Scaling and statistics of large-defect adverse pressure gradient turbulent boundary layers

International Journal of Heat and Fluid Flow 59 (2016) 109–124

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Scaling and statistics of large-defect adverse pressure gradient turbulent boundary layers A.G. Gungor a,∗, Y. Maciel b, M.P. Simens c, J. Soria d a

Faculty of Aeronautics and Astronautics, Istanbul Technical University, Maslak, Istanbul 34469, Turkey Department of Mechanical Engineering, Laval University, Quebec City, QC G1V 0A6, Canada c School of Aeronautics, Universidad Politécnica de Madrid, Madrid 28040, Spain d Department of Mechanical and Aerospace Engineering, Monash University, Melbourne 3800, Australia b

a r t i c l e

i n f o

Article history: Received 6 October 2015 Revised 11 March 2016 Accepted 16 March 2016

Keywords: Direct numerical simulation Adverse pressure gradient Turbulent boundary layer

a b s t r a c t The purpose of this article is to test similarity laws and scaling ideas, as well as characterize turbulence behaviour of large-defect adverse-pressure gradient turbulent boundary layers using six experimental and numerical databases including a new direct numerical simulation of a strongly decelerated non-equilibrium turbulent boundary layer. In the latter flow, at a moderate Reynolds number, the mean velocity profiles depart from the classical law of the wall throughout the inner region including in the viscous sublayer and they do not follow the log law. However, the agreement is excellent with the extended law of the wall that accounts for the pressure gradient for the viscous sublayer. The Reynolds stress components are not self-similar in the viscous sublayer when the velocity defect is important, but they scale reasonably well with the pressure-viscous scales. Detailed comparisons of the six different flows are made in the outer region. In order to do such comparisons, an outer region velocity scale analogous to the commonly defined free shear layer velocity scales is introduced. It is found that the investigated one-point velocity statistics in the upper half of large-defect boundary layers resemble those of a mixing layer: mean velocity defect, Reynolds stresses, turbulent kinetic energy budgets, uv correlation factor and structure parameter −uv/2k. The dominant peaks of turbulence production and Reynolds stresses are located roughly in the middle of the boundary layer. The profiles of the uv correlation factor reveal that u and v become less correlated throughout the boundary layer as the mean velocity defect increases, especially near the wall. The structure parameter is low in the large-defect disequilibrium boundary layers, similar to large-defect equilibrium flows and mixing layers and decreases as the mean velocity defect increases. All large-velocity-defect boundary layers analysed are found to be less efficient in extracting turbulent energy from the mean flow than zero-pressure-gradient turbulent boundary layers, even throughout the outer region. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Wall-bounded turbulent flows are important to understand in order to improve energy efficiency for a wide range of machines and systems associated with these turbulent flows. An important sub-group of wall bounded turbulent flows are adverse pressure gradient (APG) turbulent boundary layer (TBL) flows. These flows are found for instance around surfaces with curvature, as encountered in many aerodynamic applications such as airplane wings, cars and turbomachinery. Although a significant amount of re-

∗

Corresponding author. Tel.: +90 2122857352. E-mail address: [email protected] (A.G. Gungor).

http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.03.004 S0142-727X(16)30029-7/© 2016 Elsevier Inc. All rights reserved.

search has been devoted to understanding channel flows, pipe flows and zero pressure gradient (ZPG) turbulent boundary layer flows, which has led to a consistent theory of these canonical wall flows, the same cannot be said for APG TBLs. There exists nonetheless a wealth of theoretical, experimental and numerical studies on APG TBLs, many of which are summarized below. Among many things, these studies clearly demonstrate that the fundamental problem in APG boundary layer flow research is the lack of a recognized theoretical framework and, consequently, a lack of well-thought laboratory and numerical experiments based on such a framework. A clear and agreed understanding of which parameters are paramount for the development of the APG boundary layer does not yet exist. An overview will be presented of the most significant ideas and results which are consequential to the study reported in this article.

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The velocity components of the boundary layer flow are denoted by u, v, w in the respective spatial coordinates x, y, z for the streamwise, wall-normal and spanwise directions, respectively. For the sake of the discussion and to retain generality, the length and velocity scales of the inner and outer regions of the boundary layer are left unspecified for the moment and are denoted respectively as Li , Ui and Lo , Uo . We will limit the discussion to two regions because we can neither confirm nor refute that a third, intermediate region may exist for some categories of APG TBLs; the layer-structure and scaling of the TBL is discussed further on. In addition to the presence and sign of the pressure gradient, pressure gradient TBLs can be further distinguished with two important characteristics of shear layers: (1) the importance of the mean velocity (momentum) defect and (2) the state of dynamic equilibrium. The mean velocity defect, Ue − U (y ), where Ue is the external velocity at the edge of the boundary layer, reflects the local state of the boundary layer as a consequence of the upstream history of the flow. It gives an indication on the strength and distribution of mean shear rates and hence partly also on the local characteristics of momentum transfer and turbulence behaviour. In that respect, large-velocity-defect TBLs resulting from a strong or a prolonged adverse pressure gradient are quite distinct from ZPG TBLs and small-defect APG TBLs. In their case, mean shear rates in the outer region are no longer small in comparison to their near-wall counterparts while near the wall, the importance of viscous forces and of the wall shear stress diminishes. As a result, the near-wall turbulent kinetic energy production peak is absent or very small and the main production peak is found in the outer region of the flow (Elsberry et al., 20 0 0; Na and Moin, 1998; Skåre and Krogstad, 1994). In the case of very large defect boundary layers, for instance near separation, turbulence activity is almost absent near the wall (Elsberry et al., 20 0 0; Maciel et al., 2006b; Na and Moin, 1998; Skåre and Krogstad, 1994; Skote and Henningson, 2002) essentially because mean shear is negligible there. The shape factor H = δ ∗ /θ , where δ ∗ and θ are the displacement and momentum thicknesses respectively, and Clauser shape factor (Clauser, 1954), that is based on similarity theory and that is not as dependent on Reynolds number, are possible indicators of the importance of the mean velocity (momentum) defect. The other important characteristic is the state of dynamic equilibrium of the boundary layer. Most real TBLs are complex because under the influence of external factors such as the pressure gradient they depart from dynamic equilibrium, in the sense that there exists a streamwise variation of the relative importance of each force acting on the flow, e.g. inertia, pressure and viscous forces. In the traditional theory of equilibrium TBLs (Clauser, 1954; Rotta, √ 1950), the friction velocity uτ = τw /ρ , where τ w is the wall shear stress and ρ is the density, is implicitly assumed to be the outer velocity scale. Maciel et al. (2006a) have recast the theory in more general terms by avoiding the assumption of a priori outer scales. They showed that the main parameter that characterizes the impact of the pressure gradient on the outer region of all types of PG TBLs is a generalized form of the Rotta–Clauser’s pressure gradient parameter

βo = −

Lo dUe . Uo dx

(1)

This assumes that appropriate outer scales can be found to represent all types of TBLs (see discussion on scales below). A pressure gradient parameter also needs to be defined for the near-wall region as will be done subsequently, but it suffices for the moment to discuss the outer one. If the pressure gradient parameter β o remains constant, then the TBL is in equilibrium or quasi-equilibrium in its outer region (equilibrium is not necessarily complete because at finite Reynolds number, a constant β o is not the only condition necessary for similarity as described in Maciel et al., 2006a). In the

case of non-equilibrium TBLs, an increase (decrease) in β o leads to an increase (decrease) of the mean velocity defect. Large velocity defect and boundary layer separation can therefore be due to a sharp streamwise positive gradient of β o , even if they occur downstream in a region where dβ o /dx is no longer positive, or from a prolonged mild positive dβ o /dx. Besides studies on equilibrium TBLs, such as Clauser (1954), Stratford (1959), East and Sawyer (1980), Skåre and Krogstad (1994), and Lee and Sung (2009), there have been many studies in the past of TBLs subjected to favourable or adverse pressure gradients that lead to non-equilibrium conditions, reviewed for instance in Skote and Henningson (2002) and Maciel et al. (2006b). In most of these studies, the pressure distribution was chosen rather arbitrarily from a theoretical viewpoint, often with a specific practical application in mind. It therefore resulted in random and sometimes complex streamwise evolutions of β o whatever the assumed outer scales (Maciel et al., 2006a). It is not often recognized that the state of the boundary layer is directly attributable to the streamwise variation of β o (and β i ) and not simply to the presence of a pressure gradient. As a result, even if we have an overall knowledge of the main effects of the pressure gradient on TBLs, we do not fully understand them. Moreover, we are still unable to make a clear distinction between the local effects of pressure gradient and those resulting from the upstream history on the non-equilibrium evolution of a TBL. It is also important to discuss the mean flow structure, the various similarity laws and the scaling of TBLs subjected to pressure gradients since there exist several interpretations and theories. In the case of canonical wall-bounded turbulent flows and small-defect TBLs, a two-region structure (inner/outer) further subdivided into four layers (viscous sublayer, buffer layer, overlap layer and defect layer) is traditionally accepted even if it is still debated (Klewicki, 2010; Marusic et al., 2010). The outer region, which includes the overlap and defect layers, is usually considered to be the upper layer of the turbulent boundary layer (or pipe or channel flow) where viscous momentum transfer is negligible with respect to turbulent momentum transfer. In the viscous sublayer, the classical law of the wall for the wall-normal distribution of mean velocity is obtained by assuming that Reynolds stresses are negligible compared to viscous shear stresses

U+ =

U = y+ uτ

(2)

In the overlap layer, the matching of the inner and outer expressions for the total shear stress is usually performed in a manner that leads to the traditional log law

U+ =

1

κ

ln y+ + B

(3)

where κ is the von Karman constant. For large-defect TBLs, the mean momentum and turbulent energy balances are completely different and a widely accepted layer representation does not exist. By using asymptotic theory, Melnik (1989), Durbin and Belcher (1992), Bush and Krishnamurthy (1992) and Scheichl and Kluwick (2007) have each proposed different types of three-layer structures (inner/intermediate/outer). Other researchers, using asymptotic theory or dimensional and physical arguments, suggest various two-layer structures that represent either a gradual or an abrupt shift from the canonical structure as the velocity defect or pressure gradient increases (Afzal, 1983; 1996; Kader and Yaglom, 1978; Perry et al., 1966; 2002; Perry and Schofield, 1973; Skote and Henningson, 2002; Stratford, 1959; Townsend, 1961). In all these works, equilibrium or quasiequilibrium is usually assumed, together with various other assumptions. Some of these theories are tested in the present study. But since we cannot truly resolve the layer-structure issue, we use loosely

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here the terms “near-wall (inner) region” and “outer region” without giving them a precise definition and without defining their extent. One could however still define the outer region as the upper layer of the TBL where viscous momentum transfer is negligible with respect to all other mean momentum terms. Another important issue related to the layer structure of the TBL is to know when the classical law of the wall progressively starts not to hold anymore. In other words, what are the conditions of velocity defect or non-equilibrium that lead to a gradual breakdown of the law of the wall. Studies of equilibrium or quasiequilibrium TBLs as in Clauser (1954), East and Sawyer (1980), Skåre and Krogstad (1994), and Elsberry et al. (20 0 0) tend to confirm that the classical law of the wall holds in that case whatever the extent of the mean velocity defect, even for TBLs maintained on the verge of separation (Elsberry et al., 20 0 0). But if the boundary layer is not in dynamic equilibrium in its inner region, it is not expected by essence that a similarity law can hold, as noted already a long time ago by Clauser (1956). There are indeed several known cases of non-equilibrium APG TBLs, not necessarily close to separation, where the log law and other similarity laws have been found not to apply (Alving and Fernholz, 1995; Debisschop and Nieuwstadt, 1996; Dengel and Fernholz, 1990; Driver, 1991; Indinger et al., 2006; Kline et al., 1967; Maciel et al., 2006a; Monty et al., 2011; Na and Moin, 1998; Nagano et al., 1993; 1998; Simpson et al., 1981a; Skote and Henningson, 2002; Spalart and Watmuff, 1993). At the same time, other studies reveal that the log law still holds in milder non-equilibrium APG conditions (Coles, 1956; Marusic and Perry, 1995; Samuel and Joubert, 1974) or in part of the overlap region at high Reynolds number (Knopp et al., 2015). It is also difficult to assess accurately at what location from the wall the classical law of the wall starts to break down in laboratory experiments because near wall measurements are difficult to make. In the case of low Reynolds number DNSs, Na and Moin (1998) and Skote and Henningson (2002) showed that it breaks down even in the viscous sublayer in strongly decelerated flows. When the pressure force is a leading-order contributor to the streamwise momentum balance, many existing theories modify the classical law of the wall by incorporating the pressure gradient, see for instance Skote and Henningson (2002) or Nickels (2004). It usually leads to pressure-viscous scales being defined for the inner region of large-defect TBLs (Mellor and Gibson, 1966):



Ui = u pi = Li = L pi =

ν dp ρ dx

1/3

(4)

ν

(5)

u pi

An inner pressure gradient parameter can also be defined as in Mellor (1966)

βi =

 u 3 pi

uτ

=

ν dp ρ u3τ dx

(6)

In the viscous sublayer, by assuming that the Reynolds shear stress is negligible compared to the viscous shear stress and that momentum advection is negligible, an extended form of the law of the wall can be obtained (Patel, 1973):

U

pi

U 1  pi 2 = = y + u pi 2



uτ u pi

2

y pi

(7)

where y pi = yu pi /ν . Eq. (7) can also be expressed in friction-viscous units as

U + = y+ +



1 u pi 2 uτ

3

1 2

(y+ )2 = y+ + βi (y+ )2

(8)

The y2 term in expressions (7) and (8), which would be the only one left at separation, incorporates the leading effects of the

111

pressure gradient. Skote and Henningson (2002) found that their DNS data of an APG TBL and of a non-equilibrium separated TBL agreed with this extended law of the wall, suggesting that very near the wall, history effects can be neglected. However, the region in which the Reynolds shear stress is negligible compared to the viscous shear stress becomes infinitely small at separation. As mentioned previously, several similarity laws have been proposed for the overlap layer(s) of large defect TBLs. Under the assumption of a two-layer structure, one notable example is the following overlap law obtained by Afzal (1996) and later by Skote and Henningson (2002) using a totally different theoretical approach and assumptions:



+

U =

1

κ



+

ln y − 2 ln

1 + βi y+ + 1 2



+

2

κ

( 1 + βi y+ − 1 ) + B (9)

Townsend (1961) and Mellor (1966) had also derived equations that are fairly similar to this extended overlap law. A modified version of this law that approximately accounts for the role of nonlinear inertia also exists (Knopp et al., 2014; McDonald, 1969). In the ZPG case, when βi = 0, Eq. (9) reduces to the log law (3). In pressure-viscous units, the extended overlap law (9) becomes (Skote and Henningson, 2002)

U pi =

  2 γ   2 γ 2 + y pi + γ ln y pi − ln γ 2 + y pi + γ + C κ κ 1

(10) where γ = uτ /u pi . At separation, when γ = 0, Eq. (10) reduces to the half-power law obtained by several researchers

U pi =

2

κ

y pi + C

(11)

Note that the extended similarity laws (7)–(10) imply that a given flow is self-similar in its inner region only if β i remains constant. Another closely related issue is the choice of appropriate length and velocity scales for the different layers of large-defect TBLs. Scales for the inner region have already been considered above. For the outer region, most researchers use the boundary layer thickness, δ , as the outer length scale of the ensemble-averaged flow variables for all pressure gradient conditions. Another pertinent length scale would be a generalized version of the Rotta–Clauser length scale (Maciel et al., 2006a; Rotta, 1950): = δ∗Ue /Uo where uτ has been replaced by Uo in order to retain generality. If the Zagarola–Smits velocity scale is used as the outer velocity scale Uo (see below), then is equal to δ . Note that from a theoretical viewpoint, it can be argued that the displacement and momentum thicknesses are not acceptable length scales since it is expected that the ratio of these thicknesses to the boundary layer thickness vanishes in the limit of an infinite Reynolds number, at least for canonical wall flows. The question of the outer velocity scale is more complex. It should scale the mean velocity defect, Ue − U, and hence in combination with the outer length scale, it should represent the order of magnitude of the outer mean shear rates: Uo /Lo ∝ dU/dy. It might therefore also scale the Reynolds stresses. For canonical turbulent wall flows, the most commonly used outer velocity scale is the friction velocity uτ . However for large-defect TBLs and TBLs in strong non-equilibrium state, the friction velocity does not always scale the mean velocity defect, nor the Reynolds stresses. It suffices to think of a TBL subjected to a prolonged positive streamwise gradient of β o . In such a flow, uτ gradually decreases towards zero in the streamwise direction while on the contrary the mean velocity defect and the Reynolds stresses in the outer region are increasing. Mellor and Gibson (1966) introduced a pressure-gradient-based outer velocity scale, u po = (δ ∗ /ρ )(d p/dx ) for the mathematical

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purpose of being able to analytically study equilibrium boundary layers with large velocity defects. Indeed, by using upo instead of uτ in their analysis, the singularity at separation due to uτ → 0 was removed. But the problem of using upo as a velocity scale is that a singularity now exists whenever d p/dx = 0 in a flow. Several researchers have proposed the external velocity Ue as the outer velocity scale for the mean velocity defect and for the Reynolds normal stresses. In many APG turbulent boundary layers in nonequilibrium conditions however, Ue clearly does not scale the mean velocity defect and the Reynolds stresses (Maciel et al., 2006b). But even in small defect TBLs Ue cannot be a valid outer scale since in the limit of an infinite Reynolds number, the outer velocity scale should become infinitely small with respect to the external velocity, that is Ue itself (Maciel et al., 2006a; Tennekes and Lumley, 1972). Ue is the equivalent of the high speed velocity in free shear flows. In both wall-bounded and free shear flows, the high speed velocity is not a measure of the velocity difference across the layer (outer layer in the case of wall-bounded flows), and hence of the mean velocity defect. Zagarola and Smits (1998) proposed an outer velocity scale proportional to the momentum deficit, namely Uzs = Ue δ ∗ /δ . In the case of canonical wall flows, Panton (2005) has clearly shown that the use of uτ and Uzs as an outer scale is asymptotically equivalent, although numerically distinct. By analysing nine very different APG flow databases, Maciel et al. (2006a) found that Uzs is an appropriate outer velocity scale for APG TBLs including very-large-defect TBLs. More recently, Logdberg et al. (2008) were able to scale the mean velocity defect profiles with Uzs for three different cases of thin separation bubble flows. Outer velocity scales as mentioned above are thus derived from a boundary layer point of view, however several researchers as Simpson et al. (1977), Driver (1991) and Na and Moin (1998) have observed that the mean velocity profile and certain turbulent statistics in the outer region of large-defect TBLs behave similarly to those of a plane mixing layer. Furthermore, Bradshaw and coworkers (Chandrsuda and Bradshaw, 1981; Wood and Bradshaw, 1982; 1984) discussed the influence of the wall on turbulent mixing layers concluding that although the influence of the v = 0 condition at the wall is substantial it takes a long distance for the free shear layer to develop into a boundary layer. The persistence of a free shear layer behaviour therefore also warrants the use of a typical free shear layer scaling for large-defect TBLs. In the present study, we introduce a free-shear-layer type outer velocity scale that allows us to compare boundary layers with plane mixing layers (see Section 3.2). Turbulence properties also differ considerably between canonical wall flows and large-defect TBLs. In canonical wall flows, an intense production peak of turbulent kinetic energy is found near the wall and can be related to a large extent to the coherent streaky near-wall structures resulting from a large mean shear (Robinson, 1991). In contrast, the near-wall production peak is absent or very small when the boundary layer has a large mean velocity defect (Na and Moin, 1998; Skåre and Krogstad, 1994). The main production and Reynolds stress peaks are found instead in the outer region around the middle of the boundary layer (Elsberry et al., 20 0 0; Maciel et al., 20 06b; Na and Moin, 1998; Nagano et al., 1993; Simpson et al., 1981b; Skåre and Krogstad, 1994). It is not clear however if the change in turbulence behaviour can be solely attributed to redistribution of the mean shear. Elsberry et al. (20 0 0) suggest that the flow is affected by an inflectional instability in the outer region similar to that of mixing layers. Indeed, large-defect velocity profiles have three inflection points and the highest one is approximately in the middle of the boundary layer, close to where turbulence production and Reynolds stresses have a maxima, like in mixing layers.

In the present work, similarity theories, scaling and statistical properties of large-defect APG TBLs are investigated with the help of six databases, including a DNS of a strongly decelerated nonequilibrium TBL. This DNS was designed to produce a flow with a large streamwise increase of both β i and β o and at the highest Reynolds number possible. The other databases are those of two other large-defect APG TBLs, one in equilibrium and one in strong disequilibrium, and two canonical shear layer flows: a plane mixing layer and a ZPG TBL. 2. Description of databases 2.1. Existing databases As previously mentioned, five existing databases available in the literature are considered. Two of them are of large-defect APG TBLs coming from laboratory experiments. In order to investigate the influence of the equilibrium state of the flow, both equilibrium and strong disequilibrium cases are chosen. The equilibrium APG TBL is that of Skåre and Krogstad (1994) (SK). It has a relatively large velocity defect (H ∼ = 2) and is at high Reynolds numbers. The nonequilibrium database (MRL) is of a strongly decelerated APG TBL that eventually separates (Maciel et al., 2006b). It has a pressure distribution similar to that of an airfoil at high angle of attack, with a pronounced suction peak upstream of the measurement section. The velocity defect increases substantially in the streamwise direction and the database includes data at separation. Comparisons are also made with ZPG TBLs coming from recent DNSs of Simens et al. (2009) and Sillero et al. (2013) which cover different Reynolds number ranges. Finally, the single-stream mixing layer of Wygnanski and Fiedler (1970) (WF) is also considered because, as discussed in the introduction, the velocity statistics of the outer region of large-defect boundary layers closely resemble those of the high speed side of a mixing layer (Driver, 1991; Na and Moin, 1998; Simpson et al., 1977). The data used is in the self-preserving region, corresponding to a Reynolds number of approximately 3 × 105 , based on the initial stream velocity and the distance to the flow origin. The key characteristics and parameter range of the boundary layer databases are given in Table 1. The new non-equilibrium APG flow (GMSS) is described in the next section. The pressure gradient parameter for the outer region β o (Eq. (1)) is expressed with the new outer velocity scale Um introduced in Section 3.2 (Eq. (14)): βm = −(δ /Um )(dUe /dx ). Note that the pressure gradient parameter based on the Zagarola–Smits scales βzs = −(δ /Uzs )(dUe /dx ) is also a valid pressure gradient parameter for the outer region. β zs and β m are not equivalent but their streamwise evolutions are qualitatively similar. This can be appreciated in Fig. 2(c) for the new nonequilibrium APG flow (GMSS). The traditional outer-region pressure gradient parameter, Rotta–Clauser’s one βτ = −( /uτ )(dUe /dx ), is also shown in Table 1 for reference. β τ assumes the outer region velocity scale to be uτ , which is not the case for large-velocity defect TBLs such as the ones considered in this study. 2.2. Direct numerical simulation The information provided by the existing databases is complemented by a DNS of a strongly decelerated non-equilibrium TBL. This flow is simulated in a parallelepiped domain over a smooth no-slip wall, with spanwise periodicity and streamwise non-periodic inflow and outflow. The numerical scheme used for the simulation is similar to the ones described in Simens et al. (2009) and Sillero et al. (2013). The Navier–Stokes equations are integrated using a fractional step method on a staggered grid, with third-order Runge–Kutta time-integration, fourth order

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Table 1 Boundary layer databases and the corresponding ranges of the parameters. Flow

Reθ

H

βm

βτ

Equilibrium APG TBL Skåre and Krogstad (1994) (SK) Non-equilibrium APG TBL Maciel et al. (2006b) (MRL) Non-equilibrium APG TBL present (GMSS) ZPG TBL Simens et al. (2009) (SJHM) ZPG TBL Sillero et al. (2013) (SJM)

39,0 0 0–51,0 0 0

1.99–2.01

0.092–0.094

19.6–21.4

3350–12,691

1.72–3.85

0.099–0.043

3.2–∞

1003–4655

1.45–3.77

0–0.067

0–∞

617–2139

1.41–1.54

0

0

2780–6680

1.36–1.38

0

0

Fig. 1. Schematic in the xy plane of the numerical simulation setup showing the boundary layer thickness. The computational box dimensions are to scale. Table 2 Parameters of the turbulent boundary layer simulations. Lx , Ly , and Lz are the box dimensions along the three axes. Nx , Ny , and Nz are the collocation grid sizes. The momentum thickness θ is measured at the middle of each box. Case

Reθ

(Lx , Ly , Lz )/θ

Nx , Ny , Nz

ZPG TBL APG TBL

617–1274 1003–4638

320 × 49 × 126 118 × 37 × 28

1201 × 191 × 768 1921 × 380 × 768

compact spatial discretization for the convective and viscous terms, and second order discretization for the pressure in the directions perpendicular to the span, which is spectral. The numerical code has been verified thoroughly by Simens et al. (2009), Simens and Gungor (2013) and Sillero et al. (2013). The DNS computational setup sketched in Fig. 1 consists of two simulation domains running concurrently as described in Sillero et al. (2013). The ZPG TBL is intended to provide a realistic turbulent inflow for the APG layer. The ZPG TBL rescales the velocity fluctuations while fixing the inflow mean velocity to a prescribed profile (Simens et al., 2009). The recycling plane is located at x ≈ 398θ inlet ≈ 45δ inlet , where θ inlet and δ inlet are respectively the momentum and boundary layer thicknesses at the beginning of the auxiliary ZPG DNS. A plane located at x ≈ 268θ inlet ≈ 30δ inlet of the first domain is transferred at each time step into the inlet of the APG layer. The velocities at the outflow are estimated by a convective boundary condition, with small corrections to enforce global mass conservation (Simens et al., 2009). Table 2 summarizes the simulation parameters for both layers. For the APG DNS, the box dimensions with respect to the boundary layer thickness at the outlet are (Lx , Ly , Lz )/δexit = (11.0, 3.4, 2.6 ). The resolutions in terms of the Kolmogorov length η are ( x, y, z) < 4η except near the inlet very close to the wall where x < 8η. The specific DNS performed here was validated by several means. The ZPG TBL data was compared in detail with that of the DNS of Simens et al. (2009), with special attention paid to the mean velocity and Reynolds stresses at the transfer plane at Reθ = 1003. The agreement is very good. The adequacy of the domain size in the spanwise direction was checked by examining the three-dimensional two-point spatial correlations of (u, u), (v, v),

(w, w) and (u, v) for several wall-normal reference positions. All correlation coefficients are close to zero (within noise) at the lateral boundaries. In comparison to ZPG TBLs, the spanwise correlation lengths scaled with δ are shorter in large-velocity-defect TBLs (Simens et al., 2015). It was also checked that the convective outflow boundary condition did not introduce large pressure pulses that could affect the separated flow region. After an initial transient phase, statistics were sampled over 10 flow-through times with respect to Ue0 for the APG DNS. One-point statistics were averaged over time and spanwise direction. Before presenting the databases, it is worthwhile to mention how the parameters involving the inviscid velocity are defined. Since the mean velocity can vary with y in the external inviscid region of APG TBLs due to curvature or variable divergence of the streamlines, the mean velocity defect and the displacement and momentum thicknesses should normally be defined with the yvarying inviscid velocity Uinv (x, y) and not with a y-invariant external (freestream) velocity Ue (x), which in any case should also normally be defined using Uinv . Unfortunately, Uinv cannot always be easily computed when streamline curvature makes its y variations complex, like in the present flow. For this reason, all parameters involving the inviscid velocity are defined here with a y-invariant external velocity Ue (x). This implies that for consistency, the pressure gradient used in the definition of the pressure gradient parameters is the one at the edge of the boundary layer and not the pe e one at the wall, ddx = −ρUe dU . dx For the APG DNS, the desired pressure gradient is controlled by imposing a streamwise dependent wall-normal velocity distribution at the upper boundary of the computational domain. The streamwise and spanwise velocities at the top boundary satisfy free-slip conditions. The imposed wall-normal velocity at the top boundary V(x)/Ue0 and the resulting streamwise velocity at the same boundary U(x)/Ue0 are illustrated in Fig. 2(a) and (b) respectively. As discussed before, the definition of the external velocity Ue is not straightforward in the case of a flow with streamline curvature, like the present flow. In order to be as consistent as possible with boundary layer theory, Ue is chosen as the maximum streamwise component of velocity in the wall-normal direction. The streamwise evolutions of Ue and of the wall-friction

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Fig. 2. Streamwise evolution of mean flow parameters in the present APG DNS. (a) Imposed wall-normal velocity at the top boundary. (b) , Streamwise velocity at the , U e /U e0 ; , 10uτ /Ue0 . (c) Pressure gradient parameters: , βm; , β zs ; , βτ × 10−3 ; , β i . Vertical dashed lines in (a) denote the four top boundary; ), 2.5 ( ), 3 ( ), and 3.43 ( ). streamwise positions where H = 2 (

velocity uτ are also depicted in Fig. 2(b). The flow at the edge of the boundary layer decelerates over most of the domain but then reaccelerates at the end. As can be deduced from the evolution of uτ , the wall shear stress decreases and becomes negative in a downstream segment of the flow, which corresponds to a very thin separation bubble. The reacceleration of the flow at the end reattaches the boundary layer and thereby avoids encountering problems with the outflow boundary condition. Fig. 2(c) shows the streamwise evolution of the outer pressure gradient parameters β m , β zs and β τ , defined in Section 2.1, and of the inner pressure gradient parameter β i , given by Eq. (6). Both the inner and outer pressure gradient parameters increase importantly in different upstream portions of the flow. The positive gradient of these pressure gradient parameters is responsible for the increase in the streamwise mean momentum defect as shown in Fig. 3(b). The outer pressure gradient parameters β m and β zs start decreasing around the middle of the domain. The impact of the pressure force on the outer region is therefore diminishing, as it will be seen more clearly with the mean momentum budgets in Section 3, but this change is not strong enough to reverse the situation in terms of mean momentum defect, which keeps increasing until flow separation. Fig. 3 displays the evolutions of the skin friction coefficient, Cf and the shape factor, H of the present ZPG and APG boundary layers as functions of Reθ in semi-log plots. The numerical ZPG data of Simens et al. (2009) and Sillero et al. (2013) are also included for comparison. The decrease of Cf and increase of H with respect to Reθ is rapid, indicating the strong disequilibrated nature of the flow. 3. Mean flow In this section, the mean flow properties of the large defect TBLs are analysed in relation to known similarity and scaling laws and through comparisons with the two canonical shear flows. For the present flow, the mean flow data is taken at four streamwise positions where H = 2, 2.5, 3 and 3.43 (except in the case of Fig. 5) as depicted in Fig. 2(a). The fourth one is the last mesh position before separation (C f ≈ 10−6 ).

Fig. 3. (a) Skin friction coefficient and (b) shape factor as a function of Reθ . , ZPG TBL of SJM; , present ZPG TBL simulation; ZPG TBL of SJHM; present APG TBL simulation.

, ,

Streamwise mean velocity profiles of the various flows are shown in Fig. 4. The mean velocity defect progressively increases in the present flow. The non-equilibrium APG TBL of MRL behaves in a similar manner but only the profile at separation is shown in Fig. 4 for clarity. Excluding the wall region, the velocity profiles of the large-defect boundary layers clearly resemble the single-stream mixing layer one, with large velocity gradients, similar curvatures and the presence of an inflection point near the middle of the layer. Two of the boundary layer velocity profiles are at the same shape factor of 2, namely one for the present flow and the equilibrium one of SK, but the shapes of these profiles are very different. Since the present flow is a disequilibrium boundary layer, such a shape difference is expected. It is probably more related to the different streamwise evolution of the flows than to the dissimilarity in Reynolds number. Fig. 4 also shows two profiles at separation,

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Fig. 4. Mean velocity profiles. Present non-equilibrium APG TBL: , H = 2.0; , H = 2.5; , H = 3.0; , H = 3.43, C f = 0. , ZPG TBL of SJHM at Reθ = 20 0 0; , equilibrium APG TBL of SK; , profile at separation of the nonequilibrium APG TBL of MRL; and , mixing layer of WF.

one for the present flow and one for the MRL flow. In this case, the different behaviour near the wall may be due to differences in both streamwise evolution of the TBLs and Reynolds number. It is interesting to note that the four profiles of the present APG TBL possess three inflection points, the first one being very near the wall with y < 0.01δ (see Fig. 7). Such a behaviour has not been observed frequently in APG TBL experiments. To our knowledge, large defect profiles with three inflection points have only been obtained by Simpson et al. (1981a). In the case of the MRL and SK flows, the mean velocity profiles only show respectively one and two inflection points, but in these cases at relatively high Reynolds numbers the measurement techniques used did not allow to investigate the profiles very near the wall where another inflection point might be present. The dynamical behaviour of the mean flow of large-defect TBLs is further investigated with the help of streamwise mean momentum budgets in the present APG TBL. The following form of the streamwise mean momentum equation is used:



 1 ∂ P ∂ uv ∂ uu ∂U ∂U ∂ 2U − U +V − − − +ν 2 = 0 ∂x ∂y ρ ∂x ∂ y  ∂x  ∂y         R A

FP

Ruv

uu

(12)

D

where A is the total mean advection, FP the pressure gradient, Ruv the Reynolds shear stress gradient, Ruu the Reynolds normal stress gradient and D the viscous diffusion. Fig. 5 shows the mean momentum budgets at four streamwise positions. Note that the first three positions are different from those selected in all other figures of this section. The first one has been chosen further upstream, with H = 1.5, in order to show more fully the dynamic evolution of the flow. The budget terms are scaled with the pressure-viscous scales upi and ν /upi because it is equivalent to scaling with the pressure gradient FP . In this manner, the importance of the other budget terms with respect to the pressure force is easily visualised. Positive (negative) values indicate a local gain (loss) of streamwise mean momentum. In this strong APG flow, the pressure force is definitely a leading-order contributor to the momentum balance throughout the boundary layer. From the streamwise evolution of the momentum budget it is seen that the flow maintains an approximate dynamic equilibrium (constant relative importance of each force acting on the

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flow) only in the very near wall zone below the first inflection point of the mean velocity profile (zero-crossing of viscous diffusion). For the first and last streamwise positions of Fig. 5, the first inflection point is respectively at y = 0.004δ (y+ = 0.7) and y = 0.007δ . This very near wall zone is also a zone of negligible momentum advection. Outside this zone but below ypi ≈ 10, the pressure force gains importance with respect to all other forces in the streamwise direction. Moreover, it is clear that the viscous force does not balance the turbulent force in this near wall zone centred on the maxima like it does in ZPG TBLs, channel flows and pipe flows (Wei et al., 2005). In fact, except very close to the wall, the viscous force loses importance with respect to all other forces as the velocity defect increases. In the outer region, say y > 0.15δ , both types of Reynolds stress gradients gain importance in the streamwise direction because of the increased turbulent activity there as the velocity defect increases. The Reynolds normal stress gradient is never negligible with respect to the Reynolds shear stress gradient in the outer region when the velocity defect is large. The dynamic evolution of the mean momentum in the upper half of the boundary layer is qualitatively similar to that shown in Maciel et al. (2006b) for the MRL non-equilibrium flow: the pressure force loses importance with respect to advection and Reynolds stress gradients. Although the pressure distributions are very different in these two flows, in both cases the outer pressure gradient parameter (β m or β zs ) decreases after an initial increase, as shown in Fig. 2(c). β m and β zs indeed reflect the local impact of the pressure gradient on the mean flow in the outer region. 3.1. Near wall region With the mean flow dynamics in mind, it is now possible to investigate the correspondence between large-defect TBLs and inner similarity laws and scales. Fig. 6 shows the mean velocity profiles normalized with the friction-viscous scales (uτ and ν /uτ ) of the present flow, the equilibrium TBL of SK and the ZPG TBL of SJHM at Reθ = 20 0 0. Profiles at separation cannot be normalized with the friction-viscous scales since in this case U + → ∞ and y+ → 0. Even if at low Reynolds number, the ZPG TBL profile follows fairly closely the law of the wall throughout the inner region. As already shown by SK, the equilibrium TBL follows the logarithmic law in the overlap layer even if Cf is relatively low. This is coherent with the fact that the inner pressure gradient parameter β i is low in this flow, βi = 0.013, which is a consequence of the high Reynolds number. The equilibrium state reached by the inner region in the SK flow, with β i nearly constant, is therefore very similar to that of canonical wall-bounded flows. In contrast, the non-equilibrium TBL deviates from the law of the wall throughout the inner region and the departure increases as β i increases. The departure in the nearwall region is more clearly seen in Fig. 6(b). This confirms that the pressure force can play an important role even in the viscous sublayer in strong APG flows and that it has to be taken into account in the law of the wall (Mellor and Gibson, 1966; Skote and Henningson, 2002; Stratford, 1959), for instance in the form of the extended law of the wall (8). However, as it was just mentioned, the Reynolds number also plays a role here. If the Reynolds number of the flow was higher, β i would be lower and the departure from the law of the wall would be probably less important. In Fig. 7, the mean velocity profiles for the present APG TBL are plotted using the pressure-viscous scales and showing only the near wall region. The extended law of the wall, Eq. (7), and the square law that is expected to hold at separation are also shown in this figure. As expected from the mean momentum budgets of Fig. 5 and from the fact that β i is not constant, the near-wall profiles are not self-similar even when normalized with pressureviscous scales. At the wall vicinity, the profiles agree with the

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Fig. 5. Streamwise mean momentum budgets of the present flow normalized with the pressure-viscous scales upi and ν /upi . , total advection (A); , pressure , Reynolds shear stress gradient (Ruv ); , Reynolds normal stress gradient (Ruu ); , viscous diffusion (D). (a) H = 1.5, (b) H = 2.2, (c) H = 2.9, (d) gradient (FP ); ) are respectively at y = 0.15δ and 0.5δ . H = 3.43, C f = 0. The vertical dashed lines (

Fig. 6. Mean velocity profiles normalized with friction-viscous scales. Lines as in Fig. 4. (a) , U + = y+ .

, log law (Eq. (3)) with κ = 0.41 and B = 5.1. (b) Near-wall region only;

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Fig. 7. Mean velocity profiles in near-wall region of the present flow normalized , extended law of the with pressure-viscous scales. Solid lines as in Fig. 4; , U pi = 12 (y pi )2 ; +, near-wall inflection point of the velocity profile. wall (Eq. (7));

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Fig. 8. Velocity defect profiles normalized with Um . Lines and symbols as in Fig. 4.

extended law of the wall, as was also found by Skote and Henningson (2002). This is not surprising since the extended law is equivalent to a truncated power series expansion centred at the wall. The extended law of the wall (7) also implies that the velocity profiles have positive curvature in the vicinity of the wall due to the pressure force (positive y2 term). This is indeed the case. The + symbols in Fig. 7 show where the first inflection points of the velocity profiles are located, which corresponds to the position of zero-crossing of the viscous force in the momentum budgets shown in Fig. 5. The profile at separation (red line) follows in the vicinity of the wall the quadratic behaviour expected when uτ → 0 (β i → ∞). 3.2. Outer region Since the outer region of non-equilibrium TBLs is strongly dependent on the upstream and downstream history of the flow, in contrast to equilibrium TBLs, no similarity laws of the outer mean velocity defect profile are expected to hold in this case. It is nonetheless valuable to discuss outer region scaling and to compare the mean velocity statistics in the outer region. Outer (or high-speed region) scales common to both mixing layers and boundary layers have to be defined in order to be able to compare these two types of flows. For simplicity, we choose scales that are similar to those frequently used for free shear layers. For the boundary layers, the outer length scale remains the boundary layer thickness δ as defined previously, that is the wall-normal distance from the wall to the point where U = 0.99Ue . For the single-stream mixing layer, since the data is not accurate in the low-speed region, the outer length scale is defined only using the high-speed part of the flow:

δ = 2(y0.99 − y0.5 )

(13)

where y0.5 and y0.99 correspond to the positions where U = 0.5Ue and 0.99Ue , respectively. A new outer velocity scale Um is used for both TBLs and mixing layer and it is defined as twice the velocity defect at the middle of the shear layer:

Um = 2(Ue − U (y = 0.5δ ))

(14)

It is (only) approximately equivalent to Uzs , in the sense that both velocity scales are proportional to the mean velocity deficit of the outer flow.

Fig. 9. Corrsin shear parameter in different wall-bounded flows. Coloured lines as , channel flow at Reτ = 934 and 2003 (del in Fig. 4 for the present APG TBL. , ZPG TBLs at Reθ = 1968 (SJHM) Álamo et al., 2004; Hoyas and Jiménez, 2006). and 40 0 0 (SJM). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

The velocity defect profiles of the various flows normalized with Um are presented in Fig. 8. The defect profiles are not identical, as expected, but they are all regrouped which indicates that the choice of Um and δ as common outer scales is indeed adequate to compare velocity statistics between these flows. With the exception of the ZPG TBL, the velocity defect profiles of all other flows are actually very similar on the high-speed side, y > 0.5δ . In agreement with previous studies of Maciel et al. (2006a) and Logdberg et al. (2008), Uzs and δ are also found to be appropriate outer scales for the mean velocity defect of all boundary layers (not shown), but Uzs cannot be used for the mixing layer. 4. Turbulent statistics The Corrsin shear parameter defined as S∗ = Sq2 / , where S = ∂ U/∂ y and q2 = 2k, is the ratio between the turbulent dissipation time and the mean shear deformation time. It measures the importance of the interaction between the mean shear and the energy-containing turbulent structures (Jiménez, 2013). The local mean shear dominates and controls the evolution of the large-scale structures if S∗  1. Fig. 9 displays S∗ for the present large-defect

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Fig. 10. Reynolds stresses of the present flow and of the ZPG TBL of SJHM at Reθ = 20 0 0 in friction-viscous inner scaling. Near-wall region is shown in the inset with linear scales. Lines as in Fig. 4.

boundary layer in comparison to canonical wall-bounded flows. In the case of the turbulent mixing layer of WF, S∗ is also around 10 for most of the layer. Hence, similar to other shear flows, the large-scale turbulence of large-defect TBLs appears to be driven by the mean shear in the outer region. This suggests that similarities may exist between the turbulent production mechanisms in these flows, although inflectional instabilities might also play a role in the case of large-defect TBLs and mixing layers. This will be further discussed below when considering turbulent kinetic energy budgets. The behaviour of the Reynolds stresses near the wall is presented in Figs. 10 and 11. Note that the friction-viscous scales uτ and ν /uτ do not scale the Reynolds stresses when the velocity defect is important, even in the very near wall region. This is clearly illustrated in Fig. 10 where the Reynolds stresses of the present flow normalized with the friction-viscous scales are compared with those of the ZPG TBL of SJHM at Reθ = 20 0 0 (profiles at separation are absent since uτ = 0 in this case). The discussion of the near-wall behaviour therefore focusses on the Reynolds stresses in the present large-defect TBL in pressure-viscous inner scaling, as shown in Fig. 11. Unfortunately, the pressure-viscous scales cannot be used for ZPG TBLs since the pressure gradient is zero in these boundary layers. Fig. 11(a) shows that the near-wall peak of u2 , which is a common feature of canonical wall flows, has vanished in the large-defect region of this flow (it exists further upstream). The maximum of all Reynolds stresses is in the middle of the boundary layer as can be seen from Fig. 11(a)–(d).

It was shown in the previous section that the mean velocity profiles in pressure-viscous scaling vary considerably in this non-equilibrium flow even at the vicinity of the wall. In contrast, Fig. 11 shows that the Reynolds stresses vary more slowly in the near-wall region, in particular u2 pi . These variations can be better appreciated in the near-wall zooms of the insets of Fig. 11. The streamwise evolution of the Reynolds stresses in the near wall region varies completely from one stress component to another. In the case of u2 pi , the profiles are not self-similar but they are regrouped up to ypi ≈ 4, which corresponds to y ≈ 0.02δ for the four positions (y+ ≈ 7 in the case of the first position, green line, where friction-viscous scales are still meaningful). The transverse normal stresses v2 pi and w2 pi increase, albeit at different rates, while the Reynolds shear stress −uv pi decreases. A common inner scale for the four Reynolds stress components therefore does not exist for this non-equilibrium flow. The outer region behaviour of the Reynolds stresses is now examined. In this case, it is possible to compare all the shear flows. In Fig. 12, the Reynolds stresses are normalized with the common outer scale Um in order to be able to compare the mixing layer and the boundary layers. Note that in the case of the TBLs the trends shown in this figure remain the same when the Reynolds stresses are normalized with Uzs . The general trend observed is one of a decrease of all Reynolds stresses with increasing mean velocity defect throughout the boundary layer but especially near the wall. Since Um is proportional to the mean strain rates present in the outer region, the fact that all the Reynolds stresses scaled with Um

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Fig. 11. Reynolds stresses of the present flow in pressure-viscous inner scaling. Near-wall region is shown in the inset with linear scales. Lines as in Fig. 4.

decrease with velocity defect throughout the outer region implies that large-defect TBLs are less efficient in transferring energy from the mean flow to turbulence. The behaviour of the Reynolds stresses of the TBLs subjected to a strong APG shown in Fig. 12 is typical of that seen in other strongly decelerated flows. The maximum of all Reynolds stresses is near the middle of the boundary layer and the near-wall peak of u2  is not present. The comparison of the three large-defect flows indicates that when the shape factor is identical or similar, the level of the various Reynolds stresses normalized with Um is also similar. For instance, the Reynolds stress profiles at H = 2 from the present flow and from SK (green) differ in shape, with maxima at different locations, but are at comparable levels for all Reynolds stresses. In the case of the profiles at separation, the Reynolds stress profiles of the present flow and of the MRL flow are actually quite similar. These results suggest that the effect of the upstream history is not as important as the effect of the local mean shear, at least for these three very different large defect TBLs. Interestingly, the Reynolds stress profiles of the mixing layer are qualitatively similar to the profiles at separation of the boundary layers, but the Reynolds stresses are stronger in the mixing layer. In the following, it will be seen that other turbulence statistics are similar between these two flows. Fig. 13 presents selected terms of the turbulent kinetic energy budgets normalized with Um and δ for four flows: the present flow at three positions corresponding to H = 2.0, H = 2.5 and at separation (H = 3.43, C f = 0), the ZPG TBL of SJM, the equilibrium APG

TBL of SK and the mixing layer of WF. The budgets of the APG TBLs are very different from those of canonical wall flows like the ZPG TBL. In the latter, the maximum of production, dissipation and turbulent transport occur in the near-wall region and with values at least an order of magnitude higher than in the outer region. These facts can be partly appreciated with the ZPG TBL profiles shown in Fig. 13(a). In large-velocity-defect TBLs, maxima of production and dissipation are also found in the middle of the boundary layer, with higher levels of production than dissipation. However, the presence of these outer peaks does not necessarily mean that the levels of production and dissipation in the outer region of large-velocity-defect TBLs are higher than those of ZPG TBLs. Fig. 13 shows that these levels are in fact much lower in the lower portion of the outer region, while in the upper half of all boundary layers they are comparable or lower. By considering Fig. 13(a), (d)–(f), it is possible to follow the streamwise evolution of the budget terms in the present nonequilibrium TBL. As the velocity defect increases, the near-wall peak of production gradually decreases and disappears while the outer peak develops but loses intensity when scaled with Um and δ . The global decrease of turbulence production is consistent with the decrease of the Reynolds stresses. The mean shear ∂ U/∂ y increase in the outer region does not compensate the loss of coherence between u and v shown in Fig. 14(a). Dissipation also decreases throughout the boundary layer but especially in the near-wall region. Turbulent transport takes the turbulent kinetic energy from the middle of the boundary layer and feeds it to both the wall region and the edge of the boundary layer.

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Fig. 12. Reynolds stresses in outer scaling with Um and δ . Lines and symbols as in Fig. 4. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.).

In the outer region, the budget profiles of the equilibrium APG TBL of SK (Fig. 13(b)) are similar in shape and level to those of the present flow, but at a shape factor of H = 2.5 instead of 2.0. Recall that the SK flow is at a shape factor of 2.0. The two flows have very different upstream histories. It is therefore not surprising that their outer regions behave differently. Comparisons of the budget terms in the near-wall region are more difficult to establish since detailed measurements are not available for the SK flow, a high Reynolds number flow with a very thin inner region. The production profile of Fig. 13(b) suggests the presence of a near-wall peak of production that might be similar to the one of the present APG flow at H = 2.0, although of course narrower when plotted with outer scales. Like for the Reynolds stresses, the budget terms of the mixing layer are qualitatively similar to those of the large-velocity-defect TBLs. A resemblance between TBLs close to separation and plane mixing layers had already been noted by Na and Moin (1998). The levels of the terms are however much higher for the mixing layer when scaled with Um and δ , and the maxima are closer to the high-speed edge of the shear layer. This is consistent with the behaviour of the Reynolds stresses. Another noticeable difference is the higher level (in relative terms) of turbulent transport and advection of turbulent kinetic energy in the mixing layer. Fig. 14(a) shows the uv correlation factor for the various flows. The level of correlation between u and v is higher in the ZPG TBL than in all other flows. In the case of the APG TBLs, the correlation

of u and v is seen to deteriorate as the mean velocity defect increases, especially near the wall. Lower correlation levels for largedefect TBLs had already been noted by Simpson et al. (1981a). The correlation profiles of the mixing layer and of the APG TBL of MRL at separation are strikingly similar. They suggest that turbulence significantly loses its coherency in the low-speed side of these flows. The structure parameter −uv/2k, where k is the turbulent kinetic energy, is shown in Fig. 14(b) for all the flows for which k can be determined. Like for the uv correlation factor, the largedefect TBL and the mixing layer have a lower structure parameter than the ZPG TBL. The drop of the structure parameter for largedefect TBLs had been observed previously, see for instance Alving and Fernholz (1995). In the present flow, although it is not clearly visible with this linear plot, the structure parameter decreases near the wall as the velocity defect increases in a manner similar to the uv correlation factor. These various observations suggest that the coherent structures responsible for most of the momentum transfer and production of turbulent energy might differ between canonical wall flows and large-defect APG TBLs. In order to further investigate this, the fractional contributions from the four quadrants to the Reynolds shear stress are computed for both the ZPG TBL and the present APG flow: uvi /uv where i = 1, 2, 3, 4 denotes a quadrant in the parameter space of u and v. In Fig. 15, the results at H = 2 and separation of the present flow are compared with those of the ZPG TBL.

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Fig. 13. Selected terms of the turbulent kinetic energy budgets normalized with Um and δ : , production; , dissipation; , turbulent transport; , advection. (a) ZPG TBL of SJM at Reθ = 40 0 0, (b) equilibrium APG TBL of SK, (c) mixing layer of WF. Present APG flow: (d) H = 2.0, (e) H = 2.5, (f) H = 3.43, C f = 0. Reduced ordinate range in the inset in (f).

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Fig. 14. (a) uv-correlation factor; (b) structure parameter. Lines and symbols as in Fig. 4.

Fig. 15. Fractional quadrant contribution to the Reynolds shear stress. Positive values: , Q2; , Q4. Negative values: for the present flow at (a) H = 2.0 and (b) H = 3.43, C f = 0. Lines with circles are the ZPG TBL of SJM.

The pressure gradient is seen to increase the fractional contribution of all quadrants, except for Q2 in the outer region. The differences between the two flows are more pronounced near the wall as the defect increases. In the ZPG case, the sweeps (Q4 events) contribute the most to the Reynolds shear stress very near the wall while ejections (Q2 events) dominate slightly elsewhere. Whereas for the APG case, sweeps contribute more than ejections in the near wall region and in the lower part of the outer region as well. Correspondingly, the negative contribution of Q1 and Q3 motions increase near the wall due to the increased energy transfer through the turbulent transport toward the wall (Nagano et al., 1998). Also, as suggested in Krogstad and Skåre (1995), the strength of Q1 motions might be due to reflections of Q4 motions at the wall. The crossing between the sweep-dominated region and the ejectiondominated region takes place much further from the wall in the APG case. This point is very close to the local maximum of the Reynolds shear stress. 5. Conclusions The one-point velocity statistics, the scaling and similarity laws of large-defect APG TBLs are investigated. Three large-defect APG

, Q1;

, Q3. Lines without symbols are

TBLs with very different streamwise evolutions are used for that purpose, one equilibrium and two non-equilibrium flows. To enhance the analysis, one of the non-equilibrium TBLs was obtained through DNS. These flows are compared with two types of canonical shear flows: ZPG TBL and single-stream mixing layer. Similarity laws for the inner region are tested with the new DNS database, the only one that provides accurate near-wall data. This large-defect non-equilibrium APG TBL does not follow the classical law of the wall, even very close to the wall. In the latter region, the mean velocity profiles agree with the extended law of the wall that accounts for the pressure gradient. This implies that the viscous sublayer (approximately ypi <10) is not too sensitive to history effects, a result also obtained by Skote and Henningson (2002) with a different non-equilibrium APG TBL. In this viscous sublayer, it is also found that the various Reynolds stresses are not self-similar but scale reasonably with the pressure-viscous scales. For all large-velocity-defect TBLs, the maximum turbulence activity is found to be in the middle of the boundary layer and not near the wall. The near-wall peak of streamwise Reynolds normal stress is absent or very small and that of turbulence production is rapidly decaying in the streamwise direction. The present analysis also shows that turbulence loses coherency as the velocity defect

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increases. u and v are more decorrelated throughout the boundary layer than in ZPG TBLs. Near the wall, u and v become more and more decorrelated as the velocity defect increases. In connection to the previous observations, an important conclusion that is drawn from the present analysis is that, in part due to the important reduction of velocity gradients near the wall, large-velocity-defect boundary layers are globally less efficient in extracting turbulent energy from the mean flow than the ZPG TBL. When normalized with an adequate outer region velocity scale that reflects the global value of the mean shear in the outer region (Um or Uzs ), the Reynolds stresses and the production of turbulent kinetic energy are weaker in the large-velocity-defect boundary layers than in the ZPG TBL throughout the boundary layer, but more importantly so in its lower half. Due to the resemblance of mean velocity profiles in the outer region of large-velocity-defect TBLs to the high speed part of mixing layers, the large-defect TBLs are also compared with a singlestream mixing layer. To be able to do so, an outer region velocity scale analogous to the commonly defined free shear layer velocity scales is introduced. The resemblance between these two types of flows is striking in terms of the distributions of Reynolds stresses, production, dissipation and turbulent transport of turbulent kinetic energy, uv correlation and structure parameter −uv/2k. The outer-region turbulent statistics of TBLs close to separation appear to have a similar character to those of the single-stream mixing layer. This suggests that turbulence seems not to be significantly affected by the presence of the wall in a large-defect TBL close to separation. But the intensity of turbulence is less in relation to the mean shear in the large-defect TBLs. The turbulent energy producing mechanisms are therefore not necessarily identical between large-defect TBLs and mixing layers. The contribution of sweeps and ejections to the Reynolds shear stress is also compared between the newly simulated large-defect TBL and a ZPG TBL. The fractional quadrant decomposition of the Reynolds shear stress indicates that sweeps seem to dominate ejections in the near wall region of the large-defect boundary layer, whilst ejections dominate the outer region. The switch between these two regions is observed to take place close to the maximum of the Reynolds shear stress in the outer layer, instead of very close to the wall, as in ZPG TBLs. Acknowledgements Funded in part by the Multiflow program of the European Research Council under grant ERC-2010. AdG-20100224. YM and JS were supported in part respectively by NSERC Discovery Grant of Canada and ARC Discovery Grant of Australia DP130103621. YM thanks TUBITAK (2221 Program) for supports during the collaboration stay in Turkey. The computations were made possible by generous grants of computer time from Barcelona Supercomputing Center (Spain) under Project number FI-2015-2- 0031 and from the National Center for High Performance Computing of Turkey (UYBHM) under Grant number 1002222012. The authors would like to thank Prof. Jiménez for organizing the First Multiflow Summer Workshop and for suggesting the idea of comparing largedefect boundary layers with mixing layers. References Afzal, N., 1983. Analysis of a turbulent boundary layer subjected to a strong adverse pressure gradient. Int. J. Eng. Sci. 21 (6), 563–576. Afzal, N., 1996. Wake layer in a turbulent boundary layer with pressure gradient: a new approach. In: Gersten, K. (Ed.), IUTAM Symposium on Asymptotic Methods for Turbulent Shear Flows at High Reynolds Numbers. In: Volume 37 of Fluid Mechanics and its Applications. Kluwer Academic Publisher, pp. 95–118. Alving, A.E., Fernholz, H.H., 1995. Mean velocity scaling in and around a mild, turbulent separation bubble. Phys. Fluids 7 (8), 1956–1969.

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