# RANS multi-scale turbulence modelling and its application to high freestream turbulence boundary layers

Stefan Coroama, Jaime Vaquero, Nicolas Renard, François Chedevergne

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Stefan Coroama, Jaime Vaquero, Nicolas Renard, François Chedevergne. RANS multi-scale turbulence modelling and its application to high freestream turbulence boundary layers. ETMM15, Sep 2025, Dubrovnik, Croatia. pp.39-44, 10.5281/zenodo.17280519. hal-05333695

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Distributed under a Creative Commons Attribution 4.0 International License
RANS MULTI-SCALE TURBULENCE MODELLING AND ITS
APPLICATION TO HIGH FREESTREAM TURBULENCE
BOUNDARY LAYERS

S. Coroama¹, J. Vaquero¹, N. Renard¹ and F. Chedevergne²

¹ DAAA, ONERA, Institut Polytechnique de Paris, 92190, Meudon, France

² DMPE, ONERA, Université de Toulouse, 31000, Toulouse, France

stefan.coroama@onera.fr

**Abstract**

The present paper aims at describing the de-
velopment rationale behind a two-scale $k-\omega$
RANS (Reynolds-Averaged Navier-Stokes) turbu-
lence model. This model is designed to cope with
the difficulties of classical single-scale models on the
prediction of turbulent boundary layers under the in-
fluence of high free-stream turbulence (FST). From a
detailed literature review, two physical phenomena are
spotted and careful modelling is proposed by using the
possibilities offered by the multi-scale RANS frame-
work. The numerical results obtained with the new
model on a flat plate under strong FST with a wide
range of turbulence levels demonstrate the improved
skin friction predictions attained when compared to a
$k-\omega$ single-scale model.

# 1 Introduction

Turbulent boundary layers (TBLs) evolving under
the influence of a highly turbulent free-stream are
commonly encountered in industrial flow config-
urations such as turbomachinary flows. Indeed,
high pressure turbine blades usually operate with a
free-stream turbulence (FST) intensity up to 30%
(Maciejewski and Moffat (1992)). From now on,
the FST intensity is quantified by the parameter
$T_{u\infty} = \sqrt{u_\infty^2/U_\infty}$ where $u_\infty$ and $U_\infty$ denote
the free-stream streamwise Reynolds stress and
streamwise velocity respectively. The effects of
the interaction between a TBL and a large-scale,
high-intensity FST have been extensively studied in
the literature and it has been reported that the skin
friction and heat transfer are greatly increased when
compared to a canonical TBL (Thole and Bogard
(1996), Esteban et al (2017)). Furthermore, it has
been observed that the inner-scaled mean streamwise
velocity profile $\bar{U}^+$, where $\bar{U}$ is the mean streamwise
velocity and the superscript + refers to the classical
inner-scaling using the friction velocity $u_\tau (\doteq \sqrt{\tau_w/\rho})$
with $\tau_w$ the wall-shear stress and $\rho$ the density), is
unchanged in the viscous sublayer, buffer layer and
logarithmic regions of the TBL no matter the strength
of the FST whereas the wake region is strongly

affected (Sharp *et al* (2009), Dogan *et al* (2016)).
The penetration of high intensity FST in the TBL has
also a considerable impact on the diagonal Reynolds
stresses which tend to greatly increase in the wake
region (Stefes and Fernholz (2004)). Moreover, there
is a strong interaction between the inner and outer
region for the streamwise stress $\overline{u'^2}$ as this component
is seen to increase even in the inner layer (Dogan *et
al* (2016)) similarly to what is observed for canonical
TBLs at high Reynolds numbers (Smits *et al* (2021)).
Further insights about the behaviour of the different
turbulent scales involved in the interaction between
the TBL and the FST is gained by performing a
spectral analysis of the streamwise Reynolds stress.
Such studies highlight the distinct behaviour of the
small and large turbulent scales of wall turbulence
involved in TBL's developing under strong FST. First,
it has been pointed out that an outer peak with high
intensity emerges in the $\overline{u'^2}$ spectra when a high level
of FST is present and that this peak is associated
with high wavelength turbulent structures (Hearst
*et al* (2018), Jooss *et al* (2021)). Second, the inner
peak, associated with small wavelengths, was found
to remain unchanged by the FST intensity while a
subsequent increase in the contribution of large scales
in the inner region exists. Finally, the scale separation
computed by Dogan *et al* (2016), by applying a sharp
spectral cut-off filter, further demonstrates the distinct
behaviour of small and large scales. Indeed, their
study indicates that the large-scale turbulent structures
coming from the intense large-scale FST seem to
penetrate into the TBL all the way to the wall while
no influence of the FST is found on the small-scale
velocity variance.

Accurate numerical predictions of the skin friction
and wall heat transfer coefficient enhancement for
TBLs under a high FST intensity are highly valu-
able for design purposes of high pressure turbine
blades. The multi-scale RANS (Reynolds-Averaged
Navier-Stokes) approach to turbulence modelling
appears to offer a sound framework to cope with the
difficulties of classical one-scale turbulence models
<figure><img src="image_2.png" /><figcaption>Figure 1: Left - profiles of the inner-scaled streamwise Reynolds stress $\overline{u'^2}$ in the TBL for different values of $T_{u\infty}$. Middle - small-scale ($\lambda_x^+ < 4000$) contribution to $\overline{u'^2}$. Right - large-scale ($\lambda_x^+ > 4000$) contribution to this Reynolds stress. Data taken from Dogan *et al* (2016).</figcaption></figure>

to predict such flows (Iyer and Yavuzkurt (1999), Bush *et al* (2019)). These difficulties arise from the stringent approximations single-scale turbulence models are based on such as Kolmogorov's hypothesis which ultimately implies that the turbulence spectrum can be determined by unique length and velocity scales. Two-scale models avoid this crude hypothesis by enabling modelling of the scale segregation in small and large-scale contributions. The choice of a two-scale turbulence model to tackle the interaction between TBLs and strong FST is then motivated by the experimental observations concerning the different behaviour of the small and large scales in the TBL which cannot be reproduced by a single-scale turbulence model.

## 2 Scale separation and inactive character of the large scales

Numerous experimental studies have shed light on the strong scale separation observed in TBLs under the influence of a large-scale and intense FST (Sharp *et al* (2009), Dogan *et al* (2016), Hearst *et al* (2018)). A better glimpse at the respective contributions of small and large scales to $\overline{u'^2}$ is given by Dogan *et al* (2016) where partial integration of the streamwise Reynolds stress spectrum was carried out. The spectral cut-off was fixed at a wavelength $\lambda_x^+ = 4000$ ($\lambda_x/\delta \approx 1-2$ with $\delta$ the boundary layer thickness). Figure 1 illustrates the results obtained in their study and it is clearly shown that the small-scale contributions to $\overline{u'^2}$ are relatively universal with respect to the FST level whereas the large-scale component is greatly increased throughout the whole TBL.

Although the inner-scaled streamwise Reynolds stress substantially increases in the entire TBL when it is subjected to a high FST, the inner-scaled mean velocity and Reynolds shear stress $\overline{u'v'}$ (where $v'$ is

the wall-normal fluctuating velocity component) in the buffer and logarithmic layers are not influenced by the level of FST. These experimental observations may be linked with the concept of *active* and *inactive* motions introduced by Townsend in 1961 (Townend (1961)) which received attention in recent research concerning high-Reynolds-number TBLs (Deshpande *et al* (2021)). Originally, this hypothesis was formulated to conciliate the experimental observations about the universality of $\overline{u'v'}$ while $\overline{u'^2}$ shows $Re_\tau$-dependence in canonical TBLs (with $Re_\tau = \rho_w\delta u_\tau/\mu_w$ the friction Reynolds number where $\mu$ is the dynamic viscosity and the subscript $w$ refers to wall quantities). The distinct Reynolds stresses $Re_\tau$-dependence can be better grasped with the Attached Eddy Hypothesis (Townend (1976), Perry *et al* (1986)) which predicts a logarithmic law for the wall-parallel Reynolds stresses while $\overline{u'v'}$ and $\overline{v'^2}$ are constant in the logarithmic layer. This is attributed to the wall-blockage kinematic effect of the solid boundary on the wall normal fluctuating velocity $v'$. The pressure-strain correlation then redistributes energy from the wall-normal Reynolds stress to the wall parallel ones (Bradshaw (1994)). The analysis from Deshpande *et al* (2025) suggests that because of the wall-blockage effect exerted on the wall-normal velocity component, every attached structure only contributes to $v'$ in the vicinity of its height while the same structure enhances $u'$ and $w'$, $w'$ being the transverse velocity fluctuation, all the way to the wall (inviscid theory). As the TBL develops and its friction Reynolds number increases, additional taller structures appear and add contributions to the wall-parallel Reynolds stresses all the way to the wall while only marginally contributing to $\overline{u'v'}$ (see figure 3 of Deshpande *et al* (2025)). As such, turbulent motion in the TBL can be decomposed in an *active* and *inactive* component, the first being Reynolds-number independent as opposed to *inactive* motion (Deshpande *et al* (2021)). Panton (2007) proposes the following
decomposition :

$$
\left\{
\begin{array}{l}
u' = u'_{\text{a}} + u'_{\text{ia}}, \\
w' = w'_{\text{a}} + w'_{\text{ia}}, \\
v' = v'_{\text{a}}
\end{array}
\right.
\qquad (1)
$$

such that $\overline{u'v'} = \overline{u'_{\text{a}}v'_{\text{a}}}$. In (1), the subscript “ia” and “a” respectively refer to *inactive* and *active* motions. By inserting this decomposition in the correlation factor $R_{\overline{u'v'}}$, one obtains:

$$
R_{\overline{u'v'}} = \frac{-\overline{u'v'}}{\sqrt{\overline{u'^2}} \sqrt{\overline{v'^2}}} = \frac{-\overline{u'_{\text{a}}v'_{\text{a}}}^{+}}{\sqrt{\overline{u'^2}} + \overline{u'^2_{\text{ia}}} + \sqrt{\overline{v'^2}} + \overline{v'^2_{\text{ia}}}} \quad (2)
$$

by neglecting non-linear interactions such as amplitude modulation (Mathis *et al* (2009)). Equation (2) along with the previous analysis suggests that if the correlation factor decreases with the Reynolds number or the FST level, it is attributed to an increase in the magnitude of *inactive* motions. Plots of the correlation factor $R_{\overline{u'v'}}$ (eq (2)) are shown in figure 2 for different FST intensities and it is observed that this factor decreases as $T_{u\infty}$ rises.

<figure><img src="image_3.png" /><figcaption>Figure 2: Profiles of the correlation factor $R_{\overline{u'v'}}$ for different FST intensities. Data at $T_{u\infty} = 0\%$, 2.6% and 5.8% is taken from Hancock and Bradshaw (1989). Profiles at $T_{u\infty} = 8.3\%$ and 12.7% are reproduced from Dogan *et al* (2019).</figcaption></figure>

The decrease in correlation factor with the FST intensity observed in figure 2 along with the increase in the large-scale contribution with $T_{u\infty}$ while the small scales are unmodified by the FST (figure 1) suggest that the large-scale turbulent structures penetrating into the TBL contribute only partially to the Reynolds shear stress (Dogan *et al* 2019). The major flaw of one-scale turbulence models for predicting the effect of strong FST on TBLs could then be associated with their inability to capture the *inactive* component of the turbulent kinetic energy carried by the large scales of the flow. Thus, the small/large scale separation operated by the two-scale turbulence model developed in this work is used to capture this phenomenon.

### 3 A two-scale turbulence model

Multiple-scale turbulence models are built by partitioning the turbulent kinetic energy spectrum in multiple parts and then integrating the spectrum on each wavenumber interval (Hanjalic *et al* (1980)). Each of this wavenumber interval is then characterised by its partial kinetic energy $k^{(m)}$, the spectral fluxes $F^{(m)}$ from larger to smaller turbulent scales (figure 3) as well as a partial eddy viscosity $\mu_t^{(m)}$. The analytical work from Schiestel *et al* (1987) shows that a model transport equation can be derived for $k^{(m)}$ and different model variants are available according to the nature of the second transported partial variable. For instance, Gleize *et al* (1996) derived a two-scale model (named MKFLC2) by using the spectral fluxes as the second turbulent variable and obtained very satisfactory results for nonequilibrium boundary layers. Nonetheless, this model was not designed for TBLs interacting with strong FST so a new two-scale model has been developed.

<figure><img src="image_4.png" /><figcaption>Figure 3: Turbulent kinetic energy spectrum partitioning.</figcaption></figure>

The proposed two-scale model is based on a $k-\omega$ formulation where the characteristic turbulent frequencies $\omega^{(m)}$ are defined as

$$
\left\{
\begin{array}{l}
F^{(1)} = \beta^{(1)} k^{(1)} \omega^{(1)}, \\
F^{(2)} = \beta^{(2)} k \omega^{(2)}
\end{array}
\right.
\qquad (3)
$$

where $\beta^{(m)}$ are model constants and $k$ is the total turbulent kinetic energy $k = k^{(1)} + k^{(2)}$. This model relies on the Boussinesq hypothesis which assumes a linear relation between the Reynolds stresses and the strain-rate tensor. For multi-scale models, this assumption is extended to yield, using Favre mass-averaged quantities noted $\tilde{\cdot}$ and Einstein's convention:

$$
-\bar{\rho} \tilde{u}_i'' \tilde{u}_j''^{(m)} + \frac{2}{3} \bar{\rho} k^{(m)} \delta_{ij} = 2 \mu_t^{(m)} \left( S_{ij} - \frac{1}{3} \frac{\partial \tilde{U}_l}{\partial x_l} \delta_{ij} \right) \quad (4)
$$

where $\delta_{ij}$ is the Kronecker symbol, $\mu_t^{(m)}$ is the partial eddy viscosity and $S_{ij}$ is the strain-rate tensor for the Favre-averaged velocity. The total eddy viscosity
used in the mean flow is computed by the following summation :

$$ \mu_t = \mu_t^{(1)} + \mu_t^{(2)}. \quad (5) $$

Several numerical tests were carried out and lead to the following expressions for the partial eddy viscosities :

$$ \mu_t^{(1)} = \bar{\rho} \frac{f_{ia} k^{(1)} + k^{(2)}}{\omega^{(1)}} \quad \text{and} \quad \mu_t^{(2)} = \bar{\rho} \frac{k^{(2)}}{\omega^{(2)}} \quad (6) $$

where $f_{ia}$ is a specific function designed to account for the mainly inactive character of the large scales. This function reads :

$$ f_{ia} = \frac{\frac{1}{r_C} + r_C}{\frac{k^{(1)}}{k^{(2)}} + \frac{k^{(2)}}{k^{(1)}}} \quad (7) $$

where $r_C$ is the value of the ratio $k^{(2)}/k^{(1)}$ in the logarithmic region ($r_C = 1.3$ in our model). The compressible version of the two-scale $k-\omega$ model reads:

$$ \left\{ \begin{aligned} \bar{\rho} \frac{dk^{(1)}}{dt} &= \bar{\rho} P^{(1)} - \beta^{(1)} \bar{\rho} k^{(1)} \omega^{(1)} \\ &\quad + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_k^{(1)} \mu_t \right) \frac{\partial k^{(1)}}{\partial x_j} \right] \\ \bar{\rho} \frac{d\omega^{(1)}}{dt} &= C_{\omega 0}^{(1)} \bar{\rho} \frac{\omega^{(1)}}{k^{(1)}} P^{(1)} - C_{\omega 2}^{(1)} \bar{\rho} \left( \omega^{(1)} \right)^2 \\ &\quad + \sigma_{cd}^{(1)} \frac{\mu_t}{k^{(1)}} \frac{\partial k^{(1)}}{\partial x_j} \frac{\partial \omega^{(1)}}{\partial x_j} \\ &\quad + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_\omega^{(1)} \mu_t \right) \frac{\partial \omega^{(1)}}{\partial x_j} \right] \\ \bar{\rho} \frac{dk^{(2)}}{dt} &= \bar{\rho} P^{(2)} + \beta^{(1)} \bar{\rho} k^{(1)} \omega^{(1)} - \beta^{(2)} \bar{\rho} k \omega^{(2)} \\ &\quad + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_k^{(2)} \mu_t \right) \frac{\partial k^{(2)}}{\partial x_j} \right] \\ \bar{\rho} \frac{d\omega^{(2)}}{dt} &= C_{\omega 0}^{(2)} \bar{\rho} \frac{\omega^{(2)}}{k} (P^{(1)} + P^{(2)}) - C_{\omega 2}^{(2)} \bar{\rho} \left( \omega^{(2)} \right)^2 \\ &\quad + \sigma_{cd}^{(2)} \frac{\mu_t}{k^{(2)}} \frac{\partial k^{(2)}}{\partial x_j} \frac{\partial \omega^{(2)}}{\partial x_j} \\ &\quad + \frac{\partial}{\partial x_j} \left[ \left( \mu + \sigma_\omega^{(2)} \mu_t \right) \frac{\partial \omega^{(2)}}{\partial x_j} \right], \end{aligned} \right. \quad (8) $$

with $\bar{\rho}P^{(m)} = -\bar{\rho}u_i''u_j''^{(m)}(\partial\tilde{U}_i/\partial x_j)$. In total, the proposed model introduces 12 numerical constants that have to be specified. To this end, a calibration procedure has been followed and consists in two main steps. First, the model's behaviour was studied on two canonical flow configurations, namely the decay of homogeneous isotropic turbulence and the logarithmic layer of a zero pressure-gradient TBL in the case of an asymptotically infinite Reynolds number. For freely decaying homogeneous turbulence, it was

supposed that the turbulent kinetic energy decayed as a power law with exponent $n_k = 1$ to comply with the experimental results from Dogan et al (2016). Another calibration choice was to impose a strong scale separation in the free stream by setting $k^{(1)}/k^{(2)} = 20$ to reproduce the experimental observations described in section 2. The calibration in the logarithmic region enabled to impose the Von Karman constant as $\kappa_{VK} = 0.384$. The second step was a numerical optimisation on the zero pressure-gradient TBL skin friction coefficient evolution with the Reynolds number based on the momentum thickness $Re_\theta$ ($= \rho_\infty U_\infty \theta/\mu_\infty$ where $\theta$ is the momentum thickness and the subscript $\infty$ refers to free-stream quantities). The remaining constants were then chosen so that the current model accurately predicts the skin friction evolution of the Coles-Fernholz correlation given in (Nagib et al (2007)). This correlation was adopted for the calibration as it was specifically corrected to collapse with experimental results on TBLs with $Re_\theta$ up to 70000.

The present model was then implemented in the Navier-Stokes solver elsA developed by ON-ERA/Safran (Cambier et al (2014)) and special attention was paid to the wall boundary conditions, especially for the characteristic frequencies $\omega^{(m)}$. Indeed, by assuming their transport equations simplify to an equilibrium between viscous diffusion and dissipation, one finds that $\omega^{(m)}$ behaves as $1/y^2$ ($y$ is the wall-normal coordinate) in the vicinity of the wall. The wall values of $\omega^{(m)}$ are therefore specified by an extrapolation of their values in the first wall-normal cell in a similar fashion to Menter (1994):

$$ \left( \omega^{(m)} \right)_w = 10 \frac{6\mu}{\bar{\rho} C_{\omega 2}^{(m)} d_{w1}^2}, \quad (9) $$

with $d_{w1}$ the wall distance from the first cell adjacent to the wall.

The numerical simulations presented hereafter were initialised from a uniform flow field and the partial turbulent variables were computed by the values of $T_{u0}$ and $(\mu_t/\mu)_0$ specified by the user (the subscript "0" referring to initial conditions). After computing the values of $k_0$ and $\omega_0$ from a single-scale $k-\omega$ model, the splitting into the partial turbulent variables is realised as follows :

$$ \left\{ \begin{aligned} k_0^{(1)} &= \frac{k_0}{1 + r_{G_0}}, \quad k_0^{(2)} = r_{G_0} k_0^{(1)}, \\ \omega_0^{(1)} &= \frac{f_{ia_0} + r_{G_0} + W_{G_0} r_{G_0}}{1 + r_{G_0}} \omega_0, \quad \omega_0^{(2)} = \frac{\omega_0^{(1)}}{W_{G_0}} \end{aligned} \right. \quad (10) $$

where $r_{G_0} = 1/20$, $W_{G_0} = 0.68$ and $f_{ia_0} = (1/r_C + r_C)/(1/r_{G_0} + r_{G_0})$.
## 4 Application

Numerical simulations using the Two-Scale $k - \omega$ (named TS $k - \omega$) model have been carried out on a flat plate TBL subjected to FST levels $T_{u\infty}$ ranging from 7.7 % to 13.0 % and the results were compared to the experimental data obtained by Dogan *et al* (2016), Hearst *et al* (2018) and Jooss *et al* (2021). The current model's performance was also assessed by comparison with the numerical predictions obtained with the single-scale $k - \omega$ SST model from Menter (1994). Before the computations, the initial values of $T_{u0}$ and $(\mu_t/\mu)_0$ required by the solver were determined to hold the correct FST intensity decay as in the experiments. Figure 4 compiles all the numerical results obtained during the current study.

<figure><img src="image_5.png" /><figcaption>Figure 4: Comparison between the experimental profiles of inner-scaled longitudinal mean velocity (taken from Dogan *et al* (2016), Hearst *et al* (2018) and Jooss *et al* (2021)) and the numerical predictions obtained with the TS $k - \omega$ and the $k - \omega$ Menter SST model for $T_{u\infty}$ spanning the domain [7.7%,13.0%].</figcaption></figure>

It can be seen from figure 4 that the two-scale model with the function $f_{ia}$ proposed in this work better predicts the inner-scaled mean velocity profile. Indeed, the present model avoids the skin friction overestimation issue encountered by the single-scale

model by effectively damping the eddy viscosity in the wake region with the function $f_{ia}$.

The behaviour of $f_{ia}$ for canonical and strong FST TBLs and its beneficial impact are illustrated in figure 5. On the one hand, this function remains close to 1 in the TBL because the high values of $k^{(1)}/k^{(2)}$ do not diffuse in the TBL for the $T_{u\infty} \approx 0\%$ case. This function therefore yields no influence in low FST cases. On the other hand, the high free-stream values of $k^{(1)}/k^{(2)}$ impregnate the TBL as the turbulence intensity increases which in turn leads to a gradual decrease of $f_{ia}$ from $\approx 1$ in the logarithmic layer to its low free-stream value.

<figure><img src="image_6.png" /><figcaption>Figure 5: Profiles of $f_{ia}$ for TBLs under no FST and strong FST ($T_{u\infty} \approx 0\%$ and $T_{u\infty} = 12.7\%$ respectively). Boundary layer profiles of the ratio $k^{(1)}/k^{(2)}$ are also given for these two cases.</figcaption></figure>

## 5 Conclusion

This paper focused on the presentation of a multi-scale model designed to capture the effects of high FST on TBLs. A thorough literature review highlighted two main physical phenomena intrinsic to the interaction between a developing TBL and strong FST : the pronounced scale separation and the mainly *inactive* character of the large-scale motion in the TBL. Reproducing these observations with the present model is deemed essential and the multi-scale approach to turbulence modelling is well adapted to capture these features as opposed to classical single-scale models.

A two-scale turbulence model along with a specific function $f_{ia}$ were then proposed to cope with the struggle of classical RANS models on high FST cases. It was demonstrated that the present model yields enhanced inner-scaled mean velocity profiles predictions on the flat plate TBL when compared to the $k - \omega$ SST model on a wide range of FST intensities.
In the future, assessing the model on industrial
cases of increasing complexity, where extensive data
is available, would be an interesting study to further
validate the modelling choices presented in this paper.

Acknowledgments

The authors wish to warmly thank all the per-
sons involved in the past and present evolution of the
flow solver elsA. This work was partially supported by
the SONICE project which is a France Relance pro-
gram co-funded by the French Civil Aviation Author-
ity (DGAC). France Relance benefits from EU funding
via the NextGenerationEU initiative. The authors also
wish to thank Dr Vincent Gleize for his interest and
fruitful discussions in the early stages of this work.

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