@tþ@ �cð�uj þ use

gj Þ

� �@xj

¼ 1

ReSc

@2�c

@xj@xj�@gj

@xj(36)

Here �ui and �c denote the filtered velocity and concentration fields,respectively. The effect of all scales of motion below the filterwidth appears as additional subgrid-scale (SGS) stress terms sij inthe momentum equation, and as an SGS flux term gj in the con-centration equation. In LES implementations, the SGS terms sij

and gj are frequently modeled using an eddy viscosity approxima-tion, which relates the SGS stresses to the resolved strain rate �Sij.The SGS stresses are thus calculated from

sij �1

3skkdij ¼ �2�t

�Sij (37)

�Sij ¼1

2ð@�ui

@xjþ @�uj

@xiÞ (38)

where �t denotes the SGS eddy viscosity, which is calculatedusing the Smagorinsky model [62] as

�t ¼ CsDð Þ2 �Sj j (39)

where �Sj j indicates the magnitude of the resolved strain rate ten-sor, D represents a length scale proportional to the local grid spac-ing, and Cs denotes an empirical model coefficient that needs tobe specified

�Sj j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2�Sij

�SijÞq

(40)

D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDxDyDzÞ3

q(41)

In the above, Dx, Dy, and Dz represent the local grid spacings alongthe x-, y-, and z-directions, respectively. Analogous to the SGS



stress terms, the SGS flux terms are calculated using an eddydiffusivity model as

gj ¼ ��t

Sct

@�c

@xj(42)

Here Sct denotes the SGS Schmidt number. Both the model coeffi-cient Cs and the SGS Schmidt number Sct need to be specified ei-ther a priori, or calculated during the simulation. Usually moreaccurate LES predictions can be achieved by employing the so-called dynamic procedure [63,64], which samples the resolvedflow field to estimate the model coefficient Cs and the SGSSchmidt number Sct. Such dynamic models usually yield higheraccuracy by adjusting the dissipation provided by the SGS modelto local flow conditions, allowing it to vanish in laminar regions.

Similar to DNS, LES also yields detailed, time-dependentlarge-scale flow structures. In fact, flow statistics that dependmostly on the large-scale structures such as mean velocity andconcentration, turbulence intensities, scalar fluctuations, bed shearstress, etc., can be estimated reasonably accurately on the basis ofLES simulations. Stevens et al. [65] demonstrate that LES, at suf-ficiently high resolutions, can reproduce accurately higher orderstatistics such as even-order moments of the streamwise velocityfluctuations up to order 10. Because of the much lower computa-tional cost, LES simulations have been conducted for significantlyhigher Reynolds numbers than DNS. Ooi et al. [66] carried outLES simulations for gravity currents at Reynolds numbers of O(105� 106). These simulations show that the Kelvin–Helmholtzbillows become three-dimensional sooner than for moderateReynolds number gravity currents, and that the dissipation ratebecomes independent of time. For field scale flows, the Reynoldsnumber can become as large as O (108� 109). Though LES canbe applied at any Reynolds number for free shear flows, the pres-ence of a solid bottom boundary in gravity and turbidity currentslimits LES to Reynolds numbers lower than those field scale val-ues, because of the need to resolve the smaller but energy contain-ing eddies in the inner layer, near-wall region. There exists thepossibility of applying LES to field scale flows without resolvingthe inner layer, by representing this layer in an average sensethrough a wall-layer model. The limitations and drawbacks of var-ious such wall-layer models are discussed by Piomelli and Balaras[67]. For attached geophysical flows over both smooth and roughboundaries, Radhakrishnan and Piomelli [68] demonstrate thatequilibrium stress based wall-layer models can predict the flowevolution accurately.

4.4 RANS Simulations. This approach numerically integra-tes the RANS equations, which are obtained by ensemble- ortime-averaging the governing continuity, momentum, and concen-tration equations [69]. Unlike eddy-resolving DNS or LES meth-ods, RANS simulations only yield the mean flow field. Thegoverning RANS equations are structured similarly to the LESequations (35) and (36), except that the bar symbol now refers tothe ensemble-averaged flow field rather than the filtered field, andthe additional terms sij and gj now represent the Reynolds stressesand the scalar fluxes, respectively. Additional closure equationsfor computing the Reynolds stresses and scalar fluxes are dis-cussed by Wilcox [69]. The most common RANS closure is basedon the solution of a transport equation for the turbulent kineticenergy k and the dissipation rate e. In the context of turbiditycurrents, Choi and Garcia [70] applied the k-e model to flowspropagating down a slope. They employ the boundary layerapproximation of the governing equations to reduce the computa-tional cost, and they report the impact of the model constants onthe predicted rate of water entrainment. More recently, Sequeiroset al. [71] utilized the k-e model along with an Exner equation forthe sediment transport [72,73], to study buoyancy-reversing tur-bidity currents and the associated bed form evolution. Abd El-Gawad et al. [74] invoked the Mellor–Yamada turbulence closure

to study the evolution of turbidity currents propagating overcomplex topography in a submarine environment that includescanyons, fans, and sinuous channels in the Niger delta. Theauthors vary the inflow conditions to match the sediment dataobtained at seven piston cores, and they demonstrate that theRANS approach can be utilized to calculate the inflow conditionsfrom the core data. The simulation results allow for the predictionof the resulting sediment deposit patterns.

While the RANS approach is computationally less expensivethan LES or DNS and thus allows us to carry out simulations forkm-scale currents, its predictions generally are also less accurate.As Choi and Garcia [70] show, it may be necessary to tune themodel constants to local flow conditions in order to improve thequality of RANS predictions. These difficulties reflect the fact thatthe Reynolds stresses appearing in the RANS equations depend onthe large flow structures, which in turn vary with the boundaryand flow conditions. Hence, it is challenging to construct a univer-sal RANS model. In contrast, the SGS terms appearing in the LESequations depend mainly on the smaller, more universal unre-solved eddies which are independent of the local flow conditions.

4.5 Gravity and Turbidity Currents Propagating OverFlat Terrain. The lock-exchange setup sketched in Fig. 3 repre-sents the most popular configuration for conducting laboratoryexperiments on gravity and turbidity currents. Furthermore, itsgeometrical simplicity and straightforward initial and boundaryconditions render it ideal for depth-resolving, two- and three-dimensional simulations as well. The first DNS simulations ofcompositional gravity currents were reported by H€artel et al. [56]for a Reynolds number of 1225. In two dimensions, these authorsemployed a vorticity-streamfunction formulation in conjunctionwith a mixed spectral/compact finite difference discretization.Their three-dimensional simulations, on the other hand, relied onprimitive variables, along with a mixed spatial discretizationbased on a spectral-element collocation technique in the verticaldirection and Fourier expansions in the stream- and spanwisedirections. The temporal discretization was semi-implicit.

Figure 4 displays the temporal evolution of the flow field bymeans of three-dimensional density isosurfaces, with additionaldensity contours shown in the side plane. Beyond t¼ 10, the well-known lobe-and-cleft structure of the front can be clearly recog-nized. This structure evolves from a spanwise instability of thefront whose linear growth was analyzed in detail by H€artel et al.[75]. The shear along the interface between the two fluids giverise to pronounced Kelvin–Helmholtz vortices. Initially, thesevortices are predominantly two-dimensional and extend across thewhole channel span. Beyond t¼ 15, they lose their spanwisecoherence and acquire a strongly three-dimensional structure.These simulations provide detailed insight into the physics of thecurrent, and in particular into the evolution of the front. Up untilthen, it had been assumed that the nose of the current coincidedwith a stagnation point, in a coordinate system moving with thecurrent front. However, the numerical simulations revealed thatthe stagnation point is located below and behind the nose of thecurrent.

Necker et al. [57] extended these simulations to turbidity cur-rents and conducted a detailed comparison between compositionaland particle-laden currents. Their simulations were performed at asignificantly higher Reynolds number of 2240. These authorsfound the initial evolution of gravity and turbidity currents to bequite similar, with lobe-and-cleft structures at the front andKelvin–Helmholtz billows along the interface. During the laterstages, however, turbidity currents were seen to lose their poten-tial energy more rapidly, due to the settling of the particles. Thiscauses turbidity currents to slow down earlier as compared togravity currents. The rate at which a turbidity current loses par-ticles initially grows rapidly, as the result of its increasing length.Later on, as the particle concentration of the current decreases, thesedimentation rate decreases along with it. Figure 5 compares



deposit profiles from a two-dimensional simulation with corre-sponding experimental data of de Rooij and Dalziel [76]. Here,the deposit profile is computed as a function of time as

Dtðx1; tÞ ¼1

Lsh

ðt

0

cwðx1; sÞusds (43)

where cw is the particle concentration value at the wall. As the fig-ure illustrates, the numerically evaluated deposit profile agreeswell with the experimental data downstream of the lock region.The differences between the numerical profile and the experimen-tal data in the lock region are likely due to the initial stirring ofthe suspension in the experiments which is not represented in thesimulation, as well as to particle sedimentation before the start ofthe experiment.

Fig. 4 Full-depth, lock-exchange Boussinesq gravity current at Re 5 1 225. The flow is visual-ized at different times t by means of the three-dimensional density isosurface q 5 0.5, alongwith density contours in the side-plane. Initial parameters: L 5 23, l 5 10, H 5 2, d 5 2, andW 5 3 (Reprinted with permission from H€artel et al. [56]. Copyright 2000 by CambridgeUniversity).



In a subsequent investigation, Necker et al. [77] compared theenergy budgets of gravity and turbidity currents, cf. Fig. 6. Thefigure shows the time history of potential energy (Ep), kineticenergy (k), viscous dissipation due to the resolved motion (Ed),and viscous dissipation in the unresolved small-scale Stokes flowaround the individual particles (Es). All energy components arenormalized by the initial potential energy Ep0. Figure 7 illustratesthe development of a turbidity current via isosurfaces of particleconcentration at various instants in time. More than half of thepotential energy is converted into kinetic energy during the first2–3 units of time (cf. Fig. 7(b)) when the fluid in the lock regionstarts a convective motion from rest. As the length of the currentincreases with time (Figs. 7(c) and 7(d)), the interfacial area withstrong velocity gradients increases, which results in increasingviscous dissipation Ed with time. For the specific parameter valuesinvestigated in Ref. [77], the authors find that, by the time the cur-rent comes to rest, as much as 40% of the initial potential energyhas been lost in the unresolved Stokes flow around the particles.The spatial distribution of the viscous dissipation shows two peakregions, one in the boundary layer close to the wall and one nearthe interface between the two fluids. Close to 15% of the total dis-sipation occurs in the thin boundary layer region near the wall.

Necker et al. [77] furthermore employed Lagrangian markers tostudy the mixing between interstitial and ambient fluid, both ofwhich have the same density. Their analysis showed the perhapssomewhat counterintuitive result that the two fluids become mostthoroughly mixed for intermediate particle settling velocities. Forvery small settling velocities, the particles remain suspended inthe interstitial fluid for very long times, dragging it toward thebottom wall and preventing it from mixing. For very large settlingvelocities, on the other hand, the particles settle out before muchof their potential energy can be converted into kinetic energy,which again prevents strong mixing. For intermediate settlingvelocities, the particles remain suspended sufficiently long so thatmuch of their potential energy is converted into kinetic energyand a vigorous flow develops, but they settle out sufficiently fastso that clear, neutrally buoyant interstitial fluid forms that has suf-ficient kinetic energy left to mix with the ambient fluid.

While most lock-exchange simulations assume the lock fluid tobe at rest initially, this is usually not the case in correspondingexperiments. First of all, the removal of the partition injects a cer-tain amount of energy into the fluid. Second, for lock-exchangeexperiments involving turbidity currents the suspension usuallyhas to be stirred vigorously right before the removal of the

partition, to ensure that the particles are well mixed across thefluid column. In order to analyze the effects of initially presentkinetic energy on the flow evolution, Necker et al. [77] performedadditional simulations in which the lock fluid initially contained12.5% and 25% of the initial potential energy as kinetic energy.While this initial kinetic energy dissipates relatively quickly, itnevertheless significantly enhances the mixing within the currentand causes it to travel slightly faster during the late stages of itsevolution. More recently, Espath et al. [78] explore the depend-ence of key flow features on Re by conducting DNS simulationsof lock-exchange turbidity currents up to Re¼ 104.

The lock aspect ratio k, which is defined as the ratio of the locklength x0 to the lock height h0, plays an important role in the flowdevelopment. Bonometti et al. [79] performed two-dimensionalNavier–Stokes simulations to investigate the effect of the lock as-pect ratio on the shape and motion of the current. For a currentwith an initial density difference of 1%, Fig. 8 shows the influenceof k on the shape of the current. Within three units of lengths ofthe front, the shape of the current is seen to depend only weaklyon k. Farther away from the front, however, the height of the cur-rent body depends strongly on k, as it is larger than the headheight for k� 6.25, and smaller for k� 1. The current speed dur-ing the slumping phase is largely independent of k for k� 6.25.However, the current speed during the slumping phase decreaseswith k for k� 1.

Cantero et al. [80] study the front velocity of the gravity currentduring the acceleration, slumping, inertial, and viscous phases bymeans of two- and three-dimensional simulations at various

Fig. 5 Nondimensional particle deposit profiles as function ofthe streamwise coordinate. Results are shown for times t 5 7.3and t 5 10.95, along with the final profile (t fi ‘) after all par-ticles have settled out. Solid line: two-dimensional simulation,dashed line: experimental data of de Rooij and Dalziel [76]. Inboth cases Re 5 10,000, us 5 0.02, l 5 0.75, and H 5 d 5 2.0(Reprinted with permission from Necker et al. [57]. Copyright2002 by Elsevier).

Fig. 6 (a) Time history of potential energy Ep and kineticenergy k. (b) Time history of the dissipation Ed due to theresolved motion, and the dissipation Es due to the unresolvedStokes flow around the individual particles. All energy compo-nents are normalized by the initial potential energy Ep0. Solid(dashed) lines indicate turbidity (gravity) current results. Thedotted-dashed line in (a) gives the sum of Ep, k, Ed and Es forthe turbidity current. Re 5 2240. Initial parameters: L 5 23, l 5 1,H 5 2, d 5 2, and W 5 2 (Reprinted with permission from Neckeret al. [77]. Copyright 2005 by Cambridge University).



Reynolds numbers. They compare the numerical front velocitypredictions with corresponding experimental data and with theo-retical predictions. Figure 9 shows the temporal variation of thefront velocity for three different Reynolds numbers. During theconstant velocity phase, the lower Reynolds number currentstravel more slowly than their higher Reynolds number counter-part. For all currents, the front velocity begins to depart from theconstant velocity phase around t� 12. For the two lower Reynoldsnumber currents, the subsequent decrease of the front velocityshows good agreement with the theoretical predictions for the vis-cous phase. For the high Reynolds number current, after the con-stant velocity phase the front velocity agrees well with the inertialphase scaling by Huppert and Simpson [34].

Cantero et al. [58] performed DNS simulations of planar grav-ity currents at Reynolds numbers of 8950 and 15,000, respec-tively, for which the flow is fully turbulent. They find that theturbulence in the near-wall region shows remarkable resemblanceto the turbulence in standard wall-bounded flows. Specifically,this region exhibits several hairpin vortices, as well as low- andhigh-speed streaks similar to the ones observed in classical bound-ary layers. Near the current front, the low- and high-speed streaksare about 200 wall units apart. Farther behind the current front,where the turbulence in the near-wall region becomes fully devel-oped, the spacing between the streaks reduces to about 100 wallunits. Here a wall unit xþ is defined as xþ ¼ x � u�h=�, where

u� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðswall=qÞ

prepresents the friction velocity and swall denotes

the shear stress at the wall. In the interfacial region,Kelvin–Helmholtz vortices are generated near the front due toshear. These vortices subsequently break up and generate smallerscale turbulence behind the current front.

Ooi et al. [66] employ LES simulations to study compositionalgravity currents at Reynolds numbers ranging from 3000 to 106,where the flow is again fully turbulent. Figure 10 depicts the flowstructure in the near-wall region for Re¼ 87,750. In particular,Fig. 10(b) shows streamwise velocity contours in a plane locatedabout 11 wall units from the bottom wall. High- and low-speedstreaks in the near-wall region are clearly visible, similar to thoseobserved in turbulent boundary layers. The average width of thesestreaks is approximately 40 wall units, and their average lengthdecays from approximately 1200 wall units near the front to about800 wall units toward the end of the streaky region. Although thenear-wall flow structure resembles that of a turbulent boundarylayer, the streaks here are larger in terms of wall unit dimensionsas compared to those in traditional turbulent boundary layers. Thisdiscrepancy can be attributed to the different flow conditions ingravity currents, such as the presence of the mixing layer at theinterface. Constantinescu [82] provides a summary of recent LESsimulations for lock-exchange gravity currents.

4.6 Turbidity Currents Interacting With Seafloor Topog-raphy. With regard to gravity and turbidity currents interactingwith complex topography, one line of research addresses thechanges in the current dynamics brought about by the topography.A host of new flow phenomena can arise as a result of such inter-actions, including hydraulic jumps, reflected solitary waves, andstrong localized vortical structures. Equally interesting, however,is the question as to how the seafloor topography is altered by theturbidity currents via erosion and deposition, resulting in the for-mation of sediment waves, gullies, channels, levees, and fans.With regard to the latter, linear stability investigations can providefundamental insight into the mechanisms by which such featurescan appear on an initially flat seafloor.

Traditionally, most such analyses have been based on depth-averaged equations, such as the cyclic-step theory [83–85]. Morerecently, however, depth-resolving linear stability investigationshave provided additional insight into the coupling between thedetailed flow structure inside the current and the sediment bedbelow. The investigation by Hall et al. [86] into the formation ofchannels and gullies by turbidity currents represents an instructiveexample in this regard. The stability analysis indicates that forsuch gullies to occur via a linear instability, above the sedimentbed the suspended sediment concentration needs to decay moreslowly than the streamwise velocity. Under such conditions, anupward protrusion of the sediment bed will find itself in an envi-ronment where erosion decays more quickly than sedimentation,and so it will keep increasing. Conversely, a local valley in thesediment bed will see erosion increase more strongly than sedi-mentation, which again will amplify the initial perturbation.

This base flow effect is modulated by the perturbation of thesuspended sediment concentration and by the shear stress due to asecondary flow structure in the form of counter-rotating stream-wise vortices, cf. the eigenfunctions shown in Fig. 11. The figureshows the shape of the sediment bed surface, along with the sedi-ment concentration disturbance and the streamwise and transverseperturbation velocity fields, in a transverse plane. The shape of thesediment bed perturbation is shown in the bottom frame. Thesemicircular lines close to the sediment bed in the top and middleframes represent contours of the sediment concentration perturba-tion, with solid lines indicating positive values and dashed linesnegative values. Streamlines of the transverse velocity perturba-tion are superimposed, with arrows denoting the direction of theflow. In the top frame, gray shading reflects the perturbationstreamwise velocity, with lighter areas indicating positive valuesand darker areas negative values.

Fig. 7 “Structure of a particle-driven gravity current visualizedby isosurfaces of concentration at t 5 0 (a), t 5 2 (b), t 5 8 (c),and t 5 14 (d). Results obtained from a 3D simulation for a Reyn-olds number of Re 5 2240 and a dimensionless settling velocityof us 5 0.02. In all cases, an isovalue of 0.25 is employed.” Notethat at t 5 14 the concentration in the rear part of the channelhas almost dropped to zero. Initial parameters: L 5 23, l 5 1,H 5 2, d 5 2, and W 5 2 (Reprinted with permission from Neckeret al. [57]. Copyright 2002 by Elsevier).



Above the peaks of the perturbed sediment bed, we observe anegative sediment concentration perturbation (reduced sedimentloading), which results in lower hydrostatic pressure as comparedto the troughs of the sediment bed, where the sediment concentra-tion increases. Hence, a spanwise pressure gradient exists alongthe sediment bed surface, which drives a perturbation flow fromthe troughs to the peaks. Via the continuity equation, this span-wise perturbation flow along the sediment bed surface leads to theformation of the counter-rotating streamwise vortices visible in

the figure. Note that above the sediment bed peaks, these vorticescarry low-speed fluid, i.e., fluid with a small streamwise velocity,away from the sediment bed, while high-speed fluid from the freestream is brought toward the sediment bed at the troughs. In thisway, the shear stress at the bed surface, which primarily is a func-tion of the local streamwise velocity gradient, is enhanced abovethe troughs and lowered above the peaks. This is reflected by themiddle frame of the figure, which shows the perturbation shearthrough gray shading, with lighter areas indicating positive valuesand darker areas negative values. Thus, erosion increases in thevalleys and decreases at the peaks, which further amplifies the ini-tial sediment bed perturbation.

In a similar vein, Lesshafft et al. [87] address the formation ofdeep-water sediment waves, via the interaction of an erodiblesediment bed with a turbidity current. Their stability analysisdemonstrates the existence of both Tollmien–Schlichting and in-ternal wave modes in the stratified boundary layer. For the internalwave mode, the stratified boundary layer acts as a wave duct,whose height can be determined analytically from theBrunt–V€ais€al€a frequency criterion. Consistent with this criterion,distinct unstable perturbation wave number regimes exist for theinternal wave mode. For representative turbidity current parame-ters, the analysis predicts unstable wavelengths that are consistentwith field observations. Furthermore, for most of the unstablewave number ranges, the phase relations between the sedimentbed deformation and the associated wall shear stress and concen-tration perturbations are such that the sediment waves migrate inthe upstream direction, which again is consistent with fieldobservations.

While linear stability analysis represents a powerful tool forobtaining insight into the initial formation process of seafloor to-pography, the effects of such topography on gravity and turbiditycurrents can best be explored via laboratory experiments [88–92]and/or numerical simulations. Along these lines, Nasr-Azadaniand Meiburg [55,93] employ DNS simulations to study the inter-action of turbidity currents with a local seamount, cf. Fig. 12.Their representation of the seafloor topography is based on an

Fig. 8 “Instantaneous shape of Boussinesq density currents (q1/q2 5 1.01) at time t 5 7. Iso-lines of (q 2 q2)/(q1 2 q2) 5 0.05, 0.25, 0.5, 0.75, and 0.95 are shown.” Initial parameters: l 5 x0,H 5 10, d 5 h0 5 1, L 5 37.5 for k 5 18.75 and L 5 12.5 for all other k values (Reprinted with per-mission from Bonometti et al. [79]. Copyright 2011 by Cambridge University).

Fig. 9 “Time evolution of the front velocity for planar currentsfrom three-dimensional simulations. The plot also includesexperimental data from two of the lower Reynolds numberexperiments by Marino et al. [81] with l 5 x0 5 1 and d 5 h0 5 1.Also included are the theoretical predictions for all phases ofspreading. The viscous phase predictions are for Re 5 8950,x0 5 1 and h0 5 1” (Reprinted with permission from Canteroet al. [80]. Copyright 2007 by Cambridge University).



immersed boundary method [94–98]. Rather than terrain-following coordinates, this approach uses a Cartesian grid every-where and modifies the fluid velocity and concentration at thenodes near the bottom boundary in order to satisfy the no-slipcondition along with the appropriate boundary condition for theconcentration field.

The authors explore the effects of the seamount’s height on thedynamics and depositional behavior of bidisperse turbidity cur-rents. The seamount is seen to deflect the current laterally, whichenhances the deposition toward the sides of the seamount whilereducing it on the seamount itself. The strength of these effectsvaries with the height of the seamount, as well as with the grainsize. Especially the tall seamount tends to transform the spanwiseboundary layer vorticity into strong horseshoe vortices, while italso deforms the mixing layer vorticity into inverted horseshoevortices. In this way, a complex vortical flow structure is gener-ated in the neighborhood of the seamount, which in turn results incomplicated erosion and deposition patterns. An alternativeapproach for representing complex seafloor topography involvesthe use of finite element methods. This line of research is pursuedby Coutinho and colleagues [99–102], who apply parallel stabi-lized finite element methods with adaptive meshes to lock-exchange type gravity and turbidity currents. A spectral elementapproach toward simulating the propagation of gravity currentsover complex seafloor topography is employed by €Ozg€okmenet al. [103].

Most computational investigations to date are limited to noner-odible seafloors, although a few simulations are beginning toaccount for the erosion of sediment. Along these lines, Blanchetteet al. [104] investigate two-dimensional turbidity currents propa-gating down an erodible inclined plane. They focused on the ava-lanchelike growth and acceleration of turbidity currents via theerosion and entrainment of particles from the sediment bed, whichis simulated based on the empirical erosion model of Garcia andParker [105]. Blanchette et al. [104] analyze the influence of suchquantities as bottom slope angle, particle size, and lock height onthe growth or decay of turbidity currents, with the goal of identify-ing critical threshold conditions beyond which such currentsbecome self-sustaining. We note that the erosion model theyemploy is based on an idealized set of conditions, and that addi-tional research into the dynamics of erosion and resuspension isrequired to obtain improved quantitative predictions. In a similarvein, the simulations reported by Strauss and Glinsky [106] and

Hoffmann et al. [107] explore the nonlinear stages of sedimentformation by turbidity currents.

4.7 Gravity Currents Interacting With EngineeringInfrastructure. As briefly mentioned in the introduction, gravityand turbidity currents can pose considerable hazards to

Fig. 10 “Visualization of the flow structure in the near-wall region of a turbulent gravity cur-rent for Re 5 87,750. (a) Vertical vorticity contours near the bottom wall; (b) streamwise veloc-ity contours showing the high- and low-speed streaks in a plane located at about 11 wall unitsfrom the bottom wall. The light and dark vorticity contours in (a) correspond to xy 5 2ub/h andxy 5 22ub/h, respectively” (Reprinted with permission from Ooi et al. [66]. Copyright 2009 byCambridge University).

Fig. 11 Dominant unstable eigenfunction modes for a turbiditycurrent propagating over a plain erodible sediment bed. Shownis a plane normal to the main flow direction. “The solid anddashed lines depict positive and negative concentration pertur-bation contours, respectively. Streamlines of the transverseperturbation velocity field are superimposed, with arrowsdenoting the flow direction. In the top frame, gray shadingreflects the perturbation u-velocity, with lighter areas indicatingpositive values, and darker areas negative values. The middleframe shows perturbation shear ›u/›z through gray shading,with lighter areas indicating positive values and darker areasnegative values. The shape of the interface perturbation isshown in the bottom frame” (Reprinted with permission fromHall et al. [86]. Copyright 2008 by Cambridge University).



engineering infrastructure, such as submarine oil and gas pipe-lines, wellheads and telecommunication cables on the seafloor.Mitigation of these hazards requires detailed knowledge not onlyof the forces exerted by the currents but also of the associated fre-quencies, so that potential resonant vibrations of cables and pipe-lines can be avoided. Gravity currents can furthermore endangersubmarine infrastructure via scour, which exposes and weakensthese structures by removing the sediment around them. Comple-menting earlier laboratory experiments [108,109], computationalinvestigations in recent years have matured to the point wherethey can add substantially to our insight into the interactionbetween gravity currents and obstacles of various shapes.

Gonzalez-Juez et al. [110] employ two- and three-dimensionalLES simulations to investigate the forces acting on a bottom-mounted square cylinder during the interaction with a lock-exchange gravity current, cf. Figs. 13–15. Their numericalapproach is based on the nondissipative finite volume DNS/LEScode of Pierce and Moin [111], which uses the dynamic

Smagorinsky model to calculate the SGS stress terms. Utilizing acartesian grid, the simulations represent the obstacle via a gridblanking methodology that sets the velocity, pressure, and concen-tration inside the obstacle to zero. Consistent with the experimentsof Ermanyuk and Gavrilov [108,109], Gonzalez-Juez et al.observe impact, transient, and quasisteady stages during the cur-rent/obstacle interaction. During the impact stage, as the currentapproaches the obstacle, the drag force on the obstacle increasesexponentially and reaches an overall maximum value. During thesubsequent transient stage the drag fluctuates in time, while thefinal quasisteady stage sees a decrease in both the average dragand its temporal fluctuations.

Figure 13 shows a two-dimensional simulation for Re¼ 2000.The domain height H is five times larger than the lock height h,and the obstacle is placed three lock heights downstream of thegate. For different times, the figure displays the concentration fieldalong with instantaneous streamlines on the left, while the vortic-ity is shown on the right. During the impact stage, a small

Fig. 12 Gravity current interacting with a localized seamount. The development of the currentstructure is visualized by a sediment concentration contour for a flat seafloor (FL—top row), ashallow seamount (B1—middle row), and a taller seamount (B2—bottom row). While the cur-rent primarily flows over the shallow seamount, it mostly flows around the taller seamount.“The shading in the bottom plane indicates the magnitude of the wall shear stress, while thevertical plane to the right depicts the concentration field in the symmetry plane z 5 1.5.” Initialparameters: L 5 38, l 5 1, H 5 d 5 2, and W 5 3 (Reprinted with permission from Nasr-Azadaniand Meiburg [55]. Copyright 2014 by Cambridge University).



separation region forms above and behind the obstacle. The tran-sient stage starts as the current is deflected upward, whichincreases the size of the separation region. Some distance down-stream of the obstacle the current plunges downward and re-attaches to the bottom wall, which results in the temporarytrapping of lighter ambient fluid in the wake behind the obstacle.Subsequently, this trapped lighter fluid is gradually flushed out ofthe obstacle’s wake by the denser fluid, and the quasisteady stagebegins. The drag oscillation during the transient stage is causedprimarily by this time-dependent nature of the separation region.

The computational simulation results provide us with the op-portunity to extract detailed information about the flow field thatmight be more difficult to evaluate experimentally, such as the

individual contributions to the various forces. Along these lines,Gonzalez-Juez et al. [110] examine the temporally varying dragand lift in Fig. 14, which shows the overall drag along with itsindividual components due to the pressure force on the upstream(Fw) and downstream (Fe) faces of the cylinder, and the viscousdrag force (Fv) acting on the top surface of the cylinder. The signof the pressure force is positive when it is directed from the fluidtoward the solid wall. The overall drag is obtained asFD¼Fw�FeþFv.

The figure indicates that the overall drag is dominated by thepressure contributions, whereas the viscous contribution is negli-gible. In this graph, the impact, transient, and quasisteady stagescan be associated with the time intervals 1 < t=ðh=VÞ < 3:3;

Fig. 13 Gravity current interacting with a bottom-mounted square cylinder. Temporal evolu-tion of the concentration (left) and vorticity (right) fields. “Instantaneous streamlines in thelaboratory reference frame are superimposed onto the concentration fields.” Initialparameters: L 5 24, l 5 9, H 5 d 5 1 (Reprinted with permission from Gonzalez-Juez et al. [110].Copyright 2009 by Cambridge University).



3:3 < t=ðh=VÞ < 8:8 and t=ðh=VÞ > 8:8, respectively, where Vdenotes the front velocity. The drag increase during the impactstage can be attributed to the increase in pressure on the upstreamside of the cylinder as the gravity current approaches it. Subse-quently, the drag decreases during the time interval t=ðh=VÞ� 3:3� 4:4, due to a decrease in Fw and a simultaneous increasein Fe. This decrease in Fw is due to the formation of a recircula-tion region upstream of the cylinder, as indicated by time framet=ðh=VÞ ¼ 4:1 in Fig. 13. Concurrently, the recirculation zonebehind the cylinder is convected downstream and away from thecylinder, so that Fe increases.

During the time interval t=ðh=VÞ � 4:4� 5:3, the dragincreases and reaches a second maximum. The vorticity field att=ðh=VÞ ¼ 4:7 in Fig. 13 indicates that the clockwise vortexupstream of the cylinder has been convected past the cylinder andinto the downstream recirculation zone, triggered perhaps by thepassing of a Kelvin–Helmholtz billow above the cylinder. Thischange in flow structure around the cylinder is associated with apressure increase (decrease) on the upstream (downstream) side,resulting in the increase of the overall drag. Beyondt=ðh=VÞ � 11, the drag decreases again.

Figure 14(b) provides corresponding information for the tempo-ral variation of the lift. While the lift is negative initially, itincreases rapidly as the current arrives at the cylinder. The liftattains a maximum at the same time as the drag, when Fig. 13indicates the presence of a recirculation zone above the cylinder,which causes a local pressure decrease. Subsequently, the lift

decreases as this recirculation zone is convected downstream, anddense fluid sweeps over the cylinder during the interval3:3 < t=ðh=VÞ < 4:1. A renewed increase in the lift is observed asvorticity is convected over the cylinder from 4:1 < t=ðh=VÞ < 5,which is followed by yet another decrease from 5 < t=ðh=VÞ < 6:2, when this vorticity enters the downstream recircula-tion region. During the late stages the lift remains negative, asdense fluid covers the cylinder. Just as for the drag, the figuredemonstrates that the viscous contribution to the lift is negligible.

Gonzalez-Juez et al. [110] furthermore compare two- and three-dimensional LES simulations to the experimental data of Erma-nyuk and Gavrilov [108] for the same geometric configuration, cf.Fig. 15. This comparison indicates that, while two-dimensionalsimulations accurately describe the impact stage, they overpredictthe force fluctuations during the transient and quasisteady stages.This is a consequence of the three-dimensional nature of theKelvin–Helmholtz vortices and the vortex shedding process in theexperiments, so that it takes a three-dimensional simulation toreproduce this behavior.

For a circular cylinder mounted above the ground, Gonzalez-Juez et al. [112] investigate the effect of the gap size on the forcesgenerated during the interaction with a gravity current. They findthe gap size to have a negligible effect on the maximum drag dur-ing the impact stage, while the lift fluctuations increase with thegap size. During the quasisteady stage, the case of a gravity cur-rent flowing past a circular cylinder exhibits similarities with acorresponding constant density boundary layer flow past a

Fig. 14 Temporal evolution of the drag (thick solid line) (a) and lift (thick solid line) (b) experi-enced by a bottom-mounted square cylinder during the interaction with a gravity current.“Also shown in (a) are the pressure forces on the upstream Fw (dashed line) and downstreamFe (dashed-dotted line) faces. The viscous drag and lift components (thin solid line) are muchsmaller than the pressure components” (Reprinted with permission from Gonzalez-Juez et al.[110]. Copyright 2009 by Cambridge University).

Fig. 15 “Temporal evolution of the drag (a) and lift (b) in the experiments by Ermanyuk andGavrilov [108] (solid line, squares), three-dimensional LES results (dashed line), and two-dimensional results (dashed-dotted line).” While the two-dimensional simulation overpredictsthe force fluctuations, the three-dimensional simulation accurately captures the nature of theKelvin–Helmholtz vortices and the vortex shedding process, so that it yields good forcepredictions (Reprinted with permission from Gonzalez-Juez et al. [110]. Copyright 2009 byCambridge University).



cylinder. For both flows, there is a critical gap size above whichthe forces begin to fluctuate due to vortex shedding from the cyl-inder. For small gap sizes, the interaction of the wake vorticitywith the boundary layer vorticity can lead to the suppression ofvortex shedding. Furthermore, in contrast to the constant densityboundary layer flow, a cylinder interacting with a gravity currentexperiences lift as a result of buoyancy, and also due to the deflec-tion of the wake toward the wall.

Gonzalez-Juez et al. [113] explore the wall shear stress in theneighborhood of a circular cylinder mounted above a wall, toobtain insight into the nature of the scour that could be triggeredat various stages of the interaction between the gravity currentand the cylinder. Figure 16 shows the spanwise vorticity in themidplane at different times, for a three-dimensional simulationwith Re¼ 9000, H/h¼ 2.5, D/h¼ 0.1, and G/h¼ 0.03. Here, Hindicates the domain height, h denotes the lock height, D repre-sents the diameter of the circular cylinder, and G is the gap size.Figures 16(b) and 16(c) show the formation of a jet of dense fluidin the gap during the impact stage. This jet continues during thetransient stage (Fig. 16(d)), when the main current plunges down-stream of the cylinder. These two processes are associated withhigh levels of shear stress at the bottom wall. The maximum fric-tion velocity during the impact stage is about 60% larger than thequasisteady stage value, which suggests that scouring can besevere during the impact stage. There exists considerable span-wise variation in the friction velocity as a result of the lobe-and-cleft instability, with larger friction velocities observed near thelobes, rendering these the prime candidates for strong scour.

The above investigations are extended to arrays of cylindricalobstacles, as well as to obstacles in the form of dunes or sedimentwaves, by Tokyay et al. [114–116].

4.8 Stratification Effects. Density stratification can affectthe dynamics of gravity and turbidity currents in different ways.In the context of atmospheric or oceanic flows, a gravity currentmay propagate within a stratified ambient fluid. This scenario wasexplored by Maxworthy et al. [10] via lock-exchange experimentsand corresponding two-dimensional DNS simulations. In theirsetup, the ambient fluid was linearly stratified, so that it couldgive rise to internal waves with an intrinsic frequency and propa-gation velocity. Depending on whether the front velocity of thecurrent is larger or smaller than this wave propagation velocity,one can distinguish between super- and subcritical currents. Forsubcritical currents, the authors found that internal wave interac-tions with the current resulted in an oscillatory front velocity. Forsupercritical currents, on the other hand, the front velocity wasseen to decay monotonically. A corresponding one-layer shallowwater model was developed by Ungarish and Huppert [117],although such models by their very nature cannot incorporate theeffects of the internal waves in the ambient fluid. Detailed theoret-ical models for the related scenario of a gravity current propagat-ing into a two-layer stratified ambient are developed by Flynnet al. [29] as well as by White and Helfrich [118]. In Ref. [119],the latter authors consider arbitrary density stratifications.

The effects of density stratification may be important not just inthe ambient but also within the current itself. Again, box modelsand shallow water models for such stratified currents weredeveloped by Zemach and Ungarish [120]. This importance ofstratification within the current was explored in depth in the com-putational “turbidity current with a roof” model of Balachandarand colleagues [121]. These authors consider a downslope flowdriven by suspended particles, similar to a turbidity current travel-ing downslope along the seafloor. However, unlike turbidity cur-rents in the ocean which interact with the ambient water column,their model current (shown in Fig. 17) is bounded above by a“roof” that effectively prevents the entrainment of ambient water.The computational model encompasses the channel sectionbounded by the dashed lines. The authors employ periodic bound-ary conditions in the flow direction, based on the assumption that

the mean flow in this direction does not vary. They furthermoreassume that any sediment deposited at the bottom wall is immedi-ately re-entrained into the flow, so that the overall amount ofsuspended sediment does not vary with time.

In spite of neglecting the energy loss and the dissipation experi-enced by oceanic currents through the entrainment of ambientwater, the model provides interesting insight into the damping ofturbulence due to self-stratification. The authors investigate theinfluence of the settling velocity us on the turbulence at Re¼ 180,where the Reynolds number is based on the friction velocity andthe domain half height (Re¼ u*h/�). Figure 18 shows the rms-values of the streamwise (urms) and wall-normal (wrms) velocityfluctuations, respectively, as functions of the settling velocity,where both rms values have been normalized by the average fric-tion velocity. Both velocity fluctuations are seen to decrease nearthe bottom wall for increasing settling velocity. Figure 19 showsthat for even larger settling velocities the turbulence near the bot-tom wall is strongly damped, which is a consequence of theincreasing local sediment concentration gradient. In summary, wecan distinguish two regimes: for moderate settling velocities theflow remains turbulent, although with reduced turbulence inten-sity, whereas for large settling velocities the flow near the bottomwall completely relaminarizes.

The above configuration by Cantero et al. [121] employed a no-slip wall as the roof of the turbidity current, which continuouslygenerates turbulence. This turbulence then diffuses toward thebottom wall, where it is being damped as a result of the strongerlocal stratification. Shringarpure et al. [122] extend this line ofinvestigation and consider a free-slip boundary at the top wall, sothat no turbulence is being generated there. This results in thecomplete suppression of turbulence across the entire water columnfor sufficiently high settling velocities. Figures 20 and 21 showisosurfaces of the swirling strength kci for two settling velocities,in order to identify vortical flow structures. Here, the swirlingstrength is defined as the imaginary part of the complex eigenval-ues of the local velocity gradient tensor [123,124]. For the lowersettling velocity case (us¼ 0.026) shown in Fig. 20, vortical struc-tures such as hairpin and quasi-streamwise vortices can be seen inthe domain, although there are regions which are devoid of thesevortical structures due to stratification effects. For the slightlyhigher settling velocity case (us¼ 0.0265) shown in Fig. 21, thedomain contains far fewer vortical structures at this specificinstant in time. In fact, at later times, even these vortical structuresdisappear as the flow completely relaminarizes.

Shringarpure et al. [122] provide an explanation for the sup-pression of turbulence when the settling velocity is above thecritical value. They conduct a quadrant analysis [125] of thestreamwise (u) and wall-normal velocity (w) fluctuations, byassigning these fluctuations to respective quadrants in the u, w-plane. For instance, a second quadrant (Q2) event corresponds toa negative streamwise and a positive wall-normal velocity fluctua-tion at a location. This can be interpreted as an “ejection” event,where the low speed fluid from the near-wall region is ejected intothe bulk flow. The authors show that above the critical settling ve-locity, the stable stratification effectively suppresses such secondquadrant (Q2) events, which results in fewer and weaker turbulenthairpin vortices, thereby causing them to lose their ability toautogenerate, and suppressing the mechanism that producesturbulence.

4.9 Schmidt Number Effects. With regard to oceanic appli-cations, it is important to keep in mind that salinity in water has aSchmidt number of 700. The effective Schmidt number for sedi-ment in water depends on the grain size [126,127] and can beeven higher. As a result, salinity concentration gradients can besignificantly steeper than velocity gradients, and sediment concen-tration gradients can be even steeper. The relative magnitude ofthese length scales can be estimated via the Batchelor scale kB,which represents the smallest scale for a diffusing scalar. It is



defined as the ratio of the Kolmogorov length scale g of the turbu-lent velocity field and the square root of the Schmidt number

g ¼ �3

e

3=4

; kB ¼gffiffiffiffiffiScp (44)

where e denotes the local dissipation rate of the turbulent kineticenergy. Hence, the grid resolution required to resolve the smallestdiffusive scales can be significantly smaller than the Kolmogorovscale, which drastically increases the computational effort. Tokeep the computational cost manageable, most DNS simulationsto date assume Sc¼ 1. Necker et al. [77] find that the integralproperties of turbidity currents are independent of the actual Scvalue, as long as this value is not much smaller than one. For non-Boussinesq lock-exchange flows, Birman et al. [128] report thatthe influence of Sc-variations in the range of 0.2–5.0 is small.Bonometti and Balachandar [129] perform a parametric study inorder to gain insight into the effects of the Sc value on the dynam-ics of gravity currents. Their results show that for low Re valuesthe front velocity increases with Sc, whereas at high Re values thefront velocity becomes largely independent of Sc. The authors fur-thermore find the size of the lobe-and-cleft structures to be inde-pendent of Sc. These results suggest that for many applications it

Fig. 16 Gravity current interacting with a circular cylinder mounted above a wall. “Spanwisevorticity fields at z/h 5 0.5 and different times for Re 5 9000, D/h 5 0.1, and G/h 5 0.03: (a)t=ðh=V Þ5 8:5, (b) and (c) t=ðh=V Þ5 9:2, (d) t=ðh=V Þ5 9:9, and (e) and (f) t=ðh=V Þ5 16:3. The cyl-inder is located at x/h 5 9 2 9.1. The region near the cylinder in (b) and (e) is enlarged in (c)and (f), respectively. Note the formation of a jet of dense fluid in the gap in (b) and (c), and thesubsequent plunge of the current downstream of the cylinder in (d),” Initial parameters:L 5 28, l 5 12, H 5 2.5, and d 5 1 (Reprinted with permission from Gonzalez-Juez et al. [113].Copyright 2010 by Cambridge University).

Fig. 17 “Problem setting: the flow in the channel is driven bythe excess density of the water-sediment mixture. Since thechannel is open at both ends to the tank, there is no net pres-sure gradient acting on the flow. The flow represents an ideal-ized case of a turbidity current in which ambient waterentrainment is not allowed in the channel [121]” (Reprinted withpermission from Cantero et al. [121]. Copyright 2009 by theAmerican Geophysical Union).



Fig. 18 Turbulence statistics as a function of the wall-normal coordinate for different settlingvelocities: case 0 (us 5 0), case 1 (us 5 5 3 1023), case 2 (us 5 1022), case 3 (us 5 1.75 3 1022),case 4 (us 5 2 3 1022), case 5 (us 5 2.125 3 1022). (a) Streamwise component (c) vertical com-ponent (Reprinted with permission from Cantero et al. [121]. Copyright 2009 by the AmericanGeophysical Union).

Fig. 19 Turbulence statistics as a function of the wall-normal coordinate for different settlingvelocities: case 6 (us 5 2.3 3 1022), case 7 (us 5 2.5 3 1022), case 8 (us 5 3 3 1022), case 9(us 5 3.5 3 1022), case 10 (us 5 5 3 1022). (a) Streamwise component (c) vertical component(Reprinted with permission from Cantero et al. [121]. Copyright 2009 by the American Geo-physical Union).



can be justified to employ Sc values of unity. Note, however, thatwhen double-diffusive effects [130] become important, it will beessential to account for the different diffusivities of the variousscalars, such as in the investigation of double-diffusive sedimenta-tion by Burns and Meiburg [131].

4.10 Non-Boussinesq Currents. When the density differencebetween the light and dense fluids exceeds a few per cent, non-Boussinesq effects become influential and need to be accountedfor. Examples concern tunnel fires, powder snow avalanches andpyroclastic flows triggered by volcanic eruptions [132]. One- andtwo-layer shallow water models for such flows were developed byUngarish [133,134]. They indicate that the symmetry between thelight and dense fronts observed under Boussinesq conditions isdestroyed by non-Boussinesq effects, so that strong differencesbetween the two fronts emerge.

These findings are confirmed by the depth-resolving two-dimensional Navier–Stokes simulations of Birman et al. [128],which covered full-depth lock-exchange flows over slip walls fordensity ratios from 0.2 to 0.998. The simulations show that forstrong density contrasts the light front continues to resemble anenergy-conserving current with a thickness close to half the chan-nel height. The dense front, on the other hand, has a much smallerthickness and is strongly dissipative. Lowe et al. [135] performed

corresponding laboratory experiments for non-Boussinesq cur-rents, and compared their front velocity results with numericalsimulation data of Birman et al. [128], as well as various theoreti-cal predictions. Figure 22 illustrates the front speed of the densecurrent as a function of the density ratio c. The figure shows goodagreement between the numerical results of Birman et al. and theexperiments of Lowe et al. [135]. The simulation results of Bono-metti et al. [139] show that for strong density contrasts the dissipa-tion near the bottom wall can be an order of magnitude larger thanat the interface and near the top wall. Bonometti et al. [79] fur-thermore show that the current speed during the slumping phasefor heavy non-Boussinesq currents becomes independent of thelock aspect ratio k even at very low values of k. However, for lightnon-Boussinesq currents the current speed during the slumpingphase increases with k for k� 20.

4.11 Effects of Particle Inertia. Recall that the assumptionof negligible particle inertia enabled us to obtain the velocity ofthe sediment concentration field by superimposing the settling ve-locity onto the fluid velocity field. This approach, which has beencommonly taken in turbidity current simulations to date, has theimmediate consequence that the sediment concentration field isdivergence-free, so that particles do not get ejected from vortexcores, or accumulate near stagnation points. There are situations,however, when particle inertia will be of some importance, suchas in powder snow avalanches or pyroclastic flows. Inertial effectscan result in a host of new phenomena, as discussed for some ca-nonical flows such as mixing layers or homogeneous turbulence inRefs. [140–144] and other investigations. Balachandar and col-leagues developed an equilibrium Eulerian approach based on anexpansion for small but finite particle Stokes numbers [145] thatallows for the incorporation of the leading order effects of particleinertia, while retaining the Eulerian description of the sedimentconcentration field. Cantero et al. [146] apply this approach toexplore the influence of particle inertia in gravity and turbidity

Fig. 20 Swirling strength isosurface kci 5 22 for us 5 0.026. Atthis value of the settling velocity, strong streamwise and hairpinvortices are visible near the bottom wall, indicating the pres-ence of vigorous turbulence (Reprinted with permission fromShringarpure et al. [122]. Copyright 2012 by CambridgeUniversity).

Fig. 21 Swirling strength isosurface kci 5 22 for us 5 0.0265. Atthis slightly larger value of the settling velocity, the turbulenceis strongly damped, so that the presence of hairpin vortices isgreatly reduced (Reprinted with permission from Shringarpureet al. [122]. Copyright 2012 by Cambridge University).

Fig. 22 Front velocity of a dense, non-Boussinesq current asfunction of the density ratio c. Theoretical predictions are plot-ted as lines, while experimental and simulation data are repre-sented by symbols. The theoretical prediction of Rotunno et al.[136] is shown as a solid line, and the theoretical prediction ofLowe et al. [135] is displayed as a dash-dotted line. Diamonds:experimental results of Lowe et al. [135]; squares: experimentaldata of Gr€obelbauer et al. [137]; asterisks: results of Keller andChyou [138]; triangles: simulation data of Birman et al. [128];circles: simulation data of Bonometti et al. [139]; plus symbols:simulation data of Rotunno et al. [136]. Figure based on datafrom Lowe et al. [135] and Rotunno et al. [136].



currents. Their two-dimensional simulations confirm that particlesget ejected out of the Kelvin–Helmholtz mixing layer vortices,while accumulating near the front and closer to the body of thecurrent, which in turn affects the resulting deposit profiles. Thiscan result in elevated particle concentrations near the front andthe bottom boundaries, and thus in larger front velocities. Further-more, the shear stress at the bottom boundary is seen to increase,with potential consequences for the current’s ability to erode thesediment bed.

4.12 Effects of Rough Boundaries. Gravity and turbiditycurrent simulations to date typically have assumed smooth boun-daries. Geophysical applications, on the other hand, usuallyinvolve rough boundaries. The influence of such boundary rough-ness has recently be explored in the simulations by Bhaganagar[147], who employs an immersed boundary method to resolve theflow around roughness elements at Re¼ 4000. Here, the rough-ness elements in the near-wall region are represented as a periodicseries of crests and valleys along the streamwise and spanwisedirections. Comparisons of smooth and rough-wall gravity currentresults at identical Reynolds numbers indicate that for rough wallsgravity currents travel more slowly and are subject to strongerlobe-and-cleft instabilities. Furthermore, the Kelvin–Helmholtzbillows at the interface become three-dimensional faster, andstrong three-dimensional turbulence is observed at earlier times.€Ozg€okmen and Fischer [148] employ a spectral-element model tostudy gravity currents propagating downslope over both smoothand rough boundaries, in the presence of background stratifica-tion. Their results show that currents propagating over roughboundaries experience increased drag, which results in earlier sep-aration from the bottom boundary as compared to currents propa-gating over smooth boundaries.

4.13 Effects of Viscosity Variations. Most turbidity currentsimulations assume Newtonian rheology and constant viscosity,even in the presence of sediment. It is well known, however, thatat larger concentrations the sediment can alter the rheology of thefluid by introducing non-Newtonian behavior and/or a concentra-tion dependent viscosity ([149]), with significant consequencesfor the dynamics of the flow. A striking example of the potentialeffects of viscosity variations on the flow is provided by Govin-darajan and Sahu [150], who discuss their influence on the stabil-ity of shear flows. Along similar lines, Sameen and Govindarajan[151] study the effects of wall heating and the associated changesin viscosity, heat diffusivity and buoyancy on the stability ofchannel flows. Their results show that the flow is stabilized whenthe viscosity decreases toward the wall, while an opposite effect isobserved when the viscosity increases toward the wall. Zontaet al. [152] investigate the effects of temperature-dependent vis-cosity in Newtonian fluids for forced convection in a channelwhere the fluid density is uniform. The authors show that the tur-bulence is enhanced on the cold side of the channel where the vis-cosity is higher. Zonta et al. [153] explore the effects oftemperature-dependent viscosity on stably stratified channel flowwhen the fluid density varies with temperature. The authors findthat the flow relaminarizes on the cold side of the channel wherethe viscosity is higher.

Yu et al. [154] employ a channel flow configuration to investi-gate the effects of viscosity variations on the dynamics of sedi-ment laden flows. Toward this end, they study the following fourcases. In case 0, the flow is driven by an applied streamwise pres-sure gradient without any sediment. In case 1, the sediment indu-ces a density stratification which interacts with the flow dynamics.In cases 2 and 3, in addition to the density stratification, the sedi-ment increases the viscosity, thereby damping the turbulence. Forcases 2 and 3, the authors assume a Newtonian rheology with theviscosity varying as a function of the sediment concentration. Spe-cifically, they employ the power-law rheological model proposedby Krieger and Dougherty [155]. They choose the power-law

parameters such that the increase in viscosity for case 3 is higherthan the increase in viscosity for case 2. Figure 4(c) in Yu et al.[154] shows the effective viscosity profile for the various casesstudied. For case 3, the effective viscosity increases by 40% nearthe bed and by 15% near the top wall. This increased viscositydamps the turbulence fluctuations, as can be seen from the RMSof the velocity fluctuation profiles shown in Fig. 5 by Yu et al.[154]. The velocity fluctuations along the spanwise and verticaldirections are more strongly damped as compared to the fluctua-tions along the streamwise direction. As the turbulence becomesweaker due to the enhanced viscosity, the sediment accumulatesnear the bed and its profile starts to resemble the laminar profileshown in Fig. 4(b) by Yu et al. [154]. Yu et al. [156] study thesediment transport in an oscillatory bottom boundary layer. Whenthey consider an increase in viscosity due to sedimentconcentration, the turbulence in the flow is further damped ascompared to cases where the turbulence damping occurs primarilydue to density stratification. They also show that the enhanced vis-cosity, in addition to the density stratification, can cause laminari-zation of the bottom boundary layer.

5 Open Questions and Outlook

While the modeling of compositional gravity currents anddilute, noneroding turbidity currents has reached a certain level ofmaturity, the same cannot yet be said about dense turbidity cur-rents with significant erosion, resuspension, and bedload transport.Especially the dynamics of the near-bed region of such currents,which is characterized by high sediment concentrations, is stillpoorly understood, as it is governed by intense particle–fluid andparticle–particle interactions that give rise to strongly non-Newtonian dynamics, and to mass and momentum exchangesbetween the current and the sediment bed. As a result, insight intothe erosional and depositional behaviors of such currents, and thecoupling between the motion of the current above the sedimentbed and the fluid flow inside the bed is just beginning to emerge.From the perspective of computational research into these matters,grain-resolving simulations appear to hold great potential in thisregard. Approaches along these lines have been developed by sev-eral research groups in recent years, among them [157–159], andtheir application to turbidity current research can be expected toresult in substantial progress.

A further interesting research direction can be found in the so-called inverse modeling of turbidity currents, which refers to the aposteriori reconstruction of the turbidity current flow from sparseinformation about the turbidite deposit it formed. The practicalapplication in the context of deep-water oil exploration involvesusing incomplete deposit information obtained from isolated ex-ploratory wells, in order to reconstruct the flow field during theearly stages of the turbidity current. This early flow field can thenserve as initial condition for a forward-in-time simulation thatprovides comprehensive information on the resulting deposit. Firststeps in this direction were taken by the investigation of Lesshafftet al. [160], based on a derivative-free surrogate modelingapproach, albeit for much simplified conditions.

Several interesting aspects related to the modeling of gravityand turbidity currents have not been addressed in this review.Among these are so-called intrusions, i.e., gravity currents propa-gating horizontally along intermediate density contours in strati-fied ambient fluids [161,162]. These can be encountered both inthe ocean and in the atmosphere, and similarly to gravity currentspropagating along the base of a stratified ambient, they can triggerand interact with internal gravity waves, thereby giving rise tocomplex flow patterns and mixing dynamics [163–165]. Theeffects of rotation become important for large-scale gravity andturbidity currents in the ocean [166]. Due to Coriolis forces result-ing from Earth’s rotation, such currents can develop a significantalong slope component, rather than propagating mainly in thedownslope direction, which fundamentally alters their dynamics,as well as the deposit patterns they generate. A more detailed



discussion of intrusions, along with further references, is providedby Ungarish [2].

Gravity currents in porous media [167,168] have receivedrenewed interest in recent years, due to their importance in thecontext of CO2-sequestration in depleted oil reservoirs [169].These currents frequently are amenable to computational model-ing based on Darcy’s law and its variations, rather than theNavier–Stokes equations.

Low Reynolds number, viscous gravity currents [170] areencountered in both geophysical and industrial contexts. Examplesof the former include gravity-driven flows in magma chambers aswell as lava flows [171]. These are subject to solidification, whichgives rise to additional complications in terms of developing accu-rate theoretical and computational models. Industrial applicationsof viscous gravity currents can be found in the petroleum industry,and specifically in processes related to oil well operations.

It is abundantly clear from the above discussion that the explo-ration of gravity and turbidity currents represents a rich andvibrant research area that is characterized by fascinating fluid dy-namical phenomena covering a wide range of length and timescales. The field gives rise to numerous questions of fundamentalinterest that remain unanswered to date, and whose resolution willrequire the joint efforts of experimental, theoretical, and computa-tional scientists, along with field observations.

Acknowledgment

The authors would like to acknowledge funding from varioussources over the years, including the National Science Founda-tion, the Bureau of Ocean Energy Management via Dr. GuillermoAuad, several industrial consortia coordinated by Professor BenKneller of the University of Aberdeen, as well as ConocoPhillipsand Petrobras. Computing resources have been generously madeavailable through the Community Surface Dynamics ModelingSystem under the leadership of Professor James Syvitski of UCBoulder. Special thanks go to our collaborators and discussionpartners over the years, specifically to S. Balachandar, GeorgeConstantinescu, Mike Glinsky, Rama Govindarajan, CarlosH€artel, Tom Hsu, Leonhard Kleiser, Ben Kneller, Paul Linden,Tony Maxworthy, Jim McElwaine, Jim Rottman, Bruce Suther-land, and Marius Ungarish, as well as to the graduate students andpostdocs at UCSB, among them Vineet Birman, Francois Blanch-ette, Zac Borden, Esteban Gonzalez-Juez, Brendon Hall, NathanKonopliv, Lutz Lesshafft, and Mohamad Nasr-Azadani.

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