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.     

On a Generalized Nonlinear K–ε Model and the Use of Extended Thermodynamics in Turbulence

A resent extension of the nonlinear K–ε model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–ε model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–ε type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–ε model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed. Received 19 February 1998 and accepted 23 October 1998     

Evaluation of a Modified Reynolds Stress Model for Turbulent Dispersed Two-Phase Flows Including Two-Way Coupling

A modified Reynolds stress turbulence model for the pressure rate of strain can be derived for dispersed two-phase flows taking into account gas-particle interaction. The transport equations for the Reynolds stresses as well as the equation for the fluctuating pressure can be derived starting from the multiphase Navier–Stokes equations. The unknown pressure rate of strain correlation in the Reynolds stress equations is then modelled by considering the multiphase equation for the fluctuating pressure. This leads to a multiphase pressure rate of strain model. The extra particle interaction source terms occurring in the model for the pressure rate of strain can be constructed easily, with no noticeable extra computational cost. Eulerian–Lagrangian simulation results of a turbulent dispersed two-phase jet are presented to show the differences in results with and without the new two-way coupling terms.     

Experimental and numerical investigation of the dynamics of uniform turbulence of a stably stratified fluid

The results of experimental and numerical studies of the dynamics of the parameters of uniform turbulence of a stably stratified fluid for different molecular Prandtl-Schmidt numbers over a wide range of buoyancy times Nτ are given. The tank, the measurement apparatus used, and the experimental procedure are briefly described. The numerical modeling used a second-order model of uniform turbulence of a stratified medium. The influence of fluctuations of the turbulent mass (heat) flux q(Nτ) on the evolution of the statistical parameters of the velocity and temperature fields is analyzed, and an invariant equation is found for the parameters of the strong turbulence of the stratified fluid. It is shown that the statistical parameters of the turbulence, being smoothed with respect to the amplitude of the fluctuations, vary self-similarly with time after the collapse point. Donetsk State University, Donetsk 340055.¹Institute of Heat and Mas Transfer, National Academy of Sciences of Belarusia, Minsk 220072. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 64–75, July–August, 1998.     

Toward a Turbulence Constitutive Relation for Geophysical Flows

Rapidly rotating turbulent flows are frequently in approximate geostrophic balance. Single-point turbulence closures, in general, are not consistent with a geostrophic balance. This article addresses and resolves the possibility of a constitutive relation for single-point second-order closures for classes of rotating and stratified flows relevant to geophysics. Physical situations in which a geostrophic balance is attained are described. Closely related issues of frame-indifference, horizontal divergence, and the Taylor–Proudman theorem are discussed. It is shown that, in the absence of vortex stretching along the axis of rotation, turbulence is frame-indifferent. Unfortunately, no turbulence closures are consistent with this frame-indifference that is frequently an important feature of rotating or quasi-geostrophic flows. A derivation and discussion of the geostrophic constraint which ensures that the modeled second-moment equations are frame-invariant, in the appropriate limit, is given. It is shown that rotating, stratified, and shallow water flows are situations in which such a constitutive relation procedure is useful. A nonlinear nonconstant coefficient representation for the rapid-pressure strain covariance appearing in the Reynolds stress and heat flux equations, consistent with the geostrophic balance, is described. The rapid-pressure strain closure features coefficients that are not constants determined by numerical optimization but are functions of the state of turbulence as parametrized by the Reynolds stresses and the turbulent heat fluxes as is required by tensor representation theory. These issues are relevant to baroclinic and barotropic atmospheric and oceanic flows. The planetary boundary layers in which there is a transition, with height or depth, from a thermally or shear driven turbulence to a geostrophic turbulence is a classic geophysical example to which the considerations in this article are relevant. Received 14 October 1996 and accepted 9 June 1997     

Multiphase fluid dynamics and transport processes of low capillary number cavitating flows

To better understand the multiphase fluid dynamics and associated transport processes of cavitating flows at the capillary number of 0.74 and 0.54, and to validate the numerical results, a combined computational and experimental investigation of flows around a hydrofoil is studied based on flow visualizations and time-resolved interface movement. The computational model is based on a modified RNG k-ε model as turbulence closure, along with a vapor-liquid mass transfer model for treating the cavitation process. Overall, favorable agreement between the numerical and experimental results is observed. It is shown that the cavi- tation structure depends on the interaction of the water-vapor mixture and the vapor among the whole cavitation stage, the interface between the vapor and the two-phase mixture exhibits substantial unsteadiness. And, the adverse motion of the interface relates to pressure and velocity fluctuations inside the cavity. In particular, the velocity in the vapor region is lower than that in the two-phase region.     

Direct and large eddy simulation of stratified turbulence

Simulations of geophysical turbulent flows require a robust and accurate subgrid-scale turbulence modeling. To evaluate turbulence models for stably stratified flows, we performed direct numerical simulations (DNSs) of the transition of the three-dimensional Taylor–Green vortex and of homogeneous stratified turbulence with large-scale horizontal forcing. In these simulations we found that energy dissipation is concentrated within thin layers of horizontal tagliatelle-like vortex sheets between large pancake-like structures. We propose a new implicit subgrid-scale model for stratified fluids, based on the Adaptive Local Deconvolution Method (ALDM). Our analysis proves that the implicit turbulence model ALDM correctly predicts the turbulence energy budget and the energy spectra of stratified turbulence, even though dissipative structures are not resolved on the computational grid.     

Time evolution of structures in rotating and stratified turbulence

Rotating and stably stratified turbulence exhibit not only significant anisotropies but also dynamics, which are qualitatively different from purely rotating or stratified turbulence. Furthermore, the different time scales due to rotation, stratification and the turbulence one open up a wide field of possibilities for the temporal evolution of rotating and stratified turbulence.We analyze results from DNS with different parameters α = f/N by visualizing iso-enstrophy surfaces, the temporal evolution of velocity correlation length scales and angular energy spectra.We retrieve standard results, such as a large anisotropy for small scales in rotating turbulence and a large anisotropy for intermediate scales in the vortex mode of stratified turbulence. Furthermore, at large times we find qualitatively different phenomena for cases α = 10 and α = 0.1 such as modified cascades due to the existence of potential energy or small scale vorticity production respectively.     

稳定分层流动中湍流特性的研究综述

本文总结了近60 年来分层流动中湍流特性研究的成果. 主要从两个方面进行了综述:(1) 分层流动中湍流场的演变和混合. 在这方面主要分析稳定分层对湍流混合和湍流结构的影响, 以及混合层内湍流结构的特性和混合层的演化规律. (2) 分层流动中湍流的扩散和输运. 动量和标量的逆梯度输运特性是分层湍流研究的一个重要方向;分析分层对湍流扩散的影响. 并指出了一些值得今后进一步研究的方向.     

Turbulent flow in a channel with permeable walls. Direct numerical simulation and results of three-parameter model

Steady turbulent viscous incompressible fluid flow in a plane channel is calculated for the case of uniform blowing and suction through opposite walls. There are no experimental data for flows of this type. The flows were calculated by two methods: a direct numerical simulation method and using a three-parameter turbulence model. Direct numerical simulation was carried out using the same (apart from the boundary conditions) algorithm for numerical solution of the Navier-Stokes equations as that used earlier for calculating flows in pipes and channels with impermeable walls. In the second group of calculations the version of the model published in 1978 was used. The results obtained by the two methods are in good agreement. The difference is within the spread of the experimental data used for determining the parameters of the model. The agreement obtained makes it possible to assert that the turbulence direct numerical simulation algorithm developed can be used for the analysis of flows with quite different boundary conditions, including cases where there are no corresponding experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 18–26, November–December, 1998. The work was financially supported by the Russian Foundation for Basic Research (project Nos. 96-01-00602 and 96-01-00259).     

Turbulent flow simulation of liquid jet emanating from pressure-swirl atomizer

The performances of three linear eddy viscosity models (LEVM) and one algebraic Reynolds stress model (ARSM) for the simulation of turbulent flow inside and outside pressure-swirl atomizer are evaluated by comparing the interface position with available experimental data and by comparing the turbulence intensity profiles at the atomizer exit. It is found that the turbulence models investigated exhibit zonal behaviors, i.e. none of the models investigated performs well throughout the entire flow field. The turbulence intensity has a significant influence on the global characteristics of the flow field. The turbulence models with better predictions of the turbulence intensity, such as Gatski-Speziale’s ARSM model, can yield better predictions of the global characteristics of the flow field, e.g. the reattachment lengths for the backward-facing step flow and the sudden expansion pipe flow, or the discharge coefficient, film thickness and the liquid sheet outer surface position for the atomizer flows. The standard k–ε model predicts stronger turbulence intensity as compared to the other models and therefore yields smaller film thickness and larger liquid sheet outer surface position. In average, the ARSM model gives both quantitatively and qualitatively better results as compared to the standard k–ε model and the low Reynolds number models.     

Spectral structure of stratified turbulence: direct numerical simulations and predictions by large eddy simulation

Density stratification has a strong impact on turbulence in geophysical flows. Stratification changes the spatial turbulence spectrum and the energy transport and conversion within the spectrum. We analyze these effects based on a series of direct numerical simulations (DNS) of stratified turbulence. To facilitate simulations of real-world problems, which are usually beyond the reach of DNS, we propose a subgrid-scale turbulence model for large eddy simulations of stratified flows based on the Adaptive Local Deconvolution Method (ALDM). Flow spectra and integral quantities predicted by ALDM are in excellent agreement with direct numerical simulation. ALDM automatically adapts to strongly anisotropic turbulence and is thus a suitable tool for studying turbulent flow phenomena in atmosphere and ocean.     

Analysis and Modeling of the Turbulent Diffusion of Turbulent Kinetic Energy in Natural Convection

Buoyant flows often contain regions with unstable and stable thermal stratification from which counter gradient turbulent fluxes are resulting, e.g. fluxes of heat or of any turbulence quantity. Basing on investigations in meteorology an improvement in the standard gradient-diffusion model for turbulent diffusion of turbulent kinetic energy is discussed. The two closure terms of the turbulent diffusion, the velocity-fluctuation triple correlation and the velocity-pressure fluctuation correlation, are investigated based on Direct Numerical Simulation (DNS) data for an internally heated fluid layer and for Rayleigh–Bénard convection. As a result it is decided to extend the standard gradient-diffusion model for the turbulent energy diffusion by modeling its closure terms separately. Coupling of two models leads to an extended RANS model for the turbulent energy diffusion. The involved closure term, the turbulent diffusion of heat flux, is studied based on its transport equation. This results in a buoyancy-extended version of the Daly and Harlow model. The models for all closure terms and for the turbulent energy diffusion are validated with the help of DNS data for internally heated fluid layers with Prandtl number Pr = 7 and for Rayleigh–Bénard convection with Pr = 0.71. It is found that the buoyancy-extended diffusion model which involves also a transport equation for the variance of the vertical velocity fluctuation gives improved turbulent energy diffusion data for the combined case with local stable and unstable stratification and that it allows for the required counter gradient energy flux.     

Constraints on the “rapid” part of the pressure-strain rate correlations derived from the spectral presentation

Based on the exact spectral presentation of the “rapid” part of the pressure-strain rate correlations, semi-empirical approximations used for these correlations within the framework of the second-order closures are analyzed. Simple inequalities relating the values of the model constants, mean velocity parameters, and Reynolds tensor invariants are derived. For certain types of flows, in contrast to conditions of realizability, these inequalities allow verification of the approximations before solving differential equations. It is demonstrated that some models cannot be considered as sufficiently precise ones to describe flows with high degrees of anisotropy. In particular, the condition of non-negative determinacy of the spectral matrix is violated in a considerable region of the physically admissible range of parameters. The boundaries of this region are calculated for an irrotational three-dimensional distortion and for an arbitrary two-dimensional distortion of turbulence in channel flows. Simple constraints on model constants are obtained, which allow these violations to be avoided. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 29–39, March–April, 2008.     

Heat and mass transfer study of impinging turbulent premixed flames

 Impinging jet combusting flows on granite plates are studied. A mathematical model for calculating heat release in turbulent impinging premixed flames is developed. The combustion including radiative heat transfer and local extinction effects, and flow characteristics are modeled using a finite volume computational approach. Two different eddy viscosity turbulence models, namely the standard k–ɛ and the RNG k–ɛ model with and without radiation (discrete transfer model) are assessed. The heat released predictions are compared with experimental data and the agreement is satisfactory only when both radiative heat transfer and local extinction modeling are taken into account. The results indicate that the main effect of radiation is the decrease of temperature values near the jet stagnation point and along the plate surface. Radiation increases temperature gradients and affects predicted turbulence levels independently of the closure model used. Also, the RNG k–ɛ predicts higher temperatures close the solid plate, with and without radiative heat transfer. Received on 13 November 2000 / Published online: 29 November 2001     

Prediction of separation flows around a 6：1 prolate spheroid using RANS/LES hybrid approaches

This paper presents hybrid Reynolds-averaged Navier–Stokes (RANS) and large-eddy-simulation (LES) methods for the separated flows at high angles of attack around a 6:1 prolate spheroid. The RANS/LES hybrid methods studied in this work include the detached eddy simulation (DES) based on Spalart–Allmaras (S–A), Menter’s k–ω shear-stress-transport (SST) and k–ω with weakly nonlinear eddy viscosity formulation (Wilcox–Durbin+, WD+) models and the zonal-RANS/LES methods based on the SST and WD+ models. The switch from RANS near the wall to LES in the core flow region is smooth through the implementation of a flow-dependent blending function for the zonal hybrid method. All the hybrid methods are designed to have a RANS mode for the attached flows and have a LES behavior for the separated flows. The main objective of this paper is to apply the hybrid methods for the high Reynolds number separated flows around prolate spheroid at high-incidences. A fourth-order central scheme with fourth-order artificial viscosity is applied for spatial differencing. The fully implicit lower–upper symmetric-Gauss–Seidel with pseudo time sub-iteration is taken as the temporal differentiation. Comparisons with available measurements are carried out for pressure distribution, skin friction, and profiles of velocity, etc. Reasonable agreement with the experiments, accounting for the effect on grids and fundamental turbulence models, is obtained for the separation flows. The project supported by the National Natural Science Foundation of China (10502030 and 90505005).     

LES Study of Turbulent Boundary Layer Over a Smooth and a Rough 2D Hill Model

Large eddy simulation (LES) is carried out to investigate the turbulent boundary-layer flows over a hill-shaped model with a steep or relatively moderate slope at moderately high Reynolds numbers (Re = O(10³)) defined by the hill height and the velocity at the hill height. The study focuses on the effects of surface roughness and curvature. For Sub-grid Scale (SGS) modeling of LES, both the dynamic Smagorinsky model (DSM) and the dynamic mixed model (DMM) are applied. The behavior of the separated shear layer and the vortex motion are affected by the oncoming turbulence, such that the shear layer comes close to the ground surface, or the size of a separation region becomes small because of the earlier instability of the separated shear layer. Appropriate measures are required to generate the inflow turbulence. The methods of Lund et al. (J. Comput. Phys., 140:233–258, 1998) and Nozawa and Tamura (J. Wind Eng. Ind. Aerodyn., 90:1151–1162, 2002; The 4th European and African Conference on Wind Engineering, 1–6, 2005) are employed to simulate the smooth- and rough-wall turbulent boundary layers in order to generate time-sequential data of inflow turbulence. This paper discusses the unsteady phenomena of the wake flows over the smooth and rough 2D hill-shaped obstacles and aims to clarify the roughness effects on the flow patterns and the turbulence statistics distorted by the hill. Numerical validation is conducted by comparing the simulation results with wind tunnel experiment data for the same hill shape at almost the same Re. The applicability of DSM and DMM are discussed, focusing on the recirculation region behind a steep hill.     

Conjugate flows and smooth bores in a weakly stratified fluid

The problem of steady-state flows in a layer of a continuously stratified fluid is considered. The sufficient condition of existence of families of shear flows that are consistent with the meaning of the laws of conservation of mass, momentum, and energy with a uniform flow is given. Approximate solutions of the smooth-bore type, which describe the wave transitions for pairs of conjugate flows of the first spectral mode, are obtained. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 69–78. March–April, 1999.     

Conjugate shear flows of a weakly stratified fluid

This paper studies the problem of pairs of horizontal shear flows of weakly stratified fluids with identical mass, momentum, and energy fluxes. The initial problem is reduced to a system of two scalar equations for the main- and perturbed-flow parameters by using bifurcation methods. The existence conditions for nontrivial branches of conjugate flows close to the main flow are investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 79–88, March–April, 2009.     

Possibilities and Limitations of Computer Simulations of Industrial Turbulent Dispersed Multiphase Flows

We give an overview on the usage of computer simulations in industrial turbulent dispersed multiphase flows. We present a few examples of industrial flows: bubble columns and bubbly pipe flows, stirred tanks, cyclones, and a fluid catalytic cracking unit. The fluid catalytic cracking unit is used to illustrate the complexity of the physical phenomena involved, and the possibilities and limitations of the different approaches used: Eulerian–Lagrangian (particle-tracking) and Eulerian–Eulerian (two-fluid). In the first approach, the continuous phase is solved using either RANS simulations (Reynolds-Averaged Navier–Stokes simulations) or DNS/LES (Direct Numerical Simulations/Large-Eddy Simulations), and the individual particles are tracked. In the second approach, the dispersed phase is averaged, leading to two sets equations, which are quite similar to the RANS equations of single-phase flows. The Eulerian–Eulerian approach is the most commonly used in industrial applications, however, it requires a significant amount of modelling. Eulerian–Lagrangian RANS can be simpler to use; in particular in situations involving complex boundary conditions, polydisperse flows and agglomeration/breakup. The key issue for the success of the simulations is to have good models for the complex physics involved. A major weakness is the lack of good models for: the turbulence modification promoted by the particles, the inter-particle interactions, and the near-wall effects. Eulerian–Lagrangian DNS/LES can play an important role as a research tool, in order to get a better physical understanding, and to improve the models used in the RANS simulations (either Eulerian–Eulerian or Eulerian–Lagrangian).