S. Balachandar

On the front velocity of gravity currents

Highly resolved three-dimensional and two-dimensional simulations of gravity currents in planar and cylindrical configurations are presented. The volume of release of the heavy fluid is varied and the different phases of spreading, namely acceleration, slumping, inertial and viscous phases, are studied. The incompressible Navier–Stokes equations are solved assuming that the Boussinesq approximation is valid for small density difference. The simulations are performed for three different Reynolds numbers (Re): 895, 3450 and 8950 (this particular choice corresponds to values of Grashof number: 10 5 ,1 .5×10 6 and 10 7 , respectively). Following their sudden release, the gravity currents are observed to go through an acceleration phase in which the maximum front velocity is reached. As the interface of the current rolls up, the front velocity slightly decreases from the maximum and levels off to a nearly constant value. At higher Re, three-dimensional disturbances grow rapidly and the currents become strongly turbulent. In contrast, in two-dimensional simulations, the rolled-up vortices remain coherent and very strong. Depending on the initial Re of the flow and on the size of the release, the current may transition from the slumping to the inertial phase, or directly to the viscous phase without an inertial phase. New criteria for the critical Re are introduced for the development of the inertial phase. Once the flow transitions to the inertial or viscous phase, it becomes fully three-dimensional. During these phases of spreading, two-dimensional approximations underpredict the front location and velocity. The enhanced vortex coherence of the two-dimensional simulations leads to strong vortex interaction and results in spurious strong time variations of the front velocity. The structure and dynamics of the three-dimensional currents are in good agreement with previously reported numerical and experimental observations.

Two-dimensional and unsteady natural convection in a horizontal enclosure with a square body

A two-dimensional solution for unsteady natural convection in an enclosure with a square body is obtained using an accurate and efficient Chevyshev spectral collocation method. A spectral multidomain methodology is used to handle a square body located at the center of the computational domain. The physical model considered here is that a square body is located at the center between the bottom hot and top cold walls. To see the effects of the presence of a body on natural convection between the hot and cold walls, we considered the cases that the body maintains the adiabatic and isothermal thermal boundary conditions for different Rayleigh numbers varying in the range of 103 to 106. When the Rayleigh number is small, the flow and temperature distribution between the hot and cold walls shows a symmetrical and steady pattern. At the intermediate Rayleigh number, the fluid flow and temperature fields maintain the steady state but change their shape to the nonsymmetrical pattern. When the Rayleigh number is high, the flow and temperature fields become time dependent, and their time-averaged shapes approach the symmetric pattern again. The Rayleigh number for the fluid flow and temperature fields to become nonsymmetrical and time dependent depends on the thermal boundary conditions of a body. The variation of time- and surface-averaged Nusselt numbers on the hot and cold walls and at the body surfaces for different Rayleigh numbers and thermal boundary conditions are also presented to show the overall heat transfer characteristics in the system.

High-resolution simulations of cylindrical density currents

Three-dimensional highly resolved simulations are presented for cylindrical density currents using the Boussinesq approximation for small density difference. Three Reynolds numbers (Re) are investigated (895, 3450 and 8950, which correspond to values of the Grashof number of 10 5 , 1.5 × 10 6 and 10 7 , respectively) in order to identify differences in the flow structure and dynamics. The simulations are performed using a fully de-aliased pseudospectral code that captures the complete range of time and length scales of the flow. The simulated flows present the main features observed in experiments at large Re. As the current develops, it transitions through different phases of spreading, namely acceleration, slumping, inertial and viscous. Soon after release the interface between light and heavy fluids rolls up forming Kelvin-Helmholtz vortices. The formation of the first vortex sets the transition between acceleration and slumping phases. Vortex formation continues only during the slumping phase and the formation of the last Kelvin-Helmholtz vortex signals the departure from the slumping phase. The coherent Kelvin-Helmholtz vortices undergo azimuthal instabilities and eventually break up into small-scale turbulence. In the case of planar currents this turbulent region extends over the entire body of the current, while in the cylindrical case it only extends to the regions of Kelvin-Helmholtz vortex breakup. The flow develops three-dimensionality right from the beginning with incipient lobes and clefts forming at the lower frontal region. These instabilities grow in size and extend to the upper part of the front. Lobes and clefts continuously merge and split and result in a complex pattern that evolves very dynamically. The wavelength of the lobes grows as the flow spreads, while the local Re of the flow decreases. However, the number of lobes is maintained over time. Owing to the high resolution of the simulations, we have been able to link the lobe and cleft structure to local flow patterns and vortical structures. In the near-front region and body of the current several hairpin vortices populate the flow. Laboratory experiments have been performed at the higher Re and compared to the simulation results showing good agreement. Movies are available with the online version of the paper.

Forces on a finite-sized particle located close to a wall in a linear shear flow

To understand and better model the hydrodynamic force acting on a finite-sized particle moving in a wall-bounded linear shear flow, here we consider the two limiting cases of (a) a rigid stationary spherical particle in a linear wall-bounded shear flow and (b) a rigid spherical particle in rectilinear motion parallel to a wall in a quiescent ambient flow. In the present computations, the particle Reynolds number ranges from 2 to 250 at separation distances to the wall from nearly sitting on the wall to far away from the wall. First we characterize the structure of the wake for a stationary particle in a linear shear flow and compare with those for a particle moving parallel to a wall in a quiescent ambient [see L. Zeng, S. Balachandar, and P. Fischer, J. Fluid Mech. 536, 1 (2005)]. For both these cases we present drag and lift results and obtain composite drag and lift correlations that are valid for a wide range of Re and distance from the wall. These correlations have been developed to be consistent wit...

Static and dynamic contact angles of water droplet on a solid surface using molecular dynamics simulation.

The present study investigates the variation of static contact angle of a water droplet in equilibrium with a solid surface in the absence of a body force and the dynamic contact angles of water droplet moving on a solid surface for different characteristic energies using the molecular dynamics simulation. With increasing characteristic energy, the static contact angle in equilibrium with a solid surface in the absence of a body force decreases because the hydrophobic surface changes its characteristics to the hydrophilic surface. In order to consider the effect of moving water droplet on the dynamic contact angles, we apply the constant acceleration to an individual oxygen and hydrogen atom. In the presence of a body force, the water droplet changes its shape with larger advancing contact angle than the receding angle. The dynamic contact angles are compared with the static contact angle in order to see the effect of the presence of a body force.

Direct Numerical Simulations of Planar and Cylindrical Density Currents

The collapse of a heavy fluid column in a lighter environment is studied by direct numerical simulation of the Navier-Stokes equations using the Boussinesq approximation for small density difference. Such phenomenon occurs in many engineering and environmental problems resulting in a density current spreading over a no-slip boundary. In this work, density currents corresponding to two Grashof (Gr) numbers are investigated (10 5 and 1.5 × 10 6 ) for two very different geometrical configurations, namely, planar and cylindrical, with the goal of identifying differences and similarities in the flow structure and dynamics. The numerical model is capable of reproducing most of the two- and three-dimensional flow structures previously observed in the laboratory and in the field. Soon after the release of the heavier fluid into the quiescent environment, a density current forms exhibiting a well-defined head with a hanging nose followed by a shallower body and tail. In the case of large Gr, the flow evolves in a three-dimensional fashion featuring a pattern of lobes and clefts in the intruding front and substantial three-dimensionality in the trailing body. For the case of the lower Gr, the flow is completely two dimensional. The dynamics of the current is visualized and explained in terms of the mean flow for different phases of spreading. The initial phase, known as slumping phase, is characterized by a nearly constant spreading velocity and strong vortex shedding from the front of the current. Our numerical results show that this spreading velocity is influenced by Gr as well as the geometrical configuration. The slumping phase is followed by a decelerating phase in which the vortices move into the body of the current, pair stretch and decay as viscous effects become important. The simulated dynamics of the flow during this phase is in very good agreement with previously reported experiments.

Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction

We examine the development of a thermal plume originating from a localized heat source using direct numerical simulation. The Reynolds number of the plume, based on source diameter and the characteristic buoyancy velocity, is chosen to be 7700, which is sufficiently large so that the flow turns to a fully turbulent state. A highly resolved grid of 622 million points is used to capture the entire range of turbulent scales in the plume. Here at the source, only heat has been added with no mass or momentum addition and accordingly the vertical evolution of the mass, momentum and buoyancy fluxes computed from the simulation have been verified to follow those of a pure thermal plume. The computed vertical evolution of the time-averaged centreline velocity and temperature are in good agreement with available experimental measurements. Investigation of the time evolution of the plume shows periodic formation of vortex ring structure surrounding the main ascending column of hot fluid. The vortex ring forms very close to the heat source and even at formation it is three-dimensional. The vortex ring ascends with the plume and at an elevation of about two diameters it strongly interacts with and destabilizes the central column and subsequently a complex turbulent flow arises. Thus, relatively laminar, transitional and fully turbulent regimes of the plume evolution can be identified. In the fully turbulent regime, complex three-dimensional hairpin-like vortex structures are observed; but vestiges of the coherent vortex rolls that form close to the source can be observed in the turbulent statistics. It is shown that local entrainment consists of contraction and expulsion phases. Such instantaneous mechanisms drive the entrainment process, and the instantaneous entrainment coefficient shows large variation in both time and space with local values up to three times higher than the average entrainment level. Such findings support the view that entrainment mechanisms in plumes should be considered from an unsteady point of view. Movies are available with the online version of the paper.

Interaction of a planar shock wave with a dense particle curtain: Modeling and experiments

The interaction of a planar shock wave with a dense particle curtain is investigated through modeling and experiments. The physics in the interaction between a shock wave with a dense gas-particle mixture is markedly differently from that with a dilute mixture. Following the passage of the shock wave, the dense particle curtain expands rapidly as it propagates downstream and pressures equilibrate throughout the flow field. In the simulations, the particles are viewed as point-particles and are traced in a Lagrangian framework. A physics-based model is then developed to account for interphase coupling. Compared to the standard drag law, four major improvements are made in the present interphase coupling model to take into account: (1) unsteady force contributions to particle force; (2) effect of compressibility on hydrodynamic forces; (3) effect of particle volume fraction on hydrodynamic forces; (4) effect of inter-particle collision. The complex behavior of the dense particle curtain is due to the interp...

Direct numerical simulation of stratification effects in a sediment-laden turbulent channel flow

This work presents direct numerical simulations of sediment-laden turbulent channel flows employing an Eulerian–Eulerian approach. The flow is driven by a constant pressure gradient and self-stratifies owing to the presence of settling sediment particles. The study is performed so as to systematically vary the settling velocity of the sediment particles, which is the parameter that controls the stratification of the flow. The effect of stratification is to suppress vertical Reynolds fluxes. For the larger values of settling velocity considered here, the flow relaminarizes below the streamwise velocity maximum, completely suppressing the turbulent vertical mass and momentum transfer. For the cases that remain actively turbulent, Reynolds fluxes are partially suppressed, but remain larger than their viscous counterparts. These cases present a clear pattern of low-speed streaks with a mean separation of approximately 100 wall units. Moreover, for these cases a pattern of high concentration streaks is found to be in high spatial correlation with the low speed streaks.

Drag and lift forces on a spherical particle moving on a wall in a shear flow at finite Re

Recent research (Zeng, PhD thesis, 2007; Zeng et al ., Phys. Fluids , vol. 21, 2009, art. no. 033302) has shown that both the shear- and wall-induced lift contributions on a particle sharply increase as the gap between the wall and the particle is decreased. Explicit expressions that are valid over a range of finite Re were obtained for the drag and lift forces in the limiting cases of a stationary particle in wall-bounded linear flow and of a particle translating parallel to a wall in a quiescent ambient. Here we consider the more general case of a translating and rotating particle in a wall-bounded linear shear flow where shear, translational and rotational effects superpose. We have considered a modest Reynolds number range of 1–100. Direct numerical simulations using immersed boundary method were performed to systematically figure out the characteristics of hydrodynamic forces on a finite-sized particle moving while almost in contact with a wall. We present composite correlation for the hydrodynamic forces which are in agreement with all the available low-Reynolds-number theories.