A turbulent time scale based k–ε model for probability density function modeling of turbulence/chemistry interactions: Application to HCCI combustion

Homogenous Charge Compression Ignition (HCCI) engine technology is known as an alternative to reduce NO_x and particulate matter (PM) emissions. As shown by several experimental studies published in the literature, the ideally homogeneous mixture charge becomes stratified in composition and temperature, and turbulent mixing is found to play an important role in controlling the combustion progress. In a previous study, an IEM model (Interaction by Exchange with the Mean) has been used to describe the micromixing in a stochastic reactor model that simulates the HCCI process. The IEM model is a deterministic model, based on the principle that the scalar value approaches the mean value over the entire volume with a characteristic mixing time. In this previous model, the turbulent time scale was treated as a fixed parameter. The present study focuses on the development of a micro-mixing time model, in order to take into account the physical phenomena it stands for. For that purpose, a (k–ε) model is used to express this micro-mixing time model. The turbulence model used here is based on zero dimensional energy cascade applied during the compression and the expansion cycle; mean kinetic energy is converted to turbulent kinetic energy. Turbulent kinetic energy is converted to heat through viscous dissipation. Besides, in this study a relation to calculate the initial heterogeneities amplitude is proposed. The comparison of simulation results against experimental data shows overall satisfactory agreement at variable turbulent time scale.     

A stability condition for turbulence model： From EMMS model to EMMS-based turbulence model

The closure problem of turbulence is still a challenging issue in turbulence modeling. In this work, a stability condition is used to close turbulence. Specifically, we regard single-phase flow as a mixture of turbulent and non-turbulent fluids, separating the structure of turbulence. Subsequently, according to the picture of the turbulent eddy cascade, the energy contained in turbulent flow is decomposed into different parts and then quantified. A turbulence stability condition, similar to the principle of the energy-minimization multi-scale （EMMS） model for gas-solid systems, is formulated to close the dynamic constraint equa- tions of turbulence, allowing the inhomogeneous structural parameters of turbulence to be optimized. We name this model as the ＂EMMS-based turbulence model＂, and use it to construct the corresponding turbulent viscosity coefficient. To validate the EMMS-based turbulence model, it is used to simulate two classical benchmark problems, lid-driven cavity flow and turbulent flow with forced convection in an empty room, The numerical results show that the EMMS-hased turbulence model improves the accuracy of turbulence modeling due to it considers the principle of compromise in competition between viscosity and inertia.     

Extension of k–ε models for a turbulent falling film

An extension applicable to various k–ε models is proposed to account for the damping of turbulence due to the surface tension. It consists of a sink in the equation for $ {{\overline{D} k} \mathord{\left/ {\vphantom {{\overline{D} k} {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ and a source in the equation for $ {{\overline{D} \varepsilon } \mathord{\left/ {\vphantom {{\overline{D} \varepsilon } {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ both derived from dimensional analysis. First numerical experiments are undertaken with a commercial CFD software. Reasons are given why the tuning of the model should be performed with thermal experiments. The proposed model is practical, since it only needs the programming of source terms. On account of it’s mathematical structure it needs a very small closure coefficient. The intention of this article is to stimulate the numerical turbulence research in a way that is applicable to the engineering practice. A comparison to experimental data is not done here.     

A USM-Θ two-phase turbulence model for simulating dense gas-particle flows

A second-order moment two-phase turbulence model for simulating dense gas-particle flows (USM- model), combining the unified second-order moment two-phase turbulence model for dilute gas-particle flows with the kinetic theory of particle collision, is proposed. The interaction between gas and particle turbulence is simulated using the transport equation of two-phase velocity correlation with a two-time-scale dissipation closure. The proposed model is applied to simulate dense gas-particle flows in a horizontal channel and a downer. Simulation results and their comparison with experimental results show that the model accounting for both anisotropic particle turbulence and particle-particle collision is obviously better than models accounting for only particle turbulence or only particle-particle collision. The USM- model is also better than the k--kp- model and the k--kp-p- model in that the first model can simulate the redistribution of anisotropic particle Reynolds stress components due to inter-particle collision, whereas the second and third models cannot.The project supported by the Special Funds for Major State Basic Research of China (G-1999-0222-08), the National Natural Science Foundation of China (50376004), and Ph.D. Program Foundation, Ministry of Education of China (20030007028)     

A k–$${\varepsilon}$$ turbulence closure model of an isothermal dry granular dense matter

The turbulent flow characteristics of an isothermal dry granular dense matter with incompressible grains are investigated by the proposed first-order k–\({\varepsilon}\) turbulence closure model. Reynolds-filter process is applied to obtain the balance equations of the mean fields with two kinematic equations describing the time evolutions of the turbulent kinetic energy and dissipation. The first and second laws of thermodynamics are used to derive the equilibrium closure relations satisfying turbulence realizability conditions, with the dynamic responses postulated by a quasi-linear theory. The established closure model is applied to analyses of a gravity-driven stationary flow down an inclined moving plane. While the mean velocity decreases monotonically from its value on the moving plane toward the free surface, the mean porosity increases exponentially; the turbulent kinetic energy and dissipation evolve, respectively, from their minimum and maximum values on the plane toward their maximum and minimum values on the free surface. The evaluated mean velocity and porosity correspond to the experimental outcomes, while the turbulent dissipation distribution demonstrates a similarity to that of Newtonian fluids in turbulent shear flows. When compared to the zero-order model, the turbulent eddy evolution tends to enhance the transfer of the turbulent kinetic energy and plane shearing across the flow layer, resulting in more intensive turbulent fluctuation in the upper part of the flow. Solid boundary as energy source and sink of the turbulent kinetic energy becomes more apparent in the established first-order model.     

A new drag model for TFM simulation of gas–solid bubbling fluidized beds with Geldart-B particles

In this work, a new drag model for TFM simulation in gas–solid bubbling fluidized beds was proposed, and a set of equations was derived to determine the meso-scale structural parameters to calculate the drag characteristics of Geldart-B particles under low gas velocities. In the new model, the meso-scale structure was characterized while accounting for the bubble and meso-scale structure effects on the drag coefficient. The Fluent software, incorporating the new drag model, was used to simulate the fluidization behavior. Experiments were performed in a Plexiglas cylindrical fluidized bed consisting of quartz sand as the solid phase and ambient air as the gas phase. Comparisons based on the solids hold-up inside the fluidized bed at different superficial gas velocities, were made between the 2D Cartesian simulations, and the experimental data, showing that the results of the new drag model reached much better agreement with experimental data than those of the Gidaspow drag model did.     

A nonlinear kp-εp particle two-scale turbulence model and its application

A particle nonlinear two-scale turbulence model is proposed for simulating the anisotropic turbulent two-phase flow. The particle kinetic energy equation for two-scale fluctuation, particle energy transfer rate equation for large-scale fluctuation, and particle turbulent kinetic energy dissipation rate equation for small-scale fluctuation are derived and closed. This model is used to simulate gas–particle flows in a sudden-expansion chamber. The simulation is compared with the experiment and with those obtained by using another two kinds of tow-phase turbulence model, such as the single-scale two-phase turbulence model and the particle two-scale second-order moment (USM) two-phase turbulence model. It is shown that the present model gives simulation in much better agreement with the experiment than the single-scale two-phase turbulence model does and is almost as good as the particle two-scale USM turbulence model. The project supported by China Postdoctoral Science Foundation (2004036239).     

A droplet transport model for channel and pipe flow based on particle kinetic theory and a stress–ω turbulence model

A model for droplet transport in turbulent gas–liquid flow in horizontal channels or pipes is developed. We invoke the kinetic theory of Reeks and co-workers for inertial particles, and the Reynolds stress–ω model of Wilcox for the gas turbulence. We introduce a suitable turbulence boundary condition at the gas–liquid interface. The full kinetic model is compared to experimental data using high density SF₆ gas and Exxsol oil, and the model gives a good prediction of the droplet distribution.     

Experimental study on the effect of drag reducing polymer on flow patterns and drag reduction in a horizontal oil–water flow

In this study, a HMW anionic co-polymer of 40:60 wt/wt NaAMPS/acrylamide was used as a drag reducing polymer (DRP) for oil–water flow in a horizontal 25.4 mm ID acrylic pipe. The effect of polymer concentration in the master solution and after injection in the main water stream, oil and water velocities, and pipe length on drag reduction (DR) was investigated. The injected polymer had a noticeable effect on flow patterns and their transitions. Stratified and dual continuous flows extended to higher superficial oil velocities while annular flow changed to dual continuous flow. The results showed that as low as 2 ppm polymer concentration was sufficient to create a significant drag reduction across the pipe. DR was found to increase with polymer concentration increased and reached maximum plateau value at around 10 ppm. The results showed that the drag reduction effect tends to increase as superficial water velocity increased and eventually reached a plateau at U_sw of around 1.3 m/s. At U_sw > 1.0 m/s, the drag reduction decreased as U_so increased while at lower water velocities, drag reduction is fluctuating with respect to U_so. A maximum DR of about 60% was achieved at U_so = 0.14 m/s while only 45% was obtained at U_so = 0.52 m/s. The effectiveness of the DRP was found to be independent of the polymer concentration in the master solution and to some extent pipe length. The friction factor correlation proposed by Al-Sarkhi et al. (2011) for horizontal flow of oil–water using DRPs was found to underpredict the present experimental pressure gradient data.     

Simulation of Wind Flow Around a Building with a k–ε Model

The three-dimensional numerical simulation of airflow around a building using a k–ε two-equation turbulence model is presented in this paper. Several cases of numerical simulation of airflow around a building are carried out to estimate the influence of mesh spacing on simulated results. The accuracy of simulations is examined by comparing the predicted results with wind-tunnel experiments. It is confirmed that numerical simulations by means of the k–ε model reproduce the velocity fields well when using fine mesh resolution. In the latter part of the paper, the simulation method is applied to predict the flow field around a building with different width-to-height ratios, under light wind conditions. Received 16 June 1999 and accepted 20 July 2000     

Numerical study of Phan-Thien–Tanner viscoelastic fluid flow around a two-dimensional circular cylinder at a low Reynolds number: a new classification for drag variations regimes

Meccanica - This study numerically investigates a low Reynolds two-dimensional flow of a viscoelastic fluid over a circular cylinder using the finite volume method. The Phan-Thien–Tanner...     

A k–ℓ turbulence closure sensitized to non-simple shear flows

This paper introduces a two-equation turbulence model sensitized to deviations from simple shear flows. The closure is topography-parameter-free and is based on solving transport equations for the turbulence kinetic energy (k) and the turbulence length-scale (?). Brief model derivation details are given and test cases are presented to compare the model's performance to other closures and to experimental data. The flow examples demonstrate the advantage of the k–? model in non-simple shear flows.     

Numerical study of oscillating boundary layer flow over a flat plate using k–kL–ω turbulence model

Oscillating boundary layer flow over an infinite flat plate at rest was simulated using the k–k_L–ω turbulence model for a Reynolds number range of 32 ⩽ Re_δ ⩽ 10,000 ranging from fully laminar flow to fully turbulent flow. The k–k_L–ω model was validated by comparing the predictions with LES results and experimental results for intermittently turbulent and fully turbulent flow regimes. The good agreement obtained between the k–k_L–ω model prediction with the experimental and LES results indicate that the k–k_L–ω model is able to accurately simulate transient intermittently turbulent flow and as well as accurately predict the onset of turbulence for such oscillatory flows.     

Phase dynamics of effective drag and lift components in vortex-induced vibration at low mass–damping

In this work, we investigate the dynamics of vortex-induced vibration of an elastically mounted cylinder with very low values of mass and damping. We use two methods to investigate this canonical problem: first we calculate the instantaneous phase between the cylinder motion and the fluid forcing; second we decompose the total hydrodynamic force into drag and lift components that act along and normal to, respectively, the instantaneous effective angle of attack. We focus on the phase dynamics in the large-amplitude–response range, consisting of the initial, upper and lower “branches” of response. The instantaneous phase between the transverse force and displacement shows repeated phase slips separating periods of constant, or continuous-drifting, phase in the second half of the upper branch. The phase between the lift component and displacement shows strong phase locking throughout the large-amplitude range – the average phase varies linearly with the primary frequency – however the modulation of this phase is largest in the second half of the upper branch. These observations suggest that the large-amplitude–response dynamics is driven by two distinct limit cycles – one that is stable over a very small range of reduced velocity at the beginning of the upper branch, and another that consists of the lower branch. The chaotic oscillation between them – the majority of the upper branch – occurs when neither limit cycle is stable. The transition between the upper and lower branches is marked by intermittent switching with epochs of time where different states exist at a constant reduced velocity. These different states are clearly apparent in the phase between the lift and displacement, illustrating the utility of the force decomposition employed. The decomposed force measurements also show that the drag component acts as a damping factor whereas the lift component provides the necessary fluid excitation for free vibration to be sustained.