*2.3.1.3 Steelant and Dick model*

*Boundary Layer Flows - Theory, Applications and Numerical Methods*

**2.3 Intermittency equation transition models**

*2.3.1 Models depending on nonlocal flow variables*

*2.3.1.1 Dhawan and Narasimha model*

transition cases.

γ =

{

*2.3.1.2 Cho and Chung model*

tency transport equation is given as:

*uj*

transition onset, these models depend on the diffusion of the turbulence from freestream into the boundary layer and its interaction with the source terms of the turbulence models. For this reason, these models are more suitable for bypass transition flows. Nonetheless, due to the similarities between a developing laminar boundary layer and a viscous sublayer, their success is thought to be coincidental, and thus these modes are mostly unreliable. These models also lack sensitivity to adverse pressure gradients and convergence problems arise for separation-induced

It has been known from experiment that turbulence has an intermittent character with large fluctuations in flow variables like velocity, pressure, etc. Based on this observation, transition to turbulence has been tried to be modeled using the so-called intermittency function. One-, two- or three-equation partial differential equations have been derived to include the intermittency equation as one of the equations of the complete equation set including relevant experimental calibrations that mimic the actual physical behavior. To this end, "nonlocal" [7–9] and "local" [1, 10, 11] correlation transition models have been proposed. In the following, a systematic line of progress is presented that reveals the evolution of such models.

Dhawan and Narasimha [6] proposed a scalar intermittency function-γ that would provide some sort of a measure of progression toward a fully turbulent boundary layer. Based on the experimentally measured streamwise intermittency distributions on flat plate boundary layers, for instance, Dhawan and Narasimha [6]

0 *x* < *xt* 1.0 <sup>−</sup> *exp*[−(*<sup>x</sup>* <sup>−</sup> *xt*)

In the above function, xt is the known transition onset location, n is the turbulence spot formation rate per unit time per unit distance in the spanwise direction, σ

Cho and Chung [8] developed the k-ε-γ turbulence model that is not designed for prediction of transitional flows but for free shear flows. In this model, the intermittency effect is incorporated into the conventional k-ε turbulence model with the addition of an intermittency transport equation for the intermittency factor γ. In this model, the turbulent viscosity is defined in terms of k, ε, and γ. The intermit-

where Dγ is the diffusion term and Sγ is the source term. This model is tested for a plane jet, a round jet, a plane far-wake, and a mixing layer case. As mentioned

is a turbulence spot propagation parameter, and U is the freestream velocity.

∂γ \_\_\_ ∂*xj*

<sup>2</sup>*n* \_\_\_\_\_\_\_\_ *<sup>U</sup>* ] <sup>=</sup> 1.0 <sup>−</sup> *exp*(−0.41 <sup>ξ</sup><sup>2</sup>

) *x* ≤ *xt* (1)

= *D*<sup>γ</sup> + *S*<sup>γ</sup> (2)

introduced the following function for streamwise intermittency profile:

**104**

Steelant and Dick [7] developed an intermittency transport model that can be used with the so-called conditioned Navier-Stokes equations. In this model, the intermittency function of Dhawan and Narasimha [6] is first differentiated along the streamline direction, s, and the following intermittency transport equation is obtained:

$$\frac{d\chi}{d\pi} + \frac{\partial \rho u \chi}{d\pi} + \frac{\partial \rho v \chi}{\partial y} = \{\mathbf{1} - \chi\} \rho \sqrt{u^2 + v^2} \,\beta(\varepsilon) \tag{3}$$

In the above equation, β(s) is a turbulent spot formation and propagation term, which is seen in the exponential function part of the Dhawan and Narasimha model. Steelant and Dick tested their model for zero, adverse and favorable pressure gradient flows by using two sets of the so-called conditioned averaged Navier-Stokes equations. Although their model reproduces the intermittency distribution of Dhawan and Narasimha for the streamwise direction, a uniform intermittency distribution in the cross-stream direction is assumed. Yet, this is inconsistent with the experimental observations of, for instance, Klebanoff [26] where a variation of the intermittency in the normal direction by means of an error function formula.

#### *2.3.1.4 Suzen and Huang model*

Suzen and Huang [9] proposed an intermittency transport equation model by mixing the production terms of the Cho and Chung [8] and Steelant and Dick [7] models by means of a new blending function. An extra diffusion-related production term due to Cho and Chung is also added to the resultant equation. This model successfully reproduces experimentally observed streamwise intermittency profiles and demonstrates a realistic profile for the cross-stream direction in the transition region. This model is coupled with the Menter's k-ω SST turbulence model [27] in which the intermittency factor calculated by the Suzen and Huang model is used to scale the eddy viscosity field computed by the turbulence model. This model is successfully tested against several flat plate and low-pressure turbine experiments. However, as mentioned before, this model is not a fully local formulation, and thus it cannot be implemented in straightforward fashion in the modern CFD codes.

#### *2.3.2 Models depending on local flow variables*
