12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

MULTI-FIDELITY APPROACH FOR TRANSITIONAL BOUNDARY-LAYER

Minwoo Kim
Jiseop Lim
School of Mechanical Engineering
School of Mechanical Engineering
Gwangju Institute of Science and Technology Gwangju Institute of Science and Technology
Buk-gu, Gwangju 61005, South Korea
Buk-gu, Gwangju 61005, South Korea
mwkim3698@gist.ac.kr
wltjq06@gist.ac.kr

Solkeun Jee
Donghun Park
School of Mechanical Engineering
Department of Aerospace Engineering
Gwangju Institute of Science and Technology
Pusan National University
Buk-gu, Gwangju 61005, South Korea
Geumjeong-gu, Busan 46241, South Korea
sjee@gist.ac.kr
parkdh@pusan.ac.kr

ABSTRACTet al., 2021) flows. The NPSE method can incorporate nonlinear interactions between instabilities up to a weakly nonlinear
region where the mean flow is still laminar. LES with major
instabilities assigned at the LES inlet from NPSE can carry
the simulation from the nonlinear region up to fully turbulent
flow. Our previous study (Kim et al., 2020) shows that appropriate sub-grid-scale models are dormant in pre-transition region where all the instabilities are resolved in a given LES gridand quickly active in generating the eddy viscosity in the latenonlinear region to the turbulent flow where the given spatial
resolution is not fine enough to capture all the turbulence spectrum. Since the NPSE-coupled LES method already demonstrates DNS-like fidelity with a roughly 10% of the DNS computational cost for transitional wall-bounded flows (Kim et al.,2019; Lim et al., 2021), the current study focuses on the LESRANS coupling method to maintain the DNS-like fidelity anda further reduction in the computational cost.

A multi-fidelity computational approach is proposed fortransitional boundary layer flow in order to obtain both highfidelity and high efficiency in flow simulation. Because we cancategorize the transitional flow into three regions, i.e., laminarflow with growing instabilities, transition region, and turbulent
flow, three models are judiciously selected. A boundary layer
stability method, here nonlinear parabolized stability equations (NPSE), is chosen for the laminar region incorporating
nonlinear interactions between the instabilities. Large-eddy
simulation (LES) is applied for the transition region including the beginning of the turbulent flow. Reynolds-averaged
Navier-Stokes (RANS) simulation is used for the downstream
turbulent regime. Since the NPSE-coupled LES method has
been validated for transitional boundary layers including bothincompressible (Kim et al., 2018, 2019, 2020, 2021) and compressible (Lim et al., 2021) flows, appropriate treatments forthe interface between LES and RANS are mainly investigatedin this study. The current multi-fidelity approach is tested fora transitional boundary layer on a flat plate. Computational fidelity and cost are compared with previous high-fidelity computations. We successfully demonstrate that the proposed
multi-fidelity approach provides both high fidelity and high efficiency in transitional flow computations.

The several multi-fidelity method for a turbulent study
have been suggested. The most famous approach is the detached eddy simulation (DES) while it is not proper for thetransitional flow due to modeling the boundary-layer using
RANS. To overcome this limitation, Bader et. al suggested
the k − ω delayed DES method (Bader et al., 2022). It used
the operator developed in Vreman sub-grid-scale model (Vreman, 2004) instead of Smagorinsky operator. It predicts transition very roughly with under-resolved vortical structures. On
the other hands, the zonal RANS-LES method, i.e. embeddedLES, have been suggested and used (Quéméré & Sagaut, 2002;
Anupindi & Sandberg, 2017; Woodruff, 2019; Roidl et al.,2012). For the zonal method, two interfaces exist: RANS-toLES interface and LES-to-RANS interface. The interface fromthe RANS to LES have been widely investigated. The small
eddies which are not captured in RANS model are required for
the LES so that the inlet boundary conditions which generatesmall-scale fluctuations are studied. For example, the synthetic
eddy method was widely used to assign more physical fluctuations rather than white noises. The interface from the LES
to RANS is straightforward because a variables for RANS arereconstructed from the LES by applying the Reynolds average. Additionally, these hybrid RANS-LES method have been
generally applied in separated flow. Therefore, the detail con-Introduction
Turbulent transition in boundary layer flow is a criticalflow phenomena which is directly related to vehicle drag andheat transfer. Nonetheless, an efficient and accurate predictionof a whole transition process is challenging. A main difficultyarises from three distinct stages involved in a complete transition process: laminar flow, transition region, and turbulent
flow. Historically, flow modeling approaches have been developed mainly for each individual region. Consequently it isimperative to suggest a unified approach which provides both
high fidelity and computational efficiency for a whole transitional flow, which is the main goal of this study.
The NPSE-coupled LES method has been validated fortransitional boundary layers including both incompressible(Kim et al., 2018, 2019, 2020, 2021) and compressible (Lim1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
consider a lot of instability modes so that the computational
cost of PSE extensively increases; it will be a DNS-like computational cost. Thus, large-eddy simulation (LES) is conducted instead of PSE to capture the late transition regions
and subsequent early turbulent flow. The authors have successfully conducted turbulent transition simulation using PSE-LES
combining method (Kim et al., 2019; Lim et al., 2021). The
detail methods for the current LES and combining method are
documented in the reference (Kim et al., 2019) including thegrid information and numerical schemes.

sideration was not conducted in order to investigate sensitivityof this simple reconstruction method. Only a few studies focused on the interface from the LES to RANS (Nolin et al.,2005; Schluter et al., 2002). It should be noted that the moststudy using the zonal method is applied in turbulent flow.
In this study, we suggest a multi-fidelity approach which
combines three methods: (1) stability analysis, particularly
nonlinear parabolized stability equations (NPSE), for laminarflow with instabilities, (2) large-eddy simulation (LES) for thehighly-nonlinear transition region and the beginning of turbulent flow, and (3) Reynolds-averaged Navier-Stokes (RANS)
model for the subsequent fully-turbulent region. The PSE-LES
interface treatment was successfully validated in the authors’previous studies (Kim et al., 2019; Lim et al., 2021). Thus,
the current study focuses on the discussion of the RANS-LES
interface which have not been gained enough attention in thecommunity. It is expected that the suggested method is able toprovide DNS-like fidelity with a fraction of DNS/LES computations for a whole transitional flow.

RANS
Reynolds-averaged Navier-Stokes (RANS) method isuseful for the turbulent boundary-layer. It is well known that
RANS have been successfully validated for the simple flow
like non-separated flow or mild pressure-gradient condition.
In the current case, the flow is well attached, and the current
simulation is a canonical boundary-layer over a flat plate sothat RANS guarantees the acceptable accuracy with low-cost.
Therefore, RANS is applied to the turbulent boundary-layer
after LES in order to reduce computational cost.
The governing equations for the RANS approach are written as equation 3 and equation 4.

Computational Methods
The multi-fidelity approach is suggested in the current
study by combining NPSE, LES, and RANS as represented
in figure 1. NPSE is conducted to predict growth of instability modes in 2-D laminar boundary-layer. NPSE is able toconsider nonlinear interactions and growth of instabilities accurately with low-computational cost. Because LES is able toaccurately resolve turbulent transition with moderate cost, it isapplied from nonlinear region up to the subsequent early turbulent flow as shown in figure 1. RANS simulation is carriedout for turbulent boundary-layer following LES for further reducing the computational cost while maintaining reasonableaccuracy.
The total variable is decomposed into the mean flow
part, φ̄ , and the sub-grid-scale, residual part, φ 0 , for the LES
method.
φ̌ = φ̄ + φ 0(3)∂ τiRj 1 ∂ P
∂U j∂ 2U j∂+UiU j = ν
−
−
∂t
∂ xi
∂ xi ∂ xi
∂ xi
ρ ∂xj

(4)

The equations require the closure model for the Reynolds
stress tensor τiRj . The common approach is the Boussinesq
approximation which assumes that the stress tensor is proportional to the strain rate tensor.

(1)

τiRj = −2νt Si j

For the RANS approach, the total variable is decomposed
into the Reynolds-averaged part, Φ, and the turbulent fluctuation, φ .
φ̌ = Φ + φ

∂Ui
=0
∂ xi

(5)

Here, the eddy viscosity, νt , is still unknown so that the current
RANS uses the k-ω SST model to determine the eddy viscosity
Menter et al. (2003).

(2)

Flow Conditions and Interfaces
The current computational case to validate the suggested multi-fidelity approach is subharmonic transition ona zero-pressure-gradient flat plate following the Kachanov &Levchenko (1984)’s experiment. It showed the subsonic natural transition induced by subharmonic resonance, i.e. Htype transition, which is triggered by the Tollmien-Schlichting
wave and the subharmonic oblique waves. The subsonic natural transition has long laminar boundary-layer, short transition
length, and subsequent turbulent boundary-layer. Therefore, it
is a good candidate for showing a performance and benefits ofthe current multi-fidelity method.
Because the instability modes in long laminar flow growfrom very small scales as shown in figure 1, the computational cost becomes DNS-like when LES reproduces them.
Thus, NPSE which is advanced stability analysis is used to
predict both instabilities and nonlinear interactions between
them. The obtained instabilities are assigned into the inlet of
LES with laminar solution.

NPSE
The nonlinear parabolized stability equations (PSE) is a
powerful tool to predict growth and interactions of instability modes both in linear and nonlinear stages. It takes into
account of non-parallel effects and complex modal interactions. In the current study, PSE is applied to accurately capture the instability modes and secondary instability in laminarboundary-layer with low-cost. The nonlinear PSE code usedin the current study have been successfully validated in incompressible boundary-layer (Park & Park, 2013; Kim et al., 2019,2020; Park et al., 2019; Kim et al., 2021) and compressible
boundary-layer (Park & Park, 2016; Lim et al., 2021).

LES
The late transition regions and turbulent boundary-layer
have matured wave spectrum. In this situation, PSE should212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1.

Schematic diagram of the current multi-fidelity method.

Computational Results

Figure 2a shows the amplitude √growth of five selectedmodes. NPSE is conducted until Rex = 681 where theboundary layer is still laminar although strong nonlinear interactions occur. Thirteen modes determined by the thresholdmax (A(m,n) ) > 0.01% ∗ U∞ following the (Kim et al., 2019),
where m and n are temporal and spanwise spatial modes, respectively, are assigned into the inlet of LES with laminar solution. The thick lines which are obtained from √LES well follow
NPSE results. The amplitude is reduced after Rex = 700 due
to saturating wave-spectrum during turbulent transition. The
details of NPSE-coupled LES method are well described in theauthors’ previous studies (Kim et al., 2019; Lim et al., 2021).
RANS simulation is applied in the fully turbulent flow
after LES computation for turbulent transition. LES is able
to provide Reynolds-averaged variables. From the given variables from LES and the Boussinesq relation, the variables for
RANS simulation can be reconstructed. Figure 2b shows theinlet profiles of the RANS simulation. In the inlet of RANS,
three variables are used: velocity vector, turbulent kinetic energy, and turbulent dissipation rate. The Reynolds-averaged
velocity vector and turbulent kinetic energy can be straightforwardly obtained from the results of LES. However, the turbulent dissipation rate ω cannot be calculated directly from the
LES results. Thus, the turbulent dissipation rate is obtained using turbulent kinetic energy and eddy viscosity. The following
equations show the procedures for the calculation for ω.

ω=k̄
νt.RANS

In order to show the efficiency of the current method, thenumber of grid counts are compared with previous studies intable 1. As a result, a number of grid count is reduced by
two orders-of-magnitude compared to the simulation where
the turbulent boundary-layer is also computed by LES. While
DNS study (Lozano-Durán et al., 2018) requires 250 million
grid points and LES (Kim et al., 2019) requires 30 million
grid points for the same transitional boundary-layer, the current multi fidelity needs only 6 million grid points.

Table 1.

||τ̄irj |||| S̄i j − 1/3δi j S̄kk ||Computational methods

grid points

DNS (Lozano-Durán et al., 2018)

250×106

PSE-LES (Kim et al., 2019)30×106

Current multi-fidelity method

6×106

The computational results are shown in figure 3. Judiciously combined three methods successfully predict transitional boundary layer (see figure 3b). Because of the highfidelity and unsteady nature of LES, the instantaneous fluctuating boundary-layer and vortical structures are captured asshown in figure 3a and figure 3a. The staggered Λ−structures
which are generally observed in subharmonic transition are
well captured in LES computation (see, figure 3b). The subsequent hairpin vortices and small eddies are also reproduced.
Skin-friction coefficients of laminar, LES, and RANS
simulation are smoothly connected in figure 3c. LES predicts
that skin friction deviates from laminar value due to turbulent
transition and rapidly grows to turbulent value. RANS simulation well expects skin friction of turbulent boundary layer.
Skin friction of RANS, however, slightly overshoot near the
inlet. It is conjectured that ω which is constructed from LES
variables rather than directly obtained from the transport equation does not suit to RANS equations.
For a remedy of the discrepancy at the interface of LES
and RANS, the inlet of the RANS is slightly modified. The
profiles are shown in Figure 4a The reconstructed turbulent
kinetic energy is different from the desired value for RANS
computations, red line. Therefore, turbulent kinetic energy(6)

While eddy viscosity of LES exists, it differs from eddy viscosity of RANS because of the different physics between them.
Eddy viscosity for the RANS, which will be used for the calculation of ω assigned into the inlet of RANS, should be also
reconstructed. Here, relying on the Boussinesq approximation,
eddy viscosity for RANS simulation is calculated as shown inequation 7 (Monier et al., 2018).

νt.RANS =A number of grid points comparison.

(7)

where ||.|| denotes norm, and τ̄irj means traceless stress tensor
of LES.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2. Computational data for (a) NPSE- LES and (b) LES-RANS interfaces: (a) amplitude growth of 5 major modes from NPSE
analysis among 13 instabilities assigned at the LES inlet (b) mean velocity (U + and V + ) and turbulence quantities (k and ω) assigned
at RANS inlet from the LES computation.

Figure 3. Selected computational results: (a) Vortical structures visualization using iso-surface of Q = 20 coloured by normalized
streamwise velocity in the LES domain. (b) Normalized instantaneous streamwise velocity fields on a spanwise-normal plane; the scale
x:y=1:6 is used for the visualization purpose. (c) Averaged skin friction coefficient C f .

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4. Selected computational results from the modified RANS inlet: (a) The mean velocity (U + and V + ) and turbulence quantities(k and ω) assigned at RANS inlet. (b) Averaged skin friction coefficient C f .

taken from the pure RANS, which has the same boundarylayer thickness with the current LES results at the LES-RANS
interface, is assigned into the inlet of RANS. The steady 2-D
RANS simulation for a flow over a flat plate is additionallyconducted in order to obtain the pure RANS results.
As a result, the current modification works as intended.
There is no weird pick near the inlet in the RANS computation.
The skin friction curve becomes much smoother compared tofigure 3c. The limitation of the current method is that it demands pure RANS data. Also, the current modification mitigates the transient behavior near the RNAS inlet while small
modulation still exists. Thus, the further investigation will beneeded and conducted.

LES-RANS interface successfully removes this pick.
In the current study, the suggested multi-fidelity method isdemonstrated in the canonical case. Furthermore, the authors
plan to explore the suggested multi-fidelity approach to more
complex transitional flows. The current NPSE-LES-RANS
method can be extended to other transitional flows covering
a wide range of transition paths, because NPSE can handle
both natural and non-natural transition paths, and both LES
and RANS would resolve the subsequent turbulent flow adequately.

Acknowledgement
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2021R1A2C1006193). Also, thiswork was supported by the National Supercomputing Center with supercomputing resources including technical support(KSC-2022-CRE-0163, KSC-2021-CRE-0372, KSC-2021CRE-0153, KSC-2020-CRE-0057).

Conclusions
A multi-fidelity approach is suggested in the current
study. Three methods, NPSE, LES, and RANS, are incorporated carefully to describe a whole transitional boundary layer.
The suggested approach provides high fidelity and high efficiency in numerical computations. NPSE is used in the laminar boundary layer to predict interactions and growth of instabilities with low computational cost. LES is carried out for turbulent transition and the early turbulent flow. LES well reproduces vortical structures and boundary-layer growth. RANS
is applied to the subsequent turbulent boundary layer. The
computational cost is drastically decreased compared to previous studies. While DNS and LES require 250 million and
30 million grid points (Lozano-Durán et al., 2018; Kim et al.,2019), respectively, the current approach only needs 6 million
grid points. Also, streamwise velocity fields and the skin friction show that the current multi-fidelity approach well predicts
the whole transitional boundary layer. The initial pick near
the RANS inlet is shown due to the different solution required
for the RANS even though the solution is obtained from thehigh-fidelity computation. The additional modification for the

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Osaka, Japan (Online), July 19-22, 2022
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