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

COMPARISON OF SMOOTH- AND ROUGH-WALL NON-EQUILIBRIUM BOUNDARYLAYERS WITH FAVORABLE AND ADVERSE PRESSURE GRADIENTS
Ralph J. Volino
Department of Mechanical Engineering
United States Naval Academy
590 Holloway Rd.
Annapolis, Maryland, 21402, USA
volino@usna.edu

Michael P. Schultz
Department of Naval Architecture and Ocean Engr.
United States Naval Academy
590 Holloway Rd.
Annapolis, Maryland, 21402, USA
mschultz@usna.edu

ABSTRACT
Measurements were made in rough-wall boundary layers
subject to favorable, zero, and adverse pressure gradients.
Profiles of mean velocity and turbulence quantities wereacquired, and comparisons were made to equivalent smoothwall cases with the same freestream velocity distributions.
Outer layer similarity was observed between the rough- andsmooth-wall cases in all quantities in the favorable and zeropressure gradient regions, but large differences were observedwith adverse pressure gradients. The results suggest thatsimilarity might be achieved if cases with the same Clauserpressure gradient parameter history were compared. The results
also suggest that the equivalent sandgrain roughness may
remain constant for a surface as the pressure gradient is varied.

Many boundary layers of practical interest occur underapproximately ZPG conditions. An example is the boundary
layer on the side of a large ship. In many other cases of interest,
however, significant non-zero pressure gradients are present.
Examples include the flows around airfoils and planing
surfaces on naval vessels, and in many cases the surfaces inquestion are rough. Although some notable studies of the
combined effects of roughness and pressure gradients can befound in the literature, there has been considerably less workthan in the more canonical cases. Important questions remain to
be answered. Does ks for a surface remain constant in thepresence of a pressure gradient? Is outer layer similarity
between rough- and smooth-wall cases preserved, and if so,
what flow parameters result in similarity?
The present study investigates rough-wall boundary layerswith non-equilibrium pressure gradients under a range of
Reynolds
numberandacceleration
parameters,
K=/U∞2(dU∞/dx), where  is the kinematic viscosity, x is thestreamwise coordinate, and U∞ is the local freestream velocity.
Experiments were done in the same facility and under the same
conditions as the smooth-wall study of Volino (2020). The
present paper focuses on mean velocity and turbulence
statistics, and the extent of similarity with the smooth-wallresults.

INTRODUCTION
Both smooth- and rough-wall boundary layers have beenstudied extensively under canonical zero pressure gradient(ZPG) conditions, as documented in review articles such as
Chung et al. (2021). Roughness increases the drag on the wall,
expressed as the skin friction coefficient, Cf, or the friction
velocity, u, due to the form drag on the roughness elements.
The effect of the roughness can be quantified as the roughness
function, U+, which is the shift downward of the mean
velocity profile in inner coordinates below the canonical law of
the wall, u+=(1/)ln(y+)+A. For a fully rough surface, theroughness function is directly related to the equivalent
sandgrain roughness, ks, of the surface by the function
U+=(1/)ln(ks+)+A-8.5, based on the work of Nikuradse(1933). Although ks is a hydrodynamic quantity (a function of
the increase in u compared to the smooth wall, not a physicaldimension of the surface), it has been observed under ZPG
conditions to remain approximately constant for a given
physical roughness. Predicting ks from the physical roughness
geometry has been the subject of considerable research as
discussed in Flack and Chung (2022). Once ks is obtained for agiven rough surface, it can be used for computational
predictions at any Reynolds number as long as fully roughconditions are met (typically taken as ks+>80).
In the outer region of the boundary layer, Townsend (1976)
proposed similarity between rough- and smooth-wall boundary
layers when scaled with the boundary layer thickness, , and u.
Similarity implies that the structure in the outer part of theboundary layer is the same in rough- and smooth-wall cases,
with the wall shear serving only as a boundary condition. Some
exceptions to similarity have been observed, such as when theroughness height is a large fraction of , but in general, outer
similarity has been shown to be very robust in ZPG boundary
layers.

EXPERIMENTS
Experiments were conducted in the water tunnel describedin Volino (2020). The test section, shown schematically in Fig.
1, was 2 m long, 0.2 m wide, and 0.1 m tall at the inlet. The
lower wall was a flat plate that served as the test wall. The
boundary layer was tripped with a 0.8 mm diameter wire at theleading edge. Immediately downstream of the trip the wall wassmooth, followed by uniform roughness beginning 0.23 m
downstream of the trip. The roughness was random andmathematically generated with the parameters listed in Table 1.
This is the same positively skewed roughness used in the ZPG
study of Flack et al. (2020). The upper wall was comprised offour flat plates that were independently adjusted. The firstsection was set to provide a ZPG entry region that extended
from the inlet to x=0.6 m. The second provided a favorablepressure gradient (FPG) from 0.6 to 1.1 m. The third was set
for a ZPG recovery from 1.1 to 1.6 m, and the last section wasset for an adverse pressure gradient (APG) from 1.6 to 1.8 m.
Cases were considered for three inlet freestream velocities(U∞o=0.5, 1, and 2 m/s), and four positions of the upper wall,
including ZPG baseline cases. For the non-ZPG, K in the FPG
was equal to 2×10-6 (strong ramp), 1×10-6 (moderate), and
0.5×10 -6 (mild) for the U∞o=0.5 m/s cases with the threedifferent upper wall positions, respectively. For the higher

1

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

Figure 1. Cross section of test section in streamwise-wall normal plane. Three positions of upper wall shown: blue = strong ramp, red =moderate ramp, green = mild ramp. Numbers 1-12 indicate streamwise measurement stations.
This suggests that the adjustment to the pressure gradientshould be somewhat more rapid in the rough-wall case. The
roughness has a strong effect on the boundary layer, with  andu increased by about 30% and 50% respectively above thesmooth wall values in the ZPG and FPG, but in defectcoordinates, outer region similarity is still apparent. In the APG(Fig. 2c), the behavior changes. The profiles are much less fullin the rough-wall case, and the dimensionless velocities aremuch higher.
A potentially important difference between the rough andsmooth cases in the APG is that  was much higher for therough wall, increasing from 6.5 to 73 between stations 10 and
12 with the rough wall and from 2.7 to 6.6 with the smooth
wall. Figure 3 shows the mean velocity profiles from themoderate ramp case (symbols) and the smooth-wall resultsfrom the strong ramp case (solid lines) and the moderate ramp
case (dashed lines). In the FPG and ZPG recovery regions, therough-wall results are similar to the corresponding moderate
ramp smooth-wall profiles, in agreement with the results ofFig. 2. The profiles do not deviate as much from ZPG
conditions as those with the strong ramp. In the APG, therough-wall results fall between those of the moderate andstrong ramp. The  values were 1.3-1.9, 0.79-1.02, and 2.7-6.6,
for the rough, moderate ramp smooth, and strong ramp smoothcases, respectively. These results suggest that if the  history
were the same for rough- and smooth-wall cases, that outerregion similarity might hold.
The Reynolds stresses and higher order moments were
examined, and the same similarity between the rough- andsmooth-wall cases was observed for all quantities. The
streamwise component of the Reynolds stress, u′2, is shown in
Fig. 4 for the strong ramp cases in the same format as Fig. 2.
As expected, the near wall peak of the smooth wall profiles,
which occurs in the buffer region, is not present in the roughwall cases. The near wall peak is due primarily to viscous
effects near the wall, and this region is disrupted by theroughness elements, with the viscous drag replaced by pressuredrag on the roughness. In the outer region, there is similarity
between the smooth- and rough-wall cases in the initial ZPG
region. As with the mean velocity profiles, this similarity
largely holds through the FPG and ZPG recovery regions, but
the rough-wall cases appear to respond more quickly to thechanges in pressure gradient and reach their new equilibrium
faster. In the APG region, the similarity is lost. The boundary
layer grows and approaches separation faster on the rough wall,
resulting in a more rapid drop in u. The result is u′2/u anorder of magnitude larger in the rough wall case by the last
station in the APG.
Figure 5 shows u′2 for the moderate ramp rough-wall casewith comparison to the smooth wall cases in the format of Fig.
3. In the FPG, the moderate ramp results in less change in the
profiles than in the strong ramp case of Fig. 4. The trend is thesame however, with the acceleration suppressing the turbulencein the outer region, agreement with the corresponding,moderate ramp, smooth-wall case, and more rapid change
toward a new equilibrium with the rough wall. The behavior in
the ZPG recovery is the same. In the APG region, the roughwall and moderate ramp smooth-wall profiles do not agree,

Table 1. Rough surface statistics.
ka (m)krms (m)kt (mm)ks (mm)

Sk

Ku

ES

277

350

3.5

2.2

0.98

4.18

0.4

inlet velocities at each position, K was inversely proportional to
U∞o. Constant, positive K flows approach equilibrium on asmooth wall. In the APG region, K was constant with a value
opposite in sign and half the magnitude of K in the upstream
FPG for each case. Equilibrium in APG flows requires aconstant Clauser pressure gradient parameter, =KRe*/(Cf/2),
where Re* is the displacement thickness Reynolds number.
The  value continuously rose in the APG regions of thepresent study, so they were not in or approaching equilibrium.
Velocity profiles were acquired along the spanwise
centerline of the test section at the 12 streamwise stations
shown in Fig. 1. The profiles were measured with a TSI
FSA3500 two-component laser-Doppler velocimeter (LDV).
The LDV included a four-beam fiber optic probe with a custom
beam displacer that was used to shift one of the beams,
resulting in three co-planar beams that were aligned parallel tothe wall. A 2.6:1 beam expander was located at the exit of theprobe to reduce the diameter of the measurement volume to 45
m. The probe volume length was 340 m. The flow wasseeded with 2 m diameter silver-coated glass spheres. Data
were acquired in coincidence mode. Each profile included 43
locations extending from the roughness surface to thefreestream. Data were acquired at each location for 10,000
large eddy turnover times (U∞).
RESULTS
Under fully rough, ZPG conditions, boundary layers areinvariant with freestream velocity (i.e., Cf, and the boundary
layer thicknesses remain constant at a given streamwise
location), as explained in Pullin et al. (2017), and this was also
observed in the present non-ZPG cases. Results are shown
below for the U∞o=1 m/s cases.
Figure 2 shows mean streamwise velocity profiles in defectcoordinates for the strong ramp configuration. The method ofVolino and Schultz (2018), which utilizes the measured
Reynolds shear stress and mean velocity profiles, was used todetermine u for each profile. In Fig. 2, the present rough-walldata are shown with symbols, and the corresponding smoothwall data of Volino (2020) are shown with lines. The profiles
are depressed toward a new equilibrium by the FPG (Fig. 2a).
The progress toward the new equilibrium appears to be slightly
faster for the rough wall, but the smooth- and rough-wall cases
are otherwise quite similar. In the ZPG recovery (Fig. 2b), thesmooth and rough cases both go back to similar canonical ZPG
conditions, with the rough wall cases appearing to proceedsomewhat more rapidly to the new condition. Devenport andLowe (2022) show that the length scale for adjustment to achange in pressure gradient is 2/(Cf)=/(HU∞K), where  isthe momentum thickness and H=*/ is the shape factor. For
the smooth- and rough-wall cases of Fig. 2, , U∞, and K are
the same, and H is about 20% higher in the rough wall case.

2

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

(a)

St. 1
2
3
4
5
6

10

5

0
0

0.2

0.4

0.6

0.8

1

80

(b)

St. 6
7
8
9

10

(c)
St. 9
10
11
12

60
40

5
20
0
0

0.2

0.4

0.6

y/

0.8

1

0
0

0.2

0.4

y/

0.6

0.8

1

y/

Figure 2. Mean velocity profiles in defect coordinates, a) FPG, b) ZPG recovery, c) APG. Strong ramp case. Symbols – rough wall, lines– smooth wall. Symbol and line color match at each station.
(a)

St. 1
2
3
4
5
6

10

5

0
0

0.2

0.4

0.6

0.8

1

30

(b)

St. 6
7
8
9

10

5

(c)
St. 9
10
11
12

20

10

0
0

0.2

0.4

0.6

y/

0.8

1

0
0

0.2

0.4

y/

0.6

0.8

1

y/

Figure 3. Mean velocity profiles in defect coordinates, a) FPG, b) ZPG recovery, c) APG. Symbols – rough wall w/ moderate ramp, solidlines – smooth wall with strong ramp, dashed lines – smooth wall with moderate ramp.
5

5

120

(a)

(b)

4

4

3

3

2

2

1

1

0
0

0
0

(c)

100
80
60

0.2

0.4

0.6

0.8

1

40
20
0.2

0.4

0.6

y/

0.8

1

0
0

0.2

0.4

y/

0.6

0.8

1

y/

Figure 4. Streamwise component of Reynolds normal stress profiles, a) FPG, b) ZPG recovery, c) APG. Strong ramp case. Same legend
as Fig. 2.
5

5

12

(a)

(b)

4

4

3

3

2

2

1

1

0
0

0
0

(c)

10
8
6

0.2

0.4

0.6

0.8

1

4
2
0.2

0.4

0.6

y/

y/

0.8

1

0
0

0.2

0.4

0.6

0.8

1

y/

Figure 5. Streamwise component of Reynolds normal stress profiles, a) FPG, b) ZPG recovery, c) APG. Symbols – rough wall w/
moderate ramp, solid lines – smooth wall with strong ramp, dashed lines – smooth wall with moderate ramp. Same legend as Fig. 3.
with the values in the rough wall case approaching double
those of the smooth-wall by the end of the APG. As with themean profiles of Fig. 3, there is better agreement between themoderate ramp rough-wall profiles and the strong ramp
smooth-wall profiles.

Reynolds shear stress, -u′v′, profiles are shown in Figs. 6
and 7 in the same format as Figs. 4 and 5. As with the mean
velocity and u′2, similarity between the rough- and smooth-wallcases with the same K is seen in the FPG and ZPG recovery forboth the strong and moderate ramp cases, with the rough wall3

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

(a)

1
0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

0.2

0.4

0.6

0.8

1

0
0

20

(b)

1

(c)

15
10
5

0.2

0.4

0.6

y/

0.8

1

0
0

0.2

0.4

y/

0.6

0.8

1

y/

Figure 6. Reynolds shear stress profiles, a) FPG, b) ZPG recovery, c) APG. Strong ramp case. Same legend as Fig. 2.
(a)

1
0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

0.2

0.4

0.6

0.8

1

0
0

(c)

(b)

1

2
1.5
1
0.5

0.2

0.4

0.6

y/

y/

0.8

1

0
0

0.2

0.4

0.6

0.8

1

y/

Figure 7. Reynolds shear stress profiles, a) FPG, b) ZPG recovery, c) APG. Symbols – rough wall w/ moderate ramp, solid lines –smooth wall with strong ramp, dashed lines – smooth wall with moderate ramp. Same legend as Fig. 3.
cases proceeding somewhat more rapidly to the new
equilibrium. In the APG region, as with the quantities
presented above, the best agreement is between cases withdifferent velocity distributions but more similar  histories.
The behavior observed in the Reynolds stresses is alsopresent in the higher order moments. Profiles of u′2v′, areshown in Figs. 8 and 9 as an example. In the FPG and ZPG
regions, there is similarity between the rough- and smooth-wallcases with the same freestream velocity distribution. In the
APG flow, the rough- and smooth-wall cases again show better
agreement when cases with more similar  histories arecompared.
Another way of evaluating similarity or differences
between cases is with the diagnostic plot, which was applied torough wall boundary layers by Castro et al. (2013). Profile datafor rms u′ is normalized by the local mean velocity U, andplotted vs the local U normalized by U∞. An example is shownfor the FPG and ZPG recovery region profiles of the moderate
ramp smooth- and rough-wall case in Fig. 10a. The smoothwall data collapse and lie just above the expected smooth wallline given in Castro et al. (2013) for ZPG cases. The roughwall profiles collapse onto a different line, which also agrees
with ZPG results from Castro et al. (2013). The FPG has no
discernable effect on the profiles in these coordinates. To
collapse the all data, Castro et al. (2013) introduced modified
coordinates, plotting u′/(U+U) vs (U+U)/(U+U), whereU=uU+ is a dimensional form of the roughness function.
The profiles of Fig. 10a are shown in these coordinates in Fig.
10b, and they collapse onto the smooth wall line, as expected.
The profiles from the strong and mild ramp cases (not shown)also collapse onto the same lines.
The diagnostic plots for the APG region profiles are shownin the original coordinates in Fig. 11 and the modified
coordinates in Fig. 12. For the weak and moderate ramp cases,
there is no significant difference from the ZPG results. In thestrong ramp cases, there is a slight rise in the profiles in thesmooth-wall case, and a larger rise in the rough-wall case. It
appears that if  becomes large enough and the boundary layer
begins to approach separation, there is a departure fromsimilarity from the ZPG behavior in the diagnostic plot
coordinates.
From the mean profiles, U+ was determined and is shown
as a function of ks+ for all the cases of the present study in Fig.
13. There is considerable scatter but reasonable agreement withthe expected line for the cases with fully rough conditions. In
the APG region of the strong ramp cases, u drops to low
values, resulting in transitionally rough behavior with acorresponding drop below the fully rough correlation. In the
fully rough cases, there is no obvious trend with pressure
gradient. It appears that at least for the cases of the present
study, with their particularly pressure gradient histories, that anassumption of a constant ks with varying pressure gradient may
be justified.
The skin friction coefficient is shown in Fig. 14 as afunction of /yo for all the rough and smooth wall cases. These
coordinates were shown in Castro (2007) to result in similarity
for a large range of rough surfaces. When fully rough, yo is
proportional to ks and is the y value where the log region of themean velocity profile in inner coordinates extrapolates to u+=0.
The APG data fall below the expected ZPG correlation. Kaysand Crawford (1980) suggest an empirical correction to Cf of(1+/5). For the present results, (1+/6) provided a somewhat
better fit, and this is shown in Fig. 14b. Kays and Crawford(1980) indicate the correction is applicable for equilibrium
flows with constant , but it also works well with the present
cases. At the most downstream station of the APG in the strong
ramp cases, (1+/6) produces an overcorrection. This may be
due to the flow being in the transitionally rough regime.

4

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

1

(a)

0.5

50

0

-0.5
0

(c)

100

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

y/

0.6

0.8

1

y/

Figure 8. u′2v′ profiles, a) FPG, b) ZPG recovery, c) APG. Strong ramp case. Same legend as Fig. 2.

1

1

(a)

0.5

(c)

2

(b)

1

0.5

0
0

-0.5
0

0

0.2

0.4

0.6

0.8

-1

-0.5
0

1

0.2

0.4

0.6

y/

0.8

-2
0

1

0.2

0.4

y/

0.6

0.8

1

y/

Figure 9. u′2v′ profiles, a) FPG, b) ZPG recovery, c) APG. Symbols – rough wall w/ moderate ramp, solid lines – smooth wall with strong
ramp, dashed lines – smooth wall with moderate ramp. Same legend as Fig. 3.

0.5

0.5
(a)

(b)

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

0.2

0.4

0.6
U/U

0.8

0
0

1

0.2

0.4

0.6
U /U

0.8

1

Figure 10. Diagnostic plots for FPG and ZPG profiles of moderate ramp cases, a) standard coordinates, b) modified coordinates. Symbols– rough wall, dashed lines – smooth wall. Same legend as Fig. 2. Straight black line and dashed straight red line are from Castro et al.
(2013) for ZPG smooth and rough cases, respectively.

0.5

0.5

0.5

(a)

(b)

(c)

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0
0

0.2

0.4

0.6
U/U

0.8

1

0
0

0.2

0.4

0.6
U/U

0.8

1

0
0

0.2

0.4

0.6
U/U

0.8

1

Figure 11. Diagnostic plots for APG profiles in standard coordinates, a) strong ramp, b) moderate ramp, c) mild ramp. Symbols – roughwall, dashed lines – smooth wall. Same legend as Fig. 2. Straight black line and dashed straight red line are from Castro et al. (2013) for
ZPG smooth and rough cases, respectively.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.5

0.5

0.5

(a)

(b)

(c)

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0
0

0.2

0.4

0.6
U /U

0.8

1

0
0

0.2

0.4

0.6
U /U

0.8

1

0
0

0.2

0.4

0.6
U /U

0.8

1

Figure 12. Diagnostic plots for APG profiles in modified coordinates, a) strong ramp, b) moderate ramp, c) mild ramp. Symbols – roughwall, dashed lines – smooth wall. Same legend as Fig. 2. Straight black line from Castro et al. (2013) for ZPG cases.
By color
ZPG
Strong PG
Moderate PG
Mild PG
By shape
U o =0.5 m/s
U o =1.0 m/s
U o =2 m/s

U+ =(1/ )ln(k +s )+B-8.5
10

0.01

5

0.005

0

10 1

10 2

10 3

k +s

Figure 13. Roughness function as afunction of ks+. Open symbols – ZPG, grayfilled – FPG, yellow filled – APG.

0

10 1

10 2
/y o

10 3

0.01

0.005

0

10 1

10 2

10 3

/y o

Figure 14. Skin friction coefficient as a function of momentum thickness androughness length, a) uncorrected, b) corrected with (1+/6). Rough wall – large
symbols, smooth wall – small symbols.

CONCLUSIONS
Mean velocity and turbulence profiles were measured inboundary layers on a rough wall subject to zero, favorable, andadverse pressure gradients. Reynolds numbers and the strengthof the pressure gradient were varied. Comparisons were madeto smooth-wall cases with the same freestream velocity
distributions. The rough-wall results were invariant with
Reynolds number, in agreement with previous observations in
ZPG flows. In FPG and ZPG regions, the profiles of allquantities showed similarity between the smooth- and roughwall results, with the rough-wall cases moving somewhat more
quickly to a new equilibrium when the pressure gradient waschanged. In APG regions, large departures from similarity wereobserved. This was attributed to the more rapid growth of therough-wall boundary layer, particularly in the strongest
pressure gradient cases. When cases with different dimensionalpressure gradients, but more similar  histories were compared,
the smooth- and rough-wall cases were more similar.
Diagnostic plots were also useful for showing the similarity in
FPG and ZPG regions, and the departure from similarity withan APG. The results from all cases suggest that the equivalent
sandgrain roughness for a surface may remain constant in thepresence of a non-zero pressure gradient.

boundary layers”, Journal of Fluid Mechanics, Vol. 727, pp.
119-131.
Chung, D., Hutchins, N., Schultz, M. P., and Flack, K. A.,
2021, “Predicting the Drag of Rough Surfaces”, Annual Review
of Fluid Mechanics, Vol. 53, pp. 439-471.
Flack, K., and Chung, D., 2022, “Important Parameters fora Predictive Model of ks for Zero Pressure Gradient Flows”,
AIAA 2022-1036, Proceedings, AIAA SCITECH 2022 Forum.
Flack, K. A., Schultz, M. P., and Volino, R. J., 2020, “Theeffect of a systematic change in surface roughness skewness on
turbulence and drag”, International Journal of Heat and Fluid
Flow, Vol. 85, 108669.
Devenport, W. J., and Lowe, K. T., 2022, “Equilibrium andnon-equilibrium turbulent boundary layers”, Progress in
Aerospace Sciences, Vol. 131, 100807.
Kays, W. M., and Crawford, M. E., 1980, Convective Heatand Mass Transfer, 2nd Ed., McGraw-Hill Book Company,
New York, p. 189.
Nikuradse, J., 1933, “Laws of flow in rough pipes”, NACA
Technical Memorandum 1292 (1950).
Pullin, D. I., Hutchins, N., and Chung, D., 2017,“Turbulent flow over a long flat plate with uniform roughness,”
Physical Review Fluids, Vol. 2, 082601.
Townsend, A. A., 1976, The Structure of Turbulent Shear
Flow, 2nd Ed., Cambridge University Press, Cambridge.
Volino, R. J., and Schultz, M. P., 2020, “Determination ofwall shear stress from mean velocity and Reynolds shear stressprofiles”, Physical Review Fluids, Vol. 3, 034606.
Volino, R. J., 2020, “Non-equilibrium development in
turbulent boundary layers with changing pressure gradients”,
Journal of Fluid Mechanics, Vol. 897, A2.

REFERENCES
Castro, I. P., 2007, “Rough-wall boundary layers: mean
flow universality”, Journal of Fluid Mechanics, Vol. 585, pp.
469-485.
Castro, I. P., Segalini, A., and Alfredsson, P. H., 2013,“Outer-layer turbulence intensities in smooth- and rough-wall6