# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
###### Abstract
In their work, [PERSON] et al. [16] incorporated an additional length-scale equation into the conventional k-L-a turbulence model. By introducing two length scales, the model was designed to accommodate the intricate interactions and energy transfers taking place at various spatial scales within turbulent flows. Additionally, their study showcased that the two-length-scale model could successfully replicate anticipated growth parameters and turbulence intensities for both buoyancy-driven and shear-driven mixing. This observation suggests the efficacy of the model as a multi-scale turbulence model. [PERSON] [17] proposed an additional equation for the 'inverse time scale' (\(S\)) together with a modified form of the \(\varepsilon\)-equation and employed the same turbulence viscosity relation as that of the \(k\)-\(\varepsilon\) model. It is predicated on the modelling of the time lag between the energy cascade's initiation and its micro-scale destruction. [PERSON] [18] extended a two-equation k-\(\mathrm{R_{T}}\) model [19], in which \(\mathrm{R_{T}}\) denotes the undamped eddy viscosity (\(R\equiv k^{2}/\varepsilon\)) by writing a transport equation for \(\mathrm{R_{T}}\) for low-Re-number flows. The model enforces that in the immediate proximity of solid surfaces, the eddy time scale must not be lower than the Kolmogorov scale. [PERSON] and [PERSON] [20] also introduced a third equation for the so called 'transported strain parameter', (S) in their model together with the \(k\)- and \(\varepsilon\)-equations and employed a dumping function in the turbulence viscosity relation that is predominantly influenced by [PERSON] and [PERSON] [21], who provide a twenty-year evolution of the \(k\)-\(\varepsilon\)-\(\upsilon^{2}\) type of three-equation models focusing on the variance of wall-normal fluctuating velocity and its source \"\(\ell\)\" in their paper. Other proposals on multiple-scale closures are made by [PERSON] [22], [PERSON] and [PERSON] [23], [PERSON] and [PERSON] [24], [PERSON] and [PERSON] [25], and at the second moment closure level by [PERSON] et al. [26]. Although the extensions of eddy viscosity models are possible, the approaches at the second-moment-closure level are still costly, especially for time-dependent realistic in-cylinder engine flow simulations or complex flows involving bubbles and bubble induced turbulence [27]. Moreover, there are models developed for specific problems. In order to estimate the aerodynamic performance of iced aerofoils, [PERSON], [PERSON], and [PERSON] developed a modified three-equation turbulence model in their research. As ice formation on a wing can severely degrade its aerodynamic performance and represent a hazard to aircraft safety, this model was designed to precisely predict the stalling behavior of frozen aerofoils. The original model that they started working on was of [PERSON] \(k-\overline{v^{2}}-\omega\)[28]. The improvements [PERSON] et al. made were intended to calibrate the transitional behavior and enhance the performance of the non-equilibrium state. In Section 2, a brief account of the WFC model, which the modified model is based on, is given before the model formulation presented in Section 3.