# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
###### Abstract  
A novel three-equation turbulence model has been proposed as a potential solution to overcome some of the issues related to the \(k\)-\(\epsilon\) models of turbulence. A number of turbulence models found in the literature designed for compressed turbulence within internal combustion engine cylinders tend to exhibit limitations when applied to turbulent shear flows, such as those occurring through intake or exhaust valves of the engine. In the event that the flow is out of equilibrium where P\({}_{\mathrm{k}}\) deviates from \(\varepsilon\), the turbulence models require a separate turbulence time-scale determiner along with the dissipation, \(\varepsilon\). In the current research, this is accomplished by resolving an additional equation that accounts for turbulence time scale, \(\tau\). After presenting the rationale behind the model, its application to three types of free shear flows were given. It has been shown that the three-equation \(k\)-\(\epsilon\) model outperforms the standard \(k\)-\(\epsilon\) model as well as a number of two-equation models in these flows. Initially, the \(k\)-\(\epsilon\)-\(\tau\) model handles the issue of the plane jet/round jet anomaly in an effective manner. Secondly, it outperforms the two-equation models in predicting the flow behavior in the case of plane wake, one that is distinguished by its weak shear form. turbulence model; time scale; three-equation model; jets; wakes +
Footnote †: journal: Applied  
0000-0002-43  
requires the prediction of not only the compression effects but also the behavior of shear flows found especially during the intake stroke of an IC engine. An alternative three-equation turbulence model (the \(k\)-\(\varepsilon\)-\(\varepsilon_{d}\) model) for compressed turbulence [6] has also been developed in a deductive way from the standard k-\(\varepsilon\) model based on kinematic viscosity variation during compression, and its performance was compared with the DNS data of [PERSON] et al. [5]. However, it also requires modification for shear flows and has not been tested in real engine geometries. On the other hand, [PERSON] and [PERSON] [7] employed various closure models, namely equilibrium models, second-order Reynolds stress transport models (RSTMs), and differential models. They also introduced a non-equilibrium model for the anisotropic Reynolds stress tensor. This newly proposed model was found to be in relatively good agreement with the results obtained from the integration of the full Reynolds stress differential model (RDM). This agreement suggests that the approximations and assumptions made to derive the new non-equilibrium model closure are reasonable for internal combustion engines (ICEs) and reciprocating compressor models (RCMs). Previous multi-scale models proposed by [PERSON] and [PERSON] [8] and [PERSON] et al. [9] were predicated on the notion of segmenting the energy spectrum of turbulence into regions for production, energy transfer, and dissipation, and required the solution of four instead of three PDEs, which is adopted by the recent studies carried out by [PERSON] et al. [10] and [PERSON] [11]. [PERSON] and [PERSON] obtained the turbulence time-scale equation (kT) by integration of auto correlation in stationary flows [12]. Their kT-equation replaced the \(\varepsilon\)-equation and they worked within a two-equation framework with only one scale to represent turbulence. [PERSON] and [PERSON] [13] introduced a two-equation model that adopts [PERSON]'s hypothesis together with novel formulations of the inhomogeneous terms in the transport equations for velocity and length scales. However, they have employed \(\varepsilon\) as their length-scale determiner. [PERSON] and [PERSON] [14] modified the \(\varepsilon\)-equation by employing the time scale as \(t=\sqrt{\upsilon/\varepsilon}\) for energy-dissipating small eddies, without solving an additional equation for time scale in their model. [PERSON] and [PERSON] [15] extended this approach to low-Re-number modelling within a two-equation eddy-viscosity framework.