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

A NEW POINT OF VIEW ON SKIN-FRICTION CONTRIBUTIONS IN
ADVERSE-PRESSURE-GRADIENT TURBULENT BOUNDARY LAYERS

Marco Atzori
Department of Particulate Flow Modelling
Johannes Kepler University
4040 Linz, Austria
marco.atzori@jku.at

Alexander Stroh & Davide Gatti
Institute of Fluid Mechanics (ISTM)
Karlsruhe Institute of Technology
76131 Karlsruhe, Germany

Koji Fukagata
Department of Mechanical Engineering
Keio University
223-8522 Yokohama, Japan

Ricardo Vinuesa & Philipp Schlatter
SimEx/FLOW, Engineering Mechanics
KTH Royal Institute of Technology
100 44 Stockholm, Sweden

ABSTRACT

most relevant phenomena. Turbulent boundary layers (TBLs)
subjected to strong adverse pressure gradients (APGs) are oneof such cases. The appearance of many contributions withhigh values can also become problematic in describing control
effects. The skin-friction reduction due to uniform blowing
and skin-friction increase due to suction are convincingly
explained by the turbulent-fluctuation contribution on ZPG
TBL (Kametani et al., 2015). When control is applied on theintense APG TBL developing over a wing section, however,many terms are significantly impacted (Atzori et al., 2021). A
similar outcome is also obtained using the RD decomposition(Renard & Deck, 2016), as shown by Fan et al. (2022) onwing boundary layers with control. The RD decomposition isderived to obtain better scaling than the FIK identity when thecontributions are expressed in inner units, but the increase ofmany relevant terms in cases that are not homogeneous in thestreamwise direction remains an issue. The aim of the present
paper is to derive an alternative form of the FIK identitythat can provide results with easier interpretation in case ofstrongly non-homogeneous flows.

Skin-friction decompositions such as the so-called FIK
identity (Fukagata et al., 2002) are useful tools in identifying relevant contributions to the friction, but may also leadto results difficult to interpret when the total friction is recovered from cancellation of multiple terms with large values. We propose a new formulation of the FIK contributions
related to streamwise inhomogeneity, which is derived fromthe convective form of the momentum equation and using theconcept of dynamic pressure. We examine turbulent boundary layers subjected to various pressure-gradient conditions,
including cases with drag-reducing control. The new formulation distinguishes more precisely the roles of the free-streampressure distribution, wall-normal convection, and turbulent
fluctuations. Our results allow to identify different regimes
in adverse-pressure-gradient turbulent boundary layers, corresponding to different proportions of the various contributions,and suggest a possible direction towards studying the onset ofmean separation.

INTRODUCTIONDATA SETThe
skin-friction
coefficient,denotedbyc f = 2τw /(ρU∞2 ), is a key physical quantity in wallbounded flows. In this expression, the wall-shear stressis τw = ρν(dUt /dyn )yn =0 , and (1/2)ρU∞2 is the dynamic
pressure in the free stream. The subscripts t and n denote
the wall-tangential and wall-normal directions, respectively.
Hereafter, mean quantities and fluctuations are definedaccording to the Reynolds decomposition ũ = U + u, where
the averaging operator is U = ũ. Various identities that
express c f as a combination of different contributions havebeen derived, with the aim of explaining the mechanisms thatgenerate friction. One example is the so-called FIK identity(Fukagata et al., 2002), which is the focus of the present
paper. The results are relatively straightforward for canonicalflows such as periodic pipes and channels, where most ofterms in the mean governing equations vanish. However, inmore complex turbulent flows, skin-friction decompositions
can lead to many contributions of much higher values than
the total c f , which makes it more challenging to identify the

We focus on the high-resolution large-eddy-simulation(LES) data sets for ZPG and APG TBLs over flat plate presented by Eitel-Amor et al. (2014) and Bobke et al. (2017),
respectively, to describe pressure-gradient effects. These simulations were performed using the spectral code SIMSON(Chevalier et al., 2007). High-resolution LESs of the flow
around a NACA4412 airfoil at a Reynolds number based on
the chord length of Rec = 400, 000 and angle of attack 5 degrees, including the case with uniform blowing for drag reduction (Vinuesa et al., 2018; Atzori et al., 2021), are also
considered to describe control effects. These simulations were
performed using the spectral-element code Nek5000 (Fischer
et al., 2008). In both cases, the LES filter is based on therelaxation-term approach described by Schlatter et al. (2004).
In the airfoil simulations, only a portion of the suction side isconsidered in the present study, namely between x/c = 0.24
and x/c = 0.86, where uniform blowing is applied in the control case (note that x/c is the distance from the leading edge

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Range of pressure-gradient parameter (β ), Reynolds
number based on the momentum thickness (Reθ ), and friction
Reynolds number (Reτ ) in the considered data set. Note that
WSS and WSSB denote a portion of the wing suction side
without and with uniform blowing, respectively.

Case

β

Reθ

Reτ

ZPG

≈0

[541, 2272][218, 780]

APG1[0.2, 1, 1][556, 3249][222, 798]

APG2

[0.3, 1.6]

[569, 3400][221, 827]

WSS

[0.1, 9.6]

[450, 2155][179, 363]

WSSB

[0.1, 17][450, 2455][175, 335]

lated to theboundary-layer thickness (where Reδ = Ue δ99 /ν);R R
cTf = −4 01 (1 − η)uv dη is the turbulent-fluctuation contribution; cPf = −2 01 (1−η)2 (∂ P/∂ x) dη is the contribution thatR

includes the pressure gradient; and cDf = −2 01 (1 − η)2 Ix dηincludes the remaining terms of Eq. (1), which are zero forflows that are homogeneous in the wall-tangential direction,
i.e.:
R

Ix =

Ix∗ = U

1 2
U + P = P0
2

(3)

⇒

U

∂U
∂P
=−
,
∂x
∂x

(4)

where q = (1/2)U 2 is the dynamic pressure, and the total pressure, denoted by P0 , is constant. The same expression is also
obtained in the context of the boundary-layer approximation,and it shows a connection between cD1∗
and cPf . We then def
fine two new contributions as:

The FIK identity is derived from the mean-momentum
conservation in the wall-tangential direction, which can be
written in conservative form as:∂ 2U ∂ 2U
+ 2
∂ y2
∂x

∂U ∂ (u2 )∂U
1 ∂ 2U
+
+V
−
.
∂x
∂x
∂ y Reδ ∂ x2

This formulation leads to different contributions related theD1mean convection, denoted by cD1∗
and cD3∗
f
f , instead of c f
D3
D1∗
D3∗
and c f . The definitions of c f and c f suggest that it ispossible to rearrange the contributions in the following way.
In the irrotational region above the boundary layer, theBernoulli and Euler equations (approximately) hold:

SKIN-FRICTION CONTRIBUTIONS∂ (U 2 ) ∂ (UV )
∂P
+
=−
+ν
∂x
∂y
∂x

(2)

The integration variable, η = y/δ99 is the wall-normal coordinate normalised with the boundary-layer thickness. The
contribution cDf can be further decomposed into four terms:
D4
D2
D3
cDf = cD1
f + c f + c f + c f . These four additional contributions correspond
to
the
four
terms included in Ix , for instanceR1
2 (∂U 2 /∂ x) dη.
cD1
=
−2
(1
−
η)
0
f
D2
The first two contributions, cD1
f and c f , are directly related to the flow development in the wall-tangential direction;
D4
cD3
f is related to mean wall-normal convection; and c f is related to viscous dissipation due to streamwise inhomogeneity.
These four terms are not of the same order of magnitude beD2
D4
D3
cause cD1
f and c f tend to be much larger then c f and c f , sothat their balance determines the streamwise development ofcDf .
An alternative form of the skin-friction contributions canbe obtained if the triple integration by parts is performed on
the convective form of Eq. (1). In this case, the inhomogeneity
term Ix becomes:

measured on the chord, normalised with the chord length).
The following relevant boundary-layer parameters are reported in Table 1 for all cases examined in the paper. The
Clauser pressure-gradient parameter (Clauser, 1956) is computed as β = δ ∗ /τw dP/dxt |e . In this expression, δ ∗ is the displacement thickness, and dP/dxt |e is the pressure gradient at
yn = δ99 , where δ99 is the location of edge velocity. The valueof Ue , and thus δ99 , δ ∗ and θ , are evaluated consistently for theentire data set using the method based on the diagnostic scaling proposed by Vinuesa et al. (2016). The Reynolds number
based on the momentum thickness is defined as Reθ = θUe /ν
and thepfriction Reynolds number is defined as Reτ = δ99 uτ /ν(uτ = τw /ρ). Note that the choice of different ranges of Reθ
between the ZPG and APG TBL over the flate plate is a direct consequence of the different pressure-gradient conditions.
In particular, a positive value of β results in a higher θ andthus higher Reθ . A similar effect is also observed for uniform
blowing, in the comparison between reference (WSS) and controlled (WSSB) wing simulation. The behaviour of the friction
Reynolds number, on the other hand, is more complex, because
APG and uniform blowing increase the boundary-layer thickness but also reduce friction at the same time. In the flat-platecases considered here, the net effect of the pressure gradient isthat of increasing Reτ , but on the suction side of the wing, atvery high values of β (Vinuesa et al., 2018) or in the case of
blowing (Atzori et al., 2021), Reτ eventually decreases. Dueto this effect, Reτ cannot be considered a reliable indicator ofscale separation in cases with intense APGs.

D1∗
cDP+ cPf = −2
f = cf∂ uv ∂ u2
+
.
∂y
∂x
(1)
In this expression, U = Ut and V = Vn denote the walltangential and wall-normal mean velocity components, respectively, and x = xt and y = yn denote the wall-tangential andwall-normal coordinates, respectively. Note that we will focus on cases where wall-curvature effects are absent or negligible. Applying a triple integration by parts to Eq. (1),one can obtain: c f = cδf + cTf + cDf + cPf . In this expression,∂ (U 2 ) ∂ (u2 ) ∂ (UV )1 ∂ 2U
.
+
+
−
∂x
∂x
∂y
Reδ ∂ x2



+

D3∗
cDV
= −2
f = cf

Z 1
0

Z 1



∂U ∂ P+dη,(5)(1 − η)2 U∂x∂x0(1 − η)2V

∂U
dη.
∂η

(6)

The first one can be written in terms of the contributions deD1
P
rived from the conservative form as cDP
f = (1/2)c f + c f ,DV
D1
D3
while the latter can be written as c f = (1/2)c f + c f .
The sign of the cDP
f contribution shows in which direction the turbulent boundary layer departs from the conditions
under which the Bernoulli equation holds. Note that we willcδf = 4(1 − δ99 /δ ∗ )/Reδ denotes a contribution directly re-2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
contribution. The positive sign of cDP
f also shows that the dynamic pressure is decreasing faster within the turbulent boundary layer than in a corresponding irrotational flow.

focus on APG TBLs, where ∂ P/∂ x > 0. In the free stream, theintegrand of cDP
f is zero. In the case that U(∂U/∂ x) > ∂ P/∂ x,i.e. the rate of change in the streamwise direction of the dynamic pressure is higher than in a corresponding irrotational
flow, cDP
f is negative, leading to skin friction reduction. On theother hand, if U(∂U/∂ x) < ∂ P/∂ x, i.e. the rate of change ofthe dynamic pressure is lower than in a corresponding irrotational flow, then cDP
f is positive. Note that these inequalities
do not necessary hold at every wall-normal distance becauseof the integration along η, but are indications of the generaltrend.
The contribution that includes the wall-normal convection, cDV
f , can be further decomposed by introducing the mean
spanwise vorticity (denoted by Ωz = ∂V /∂ x−∂U/∂ y): cDVf =APG TBLs exhibit a more complex behavior and cPf , cD1
f ,have
absolute
valuesmuch
higherthanc.
Thebaland cD3
ff
ance between these terms is not obvious, at a first sight. InP
D3fact, cD1
f is positive while c f and c f are negative, and they all
become rapidly more relevant in terms of relative contribution
to c f as β increases. Because all terms connected to streamwise inhomogeneity have similar evolution, it is not possible
to readily distinguish different regimes for the various valuesof β . The high magnitude of these contributions also suggests
the counter-intuitive conclusion that turbulent fluctuations playa only minor role in determining friction, further highlighting
the necessity of re-grouping different terms.

cDVV
+ cDΩ
f . The first and second term in this expression inf
clude V (∂V /∂ x) and −V Ωz , respectively:

If the convective formulation is used, cTf becomes themost relevant contribution. Comparing its absolute magnitudein APG and ZPG (not shown here), cTf is even higher in APG
than in ZPG TBLs at similar Reθ or Reτ , despite the lower c f
in the former. This fact is due to the intense turbulent fluctuations in the outer region of the APG TBLs. Both dynamicpressure and vorticity-convection contributions have lower intensities than cTf , contrary to the inhomogeneity contributionsof the conservative FIK formulation. The dynamic-pressurePcontribution cDP
f describes directly the balance between c f and
D1DΩ
c f , and the contribution c f is always negative and rapidly
becomes smaller in the region of quickly-increasing β , butlater increases (i.e. its modulus decreases, thus approachingzero from negative values). The streamwise development ofDΩ allows to identify different streamwise regions,cDP
f and c fcorresponding to changes in the development of the Clauserpressure-gradient parameter β and the total c f . At low valuesof β , cDP
f is in good agreement between APG and ZPG. As βcontinues to increase, cDP
f decreases up to changing sign and it
eventually reaches an equilibrium value if β is approximately
constant for a portion of the domain that is long enough. Thechange of sign of cDP
f happens to be at a similar location as
the maximum value of β (which is the location examined inTable 2). Note that the vorticity-convection contribution is immediately affected by the change of β , causing the decreaseof c f in the first portion of the flat plate. Only downstream,
the dynamic-pressure contribution also contributes to reducing c f . The sign of cDP
f shows that the TBL is moving from astate where the dynamic pressure is increasing at a lower rate
than for a corresponding irrotational flow, to a state where thedynamic pressure is increasing at a higher rate.

Z 1



∂V
2
cDVV
=
−2
(1
−
η)
V
dη,
f
∂x
0
cDΩ
= +2
f

Z 1
0

(1 − η)2V Ωz dη.

(7)
(8)

The introduction of these new terms is motivated by the factthat ∂U/∂ y tends to be much larger than ∂V /∂ x in TBLs, soDΩ
is negligible and cDVthat cDVV
f can be approximated with c f
f
in most cases. Assuming that V is positive, the sign of cDΩ
fis negative. The sign of cDΩ
shows
that
wall-normal
convecftion is always associated with skin-friction reduction, which
is a natural consequence of the fact that it reduces near-wallmomentum.

RESULTSIn the present section, we will examine contributions tothe skin friction, as evaluated from the two formulations of the
FIK identity discussed above. Our objective is to illustrate how
the convective formulation is more sensitive to the streamwise
development of the flow than the conservative one, and more
effective in highlighting the relevant contributions. Later on,
the more detailed information obtained with the new formulation is used to discuss how increasingly intense APGs maylead to separation.

Flat-plate boundary layers
We first consider the ZPG TBL at friction Reynolds numbers between Reτ ≈ 200 and 570, and the two APG TBLs atsimilar Reτ but under different pressure-gradient conditions,as shown in Fig. 1. In this figure, we focus on the most relevant relative FIK contributions, which are normalised with thetotal c f . In the ZPG TBL, relative skin-friction contributions
exhibit only a weak dependence on the Reynolds number as
x increases. The turbulent term is the most relevant one, andmaintains a value of cTf ≈ 70%c f (values at x = 750 are reported in Fig. 2). If the conservative FIK formulation is used,cPf vanishes, but the two contributions including the mean velocity components have still magnitudes of cD1
f ≈ 48%c f and
cD3
≈
−28%c
.
If
the
convective
formulation
is used, the conf
ftribution cDΩ
isnegligible,and
thisresultimmediatelyshowsfthe limited relevance of wall-normal convection in the ZPG
case. At the same time, cDP
f ≈ 24%c f so that the change ofdynamic pressure is identified as the dominant inhomogeneity

An interesting observation is possible from both FIK formulations, which is that a significant change of state of theboundary layer occurs not as a result of β reaching its maximum value, but rather as a results of the change of rate at
which β is increasing before reaching its maximum, at a lower
x. For both β distributions, the relative contributions due toconvection peak before the maximum β , where c f is rapidly
decreasing as compared to the ZPG case. The relative contributions later decrease and, at much higher x, where β is almost
uniform and decreasing, they eventually reach a more stablevalues. The conservative formulation provides a precise description of the transitional regime between these two states,
which is captured by cDV
f . The relatively long transition between the two states that appears in Fig. 1, compared with theβ distribution, is another facet of history effects (Bobke et al.,2017).

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

Figure 1. Comparison between ZPG and (top row) APG1 and (bottom row) APG2. Solid and dashed lines denote the ZPG and APG
data, respectively. (Left column) Clauser pressure-gradient parameter, denoted by β . (Center-left column) Skin-friction coefficient,denoted by c f . (Center-right column) Relative skin-friction contributions from the conservative formulation: ( , cTf /c f ) turbulent
D3
fluctuations, ( , cD1
f /c f ) mean-velocity streamwise-development, ( , c f /c f ) mean wall-normal convection of momentum, and ( ,Pc f /c f ) pressure-gradient. (Right column) Relative skin-friction contributions from the convective formulation: ( , cTf /c f ) turbulentDΩ
fluctuations, ( , cDP
f /c f ) dynamic pressure, and ( , c f /c f ) mean wall-normal convection of vorticity. The turbulent-fluctuation
D1
P
DV
D1
D3
contribution is the same in both the center-right and right columns. Note that cDP
f = 1/2c f + c f and c f = 1/2c f + c f . The circles
denote the location with highest β . Notice the different scales in the right-hand panels.
Table 2. Relative skin-friction contributions at a selected location for the ZPG, APG1, and APG2. For the APG1 and APG2, thevalues are at the location of maximum β (highlighted in Fig. 1). For the ZPG, at x = 750. The red square denote the most intense
relative contribution or relative control effect.

Wing boundary layers and control effects

x/c = 0.75 (Fig 3), and even higher values further downstream.

A qualitative difference between how both FIK formulations describe pressure-gradient effects is also evident in theidentification of the most relevant contribution in TBL developing over the NACA4412 suction side (Figs. 2 and 3).
Due to the relatively high values of the pressure gradientand the mean-velocity components, turbulent fluctuations appear again to have a secondary role in determining the value ofthe total c f , if the conservative formulation is employed. The
cause for decreasing c f is also not directly evident because theD3three highest contributions, cTf , cD1
f , and c f , have all a similar
streamwise development and are much larger than c f . For instance, cD1
f reaches a value of more than 9 times the total c f at

On the other hand, if the convective formulation is used,cTf becomes the skin-friction contribution with highest absolute value over most of the profile, up to the region of verystrong APG at x ≈ 0.8. In this case, cDP
f is always positiveand follows the development of β . Subsequently, the decreaseof c f , which would eventually lead to separation for a higher
angle of attack, is entirely due to the enhanced vorticity convection in the wall-normal direction. This phenomenon is notdirectly evident when the conservative FIK formulation is considered.
The streamwise development of the relative contributions
on the airfoil highlights similarities with the first portion of4

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

Figure 2. Comparison between (solid lines) suction side of a NACA4412 at Rec = 400, 000, and (dashed lines) the same case withuniform blowing. Color code for skin-friction contributions as in Fig. 1. The circles denote the streamwise location x/c = 0.75.
Table 3. Skin-friction contributions and control effects normalised with the total c f at x/c = 0.75 on suction side of the airfoil, at thelocation highlighted in Fig. 2. The red squares denote the most intense relative contribution or relative control effect.

the flat plate, where β is also rapidly increasing. In particular,DΩcDP
f /c f is positive and c f /c f becomes progressively lower.
However, contrary to the flat-plate cases, β is increasing overthe entire streamwise extent, eventually leading to both cDPfand cDΩ
having a higher absolute value than cTf , which wasf
not observed for the flat plate, where β does not increase fastenough and for a long enough portion of the domain. Theseobservations were not obvious using the conservative FIK formulation, due to the high values of all terms, which makes
more difficult to distinguish different state of the boundary
layer.
The new FIK formulation also simplifies the descriptionof control effects, here discussed in terms of control using walltranspiration. In fact, uniform blowing is known to reduce skinfriction and has non-linear interactions with APG (see e.g.Eto
et al., 2019; Atzori et al., 2021). Applying the FIK decomposition, blowing significantly increases cTf and cD3
f , and decreases
P . In particular, cD1 and cD3 are increased and decD1
and
c
f
f
f
f
creased, respectively, by an amount which is almost equal toDΩ
2c f . However, the modifications of cDP
f and c f are less dramatic, so that the conservative formulation allows to identify
immediately that the skin-friction reduction is due to the increased mean-vorticity convection, which counterbalances theincrease of dynamic-pressure and turbulent contributions.

tle reduction of c f due to the boundary-layer development innear-equilibrium conditions Bobke et al. (2017). On the other
hand, if c f is obtained as a result of cancellation of terms with
opposite sign and changing at different rates, the corresponding relative contributions will progressively increase. In fact,
contributions that are decreasing slower than c f or increasing
will virtually diverge at c f = 0. Under the conditions of adverse pressure gradient, the c f reduction is indeed the result ofa change of absolute as well as relative skin-friction contributions, i.e. a modification of how different phenomena concur indetermining the total friction. In particular, we identified threedifferent regimes: 1) rapidly increasing values of β , where cDPf
is positive and increasing but cDΩ
isdecreasing(i.e.
becomingfmore negative) faster than cDP
f increases, leading to a rapid
reduction of c f ; 2) decreasing values of β , where both cDPfand cDΩf are negative but with approximately constant relative
contributions, associated with a slow reduction of c f ; and 3)
an intermediate region, with values of β that are either slowlyDΩ
increasing or decreasing, where cDP
f and c f tend to decrease
and increase, respectively, towards their values in regime 2).
The similarities between the second regime and the ZPG
case suggest that mean separation will not occur under these
conditions. The data extracted from the wing boundary layer,on the other hand, suggest that separation will occur in a state
similar to that of the first regime. We can thus formulate thehypothesis that in the specific case considered here, namely
β increasing from a low value as the boundary layer develops, there exists a critical rate of change of β above which
the adverse pressure gradient will eventually cause separation,and below which separation will not occur. This hypothesis isalso corroborated by the fact that turbulent boundary layers at
higher Reynolds number are less sensitive to pressure gradients. There is thus a distinction between a TBL that is leading

Path towards separation
In principle, the onset of mean separation, corresponding
to c f = 0, can be approached from any combination of relative contributions. In the particular case in which all terms
decrease at the same rate, c f will also decrease accordingly.
This is what we observed in the ZPG case, where the relative contributions are approximately constant, despite the gen-5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022of the relative contributions. It may be worth to further develop these ideas, examining a larger sample of data. A refined
derivation of the new formulation of FIK identity will be also
considered, to highlight curvature effects. A similar aggregation between contributions may also be introduced in the RD
decomposition (Renard & Deck, 2016).

towards separation, where the rate at which β is increasing isfast enough to overcome the growth of the boundary layer, andthe opposite case where, even though β remains positive andstill increasing, the boundary layer approaches a state progressively farther from separation.
Key questions to address in this description of the path
towards separation remain: 1) whether it is actually enoughto observe the onset of the intermediate regime, to determinethat separation will not occur, and 2) what is the impact of theinitial value of β . In fact, in cases such as that of an initiallyvery high but constant or even decreasing β , the increase of
Reynolds number may not be enough to avoid separation.

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Flow 72, 86–99.

CONCLUSIONS AND OUTLOOKWe described a new formulation of FIK skin-friction decomposition with contributions that are related to streamwise
inhomogeneity, and we discussed how the results of the decomposition change in TBLs subjected to various pressuregradient conditions, including one flow-control case with uniform blowing. The new formulation is derived from the convective form of the mean streamwise momentum conservation,and the concept of dynamic pressure is used to re-group thepressure gradient and wall-tangential derivatives. The implicit
cancellations in the definition of the new dynamic-pressureand vorticity-convection contributions allow these two terms to
be more sensitive to pressure-gradient conditions, and to identify immediately the main phenomena that govern the development of c f . In particular, the new formulation distinguishes
more easily between adverse-pressure gradients of different
intensities, and between mean-flow and turbulent-fluctuations
effects.
We identified that APGs can lead to different states ofthe boundary layers, depending on the flow history as wellas the value and the local rate of the change of the pressuregradient parameter, β . Intense and quickly-increasing β leads
to a regime where the dynamic-pressure contribution is positive and increasing, but such an increase is counteracted by thefast decrease of the vorticity-convection contribution, resulting
in a fast reduction of c f and eventually to separation. A uniform or decreasing β instead, causes both dynamic-pressureand vorticity-convection contributions to be negative but almost uniform, resulting in a lower reduction rate for c f . This
regime is in fact similar to that of a ZPG TBL. The trends observed on flat-plate and wing TBLs suggested that it is possible
to employ the skin-friction contributions to study how APGslead to separation. Finally, this study confirm once again theimportance of history effects in determining the local state of
the flow, as clearly shown by the postponed evolution of theskin-friction contributions, with respect to that of β .
A few consideration are left for future studies. Our results
suggest that the rate of the change of β is critical in determining whether a progressively stronger adverse pressure gradientcan cause separation, and that the state of the flow that precedes separation is characterised by specific rates of change

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