Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k–ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.     

Marangoni Mixed Convection Boundary Layer Flow

The paper deals with a steady coupled dissipative layer, called Marangoni mixed convection boundary layer, which can be formed along the interface of two immiscible fluids, in surface driven flows. The mixed convection boundary layer is generated when besides the Marangoni effects there are also buoyancy effects due to gravity and external pressure gradient effects. We shall use a model proposed by Golia and Viviani (L’ Aerotecnica missili e Spazio 64 (1985) 29–35, Meccanica 21 (1986) 200–204) wherein the Marangoni coupling condition has been included into the boundary conditions at the interface. The similarity equations are first determined, and the pertinent equations are solved numerically for some values of the governing parameters and the features of the flow and temperature fields as well as the interface velocity and heat transfer at the interface are analysed and discussed.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

A two‐scale low‐Reynolds number turbulence model

In this study, a two‐scale low‐Reynolds number turbulence model is proposed. The Kolmogorov turbulence time scale, based on fluid kinematic viscosity and the dissipation rate of turbulent kinetic energy (ν, ε), is adopted to address the viscous effects and the rapid increasing of dissipation rate in the near‐wall region. As a wall is approached, the turbulence time scale transits smoothly from a turbulent kinetic energy based (k, ε) scale to a (ν, ε) scale. The damping functions of the low‐Reynolds number models can thus be simplified and the near‐wall turbulence characteristics, such as the ε distribution, are correctly reproduced. The proposed two‐scale low‐Reynolds number turbulence model is first examined in detail by predicting a two‐dimensional channel flow, and then it is applied to predict a backward‐facing step flow. Numerical results are compared with the direct numerical simulation (DNS) budgets, experimental data and the model results of Chien, and Lam and Bremhorst respectively. It is proved that the proposed two‐scale model indeed improves the predictions of the turbulent flows considered. Copyright © 2000 John Wiley & Sons, Ltd.     

NUMERICAL PREDICTION OF TWO-PHASE FLOW IN BUBBLE COLUMNS

A numerical model is described for the prediction of turbulent continuum equations for two-phase gas–liquid flows in bubble columns. The mathematical formulation is based on the solution of each phase. The two-phase model incorporates interfacial models of momentum transfer to account for the effects of virtual mass, lift, drag and pressure discontinuities at the gas–liquid interface. Turbulence is represented by means of a two-equation k–ϵ model modified to account for bubble-induced turbulence production. The numerical discretization is based on a staggered finite-volume approach, and the coupled equations are solved in a segregated manner using the IPSA method. The model is implemented generally in the multipurpose PHOENICS computer code, although the present appllications are restricted to two-dimensional flows. The model is applied to simulate two bubble column geometries and the predictions are compared with the measured circulation patterns and void fraction distributions.     

Statistical models for predicting the effect of bidisperse particle collisions on particle velocities and stresses in homogeneous anisotropic turbulent flows

The purpose of this paper is to present and compare two statistical models for predicting the effect of collisions on particle velocities and stresses in bidisperse turbulent flows. These models start from a kinetic equation for the probability density function (PDF) of the particle velocity distribution in a homogeneous anisotropic turbulent flow. The kinetic equation describes simultaneously particle–turbulence and particle–particle interactions. The paper is focused on deriving the collision terms in the governing equations of the PDF moments. One of the collision models is based on a Grad-like expansion for the PDF of the velocity distributions of two particles. The other model stems from a Grad-like expansion for the joint fluid–particle PDF. The validity of these models is explored by comparing with Lagrangian simulations of particle tracking in uniformly sheared and isotropic turbulent flows generated by LES. Notwithstanding the fact that the fluid turbulence may be isotropic, the particle velocity fluctuations are anisotropic due to the impact of gravitational settling. Comparisons of the model predictions and the numerical simulations show encouraging agreement.     

Generalized detection of a turbulent front generated by an oscillating grid

This report presents experimental results on the propagation of a turbulent front induced by an oscillating grid starting from rest. The purpose of this preliminary investigation is to implement and validate detection methods of the turbulent/non-turbulent interface, which are based on flow measurements (velocity and vorticity) and scalar intensity, for oscillating grid turbulence. This is done using particle image velocimetry (PIV) and fluorescent dye visualization, separately. The results of both techniques describe the spreading of the turbulent front, confirming the known dependency of the front location, H, on time, t. It is demonstrated, that the level-based detection of a turbulent front can be applied to an unsteady flow, such as grid turbulence advancing into a fluid at rest.     

Numerical simulation of turbulent free-surface flow in curved channel

This paper presents first results of numerical simulation of turbulent free-surface flow. Simple implementation of surface capturing method is based on the variable density approach. The flow is treated as if there is only one fluid, but with variable material properties (density, viscosity). The switch in these values is done by a function resulting from the mass conservation principle. This approach simplifies the implementation of turbulence model. In this case the SST k−ω model was chosen in modification given by Hellsten.Numerical solution was carried out by finite-volume method with explicit Runge-Kutta time-integration. The artificial compressibility method was used for time-marching search for steady state solution. The whole model was tested on horizontally placed square-sectioned 90^∘ bend, which was partially filled by the water. The main goal of this study was to demonstrate the applicability of this model and solution method for capturing the water-air interface as well as for predicting the turbulent effects in both fluids.     

Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for <Emphasis Type="Italic">k</Emphasis> and <Emphasis Type="Italic">ε</Emphasis> Transport Equations

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface.     

Direct numerical simulation of a freely decaying turbulent interfacial flow

Whereas Large Eddy Simulation (LES) of single-phase flows is already widely used in the CFD world, even for industrial applications, LES of two-phase interfacial flows, i.e. two-phase flows where an interface separates liquid and gas phases, still remains a challenging task. The main issue is the development of subgrid scale models well suited for two-phase interfacial flows. The aim of this work is to generate a detailed data base from direct numerical simulation (DNS) of two-phase interfacial flows in order to clearly understand interactions between small turbulent scales and the interface separating the two phases. This work is a first contribution in the study of the interface/turbulence interaction in the configuration where the interface is widely deformed and where both phases are resolved by DNS. To do this, the interaction between an initially plane interface and a freely decaying homogeneous isotropic turbulence (HIT) is studied. The densities and viscosities are the same for both phases in order to focus on the effect of the surface tension coefficient. Comparisons with existing theories built on wall-bounded or free-surface turbulence are carried out. To understand energy transfers between the interfacial energy and the turbulent one, PDFs of the droplet sizes distribution are calculated. An energy budget is carried out and turbulent statistics are performed including the distance to the interface as a parameter. A spectral analysis is achieved to highlight the energy transfer between turbulent scales of different sizes. The originality of this work is the study of the interface/turbulence interactions in the case of a widely deformed interface evolving in a turbulent flow.     

Experiments on entrainment in an annulus with and without velocity gradient across the density interface

The entrainment process in a two layer density stratified fluid column was studied experimentally by imposing external shear stress on one or both layers. The experiments have been conducted in an annular tank containing two water layers of different salt concentration and the shear stress was applied by means of rotating screens. The following quantities were measured: the screen velocity (which was kept constant during each experiment), the stress at the upper screen, and vertical profiles of circumferential velocity and density at different radial locations. When equal stress was imposed at the surface of the upper layer and at the bottom of the lower layer, entrainment took place from the two sides of the density interface at equal rate so that the interface was stationary in the central position between the two screens and there was no velocity gradient across the interface. The dependence of the entrainment coefficient on Richardson number obtained in these experiments was similar in form to that obtained in the shear-free experiments with an oscillating grid (e.g. Nokes 1988). When a shear stress was applied at the upper surface only, the upper layer depth increased with time and a velocity gradient existed at the interface. The influence of the interfacial velocity gradient on the entrainment rate was studied by comparing the rates obtained with and without this velocity gradient. The entrainment rates were approximately the same for high values of the Richardson number while at low Richardson number the entrainment rate was much larger when a velocity gradient existed across the interface. The main results of this work are as follows:

1. Despite the curved geometry of the annular system, the dependence of the entrainment coefficient on Richardson number for shear-free interface experiments was found to be similar in form to that obtained for oscillating grid experiments.
2. The entrainment across the interface is due to turbulent energy generated at some distance from the interface by an external source (i.e. shear stress induced by a screen) and due to turbulence produced locally at the interface by a velocity gradient. The relative contribution of each turbulence source to the total entrainment was found to depend on the stability of the interface.

     

Direct numerical simulation of thermocapillary flow based on the Volume of Fluid method

A numerical method for direct simulation of thermal Marangoni effects at dynamically deformable interface of two-phase incompressible fluids is developed. The approach is based on the Volume of Fluid (VOF) method with special focus on the numerical treatment of the temperature surface gradient because of its decisive role as the driving force of the flow. The surface gradient calculation is split into computing its length and direction in order to satisfy the correct thermal boundary condition at the interface without losing mobility of the interface. The method is applied to three different types of thermocapillary flow, namely thermocapillary migration of a droplet in an ambient fluid with linear temperature gradient, thermocapillary convection in a liquid layer under linear temperature gradient along the interface, and Marangoni convection due to Bénard–Marangoni instability. In the first case, different aspects of the dynamics of the migration are considered for validation such as the terminal migration velocity, the initial acceleration and quantification of the wall effects. The simulation also considers high convective heat transfer and covers a wide range of Marangoni numbers up to 5000, where good agreement with both theoretical and experimental results is achieved. In the second case, the convection velocity in the liquid layer is compared with an analytical result. In the final application, pattern formation due to the Bénard–Marangoni instability in a liquid layer in square geometry of small aspect ratio is investigated for realistic Biot number and dynamically deformable fluid interface. The results show good agreement with experiments from literature, where our numerical simulation also predicts cell pattern for a particular aspect ratio which is outside the limitation of the above cited experimental work.     

Turbulence and cavitation models for time-dependent turbulent cavitating flows

Cavitation typically occurs when the fluid pressure is lower than the vapor pressure at a local thermodynamic state,and the flow is frequently unsteady and turbulent.To assess the state-of-the-art of computational capabilities for unsteady cavitating flows,different cavitation and turbulence model combinations are conducted.The selected cavitation models include several widely-used models including one based on phenomenological argument and the other utilizing interface dynamics.The kε turbulence model with additional implementation of the filter function and density correction function are considered to reduce the eddy viscosity according to the computed turbulence length scale and local fluid density respectively.We have also blended these alternative cavitation and turbulence treatments,to illustrate that the eddy viscosity near the closure region can significantly influence the capture of detached cavity.From the experimental validations regarding the force analysis,frequency,and the cavity visualization,no single model combination performs best in all aspects.Furthermore,the implications of parameters contained in different cavitation models are investigated.The phase change process is more pronounced around the detached cavity,which is better illus-trated by the interfacial dynamics model.Our study provides insight to aid further modeling development.     

Numerical simulations of negatively buoyant jets in an immiscible fluid using the Particle Finite Element Method

Negatively buoyant jets consist in a dense fluid injected vertically upward into a lighter ambient fluid. The numerical simulation of this kind of buoyancy‐driven flows is challenging as it involves multiple fluids with different physical properties. In the case of immiscible fluids, it requires, in addition, to track the motion of the interface between fluids and accurately represent the discontinuities of the flow variables. In this paper, we investigate numerically the injection of a negatively buoyant jet into a homogenous immiscible ambient fluid using the Particle Finite Element Method and compare the two‐dimensional numerical results with experiments on the injection of a jet of dyed water through a nozzle in the base of a cylindrical tank containing rapeseed oil. In both simulations and experiments, the fountain inlet flow velocity and nozzle diameter have been varied to cover a wide range of Froude Fr and Reynolds Re numbers ( 0.1 < Fr < 30, 8 < Re < 1350), reproducing both weak and strong laminar fountains. The flow behaviors observed for the different numerical simulations fit in the regime map based on the Re and Fr values of the experiments, and the maximum fountain height is in good agreement with the experimental observations, suggesting that particle finite element method is a useful tool for the study of immiscible two‐fluid systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulations of non‐equilibrium turbulent boundary layer flowing over a bump

Large‐eddy simulation (LES) and Reynolds‐averaged Navier–Stokes simulation (RANS) with different turbulence models (including the standard k?ε, the standard k?ω, the shear stress transport k?ω (SST k?ω), and Spalart–Allmaras (S–A) turbulence models) have been employed to compute the turbulent flow of a two‐dimensional turbulent boundary layer over an unswept bump. The predictions of the simulations were compared with available experimental measurements in the literature. The comparisons of the LES and the SST k?ω model including the mean flow and turbulence stresses are in satisfied agreements with the available measurements. Although the flow experiences a strong adverse pressure gradient along the rear surface, the boundary layer is unique in that intermittent detachment occurring near the wall. The numerical results indicate that the boundary layer is not followed by mean‐flow separation or incipient separation as shown from the numerical results. The resolved turbulent shear stress is in a reasonable agreement with the experimental data, though the computational result of LES shows that its peak is overpredicted near the trailing edge of the bump, while the other used turbulence models, except the standard k?ε, underpredicts it. Analysis of the numerical results from LES confirms the experimental data, in which the existence of internal layers over the bump surface upstream of the summit and along the downstream flat plate. It also demonstrates that the quasi‐step increase in skin friction is due to perturbations in pressure gradient. The surface curvature enhances the near‐wall shear production of turbulent stresses, and is responsible for the formation of the internal layers. The aim of the present work is to examine the response and prediction capability of LES with the dynamic eddy viscosity model as a sub‐grid scale to the complex turbulence structure with the presence of streamline curvature generated by a bumpy surface. Aiming to reduce the computational costs with focus on the mean behavior of the non‐equilibrium turbulent boundary layer of flow over the bump surface, the present investigation also explains the best capability of one of the used RANS turbulence models to capture the driving mechanism for the surprisingly rapid return to equilibrium over the trailing flat plate found in the measurements. Copyright © 2010 John Wiley & Sons, Ltd.     

Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion

A numerical study based on the Eulerian–Lagrangian formulation is performed for dispersed phase motion in a turbulent flow. The effect of spatial filtering, commonly employed in large-eddy simulations, and the role of the subgrid scale turbulence on the statistics of heavy particles, including preferential concentration, are studied through a priori analysis of DNS of particle-laden forced isotropic turbulence. In simulations where the subgrid scale kinetic energy attains 30–35% of the total we observe the impact of residual fluid motions on particles of a smaller inertia. It is shown that neglecting the influence of subgrid scale fluctuations has a significant effect on the preferential concentration of those particles. A stochastic Langevin model is proposed to reconstruct the residual (or subgrid scale) fluid velocity along particle trajectories. The computation results for a selection of particle inertia parameters are performed to appraise the model through comparisons of particle turbulent kinetic energy and the statistics of preferential concentrations.     

Non isobaric boundary layers related to Marangoni flows

Summary The paper deals with the dissipative layers, called Marangoni boundary layers, that can be formed, along the interface of two immiscible fluids, in surface driven flows. Under the hypothesis that the flow fields of the two interfacing fluids are uncoupled, similar solutions are studied for the case in which an external pressure gradient is present. The similarity class is derived and the pertinent equations are solved numerically by mean of an algorithm based on a Quasi-Linearization Technique.

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Sommario Si considerano Strati Limiti che possono formarsi, lungo l'interfaccia di due fluidi immiscibili, in flussi indotti da gradienti di tensione superficiale, noti come strati limiti alla Marangoni. Soluzioni simili vengono derivate per il caso in cui sia presente un gradiente di pressione esterno, nell'ipotesi che i campi di moto nei due fluidi interfaccianti siano disaccoppiati. Le equazioni sono integrate numericamente con un algoritmo basato su una tecnica di Quasi-Linearizzazione.

     

An Analysis of Turbulent Motions In and Around a Differentially Forced Density Interface

The Rapid-Distortion-Theory-based analysis proposed by Fernando and Hunt [1] is extended to study the nature of turbulence in and around a density interface sandwiched between turbulent layers with dissimilar properties. It is shown that interfacial motions consist of low-frequency, resonantly excited, nonlinear internal waves and high-frequency, linear internal waves driven by background turbulence. Based on the assumptions that (i) all resonant waves and some nonresonant waves having frequencies close to the resonant frequencies grow rapidly, break, and cause interfacial mixing, (ii) the spectral amplitude of the vertical velocity in the wave-breaking regime is constant, and (iii) kinetic energy is equipartitioned between linear and nonlinear breaking wave regimes, the r.m.s. vertical velocity at the interface and the turbulent kinetic energy flux into the interface are calculated. The migration velocity of the interface is calculated using the additional assumption that the buoyancy flux into a given turbulent layer is a fixed fraction of the turbulent kinetic energy flux supplied to the interface by the same layer. The calculations are found to be in good agreement with the entrainment data obtained in previous laboratory experiments in the parameter regime where the interface is dominated by internal wave dynamics. Received 23 July 1997 and accepted 8 January 1999