Available via license: CC BY 4.0
Content may be subject to copyright.
J. Fluid Mech. (2021), vol.922, A29, doi:10.1017/jfm.2021.510
Mean ﬂow structure and velocity–bed shear
stress maxima phase diﬀerence in smooth wall,
transitionally turbulent oscillatory boundary
layers: experimental observations
Jose M. Mier1,‡, Dimitrios K. Fytanidis1,†and Marcelo H. García1
1Ven Te Chow Hydrosystems Laboratory, Department of Civil and Environmental Engineering, The
Grainger College of Engineering University of Illinois at UrbanaChampaign, Urbana, IL 61801, USA
(Received 3 September 2020; revised 21 April 2021; accepted 4 June 2021)
Oscillatory boundary layer (OBL) ﬂows over a smooth surface are studied using laser
Doppler velocimetry in a large experimental oscillatory ﬂow tunnel. The experiments
cover a range of Reynolds numbers in the transitional regime (Reδ=254–1315).
Motivated by inconsistencies in the literature, the focus is to shed light regarding the phase
shift Δφbetween the bed shear stress and the free stream velocity maxima. Details of
the mean ﬂow structure and turbulence characteristics in transitional OBL ﬂows indicate
the emergence of a logarithmic proﬁle, which for Reδ=763 appears at the middle of
the deceleration and as the Reδincreases, it appears for a longer part of the period and
for a larger region of the boundary layer. Turbulence statistics proﬁles approach those of
equilibrium, unidirectional boundary layer ﬂows with similar Reθ, deﬁned using the local
free stream velocity and momentum thickness θ. Analysis of the ensembleaverage bed
shear stress variation reveals that for Reδ<552 a single peak, associated with the laminar
regime, occurs during the acceleration phase. For Reδ=552 a second peak, associated
with the transition to turbulence, appears towards the middle of the deceleration phase.
This turbulence peak becomes larger than the ‘laminar’ one for Reδ∼763 and lags with
respect to the free stream velocity maximum. For Reδ>1036 the laminar peak disappears
under the effect of the turbulence peak. The presence of the phase lag is discussed using
data from this study and the literature, and a revised Δφdiagram is introduced for the
whole range of ﬂows, from laminar to fully turbulent.
Key words: coastal engineering, turbulent boundary layers, boundary layer structure
†Email address for correspondence: fytanid2@illinois.edu
‡Present address: Engineering Division, Port Authority of Santander, Cantabria, 39009, Spain.
© The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article,
distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/
licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium,
provided the original work is properly cited. 922 A291
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
J.M. Mier, D.K. Fytanidis and M.H. García
1. Introduction
Oscillatory boundary layer (OBL) ﬂows have received great attention in the past owing
to their large range of applications in both nature and engineered systems. Of particular
interest are wave boundary layer ﬂows in shallow and moderate waters which play an
important role on coastal engineering, sediment transport and seabed mechanics (Sleath
1984; Fredsøe & Deigaard 1992; Nielsen 1992; Sumer 2014).
Many studies are available in the literature that deal with the bottom boundary layer.
On the experimental side, the pioneering works of Hino, Sawamoto & Takasu (1976),
Hino et al. (1983), Jensen, Sumer & Fredsøe (1989), Akhavan, Kamm & Shapiro (1991a),
Sarpkaya (1993), Carstensen, Sumer & Fredsøe (2010)and van der A, Scandura &
O’Donoghue (2018) among others, summarize current knowledge regarding the oscillatory
boundary layer structure and possible ﬂow regimes in oscillatory ﬂow over ﬂat, smooth
beds; while on the numerical side, highﬁdelity direct numerical simulation (DNS) and
largeeddy simulation works have investigated the same family of ﬂows, which has
enhanced our current understanding in terms of ﬂow structure (Spalart & Baldwin 1989;
Vittori & Verzicco 1998; Salon, Armenio & Crise 2007; Pedocchi, Cantero & García
2011; Ozdemir, Hsu & Balachandar 2014; Scandura, Faraci & Foti 2016; Bettencourt &
Dias 2018; Ebadi et al. 2019), stability analysis (e.g. Akhavan, Kamm & Shapiro 1991b)
and coherent structures (Costamagna, Vittori & Blondeaux 2003; Mazzuoli, Vittori &
Blondeaux 2011). However, despite of all these advances, most of the stateoftheart
simpliﬁed models fail to accurately predict the underlying physics related to the turbulent
ﬂow–bed interaction (e.g. see Guizien, DohmenJanssen & Vittori 2003; Blondeaux,
Vittori & Porcile 2018); this is especially true when it comes to the prediction of friction
coefﬁcients (deﬁned later in the text), which are of high importance for the estimation
of sediment transport (Fredsøe & Deigaard 1992; Nielsen 1992; Liu, García & Muscari
2007; García 2008) as well as the phase difference of the maximum bed shear stress
with respect to the maximum free stream velocity. This fact highlights the need for the
development of better numerical models for nonequilibrium and transitional ﬂows but
also may be a sign of an incomplete understanding of the OBL behaviour, especially
in the transitional regime as will be shown herein. Hino et al. (1983) categorized the
OBL ﬂows literature into three categories, as follows: (a) works relevant to the ﬂow
resistance under oscillatory/wave condition;(b)works relevant to the identiﬁcation of
critical conditions for the transition between laminar and turbulent oscillatory ﬂow;and
(c) studies examining the ﬂow structure under oscillatory ﬂow conditions. The present
work bridges the gaps between these different categories and associates the ﬂow structure
effect on the wave friction for a range of ﬂow conditions varying from laminar to fully
turbulent. Special effort is placed in examining the ﬂow structures and resistance through
the transitional/intermittent turbulent regime.
Theoretical, experimental and numerical studies are available in the literature for
oscillatory (zero mean velocity) and pulsatile (with nonzero mean velocity) ﬂows.
This analysis focuses on pure reciprocating (zero mean ﬂow) OBL ﬂows which can
be characterized based on an oscillatory Reynolds number Reδ, commonly deﬁned
as Reδ=Uoδ/ν, where δis the Stokes layer thickness (δ=√2ν/ω), Uois the
amplitude of the free stream velocity oscillation (U∞=Uosin (ωt)), νis the kinematic
viscosity of the ﬂuid, ωis the angular frequency of the wave (ω=2π/T)andT
is the period of the oscillation. Interested readers can refer to studies of pulsatile
ﬂows, such as the works of Tu & Ramaprian (1983), Ramaprian & Tu (1983),
Tardu, Binder & Blackwelder (1994) and Lodahl, Sumer & Fredsøe (1998), among
others.
922 A292
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
Flow structure and phase difference diagram in OBL ﬂows
Depending on the duration of the period and the amplitude of this sinusoidal movement,
OBL ﬂows are categorized into four distinct regimes (see Akhavan et al. 1991a; Pedocchi
et al. 2011; Ozdemir et al. 2014): (i) the laminar regime (Reδ<Reδcr 1), corresponds
to Stokes’ second problem for which an analytical solution exists (Batchelor 1967);
(ii) the disturbed laminar regime (Reδcr1<Reδ<Reδcr2), in which the ﬂow behaves
like in the laminar regime but small perturbations are superimposed on the OBL ﬂow.
These disturbances are not sufﬁciently strong to alter the mean velocity proﬁle and
are caused by the formation of linear instability related features (Carstensen et al.
2010); (iii) the intermittent turbulent regime (Reδcr2<Reδ<Reδcr3), in which the ﬂow
tends to remain laminar during the acceleration phase.However turbulent bursts are
observed at the beginning of the decelerating phase after the maximum velocity, when the
pressure gradient is adverse to the ﬂow before laminarizing again during the acceleration
phase (Merkli & Thomann 1975;Hinoet al. 1983; Akhavan et al. 1991a,b; Carstensen
et al. 2010); (iv) the fully turbulent regime (Reδ>Reδcr3), in which turbulence is
observed during the whole cycle of the oscillation while the characteristic feature of the
unidirectional turbulent ﬂow, the logarithmic layer, is observed in the OBL for most of the
time during the oscillation cycle excluding a period close to the ﬂow reversal (Jensen et al.
1989).
Identifying the exact value of Reδcr1,Reδcr2and Reδcr3has become the subject of many
studies. In depth reviews of the available instability related work can be found in the
works by Akhavan et al. (1991a,b), Sarpkaya (1993), Blondeaux & Vittori (1994), Ozdemir
et al. (2014) and Thomas et al. (2015). A commonly accepted value for Reδcr1is usually
close to 85 (Blondeaux & Seminara 1979; Akhavan et al. 1991b). However, it is worth
pointing out that this theoretically derived value is the result of an analysis predicting
that the instability occurs at a time instance close to the beginning of the acceleration
phase. This ﬁnding is not in agreement with the experimental observations of Merkli
& Thomann (1975), Hino et al. (1976) and Fishler & Brodkey (1991) for pipes,and
Jensen et al. (1989) for rectangular channels, who observed the incipient turbulence
occurs during the deceleration phase. Wall imperfections (Blondeaux & Vittori 1994)
and highfrequency ‘noise’ (Thomas et al. 2015) have been used in theoretical studies to
explain the discrepancies between theory and experiments. Higher values of 260–280 have
been reported for the heightlimited case of ﬁnite oscillatory pipe ﬂow (Hino 1975; Merkli
& Thomann 1975). While laminar ﬂow behaviour has been observed for signiﬁcantly
higher Reδvalues in the lab (Kamphuis 1975;Jensenet al. 1989), Reδcr2values of 500–550
are reported both experimentally and numerically (Hino et al. 1976;Jensenet al. 1989).
However, the exact value of Reδcr2seems to be affected by the background turbulence
levels (Ozdemir et al. 2014). Finally, a Reδcr3value of 3460 was reported by Jensen et al.
(1989). Experimental observations showed that the ﬂow regime plays an important role on
bed friction (e.g. Kamphuis 1975;Jensenet al. 1989;Sarpkaya1993).
The early works by Kajiura (1964), Yalin & Russell (1966), Jonsson (1966), Riedel,
Kamphuis & Brebner (1973) and Kamphuis (1975) were focused on the estimation of
the ﬂow resistance under wave conditions, aiming mainly on setting up graphs for the
prediction of the friction factor (fw=2τ/ρU2) for various ﬂow and bed roughness
conditions. Kajiura (1964,1968) and Jonsson (1966) developed analytical formulae for
the prediction of friction factors based on some assumptions related to the velocity
proﬁle distribution. Riedel et al. (1973) and Kamphuis (1975) performed extensive sets
of experiments on ﬂat beds with glued sand particles and presented some of the ﬁrst
comprehensive plots for the friction factor for various bed roughness values. Jensen
et al. (1989) examined the velocity and turbulent structure of the OBL and identiﬁed the
922 A293
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
J.M. Mier, D.K. Fytanidis and M.H. García
transition to turbulence in terms of the friction coefﬁcient fwfor laminar, transitional and
turbulent ﬂows. Jensen et al. (1989) reported values of the friction coefﬁcient as well as
the phase difference (Δφ) between the instance when the maximum of the bed shear stress
occurs with respect to the maximum of free stream velocity. Sarpkaya (1993) studied the
OBL ﬂow structures using laserinduced ﬂuorescence (LIF) and shear force measurements
using straingauge sensors,and reported values of the friction coefﬁcient for a wide range
of ﬂows ranging from laminar to fully turbulent. More recently, Carstensen et al. (2010)
obtained similar results to those of Jensen et al. (1989) and Sarpkaya (1993). It is worth
mentioning that even though the experimental values for the transitional regime reported
by these authors are similar to those reported by Spalart & Baldwin (1989) using DNS,
they deviate from those of Kamphuis (1975) by 20%. In addition, in all these studies
(Jensen et al. 1989;Sarpkaya1993; Carstensen et al. 2010), the reported results show a
phase lead of the maximum bed shear stress with respect to the velocity maximum value.
For a laminar OBL, a constant phase lead of 45◦can be expected and derived from
the classic laminar OBL solution (Batchelor 1967). At the limit when Reδapproaches ∞
the phase difference Δφapproaches zero at a rate of approximately 1/log[Reδ] (Spalart
& Baldwin 1989). However, in the fully turbulent regime and for a large but ﬁnite Reδ
value, Fredsøe (1984) developed a semiempirical formula for the prediction of phase lead
with the values ranging below 10◦(see the paper by Fredsøe (1984), p. 1110, table 2).
These two asymptotic behaviours, when Reδapproaches zero (low values) and inﬁnity
(high values),have led researchers to assume that in the narrow range of Reδbetween
approximately 300 and 1000 the commonly reported behaviour is that the phase difference
Δφdecreases rapidly from the 45◦,when Reδ≤300, to nearly 10◦when Reδ≈1450.
The abovedescribed behaviour is shown in ﬁgure 1.Owing to the fact that some works
have used a different Reynolds number, Rew, deﬁned using half of the oscillation excursion
instead of δ,Rew=Uoα/ν (note the explicit relationship Rew=Re2
δ/2), a second abscissa
axis is added showing the values of Rew.Thiskindofdiagram is included in coastal
engineering handbooks (e.g. p. 32 of Fredsøe & Deigaard 1992) to show the bed shear
stress phase lead. Herein, it is shown that this is not the actual behaviour. A revised
phase shift diagram is advanced and ﬂow structure changes across the different regimes
are presented.
Nearbed velocity measurements by Hino et al. (1976) and Fishler & Brodkey (1991)
indicate the presence of violent turbulent bursts during the deceleration of an oscillation.
These turbulencerelated velocity spikes become dominant for ﬂows in the transitional
regime and are consistent over different periods. These increased velocity ﬂuctuations
may result in an increase of ensembleaveraged, wall shear stress during the deceleration.
A close observation of the measurements by Hino et al. (1976) shows that the phase of
the cycle when these spikes appear happens earlier as the Reδvalue increases. Later,
Hino et al. (1983) (p. 373, ﬁgure 10) presented the phase variation of wall shear stress
results for a Reδvalue of 876. From their measurements, it can be seen that the maximum
bed shear stress value occurs at the deceleration phase, i.e. lags compared with the
maximum free stream velocity. However, no analysis is presented in their work for the
phase difference variation with different Reδ, nor is a discussion about the presence of
the phaselag itself included. It is important to mention here that in ﬁgure 1, the data by
Hino et al. (1976) are plotted with positive Δφwhich corresponds to the smaller peak
during the acceleration phase rather than the maximum bed shear stress over the period
(this will be further discussed in §3.3.1). Similar behaviour has been observed in the
instantaneous bed shear stress measurements in oscillatory channel ﬂows for Reδbetween
922 A294
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
Flow structure and phase difference diagram in OBL ﬂows
102
104105
Laminar solution Fredsøe’s (1984) theoretical solution
Carstensen et al. (2010)
Spalart and Baldwin (1989)
Hino et al. (1983)
Jensen et al. (1989)
Sarpkaya (1993)
106
Reδ
Rew
107
103104
0
Phase lead φ (deg.)
Phase lead φ (rads)
0
π/4
φ
U∞
τb
45
π
0
Figure 1. Typical phase lead Δφdiagram as a function of Reδand Rew(adapted from Jensen et al. 1989).
616 and 898 by Carstensen et al. (2010). However, owing to the fact that only instantaneous
values are presented in such works, no solid conclusion can be reached regarding the
ensembleaverage bed friction behaviour and the phase difference of its maximum value
with respect to the maximum free stream velocity. Once again, no analysis is presented
explaining the presence of a phase lag in the data set, but instead a phase difference
diagram showing phase lead values is included (Appendix, p. 203, ﬁgure 21) by the
authors. The bed shear stress measurements of Jensen et al. (1989) also include phaselag
observations for Reδof 762. In their measurements phase lag turns to phase lead for an
increased value of Reδof 1140 as well as for a decreased value of Reδ=566. Although no
discussion is included in the paper by Jensen et al. (1989), these observations suggest that a
threshold value at which phase lag begins to occur may exist. However, no detailed analysis
of the phase difference between the bed shear stress and free stream velocity maxima
is included in the literature on: (i) how slowly enhanced levels of turbulence as the Reδ
number increases within the transitional regimes (from disturbed laminar to intermittent
turbulent regimes) modify the friction on the bed; and (ii) how do corresponding changes
in ﬂow structure affect the phase difference values.
The present work focuses on the examination of bed shear stress, friction factor and
phase difference in the range of 254 ≤Reδ≤1315. Special attention is given to the
identiﬁcation of a threshold value of a Reδfor which a phase lag exists. In addition, the ﬂow
structure variation across the different ﬂow regimes is examined in an effort to evaluate
the effect of ﬂow structure on friction velocity and bed shear/free stream velocity maxima
phase difference. An effort is made to bridge the remaining gaps in knowledge from the
previous experimental works of Hino et al. (1983), Jensen et al. (1989)and Akhavan et al.
(1991a) regarding the ﬂow structure in OBL for various ﬂow regimes and especially in
the intermittent turbulent regime where there is a scarcity of observations close to a wall,
within the boundary layer.
922 A295
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
J.M. Mier, D.K. Fytanidis and M.H. García
The analysis herein focuses on oscillatory ﬂows over smooth walls. However, oscillatory
ﬂows in nature commonly involve rough bottoms. Although additional analysis is
needed for the case of a rough wall, the results and conclusions from the exclusively
smoothwalled cases considered in the present analysis may be relevant for OBL ﬂows
over rough walls. For example Nielsen & Guard (2010) and Nielsen (2016) suggested that
the normalized Stokes length √2ν/ω/α is roughly interchangeable with 0.09√αks/α.
This equivalence between the viscous and roughness scales is similar to that proposed
by Colebrook (1939) for unidirectional ﬂows, for which ks/30 is equivalent to 0.11ν/u∗
(where u∗is the shear velocity). A more recent analysis regarding the roughness scaling
in the transition from smooth to fully rough conditions is provided by Pedocchi & García
(2009a).
2. Experimental apparatus and data analysis
2.1. Large Oscillatory Water and Sediment Tunnel (LOWST)
Experiments were conducted in the Large Oscillatory Water and Sediment Tunnel
(LOWST) housed in the Ven Te Chow Hydrosystems Laboratory of the University of
Illinois at UrbanaChampaign (ﬁgure 2). The test section is 12 m long and the internal
dimensions of the crosssection are 0.8 m wide by 1.2 m high. A false bed was placed at
the middle of the crosssection reducing the height of the water tunnel to 0.6 m. Special
attention was given to keeping the smooth PVC bottom ﬁxed rigidly at the middle of the
section. External disturbances were kept to a minimum via insulation of the ﬂume from
the laboratory ﬂoor. The oscillatory motion of the water is driven by three pistons that run
inside 0.78m diameter cylinders with a maximum stroke of 1.37 m. At the opposite end of
the tunnel, a 1.0 by 2.0 m holding tank open to the atmosphere acts as a passive receiver
for the water displaced by the pistons. Three servo motors, controlled by a computer,
drive the pistons using a screwgear system. Although unidirectional ﬂow was not used
in this study, the facility also has two centrifugal pumps that allow for the superposition
of a unidirectional current of up to 0.5 m s−1onto the oscillatory motion through a pipe
recirculation system. Flow straighteners and sediment traps are available at both ends of
the main test section. No sediment particles were used for the present study. LOWST can
generate oscillatory ﬂows with time periods between 5 to 15 s and maximum horizontal
velocities of up to 2 m s−1. A more detailed description of the facility can be found in the
paper by Pedocchi & García (2009b).
Instantaneous velocity measurements were conducted using a threedimensional laser
Doppler velocimetry (LDV) system from TSI Inc., with an Arion 6W multiline laser
(model Stabilite 2017, from SpectraPhysics) generating a light beam which in turn is
directed towards a FiberLightTM multicolour beam separator box (model FBL3). The
LDV technique was adopted owing to its high temporal resolution (up to 10 000 Hz),
provided that appropriate seeding is achieved in the large oscillatory ﬂow tunnel. This high
rate of data sampling (samples per second) ensures that the high frequencies of the ﬂow are
preserved, which allows for the analysis of turbulence characteristics, especially within the
boundary layer. A preliminary study examined different kinds of seeding particles, which
included hollow glass spheres (HGS) and silvercoated hollow glass spheres (SHGS)
of various densities and diameters, as well as different concentrations of particles to
ensure a maximum recording rate for the LDV system (Mier 2015). The particles used
in the experiments were the HGS particles (with a density of 1.1 g cm−3and diameter of
11 μm) which are close to neutrally buoyant and are big enough to generate highintensity
backscatter signals, and light enough to meet the turbulence criteria. Preliminary analysis
922 A296
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
Flow structure and phase difference diagram in OBL ﬂows
Holding
tank Sediment
trap Main window/
Measurement section
0.8 m
2 m 1.4 m 12 m 1.4 m 3.25 m
False PVC
bottom
0.6 m
1.2 m
AA
A
A
Sediment
trap
Pistons
Figure 2. Large Oscillatory Water and Sediment Tunnel (LOWST).
indicated that the optimum concentration (number of particles per unit volume) to ensure a
maximum data rate was approximately N=0.1–0.2×Vm, where Vmis the measurement
volume. This analysis took into consideration the effect of light attenuation through the
penetration length dwwhich was equal to 0.4m (N=0.4–0.5×Vmeαdw, where αis the
attenuation coefﬁcient with values of 7.86 m−1for HGS and 5.75 m−1for SHGS). More
information can be found in the paper by Mier & García (2009). An average value of the
diameter of the measurement volume was 0.1 mm and an average value of its length was
approximately 1 mm, which resulted in a very small measurement volume (approximately
0.01 mm3).
Velocity proﬁles were measured from a series of vertically distributed pointwise LDV
measurements. The LDV probe was mounted on a 3axis traverse, driven by a microstep
controller, capable of providing a spatial resolution of 0.01 mm in all three directions,
which was essential for the ﬁne geometric requirements needed inside the boundary layer.
The displacement range was approximately 50 cm in all three directions, which allowed
for taking measurements across the tunnel. Special attention was given to deﬁne the level
of the wall where y=0m(i.e.noslip boundary condition).
A set of magnets, one mounted on the moving pistons and one on the enclosing cylinders
of the ﬂume, was used to synchronize the time instances that deﬁne the beginning of
each cycle. The present work focuses on the examination of OBL ﬂows with a period of
10 s, which is a typical period for coastal wave applications. In the present work, 130
cycles, in each test, were used for the estimation of turbulence statistics for each phase.
Sleath (1987) argued that 50 periods are enough for the statistics to converge. Jensen et al.
(1989) performed a similar analysis conﬁrming Sleath’s ﬁndings. A similar analysis of our
results shows that negligible variations (typically less than 1 %) were observed for a higher
number of cycles.
A summary of the examined cases is presented in table 1. Temperature measurements
were conducted to estimate any signiﬁcant viscosity or density variations. The temperature
922 A297
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
J.M. Mier, D.K. Fytanidis and M.H. García
Exp.
no
Temperature
TC(◦C)
Excursion of
oscillation
2α(m) Uo(m s−1)u∗max (cm s−1)Reδ=Uoδ/ν Rew=Uoα/ν
1 18.0 0.468 0.147 1.1 254 3.2×104
2 16.6 0.761 0.239 1.4 405 8.2×104
3 23.2 0.958 0.301 1.6 552 1.5×105
4 23.6 1.159 0.364 1.7 671 2.3×105
5 27.5 1.261 0.396 1.8 763 2.9×105
6 27.0 1.362 0.428 2.3 819 3.4×105
7 26.5 1.566 0.492 2.4 937 4.4×105
8 24.5 1.770 0.556 2.6 1036 5.4×105
9 18.1 2.069 0.650 3.7 1123 6.3×105
10 20.0 2.368 0.744 3.9 1315 8.6×105
Table 1. Test conditions of pure oscillatory ﬂow. Period of the motion T=10 s. Amplitude of the oscillation
α=UoT/ν. Kinematic viscosity ν=1.79 ×10−6/(1+0.03368TC+0.00021T2
C)and density ρ=1000(1−
(TC+288.9414)(TC−3.9863)2/508929.2(TC+68.1293)).
of the water was kept constant over the time of each experiment. The measured
temperatures are also reported in table 1.
Ensemble averaging was used to estimate the mean values of all quantities as
¯u(y,ωt)=1
N
N
k=0
u(y,ω(t+kT ))) (2.1)
The instantaneous ﬂuctuations were calculated as
u(y,ωt)=u(y,ω(t+kT)) −¯u(y,ωt)) (2.2)
The rootmeansquare (r.m.s.) of the velocity ﬂuctuations and Reynold shear stresses were
calculated as
(u2)1/2(y,ωt)=1
N
N
k=0
u2(y,ω(t+kT ))1/2
(2.3)
−uv(y,ωt)=1
N
N
k=0
u(y,ω(t+kT ))v(y,ω(t+kT )) (2.4)
3. Results and discussion
3.1. Mean ﬂow structure and boundary layer properties
3.1.1. Flow regimes
Akhavan et al. (1991a) and Ramaprian & Tu (1983) used dimensional analysis and
examined the similarity laws of oscillatory and pulsatile pipe ﬂows, respectively. They
considered that the OBL ﬂows can be categorized into four regimes based on three length
scales: a geometrical length scale based on the diameter of the pipe R, an inertia length
scale δt=u∗/ω and a viscous length scale δv=ν/u∗. It is worth noting that the Stokes
length scales with the geometric mean of inertia and viscous length scales (δ∼√δtδv).
922 A298
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
Flow structure and phase difference diagram in OBL ﬂows
Akhavan et al. (1991a) showed the dimensional necessity for a logarithmic layer to exist
when two or more of the scales R,δtand δvare widely separated.
Based on the above scales, four different cases of oscillatory pipe ﬂows are deﬁned
(Akhavan et al. 1991a): (a) Case I, the pipe diameterlimited, ‘quasisteady’ turbulent
behaviour for which δtRδv(i.e. u∗/(ωR)1, Ru ∗/ν 1), where the ﬂow
behaves in a quasisteady way and a universal logarithmic law is valid; (b) Case II,
which can in a way be considered as a special version of Case I for which δt∼Rδv
(i.e. u∗/(ωR)∼1, u∗R/ν 1), for which the ﬂow obeys a modiﬁed version of the
loglaw where the universal slope expressed by von Kármán constant κmay be constant
(κ=0.41).However, the value of constant Avaries over time (A(ωt)=f(u∗/(Rω)));
(c) Case III, for which Rδtδv(i.e. u∗/(ωR),u2
∗/(ων) 1) and a logarithmic law
is valid for y<δ
t. However, in the outer layer, where y/δt→∞(i.e. δt=u∗/ω →0),
the ﬂow behaves in an ‘inviscid way’ similar to the case when u∗→0 (assuming that
ωis ﬁnite). The mean velocity and turbulent moments proﬁles depend only on Rand ω
values; (d) Case IV, which again can be considered to be a special version of Case III, for
which Rδt∼δv(i.e. u∗/(ωR),u2
∗/(ων) ∼1) and a logarithmic proﬁle is once again
valid with Asvarying over the cycle. Akhavan et al. (1991a) presented results of pipe ﬂow
of case II. Because coastal/wave ﬂow conditions are of interest, ﬂows in the current study
belong to the nondiameterlimited cases III and IV but for a closed channel. Considering
half the height of the channel (or the hydraulic radius of the channel) as equivalent to R,
Ru∗/ω (or u∗/(ωR)1 except from the shear stress reversal when u∗is zero) for all
the ﬂows considered in the present study.
The structure of the OBLs was examined by Jensen et al. (1989) for a wide range of
Reδ(Reδbetween 257 and 3464). Jensen et al. (1989) noticed a distinct difference in the
boundary layer structure for Reδof 762 (expressed in the original work as Rew=2.9×
105). This ﬂow exhibited an intermittent turbulent behaviour for which the logarithmic
distribution, u+=(1/κ) ln y++5.1, was valid after ωt=6π/9(120
◦). An explanation
for this different behaviour, given by the authors, indicated that the ﬂow experiences
transitional conditions for most of its period. However, no detailed explanation was given
about the effect of Reδon the ﬂow structure and consequently its effect on the bed shear
stress especially at the transition from the laminar to transitional and to turbulent ﬂow
regime. Hino et al. (1976) studied an OBL for Reδof 876 and R/δ of 12.8; however once
again, the effect of Reδvariation was not clearly shown as only results from a single ﬂow
case were presented. Recently, Kaptein et al. (2019)usedlargeeddy simulation to examine
the effect of the h/δ ratio (where his the height of their domain representing the water
depth on oscillatory ﬂows over a ﬂat plate) on the phase difference between free stream
velocity and bed shear stress maxima. Their results showed that for h/δ ≥40, velocity,
turbulent characteristic and bed shear stress results converged to those for h/δ →∞.In
the present study R/δ is of the order of 250, which was consider large enough to represent
the coastal boundary layer conditions for which R/δ →∞.
3.1.2. Laminar ﬂow
To test the accuracy of our measurements, the lowest Reδcase was examined (experiment
1withReδ=254) and it was compared with an analytical solution. The velocity proﬁle
for the laminar regime can be calculated using the following analytical solution:
u(y,ωt)=Uo(sin(ωt)−e−y/δ sin(ωt−y/δ)) (3.1)
922 A299
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
J.M. Mier, D.K. Fytanidis and M.H. García
10–4
–0.05 0 0.05
Acceleration
0.10 0.15
10–3
10–2
y (m)y (m)
10–1
100
10–4
–0.05 0 0.05 0.10 0.15
10–3
10–2
10–1
100
–0.10 0
0.05
0.10
0.15
0.20
–0.05
0
0.05
0.10
0.15
0.20
0.25
0
0π
–
12
5π
–
12
π
–
6
π
–
4
π
–
2
π
–
3
0,π
π/4 π/2 3π/4 π
3π
–
2
π
–,
2
7π
–
4
3π
–
,
4
5π
–
4
π
–
,
4
10–4
–0.05 0 0.05
Deceleration
0.10 0.15
10–3
10–2
10–1
100
π11π
–
12
7π
–
12
5π
–
6
3π
–
4
π
–
2
2π
–
3
u
– (m s–1)u
– (m s–1)
u
– (m s–1)
τb (N m–2)
U∞ (m s–1)
ωt (rads)
φ
(b)(a)
(c)(d)
Figure 3. Comparison of measurements against analytical solution for laminar regime (Test 1, Reδ=254):
(a)streamwise velocity proﬁles during acceleration, ◦measurements,(—–,black)analytical solution;
(b) streamwise velocity proﬁles during deceleration;(c) comparison between positive and negative parts of
the period, ◦measurements in the positive part, ×measurement in the negative part multiplied by −1.0,
(—–, black) analytical solution;(d) measurements of bed shear stress τband free stream velocity U∞,◦
measurements of bed shear stress,(•,grey) measurements of free stream velocity, (—–, black) analytical
solution for bed shear stress τb,–––U∞(t)=Uosin(ωt).
by differentiating (3.1) and using the deﬁnition of viscous shear stress (τ=ρν∂u/∂y)we
can estimate the shear stress variation as τ(y,ωt)=√2ρ(U2
o/Reδ)e−y/δ sin(ωt−y/δ +
π/4)and the wall shear stress τbcan easily be calculated for y=0as
τb
ρ=√2U2
o
Reδ
sin(ωt+π/4)(3.2)
In ﬁgures 3(a)and3(b), the analytical proﬁles for various time instances are plotted
for the acceleration and deceleration phases, respectively, together with the experimental
observations. The comparison between the analytical and experimental values agrees well.
In addition, to evaluate the symmetry of the imposed oscillation from the pistons of the
experimental facility, a comparison between the positive and negative parts of the cycle
was conducted. Such comparison of these proﬁles is shown in ﬁgure 3(c), in which the
measurement of the negative part of the period is multiplied by −1.0. No signiﬁcant bias
or skewness towards the positive or negative direction was observed in our measurements.
Finally, ﬁgure 3(d) shows a comparison with the bed shear stress measurements, estimated
922 A2910
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
Flow structure and phase difference diagram in OBL ﬂows
as τb=ρν∂u/∂yb. Once again, the experimental results agree well with the analytical
solution above.
3.1.3. Transitional ﬂow
In their work, Hino et al. (1983) examined the ﬂow structure for a ﬂow with Reδ=876.
They presented data for the mean ﬂow and turbulence characteristics for this Reynolds
number but owing to the fact that only a single ﬂow was analysed, the change of the
mean ﬂow characteristics as Reδincreased and the ﬂow went through a transition remains
unknown. In ﬁgure 4, the ensemble average velocity proﬁles for three characteristic
instances of the period (π/4, π/2and3π/4) are shown for all the examined ﬂows.
In ﬁgure 4(a–c), the velocity proﬁles are presented in wall units (where y+=u∗y/ν,
u+=¯u/u∗and u∗is the shear velocity u∗=√(τb/ρ)). The orange dashed lines show
the ﬁt of a logarithmic proﬁle similar to the ‘universal loglaw’ for turbulent equilibrium
boundary layers. Figures 4(d–f)and4(g–i) show the velocity defect normalized using the
free stream velocity U∞and shear velocity, respectively. The arrows show the general
trends of the velocity proﬁles. Jensen et al. (1989) have shown that for high enough Reδ
values the velocity proﬁles should approach the universal logarithmiclaw for a smooth
wall:
U+=1
κln y++As(3.3)
with κ≈0.41 and As≈5.1. For equilibrium boundary layers, (3.3) is valid only for
the part of the velocity close to the wall, while far from the wall additional adjusting
parameters need to be used to describe the velocity proﬁle, e.g. law of the wake (Krug,
Philip & Marusic 2017; Jimenez 2018). Akhavan et al. (1991a) showed that for Reδin the
transitional regime (when u∗/ων ∼1.)(3.3) is modiﬁed to U+=(1/κ) ln(y+)+As(ωt),
Aschanges for different phases of the period. Hino et al. (1983) also showed that Asvaries
for a transitional ﬂow (Reδ=873). In ﬁgures 4(b), 4(f)and4(i), and to an extent in
ﬁgures 4(a)and4(h), it can be observed that the mean proﬁle in the transitional ﬂows and
especially for Reδ=763 (experiment 5) deviate signiﬁcantly from both the logarithmic
proﬁles, which are observed for higher Reδcases, and the laminar proﬁles. However, as
Reδincreases there is a clear trend towards the equilibrium logarithmic law (As≈5.1in
(3.3)). The arrows in ﬁgure 4(c–i) show this transition.
To evaluate the ﬁt of the logarithmic proﬁles, the loglaw diagnostic function Ξ(Ξ=
y+(∂ ¯u+/∂yy)) is plotted in ﬁgure 5 for three Reδ(763, 937 and 1315) for ωt=π/2to
5π/6. The Ξfunction should approach 1/κ for zeropressure gradient boundary layers
in regions where the loglaw occurs (Nagib, Chauhan & Monkewitz 2007). In addition
to the equilibrium value 1/κ,the1/κ (ωt)values are also plotted for each proﬁle. It can
be seen that for Reδ=763 (experiment 5) the part of the proﬁle where a logarithmic
equation may ﬁt is smaller compared with the higher Reδcases. For this ﬂow, the slope
of the logarithmic proﬁle will be larger than 1/0.41. As Reδincreases to 937 and 1315 we
can observe that the log proﬁle slope 1/κ(ωt)starts to approach 1/0.41 for parts of the
deceleration. Furthermore, the region where a logarithmic proﬁle may ﬁt increases in size.
Finally, for Reδ=1315 the proﬁles seem to agree well with the 1/0.41 slope, although
the slope becomes smaller towards ωt=5π/6. It is important to note that in our work
the use of κ(velocity proﬁle’s slope) and As(velocity proﬁle’s intersect) for the parts of
the ﬂow that are not in equilibrium (e.g. the values for ωt<π/2) is merely to provide us
with a diagnostic parameter for the development of a true logarithmic proﬁle. This same
approach has been used in the past speciﬁcally for the case of OBL ﬂows by Hino et al.
922 A2911
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
J.M. Mier, D.K. Fytanidis and M.H. García
100101
y/δ
u
–+
103
102104
Log. fit
π
–
4
100101102
y+
y+
103104
30
20
10
0
30
20
10
0
π
–
2
100101102
y+
103104
30
20
10
0
3π
–
4
10–1 100101
(U∞ – u
–)/U∞
(U∞ – u
–)/u∗
102
π
–
4
1.0
0.5
0
y/δ
10–1 100101102
1.0
0.5
0
y/δ
Reδ
10–1 100101102
1.0
0.5
0
π
–
2
3π
–
4
π
–
4
100101102103104
30
20
10
0
100101102103104
30
20
10
0
100101102103104
30
20
10
0
π
–
2
254
405
552
671
763
820
937
1036
1123
1315
3π
–
4
y+y+y+
(e)
(b)
(a)(c)
(h)
(g)(i)
(d)(f)
Figure 4. Reynolds number effect. Ensemble average velocity proﬁles in wall units for:(a)ωt=π/4;(b)ωt=
π/2;(c)ωt=3π/4. Ensemble average velocity defect proﬁle normalized with free stream velocity U∞and
δfor:(d)ωt=π/4;(e)ωt=π/2;(f)ωt=3π/4. Ensemble average velocity defect proﬁle normalized with
shear velocity u∗for:(h)ωt=π/4;(i)ωt=π/2;(g)ωt=3π/4. Dashed orange lines show logarithmic ﬁt for
the cases with Reδ≥763. The arrows show the increasing Reδpath.
(1983)(ﬁgures 7and 9 in their original work) and by Akhavan et al. (1991a) (ﬁgures 19 and
23 in their original work). The values of κand Asare obtained by ﬁtting the logarithmic
law in a region of approximately 30 ≤y+≤150. The region where a logarithmic layer
exists varies over time and for different Reδvalues. However, the region of the ﬁt was
chosen with the aims to maximize the region of the ﬁtting but also to avoid the wake
effects (Krug et al. 2017).
Akhavan et al. (1991a) argued that Asshould approach an equilibrium value for
oscillatory pipe ﬂows when u2
∗/ων 1 (and u∗/ωR1). Their analysis did not
include cases for u2
∗/ων 1. Instead they referred to the works of Mizushina,
922 A2912
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
Flow structure and phase difference diagram in OBL ﬂows
10
8
6
4
2
0104
103
102
101
100
Reδ = 1315
y+
= y+∂u
–+/∂y+
1/
κ
1/
κ
(ωt)
10
8
6
4
2
0104
103
102
101
100
Reδ = 937
= y+∂u
–+/∂y+
1/
κ
1/
κ
(ωt)
10
8
6
4
2
0104
103
102
101
100
Reδ = 763
= y+∂u
–+/∂y+
1/
κ
1/
κ
(ωt)
π
/
2
7π
/1
23π
/4
2π
/3
5π
/6
(b)
(a)
(c)
Equilibrium logarithmic law 1/
κ
log y+ + As(ωt) : 1/
κ
Logarithmic profile 1/
κ
(ωt) log y+ + As(ωt) : 1/
κ
(ωt)
Figure 5. Loglaw diagnostic function Ξduring the deceleration phase for Reδvalues of 763, 937 and 1315.
Maruyama & Shiozaki (1974) and Ramaprian & Tu (1983) who examined conditions
of u2
∗/ωR≈0.1andu2
∗/ων ≈100. The present analysis extends signiﬁcantly the
ranges of Akhavan et al. (1991a), Mizushina et al. (1974) and Ramaprian &
Tu (1983).
3.2. Boundary layer thickness
Different characteristic length scales have been proposed in the literature to characterize
the thickness of oscillatory boundary layers. Sumer, Jensen & Fredsøe (1987) deﬁned the
thickness of the boundary layer δπ/2based on the velocity maximum at ωt=π/2. Similar
deﬁnitions have been used by Sleath (1987) and Jonsson & Carlsen (1976)forωt=π/2
but instead of the maximum velocity they used the 5% defect of the velocity with respect
922 A2913
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
J.M. Mier, D.K. Fytanidis and M.H. García
102
104105106107
0
0.01
0.02
0.03
0.04
0.05
0.06
Reδ
Rew
103104
10–1
10–2
10–3
10–40 0.5 1.0
ωt = π
/
2
δπ
/
2
/
α
δπ
/
2
/
α
y
/
α
u
–/Umax

π
/
2
Laminar solution
Fredsøe’s (1984) theoretical solution
Spalart and Baldwin (1989)
Present study
Hino et al. (1983)
Jensen et al. (1989)
Figure 6. Normalized oscillatory boundary layer thickness δπ/2/α as a function of Reδor Rew.
to the free stream value and the ﬁrst yposition from the wall where ¯uequals the free
stream velocity, respectively. Jensen et al. (1989) plotted their results of δπ/2for two
ﬂow conditions (Reδof 1789 and 3464). They also compared their results with those of
Hino et al. (1983) and Spalart & Baldwin (1989). In ﬁgure 6 the boundary thickness is
plotted based on the deﬁnition of Sumer et al. (1987). The values are normalized using
the amplitude of the oscillation α. The deﬁnition of δπ/2/α is also shown in the inset
of the plot. The prediction of the analytical solution δπ/2/α =(3π/4)(4/Re2
δ)1/2and the
solution by Fredsøe (1984) are also plotted together with the previous data of Jensen et al.
(1989), Hino et al. (1983) and Spalart & Baldwin (1989). The experimental observations
of the present work match reasonably well with the laminar solution for Reδof 254 and 405
(experiments 1 and 2). The rest of the data (experiments 3–10) connect the laminar with
the turbulent regimes. Speciﬁcally, as the Reδincreases,δπ/2/α seems to increase until
Reδ≈1500 when the turbulent solution of Fredsøe (1984) predicts well the behaviour of
the experiments by Jensen et al. (1989). The results of the present study agree well with
the results of Hino et al. (1983) and Spalart & Baldwin (1989), which are in a similar range
of Reδvalues.
For their analysis, Jensen et al. (1989) used the maximum velocity of each
ensembleaveraged proﬁle to deﬁne the boundary layer thickness ymax for each phase (for
this location also shear stress is ¯τ≈0). For this analysis, the same approach used by
Jensen et al. (1989) was adopted. No signiﬁcant changes in the results of the analysis
were observed when ¯τ≈0 was used instead of ¯umax for the deﬁnition of the boundary
layer. A plot of boundary layer thickness for all the examined cases together with the
ensembleaveraged contours of streamwise velocity are shown in ﬁgure 7. The results
are made dimensionless using the Stokes length δ. In addition to the boundary layer
thickness, the displacement thickness δ∗and momentum thickness θare plotted in ﬁgure 7,
922 A2914
https://doi.org/10.1017/jfm.2021.510 Published online by Cambridge University Press
Flow structure and phase difference diagram in OBL ﬂows
H
H
H
H
H
y
/
δy
/
δy
/
δy
/
δy
/
δ
10–1
100
101
1025.0
2.5
0
Reδ = 254
π
/
20 π
10–1
100
101
1025.0
2.5
ymax/δ
H
H = 1.4
δ∗/δ
θ/δ
0
Reδ = 405
π
/
20 π
10–1
100
101
1025.0
2.5
0
Reδ = 552
π
/
20 π
10–1
100
101
1025.0
2.5
0
Reδ = 671
π
/
20 π
10–1
100
101
1025.0
2.5
0
Reδ = 1123
π
/
20 π
10–1
100
101
1025.0
2.5
0
Reδ = 1315
π
/
20 π
10–1
100
101
1025.0
2.5
0
Reδ = 763
π
/
20 π
10–1
100
101
1025.0
2.5
0
Reδ = 820
π
/
20 π
10–1
100
101
1025.0
2.5
0
Reδ = 937
π
/
20 π
10–1
100
101
1025.0
1.0
0.5
0
2.5
0
Reδ = 1036
π
/
20 π
u
–/Uo
ωt (rads) ωt (rads)
(e)
(b)(a)
(c)(d)
(g)(h)
(i)(j)
(f)
Figure 7. Contour maps of normalized velocity proﬁles ¯u/Uo. Dimensionless boundary layer thickness ymax/δ,
displacement thickness δ∗/δ, momentum thickness θ/δ and shape factor H. The typical value of H=1.4for
fully developed unidirectional ﬂow is also shown.
which are deﬁned as
δ∗(ωt)=ymax
0
U∞(ωt)−¯u(y,ωt)
U∞(ωt)dy,(3.4)
θ(ωt)=ymax
0¯u(y,ωt)