On the Decay of Freestream Turbulence Predicted by Two-equation Eddy-viscosity Models

As reported before by Spalart and Rumsey (2007), the decay of freestream turbulence in two-equation eddy-viscosity turbulence models is overestimated. In simulations of external flows at high Reynolds numbers that assume ”fully turbulent” flow, this excessive decay has a negligible impact on the solution. The laminar regime is confined to a very small region near the leading edge of the body and so the correct prediction of transition is not relevant. On the other hand, in model scale experiments it is common pratice to force transition through the use of trip wires or roughness. Therefore, as illustrated for example in Ec¸a and Hoekstra (2006), the incorrect prediction of the transition location by two-equation eddy-viscosity models becomes an asset instead of a problem.
Nowadays there is an increasing focus on low Reynolds number applications such as underwater gliders, UAVs and wind turbines, which operate in the Reynolds number range of 10⁵ to 10⁶. In such cases, the correct prediction of transition from laminar to turbulent flow becomes essential and so standard two-equation models are not adequate to perform such simulations.
This known shortcoming of the most common turbulence models led to the development of transition models, as for example the γ − Re_θ model by Langtry and Menter (2009), which supplements the two-equation eddy-viscosity SST k − ω model proposed in Menter (1984), Menter (2003). Its local formulation and capability to account for several transition mechanisms such as natural transition, bypass transition and separation-induced transition make it an attractive option.
The onset of transition is strongly dependent on the level of the free-stream turbulence intensity, which makes the excessive decay of turbulence predicted by the SST k − ω model troublesome. In most of the applications of the γ − Re_θ published in the literature, this shortcoming of the SST k − ω model is overcomed by specifying values of the inlet eddy-viscosity (ν_t)_in significantly larger than the kinematic viscosity ν. It is not unusual to find values of (ν_t)_in = 100ν or even larger, which is at least questionable. On the other hand, when (ν_t)_in is kept at values smaller or equal than ν, the dissipation terms of the k and ω transport equations must be modified to obtain the desired level of turbulence intensity at the leading edge of the body, as discussed in Spalart and Rumsey (2007) and Ec¸a et al. (2016).
The goal of this work is twofold:
1. Demonstrate that the onset of transition is essentially determined by the local value of the turbulence kinetic energy k.
2. Modify the SST k − ω turbulence model to obtain a decay of free-stream turbulence that does not require the use of awkward values of (ν_t)_in to obtain the correct location of the onset of transition. This modification requires two steps (a) Recalibrate the constants of the dissipation terms of the k and ω transport equations for freestream turbulence using the data available from the ERCOFTAC database. (b) Introduce a new blending function that guarantees the use of the standard constants in the viscous region of the flow and the new constants in the free-stream. Identification of both regions is made from the value of the dimensionless total head. In this paper we present the first step of this development that is performed for the flow over a flat plate, for which there are experimental data available for transitional flows with different levels of freestream turbulence in the ERCOFTAC Classic Database.