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

INFLOW TURBULENCE GENERATION USING EQUIVALENT BOUNDARYLAYER

Haining Wang
Weixi Huang
Department of Engineering Mechanics
Department of Engineering Mechanics
Tsinghua University
Tsinghua University
Haidian District, Beijing, 100084, P. R. China Haidian District, Beijing, 100084, P. R. China
whn17@mails.tsinghua.edu.cn
hwx@tsinghua.edu.cn

Chunxiao Xu
Department of Engineering Mechanics
Tsinghua University
Haidian District, Beijing, 100084, P. R. China
xucx@tsinghua.edu.cn

ABSTRACT

flow turbulence generation. The basic idea of LWS method
is to rescale the downstream flow based the self-similarity of
TBL and reintroduce it to the inlet. Its key assumption is thatthe inner region of TBL flow scales in y+ and the outer region in y/δ , where y+ and y/δ are the wall-normal coordinates
measured by inner and outer length scales respectively. Therefore, after piecewise rescaling and appropriate matching, flow
at a downstream position can be recycled to serve as boundary
condition at the inlet.
Despite the major success achieved, LWS method is subject to some drawbacks. The method is artificial in three respects. 1. It relies on the streamwise developing law of TBL,
which is derived from the log law or power law of the mean
velocity. 2. It explicitly assumes the form of self-similarity, including the universal law of the wall in the inner region and defect law in the outer region, which, though being fairly precise
for the mean streamwise velocity, are not satisfactory for themean wall-normal velocity and velocity fluctuations; Moreover, the range of applicability of these two laws are not precisely clear, which leads to the third problem: 3. The weighting function and its parameters to match the two regions are
empirically chosen, reflecting the arbitrariness in applying theinner and outer laws. Another difficulty concerns the choiceof the recycling position. The recycling plane should not be
too close to the inlet, because it must fall in healthy flow
sections possessing self-similarity, avoiding unphysical adjustment zones near the inlet and outlet; Besides, sufficient natural evolution of turbulence structures should be accommodated
between the inlet and recycling plane, especially for highReynolds-number turbulence which contains abundant largescale structures. On the other hand, the recycling plane should
not be too far from the inlet, because the spanwise and temporal scales need also be matched aside from the wall-normal
rescaling.
The above discussion has revealed the shortcomings ofthe traditional method in generating TBL-type inflow turbulence. In the current study, we propose a new method to generate TBL-type inflow turbulence fluctuations. This method isbased on the open-channel model, therefore naturally applying the streamwise periodic boundary condition to simulation

Eddy-resolved simulation of external flow usually requires inflow conditions representing a zero-pressure-gradientturbulent boundary layer (ZPGTBL) flow, and the quality of
the inflow conditions directly impact the accuracy of the simulation. The present study proposes a new method to generate
ZPGTBL-type inflow turbulence, i.e. the equivalent boundary layer (EBL). Based on half channel model, EBL approximates ZPGTBL flow by recovering the mean momentum balance with driving force. It simulates streamwise equilibrium
turbulence, applying periodic boundary conditions, and thus
overcomes the complexity and artificiality incurred by the classic recycling-rescaling methods. The current paper discusses
the difference between turbulent channel and boundary layer
flows from the equation point of view, and designs the driving force corresponding to the mean inertial force of boundary layer. Also, the total shear stress models for obtaining thedriving force are validated both a priori and a posteriori. Direct numerical simulations (DNS) are carried out for EBLs at
Reθ = 1000, 1420 and 2000 (where Reθ is the Reynolds number based on momentum thickness), showing that EBL well reflects the statistical characteristics of ZPGTBL at corresponding Reynolds numbers. The application of EBL to the generation of inflow conditions is also demonstrated by DNS of turbulent boundary layers with inlet Reθ = 1000, 1420 and 2000.
The computational results agree well with generally acknowledged DNS data published in literature, in terms of streamwise
developing statistics and profiles and energy spectra at characteristic cross-sections. Judging from the mean velocity, the adjustment section is shorter than one boundary layer thickness.

1

INTRODUCTION

The present study focuses on the generation of inflow
turbulence fluctuations representing the zero-pressure-gradientturbulent boundary layer (TBL). A classic approach to thisgoal is the ’LWS’ method, proposed by Lund et al. (1998) andadapted from the coordinate transformation method by Spalart(1988). Due to its simplicity and low cost in computation,
LWS method has been one of the most popular methods in in-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
pressure gradient and the total shear stress (sum of the meanviscous shear stress and the Reynolds shear stress), while thatin TBL is between the mean inertial force and the total shear
stress. The mean pressure gradient ∂ P/∂ x in channel flow isconstant throughout the whole field, while the inertial force
U∂U/∂ x + V ∂U/∂ y in TBL (at a fixed streamwise position)
is varying along the wall-normal direction: It is zero at thewall and outside the boundary layer, and peaks somewhere inside the boundary layer (see Figure 1(c)). This observationof the equations suggests that, the wall-normal distribution of
the driving force counterbalancing the fluid shear is key to thestatistical differences between channel flows and TBL.
The instantaneous streamwise equation of motion forchannel flows isof the spatially homogeneous flow. Meanwhile, in order to fill
the gap between channel flow and TBL, external force is added
to the open-channel flow to drive it closer to the real TBL. The
rest of the paper is organized as follows. In Section 2 we introduce the mathematical formulations including the theoreticbasis of EBL and how the driving force can be modelled. In
Section 3 the idea of using EBL to generate inflow turbulence
for TBL is validated by DNS a priori and a posteriori. Finally,in Section 4, we conclude the present work.

2

FORMULATIONThe computational models in the current study are openchannel and TBL, whose symbols and notations are defined asfollows. x, y, z represent the streamwise, wall-normal and spanwise directions respectively, and u, v, w are the corresponding velocity components (also denoted by their vector forms
x , u ). The boundary layer thickness, displacement thickness
and momentum thickness are denoted by δ , δ ∗ , θ respectively.
The Reynolds number Re = U∞ δ /ν is defined by the boundary layer thickness, free velocity U∞ and kinetic viscosity ν.
Also frequently used are the Reynolds number based on the
momentum thickness Reθ = U∞ θ /ν and the friction Reynolds
number Reτ = uτ δ /ν, where uτ is the friction velocity. The
superscript ‘+’ represents normalization by the wall viscous
units, and η = y/δ is the wall-normal coordinate scaled byouter boundary layer thickness. The prime symbol ‘(·)0 ’ represents fluctuations and the angle bracket ‘h·i’ or capital lettersrepresent mean quantities averaged along temporal and spanwise (and streamwise if the flow is streamwise equilibrium)
directions.

2.1

∂u
1
+ u · ∇u = −
∂t
ρ


+ ν∆u



∂U
∂U
∂V
∂U
Fx = − U
+V
=U
−V
∂x
∂y
∂y
∂y

(3)

(4)

and use it in place of the mean pressure gradient term in Equation 3, i.e.
1 ∂ p0∂u+ u · ∇ u = Fx −
+ ν∆u
∂t
ρ ∂x

(5)

Simulating Equation 5 with open-channel configuration, applying streamwise periodic boundary conditions, the mean
momentum balance in TBL at this Reynolds number should
be recovered. We term such a streamwise equilibrium turbulent flow as the equivalent boundary layer (EBL).

EQUIVALENT BOUNDARY LAYER

2.2

DRIVING FORCE

Before simulating EBL, the wall-normal distribution of
the driving force needs to be known. Equation 2 shows
the driving force (mean inertial force) to be balanced withthe wall-normal derivative of the total shear stress under theboundary layer assumption. Therefore, we consider modelingthe total shear stress. Figure 1(a) shows the wall-normal distributions of the total shear stress of TBL at various Reynolds
numbers. It can be seen that the total shear stress is Reynoldsnumber-independent if scaled by the wall shear stress τw andthe boundary layer thickness δ . Since the driving force is equal
to the wall-normal derivative of the total shear stress, it can bescaled by τw /δ and δ , as in Figure 1(c). Researches have attempted to model the total shear stress provided its universality. Chen & She (2016) modelled the total shear stress in TBL
as(1)

Under the assumption of boundary layer, the mean equationsof motion for TBL are



∂U
∂U
d < u0 v0 >
d 2U


U
+V
=−
+ν 2

∂x
∂y
dy
dy
0 v0 >

1
∂
P
d
<
v


=−

ρ ∂y
dy

∂ P ∂ p0+∂x∂x

We define the driving force in TBL (at a given Reynolds number) as the mean inertial force

Existing researches have explained the difference between channel and TBL flows from the aspect of physicalmechanisms. Here we try to look for another interpretationfrom the equation point of view. The theory of turbulenceenergy hierarchy has made clear that the energy is first extracted from the mean motion, transferred from large to small
scales, and eventually transformed into heat at dissipation
scales. Therefore, the mean motion plays an important rolein differentiating one flow from another. Mathematically, thetwo flows are both governed by the Navier-Stokes (NS) equations, but the different geometric symmetry leads to different
simplifications of the equations.
The mean equations of motion for channel flows are
1 ∂P
d < u0 v0 >
d 2U


+ν 2
ρ ∂x = −dydy
0 v0 >

1
∂
P
d
<
v


=−
ρ ∂y
dy



+
τtot
(η) = 1 − η 3/2(6)

Kumar & Krishnan (2021) argued that the total shear stressis function of the shape factor H and normalized mean wallnormal velocity V /V∞ , where V /V∞ is universal (Wei &Klewicki, 2016) while H slowly varies with the Reynolds
number. They proposed a model in the form of(2)

A comparison the above equations reveals that the streamwise
momentum balance in a channel flow is between the mean+τtot(η) = H(1 −2V) + (H − 1)(η − 1)V∞

(7)

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

(b)

Figure 1. Total shear stress and driving force in TBL. (a) Total shear stress in TBL, from DNS (Schlatter & Örlü, 2010)of TBL at Reθ = 677 − 4061; (b) Models (Chen & She, 2016;
Kumar & Krishnan, 2021) of TBL total shear stress; (c) Mean
inertial force in TBL, equal to the driving force in EBL (legends as in panel a); (d) Driving force obtained from total stressmodels (legends as in panel b).

Figure 2. Instantaneous flow fields of EBL. (a) EBL1000; (b) EBL-2000. Visualized are iso-surfaces of Q ≡−0.5(∂ ui /∂ x j )(∂ u j /∂ xi ) at Q = 1, colored by the streamwise
velocity u.

posed by Kim et al. (2002) is used to decouple the momentum and continuity equations. This projection method ensures
the second-order precision in both space and time, and moreover, it requires no pressure boundary condition because thedivergence-free condition is handled directly by the ‘projection’ step. For details of the algorithm the readers are referredto the original papers (Perot, 1993; Kim et al., 2002).

where the universal function V /V∞ can be fitted by a hyperbolic tangent function V /V∞ = tanh (aη + bη 3 ), with the fitting parameters chosen as a = 0.5055, b = 1.156. The above
two models are not bounded when η → ∞, so in practice η isreplaced by a rescaled wall-normal coordinateη̄ =η
(1 + η α )1/α

(8)

3.1

This rescaled wall-normal coordinate satisfies η̄ ≈ η when ηis small, and η̄ → 1 when η → ∞. The parameter α in thefunction controls the damping rate, and the values fitting the
DNS data best are α = 17 for the Chen & She (2016) modeland α = 8 for the Kumar & Krishnan (2021) model.
Figure 1(b,d) shows the modelled total shear stress andits derivative at Reθ = 1420. The models both agree well with
the DNS data in terms of the total shear stress, but the errors
are magnified after taking the derivative. The model of Chen
& She (2016) behaves well near the wall, but shows a needfor higher-order approximations at the edge of boundary layer.
The model of Kumar & Krishnan (2021) agrees better with the
DNS data in the outer region, but violates (∂ τtot /∂ y)w = 0 at
the wall, which is obvious since the derivative of Equation 7,+∂ τtot/∂ η = H 1 − (a + 3bη 2 )/ cosh (aη + bη 3 ) −1 does notreach zero for η = 0. In sum, errors in the driving force must
be induced by taking derivative of the modelled total shear
stress. But as volume force, the driving force’s integral properties is more important, while the errors in its specific distribution have only secondary influences. This will be confirmed
by numerical simulations in Section 3.1.

3

Simulation of EBL

DNS is carried out for EBL at three Reynolds numbers.
The dimensionless governing equations are11
 ∂ u + u · ∇ u = −∇∇ p + ∆uu + C f Reτ Fx+ e 1
∂t
Re
2

∇·u =0(9)

The boundary conditions of open-channel are applied, namely
periodic boundary conditions for the streamwise and spanwise
directions, no-slip wall boundary condition on the bottom, andfree-slip boundary condition (∂ u/∂ y = ∂ w/∂ y = 0, v = 0) on
the top. The parameters of the cases are listed in Table 1. The
domain sizes are chosen in consideration of the size of VLSM
(about 1 − 2δ wide and O(10δ ) long (Smits et al., 2011)).
The grid resolution is chosen to be higher than that used in
the DNS research by Schlatter et al. (2009). The grid points
are stretched following a hyperbolic tangent distribution in thewall-normal direction. The cases EBL-1000, EBL-1420 and
EBL-2000 are preliminary examinations of EBL, in which wewish to exclude the influence related to error in the driving
forces, so C f and Reτ Fx+ appearing in the expression of driving force are interpolated from the DNS results by Schlatter &Örlü (2010), which are deemed exact. Other than that, we test
the performances of driving force models in EBL simulations
through cases EBL-1420-KK21 (using Equation 7) and EBL1420-CS16 (using Equation 6). These two cases have the same
configurations to case EBL-1420, except for the driving force.
Figure 2 visualizes the instantaneous flow fields of EBL1000 and EBL-2000. It is evident that the flow fields are populated by vortical structures typical to wall turbulence. The

NUMERICAL TESTSIn this section DNS will be carried out to examine the
EBL model and to evaluate its performance in generating inflow boundary conditions for TBL. The numerical schemes
are described as follows. The second-order central difference
and the Crank-Nicolson schemes are used for spatial and temporal discretization respectively. The projection method pro-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Parameters of the EBL cases. The case names are numbered by Reθ of their corresponding TBLs. The last two cases arelabelled by their driving force models (‘KK21’ (Kumar & Krishnan, 2021) and ‘CS16’ (Chen & She, 2016)). The U∞ , δ involved in
the parameters (e.g. the Reynolds numbers and domain sizes) are all based on their nominal values. ∆y+e represents the inner-scaled
wall-normal grid interval at the edge (y = δ ) of the boundary layer.

Case nameReReτ

Lx , Ly , Lz

Nx , Ny , Nz++
∆x+ , ∆y+min , ∆ye , ∆z

EBL-1000

7783

359

(8, 6, 5)δ

385, 325, 385

7.5, 0.14, 6.6, 4.7

EBL-1420

11168

492

(8, 4, 4)δ

385, 325, 385

10.3, 0.10, 9.1, 5.1

EBL-2000

15954

671

(8, 4, 4)δ

513, 433, 513

10.5, 0.07, 9.9, 5.2

EBL-1420-KK21

11168

492

(8, 4, 4)δ

385, 325, 385

10.3, 0.10, 9.1, 5.1

EBL-1420-CS16

11168

492

(8, 4, 4)δ

385, 325, 385

10.3, 0.10, 9.1, 5.1

structures have smaller scales near the wall, and becomeslarger further away from the wall. The vortices show streaky
patterns, and the spanwise scale of these streaks is about one
boundary layer thickness estimated from Figure 2(a). Similar
to that reported by Jiménez et al. (2010), no ordered hairpin‘forest’ as described by Wu & Moin (2009) can be observedin the present visualization. This is probably because the fully
developed periodic flow in EBL has lost any transitional effect.
A comparison of Figure 2(a) and (b) shows that more finerscaled structures emerge as the Reynolds number gets higher.
The quantitative characteristics of EBL flow fields areinspected. The first- and second-order statistics of EBL are
shown in Figure 3, and compared with the authentic TBL andchannel flows at similar Reynolds numbers. It is found in Figure 3(a) that the mean velocity profiles of EBL and TBL agree
well under y+ = 100 scaled by inner units. In the wake region, the mean velocity of TBL is higher than that of channel.
This is because the high-speed potential flow in the intermittent region injects momentum to the boundary layer (Jiménez
et al., 2010). Though the wake of EBL is also higher than thatof channel, it is lower than that of TBL, leading to a more
distinct discrepancy: the velocity of potential flow in EBL
does not reach that of TBL (U|yδ = U∞ ), but stays aroundU|yδ ≈ 0.92U∞ . The reason for such a discrepancy is still
not clear, but it is suspected to be the top boundary condition
and the lack of mean wall-normal velocity. The streamwise
fluctuation of TBL does not differ much from that of channel,
but the spanwise and wall-normal fluctuations and Reynolds
shear stress in TBL are significantly higher than that in channel, as shown in Figure 3(b,d). Note that the Reτ of the channel
flow (CHN-540) shown in Figure 3 is about 50 higher than thatof the TBL (REF-1420), but this is not the reason for the differences in their statistics. Since the fluctuation intensity of wall
turbulence grows with Reτ increasing, the two kinds of flows
would differ even more if they had the same Reτ . The above
observations are consistent with that by Jiménez et al. (2010).
For statistics which differentiate TBL from channel, our EBL
model manages to give results that are closer to TBL. Therefore, though periodic open-channel model is used, the driving
force applied succeeds in recovering the characteristics of turbulent fluctuations in authentic TBL.

Figure 3. First- and second-order statistics of EBL. (a) Mean
streamwise velocity profile; (b) Streamwise fluctuation intensity; (c) Reynolds shear stress; (d) Spanwise (upper cluster)
and wall-normal (lower cluster) fluctuation intensities. Blackdashed lines (CHN-540) represent channel flow (Lee & Moser,2015) at Reτ = 543. Black solid lines (REF-1420) represent
authentic TBL (Schlatter & Örlü, 2010) at Reθ = 1420, i.e.
Reτ = 492. Red and blue lines represent results of the current
simulations.

Figure 4. Fluctuations of pressure and vorticity, and highorder statistics of the streamwise fluctuation. (a) Pressure fluctuation intensity; (b) Vorticity fluctuation intensity; (c) Thirdorder moment of the streamwise fluctuation; (d) Forth-ordermoment of the streamwise fluctuation. The legends in panel(a) have the same meanings to that in Figure 3. The grey band
(CHN-395-590) in panels (c) and (d) represent channel flows(Moser et al., 1999) within Reτ = 395 − 590.

Pressure and vorticity fluctuations, and high-order statistics of velocity fluctuations, are also checked, as shown in Figure 4. Pressure fluctuations in TBL are different from that inchannels throughout the boundary layer. Vorticity fluctuations
are also different near the wall. The EBL results are all closer
to TBL for these quantities. Figure 4(c) and (d) show the thirdand forth-order moments of the streamwise fluctuation. Since
the DNS data of channel flows available from literature do not412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
directly correspond to the Reynolds number of EBL, we takethe channel flow statistics within a small range of Reynolds
numbers around the target one. The high-order moments ofstreamwise fluctuations are not significantly different for thechannel and TBL, while they are also reasonably captured by
the EBL. These results further confirms that the fluctuations inauthentic TBL can be correctly represented by EBL.
Also demonstrated in Figure 3 is that EBL gives similar results using either accurate or modelled driving force.
The models of Kumar & Krishnan (2021) and Chen & She(2016) perform nearly identically, but the model of Chen &She (2016) has a simpler form. The model of Kumar &Krishnan (2021) contains the shape factor H slowly varying
with the Reynolds number. Whether the introduction of sucha term improves the model performance at higher Reynolds
numbers is not clear, but judging from the Reynolds number range (Reθ < 4000) covered in Figure 1, the Reynoldsnumber-independence of driving forces is a fairly good approximation. Moreover, even if the modelled driving force
differs from the accurate one (see Figure 1(d)), consistent results can be obtained. This means that EBL is not sensitive
the specific distribution of the driving force, as long as its integral characteristic, i.e. the total shear stress (especially on thewall), is correct.

3.2

×10−3

Cf

5

TBL-2000

TBL-1420

DNS: Jiménez et al. (2010)
DNS: Schlatter & Örlü (2010)
DNS: Lee & Sung (2011)

4
3
1.55

(b)

LES: Eitel-Amor, Örlü & Schlatter (2014)

H12

1.50

Experiment: Örlü & Schlatter (2013)

1.45
1.40
1.35
1000

(c)

Reτ

750
500

Reτ = Reθ /3.27 + 70 (Chen & She 2016, with fitted offset)
250

(d)

(u′rms /U∞ )peaku′i /U∞

0.15′
2 × (vrms/U∞ )peak′(wrms/U∞ )peak

0.10
0.05
1000

1250

1500

1750

2000

2250

2500

2750

3000

Reθ

Figure 5. Streamwise developing statistics of TBL. (a) Friction coefficients; (b) Shape factor H12 = δ ∗ /θ ; (c) Friction
Reynolds number; (d) Peak values of velocity fluctuations.

Nevertheless, the prediction expression by Chen & She (2016)is based on data of a quite large Reynolds numbers range(Reτ ≈ 300 − 20000), and when Reτ > 7000, the bias 70 here
leads to a relative error of merely 1%, which is acceptable.
Figure 5(d) shows the peaks of fluctuation intensity of threevelocity components. As the Reynolds number gets higher,
the peaks have lower values and are closer to the wall, scaledin outer units (δ ,U∞ ).
Notably, the inflow turbulence generated by the EBL proposed by the present study has excellent performance. The
statistics of each TBL simulation agree well with the referencedata, and follow a natural trend of development from the very
beginning of the streamwise domain. Though minor differences and jumps are present, they are all within the scatter ofthe reference data, especially those DNS data points that are
deemed the most precise among researches on TBL. As mentioned in Section 1, the rescaling-recycling based LWS methodcommonly set the recycling plane O(10δin ) downstream of theinlet in order to skip the unphysical adjustment region. But thefurther the recycling plane is away from the inlet, the longer
the adjustment region will be. Simens et al. (2009) measured
the adjustment region to be 300θin long by the peak valuesof fluctuation intensities of three velocity components. In contrast, these quantities are plotted in Figure 5(d), and no distinct
adjustment can be observed. In sum, the inflow turbulence
generation by EBL is advantageous in that almost no adjustment region is introduced, and since the computation of EBL
is completely independent of the main simulation, it is not affected by the feedback of the error in the main simulation.
To further validate the simulation results near the inlet of
TBL simulations, particular streamwise locations are examined. The statistics near the inlets of the three TBL cases areplotted in Figure 6. Because the streamwise locations are very
near to the inlets, they can be considered to have the same
Reynolds number as the inlets’, that is, Reθ = 1000, 1420 and2000 for the three cases, respectively. As reference, the DNS
results of Schlatter & Örlü (2010) at the same Reynolds numbers are plotted in the figure. The mean velocity profiles ofthe present simulations are found to agree with the reference
even very close to the inlet, and though some errors exist forthe streamwise fluctuation intensities, they are of fairly small

Simulation of TBL

In order to verify the effectiveness of EBL in generating
inflow boundary conditions, DNS of TBL at various Reynolds
numbers are carried out using auxiliary simulations of EBL to
supply inflow boundary conditions in real time. The governing
equations for the main simulation are the N-S equations nondimensionalized by ρ,U∞ and the boundary layer thickness at
the inlet, δin .

1
 ∂ u + u · ∇ u = −∇∇ p + ∆uu∂tRe∇·u =0

TBL-1000

(a)

(10)

Periodic boundary conditions are applied to the spanwise
direction, no-slip wall boundary conditions are applied tothe bottom, far-field boundary conditions (u = U∞ , ∂ v/∂ y =∂ w/∂ y = 0) are applied to the top, and convective boundary conditions with boundary layer development considered
are applied to the outlet. The inflow boundary conditions aredecomposed into the mean and fluctuation parts. The prescription of the mean velocity at the inlet is less challenging,and empirical laws or interpolation from existing experimental/numerical data will suffice (Wu, 2017). The mean streamwise and wall-normal velocity used in the present study areinterpolated from DNS data of TBL in the public database(Schlatter & Örlü, 2010). The velocity fluctuations at the inlet
are extracted from the auxiliary simulation of EBL. The parameters of TBL cases are summarized in Table 2.
In order to validate the simulation results of TBL, thestreamwise developing statistics are examined. The three TBL
simulation cases cover Reynolds number ranges overlapping
each other, so their statistics varying with Reθ are plotted in
Figure 5. Figure 5(a,b) show the friction coefficients and shapefactors. Their agreement with results in the literature demonstrates the correctness of the computation. Figure 5(c) compares the variation of Reτ with Reθ with the analytic prediction Reτ = Reθ /3.27 proposed by Chen & She (2016). The
slopes of the present results are very close to 3.27 in the prediction expression, but the intersection shows a bias of 70.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 2. Parameters of the TBL cases. The case names are numbered by Reθ at the inlet. ∆y+e represents the inner-scaled wall-normal
grid interval at the edge (y = δ ) of the boundary layer.

Growth(δout /δin )Lx Ly Lzδin , δin , δinNx , Ny , Nz++
∆x+ , ∆y+min , ∆ye , ∆z
(at inlet)

TBL-1000 EBL-1000 7783 1000 ∼ 1540 359 ∼ 540

1.61

36, 8, 51945, 433, 385

6.6, 0.14, 5.1, 4.7

TBL-1420 EBL-1420 11168 1420 ∼ 2140 492 ∼ 720

1.56

36, 6, 41945, 433, 385

9.1, 0.10, 7.3, 5.1

TBL-2000 EBL-2000 15954 2000 ∼ 2940 671 ∼ 960

1.52

36, 6, 42305, 513, 513 10.5, 0.07, 9.0, 5.2

Case name

Auxiliary
simulation

Rein

Reθ

Reτ

ics & Astronomy 59 (11).
Jiménez, Javier, Hoyas, Sergio, Simens, Mark P & Mizuno,
Yoshinori 2010 Turbulent boundary layers and channels at
moderate Reynolds numbers. Journal of Fluid Mechanics
657, 335–360.
Kim, Kyoungyoun, Baek, Seung-Jin & Sung, Hyung Jin 2002
An implicit velocity decoupling procedure for the incompressible Navier–Stokes equations. International Journal
for Numerical Methods in Fluids 38 (2), 125–138.
Kumar, Praveen & Krishnan, Mahesh 2021 Simple model formean stress in turbulent boundary layers. Physical Review
Fluids 6 (2), 24603.
Lee, Myoungkyu & Moser, Robert D. 2015 Direct numerical simulation of turbulent channel flow up to Reτ ≈ 5200.
Journal of Fluid Mechanics 774, 395–415.
Lund, Thomas S., Wu, Xiaohua & Squires, Kyle D. 1998 Generation of Turbulent Inflow Data for Spatially-Developing
Boundary Layer Simulations. Journal of Computational
Physics 140, 233–258.
Moser, Robert D., Kim, John & Mansour, Nagi N. 1999 Direct numerical simulation of turbulent channel flow up toReτ=590. Physics of Fluids 11 (4), 943–945.
Perot, J Blair 1993 An Analysis of the Fractional Step Method.
Journal of Computational Physics 108 (1), 51–58.
Schlatter, Philipp & Örlü, Ramis 2010 Assessment of direct
numerical simulation data of turbulent boundary layers.
Journal of Fluid Mechanics 659, 116–126.
Schlatter, Philipp, Örlü, Ramis, Li, Qiang, Brethouwer, Geert,
Fransson, J H M, Johansson, A V, Alfredsson, P H &Henningson, Dan S 2009 Turbulent boundary layers up toReθ = 2500 studied through simulation and experiment.
Physics of Fluids 21 (5).
Simens, Mark P, Jiménez, Javier, Hoyas, Sergio & Mizuno,
Yoshinori 2009 A high-resolution code for turbulent boundary layers. Journal of Computational Physics 228 (11),4218–4231.
Smits, Alexander J, McKeon, Beverley J & Marusic, Ivan 2011
High–Reynolds Number Wall Turbulence. Annual Review
of Fluid Mechanics 43 (1), 353–375.
Spalart, Philippe R 1988 Direct simulation of a turbulent
boundary layer up to Rθ = 1410. Journal of Fluid Mechanics 187, 37.
Wei, Tie & Klewicki, Joseph 2016 Scaling properties ofthe mean wall-normal velocity in zero-pressure-gradientboundary layers. Physical Review Fluids 1 (8), 1–8.
Wu, Xiaohua 2017 Inflow Turbulence Generation Methods.
Annual Review of Fluid Mechanics 49 (1), 23–49.
Wu, Xiaohua & Moin, Parviz 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradientflat-plate boundary layer. Journal of Fluid Mechanics 630,
5–41.

Figure 6. Statistics near the inlets of the present TBL simulation cases. (a,d) Case TBL-1000; (b,e) Case TBL-1420;(c,f) Case TBL-2000. (a-c) Mean streamwise velocity profiles; (d-f) Streamwise fluctuations. The red lines are results at
x/δin = 1, 2 and 4 of each case of the present simulations, andthe black and grey lines are DNS results as reference for the inlet (x = 0) of each case. Black lines (Ref.S) are from Schlatter& Örlü (2010), and grey lines (Ref.J) are from Jiménez et al.
(2010).

magnitude. The DNS results of Jiménez et al. (2010) using the
LWS method are also shown in Figure 6. Note that Reynolds
number Reθ = 1000 for the statistics shown in Figure 6(a,d)
actually falls out of the useful range (Reθ = 1100 − 2050)
claimed in their paper. Therefore, evident errors between theirresults and that of Schlatter & Örlü (2010) are observed in Figure 6(a,d). Such errors are just a downstream effect of the errors in the inflow boundary condition (at Reθ = 620) generatedusing the LWS method. In contrast, the inflow turbulence generated using the present EBL are quite accurate at the inlet,
hence leading to correct results even close downstream. Judging from the mean velocity, the adjustment region induced by
our inflow boundary condition is less than 1δin , if exists at all.

4

Concluding remarks

The present study proposes a new method to generate
TBL-type inflow turbulence, namely the EBL. It is based on
the open-channel model, applying driving force to recover themean momentum balance in TBL, and thus fixes the error inapproximating TBL by open-channel. EBL is homogeneous in
the streamwise direction, so the periodic boundary condition
can be applied, overcoming the complexity and arbitrarinessof the LWS method.

REFERENCES
Chen, Xi & She, Zhen-Su 2016 Analytic prediction for planar
turbulent boundary layers. Science China Physics, Mechan-6