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

VELOCITY SPECTRA AND SCALE DECOMPOSITION OF ADVERSEPRESSURE GRADIENT TURBULENT BOUNDARY LAYERS

Sylvia K. Romero
Department of Mechanical Engineering
University of Melbourne
700 Swanston St., Parkville, Victoria 3010 Australia
sylviar@student.unimelb.edu.au

Jimmy Philip
Department of Mechanical Engineering
University of Melbourne
700 Swanston St., Parkville, Victoria 3010 Australia
jimmyp@unimelb.edu.au

Spencer J. Zimmerman
Department of Mechanical EngineeringJohns Hopkins University
3910 Keswick Road, Baltimore, MD 21218 USA
zimmerman@jhu.edu

Joseph C. Klewicki
Department of Mechanical Engineering
University of Melbourne
700 Swanston St., Parkville, Victoria 3010 Australia
klewicki@unimelb.edu.au

ABSTRACTof the success of this velocity scale is in some sense guaranteed by its construction, which is based on the dominant terms
in the mean stress balance most responsible for balancing the
turbulent momentum flux.
The premultiplied (co)spectra normalized by viscous
scales (kx u−2τ E) are compared to the premultiplied (co)spectra
normalized by uhyb (i.e., kx u−2
The premultiplied
hyb E).
(co)spectra are presented in terms of streamwise wavelength
λx normalized by viscous scales ν /uτ (to obtain λx+ ) andouter scales δ (to obtain λx /δ ), where ν is the kinematic viscosity and the boundary layer thickness is δ . The effects ofuτ normalization and uhyb normalization on the magnitude of
the premultiplied streamwise, wall-normal, and (co)spectra ofan adverse pressure gradient turbulent boundary layer (APG
TBL) are compared to a zero pressure gradient (ZPG) casefrom Zimmerman (2019); Zimmerman et al. (2019) at matching Reτ = δ uτ /ν . The spectra are not shown in Zimmerman
et al. (2019), but the dataset is qualified and described.
The two-dimensional x-momentum Reynolds-averaged
Navier-Stokes (RANS) equation for a statistically stationary
APG TBL is

Adverse pressure gradient turbulent boundary layer velocity spectra at high Reynolds number (7100 ≤ Reτ ≤ 8600)
are compared to a zero pressure gradient case from Zimmerman (2019); Zimmerman et al. (2019) at matching Reynolds
number. Wall-distance scaling observed in the zero pressure
gradient spectra is also observed in the adverse pressure gradient spectra. Effects of varying Reτ on the velocity variances
and Reynolds shear stress are studied by comparing the present
data with the data of Monty et al. (2011) at lower Reτ (≈ 1900)
and varying Clauser pressure gradient parameter β . Cases atsimilar β and Reynolds number, but with varying flow history
are also studied. Increases are observed in the outer peak of++
u2 , v2 , and −uv+ . Under a scaling that uses a hybrid velocity uhyb there is similarity between the outer region profiles ofspectrally-decomposed variances and Reynolds shear stress.

INTRODUCTION
The behavior of the streamwise variance u2 , wall-normal
variance v2 , and Reynolds shear stress −uv as well as theirpremultiplied (co)spectra kx E are analyzed to see how the energy distributions are affected by an increasing pressure gradient. Here, kx = 2π f /U is the streamwise wavenumber, where
f is the frequency and U is the mean streamwise velocity ata particular wall-normal location y. E is the spectral density of the velocity fluctuations. The integration of E at awall-normal location across all wave numbers results in the(co)variance at that wall-location. To study the relationshipbetween the premultiplied (co)spectra, wavelength, and walldistance, in the present study the premultiplied (co)spectra areplotted as contour-maps that are a function of wavelength andwall-distance.
In the present study two velocity scales are studied: thefriction velocity scale uτ and a velocity scale uhyb introducedin Romero et al. (2022a). This velocity scale, u2hyb ≡ u2τ + ρy dPdx(with ρ the fluid density) exhibited reasonable success in scaling the Reynolds stress over a range of β and Reynolds number
relative to the y-independent velocity scales studied. A degree∂ 2U +++∂ (−uv+ )
∂ U∞+
+ ∂U
+ ∂U
+
−U
−V
+U∞+ +
= 0.
+
+
+
2
∂
y
∂
x
∂
y
∂x
∂y
(1)
This is also called the mean momentum equation. Here a superscript plus denotes normalization by ν , and uτ . Note that
∂ (−uv+ )/∂ y+ , also known as the turbulent inertia (TI) termin the RANS equation in (1), is related to the differentiatedcospectra of −uv as follows:+ +∂ (−uv)
=
∂y

Z ∞
0

∂ Euv
∂y



dkx .

(2)

The differentiated cospectra are also analyzed in this paper
as this facilitates consideration of the length scales associatedwith net momentum increase (sources) versus loss (sinks) andtheir variations with distance from the wall. Furthermore, the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Summary of present experiments. The distance x[m] is the distance downstream from the start of the APG ramp
section. The APG ramp ends at x=5.3m. Arrows indicate
whether β is increasing or decreasing with distance x. Z19
refers to Zimmerman (2019) and M11 to Monty et al. (2011).

zero-crossing of the TI term corresponds to the peak locationof the Reynolds shear stress. This location is of particular importance since the zero-crossing of the TI term marks the meantransition from momentum source to momentum sink for theflow as a whole.
In the present study we will also show the effects ofuτ normalization and uhyb normalization on the spectrally
decomposed streamwise variance, wall-normal variance, andReynolds shear stress profiles of an APG TBL as compared toa ZPG case. The behavior of u2 is also compared using cases
from Monty et al. (2011). From observing the viscous-scaled
mean statistics, it has been previously noted by Bobke et al.
(2017) and Sanmiguel Vila et al. (2017) that flows with increasing β exhibit behavior of an upstream smaller β , whileflows with decreasing β exhibit behaviors of an upstream
larger β . To further analyze this behavior, cases at similar β
and Reynolds numbers, but with varying flow history are also
compared.

Sourcepresent

Z19

M11

Ramp Structure

2

Uo

0
−10

−5

0

5

β

U∞ [m/s]uτ [m/s]

δ [m]

Reτ

∆ts+

0.80

0.9 ↑

7.91

0.26

0.42

7100

0.30

5.13

1.8 ↑

7.11

0.22

0.57

7770

0.20

6.86

0.8 ↓

7.05

0.23

0.59

8600

0.22

-

0

6.47

0.23

0.50

7880

0.24

-

0

-

-

-

1820

-

-

0.8 ↑

-

-

-

1860

-

-

1.8 ↑

-

-

-

1940

-

hot-wire probe based on the design of Kawall et al. (1983).
This probe consists of a single wire orthogonal to the mean
flow direction and two wires arranged as an ×-array, withthe single wire located between two ×-array wires. The nondimensional wire length of the single wire is l + = luτ /ν =17.6, while the length√ of the ×-array wires projected onto they − z plane is l + / 2.
The effects of pressure gradients on TBLs are often parameterized in terms of the Clauser pressure-gradient parameter β = (δ ∗ /τw )(dP/dx), where δ ∗ is the displacement thickness and τw is the wall shear stress. Positive and negative values of β respectively reflect APG and FPG conditions, while
β = 0 reflects ZPG conditions. Self-preserving APG TBLs
are characterized by nominally constant β and Reτ , (e.g.,
Townsend, 1956; Tennekes & Lumley, 1972). In the present
study β increases mildly to about 2 while Reτ grows slowly. A
summary of the experiments is given in table 1. U∞ = U(y =δ )/0.99 is the freestream velocity, and ∆ts+ = (1/ fs )uτ2 /ν isthe non-dimensionalised sample interval, where fs is the sampling frequency.

EXPERIMENTS
Facility and ramp

h [m]

x [m]

10

x [m]

Figure 1. Current flow geometry and measurement locations.
Area shaded in green is along APG ramp. Arrow points in flow
direction. Measurement locations given in table 1.

The present measurements were collected in the Flow
Physics Facility (FPF) at the University of New Hampshire.
The FPF test section features a 2.8m × 6m inlet cross-section
and a downstream development fetch of 72m. Streamwisepressure gradients for the present measurements were generated via ceiling panels installed ≈31m downstream of the windtunnel inlet, cf. figure 1. This ≈15m long ceiling insert consisted of an initial 3.1m favorable pressure gradient (FPG) region, followed by a 7.0m ZPG region (where the flow relaxes
back to a nearly canonical state), and finally a 5.3m APG region. The flow was then allowed to develop downstream under
ZPG conditions for the remaining 20m of tunnel fetch. Pressure distribution measurements support that the ZPG ramp issufficiently long since the streamwise pressure gradient dP/dx
versus x equilibrates along the ZPG section. The relatively
long ZPG ramp ensures approximately ZPG flow (or at worst
a very mild FPG) before the APG ramp. Measurements along
the APG ramp where β is increasing are compared to measurements taken downstream of the ramp insert where β is decreasing. The data in the present paper is a subset of a larger
data set. Part of this data set has been previously published in
Romero et al. (2021, 2022b). The present data locations are
given in figure 1.

SPECTRA AND SCALE DECOMPOSITION
Constant Reτ

−2
The ZPG TBL and APG TBL kx u−2
τ E and kx uhyb E arecompared in figures 2 and 3, respectively. The ZPG cases areplotted in figures 2(a-d) and 3(a-c). The APG TBL plots underuτ scaling are presented in figure 2(e-p), while the plots underuhyb scaling are shown in figure 3(d-1).

Spectra under uτ scaling: In comparison to the
ZPG case the outer region of the APG case gets energized relative to the inner region when normalized by uτ . This behavior is observed in the viscous-scaled premultiplied streamwise+ ), wall-normal spectra (k+ E + ), and cospectra
spectra (kx+ Euu
x vv
+

+

+ ). Similar behavior are reflected in the u2 , v2 , and(kx+ Euv−uv+ profiles, plotted in solid green lines, in figure 4(b), figure5(a), and figure 5(c), respectively, where a distinct outer peakforms with increasing β . This reflects the decoupling of therate of momentum flux from the wall shear stress (that otherwise characterizes ZPG BLs) with the addition of the non-zeropressure gradient term.
+ the inner peak is fixed in viscous
For the ZPG TBL kx+ Euu++
units at y ≃ 15 and λx ≃ 1000, (e.g., Hutchins & Marusic,
2007; Harun, 2012). Although the inner peak is not completely+ , it appears that the inner peakcaptured in the APG TBL kx+ Euu+
remains at λx ≃ 1000; see figures 2(e, i, m). For the ZPG TBL

Measurement Probe and Parameters
Hot-wire measurements were obtained along the APG
section and downstream of the ramp insert using a three-wire

2

100

104

10−2

102

λx /δ

(d) : β = 0(c) : β = 0

102

106

100

104

10−2

(e) : β = 0.9 ↑( f ) : β = 0.9 ↑(g) : β = 0.9 ↑(h) : β = 0.9 ↑

102

102

106

100

104

10−2

λx /δ

(b) : β = 0(i) : β = 1.8 ↑( j) : β = 1.8 ↑(k) : β = 1.8 ↑(l) : β = 1.8 ↑

102

102

106

100

104

10−2
10−2

(m) : β = 0.8 ↓
y/δ

100

10−2

(n) : β = 0.8 ↓
y/δ

100

10−2

(o) : β = 0.8 ↓
y/δ

100

(p) : β = 0.8 ↓
103

102

λx+

(a) : β = 0

λx+

106

λx+

102

λx+

λx /δ

λx /δ

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

102
104

y+

Figure 2. Premultiplied energy spectra normalized by uτ . Starting from left, first column: streamwise spectra, second column:
wall-normal spectra, third column: cospectra of −uv, and fourth column: derivative of cospectra. (a-d) β = 0 at Reτ = 7880 from
Zimmerman (2019), (e-h) β = 0.9 at Reτ = 7100, (i-l) β = 1.8 at Reτ = 7800, and (m-p) β = 0.8 at Reτ = 8600. Thick black dashedline at λx /δ = 1. Grey solid lines at λx+ = 1000 and λx /δ = 3. Blue solid line at λx = 2y. Blue dashed line at λx = 20y. The circles
denote 20W (y+ ). Cases given in table 1.
+ also appears near λ + ≃ 1000. This
the inner peak of kx+ Euv
x+ in figures 2(g, k, o).
trend continues in the APG TBL kx+ Euv+ for both the ZPG and APG
In the outer region the kx+ Euu
cases, the most energetic motions occur near λx /δ ≈ 3, as seenin figures 2(a, e, i, m). For the ZPG case, the outer peak of+ also seems to appear near λ /δ ≈ 3, but perhaps at akx+ Euvx
slightly smaller λx /δ , as seen in figure 2(c). This behavior isalso seen in the APG TBL; see figures 2(g, k, o).
The most energetic motions reflected in the wall-normal+ follow the λ = 2y trend line for both the ZPG
spectra kx+ Evv
x
and APG cases. This wall-distance scaling is also exhibited in
the cospectra for y/δ / 0.15, where the areas of high magnitude follow the λx = 20y trend line. Despite the differences inenergy density relative to the local wall-shear-stress scale, thepresence of the pressure gradient has evidently little effect on
the relative scales of modes that are set by wall-distance or theboundary layer thickness.

The derivative of the cospectra are plotted in figures 2(d,+ corresponds to the zeroh, l, p). The local peak in kx+ Euv+ )/∂ y+ , which also roughly follows the
crossing of ∂ (kx+ Euv
λx = 20y trend line. This emphasizes the robustness of walldistance scaling in the transition from momentum source tomomentum sink like motions as the wavelength changes. It appears that eddy structures that are approximately smaller thanλx = 20y contribute to a negative momentum flux (or sinks)
whereas larger λx leads to a momentum source. Here the analogy with viscous stress gradient, which is always a sink, is particularly useful. At smaller scales turbulence could be assumed
to behave in an almost random manner, which is analogous to
the random molecular motion (leading to viscosity), resulting
in a sink of momentum. On the hand, larger scales are only capable of bringing momentum to the wall (likely from the outerregion) rather than taking momentum away from the wall (because the wall will restrict the existence of large structures that3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(2009), where λx−cut = δ is used since it appears to separate+ of athe large and small-scale velocity components of kx+ Euu
ZPG TBL at Reτ = 7300. This also seems to be an appropriate
cut-off wave-length of the present data, where a thick dashedline plotted at λx /δ = 1 separates the inner and outer energy
peaks, as seen in figures 2(a, e, i, m). For completeness, thewall-normal variance and Reynolds shear stress are also decomposed using the same cut-off wave-length λx−cut = δ tosee where attributes of the streamwise measurements appear
in the wall-normal variance and Reynolds shear stress profiles.
The behavior of the decomposed streamwise variance+∗profiles u2 versus u2 are compared in figure 4. The decom+∗
posed wall-normal variance profiles v2 versus v2 and de+∗composed Reynolds shear stress −uv versus −uv are compared in figure 5. The original APG TBL profiles are plotted insolid green lines, while the small-scale components are plottedas blue dashed-dot lines and the large-scale components areplotted as orange dashed lines. The respective ZPG profiles
are plotted in black.
∗In the outer region of the large-scale components of u2 ,∗v2 , and −uv∗ there is stronger similarity between the APG
cases where β is increasing (β = 0.9 and 1.8) than to the casewhere β is decreasing (β = 0.8), see figure 4(d), figure 5(b),and figure 5(d), respectively. Whether scaled by uτ or uhyb ,in the inner region of the large-scale components, the cases atsimilar β (= 0.8 and 0.9) behave more like one another than theβ = 1.8 case.
Now looking at the small-scale components, there is verystrong similarity between the two increasing β cases (β = 0.9∗
and 1.8) throughout the u2 profile and in the −uv∗ profile for∗y/δ ' 10−2 . There is slightly more scatter in the v2 profiles.
Overall under uhyb scaling, the decreasing β measurements have a slight increase towards the ZPG case in the outerregion of the profiles, but still behave similarly to the ramp
measurements.

can carry momentum away from the wall) making them anoverall momentum source.
For the ZPG TBL, a length scale distribution W + thatallows the mean momentum equation to be written in a selfsimilar form is defined (e.g., in Morrill-Winter et al., 2017a)
as:

W + (y+ ) =∂ 2U +∂ y+ 2

−1/2

.

(3)

The distribution of 20W + (y+ ) markers are plotted in figures2(d, h, l, p) as circles. Due to the difficulty of taking thesecond derivative of the experimental mean velocity profiles,
these profiles are not definitive, but do appear to exhibit walldistance scaling. In Morrill-Winter et al. (2017b) it is noted
for a ZPG TBL that the distribution of 20W + (y+ ) roughly coincides with the zero-crossing of the differentiated cospectra.
This is also observed in the present APG TBL plots.

Spectra under uhyb scaling: Figure 3 shows
spectra in uhyb scaling, where we observe that for y/δ ' 0.1−2−2
the kx u−2
hyb Euu , kx uhyb Evv , and kx uhyb Euv magnitudes decrease
relative to the ZPG case. Similarity is evidenced between the∗
APG cases in this region. This is also observed for the u2 in∗figure 4(d), v2 in figure 5(b), and −uv∗ for figure 5(d). The
profiles of the APG cases fall below the ZPG case in the outerregion. Here superscript ∗ represents uhyb scaling.
Stronger β trends are observed for y/δ . 0.1. In this−2
−2
region kx u−2
hyb Euu , kx uhyb Evv , and kx uhyb Euv increase with in++creasing β . Like the kx Euu , the energetic outer region ofkx u−2hyb Euu occurs in the vicinity of λx /δ ≈ 3, but has moved
inwards towards the wall. Near the wall (y/δ . 10−2 ), thespectra under uhyb scaling behave similar to the spectra underuτ scaling since uhyb approaches uτ at the wall by construction.
For y/δ & 10−2 the areas of high magnitude in kx u−2
hyb Evv
exhibit wall-scaling near the λx = 2y line; see figures 3(e, h, k).
Again, wall-distance scaling is also observed in the cospectra,
but along λx = 20y; see figures 3(f, i, l).

Changes with Reτ
The present measurements at 7100 ≤ Reτ ≤ 8600 (in figures 4(b, d)) are compared to profiles at matching β fromMonty et al. (2011) at Reτ ≈ 1900 (in figures 4(a, c)).
The results of figures 4(a, c) indicate that at low Reynolds
number there is relatively little change with β in the smallscale components near the inner peak for both uτ and uhyb scaling. While the inner peak is not completely captured in figures4(b, d), there is a larger increase in the small-scale components
in the near-wall region from the ZPG to the APG cases. The
outer flow appears to have a stronger influence on the inner
flow at these large Reynolds numbers.
Both the small-scale and large-scale components underuτ scaling have increased from the ZPG case to APG casebut, larger increases are seen in the large-scale components
in the outer region. In the ZPG case, the large-scale components exhibit an outer peak that is denoted by a vertical line aty/δ ≈ 0.1 in figures 4(a-d). Under uτ scaling the outer region
increases with increasing β (see figures 4(a-b)), while underuhyb scaling there is strong similarity between the APG casesin the outer region (see figures 4(c-d)).

Changes with flow history: The effectiveness ofuhyb for APG cases at similar β and Reynolds number, butwith varying flow history, are now compared. In this section acase where β is just increasing from 0 (the β = 0.9 ↑ case), is
compared to a case where β is decreasing and the flow has had
longer time to develop (the β = 0.8 ↓ case).
Now looking at kx u−2
hyb Euu , in figure 3(d) β is just increasing from 0, while in figure 3(j) β is decreasing and the flow has
had longer time to develop. The APG case in figure 3(j) whereβ is decreasing has a slightly higher outer region than the casewhere β is increasing in figure 3(d). This is similarly observed∗in the outer region of the u2 profiles in figure 4(d). In the inner∗
region (y/δ / 10−2 ) of the u2 profile the decreasing β profileshows stronger similarity to the ZPG case than the increasing
β profile, see figure 4(d). This suggests that agreement between ZPG profiles and APG profiles subjected to uhyb scaling
may improve as the development length/time of the APG caseincreases.
To quantify the contribution of the small-scale and largescale motions, the streamwise variance is decomposed into
small-scale and large-scale components by using a sharp spectral cut-off. The choice of the appropriate cut-off wave-length
λx−cut in the present analysis follows the work of Mathis et al.

CONCLUSIONS+ , k+ E + ,Increases are seen in the magnitude of kx+ Euu
x vv
+
+
+
+
2
2
kx Euv and in the outer peak of u , v , and −uv+ as βincreases. Under uhyb scaling, the APG TBL premultiplied4

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

10−2

100

104
102

(c) : β = 0

102

106

100

104

(d) : β = 0.9 ↑(e) : β = 0.9 ↑( f ) : β = 0.9 ↑

102

102

106

100

104

10−2

(g) : β = 1.8 ↑(h) : β = 1.8 ↑(i) : β = 1.8 ↑

102

102

106

100

104

10−2
10−2

( j) : β = 0.8 ↓
y/δ

100

10−2

(k) : β = 0.8 ↓
y/δ

100

10−2

(l) : β = 0.8 ↓
y/δ

λx+

(b) : β = 0(a) : β = 0

λx+

100

λx+

λx /δ

100

106

10−2

λx /δ

10−2

102

10−2

λx /δ

100

λx+

λx /δ

10−2

102

100

Figure 3. Premultiplied energy spectra normalized by uhyb . Starting from left, first column: streamwise spectra, second column: wallnormal spectra, and third column: cospectra of −uv. (a-c) β = 0 at Reτ = 7880 from Zimmerman (2019), (d-f) β = 0.9 at Reτ = 7100,(g-i) β = 1.8 at Reτ = 7800, and (j-l) β = 0.8 at Reτ = 8600. Thick black dashed line at λx /δ = 1. Grey solid lines at λx+ = 1000 andλx /δ = 3. Blue solid line at λx = 2y. Blue dashed line at λx = 20y. Cases given in table 1.

REFERENCES(co)spectra and (co)variances behave similarly to one another,but decrease relative to the ZPG case in the outer region. The
small-scale and large-scale components of the decomposed∗∗
u2 , v2 , and −uv∗ APG profiles also exhibit similarity inthe outer region over a range of Reτ . Based on the similarity between APG cases, these findings indicate that uτ scaling
performs well in the inner region, while uhyb scaling is more
appropriate in the outer region regardless of scale.

Bobke, A., Vinuesa, R., Örlü, R. & Schlatter, P. 2017 History
effects and near equilibrium in adverse-pressure-gradientturbulent boundary layers. J. Fluid Mech. 820, 667–692.
Harun, Z. 2012 The structure of adverse and favourable pressure gradient turbulent boundary layers. PhD thesis.
Hutchins, N. & Marusic, I. 2007 Large-scale influences innear-wall turbulence. Philosophical Transactions of theRoyal Society A: Mathematical, Physical and Engineering
Sciences 365 (1852), 647–664.
Kawall, J. G., Shokr, M. & Keffer, J. F. 1983 A digital technique for the simultaneous measurement of streamwise andlateral velocities in turbulent flows. J. Fluid Mech. 133, 83–112.
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent
boundary layers. J. Fluid Mech. 628, 311–337.
Monty, J.P., Harun, Z. & Marusic, I. 2011 A parametric study

Wall-distance scaling observed in the wall-normal spectra
appears to contribute to the wall-distance scaling seen in thecospectra. Observations of the differentiated cospectra highlight that the transition from momentum source to sink scaleswith wall-distance for both increasing and decreasing β cases.
The robustness of wall-distance scaling is further emphasized
by the behavior of W + profiles for the APG cases studied.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
10

8

8

6

6

u2

u2

∗

+

10

4

4

2

2

0

(a)
10−2

(b)
0

y/δ

10

10−2

0

y/δ

(c)
10−2

100

(d)
y/δ

100

10−2

y/δ

100

Figure 4. Decomposition of the broadband streamwise variance profile (a-b) scaled by u2τ and (c-d) scaled by u2hyb . (a, c) β = 0, 0.8,
and 1.8 at Reτ ≈ 1900 and (15 ≤ l + ≤ 17) from Monty et al. (2011). (b, d) β = 0, 0.9, 1.8, and 0.8 at 7100 ≤ Reτ ≤ 8600 and(16 ≤ l + ≤ 20). APG profiles plotted in color, respective ZPG profiles plotted in black. Total profile for APG cases plotted as solid

green lines. Small-scale component (λx < δ ) plotted as blue dashed-dot lines and large-scale component (λx > δ ) plotted as orange
dashed lines. Line thickness increases with increasing β . The present β = 0.8 ↓ case is plotted in lighter shades to differentiate fromthe β = 0.9 ↑ case. Vertical line plotted at y/δ = 0.1.
3

2

(a)

(b)

(c)

(d)

−uv∗

−uv+

v2

v2

∗

+

2
1

1

0

0
10−2

y/δ

100

10−2

y/δ

100

10−2

y/δ

100

10−2

y/δ

100

Figure 5. Decomposition of the broadband (a-b) wall-normal variance and (c-d) Reynolds shear stress profiles. (a, c) scaled by u2τ (b,d) scaled by u2hyb . APG profiles plotted in color, respective ZPG profiles plotted in black. Total profile for APG cases plotted as solid
green lines. Small-scale component (λx < δ ) plotted as blue dashed-dot lines and large-scale component (λx > δ ) plotted as orange
dashed lines. Line thickness increases with increasing β . β = 0, 0.9, 1.8, and 0.8 at 7100 ≤ Reτ ≤ 8600 and (16 ≤ l + ≤ 20). The
present β = 0.8 ↓ case is plotted in lighter shades to differentiate from the β = 0.9 ↑ case. Vertical line plotted at y/δ = 0.1.
of adverse pressure gradient turbulent boundary layers. Int.
J. Heat Fluid Flow 32 (3), 575–585.
Morrill-Winter, C., Philip, J. & Klewicki, J. 2017a An invariant representation of mean inertia: theoretical basis for alog law in turbulent boundary layers. J. Fluid Mech. 813,
594–617.
Morrill-Winter, C., Philip, J. & Klewicki, J. 2017b Statistical evidence of anasymptotic geometric structure to the momentum transporting motions in turbulent boundary layers.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375 (2089),
20160084.
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