Role of head of turbulent 3-D density currents in mixing during slumping regime

Transcription

1 Role of head of turbulent 3-D density currents in mixing during slumping regime 1, a) Kiran Bhaganagar Associate Professor, Department of Mechanical Engineering, 1 UTSA Blvd, San Antonio, TX 78248, USA (Dated: 30 September 2016) 1

2 Afundamentalstudywasconductedtoshedlightonentrainmentandmixingin buoyancy-driven Boussinesq density currents. Large-eddy simulation (LES) was performed on lock-exchange (LE) release density currents - an idealized test bed to generate density currents. As dense fluid was released over a sloping surface into an ambient lighter fluid, the dense fluid slumps to the bottom and forms a characteristic head of the current. The dynamics of the head dictated the mixing processes in LE currents. The key contribution of this study is to resolve an ongoing debate on mixing: We demonstrate that substantial mixing occurs in the early stages of evolution in an LE experiment and that entrainment is highly inhomogeneous and unsteady during the slumping regime. Guided by the flow physics, entrainment is calculated using two di erent but related perspectives. In the first approach, the entrainment parameter (E) is defined as the fraction of ambient fluid displaced by the head that entrains into the current. It is an indicator of the e ciency in which ambient fluid is displaced into the current and it serves as an important metric to compare the entrainment of dense currents over di erent types of surfaces, e.g. roughness configuration. In the second approach, E measures the net entrainment in the current at an instantaneous time t over the length of the current. Net entrainment coe cient is a metric to compare the e ects of flow dynamical conditions i.e. lock-aspect ratio that dictates the fraction of buoyancy entering the head, and also the e ect of the sloping angle. Together, the entrainment coe cient and the net entrainment coe cient provide an insight on the entrainment process. The active head of the current acts as an engine that mixes the ambient fluid with the existing dense fluid, the 3-D lobes and clefts on the frontal end of the current causes recirculation of the ambient fluid into the current, and Kelvin-Helmholtz rolls are the mixers that entrain the ambience into the current. Buoyancy and shear production occur at the interface in the head region of the current, and transport of TKE by Reynolds stresses result in high TKE. a) kiran.bhaganagar@utsa.edu 2

3 I. INTRODUCTION Buoyancy-driven Boussinesq density currents are an important class of stratified flows and are of great significance in oceanic flows and atmospheric pollution problems 1.Therelease of high-density fluid from a source into the ambient atmosphere (such as in atmospheric pollution) or the mixing of dense waters with ambient waters (such as in coastal regions) results in formation of a characteristic horizontally traveling turbulent plume. The plume carries the dense fluid and vigorously mixes it with the ambiance in a process referred to as entrainment 2,3. Turbulence mixing is the key physics that dictates entrainment and thus drives these fronts. The classic LE release method, wherein a fixed volume of heavy fluid is released into a lighter ambient fluid, is an idealized test bed to generate buoyant turbulent currents. In this method, the density di erence between the two fluids, though very small, creates substantial buoyancy force to generate highly turbulent currents that propagate at high Reynolds numbers 4.Thefundamentalinstabilitythatgeneratesturbulenceresultsfrom shear between the current, and the ambiance that results in classic 2-D Kelvin-Helmholtz instabilities 5 excites the turbulence modes and leads to 3-D lobe-cleft instabilities that result in span-wise inhomogeneity 6,7. LE release dense currents have been studied in considerable detail. Though significant progress has been made in our understanding of LE front characteristics 5,8,9 and their dynamics and though Froudes and Reynolds numbers 14,15 dictate the current dynamics, important questions remain regarding entrainment. When dense fluid is released over a sloping surface into an ambient lighter fluid, the dense fluid slumps to the bottom and forms acharacteristicheadofthecurrent. SimpsonandBritter s 16 shadow-graph revealed that the frontal zone of the current referred to as the head is the most characteristic feature of the current, with intense mixing occurring near the leading edge. Britter and Simpson 11 and Britter and Linden 5 have determined current characteristics in terms of the Froude s number, based on depth and speed of the head. With progress in experimental techniques 6,17 19 and high performance computing, complex features 7,20 25 in the head of the current have been revealed. The dynamics of the head dictates the mixing processes in LE currents. The foremost point of the head is slightly elevated from the surface, and the velocity is termed the front velocity. After the initial transience, the current travels with a nearly constant Froude s number to the regime referred to as the slumping phase, where the inertial forces 3

4 balance the buoyancy forces (e.g. Simpson 1 ) Active debate persists on the extent to which mixing in the head of the current occurs during the slumping phase. Key experiments have demonstrated dense fluid in the current mixes, with ambient fluid leading to dilution in the current and mixing being significantly confined to the head region. Hallworth et al. 26 claimed that the head remains unmixed and that entrainment is negligible in the slumping phase. Hacker, Linden, and Dalziel 8 reported mixing in the head region immediately after the collapse of the current. Recently, Fragoso, Patterson, and Wettlaufer 27 concluded from their resolved optical transmission experiments that entrainment occurs throughout the slumping regime. The aim of this study is to shed light on the entrainment process during the slumping regime. The concept of mixing within the context of buoyancy plumes was introduced by Morton, Taylor, and Turner 3 in terms of an entrainment parameter, E, defined as E = w E /V,where w E is the entrainment velocity of the ambient fluid into the plume and V is the local velocity scale. There is significant experimental evidence that the head of the current plays adynamicroleinthemixingprocess:mcelwaine s 28 theory suggests a vortex with rotation in the current in the head of the flow, and the presence of strong circulation near the head of the current is found in particle image velocimetry measurements by Thomas, Dalziel, and Marino 29 and Kneller, William & McCa rey 30.Recently,SamasiriandWoods 31 conducted experiments on 2-D axi-symmetric turbulent lock-release density currents define entrainment as the fraction of ambient fluid displaced by the head that enters the current. They found that axisymmetric LE currents entrain a fraction of 0.33 of the ambience. Motivated by this study, we used highly resolved LES to obtain instantaneous mixing in 3-D currents in asimilarmanner. In light of our current understanding of entrainment in dense currents, a key question of the entrainment process during the slumping regime needs to be resolved. We must quantify the inhomogenous process and unsteady manner in which entrainment and mixing occur within the head of the current. Turbulence plays an important role in mixing within the current. Turbulence kinetic energy (TKE) is generated from buoyancy production and shear production. TKE production mechanisms provide fundamental insights into the mixing process, though it has been accepted that mixing occurs at the interface through 2-D Kelvin-Helmholtz instabilities and due to convective instability that results in a 3-D lobecleft structure at the head 1,4. To date, very little is known about turbulence processes at 4

5 the interface. Understanding the TKE process will provide further insight to the mixing processes in LE flows. LES is used as a tool to generate high-resolution, spatial- and time-accurate LE release turbulent currents at high Reynolds numbers. LES has previously been demonstrated as a valid tool to understand flow physics 32,33.Theanalysisincludesdetailedflowvisualizations to capture, for the first time, the three- dimensional features of the current as the front travels down the slope into the ambiance. As entrainment is a highly unsteady and non-linear process that is guided by the flow physics, it is calculated using two di erent but related perspectives. In the first approach, entrainment is quantified as a fraction of the ambient fluid displaced by the head; hence, it is scaled by the area covered by the head as it advances into the ambiance during the time interval dt. The increase in volume flux during this time interval is scaled by the volume of the displaced ambient fluid. The entrainment parameter obtained by this method indicates the fraction of displaced ambient fluid that entrains into the current. In the second approach, net entrainment is calculated to estimate the overall entrainment in the current by averaging the increase in the volume of the current at a time twithrespecttooriginallockvolumeoverthelengthofthecurrent.theconclusionsbased on the above interpretations will guide us to understand the in-homogenous and unsteady entrainment process during the slumping phase. The paper is organized as follows: The details of the LES formulation are presented in Section II. Results are presented in Section III, and the conclusions are presented in Section IV. II. LES METHODOLOGY The LES solver is used to solve 3-D in-compressible Navier-Stokes equations using the Boussinesq approximation. Transport equations for momentum and scalar density are solved in finite-volume formulation. A conservative form of the Navier-Stokes equation is solved on non-uniform Cartesian meshes. Considering Boussinesq approximation, the constantdensity filtered Navier-Stokes equations and the advection-di usion equation for the density are solved. The filtered governing equations are: 5

6 Continuity Equation where u i is the filtered velocity to be used in LES. Momentum i =0 (u i u j ) = + ij + tij + g i j i j 0 where tij is the shear stress due to viscous e ect caused by turbulent eddies and: ij = µ i i 2 k ij # (3) where µ is the molecular dynamic viscosity. Substituting the expression for ij in Equation (2), assuming constant molecular viscosity µ = µ 0 we (u i u j ) = 1 + j i i 2 k ij j + 0 g i (4) where 0 = µ 0 / 0 and R ij is SGS stress tensor that replaces tij. Applying continuity constraint for incompressible flow we (u i u j ) + j j Substituting Equation (1) in Equation (5) we (u i u j ) + j j (u i u j j + 0 g i j p + 2! 0 3 k +( 0 + j 2 3 k ij + 0 g i i + g i j 0 The additional term within the parenthesis of pressure gradient in-e ect leads to the calculation of modified or resolved kinematic pressure. Nevertheless, the contribution of the isotropic part of Reynolds stresses is negligible with respect to the static pressure field. Thus the additional term clubbed to the pressure field projects negligible influence. Boussinesq approximation is realized in the last term of Equation (7) where density variations are assumed to contribute only in the gravity term. Henceforth the sum ( t + 0 )isdenotedas eff. 6

7 The scalar transport equation is solved, where in the variation in density field is evaluated with the help of an additional scalar ( u j j! (8) Where apple eff = t Sc t + 0 Sc Dynamic Smagorinsky eddy viscosity model is used to solve the turbulence equations. It gives more robust and accurate estimation of turbulence and eddies as the Smagorinsky constant (Cs) is recalculated dynamically in every iteration as a function of space and time to follow the changes in the fluid. Besides, in near wall region the dynamic Smagorinsky model is able to damp the Smagorinsky coe cient due to use of the dynamic procedure, which relaxed the need for additional damping function (i.e. Van Driest damping like in constant Smagorinsky model). Özgökmen et al.33 also showed that the dynamic Smagorinsky model lead in significant improvement of accuracy in the prediction of mixing compared to constant-coe cient Smagorinsky model in lock-exchange density current problem. All equations are discretized with a second-order bounded implicit central-di erence Crank-Nicholson scheme. The convective terms have been discretized using a second-order bounded central-di erence scheme. A Gauss scheme with linear interpolation was used to discretize the di usive terms in the momentum equation. At the bottom (wall), a no-slip boundary condition is applied for the velocity and zero flux conditions of the density field. Azeronormalgradientforvelocityanddensityfieldsisemployedatthetopfreesurface,at the inlet, and at the outlet boundary. The flow is assumed to be periodic in the spanwise direction. The flow field is initialized with zero velocity throughout the domain, and a random disturbance (1%) was applied to the density field to trigger turbulence. The serial code has been parallelized using MPI based on the domain decomposition technique.. The time step (dt) was estimated amidst the simulation run to maintain the courant number within the range of with excellent accuracy of capturing mixing physics by optimum computational cost. The Gauss scheme with linear interpolation had been employed for di usive terms (Laplacian) in momentum equation. The simulations were performed with grid spacing of 10-3, and mesh of size 700x125x100. Grid convergence has been obtained at this mesh resolution. Qualitative and quantitative validations were performed with a well-established direct 7

8 FIG. 1: Physical configuration of LE release currents over sloping bottom FIG. 2: Physical configuration of LE release currents over sloping bottom numerical simulation (DNS) solver developed by Bhaganagar 34. Front evolution with time from the LES solver was compared with the DNS solver for dense currents over horizontal surface, as shown in FIG. 2. Further, at the Reynolds number (based on lock-height and front velocity) of 4000, Froude s number (based on local front velocity and initial reduced gravity) was estimated as 0.59, a close match with DNS studies by Bhaganagar 34 and Härtel et al. 20 where they obtained Fr for the smooth wall as 0.6 and 0.61, respectively. In addition to the front conditions, instantaneous 2-D and 3-D density and velocity structures demonstrated a good match with DNS results. 8

9 III. RESULTS The physical problem considered is as follows: As shown in FIG. 2, two liquids with very slightly di erent densities ( 1 is heavier fluid and 2 is lighter fluid) are placed into a rectangular channel of height H =0.1m, Spanb =0.25m, lengthl =3m that is initially separated by a full lock of length L =0.5mandthesameheight as that of the channel. The bottom wall of the channel is tilted at an angle,, with respect to the horizontal. The same initial conditions are used for all cases, with an initial reduced gravity of g0 0 =0.05. Initial available buoyancy for the currents is B 0 =(g0lhb) 0 (initialreducedgravitytimesthe volume of the lock). Upon removal of the lock at time t =0,adensitycurrentformsoverthe bottom surface, consisting of lock fluid that entrains the surrounding ambient fluid and grows in size as it moves forward. The four simulated cases are shown in Table 1. Simulations are performed in the slumping phase, where the front travels with nearly constant velocity (u f ) as shown in Table 1. As dense currents over sloping surfaces travel faster with increasing velocity and increasing slope, a bulk Reynolds number was calculated using averaged front velocity and front height in the slumping phase. As seen in Table 1, for the same initial conditions, the bulk Reynolds number increased from 1800 to 2400 for a slope varying from 0 o to 10 o.themeanheightofthecurrentincreaseswithincreasingslopeangle.birmanet. al. 10 observed that front height increases with the angle of the slope, as the front entrains more fluid as the slope angle increases. Case Slope u f h m f Re b Fr b Fr TABLE I: LE release currents in domain of size 3m 0.1m 0.25m. Thelocklengthis 0.5m, andlockheightis0.1m. ThelockAR is 5. Bulk Reynolds number and bulk Froude s number are based on slumping velocity and averaged front height (h m f )duringthe slumping phase. Fr 0 is based on current velocity and initial reduced gravity. To keep this manuscript at a manageable length, the flow visualization is shown for 9

10 the case of dense currents over 2.5 o sloping surfaces. The di erences resulting from slope variation are discussed. Figure 2 shows the evolution of the 3-D density contours over time as the front propagates forward for dense currents over a 2.5 o slope. Upon release of the gate, the front is generated with original lock fluid. As seen in FIG. 3a, 2-D shear-driven Kelvin-Helmholtz billows appear due to the shear between the front and the ambiance. As the front advances, span-wise in-homogeneity begins at the leading edge with 3-D lobe-cleft instabilities, as seen in FIG 3b. The 3-D (convective) instability 4 results in clefts at the base of the head, cause mixing with the ambiance. Both dominant modes of instabilities intensify as the current advances, as seen in FIG. 3c. The lobe-clefts are highly unsteady, as seen in FIG 3d3h. The 3-D velocity vectors at an instantaneous time (front has reached x f =1.7m from the lock) during the slumping phase are shown in FIG 4a. At this time, the current has a nearly constant head region. Dense fluid is being supplied to the rear of the head at a velocity higher than the front velocity. A circulation develops near the head with essentially a shear layer between two boundaries. As the head advances into the ambiance, it displaces the ambient fluid, a fraction of which enters the current and mixes with the dense fluid, and the remainder is pushed in an opposite direction from the front, as clearly seen in FIG. 4b. Regions corresponding to Kelvin-Helmholtz billows also show circulation at the current/ambient interface that indicates mixing. These findings are consistent with those of Kneller, Bennett, and McCa rey 30,whousedlaserDoppleranemometrytorevealavortex near the head with strong circulation within the head region and away from the surface. The v w velocity vectors at time corresponding to location x f =7H and 9H are shown in FIG. 4c and FIG. 4d respectively. The recirculating vortex generated near the leading edge entrains the fluid in region x =0 5H, whichisthe active regionofthehead.the active head region refers to the frontal portion of the front which is dynamically active, and where the entrainment occurs. It is interesting to note entrainment is unsteady and in-homogenous process, as it dominant between x =3H 4H when front is at x f =7H (FIG. 4c), whereas, it is dominant between x =2.5H 4H when front is further downstream at x f =9H respectively 10

11 (a) (b) (c) (d) (e) (f) (g) (h) FIG. 3: Time evolution of density current over 2.50 sloping surface, as front is at xf of (a) 0.5H, (b) 0.75H, (c) 3H, (d) 5H, (e) 7H, (f) 8H, (g)10h, (h) 12H from the lock 11

12 (a) (b) (c) (d) FIG. 4: (a) 3-D vector around the head region of the density current overlaid on the density contour, (b) u-v vectors in x-y plane as the front is at x = 1.7m from the lock gate for current over a 2.5o sloping surface. A re-circulation vortex forms at frontal end of the current, and the ambient fluid displaced by the front enters the currents through lobes and clefts at the head and through Kelvin-Helmholtz billows in active head region of the front. Ambient fluid above current moves opposite to the front direction, v-w velocity vectors in head region x/h = 0 5 and y/h = 0 Hmax at t corresponding to (c) xf = 7H from lock, (d) xf = 9H from lock. A. Mixing Processes and Entrainment Coefficient: Two di erent approaches are used to evaluate the entrainment coefficient. The first approach builds on the model of Kneller et al.30 and Sher & Woods35 : As the front moves forward, the current displaces the ambient fluid. The entrainment coefficient is a fraction of the volume of the fluid that entrains into the current. Let us assume that the front moves from its position at t1 to t2, as shown in FIG. 5. The volume of ambient fluid displaced by 12

13 FIG. 5: Model for calculation of the bulk entrainment coe of Kneller et al. 30. cient using the first approach the current during the time (t 2 t 1 )asthefronttravelsadistancedx, andtheheightofthe current is h m,ish m u f.duringthistimedt, thechangeofspan-wiseaveragedvolumeofthe fluid per unit span in the current is dv/dt. During the slumping phase, as the front travels with a constant velocity u f,thebulkentrainmentcoe E = cient(e)isdefined: dv/dt h m (t)u f (9) The volume of the fluid V (t) inthecurrentisestimatedbyintegratingthedensityheight profiles from tail to nose of the current, expressed as: V (t) = Z xf 0 h(x, t)dx (10) Where, h(x, t) istheheightoffluidinthecurrentuptoh m,themaximumheightofthe front based on 10% threshold criterion. To obtain the height of the front, the interface of the mixed and the ambient fluid is identified as the height at which the density of the fluid is 10% of the ambient density. 10 % threshold was used as it captured all the mixed fluid in the current, including the K-H rollers. FIG. 6 shows E plotted against the front location using Eq. 9. For the dense currents flowing over the horizontal, after an initial transient value of E =0.5, the entrainment coe cient is 0.4 uptox =7H from the lock and settles to 0.2 forrestoftheslumping phase. The higher fraction of ambient fluid that is displaced by the head entrains during the earlier stages of slumping rather than the later stages. Samasiri and Woods 31 for lockexchange currents over horizontal surface at Fr in the range of obtainede as 0.33, which in comparable in value to that obtained in the present study. However, in their experiment, entrainment fraction reaches a steady-state value fairly quickly as the height to 13

14 FIG. 6: Entrainment parameter calculated using Eq. Entraiment1 plotted vs. front location scaled by the height of the channel the length ratio of the lock is in the range Whereas, in the present simulation with the lock length five times that of the lock-height, the entrainment process is unsteady and spatially in-homogenous during slumping phase. It should be noted that lock-aspect ratio is a critical parameter that determines the time-scales for the entrainment to reach steadystate equilibrium condition. As the lock-aspect ratio controls the amount of initial buoyancy that is carried by the head. E is higher for dense currents over a sloping surface, indicating higher entrainment. For dense currents over a 2.5 o sloping surface, E varies between 0.4 and 0.5. It increases to a higher fraction of 0.4to0.8fora5 o sloping surface. A distinct di erence between dense currents over horizontal and sloping surfaces is that a higher fraction entrains earlier in the slumping phase for the horizontal surface, whereas higher entrainment occurs at the later stages of slumping for dense current over sloping surfaces.. The entrainment coe cient is an indicator of e ciency in which ambient fluid is displaced into the current, and hence an important metric to compare the di erences in entrainment that arises due to di erences in boundary conditions, such as surface roughness. In the second approach to calculating entrainment, we evaluate the net entrainment parameter as the entrainment per length of the current (E l ). For this purpose, the net entrainment parameter is defined as increase in the volume of fluid in the current from the initial lock volume scaled by a 14

15 length scale, length of the density current (l = x b x f ), and a velocity scale, front velocity, to calculate entrainment at every time step. This is a representation of the total increase in the volume of the mixed fluid in the current at that instant of time. The volume change in the system represents the amount of ambient fluid that is entrained into the current. Entrainment (E l )isgivenas: E l = V (t) V 0 lu f t Here, the volume of original dense fluid is V o,andthevolumeoffluidinthecurrentv (t) is estimated by integrating the density height profiles from the tail to the nose of the current, expressed as: V (t) = Z xf where h(x, t) istheheightof10%thresholdandx f (11) x b h(x, t)dx (12), x b are the positions of the leading edge and the tail of the current and lock locations, respectively. FIG. 7a and 7b show the total volume in the current and the unmixed dense fluid in the current calculated using Eq. 12. The volume has been scaled by the initial lock volume, which is the same for all cases considered in this study. The total volume in the current increases as the front advances and the unmixed dense fluid decreases due to mixing. At the beginning of the simulation, the entire volume of lock fluid is unmixed with V o being 1.0. As the current advances, ambient fluid enters the current and mixes with the dense fluid. Thus, the volume of the unmixed fluid is reduced, as seen in FIG. 7b. The higher the slope, the faster the rate at which dense fluid exits the lock and mixes with the entrained fluid. Hence, the volume of the current increases with the angle of the slope. For both horizontal and inclined flowing currents, the volume increases at a higher rate in the early stages of slumping than in later stages. FIG. 8 shows the entrainment parameter calculated using Eq. 11. The entrainment parameter decreases with time: As the current flows down the slope, its contribution to overall entrainment parameter decreases as the length of the current increases. The entrainment parameter is high in the beginning, as it is associated with the formation of the head 33. During the slumping phase, the length of the current increases at a uniform rate as the current travels with constant velocity. However, since the slope of entrainment is less steep during the early stages of slumping, E drops much faster toward the end of the slumping phase. E has not yet reached a steady state, and it continues to drop for the domain considered. The final value at the end of the domain increases from 0.01 to 0.04 as the slope increases from horizontal 15

16 (a) (b) FIG. 7: (a) Non-dimensional total volume of the currents vs. front location; (b) Non-dimensional volume of the unmixed dense fluid vs. front location. Non-dimensional volume is calculated with respect to initial lock volume. 16

17 FIG. 8: Entrainment parameter per unit length calculated using Eq.11. to 10 o. This is the picture we have so far for bottom-flowing LE flows: On the release of the lock, heavier lock fluid slumps to the bottom and a frontal region with a well-defined head appears. The head carries a fraction of volume of the original fluid in the lock. As the front advances, it displaces the lighter ambient fluid. Depending on the angle of inclination over which the density currents flow, a fraction of this displaced fluid enters the current. The entrainment begins in the early stages of the slumping phase. For horizontally flowing density currents, the initial fraction of the ambient fluid that entrains the current is a high of 0.4. In the later stages of slumping, it settles to 0.2. For dense currents over sloping surfaces, E is higher for higher slopes and varies from depending on slope. The entrained ambient fluid mixed with the dense fluid in the active head region of the current. The active head region is that with strong Kelvin-Helmholtz billows at the interface. Heavy fluid from the lock enters the active head region with velocity timesthefront velocity, and dense fluid travels much faster for higher slopes. The net entrainment factor, entrainment per unit length of the current is high during the formation of the head (before the slumping regime) and it decreases during the slumping regime, as entrainment is not uniform. As the front advances, the increase in volume in current does not commensurate with increase in length of the current, indicating nonactive tail region of the current does not contribute to increase in mixed volume. Finally, to complete our analysis of the mixing process, we investigate turbulence gener- 17

18 ation processes, which are the source of mixing. Contour of the mean velocity is shown in FIG. 9a. While a high-velocity core supplies dense fluid to the head, velocity decreases away from the head. Turbulence production results from buoyancy and shear-driven production. In turbulence mixing, energy is extracted from the mean flow through the buoyancy flux and shear stress and then transferred to turbulence mixing. Turbulence at the interface causes the ambient fluid to entrain into the dense current and dilute it. The TKE equation is given as: where u 0 {z } P S + g 0 0 v 0 {z } P B P S =TKEproductionfromshear P B =TKEproductionfrombuoyancy 2=DissipationofTKE D =verticalturbulenttransportoftke!! v 0 u0 u 0 2 {z } {z } 2 D Turbulence production is a contribution from buoyancy-driven (P B )andshear-driven(p S ) production. In the turbulence mixing process, energy is extracted from the mean flow through the shear stress (u 0 v 0 )andtransferredtoturbulencemixing. Thebuoyancyproduction transfers potential energy into the turbulent kinetic energy. Turbulence at the interface causes the ambient fluid to entrain into the dense current and dilute it. TKE is consumed by dissipation (2) at the molecular scales.the vertical transport of TKE term(d) acts to redistribute the TKE vertically. The time-averaged TKE shear production, and TKE buoyancy productions and TKE are shown in FIG 9a to 9d, respectively. Vertical Shear from the mean velocity is the primary source of turbulence in a stratified density current system, which contributes to mixing in the interface region. TKE production due to shear P S and buoyancy P B is dominant behind the head, maximum around the head, and gradually decreases away from the head. The TKE production occurs in the head region 0 5H from the nose. However, due to the transport of energy by shear stress and buoyancy flux, we observe strong mixing far upstream of the current s head. Although mean shear, TKE shear, and buoyancy production occur at the 18

19 (a) (b) (c) (d) FIG. 9: (a) Mean velocity contours (b) Shear production of TKE, (c) Buoyancy production of TKE, (d) TKE for dense currents over 2.5 o slope plotted in the x-y plane using spanwise-and time-averaged fields. 19

20 interface, maximum TKE is generated near the bottom wall due to the transport of TKE by Reynolds stress, which is in accordance with Kneller et al. 30.Theheadofthecurrentcarries afractionoftheinitiallockfluid,undergoesmixingveryquicklyafterthelockrelease,and does not undergo further dilution as dense fluid is constantly supplied to the head region. As was experimentally demonstrated by Beghin, Hopfinger & Britter 36,thefractionofbuoyancy carried by the head is dependent on initial lock conditions. As all of the cases considered in this study have the same initial conditions, there are no di erences in the fraction of initial buoyancy carried by the head. TKE production due to shear at the interface and buoyancy production (the density gradient between the fluid in the current and the ambiance) occur in active head region of the current. Both shear and buoyancy productions contribute to TKE production. Though TKE production is dominant only at the interface, areas of TKE occur inside the current due to the transport of TKE by Reynolds stress, which results in mixing. IV. CONCLUSIONS When higher-density fluid from a source enters an ambiance of lighter fluid, the density di erenceeven though smallgenerates substantial buoyancy forces that develop a turbulent front, a density current that propagates horizontally. This process entrains the ambient fluid into the current and results in mixing. Details of turbulent mixing and entrainment processes in density currents are still unclear. A fundamental study was conducted to investigate the entrainment and mixing in LE density currents flowing down a slope, which comprises an idealized test bed to study buoyancy-driven turbulent stratified flows. While experiments and numerical models developed over the past decade have improved our understanding of LE flows, entrainment is still debatable and controversial. In this experiment, LES was used as a tool to simulate LE currents in the slumping phase, where the inertial and buoyancy forces balance each other to result in currents flowing with uniform velocity. Motivated by the theory of Kneller et al. 30 and experiments of Samasiri & Woods 31 we generated flow animations during the slumping regime to investigate cause of mixing in head region. The flow animations provide a substantial proof, for the first time, for Kneller el al s 30 theory. In a horizontally flowing current, an entrainment fraction corresponding to 0.3 to0.4 ofthe displaced ambient fluid mixes with the dense fluid in the head. Experiments of Samasiri & 20

21 Woods 31 for lock-exchange currents over horizontal surface with lock aspect ratio varying from andfr in the range of obtainedentrainmentfractionof0.33, which is comparable to that obtained in the present study. However, a distinct notable di erence is that entrainment fraction reaches a steady-state value fairly quickly as the height to the length ratio of the lock is in the range 0.5 1, whereas in the present study where the lock length is five times that of the lock-height, the entrainment process is unsteady and spatially in-homogenous during slumping phase. This is reasonable, as the head contains an initial fraction of the buoyancy of the lock, and is constantly supplied by the lock fluid. For the same initial conditions, a sloping surface entrains a larger fraction of the fluid, with E being 0.5 and0.8 astheslopeincreasesto2.5 o and 5 o,respectively. Overall,theentrainment coe cient is an indicator of e ciency in which ambient fluid is displaced into the current. It serves as an important metric to compare the e ciency of dense currents over di erent types of surfaces, e.g. roughness configuration, and di erent flow conditions, e.g. Slope, Reynolds number. Lock aspect ratio is an important parameter that dictates the time-scales of entrainment process. To estimate the spatially in-homogenous and unsteady entrainment, net entrainment coe cient, which is entrainment per unit length of the current, is evaluated. Averaged over a length the current, as the current advances forward and the length of the current increases, the net entrainment coe cient decreases in time indicating that the contribution of entrainment to the overall entrainment decreases in time. Unsteady, and in-homogenous entrainment occurs during the slumping regime of the density currents. Net entrainment coe cient is a metric to compare the e ects of flow dynamical conditions i.e. lock-aspect ratio that dictates the fraction of buoyancy entering the head, and also the e ect of the sloping angle. Together, the entrainment coe cient and the net entrainment coe cient provide an insight on the entrainment process. As front travels down a slope, higher-velocity dense fluid enters the trailing edge of the head region supplying the dense fluid. The front displaces the ambient air, a fraction of which enters the currents due to Kelvin-Helmholtz mixing. A recirculation vortex forms at the head of the current due to 3-D lobe-cleft features at the frontal end of the current, and a strong shear layer develops between the two boundaries (forward-moving dense fluid near the bottom and reverse-moving ambient fluid on the top). The ambient fluid above the current is pushed backward. As the front advances into the ambience, the current displaces the volume of the ambient fluid, which is the active head region 21

22 of the current. As the current advances forward, the net entrainment coe cient decreases indicating entrainment occurs substantially in active head region. Further, higher is the slope, higher are both the entrainment coe cient and net entrainment coe cient indicating more mixing occurs with increasing slope. Both the entrainment estimation approaches indicate entrainment is higher during the beginning of the slumping phase for dense currents over slope as the lock fluids travels with higher velocity into the head region of the current and resulting in higher entrainment and mixing. The active head of the current (0 5H from the nose of the current) acts as an engine that mixes the ambient fluid with the existing dense fluid, the 3-D lobes and clefts on the frontal end of the current causes re-circulation of the ambient fluid into the current, and Kelvin-Helmholtz rolls are the mixers that entrain the ambiance into the current. Vertical shear is generated in the head region that stratifies the flow resulting in buoyancy and shear production of turbulence that occurs at the current interface, and TKE thus generated is transported by Reynolds stresses inwards resulting in high TKE within the current. This work is o ered as a contribution to improve understanding of the entrainment of LE density currents and to a rm the theory that the head of an LE density current drives the entrainment process during the slumping regime, when the viscous forces are not important and the inertial forces balance the buoyancy forces. Tribute: This manuscript on stratified turbulence is o ered as a personal tribute to my teacher as buoyancy-driven turbulence has been one of his favorite topics. The final exam question posed by Lumley in graduate turbulence class at Cornell University was - During acoldchillyeveninginithaca,coldairentersyourheatedapartmentthroughasmallslit opening. Develop fundamental scaling laws to estimate the time-scales for mixing. At that time, I have not realized, but what he was looking for is density currents form as heavy cold air enters the room with lighter ambient room, and the problem that he asked was mixing e ects in typical lock-exchange dense currents. Prof. Lumley I have finally solved the mixing problem and present the solution as a tribute in this special issue Legacy of John Lumley. 22

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