Direct numerical simulation of low Reynolds number flows in an open‐channel with sidewalls

A direct numerical simulation of low Reynolds number turbulent flows in an open‐channel with sidewalls is presented. Mean flow and turbulence structures are described and compared with both simulated and measured data available from the literature. The simulation results show that secondary flows are generated near the walls and free surface. In particular, at the upper corner of the channel, a small vortex called inner secondary flows is simulated. The results show that the inner secondary flows, counter‐rotating to outer secondary flows away from the sidewall, increase the shear velocity near the free surface. The secondary flows observed in turbulent open‐channel flows are related to the production of Reynolds shear stress. A quadrant analysis shows that sweeps and ejections are dominant in the regions where secondary flows rush in toward the wall and eject from the wall, respectively. A conditional quadrant analysis also reveals that the production of Reynolds shear stress and the secondary flow patterns are determined by the directional tendency of the dominant coherent structures. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessment of the validity of a log-law for wall-bounded turbulent bubbly flows

There has been considerable discussion in recent years concerning whether a log-law exists for wall-bounded, turbulent bubbly flows. Previous studies have argued for the existence of such a log-law, with a modified von Kármán constant, and this is used in various modelling studies. We provide a critique of this idea, and present several theoretical reasons why a log-law need not be expected in general for wall-bounded, turbulent bubbly flows. We then demonstrate using recent data from interface-resolving Direct Numerical Simulations that when the bubbles make a significant contribution to the channel flow dynamics, the mean flow profile of the fluid can deviate significantly from the log-law behaviour that approximately holds for the single-phase case. The departures are not surprising and the basic reason for them is simple, namely that for bubbly flows, the mean flow is affected by a number of additional dynamical parameters, such as the void fraction, that do not play a role for the single-phase case. As a result, the inner/outer asymptotic regimes that form the basis of the derivation of the log-law for single-phase flow do not exist in general for bubbly turbulent flows. Nevertheless, we do find that for some cases, the bubbles do not cause significant departures from the unladen log-law behaviour. Moreover, we show that if departures occur these cannot be understood simply in terms of the averaged void fraction, but that more subtle effects such as the bubble Reynolds number and the competition between the wall-induced turbulence and the bubble-induced turbulence must play a role.     

Turbulence kinetic energy budget in bubbly flows in a vertical duct

Understanding turbulence kinetic energy (TKE) budget in gas–liquid two-phase bubbly flows is indispensable to develop and improve turbulence models for the bubbly flows. In this study, a molecular tagging velocimetry based on photobleaching reaction was applied to turbulent bubbly flows with sub-millimeter bubbles in a vertical square duct to examine the applicability of the k–ε models to the bubbly flows. Effects of bubbles on TKE budget are discussed and a priori tests of the standard and low Reynolds number k–ε models are carried out to examine the applicability of these models to the bubbly flows. The conclusions obtained are as follows: (1) The photobleaching molecular tagging velocimetry is of use for validating turbulence models. (2) The bubbles increase the liquid velocity gradient in the near wall region, and therefore, enhance the production and dissipation rates of TKE. (3) The k–ε models can reasonably evaluate the production rate of TKE in the bubbly flows. (4) The modulations of diffusion due to the bubbles have different characteristics from the diffusion enhancement due to shear-induced turbulence. Hence, the k–ε models fail in evaluating the diffusion rate in the near wall region in the bubbly flows. (5) The k–ε models represent the trends of the production, dissipation, and diffusion rates of ε in the bubbly flow, although more accurate experimental data are required for quantitative validation of the ε equation.     

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 gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

An embedded boundary framework for compressible turbulent flow and fluid–structure computations on structured and unstructured grids

The finite volume method with exact two‐phase Riemann problems (FIVER) is a two‐faceted computational method for compressible multi‐material (fluid–fluid, fluid–structure, and multi‐fluid–structure) problems characterized by large density jumps, and/or highly nonlinear structural motions and deformations. For compressible multi‐phase flow problems, FIVER is a Godunov‐type discretization scheme characterized by the construction and solution at the material interfaces of local, exact, two‐phase Riemann problems. For compressible fluid–structure interaction (FSI) problems, it is an embedded boundary method for computational fluid dynamics (CFD) capable of handling large structural deformations and topological changes. Originally developed for inviscid multi‐material computations on nonbody‐fitted structured and unstructured grids, FIVER is extended in this paper to laminar and turbulent viscous flow and FSI problems. To this effect, it is equipped with carefully designed extrapolation schemes for populating the ghost fluid values needed for the construction, in the vicinity of the fluid–structure interface, of second‐order spatial approximations of the viscous fluxes and source terms associated with Reynolds averaged Navier–Stokes (RANS)‐based turbulence models and large eddy simulation (LES). Two support algorithms, which pertain to the application of any embedded boundary method for CFD to the robust, accurate, and fast solution of FSI problems, are also presented in this paper. The first one focuses on the fast computation of the time‐dependent distance to the wall because it is required by many RANS‐based turbulence models. The second algorithm addresses the robust and accurate computation of the flow‐induced forces and moments on embedded discrete surfaces, and their finite element representations when these surfaces are flexible. Equipped with these two auxiliary algorithms, the extension of FIVER to viscous flow and FSI problems is first verified with the LES of a turbulent flow past an immobile prolate spheroid, and the computation of a series of unsteady laminar flows past two counter‐rotating cylinders. Then, its potential for the solution of complex, turbulent, and flexible FSI problems is also demonstrated with the simulation, using the Spalart–Allmaras turbulence model, of the vertical tail buffeting of an F/A‐18 aircraft configuration and the comparison of the obtained numerical results with flight test data. Copyright © 2014 John Wiley & Sons, Ltd.     

Events of High Polymer Activity in Drag Reducing Flows

The mechanism of drag reduction in turbulent flows due to polymers has been investigated with help of a direct numerical simulation. In particular, we consider the interaction between turbulent velocity fluctuations and polymers in terms of elastic energy that can be stored in the polymer. To this end all the terms of the elastic energy budget have been computed. The most interesting term is the production of elastic energy due to turbulent fluctuations, because it describes the interaction between polymers and turbulence. Although this term appears to be small in the average, it turns out that it can reach very large values instantaneously and intermittently, and the energy transfer from polymer to turbulence is located in very well defined areas inside the channel. This implies that locally there is a strong interaction between the polymer and the turbulent flow structure, and this strong interaction is mostly seen in areas of high velocity fluctuations.     

A numerical scheme for Euler–Lagrange simulation of bubbly flows in complex systems

An Eulerian–Lagrangian approach is developed for the simulation of turbulent bubbly flows in complex systems. The liquid phase is treated as a continuum and the Navier–Stokes equations are solved in an unstructured grid, finite volume framework for turbulent flows. The dynamics of the disperse phase is modeled in a Lagrangian frame and includes models for the motion of each individual bubble, bubble size variations due to the local pressure changes, and interactions among the bubbles and with boundaries. The bubble growth/collapse is modeled by the Rayleigh–Plesset (RP) equation. Three modeling approaches are considered: (a) one‐way coupling, where the influence of the bubble on the fluid flow is neglected, (b) two‐way coupling, where the momentum‐exchange between the fluid and the bubbles is modeled, and (c) volumetric coupling, where the volumetric displacement of the fluid by the bubble motion and the momentum‐exchange are modeled. A novel adaptive time‐stepping scheme based on stability‐analysis of the non‐linear bubble dynamics equations is developed. The numerical approach is verified for various single bubble test cases to show second‐order accuracy. Interactions of multiple bubbles with vortical flows are simulated to study the effectiveness of the volumetric coupling approach in predicting the flow features observed experimentally. Finally, the numerical approach is used to perform a large‐eddy simulation in two configurations: (i) flow over a cavity to predict small‐scale cavitation and inception and (ii) a rising dense bubble plume in a stationary water column. The results show good predictive capability of the numerical algorithm in capturing complex flow features. Copyright © 2010 John Wiley & Sons, Ltd.     

An unsteady incompressible Navier–Stokes solver for large eddy simulation of turbulent flows

An unsteady incompressible Navier–Stokes solver that uses a dual time stepping method combined with spatially high‐order‐accurate finite differences, is developed for large eddy simulation (LES) of turbulent flows. The present solver uses a primitive variable formulation that is based on the artificial compressibility method and various convergence–acceleration techniques are incorporated to efficiently simulate unsteady flows. A localized dynamic subgrid model, which is formulated using the subgrid kinetic energy, is employed for subgrid turbulence modeling. To evaluate the accuracy and the efficiency of the new solver, a posteriori tests for various turbulent flows are carried out and the resulting turbulence statistics are compared with existing experimental and direct numerical simulation (DNS) data. Copyright © 1999 John Wiley & Sons, Ltd.     

Large eddy simulation and a simple wall model for turbulent flow calculation by a particle method

A turbulent channel flow and the flow around a cubic obstacle are calculated by the moving particle semi‐implicit method with the subparticle‐scale turbulent model and a wall model, which is based on the zero equation RANS (Reynolds Averaged Navier‐Stokes). The wall model is useful in practical problems that often involve high Reynolds numbers and wall turbulence, because it is difficult to keep high resolution in the near‐wall region in particle simulation. A turbulent channel flow is calculated by the present method to validate our wall model. The mean velocity distribution agrees with the log‐law velocity profile near the wall. Statistical values are also the same order and tendency as experimental results with emulating viscous layer by the wall model. We also investigated the influence of numerical oscillations on turbulence analysis in using the moving particle semi‐implicit method. Finally, the turbulent flow around a cubic obstacle is calculated by the present method to demonstrate capability of calculating practical turbulent flows. Three characteristic eddies appear in front of, over, and in the back of the cube both in our calculation and the experimental result that was obtained by Martinuzzi and Tropea. Mean velocity and turbulent intensity profiles are predicted in the same order and have similar tendency as the experimental result. Copyright © 2012 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.     

Turbulent shear stress profiles in a bubbly channel flow assessed by particle tracking velocimetry

Particle tracking velocimetry (PTV) is applied to a bubbly two-phase turbulent flow in a horizontal channel at Re = 2 × 10⁴ to investigate the turbulent shear stress profile which had been altered by the presence of bubbles. Streamwise and vertical velocity components of liquid phase are obtained using a shallow focus imaging method under backlight photography. The size of bubbles injected through a porous plate in the channel ranged from 0.3 to 1.5 mm diameter, and the bubbles show a significant backward slip velocity relative to liquid flow. After bubbles and tracer particles are identified by binarizing the image, velocity of each phase and void fraction are profiled in a downstream region. The turbulent shear stress, which consists of three components in the bubbly two-phase flow, is computed by analysis of PTV data. The result shows that the fluctuation correlation between local void fraction and vertical liquid velocity provides a negative shear stress component which promotes frictional drag reduction in the bubbly two-phase layer. The paper also deals with the source of the negative shear stress considering bubble’s relative motion to liquid.     

Introduction of turbulent model in a mixed finite volume/finite element method

The aim of this work is to present a new numerical method to compute turbulent flows in complex configurations. With this in view, a k-? model with wall functions has been introduced in a mixed finite volume/finite element method. The numerical method has been developed to deal with compressible flows but is also able to compute nearly incompressible flows. The physical model and the numerical method are first described, then validation results for an incompressible flow over a backward-facing step and for a supersonic flow over a compression ramp are presented. Comparisons are performed with experimental data and with other numerical results. These simulations show the ability of the present method to predict turbulent flows, and this method will be applied to simulate complex industrial flows (flow inside the combustion chamber of gas turbine engines). The main goal of this paper is not to test turbulence models, but to show that this numerical method is a solid base to introduce more sophisticated turbulence model.     

Application of photobleaching molecular tagging velocimetry to turbulent bubbly flow in a square duct

To evaluate turbulence energy budget in bubbly flows, an image processing method in a photobleaching molecular tagging velocimetry is improved for accurate evaluation of velocity gradients. Turbulence properties in single-phase and two-phase dilute-bubbly flows in a square duct are measured using the improved method. As a result, the following conclusions are obtained: (1) The axial velocity and axial turbulent intensity measured by the present method agree well with those measured by laser Doppler velocimetry not only for the single-phase flow but also for the dilute-bubbly flow. (2) The present method can measure velocity components and velocity gradients in the vicinity of the wall, and therefore the present method is of great use in understanding the mechanism of turbulence generation and dissipation near the wall. (3) The present method can provide detailed information on turbulence structure such as turbulence kinetic energy budget. (4) Bubbles tend to increase not only the turbulence production but also the turbulence dissipation.     

Large eddy simulation of turbulent incompressible flows by a three‐level finite element method

The variational multiscale method provides a methodical framework for large eddy simulation of turbulent flows. In this work, a particular implementation in the form of a three‐level finite element method separating large resolved, small resolved, and unresolved scales is proposed. Residual‐free bubbles are used for the numerical approximation of the small‐scale momentum equation. A stabilizing term is added, in order to take into account the effect of the small‐scale continuity equation. This implementation guarantees the stability of the method without further provisions and offers substantial computational savings on the small‐scale level. Furthermore, it is accounted for the unresolved scales by a specific dynamic modelling procedure. The method is tested for two different turbulent flow situations. Copyright © 2005 John Wiley & Sons, Ltd.     

Investigation of flux formulae in transonic shock wave/turbulent boundary layer interaction

The aim of the present study is to examine the accuracy and improvement of various numerical methods in the solution of the transonic shock/turbulent boundary layer interaction problem and to show that a significant source of numerical inaccuracies in turbulent flows is not only the inadequacy of the turbulence model but also the numerical discretization. Comparisons between a Riemann solver and a flux-vector-splitting method as well as between various numerical high-order extrapolation schemes with corresponding experimental results are presented.     

Bubble and liquid turbulence characteristics of bubbly flow in a large diameter vertical pipe

The bubble and liquid turbulence characteristics of air–water bubbly flow in a 200 mm diameter vertical pipe was experimentally investigated. The bubble characteristics were measured using a dual optical probe, while the liquid-phase turbulence was measured using hot-film anemometry. Measurements were performed at six liquid superficial velocities in the range of 0.2–0.68 m/s and gas superficial velocity from 0.005 to 0.18 m/s, corresponding to an area average void fraction from 1.2% to 15.4%. At low void fraction flow, the radial void fraction distribution showed a wall peak which changed to a core peak profile as the void fraction was increased. The liquid average velocity and the turbulence intensities were less uniform in the core region of the pipe as the void fraction profile changed from a wall to a core peak. In general, there is an increase in the turbulence intensities when the bubbles are introduced into the flow. However, a turbulence suppression was observed close to the wall at high liquid superficial velocities for low void fractions up to about 1.6%. The net radial interfacial force on the bubbles was estimated from the momentum equations using the measured profiles. The radial migration of the bubbles in the core region of the pipe, which determines the shape of the void profile, was related to the balance between the turbulent dispersion and the lift forces. The ratio between these forces was characterized by a dimensionless group that includes the area averaged Eötvös number, slip ratio, and the ratio between the apparent added kinetic energy to the actual kinetic energy of the liquid. A non-dimensional map based on this dimensionless group and the force ratio is proposed to distinguish the conditions under which a wall or core peak void profile occurs in bubbly flows.