Kelvin–Helmholtz instabilities and mixing in surface-propagating gravity currents

An analytical vorticity-based model is introduced for steady-state inviscid Boussinesq gravity currents in sheared ambients. The model enforces the conservation of mass and horizontal and vertical momentum, and it does not require any empirical closure assumptions. As a function of the given gravity current height, upstream ambient shear and upstream ambient layer thicknesses, the model predicts the current velocity as well as the downstream ambient layer thicknesses and velocities. In particular, it predicts the existence of gravity currents with a thickness greater than half the channel height, which is confirmed by direct numerical simulation (DNS) results and by an analysis of the energy loss in the flow. For high-Reynolds-number gravity currents exhibiting Kelvin–Helmholtz instabilities along the current/ambient interface, the DNS simulations suggest that for a given shear magnitude, the current height adjusts itself such as to allow for maximum energy dissipation.

We present experimental results on the development of gravity currents moving onto sloping boundaries with slope angles $\unicode[STIX]{x1D703}=7^{\circ }$, $10^{\circ }$ and $15^{\circ }$. Different regimes of flow development are observed depending on the slope angle and on the initial velocity and density profiles, characterized by the Richardson number $J_{i}=\unicode[STIX]{x1D6FF}_{i}{g_{0}}^{\prime }/\unicode[STIX]{x0394}u_{i}^{2}$, where $\unicode[STIX]{x1D6FF}_{i}$, $\unicode[STIX]{x0394}u_{i}$ and $g_{0}^{\prime }$ are, respectively, the velocity interface thickness, the maximum velocity difference and reduced gravity at the beginning of the slope. For $J_{i}>0.7$ and the larger slope angle, the flow strongly accelerates, reaches a maximum at the beginning of the Kelvin–Helmholtz instability, then decelerates and re-accelerates again. For $0.3<J_{i}<0.6$, instability occurs earlier and velocity oscillations are less. When $J_{i}\leqslant 0.3$ the increase in velocity is smooth. The magnitude of velocity oscillation depends on the combined effect of $J_{i}$ and slope angle, expressed by an overall acceleration parameter $\overline{T_{a}}=(\unicode[STIX]{x1D6FF}_{i}/U_{i})((U_{c}-U_{i})/x_{c})$, which, to first order, is given by $J_{i}\sin \unicode[STIX]{x1D703}$, where $U_{c}$ and $x_{c}$ are, respectively, the velocity and position at instability onset. The velocity increases smoothly up to an equilibrium state when $\overline{T_{a}}\leqslant 0.06$ and exhibits an irregular behaviour at larger values of $\overline{T_{a}}$. The critical Richardson number $J_{c}$ decreases with increasing $J_{i}$ (increasing $\unicode[STIX]{x1D6FF}_{i}/h_{i}$) which is due to wall effects and $\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\neq 1$. After the beginning of Kelvin–Helmholtz instability, entrainment rates are close to those of a mixing layer, decreasing to values of a gravity current after the mixing layer reaches the boundary. It is shown here that the interfacial instability during current development affects the bottom shear stress which can reach values of $c_{D}\approx 0.03$ regardless of initial conditions. By solving numerically the depth integrated governing equations, the gravity flow velocity, depth and buoyant acceleration in the flow direction can be well predicted for all the performed experiments over the full measurement domain. The numerical results for the experiments with $J_{i}>0.3$ predict that the current requires a distance of at least $x_{n}\approx 40h_{i}$ to reach a normal state of constant velocity, which is much larger than the distance $x_{n}\approx 10h_{i}$ required in the case of a current with $J_{i}\leqslant 0.3$ that is commonly assumed for downslope currents.

The results of a numerical study of two- and three-dimensional Boussinesq density currents are described. They are aimed at exploring the role of the Schmidt number on the structure and dynamics of density driven currents. Two complementary approaches are used, namely a spectral method and a finite-volume interface capturing method. They allow for the first time to describe density currents in the whole range of Schmidt number 1 ≤ Sc ≤ ∞ and Reynolds number 102 ≤ Re ≤ 104. The present results confirm that the Schmidt number only weakly influences the structure and dynamics of density currents provided the Reynolds number of the flow is large, say of O(104) or more. On the contrary low- to moderate-Re density currents are dependant on Sc as the structure of the mixing region and the front velocities are modified by diffusion effects. The scaling of the characteristic density thickness of the interface has been confirmed to behave as (ScRe)−1/2. Three-dimensional simulations suggest that the patterns of lobes and clefts are independent of Sc. In contrast the Schmidt number is found to affect dramatically (1) the shape of the current head as a depression is observed at high-Sc, (2) the formation of vortex structures generated by Kelvin–Helmholtz instabilities. A criterion is proposed for the stability of the interface along the body of the current based on the estimate of a bulk Richardson number. This criterion, derived for currents of arbitrary density ratio, is in agreement with present computed results as well as available experimental and numerical data.

The dynamics of a sea breeze front interacting with the heavily urbanized New York City area are examined. In addition, we investigate the impact of the urban‐influenced sea breeze front on transport and diffusion of simulated passive tracer plumes. We employ the U. S. Navy's Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS®—a registered trademark of the Naval Research Laboratory) to perform a nested simulation with data assimilation for the sea breeze event of 9 August 2004. Available surface and upper‐air observations are used to validate the simulation. We also perform a sensitivity study in which the urban influence is removed (no‐urban).The sea breeze front has characteristics of a density current, including an elevated head at the leading edge. The density current moves slowly and unevenly across the city. Kelvin–Helmholtz billows form in the region of the density current head, and the results show evidence of the occurrence of Kelvin–Helmholtz instability (KHI). The density current head is greatly elevated owing to the enhanced surface drag of the urban area. This urban influence is further explored in the no‐urban simulation, in which the head of the density current is not elevated to the same degree and KHI does not occur.The sea breeze/density current has a large impact on transport and diffusion of simulated tracer plumes, not only changing the direction of plume motion due to the wind shift but also redistributing tracer material in the vertical so as to produce dramatic, rapid changes in near‐surface concentration as the front passes. In particular, large upward vertical velocity at the head of the density current advects tracer material to large elevations, greatly reducing near‐surface concentration. After passage of the front, tracer is released into the shallow density current and confined near the surface, enhancing near‐surface concentrations. KHI results in turbulent mixing at the upper surface of the plume, allowing for a small reduction in near‐surface concentration. Published in 2007 by John Wiley & Sons, Ltd.

2D numerical simulation of density currents using the SPH projection method

Small differences in the densities of a river confluence's tributaries (i.e. 0.5 kg m $^{-3}$ ) have been proposed to cause coherent streamwise-oriented vortices (SOVs) in its mixing interface. These secondary flow structures are thought to result from density-driven gravity currents being laterally confined between the converging flows. However, empirical evidence for density SOVs and the confined gravity current mechanism is lacking. To this end, experiments are carried out in a laboratory confluence permitting a spectrum of thermal density differences between its tributaries. Particle image velocimetry and laser-induced fluorescence are used simultaneously to study the mixing interface's dynamics. The sensitivity of the mixing interface's secondary flow structure to the confluence's momentum ratio and the magnitude of the density difference is evaluated. Density SOVs are confirmed in the mixing interface and are caused by the gravity currents being confined laterally as the opposing flows merge and accelerate downstream. The SOVs are largest and most coherent when the momentum of the dense channel is greater than that of the light channel. The dynamics of these secondary flow structures is strongly coupled to periodic vertically orientated Kelvin–Helmholtz instabilities. The striking similarities between the empirically reproduced SOVs herein and those recently observed at the Coaticook-Massawippi confluence (Quebec, Canada), despite a two-order magnitude difference in physical scale, suggest density SOVs are a scale-independent flow structure at confluences when specific, yet relatively common, hydraulic and density conditions align.

In the present study the velocity profiles and the instability at the interface of a two phase water-oil fluid were investigated. The main aim of the research project was to investigate the instability mechanisms that can cause the failure of an oil spill barrier. Such mechanisms have been studied before for a vast variety of conditions (Wicks in Fluid dynamics of floating oil containment by mechanical barriers in the presence of water currents. In: Conference on prevention and control of oil spills, pp 55–106, 1969; Fannelop in Appl Ocean Res 5(2):80–92, 1983; Lee and Kang in Spill Sci Technol Bull 4(4):257–266, 1997; Fang and Johnston in J Waterway Port Coast Ocean Eng ASCE 127(4):234–239, 2001; among others). Although the velocity field in the region behind the barrier can influence the failure significantly, it had not been measured and analyzed precisely. In the present study the velocity profiles in the vicinity of different barriers were studied. To undertake the experiments, an oil layer was contained over the surface of flowing water by means of a barrier in a laboratory flume. The ultrasonic velocity profiler method was used to measure velocity profiles in each phase and to detect the oil–water interface. The effect of the barrier geometry on velocity profiles was studied. It was determined that the contained oil slick, although similar to a gravity current, can not be considered as a gravity current. The oil–water interface, derived from ultrasonic echo, was used to find the velocity profile in each fluid. Finally it was shown that the fluctuations at the rearward side of the oil slick head are due to Kelvin–Helmholtz instabilities.

A novel method for combined particle image velocimetry and laser induced fluorescence is described and results from an experiment in a stratified flow are presented. A standard two-dimensional, one camera particle image velocimetry configuration is used, acquiring images of the seeding particles and the dye marking the current simultaneously and separating the two fields digitally. The implementation of the postprocessing method, its capabilities and the necessary conditions for its use are discussed in detail. The proposed method is applied to an arrested density current front. The front is made stationary by opposing a uniform velocity profile, obtained from the combination of a moving floor and the recirculation of fresh water in the channel. To improve the quality of the images, the current is made optically homogeneous by matching the refractivity index throughout the domain. Instantaneous and time averaged fields are obtained for both velocity and density. Simultaneous measurements of these fields provide insight in the mixing processes at the front of the density current. In particular persistent billow generation, similar to that found in shear layers and associated with Kelvin–Helmholtz instabilities, is observed.

We present semi-implicit (implicit-explicit) formulations of the compressible Navier–Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge–Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit Runge–Kutta method for all the tests studied, especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including inertia-gravity waves, density current (i.e., Kelvin–Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of nonreflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and nonreflecting boundary conditions (NRBCs). It is shown that the NRBCs exert a strong impact on the accuracy and efficiency of the models.

Density currents in a rotating fluid are produced by releasing a volume of buoyant fluid from a lock at one end of a long rotating channel. Coriolis forces hold the current against one wall. It is observed that the velocity and depth of the nose decrease exponentially in time, implying that the nose can effectively come to a halt at a finite distance from the lock. In reality though, the flow regime eventually changes and a viscous wedge-shaped intrusion continues. The high-Reynolds-number currents contain three-dimensional turbulence a short distance behind the nose, but the influence of rotation causes this to become quasi-two-dimensional further upstream. The intrusion and turbulent motions represent a forcing to the lower layer that produces vortex and wave-like motions which penetrate deep into the lower-layer fluid. It is shown that the exponential decay can be attributed to radiation of momentum by these inertial waves. The width l of the turbulent current varies with distance behind the nose, from 0.6 times the local time-dependent deformation radius at the ‘head’ to l ≈ R0 far upstream, where R0 is the initial deformation radius in the lock. The nose of the boundary current is unstable, with billows appearing near the tip of the intruding nose and leading to an intermittent breakup of the ‘head’ structure and oscillations of the nose velocity. These oscillations are rapid, often having frequencies much greater than f (where f = 2Ω is the Coriolis parameter), and, along with the production of the turbulence that is so characteristic of the currents, are attributed to a Kelvin–Helmholtz instability. Rotationally dominated baroclinic waves appear only a very large distance behind the nose.

Hydrostatic vs. non-hydrostatic modelling of density currents developing two dimensionally on steep and mild slopes

In this work we address the frontal instability of gravity currents. The planar laser-induced fluorescence (PLiF) flow visualization is utilized to analyze the detailed dynamics of the current, which are generated in a lock-exchange Perspex tank. We believe that two dominant modes of instability determine the complex structures at the head of the flow. The first one resembles Kelvin–Helmholtz instability, which results in Kelvin–Helmholtz billows rolling up in the shear zone above the head. The other, categorized as convective instability known as "lobes and clefts", which stems from ground friction as well as unstable inverse density stratification, and is considered to be the cause for the disruption of the span-wise symmetry of Kelvin–Helmholtz billows. Moreover, our observations indicate that the convective instability also contributes to a secondary instability associated with Kelvin–Helmholtz vortex breakdown. These instabilities not only play a central role in shaping the three-dimension characteristics of the currents, but also govern the mixing and entrainment mechanisms. Therefore, more precise measurement of the positions of the frontal instability and the flow structures, especially the turbulent structures is indeed necessary.

The frontal instability of lock-exchange density currents is numerically investigated using dissipative particle dynamics (DPD) at the mesoscopic particle level. For modeling two-phase flow, the “color” repulsion model is adopted to describe binary fluids according to Rothman–Keller method. The present DPD simulation can reproduce the flow phenomena of lock-exchange density currents, including the lobe-and-cleft instability that appears at the head, as well as the formation of coherent billow structures at the interface behind the head due to the growth of Kelvin–Helmholtz instability. Furthermore, through the DPD simulation, some small-scale characteristics can be observed, which are difficult to be captured in macroscopic simulation and experiment.

Abstract. Soft-sediment deformation structures can provide valuable information about the conditions of parent flows, the sediment state and the surrounding environment. Here, examples of soft-sediment deformation in deposits of dilute pyroclastic density currents are documented and possible syn-eruptive triggers suggested. Outcrops from six different volcanoes have been compiled in order to provide a broad perspective on the variety of structures: Soufrière Hills (Montserrat), Tungurahua (Ecuador), Ubehebe craters (USA), Laacher See (Germany), and Tower Hill and Purrumbete lakes (both Australia). The variety of features can be classified in four groups: (1) tubular features such as pipes; (2) isolated, laterally oriented deformation such as overturned or oversteepened laminations and vortex-shaped laminae; (3) folds-and-faults structures involving thick (>30 cm) units; (4) dominantly vertical inter-penetration of two layers such as potatoids, dishes, or diapiric flame-like structures. The occurrence of degassing pipes together with basal intrusions suggest fluidization during flow stages, and can facilitate the development of other soft-sediment deformation structures. Variations from injection dikes to suction-driven, local uplifts at the base of outcrops indicate the role of dynamic pore pressure. Isolated, centimeter-scale, overturned beds with vortex forms have been interpreted to be the signature of shear instabilities occurring at the boundary of two granular media. They may represent the frozen record of granular, pseudo Kelvin–Helmholtz instabilities. Their recognition can be a diagnostic for flows with a granular basal boundary layer. Vertical inter-penetration and those folds-and-faults features related to slumps are driven by their excess weight and occur after deposition but penecontemporaneous to the eruption. The passage of shock waves emanating from the vent may also produce trains of isolated, fine-grained overturned beds that disturb the surface bedding without occurrence of a sedimentation phase in the vicinity of explosion centers. Finally, ballistic impacts can trigger unconventional sags producing local displacement or liquefaction. Based on the deformation depth, these can yield precise insights into depositional unit boundaries. Such impact structures may also be at the origin of some of the steep truncation planes visible at the base of the so-called "chute and pool" structures. Dilute pyroclastic density currents occur contemporaneously with seismogenic volcanic explosions. They can experience extremely high sedimentation rates and may flow at the border between traction, granular and fluid-escape boundary zones. They are often deposited on steep slopes and can incorporate large amounts of water and gas in the sediment. These are just some of the many possible triggers acting in a single environment, and they reveal the potential for insights into the eruptive and flow mechanisms of dilute pyroclastic density currents.

A detailed experimental study was conducted to investigate the hydraulics, interfacial instability, and mixing in the supercritical region of exchange flows downstream of a sill crest. Measurements of the velocity field and the interface position were obtained using flow visualization and particle image velocimetry. Large periodic fluctuations in the measurements of the flow rate and interface position were caused by the Kelvin–Helmholtz (KH) instabilities at the interface as well as the internal seiche. These KH instabilities caused entrainment of fluid from the upper layer into the lower layer, with the entrainment coefficient considerably larger than the values for gravity currents.