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

QUADRATIC CONSTITUTIVE RELATION FOR A CORNER FLOWAND ITS APPLICATION TO A WING-BODY JUNCTURE FLOW
Hiroyuki Abe
Taisuke Nambu
Japan Aerospace Exploration Agency
Japan Aerospace Exploration Agency
Chofu, Tokyo 182-8522, Japan
Chofu, Tokyo 182-8522, Japan
abe.hiroyuki@jaxa.jp
nambu.taisuke@jaxa.jp
Yasuhiro Mizobuchi
Japan Aerospace Exploration Agency
Chofu, Tokyo 182-8522, Japanmizobuchi.yasuhiro@jaxa.jp
ABSTRACTIn the present study, we have developed a quadratic
constitutive relation (QCR) for a nonlinear two-equation k-ε
model (AMM-QCRcorner) by focusing on the behavior of themean streamwise vorticity in a corner flow. The present QCR
consists of three terms (i.e. SΩ, SS and ΩΩ terms). Results
in a square duct flow indicate that the inclusion of the ΩΩterm improves the Reynolds normal stress anisotropy near acorner and strengthens the mean streamwise vorticity there.
We have then tested AMM-QCRcorner against the wingbody juncture flow in an airfoil (i.e. NASA Juncture Flow),
where corner separation occurs in the trailing edge. In thistesting, the modification is made for the expression of the
turbulent eddy viscosity by introducing a parameter S ij2 − Ω ij2 ,representing the acceleration and deceleration of the mean
flow. This is effective for improving the overprediction of
the eddy viscosity in the trailing edge of a corner. Results
highlight that AMM-QCRcorner reproduces a separation
bubble whose size is comparable to that of the experiment.

turbulent flows with a view to predicting separated turbulent
flows in the aeronautical applications (Abe, Mizobuchi andMatsuo 2019; 2020). For improving the Reynolds normal
stress anisotropy, two QCRs (AMM-QCR) were developedwith the use of DNS data of a turbulent separation bubble(Abe 2017), i.e.
2ν2−uiu j = − kδ ij + 2ν t Sij − C1 ' t Ωik S jk + Ω jk Sik 
k
3
, (2)
ν t2 
1

− C2 '  Sik Skj − S mn S mnδ ij 
k 
3

2
k
−ui u j = − kδ ij + 2ν t Sij − C1 ν t  Ωik S jk + Ω jk Sik 
ε
3
(3)
k 
1

− C2 ν t  Sik S kj − S mn S mnδ ij ε 3with C1 ' = C2 ' = 3.2 and C1 = C2 = 0.6 . The application to
the NASA Juncture Flow however led to a smaller cornerseparation bubble than that of the experiment (Abe et al.
2020). One of the reasons is most likely due to theinsufficient predictions of Eqs. (2) and (3) in a corner flow.
Here, we consider the non-zero value of the mean
streamwise vorticity Ω x in a corner flow where theReynolds normal stress anisotropy plays a crucial role
(Bradshaw 1987). We then propose a new QCR (i.e. AMMQCRcorner) by adding a ΩΩ term to Eq. (3), i.e.

INTRODUCTION
The prediction for a corner flow (i.e. wing-body junctureflow) is one of the challenging issues in the engineering andaeronautics. While eddy viscosity models (i.e. one-equation
SA (Spalart and Allmaras 1994) model and two-equation kω (Wilcox 1988), and SST (Menter 1994) models) are
widely used in practical applications, a linear eddy viscosity
does not intrinsically yield a secondary flow near a corner
since the Reynolds stress anisotropy is not resolved there. In
the k- ε model, Nishizima and Yoshizawa (1987) and Myongand Kasagi (1990) proposed QCRs and Craft et al. (1996)
proposed a cubic constitutive relation for the eddy viscosity.
On the other hand, Spalart (2000) proposed a simple QCR
for the SA model (denoted as SA-QCR2000), i.e.,(1)−ui u j = 2ν t Sij − 2ν t Cnl1 Oik S jk + O jk Sik  .

2k−ui u j = − kδ ij + 2ν t S ij − C1 ν t  Ω ik S jk + Ω jk S ik 
ε
3
(4)
k 
1
k 
1


− C 2 ν t  S ik S kj − S mn S mnδ ij  − C3 ν t  Ω ik Ω kj + Ω mn Ω mnδ ij 
3
3
ε 
ε 



with C1 = 0.6, C2 = 0.2, C3 = −0.3 . The latter coefficients
have been determined using DNS data in the channel andsquare duct flows.
The objective of the present study is to clarify the effectof Eq. (4) in a corner flow by comparing with those of Eqs.
(1) to (3). For this purpose, we examine two corner flows.
One is a square duct flow, where the predictions areevaluated against the DNS data (Pirozzoli et al. 2018). The
other is NASA Juncture Flow, which is a wing-body juncture
flow and involves corner separation in the trailing edge of thewing near a wing-root junction (i.e. side-of-body separation)
due to the adverse pressure gradient (see Rumsey et al. 2019).
For the latter flow, the measurement was made by Kegerise
and Neuhart (2019) at subsonic conditions. The Reynolds
number based on crank chord is 2.4 million. The Mach
number is 0.189. The experimental data are available inKegerise and Neuhart (2019), as well as on the NASA
Turbulence
Modeling
Resource(TMR)
website(https://turbmodels.larc.nasa.gov), which are compared with

Here, ui u j denotes the Reynolds stress; ν t is the turbulent
eddy viscosity; a model constant Cnl1 is equal to 0.3;(

Oik ≡ (U i ,k − U k ,i ) / U m ,n U m ,n = 2Ωik / U m,n U m,n) , S ≡ (U + U ) / 2
ij

i, j

j ,i

and Ωij ≡ (U i , j − U j ,i ) / 2 denote the mean strain and vorticity
tensors; U i represents the mean velocity; the subscript
i=1,2,3 refers to the streamwise (x), wall-normal (y) andspanwise (z) directions, respectively. Spalart (2000) showedthat Eq. (1) reproduces a secondary flow in a square duct
successfully. This QCR is currently used in aeronautics forpredicting a corner flow.
On the other hand, we are developing a non-linear k- εmodel (the AMM model) using DNS data in wall-bounded1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
the predictions by AMM-QCRcorner (the AMM model with
Eq. (4)) and SA-QCR2000 (the SA model with Eq. (1)).

the normalized mean streamwise velocity U in a turbulent
channel flow at the friction Reynolds number
Reτ=Uτh/ν=2000 (Uτ and h denote the friction velocity andthe channel half-width, respectively). The AMM result
agrees well with the DNS data (Jimenez and Hoyas 2006) as
well as those of the AKN (Abe, Kondoh and Nagano 1994)
and MK (Myong and Kasagi 1990) models. Figure 1b shows
the predictions of the AMM model with Eqs. (2) to (4) for
the Reynolds normal stresses. Equation (3) gives a better
prediction for the Reynolds normal stresses than Eq. (2), andprovides the prediction that vv and ww agree well with the

AMM MODELThe AMM model is a low Re k-ε model (Abe, Mizobuchi
and Matsuo 2019, 2020), which is developed with a view topredicting separated turbulent flows. In this context, AKN
model (Abe, Kondoh and Nagano 1994) is often used in thek-ε model, where the near-wall model functions weredeveloped on the basis of the Kolmogorov scale. In the
AMM model, the model functions are expressed as2 R   
5  R   
f μ = 1 − exp  − y   1 + 3/ 4 exp  −  t    , 120    Rt  200   2
 R    2
  R  fε = 1 − exp  − y   1 − exp  −  t    .
 12    9  6  (5)

DNS data except close to the wall, although uu isunderpredicted by 30 percent there. On the other hand, Eqs.
(3) and (4) yield essentially the identical behaviors,
highlighting that the inclusion of the ΩΩ term in Eq. (4) does
not affect the predictions for the two-dimensional wall
turbulence. The difference between Eqs. (3) and (4) is
however of importance for improving the predictions for acorner flow, which will be discussed in the next section.

(6)Here, Ry is the Reynolds number based on the square-root of
the turbulent kinetic energy k and the distance from thewall y. Eqs. (5) and (6) are applicable to separated turbulent
flows. The original AMM model constants were determinedusing the DNS data of a turbulent channel flow, which are
expressed as Cμ = 0.09 , σ k = σ ε = 1.4 , Cε = 1.5 , andCε = 1.9 . In the present study, we use the modified diffusion
coefficients, i.e., σ k = 1.1 and σ ε = 1.25 . These values havebeen obtained on the basis of the mathematical analysis ofCazalbou, Spalart and Bradshaw (1994) by Abe and Spalart(2020).
Figure 1a shows the predictions of the AMM model for

RESULTS AND DISCUSSION
Square Duct
We first examine the predictions of AMM-QCRcorner (Eq.
(4)) in a square duct by comparing with those of the AMM
model with different QCRs (i.e. Eq. (2) and (3) ). The flow isdriven by a constant mean pressure gradient, which is solved
by an incompressible flow code with a second-orderaccuracy in space. The friction Reynolds number isReτ=1000. The number of grid points are 448 × 448 in thewall-normal (y) and spanwise (z) directions (note that thegrid is clustered near the wall).
Figure 2 shows the contours in the y-z plane of thenormalized mean streamwise velocity U with velocity
vectors, predicted by the AMM model with Eqs. (2) to (4).
Indeed, use of the QCR reproduces the secondary flow near acorner. There are however some differences in thepredictions. The AMM model with Eq. (2) predicts a weaker

12

Reτ=2000

U+

20

10

(a)

AMM
MK
AKN
DNS (Reτ=2003, HJ)++
U =(1/0.41)ln(y )+5.0

0 0
10

101

102

+

y

103

10
Reτ=2000

8

AMM with Eq. (4)AMM with Eq. (3)AMM with Eq. (2)DNS (Reτ=2003, HJ)

6

u+u+

4
2
0 1
10

(a)

(b)

(c)

(d)

w+w+
v+v+

2

3
10
y+ 10
(b)
Figure 1. Predictions for normalized mean velocity
U and the Reynolds normal stresses uu , vv , ww in a+turbulent channel flow at Reτ =2000: (a) U ; (b) u +u + ,v + v + , w+ w+ (the superscript + denotes normalization bywall units). AMM with Eqs. (2) to (4) have been tested.
Note that for AMM with Eq. (2), the u +u + data havebeen exclusively plotted (v, w components are notshown). The calculated data have been compared with
the DNS data of Hoyas and Jimenez (2006).

Figure 2. Contours in the y-z plane of normalized mean
streamwise velocity with velocity vectors in a square duct at
Reτ=1000, calculated with AMM model using various QCRs:(a), (b) AMM with Eq. (4) (AMM-QCRcorner); (c) AMM with
Eq. (3) (AMM-QCR) ; (d) AMM with Eq. (2) (AMM-QCR).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
secondary flow than that with Eq. (3). This is attributed tothe increased effect of the damping function fμ in the timescale term ( ν t / k ) of Eq. (2). On the other hand, the AMM
model with Eq. (4) yields a stronger secondary flow than thatwith Eqs. (2) and (3), which is essentially owing to the effectof the ΩΩ term in Eq. (4).
Figure 3 shows the contours in the y-z plane of thenormalized mean streamwise vorticity Ω x in a square duct
near a corner. Indeed, we see a large magnitude of Ω x in the(a)prediction of the AMM model with Eq. (4), which agrees
reasonably with the DNS result by Pirozzoli et al. (2018). On
the other hand, there is a weaker magnitude of Ω x for theprediction with Eq. (3) than that with Eq. (4), indicating thatthe inclusion of the ΩΩ term in Eq. (4) plays a role instrengthening the mean streamwise vorticity near a corner.
Figures 3(b) and (c) show the comparison in the predictions
obtained from the AMM model with Eq. (4) and that with Eq.
(1), the latter QCR being widely used in the aeronauticalapplications. Indeed, the prediction of the AMM model with
Eq. (4) (AMM-QCRcorner) yields a larger magnitude of Ω xthan AMM with Eq. (1).
An inspection revealed that the predictions of the AMM
model with Eq. (4) for vv (not shown here) and ww (figure4b) are closer to the DNS data than those with the AMM
model with Eq. (3), indicating that the Reynolds normal
stress anisotropy is improved in Eq. (4). This is due to theterm C3 (k / ε )ν t Ω32Ω32 = −C3 (k / ε )ν t Ω 2x > 0 in Eq. (4).
Figure 4 shows the normalized wall shear stress τ w in asquare duct near a corner at y/h=0.01. Indeed, the AMM
model with Eq. (4) leads to a larger magnitude of τ w . The
latter distribution agrees well with the DNS data of Pirozzoli
et al. (2018). This result highlights the importance ofincluding the ΩΩ term in the QCR for improving theprediction in a corner flow.

(b)

(c)

NASA Juncture Flow
We next examine the predictions for the NASA Juncture
Flow (JF) with a hone. The full span JF model consists of afuselage and two wings, where the wing span is 3397.250
mm, and the fuselage length is 4839.233 mm (see the details
in Rumsey et al. 2019). In this flow, a separation bubble (i.e.
side-of-body separation) occurs near the trailing edge of thewing near a wing-root junction where the adverse pressure
gradient is large. We thus need to predict not only aseparated flow but also a secondary flow induced by the
Reynolds stress as accurately as possible. It was noted thatthe prediction with a linear eddy viscosity model yields alarger separation bubble near a corner, while use of the QCR
reduces the size of the separation bubble (see Yamamoto etal. 2012). Particular attention is therefore given to theprediction for the corner separation bubble with the QCR.
The flow solver used in the present study is FaSTAR
developed by JAXA (Hashimoto et al. 2012), which is an
unstructured-grid compressible flow solver with the secondorder accuracy. The inviscid flux scheme HLLEW (Obayashi
and Guruswamy 1995) is used for the convective terms. The
numerical flux has been calculated at a cell vertex. The
details on the numerical methodology are given in
Hashimoto et al. (2012).
The grids used in the present study are those provided by
the TMR website (https://turbmodels.larc.nasa.gov). The grid
dependency on the JF computation with the AMM model has
been examined by Abe et al. (2020) using three unstructured
grids for free-air (i.e. without considering the mast and stingsupport system), where the number of grid points are
12,312,544 (coase), 39,121,991 (MED) and 160,761,294(FINE). It was shown that the difference in the predictions
between the coarse and MED grids is greater than thatbetween the MED and FINE grids. In the present study, we
therefore use two grids, i.e. coarse and MED grids forclarifying the effect of the QCR on the prediction of a cornerseparation bubble. Referring to the comprehensive study of(d)Figure 3. Distributions of Ω x normalized by U b / h in asquare duct for Reτ=1000, calculated using AMM model withvarious QCRs; (a) AMM with Eq. (3) (AMM-QCR); (b) AMM
with Eq. (4) (AMM-QCRcorner); (c) AMM with Eq. (1); (d)DNS by Pirozzoli et al. (2018) .
1

τw/τw

0.8

Reτ=1000

0.6
AMM with Eq. (4)AMM with Eq. (3)AMM with Eq. (2)AMM with Eq. (1)AMM (Linear)
DNS (Pirozzoli et al. 2018 )

0.4
0.2
0

(a)

0.02

0.04

0.06

0.08 z/h 0.1

w+w+

2

1
AMM with Eq. (4)AMM with Eq. (3)AMM with Eq. (2)DNS (Pirozzoli et al. 2018 )

0
0.05
0.1
z/h
(b)
Figure 4. Normalized local wall shear stress τ w and spanwise
Reynolds normal stress in a square duct for Reτ =1000,
calculated with AMM using various QCRs. The enlarged viewin the region y/h=0 to 0.1 is shown. The calculated data havebeen compared with the DNS data of Pirozzoli et al. (2018).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
Spalart and Rumsey (2009), the turbulence intensity Tu andthe viscous ratio νt/ν for the AMM model is 0.08 and 0.1.
The attack of angles computed in the present study are in therange α= -2.5 to 5 ° .
As noted in the introduction, our earlier study in this flow
(Abe et al. 2020) reported that a smaller corner separation
bubble is predicted compared with that of the experimentwhen the AMM model with Eq. (3) is used. A close
inspection revealed that this is intrinsically associated withthe overestimation of the eddy viscosity when the adverse
pressure gradient is large (see the contours of ν t in Fig. 7b).
In the current study, we therefore modify the expression forν t by incorporating a parameter S ij2 − Ω ij2 , representing theacceleration and deceleration of the mean flow, into theturbulence time scale Tμ using the augmented time scale
procedure by Yoshizawa et al. (2006), i.e.,
2 1/ 2

ν t = Cμ f μ k Tμ / 1 + CSO ( Sij2 − Ωij2 ) ( k / ε )  
2





 

,

that (=1.0) of the original limier in the AMM model (see Abe
et al. 2020). This modification is done by consideringcomputations for practical applications.
Figure 6 shows the predictions of AMM-QCRcornerwith/without the modification of Eq. (7) for the skin friction
coefficient Cf and the streamlines. Also shown in Figure 7
are the contours of ν t in the trailing edge of the wing near acorner. The attack of angle is α = 5 ° . The coarse grid is used.
When the original eddy viscosity expression ν t = Cμ f μ k Tμ
is used (Figs. 6b and 7b), the magnitude of ν t becomes large
in the trailing edge of the wing-body juncture so that the
separation bubble predicted is approximately twice smaller
than that of the experiment. Use of Eq. (7) indeed improves
the prediction near a corner where the magnitude of ν t
becomes smaller than the prediction with ν t = Cμ f μ k Tμ sothat the size of the separation bubble is comparable to that of
the experiment. This result underlines that Eq. (7) is effective
for improving the overprediction of ν t in the trailing edge
near a corner and yielding a larger size of the separation
bubble; S ij2 − Ω ij2 is an important parameter for characterizing
turbulent flows with a mean pressure gradient.
Figure 8 shows the enlarged view of the cornerseparation bubble, predicted by the AMM, AMM-QCRcornerand SA-QCR2000 (noft2) models at α = 5 ° . The MED gridis used. When a linear model (i.e. the AMM model) is used,
there appears a large separation bubble near a corner. Use of
the QCR (i.e. the AMM-QCRcorner model) reduces the size ofthe corner separation bubble (see Figs. 8a and b; see also
Table 1 where the size of the bubble is compared with that ofexperiment). This behavior is consistent with that of
Yamamoto et al. (2012) on the NASA CRM wing-bodycomputations. A comparison between AMM-QCRcorner and(7)

where CSO is a model constant. The value of CSO =1 is used
after testing the values of 0.5, 1 and 2 in the current flow. We
also modify the near-wall behavior in the diffusion
coefficients as follows, i.e.,σ k = 1.1 / f sig and σ ε = 1.25 / f sig ,

where

(

f sig = 1 + 6 − exp ( Ry / 20 )

2

).

(8)
(9)

Other modification is the realizability limiter. We use that
proposed by Medic and Durbin (2002) on turbine blades, i.e.


k
0.6
.
Tμ = min  ,1/2ε6Cμ ( Sij2 ) 


(10)

The value of the numerator (=0.6) in Eq. (10) is smaller than

z

x
y
(a)

(a)

(b)

(b)

Figure 7. Contours of the skin friction coefficient Cf (color),
streamlines (black line) near a corner together with those ofνt for AMM-QCRcorner, obtained with the attack of angle α =5 ° degrees and the coarse grid: (a) Eq. (7) is used for ν t ;(b) ν t = Cμ f μ k Tμ is used for ν t .

Figure 6. Contours of the skin friction coefficient Cf (color)and streamlines (black line) for AMM-QCRcorner, obtainedwith the attack of angle α = 5 ° and the coarse grid: (a) Eq.
(7) is used for ν t ; (b) ν t = Cμ f μ k Tμ is used for ν t .

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
SA-QCR2000 (Figs. 8b and c) indicates that AMM-QCRcorner
yields a smaller separation bubble than SA-QCR2000, theformer model predicting a larger Ω x near a corner where
AMM QCRcorner reproduces a stronger secondary flow than
SA-QCR2000 (see Fig. 9). This behavior is consistent withthat observed in a square duct (see Fig. 3). In the wing-bodyjuncture, the Cp distribution for AMM-QCRcorner also agrees
well with the experimental data (see Fig. 10 where AMM
QCRcorner predicts better than AMM and SA-QCR2000). An
inspection indicates that the size of the bubble for AMM-

QCRcorner with α=5 is within 10 percent of the experimental
results (see Table 1).
Figure 11 shows the distributions of the bubble length L
and width W as a function of α. Indeed, AMM-QCRcornerpredicts W and L reasonably at each α, where the rate ofincrease for both W and L agrees well with that of the

AMM-QCRcorner

(a)

(a)

SA-QCR2000

(b)

(b)
Figure 9. Contours of the skin friction coefficient Cf (color)and streamlines (black line) together with those of the mean
streamwise vorticity Ω x and the velocity vectors near acorner for AMM, AMM-QCRcorner and SA-QCR2000,
obtained with the attack of angle α = 5 ° and the MED grid:(a) AMM-QCRcorner ; (b) SA-QCR2000. For AMMQCRcorner, Eq. (7) is used for ν t .

(c)Figure 8. Contours of the skin friction coefficient Cf (color)and streamlines (black line) near a corner, obtained with theattack of angle α = 5 ° and the MED grid: (a) AMM ; (b)AMM-QCRcorner ; (c) SA-QCR2000. For AMM and AMMQCRcorner, Eq. (7) is used for ν t .

y=-254 [mm]1
-Cp

Table 1. The size of the corner separation bubble at α=5 ° ,obtained with the MED grid.

AMM-QCR corner
AMM
SA-QCR2000
Expt.

0

LengthWidthExperiment(Error bar)

119mm(±15.5mm)

43mm(±6.8mm)

AMM-QCRcorner

129mm

47mm

AMM

173mm

71mm

SA-QCR2000

146mm

52mm

-1

2000 2200 2400 2600 2800 3000
x [mm]

Figure 10. Distributions of the wall-pressure coefficient Cp
at y=-254 mm with α = 5 ° for AMM, AMM-QCRcorner and
SA-QCR2000, obtained with the MED grid.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
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experiment. In particular, AMM-QCRcorner predicts L better
than SA-QCR2000, highlighting the usefulness of Eq. (4) inpredicting the corner separation bubble.
200

L [mm]

AMM-QCRcorner
SA-QCR2000
Expt.

100

(a)

0

0

α [deg]5

α [deg]5100

W [mm]

80
60

AMM-QCRcorner
SA-QCR2000
Expt.

40
20

(b)

0

0

Figure 11. Distributions of the bubble length L and width W
for AMM-QCRcorner compared with those for SA-QCR2000
and the experimental data: (a) distributions of L as afunction of α ; (b) distributions of W as a function of α .
The experimental data of Kegerise and Neuhart (2019) arealso plotted for comparison.
CONCLUSIONS
The main conclusions are summarized as follows.
1) We have developed a new QCR (Eq. 4) by considering
the non-zero value of Ω x in a corner flow.
2) In a square duct, AMM-QCRcorner (AMM with Eq. (4))
reproduces a strong secondary flow near a corner withlarge Ω x , and the prediction agrees reasonably well with
the DNS data.
3) In NASA Juncture Flow, Eq. (7) is effective for
improving the overprediction of ν t in the trailing edge
near a corner. Use of the QCR reduces the size of thecorner separation bubble and yields a better prediction
than without QCR. Indeed, AMM-QCRcorner (AMM with
Eq. (4)) predicts a corner separation bubble whose size iscomparable to that of the experiment.
REFERENCES
Abe, H., 2017, “Reynolds-number dependence of wallpressure fluctuations in a pressure-induced turbulent
separation bubble,” J. Fluid Mech., Vol. 833, pp. 563–598.
Abe, H., Mizobuchi Y. and Matsuo, Y., 2019,“Prediction of turbulent flow around an airfoil with anonlinear k-ε model,” Proc. of 51th Fluid Dynamics
Conference / 37th Aerospace Numerical Simulation
Symposium, 2E07 (in Japanese).
Abe, H., Mizobuchi, Y. and Matsuo, Y., 2020, “Effect ofa Quadratic Constitutive Relation on Juncture Flow
Computations,” AIAA paper 2020-2752.
Abe, K., Kondoh, T. and Nagano, Y., 1994, “A new
turbulence model for predicting fluid flow and heat transfer
in separating and reattaching flows–I. Flow field

ACKNOWLEGDEMENTS
Computations performed on the JAXA Supercomputer
System are gratefully acknowledged. HA would like to thankProf. Y. Matsuo for the earlier collaboration, Drs. P. R.
Spalart and K. Yamamoto for the discussion on the QCRs
and Dr. M. Hishida and Mr. K. Hayashi for their
computational support throughout this work.

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