of 14
PHYSICAL REVIEW FLUIDS
6
, 114604 (2021)
Error estimation of a homogenized streamwise periodic boundary layer
Joseph Ruan
and Guillaume Blanquart
California Institute of Technology, 1200 E California Blvd, Pasadena, California 91125, USA
(Received 5 March 2021; accepted 5 October 2021; published 29 November 2021)
An analysis was conducted of the transpiration velocity of the streamwise periodic
simulation of the turbulent flat plate boundary layer. As an often imposed quantity in
numerical simulation, the transpiration velocity plays an important role in the shape of
the wall-normal profile near the outer layer. Unlike other simulation frameworks which
impose the transpiration velocity, the recently proposed framework [Ruan and Blanquart,
Phys. Rev. Fluids
6
, 024602 (2021)
] relies on a single-scale rescaling of the wall-
normal coordinate to perform streamwise periodic boundary layer simulations. The current
manuscript highlights that any error in the transpiration velocity from these simulations is
due to a difference in inner and outer layer growth rates. A new multiscale framework
to compensate for these differing growth rates is proposed and verified but ultimately
has negligible impact on the mean profiles and turbulent intensities. These remain in
excellent agreement with previously published values. It is shown that any error in the
mean continuity equation expresses itself primarily as an error in transpiration velocity,
which decreases with Reynolds number. Overall, the error in the transpiration velocity can
be used to quantify the error in single-scale, streamwise periodic simulations.
DOI:
10.1103/PhysRevFluids.6.114604
I. INTRODUCTION
Turbulent boundary layers have persistently maintained the interest of the fluids community at
large due to the abundance of turbulent phenomena within the flow. From the uniform momentum
zones [
1
], to the large-scale motions [
2
], to asymptotic Reynolds number behavior [
3
], the flat
plate turbulent boundary layer has several unique features that unfortunately only fully emerge at
high Reynolds number. To thoroughly investigate them from a computational perspective requires
simulations at Reynolds numbers at least an order of magnitude greater than the current state-of-
the-art [
4
].
Recently, Ruan and Blanquart [
5
] provided a method for streamwise periodic direct numerical
simulation (DNS) of boundary layers that reduces the required domain size by an order of magni-
tude. To do so, a
single-scale
wall-normal rescaling was applied to the boundary layer. The resulting
simulations produced accurate global quantities and mean profiles. Alongside the domain reduction,
the simulation framework also avoided imposing the transpiration velocity, which most growing
boundary layer simulations require [
6
–
8
].
The transpiration velocity (
V
∞
) is an often overlooked boundary layer quantity. Because it is
small compared to the free-stream velocity (
V
∞
∼
0
.
005
U
∞
), it is difficult to resolve in experimental
data and overall difficult to validate numerically. Nevertheless, it is directly related to the growth
of the boundary layer and even the inner layer dynamics [
9
]. Specifically, integration of continuity
immediately gives that
V
∞
=
U
∞
d
δ
∗
/
dx
, where
δ
∗
is the displacement thickness. It has also been
shown that near the free stream (
y
≈
δ
99
), the mean wall-normal profile
V
/
V
∞
is independent of
Reynolds number [
9
]. This fact was crucial in modeling the mean stress profiles in the outer layer
[
10
]. Consequently, the transpiration velocity is crucial to the stream-wise development of skin
2469-990X/2021/6(11)/114604(14)
114604-1
©2021 American Physical Society
JOSEPH RUAN AND GUILLAUME BLANQUART
friction coefficient and shape factor. In all, despite being relatively absent in many experimental
databases, its presence is felt throughout key turbulent quantities and profiles.
The mean fields of turbulent boundary layers are two-dimensional and in order to conduct
simulations, one needs to specify the stream-wise inlet, stream-wise outlet, and wall-normal outlet
for numerical well-posedness. Although much of the current literature on boundary layer simulation
has been focused on the inflow conditions [
6
–
8
,
11
–
13
], the majority of the stream-wise growing
boundary layer simulations [
6
,
7
,
11
–
13
] impose a
V
∞
(
x
) profile at the top of the domain. As men-
tioned earlier, doing so is equivalent to imposing
d
δ
∗
/
dx
. Consequently, the agreement of their other
global quantities with empirical fits is only expected. However, the stream-wise periodic framework
[
5
] imposed no such transpiration velocity, and yet the resultant global quantities of interest were as
good as if not better than those of stream-wise growing simulations. These surprisingly good results
form the basis for the present study. Since no transpiration velocity was imposed, the resultant
transpiration velocity within the periodic simulation can be used as a quantification of numerical
error. Given how deeply tied the transpiration velocity is to boundary layer dynamics, it is necessary
to (1) investigate and quantify the error in transpiration velocity and (2) look at the impact of multi
scale vs single-scale simulations for a range of Reynolds numbers numerically and analytically.
First, Sec.
II
gives a quick overview of Ref. [
5
] and theorizes how errors in the transpiration
velocity might develop. Then we investigate the multiscale nature of the boundary layer,
apriori
in Sec.
III
and
a posteriori
in Sec.
IV
. Finally, Sec.
V
provides an interrogation of the continuity
equation via an error budget.
II. ANALYSIS OF STATIONARY BOUNDARY LAYER
A. A review of streamwise periodic boundary layer simulations
We only provide a brief overview of the stream-wise periodic boundary layer framework. The
reader is referred to Ruan and Blanquart [
5
] for a complete derivation of Eqs. (
2
)–(
4
) (below).
A flat plate boundary layer under Cartesian coordinates (
x
1
,
x
2
,
x
3
), has velocity components in
the stream-wise (
x
1
), wall-normal (
x
2
), and span-wise (
x
3
) directions, respectively. We define the
following new coordinates through a wall-normal rescaling:
ξ
1
=
x
1
,ξ
2
=
x
2
q
0
q
(
x
1
)
,ξ
3
=
x
3
,
(1)
where
q
=
q
(
x
) is a single length scale meant to parallelize the boundary layer locally. This
coordinate change is applied to the Navier-Stokes equations. Two critical assumptions are further
applied:
(1) There exists a function
q
(
x
1
) such that ensemble-averaged quantities are both statistically
stationary and statistically homogeneous in the
ξ
1
,ξ
3
directions.
(2) The rescaled governing equations are evaluated at the stream-wise location where
x
1
=
x
0
,
q
(
x
1
)
=
q
0
and are assumed to be valid for a narrow stream-wise domain centered at
x
1
=
x
0
.
After an
apriori
and
a posteriori
analyses, all terms with the second streamwise derivative (
q

)
and the square of the first streamwise derivative (
q

2
) are neglected, and the following equations are
obtained:
∂
u
j
∂ξ
j
=
ξ
2
q

q
∂
u
1
∂ξ
2



source term
,
(2)
∂
u
i
∂
t
=−
u
j
∂
u
i
∂ξ
j
−
1
ρ
∂
P
∂ξ
i
+
ν
∂
2
u
i
∂ξ
2
j
+
ξ
2
q

q
u
1
∂
u
i
∂ξ
2



source term
.
(3)
The additional source terms within Eqs. (
2
) and (
3
) continuously rescale the boundary layer to
account for its overall growth. In the regular
/
conventional spatially growing boundary layer, the
114604-2
ERROR ESTIMATION OF A HOMOGENIZED STREAMWISE ...
Reynolds shear stress in the stream-wise momentum equation is primarily balanced by the normal
stress term in the outer layer. Similarly, in Eq. (
3
), the Reynolds shear stress is likewise balanced
by the source term in the outer layer. However, this source term is homogeneous in the stream-wise
direction and thereby removes the stream-wise growth of the boundary layer.
Each of the source terms contain a ratio
q

/
q
owing to the coordinate change, and the term
requires closure. Assumption 2 implies that the closure equation for
q

/
q
must simultaneously bal-
ance the time, span-wise, and stream-wise averaged continuity and momentum equations throughout
the boundary layer. To satisfy this constraint, the following closure was directly derived from the
wall-normal integrated mean continuity and stream-wise momentum equations:
q

q
=
τ
w
/ρ

∞
0

u
1
,
∞

u
1

ξ
1
,ξ
3
−
u
1
u
1

ξ
1
,ξ
3

d
ξ
2
,
(4)
where
τ
w
=
ρν

∂
u
1
/∂ξ
2

ξ
1
,ξ
3
|
ξ
2
=
0
is the wall shear stress,
ρ
is the density,
ν
is the kinematic
viscosity,
u
1
,
∞
is the free-stream velocity, and
·
ξ
1
,ξ
3
denotes averaging in the homogeneous
stream-wise (
ξ
1
) and span-wise (
ξ
3
) directions. In practice, the simulation solves the governing
equations with periodic stream-wise and span-wise directions, and the closure equation is calculated
instant- aneously.
The solutions resulting from these simulations were found to produce global quantities in good
agreement with established empirical and prior DNS values. The shape factor (
H
12
=
δ
∗
/θ
) and
skin-friction coefficient (
C
f
=
2
τ
w
/ρ
u
2
1
,
∞
) were within 0.4% and 1.6% of empirical fits [
3
,
14
],
respectively, where
δ
∗
is the displacement thickness, and
θ
is the momentum thickness. These global
quantities were comparable to those of stream-wise growing boundary layer simulations. It should
be noted that these simulations were conducted on domains an order of magnitude smaller than
those of spatially growing simulations [
6
,
13
].
B. Wall-normal velocity profile
The governing equations in Eqs. (
2
)–(
4
) rely only on a single scaling. Because
q
broadly rescales
the entire boundary layer and since the majority of the boundary layer is covered by the outer
layer, it can be assumed that
q
is an outer-scale. This was confirmed in Ruan and Blanquart [
5
]
and
q

/
q
was found to be well-approximated by
θ

/θ
. However, in the near-wall region (
x
+
2
<
10),
the relevant near-wall length scale is
δ
ν
=
ν/
√
τ
w
/ρ
. One might expect the relevant source term to
be proportional to
δ

ν
/δ
ν
rather than
q

/
q
. One can show that this may lead to a discrepancy in the
transpiration velocity profiles of growing and periodic boundary layers.
Consider the ensemble averaged continuity equation and its wall-normal integrated form
∂
u
2
∂ξ
2
=
ξ
2
q

q
∂
u
1
∂ξ
2
,
(5)
u
2
(
ξ
2
)
=
ξ
2
0
q

q
ξ
∗
2
∂
u
1
∂ξ
∗
2
d
ξ
∗
2
,
(6)
where
·
defines ensemble-averaged quantities. Figure
1
compares the wall-normal velocity profiles
between a stream-wise growing [
13
] and periodic [
5
] boundary layer at Re
δ
∗
=
1460. Both profiles
agree very well in the outer layer. The periodic profile overshoots the non-periodic profile in the
inner layer and as a direct consequence, overshoots the final free stream value by about 5%. This
result is consistent with the fact that the inner layer grows much more slowly than the outer layer and
so the broadband use of
q

/
q
as a rescaling parameter throughout the boundary layer provides small
inaccuracies in the transpiration velocity. A wall-normal varying value of
q

/
q
may be necessary to
capture appropriately the evolution of
u
2
,
∞
. Obtaining a wall-normal varying metric source term of
q

/
q
would require rescaling the wall-normal coordinate by
q
ms
(
x
1
,
x
2
) instead of
q
(
x
1
).
114604-3
JOSEPH RUAN AND GUILLAUME BLANQUART
FIG. 1. Profiles of normalized wall-normal velocity
u
2
/
u
1
,
∞
and for Re
δ
∗
=
1460. Colors: (red) stream-
wise developing DNS [
15
], (blue) Periodic DNS [
5
].
III.
A PRIORI
MULTISCALE ANALYSIS
The current section analyzes the multiscale nature of the boundary layer and investigates the
impact of multi-scale terms on the mean velocity profile.
A.
Apriori
analytical multicale behavior
The discussion in the previous section suggested that a rescaling by
q
ms
=
q
ms
(
x
1
,
x
2
) might be
necessary to better capture stream-wise growth effects on the mean flow. For example, after applying
a wall-normal rescaling by
q
ms
to
∂

u
1

x
3
,
t
/∂
x
1
, one would expect the following relation:
∂

u
1

x
3
,
t
∂
x
1
=−
ξ
2
q
ms
x
q
ms
∂

u
1

ξ
3
,
t
∂ξ
2
≈−
x
2
q
ms
x
q
ms
∂

u
1

x
3
,
t
∂
x
2
.
(7)
Boundary layer data from spatially growing DNS [
13
] can be used to evaluate the LHS and RHS
of Eq. (
7
) to extract a profile for
q
ms
x
/
q
ms
. Figure
2
compares the resulting extracted profiles against
(a)(b)
FIG. 2. Normalized profiles of
q
ms
x
/
q
ms
extracted from Ref. [
13
]forRe
δ
∗
=
1460 (a), 3550 (b). Lines:
(blue)
δ
∗
q

/
q
[Eq. (
4
)]; (black) Extracted
δ
∗
q
ms
x
/
q
ms
[Eq. (
7
)]; (green) Blending
δ
∗
q
ms
x
/
q
ms
[Eqs. (
17
), (
19
)].
114604-4
ERROR ESTIMATION OF A HOMOGENIZED STREAMWISE ...
extracted values of
q

/
q
using Eq. (
4
) for two different Re
δ
∗
. It is clear that
q
ms
x
/
q
ms
varies throughout
the buffer and mesolayer, up to
ξ
+
2
≈
100. In both the near-wall region (
ξ
+
2
<
10), and in the outer
layer (
ξ
2
/θ

1),
q
ms
x
/
q
ms
is approximately constant.
The near-wall plateau can be understood by invoking the law of the wall (
u
+
1
≈
ξ
+
2
for
ξ
+
2
<
10).
The right- hand side of Eq. (
7
) becomes
q
ms
x
q
ms
ξ
2
∂

u
1

ξ
3
,
t
∂ξ
2
=
q
ms
x
q
ms
ξ
+
2
∂
(
u
+
1
u
τ
)
∂ξ
+
2
≈
q
ms
x
q
ms
u
τ
ξ
+
2
=
ξ
2
q
ms
x
q
ms
τ
w
μ
,
(8)
where inner scale notation has been used:
u
+
=
u
1

ξ
3
,
t
/
u
τ
,
ξ
+
2
=
ξ
2
/δ
ν
,
u
τ
=
ν/δ
ν
. A Taylor
expansion of the LHS of Eq. (
7
) around
ξ
2
=
0 yields
−
∂

u
1

ξ
3
,
t
∂
x
1
≈−
ξ
2
∂
∂
x
1
∂

u
1

ξ
3
,
t
∂ξ
2
ξ
2
=
0
=−
ξ
2
τ

w
μ
.
(9)
Finally, Eq. (
7
) simplifies to
q
ms
x
q
ms
inner
≈−
τ

w
τ
w
=−
C

f
C
f
=
2
δ

ν
δ
ν
.
(10)
The outer plateau can be understood by using the law of the wake (
u
+
=
U
+
∞
+
f
o
(
ξ
2
/
), for some
universal profile
f
o
, where
=
δ
∗
U
+
∞
is the Clauser thickness). The LHS of Eq. (
7
)gives
−
∂

u
1

ξ
3
,
t
∂
x
1
=−
u

τ
f
o
+

ξ
2
u
τ
f

o
≈

ξ
2
u
τ
f

o
,
(11)
where
|
u

τ
/
u
τ
|

/
has been used to simplify the equation. The RHS of Eq. (
7
)alsogives
ξ
2
q
ms
x
q
ms
∂

u
1

ξ
3
,
t
∂ξ
2
=
q
ms
x
q
ms
ξ
2
u
τ
f

o
.
(12)
Finally, Eq. (
7
) simplifies to
q
ms
x
q
ms
outer
≈

.
(13)
B. Budget analysis of multiscale behavior
The following analysis focuses on the effects of employing the wall-normal rescaling by
q
ms
(
x
1
,
x
2
)onthe
u
-momentum equation in contrast to the effects of employing a constant wall-
normal rescaling by
q
(
x
1
).
The following governing equations are obtained after applying the same two assumptions of
statistical homogeneity and a narrow stream-wise domain, and after applying the same order-of-
magnitude simplifications (see Sec.
II A
):
∂
u
i
∂ξ
i
=
ξ
2
q
ms
x
q
ms
∂
u
1
∂ξ
2
,
(14)
∂
u
i
∂
t
=−
u
j
∂
u
i
∂ξ
j
−
1
ρ
∂
P
∂ξ
i
+
ν
∂
2
u
i
∂ξ
2
j
+
ξ
2
q
ms
x
q
ms
u
1
∂
u
i
∂ξ
2
.
(15)
An
apriori
analysis of the stream-wise momentum equation is now conducted. To do so, span-
wise and temporal averages are applied to Eq. (
15
) to obtain
u
j
∂
u
i
∂ξ
j

ξ
3
,
t



convective terms
=
ν
∂
2

u
i

ξ
3
,
t
∂ξ
2
j



viscous terms
+
ξ
2
q
ms
x
q
ms
u
1
∂
u
i
∂ξ
2

ξ
3
,
t



source term
(16)
114604-5
JOSEPH RUAN AND GUILLAUME BLANQUART
FIG. 3. Streamwise momentum magnitude budget from DNS data (Ref. [
13
]) at Re
δ
∗
=
5600. Lines:
(solid blue) Convective terms; (solid magenta) Viscous terms; (dashed black)
|
q

q
ξ
2
u
1
∂
u
1
∂ξ
2

ξ
3
,
t
|
; (dashed cyan)
|
q
ms
x
q
ms
ξ
2
u
1
∂
u
1
∂ξ
2

ξ
3
,
t
|
.
A wall-normal budget of the stream-wise momentum equation is computed using DNS data [
13
]
at Re
δ
∗
=
5600 and the results are shown in Fig.
3
. On the same budget, the original source term

ξ
2
(
q

/
q
)
u
1
∂
ξ
2
(
u
1
)

ξ
3
,
t
is shown for direct comparison against the new source term from Eq. (
16
).
The original and new metric source terms agree very well in the outer layer and balance the
Reynolds shear stress. As expected, they differ by a factor of
∼
4 in the inner layer. However, they
both remain orders of magnitudes smaller than the convective and viscous terms. This observation
is consistent with Klewicki
et al.
(2007) who noted that in the near wall region, the viscous and
shear stress terms balance nearly completely.
In the near-wall region, the contribution of the multi-scale source term will always remain many
orders of magnitude smaller than all other terms, regardless of Reynolds number. This is directly
due to the slow growth rate of the flat-plate boundary layer (e.g., in the inner layer the source term
is small). It can be shown that the viscous term scales like
u
2
τ
/δ
ν
in the inner layer. The multiscale
metric term scales like
u
2
τ
q
ms
x
/
q
ms
. Hence, the ratio of the multiscale metric term to the viscous term
scales like
δ
ν
q
ms
x
/
q
ms
∼
δ

ν
, which is monotonically decreasing with Reynolds number. Similarly,
the single-scale metric term scales like
u
2
τ
q

/
q
and its ratio to the viscous term scales like
δ
ν
q

/
q
≈
δ
ν
θ

/θ
∼
C
f
/
Re
τ
. This ratio is also monotonically decreasing with Reynolds number. In both cases,
neither metric source term is significant in the near wall region with respect to the viscous term.
In summary, from an
apriori
point of view, it appears that implementing a multiscale rescaling
function
q
ms
x
/
q
ms
should not provide significant benefits on the mean profile of stream-wise velocity.
However, an
a posteriori
perspective is still needed to clarify impacts on other turbulent quantities.
IV.
A POSTERIORI
ANALYSIS OF MULTISCALE SIMULATIONS
In this section, simulations are performed using the a model for
q
ms
x
/
q
ms
. The key purpose of this
section is to conduct an
a posteriori
analysis of multiscale effects.
A. Fitted blending function
Conducting
a posteriori
analyses requires imposing a functional form for
q
ms
x
/
q
ms
similar to
methods used in Ref. [
16
]. Section
III A
has shown that near the wall
q
ms
x
/
q
ms
≈
2
δ

ν
/δ
ν
, and far
from the wall,
q
ms
x
/
q
ms
≈

/
. To transition smoothly between the two regions, a one-parameter
114604-6
ERROR ESTIMATION OF A HOMOGENIZED STREAMWISE ...
TABLE I. DNS parameters for the turbulent boundary layer simulation cases. * indicates simulations taken
from Ref. [
5
].
Dataset
Re
δ
∗
Governing equations
Nx
×
Ny
×
Nz
Sample Time
δ
99
/
u
τ
BL1460*
1460
Eqs. (
2
)–(
4
)
300
×
120
×
160
30
BL3550*
3550
Eqs. (
2
)–(
4
)
648
×
230
×
338
15
BL1460MS
1460
Eqs. (
14
), (
15
), (
19
)
300
×
120
×
160
30
BL3550MS
3550
Eqs. (
14
), (
15
), (
19
)
648
×
230
×
338
15
smoothed step function is employed.
q
ms
x
q
ms
=

g
(
ξ
2
)
+
2
δ

ν
δ
ν
[
1
−
g
(
ξ
2
)
]
=


g
(
ξ
2
)
+
2(1
−
g
(
ξ
2
))
κ
u
+
1
,
∞
+
1

,
(17)
where the log-law has been used to relate
δ

ν
/δ
ν
to

/
, the Kármán constant is approximated by
κ
=
0
.
41, and
g
(
ξ
2
) is a smooth step function defined by
g
(
ξ
2
)
=
⎧
⎨
⎩
0
ξ
+
2

10
6
r
5
−
15
r
4
+
10
r
3
ξ
2
∈
[10
δ
ν
,
0
.
1
δ
99
]
,
1
ξ
2

0
.
1
δ
99
(18)
where
r
=
ln(
ξ
+
2
/
10)
/
ln(0
.
1
δ
99
/δ
ν
). Here, it has been assumed that the near-wall region extends
up to 10
δ
ν
, and the wake region extends down to 0
.
1
δ
99
. Thus, Eq. (
17
) provides a closure equation
for
q
ms
x
/
q
ms
in terms of

/
.
To complete the closure for

/
,the
u
-momentum and continuity equations were integrated in
the wall-normal direction and then averaged in the statistically homogeneous directions (
ξ
1
,ξ
3
)to
provide the following closure equation:
τ
w
ρ

=
∞
0

g
(
ξ
2
)
+
2(1
−
g
(
ξ
2
))
κ
u
+
1
,
∞
+
1

ξ
2
∂
∂ξ
2


u
2

ξ
1
,ξ
3
,
t
−
u

ξ
1
,ξ
3
,
t
u
1
,
∞

d
ξ
2
.
(19)
Overall, Eqs. (
17
)–(
19
) provide a fitted function to the actual
q
ms
x
/
q
ms
extracted via Eq. (
7
). A
comparison between the fitted and extracted profiles of
q
ms
x
/
q
ms
is shown in Fig.
2
for Re
δ
∗
=
1460
and 3550. The fit agrees very well with the extracted function in both the outer layer plateau all
the way down to
ξ
+
2
=
10. There is a 10% relative error in the outer layer plateau value for the
lower Reynolds number comparison, but only a 3% relative error for the higher Reynolds number
comparison. The inner layer plateau is also within 5%. This difference is primarily due to the use
of the log-law for low Reynolds number flows and these differences are expected to decrease with
increased Reynolds number.
It should be noted that Eq. (
19
) will be solved in real time for simulation purposes. Thus, the
governing equations are completely closed.
B. Simulation parameters and numerical methods
The following cases were simulated and are summarized in Table
I
. Cases BL1460MS and
BL3550MS solve Eqs. (
14
) and (
15
) with closure Eq. (
19
). All cases have periodic span-wise and
streamwise boundary conditions, and have nonperiodic wall-normal directions. The bottom of the
domain has a no-slip boundary condition, and a Neumann boundary condition is applied to the top of
the computational domain. Mass conservation is conducted at the wall-normal outlet. Wall-normal
integration of Eqs. (
2
) and
14
directly shows that any closure for
q

/
q
and
q
ms
x
/
q
ms
, respectively,
provides a value for
u
2
,
∞
. Closure equations Eqs. (
4
) and (
19
) are evaluated instantaneously at each
time step of the simulation.
114604-7
JOSEPH RUAN AND GUILLAUME BLANQUART
(a)(b)
FIG. 4. (a) Shape factor
H
12
as a function of Reynolds number Re
δ
∗
. Solid line represents empirical fit
by Ref. [
3
], and dashed lines indicate
±
1%. (b) Skin-friction as a function of Re
δ
∗
. Solid line represents
the extended Coles-Fernholz relation with
κ
=
0
.
384
,
C
=
3
.
3
,
D
0
=
182
,
D
1
=−
2466 [
14
]. Dashed lines
indicate
±
3%. Symbols:
(red) DNS [
13
];

(green) Cases BL1460MS and BL3550MS (DNS);
◦
(blue)
Cases BL1460 and BL3550 (DNS) [
5
].
The domain size, (
L
x
,
L
y
,
L
z
), is determined primarily by the sizes of large-scale motions (LSMs).
Pressure fluctuations can extend to wall-normal heights of 2
.
4
δ
99
[
11
], setting the minimum require-
ment for wall-normal height. Consequently, we set our domain height to 18
δ
∗
∼
3
δ
99
. We opt for a
span-wise width of 14
δ
∗
∼
2
.
5
δ
99
, since low-momentum streaks are approximately 0
.
5
δ
99
in width
[
7
,
17
]. Finally, the LSMs corresponding to bulges or hairpin packets have a maximum stream-wise
length of 3
δ
99
[
18
–
22
]; we opt for a domain of 7
δ
99
in stream-wise length. BL1460MS and
BL3550MS have domain sizes identical to cases BL1460 and BL3550 in Ruan and Blanquart [
5
].
In general, the very-large-scale motions tend to have sizes much larger than the current simulation
domain size [
19
]. However, as has been argued in shorter channel flow simulations Refs. [
23
,
24
],
these structures still exist in these smaller domains, and are just aliased onto smaller turbulent
structures. For channel flow simulations, they ultimately have minimal effects on one-point statistics
[
23
,
24
].
The resolution is chosen so that the smallest turbulent structures can be resolved adequately. The
stream-wise and span-wise grids are uniform with
ξ
+
1
=
9 and
ξ
+
3
=
6, which is comparable to
the resolution parameters of Orlu
etal.
[
13
](
ξ
+
1
≈
8
.
5,
ξ
+
3
≈
4). The wall-normal domain uses a
hyperbolic stretching with eight points in the viscous sublayer, (
ξ
+
2
<
5), with
ξ
+
2,min
≈
0
.
3. This
is comparable to the wall-normal resolution of Orlu
et al.
[
13
] who had 10 points in the viscous
sublayer at their lowest Reynolds number.
The governing equations are solved using NGA [
25
]. The numerical code solves the
conservative-variable formulation of the low-Mach Navier-Stokes equations with staggered finite
difference operators. It uses a fractional step method to enforce continuity and is fully secomd order
in time and second order in space.
C. Simulation results
Figure
4
compares calculated skin-friction coefficients and shape factors from simulations
BL1460MS and BL3550MS against those from single-scale simulations values [
5
], growing sim-
ulations [
13
], and empirical fits [
3
,
14
]. The multiscale simulations have shape factors that are
within 0.1% of the single-scale values and are within 0.4% of the stream-wise growing DNS
values. The multiscale skin-friction coefficients are within 0.6% of the single-scale values and are
within 2% of the stream-wise growing DNS values. The multiscale simulation shape factor and
114604-8
ERROR ESTIMATION OF A HOMOGENIZED STREAMWISE ...
(a)(b)
(c)(d)
FIG. 5. (a)
u
+
1
(b)
u
+
1
,
rms
(c)
−
u
+
1
u
+
2
(d)
u
+
2
vs
ξ
+
2
for Re
δ
∗
=
1460
,
3550. Legend: (red) DNS [
13
]; (blue)
Single-scale cases BL1460, BL3550 [
5
]; (green) Multiscale DNS cases BL1460MS and BL3550MS;

(black)
[
13
] experimental data.
skin-friction coefficient both remain well within 0.5% and 2.2% of the corresponding empirical
values, respectively. More importantly, the differences between the single-scale and multiscale
simulation results are significantly less than between either and the growing simulation results.
Figure
5
shows the inner-scaled mean stream-wise velocity profiles of the different Reynolds
number simulation results in contrast with those of Ref. [
13
], which had a stream-wise growing
boundary layer, and the single-scale simulation results of Ruan and Blanquart [
5
]. The good
agreement of shape factors and skin-friction coefficients directly implies good agreement of the
mean stream-wise velocity profiles for both Re
δ
∗
=
1460 and 3550. The normalized Reynolds
stress and rms profiles are also plotted in Fig.
5
. The near-wall peak at
ξ
+
2
≈
15 and the general
good agreement in the wake confirm that the multiscale rescaling effects appear to be minimal
on both the measured quantities of mean stream-wise velocity and rms. Differences in the mean
wall-normal velocity profile do have an impact on the behavior of the Reynolds shear stress and
second-order statistics. In the mean stream-wise momentum balance, an increased source term
directly corresponds to an increased Reynolds shear stress. Similar conclusions have been pointed
out in Ref. [
26
]. The differences in secondorder statistics are within 5% across the boundary
layer, and the impact appears to decrease with increasing Reynolds number. Overall, the multiscale
variance profiles are consistently larger than their single-scale counterparts.
Figure
5(d)
shows the wall-normal velocity profiles. There is marked improvement in both
the near-wall behavior and free-stream value. Specifically,
u
+
2
,
∞
varies by about 5% between the
multiscale and single-scale cases, and the near wall behavior has a relative deviation of 3% between
the multiscale and single-scale cases as well. The deviation between the single-scale simulation
and multiscale simulation decreases with increasing Reynolds number. From Fig.
5
, it can be
extrapolated that with the multiscale correction, the mean wall-normal velocity profile decreases
114604-9
JOSEPH RUAN AND GUILLAUME BLANQUART
in magnitude, which corresponds to an increase in magnitude of the second-order statistics.
Asymptotically, as Re
τ
→∞
, the inner layer vanishes, and thus the wall-normal region where
q
ms
x
/
q
ms
q

/
q
is small. Consequently, as the mean wall-normal velocity profile is an integration
of
q
ms
x
/
q
ms
ξ
2
∂
u
1
/∂
ξ
2
, the differences between the multi-scale and single scale
u
2
are expected to
vanish asymptotically.
The variance and wall-normal velocity profiles are dependent on the choice of the parameters fit
used in Eq. (
17
). Specifically, the value of
q
ms
x
/
q
ms
in the near-wall region is inversely proportional
to the value of
κ
. Consequently, with increasing
κ
, the source term in continuity is increased in
magnitude in the inner layer and thus, following arguments made in Sec.
II B
, the wall-normal
velocity profile in the outer layer and the overall transpiration velocity will be larger in magnitude.
BasedonFig.
5
, increases in mean wall-normal velocity profiles correspond to decreases in second-
order statistics magnitude. Overall, with increasing values of
κ
, one expects decreasing magnitudes
of the second-order statistics.
V. DISCUSSION OF REYNOLDS NUMBER CONTRIBUTION
A. Reynolds number dependence
Classical descriptions of the boundary layer require two different scalings for the inner and outer
layers [
27
]. To avoid the controversy of suggesting what velocity and length scales are involved
[
28
,
29
], we will appeal to analysis via nondimensional groups. For flat plate boundary layers, the
ensemble-averaged mean stream-wise velocity is given by
u
1
u
1
,
∞
=
f
x
2
q
,
Re
q
.
(20)
Consequently, the stream-wise derivative is given by
∂
u
1
∂
x
1
=−
x
2
q

q
∂
u
1
∂
x
2



source term
+
q

q
Re
q
∂
u
1
∂
Re
q



error term
.
(21)
Note that the last term in Eq. (
21
) is omitted in Eq. (
2
) as a consequence of Assumption 1.
Correspondingly, Eq. (
2
)–(
4
) lack any Reynolds number partial terms. Note that the neglecting of
explicit
Reynolds number partial derivative terms does not remove the Reynolds number dependence
of the mean quantities, i.e., the viscous terms still provide an
implicit
dependence on the Reynolds
number. The evolution of the shape and skin-friction coefficient (see Fig.
4
) are illustrative examples
of this Reynolds number dependence. The good agreement of the global quantities intuitively
implies that the exclusion of the Reynolds number partial terms has a negligible impact on the
mean stream-wise velocity for the current range of Reynolds numbers. However, it is unclear if this
result persists for high Re
δ
∗
.
The “error” term is a consequence of zero pressure gradient boundary layer flow relying on two
distinct length-scales. Specifically, the second key, near-wall, non-dimensional group is Re
q
.For
exactly parallel boundary layer flow (sink flow), the flow is dependent exactly on a single-length
scale:
δ
99
. In this case, the imposed nondimensionalized pressure gradient is the key nondimensional
group. In this case, the “error” term is exactly zero and the HNSE exactly follow sink flow behavior
provided appropriate boundary conditions and pressure gradient terms are retained.
The objective is to determine the magnitude of this “error” term over a large range of Reynolds
numbers and compare its magnitude to that of the source term. First, we start by recognizing that
the source term in the continuity equation and in Eq. (
21
) is nothing more than the log-indicator
function
=
x
2
∂
u
1
/∂
x
2
rescaled by
q

/
q
. It can be computed easily from any DNS data. Figure
6(a)
presents the rescaled source term calculated from various DNS and experimental databases [
5
,
6
,
13
]
for Re
θ
=
4000. All curves feature two peaks: one in the inner and one in the outer layer. Overall,
114604-10
ERROR ESTIMATION OF A HOMOGENIZED STREAMWISE ...
(a)(b)
FIG. 6. (a) Scaled source term
/
U
∞
=
x
2
∂
(
u
1
/
u
1
,
∞
)
/∂
x
2
(b) “error” term Re
θ
∂
(
u
1
/
u
1
,
∞
)
∂
Re
θ
at Re
θ
≈
4000. Symbols indicate experiments. Colors: (black) Composite fit at Re
θ
≈
4000. [
3
]; (magenta) [
6
]; (blue)
[
5
]; (red) [
13
].
there is good agreement between experimental and DNS profiles, with the experimental values
featuring scatter near the wall due to resolution [
13
].
The “error” term can be extracted from stream-wise growing boundary layers using Eq. (
21
).
Unfortunately, experimental data for this particular quantity are not available. Similarly, the “error
term” is also not immediately accessible from single-scale periodic simulations. The resulting
profiles are shown in Fig.
6(b)
. Both profiles agree on a near-wall peak and its location, and show
only small deviations on its magnitude. Further from the wall, the extracted profiles are close to
zero and are very noisy. From Eq. (
21
), the low magnitude of the “error” term in the outer layer
indicates that the majority of the stream-wise variation of
u
1
is captured by the source term. Since
q
≈
θ
is an outer scale, and since the source term describes the growth of the boundary layer due to
the growth of
q
, it is expected that the source term should capture most of the stream-wise variation
of the outer layer. In contrast, the relatively large magnitude of the “error” term in the inner layer
indicates that the majority of the stream-wise variation of
u
1
in the inner layer is not captured by the
source term. This is consistent with the results shown in Fig.
2
. It was found that the metric term
q

/
q
overshoots the actual value of 2
δ

ν
/δ
ν
. Because the source term and “error” term have opposite
signs in Eq. (
21
), it can be understood that in the inner layer, the source term overestimates the
actual stream-wise variation and the “error” term accounts for the overshoot. One could conclude
that the effect of the “error” term is primarily concentrated in the inner layer.
To extrapolate these results to higher Reynolds numbers, we turn to a composite fit [
3
]. The
log indicator function is extracted from composite fit profiles and compared to the experimental and
numerical profiles. There is good agreement throughout the inner and outer layers. The composite fit
overshoots the outer layer peak magnitude by
∼
5%, and this may be due to low-Reynolds number
effects. Overall, the composite fit agrees with experimental and DNS results quite satisfactorily.
Similarly, the composite fit is used to compute the “error” term and the resulting profile is shown in
Fig.
6(b)
. Once again, the profile agrees with the extracted DNS profiles on the location of the inner
peak at
x
2
/θ
≈
0
.
1. They only differ in the peak value by 16%.
The “error” term is plotted over a range of Reynolds numbers in Fig.
7(a)
. The magnitude
of the term decreases slowly with increasing Reynolds number (a factor of 2 from Re
θ
=
10
3
to Re
θ
=
10
8
) and the peak appears to be fixed within the inner layer. Consequently, it will be
admittedly difficult to completely capture the mean streamwise velocity gradient
∂
u
1
/∂
x
1
in a
point-wise fashion. However, in Sec.
VB
, we will explore how this error term affects wall-normal
114604-11
JOSEPH RUAN AND GUILLAUME BLANQUART
(a)(b)
FIG. 7. (a) “Error” term Re
θ
∂
(
u
1
/
u
1
,
∞
)
∂
Re
θ
, and (b) scaled source term
/
u
1
,
∞
=
x
2
∂
(
u
1
/
u
1
,
∞
)
/∂
x
2
,
predicted with the composite fit [
3
], for a range of Re
θ
=
10
3
−
10
8
.
integrated quantities. For comparison, the source term contribution is plotted in Fig.
7(b)
for the
same series of Reynolds numbers.
B. Global quantities
The previous Reynolds number dependence of the mean velocity profiles can be expressed by
analyzing the differences between the solutions to the original Navier-Stokes equations and the
solutions to Eqs. (
2
)–(
4
). One can equivalently subtract the ensemble average of Eq. (
2
)from
Eq. (
21
) and evaluate at
x
1
=
x
0
to obtain
x
2
∂
(
u
1
−
u
∗
1
)
∂
x
2
−
q
q

∂
(
u
2
−
u
∗
2
)
∂
x
2
=
∂
u
1
∂
Re
q
x
0
Re
q
,
(22)
where
q

/
q
has been divided out and asterisks denote solutions to the transformed governing
equations. Stated differently, the difference between the solutions to the Cartesian (
u
1
,
u
2
) and
transformed governing equations (
u
∗
1
,
u
∗
2
) is given by the Reynolds number derivative of the mean
velocity profile. Equation (
22
) can now be used to investigate the impact of this “error” term onto
global quantities like the skin-friction coefficient and shape factor.
Integrating Eq. (
22
) in the wall-normal direction and dividing by
θ
gives
H
12
−
H
∗
12



H
12
Error
−
1
q

u
1
,
∞
(
u
2
,
∞
−
u
∗
2
,
∞
)
=
∞
0
Re
θ
∂
u
1
/
u
1
,
∞
∂
Re
θ
x
0
d
x
2
θ
=−
Re
θ
∂
H
12
∂
Re
θ



“Error” Term
.
(23)
To leading order of ln(Re
θ
), the shape factor in the limit of large Reynolds number is approximated
by
H
12
−
1
∼
7
.
11
κ
ln(Re
θ
)
−
1
[
14
]. The RHS of Eq. (
23
) can then be approximated as
−
Re
θ
∂
H
12
∂
Re
θ
∼
7
.
11
κ
ln(Re
θ
)
−
2
.
(24)
For cases BL1460 and BL3550, this “error” term is nearly 6% of the shape factor value. Since
the relative error in shape factors from the single-scale simulations [
5
] was less than 1%, one can
conclude that the majority of the “error” term is instead balanced by the second term on the LHS of
Eq. (
23
). This second term describes a difference in transpiration velocities. For cases BL1460 and
BL3550, it can be seen from Fig.
5(d)
that the transpiration velocities of the single-scale simulations
was greater than that of the growing simulation by at most 5%. The error in the transpiration velocity
114604-12
ERROR ESTIMATION OF A HOMOGENIZED STREAMWISE ...
to leading order of Re monotonically decays with the log of the Reynolds number. For experimental
Reynolds numbers, the agreement in shape factor is only expected to improve.
VI. CONCLUSION
In this work, we investigated the transpiration velocity of single-scale periodic simulations.
Unlike other simulations of turbulent boundary layers, the single-scale periodic boundary layer
simulation does not impose a transpiration velocity. In this way, the transpiration velocity character-
izes the numerical error in the stream-wise periodic simulation. More specifically, after integrating
continuity, the transpiration velocity error was primarily caused by a difference in inner and outer
layer growth rates. A second stream-wise periodic numerical framework was formulated to account
for the differing layer growth rates. Doing so required generating a fit to a smooth transition from
2
δ

ν
/δ
ν
in the inner layer to

/
in the outer layer. Under this framework, we verified that the
transpiration velocity improved from at most 5% to 1% but the remaining mean and turbulent
intensities were virtually unchanged. The stream-wise periodic simulations appears insensitive to
the transpiration velocity, and the “improved” numerical framework may be superfluous.
Finally, through an error budget of the mean continuity equation, the errors in the mean continuity
equation must manifest either as errors in the shape factor or as errors in the transpiration velocity.
Based on low Reynolds number results, the former error is orders of magnitude smaller than the
latter error. Furthermore, using a composite fit [
3
], it can be shown that this error vanishes with
increasing Reynolds number. Thus it can be concluded that for larger and more practical Reynolds
numbers, the use of single-scale stream-wise periodic simulations remains robust.
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114604-13