Hairpin vortices in turbulent boundary layers

The present work addresses the question whether hairpin vortices are a dominant feature of near-wall turbulence and which role they play during transition. First, the parent-offspring mechanism is investigated in temporal simulations of a single hairpin vortex introduced in a mean shear flow corresponding to turbulent channels and boundary layers up to Reτ = 590. Using an eddy viscosity computed from resolved simulations, the effect of a turbulent background is also considered. Tracking the vortical structure downstream, it is found that secondary hairpins are created shortly after initialization. Thereafter, all rotational structures decay, whereas this effect is enforced in the presence of an eddy viscosity. In a second approach, a laminar boundary layer is tripped to transition by insertion of a regular pattern of hairpins by means of defined volumetric forces representing an ejection event. The idea is to create a synthetic turbulent boundary layer dominated by hairpin-like vortices. The flow for Reτ < 250 is analysed with respect to the lifetime of individual hairpin-like vortices. Both the temporal and spatial simulations demonstrate that the regeneration process is rather short-lived and may not sustain once a turbulent background has formed. From the transitional flow simulations, it is conjectured that the forest of hairpins reported in former DNS studies is an outer layer phenomenon not being connected to the onset of near-wall turbulence.


Introduction
Notable progress has been made over the last decades in understanding the physical processes involved in turbulent boundary layers (TBL). We have a relatively complete understanding of the kinematics of the whole TBL, from the near-wall or buffer region to the outer region. The statistics of velocity and vorticity in turbulent flows have been extensively studied, and were reviewed recently in a number of articles, see e.g. Refs. [23,25,35]. However, our understanding of the dynamics of TBL is more limited [15].
The region of the TBL where we have the most complete understanding of the dynamics of the flow is the buffer region. Since their first visualisation by Kline et al. [24] the near-wall streaks with alternating low and high streamwise momentum have been studied thoroughly, confirming their spanwise spacing of about 100 viscous units ℓ ⋆ . Jeong et al. [14] applied the λ 2 criterion for vortex eduction to this near-wall region, to conclude that the dominant nearwall structures are staggered quasi-streamwise vortices, surrounding the low-speed streaks. The processes that relate vortices and streaks were studied using a number of different techniques and models [10,16,17,34]. There is now general agreement in that the streaks and the vortices are involved in a self-sustaining non-linear cycle, with a period of about 400 viscous time units. The sinuous instability of the streaks results in the formation of new quasi-streamwise vortices, that interact with the mean shear to generate new streaks. This non-linear cycle is autonomous and solver is employed in perturbation-mode which allows to maintain the base flow for all times and to choose between nonlinear and linear computations. A channel flow as well as a boundary layer with a turbulent mean profile are considered. The hairpin vortex is created by a localized volume forcing which offers more flexibility for follow-up studies. The observed dynamics will provide new details about the lifetime of the streamwise and spanwise vortex constituents in a realistic environment. Additionally, possible regeneration and package-creation processes will be investigated in the context of TBL. To investigate the effects of background noise, an eddyviscosity approach is followed in this work which has been proposed by Flores [8]. In that regard, the first part of the present work can be viewed as an extension of published work on the evolution of hairpins in laminar flows [1,9,21,40].
In the second part, the transitional region of a boundary layer is simulated using an adapted trip forcing which generates a regular pattern of hairpins with low-amplitude random modulations.
The approach is partly similar to the so-called "synthetic turbulence" experimentally investigated by Coles and Barker [7] and Savas and Coles [30]. The obtained data is then analysed to answer a number of questions, e.g. how long individual hairpin vortices exist, whether span-/streamwise connection mechanisms can be observed (see [2]), and how the so-called forest of hairpins develops. Furthermore, the proposed set-up may serve as a standard flow case providing a reproducible boundary layer flow for further studies of transitional flow.

Hairpin vortices in parallel shear flow
The behaviour of a hairpin vortex in a mean shear flow is studied in the following. First, the numerical set-up and the employed forcing is described. Then, results are presented for a hairpin being tracked in an undisturbed mean flow extending the former study by Zhou et al. [40] with respect to higher Reynolds numbers and longer observation times. Furthermore, a temporal boundary layer simulation was performed to examine the effect of a slightly varied mean flow. In a second step, the effect of a turbulent background is included, whereas, in contrast to Kim et al. [21], the mean dissipative effect of turbulent fluctuations is accounted for. Both linear and nonlinear simulations are considered to examine the importance of nonlinearity for the regeneration process.

Numerical setup
In order to reproduce the hairpin regeneration process, a simulation setup similar to the one used by Zhou et al. [40] was used. The streamwise, wall-normal, and spanwise directions are x, y, and z respectively. All computational domains were homogeneous in the streamwise and spanwise direction, i.e. periodic boundary conditions were imposed and Gauss-Lobatto collocation points were used in the wall-normal direction. The details of the computational domains are given in Tab. 1. Tests with an increased box size clearly showed that this change had no remarkable influence on the results. Numerical integration was performed using the spectral code SIMSON [6] in perturbation formulation (see Appendix), such that a turbulent mean profile could be set as a time-invariant base flow. Mean velocity profiles were taken from DNS data [20,26], i.e. the simulation was started as a undisturbed flow with modified velocity profile. It should be stressed that the mean flow was fixed in the present study in contrast to former simulations by Zhou et al. and Kim et al. [21,40].
The disturbance, which is required to develop a hairpin vortex, was introduced by a transient local volume forcing, causing a deceleration and ejection of fluid near the wall. This distinguishes the present study from the one by Zhou et al. [40] who employed a conditionally averaged Q2 event extracted from DNS data. However, the velocity amplitudes produced by the present forcing are quite similar to those reported by Zhou et al. [40], and the vortex structures possesses all the features described in their work. The exact form of the forcing term was derived in two   steps. First, the amplitudes were chosen in order to resemble the velocity perturbation described by Zhou et al.. In a second step, the spatial and temporal dimensions and gradients were tuned until a loop-vortex developed quickly from the initial counter-rotating vortex pair. It was found that a very short and intense pulse with an an increased shear on the top layer gave the best results. The final form of the forcing is given by with C + = (−4.84, 2.42, 0) T , L + x = 34, L + y = 75.6, L + z = 54, and T + = 1.4. The values x 0 /h = 0.5, y 0 = 0, and z 0 = 0 define the location of maximum forcing, where h is the channel half-height. The factor in the first parenthesis in Eq. 1 causes a weak inverse forcing on the lateral sides of the main peak which supports the onset of a counter-rotating vortex pair (compare Fig. 1). The fourth-order decay in y-direction leads to a stronger shear enhancing the evolution of the first hairpin head. Changing the parameters within a range of ±10% yields very similar structures with minor differences regarding the shape of the hairpin head and the subsequent regeneration process. However, an insufficient amplitude C prevents the initial hairpin generation, whereas the threshold value strongly depends on the time coefficient T . It should be noted that the vortex evolution discussed below is not specific to the exact choice of parameters, but rather very robust.

Undisturbed background
In the following, results are presented for a single hairpin vortex in a shear flow using both nonlinear and linear solution methods. First, the nonlinear case is discussed. In Fig. 2 the vortical structures resulting from the above described forcing are shown for Reynolds numbers Re τ = 180 and 590 at t + = 300. The origin of the time axis was chosen to be t 0 , at which the initial forcing peaks (see Eq. 1). For both Reynolds numbers, the largest loop-like vortex is the one caused by the initial perturbation and is therefore denoted as primary vortex. The primary vortex is stretched by the mean shear and grows away from the wall. At Re τ = 180, the vortex head extends above the channel's centreline at y + = 180 and is therefore bent backward. Each of the primary vortices is followed by two smaller upstream vortices induced by the first one. Using the scaled forcing described above, the obtained vortical structures also scale perfectly in inner units as evident from the figure.
Similar simulations were performed using a mean velocity profile for a zero-pressure gradient turbulent boundary layer. Here, the mean velocity was extracted from a DNS of a spatially developing TBL [33] at Re τ = 500 and implemented as a base flow with with U (y) and V = 0. Since the boundary layer is not growing in the y-direction, incompressibility is fulfilled. In contrast to the former simulations, a free-stream boundary condition was imposed at the upper y-boundary. As shown in Fig. 3, the resulting structures look very similar to those from the  channel case. Slight deviations are due to differences in the convection velocity between channel and boundary layer, with the difference becoming more evident further downstream. To quantify the temporal change of the vortical structures, the rms of the spanwise vorticity ω z , computed for the whole domain, may serve as an indicator for the strength of the vortex heads. For the channel flow at Re τ = 590, ω z,rms is plotted over time in Fig. 4(a). Directly after the injection of wall-normal momentum, there is a short increase of ω z,rms , which corresponds to the connection of the initial counter-rotating vortex pair. Later on, ω z,rms drops monotonically but the curve flattens t + > 300, where the hairpin regeneration has come to an end.
The ensemble averaged Reynolds stress −u ′ v ′ passes a minimum at about t + = 30 and then grows until it reaches a maximum at about 1000 viscous time units (see Fig. 4(b)). The transient hairpin regeneration is notable from a distortion in the growth of −u ′ v ′ around t + = 100. For times t + > 1000, −u ′ v ′ slowly decays whereas the lifetime of the vortical structures is large compared to, e.g. , the period of the bursting cycle in the buffer layer which is known to be about 400 viscous time units [17]. It is likely that the lifetime of the investigated vortical structures would decrease in a turbulent flow field where interactions with surrounding vortices prohibit an undisturbed growth. It should be remarked that Zhou et al. [40] observed a monotonic growth in −u ′ v ′ since their simulations covered a much shorter period of t + = 300. However, the initial behaviour was very similar to the present findings.
Repeating the above described simulation in linear mode proves that the hairpin regeneration is a nonlinear mechanism (cf. Figs. 4(a) and 4(b)). That is, only one initial hairpin is formed in the linear case, which propagates downstream. The vortex is elongated in the streamwise direction and no self-induction takes place. Therefore, after an initial peak ω z,rms and −u ′ v ′ decay exponentially.

Dissipative background
Since in a developed turbulent flow each vortical structure interacts not only with the mean shear and itself but also with all surrounding fluctuations, a more realistic investigation of the regeneration process should invoke those effects. Kim et al. [21] extended the study of Zhou et al. [40] by including random noise in their channel simulations. Additionally, the initial hairpin was released in a velocity field extracted from DNS. However, the amplitude of the random noise was only about 15 % of the rms known from DNS and the disturbances were not purely dissipative. In the case of the DNS flow field, the observation period was strongly limited, such that only the early stages of regeneration could be observed. Nevertheless, the authors concluded that hairpin regeneration is a robust and sustaining process also in fully developed turbulent flow.
In Sec. 2.2 it has been shown in nonlinear simulations that a hairpin packet is formed after a rather short period of time (t + < 200) with a slow subsequent decay. It appears most likely that a dissipative turbulent background would accelerate the decay and, furthermore, nonlinear effects giving rise to hairpin regeneration should be damped or even inhibited. To investigate which influence a turbulent environment has on the hairpin regeneration process it is feasible to take into account the average dissipative effect of turbulent stresses, namely the turbulent viscosity ν t = − u ′ v ′ ∂U/∂y. This method is limited in the sense that backscatter of turbulent energy from small to large scales is not accounted for.
In the following, the channel flow at Re τ = 590 from Sec. 2.2 is re-simulated with a total kinematic viscosity of ν(y) = ν 0 + aν t (y), where ν 0 is the molecular viscosity, ν t is extracted from DNS [26], and a is a factor between 0 and 1. The respective trend of −u ′ v ′ is plotted in Fig. 5 for a gradually increasing factor a. For comparison, a linear simulation was performed in each case as well. It is clearly evidenced that an increasing turbulent viscosity strongly damps the hairpin vortices. For a = 0.02 one hairpin regeneration takes place, but the structures decay rapidly. Only 10 % of ν t are sufficient to completely suppress the regeneration process. The nonlinearity of the regeneration process is underlined by the fact that for increasing values of a the curves for nonlinear computations more and more approach the linear solution in which no regeneration was observed. For a ≥ 0.1 the curves for −u ′ v ′ almost collapse for the nonlinear and linear case (see Fig. 5). Relating back to the results of Zhou et al.and Kim et al. [21,40], it appears that the lifetime of hairpin packets and the importance of the parent-offspring process have been overestimated in former studies. It is rather likely that hairpin reproduction occurs only for exceptionally strong loop-vortices on a very short timescale and that these structures are not eligible to sustain turbulent motion.

Transitional boundary layer
To investigate how hairpin vortices contribute to transition in a boundary layer and how they evolve downstream, a regular pattern of hairpins is introduced into a laminar boundary layer. To this end, a spatially developing boundary layer is simulated in a computational domain of size 500 δ ⋆ 0 × 40 δ ⋆ 0 × 60 δ ⋆ 0 , with δ ⋆ 0 being the displacement thickness at the inlet, and 1024 × 257 × 256 spectral collocation points. At the inflow, a Blasius profile is prescribed and a fringe region at the end of the domain ensures streamwise periodicity. The numerical approach is identical to the one used in former DNS studies [31,33] and recently Schlatter andÖrlü reviewed the effect of several tripping techniques and inflow lengths on the development of wall turbulence [32]. In the present study, the trip forcing is modified to place a space-filling spanwise row of six "hairpin emitters"close to the inlet at x = 25 δ ⋆ 0 , using the ejection scheme described in Sec. 2.1. To be more precise, at each spanwise position an ejection was generated repeatedly with a time period T hp = 38 δ ⋆ 0 /U ∞ , where U ∞ represents the free stream velocity. The time interval T hp was chosen long enough to produce separated hairpin packets with a slight random variation of the ejection time which breaks the symmetry of the simulation. The time of ejection was allowed to vary by an uncorrelated random shift t s with |t s | < T hp /3 for each hairpin created. The forcing parameters according to Eq. 1 are taken from the simulation of a temporal boundary layer in Sec. 2.2 and, in units of δ ⋆ 0 and U ∞ , read: C = (−0.19, 0.96, 0) T · (ρU 2 ∞ /δ ⋆ 0 ), L x = 1.80 δ ⋆ 0 , L y = 3.78 δ ⋆ 0 , L z = 2.70 δ ⋆ 0 , and T = 1.56 δ ⋆ 0 /U ∞ . The developed flow field after approximately three turnover times 500δ ⋆ 0 /U ∞ is shown in Fig. 6. At the inlet, where the flow is still laminar, the six lines of initial hairpins are clearly visible. They undergo a short period of regeneration with secondary and tertiary hairpins while being elongated in the direction of the mean flow. The vortex heads immediately grow towards the edge of the boundary layer and even further. At this stage, which is located around Re θ = 300, the initial hairpin heads almost disappeared (see Figs 7a and 7b). While the initial  structures are dissipated in the outer layer, there is new dynamics emerging beneath. Namely, secondary low-speed regions, which develop between the original hairpin trails, become sinuously unstable for Re θ > 300 and burst violently around Re θ = 370 (see in Figs. 6b and 6c). These unstable streaks shed a large number of vortical structures that grow to the wake region of the boundary layer. At the same time, the near-wall streaks reorganise and establish the pattern well-known for wall turbulence with a streak spacing of about 100 viscous units. From Fig. 6(b), it becomes obvious that the vortical structures near the wall are oriented in a quasi-streamwise manner, while the outer layer structures possess an arch-like shape with a certain amount of spanwise rotation. Depending on the perspective of the observer and on the contour level used for vortex visualisation, these outer structures might give the impression of a forest of hairpins (compare Figs. 7c and 7d). A closer look reveals that the arch-like structures in the outer layer are neither connected to the wall nor to the quasi-streamwise vortices in the near-wall region. Hence, a qualitative analysis of the present data implies that the flow is not dominated by wall-attached hairpins beyond Re θ = 350, which shall be further investigated in the following. A snapshot of the fluctuations of the spanwise vorticity ω ′ z = ω z − ω is presented in Fig. 8. To reveal the different behaviour of the flow in the initial hairpin trails and between those, two xy-planes are shown at the respective positions. The plane at z/δ ⋆ 0 = 5 is centred in one of the hairpin emitters and heads of the corresponding hairpins can be traced up to x/δ ⋆ 0 ≈ 125, where a connection to the wall is no longer evident (compare Fig. 7b). At z = 0, the secondary lowspeed streak, visible as a red area close to the wall, is fully developed at x/δ ⋆ 0 = 100 (Re θ = 255) and its instability reaches a maximum amplitude at approximately x = 175 δ ⋆ 0 (Re θ = 360). New arch-like vortices are shed from this point into the outer region, which becomes much   more intermittent. Close to the wall, the newly formed streaks produce numerous wall-attached eddies.
Averaging the flow field in time and spanwise direction yields the distribution of the turbulence intensity u ′ u ′ , the Reynolds stress −u ′ v ′ , and the variance of the spanwise vorticity ω ′ z ω ′ z , which are summarized in Fig. 9. The upfloating vortex heads at the inflow are visible in all three quantities as a weak outer maximum for 50 < x/δ ⋆ 0 < 100. Especially the enstrophy contour indicates that the initial hairpins get lost in the wake region. Crucial for the subsequent transition are the streaks near the wall and their growing instability which manifests itself in a global maximum of u ′ u ′ at x/δ ⋆ 0 ≈ 150. In turn, the vortices emitted in this region produce a maximum of the Reynolds stress −u ′ v ′ at x/δ ⋆ 0 ≈ 180 and from that point the boundary layer thickness increases rapidly. For x/δ ⋆ 0 > 200, no remarkable activity is found in the wake region and u ′ u ′ and −u ′ v ′ approach the expected y-dependence for developed wall turbulence. It can be concluded from Fig. 9 that the near-wall cycle sets in at x/δ ⋆ 0 ≈ 200 and that the arch-like vortical structures observed on top of the boundary layer (cf. Fig. 7c) are very weak compared to the near-wall dynamics. Focusing on integral and mean quantities, it becomes apparent that the boundary layer passes through transition rather quickly. Figs. 10(a) and 10(b) show the growth of the momentum thickness and the development of the mean velocity profile. Beyond x/δ ⋆ 0 = 200, the turbulent mean velocity is well developed and Re θ (x) increases linearly up to approximately x/δ ⋆ 0 = 400 where the fringe regions starts to take effect. The shape factor H 12 = δ ⋆ /θ, shown in Fig. 11, confirms that the mean velocity profile possesses the turbulent characteristics for x/δ ⋆ 0 > 200 whereas the friction coefficient c f = 2τ w /(ρU 2 ∞ ) approaches the value expected for wall turbulence asymptotically after a brief overshoot.
Interestingly  are formed (60 < x/δ ⋆ 0 < 110) and c f grows further as the streaks become unstable.

Conclusions
Simulations of a hairpin vortex in a mean shear flow evidenced the regeneration of hairpin vortices and proved the nonlinear nature of the parent-offspring mechanism. In contrast to former studies, the observation time was largely extended, revealing that nonlinear self induction is very short-lived, i.e. for convection times greater than t + = 300, the structures decay constantly. The assumption that hairpin regeneration cannot sustain in turbulent flows is further supported by simulations including a turbulent viscosity. Even a small amount of 10 % background dissipation completely suppresses the generation of new hairpins. In a second part, a transitional boundary layer with an adapted hairpin trip forcing was set up. Using various flow field visualisations, it was shown that the initial hairpins do regenerate briefly before they are absorbed in the wake region. Therefore, it is very unlikely that hairpin vortices persist in fully developed turbulent boundary layers. Instead, the actual transition is driven by secondary low-momentum streaks which develop a growing instability and finally initiate the near-wall dynamics that sustain turbulence. This behaviour complies with mechanisms known from stability theory, which predicts transient growth of sinuous instabilities while varicose perturbations are less stable [3,34]. Arch-like vortices were observed in the wake region, which may be associated with the reported phenomenon of a forest of hairpins. However, how these vortices are formed and what causes their absence at higher Reynolds numbers remain open questions.
Since the boundary conditions and the trip forcing used are reproducible, the presented boundary layer simulation could be a valuable tool for further investigations of early stages of transition in the sense of a synthetic boundary layer flow. Variation of the streamwise position of forcing and the number and strength of initial hairpin vortices would allow for a more detailed study of the processes that lead to the onset of wall-turbulence and the formation of large-scale structures.

Acknowledgments
This work was supported in part by the Multiflow program of the European Research Council. The authors would like to thank Adrián Lozano-Durán for his insightful comments on the manuscript. Additional financial support is acknowledged from the Göran Gustafsson Foundation. Computer time was provided by SNIC (Swedish National Infrastructure for Computing).

Appendix: Perturbation mode formulation
In the following, the concept of the perturbation-mode simulations is discussed in more detail. The basic principle is to decompose the velocity field into a constant base flow on the one hand and time dependent deviations from the base flow (perturbations) on the other hand. This approach eases stability analyses since distinct perturbations and their evolution can be studied in an any kind of base flow. In the case of laminar channel flow, the Poiseuille profile would serve as the physically correct base flow and at the same time it represents as a stable (attracting) state of the system. That is, any artificial perturbation will be damped until the laminar profile is restored. If the Reynolds number is high enough and the initial perturbations possess certain characteristics, another stable state of the system, the turbulent regime, can be established and maintained. It is important to note that, in the turbulent state, the base flow is no longer equivalent to the mean flow. As long as, for a given geometry, the base flow is the solution of the (steady) Navier-Stokes equations, the perturbation formulation is identical to the original non-decomposed equations of motion. However, in the present work the turbulent mean profile was used as a base flow, which is not a solution of the steady Navier-Stokes equations since the Reynolds stress terms are missing. To justify this approach, we will explain the basic concept of the perturbation formulation and discuss the consequences of using a non-physical base flow.
In the incompressible Navier-Stokes equations, the velocity can be split into a base flow U i and a perturbation u ′ i component Rewriting Eq. (A.1) for this decomposed velocity field yields Subtracting the pure base flow contributions, we end up with the equations of motion in perturbation form If the time-invariant base flow U i (x) fulfills then Eq. (A.4) describes the same physics as Eq. (A.1). It should be remarked that nonlinear interactions can easily be cancelled out by neglecting the second term on the left-hand side of Eq. (A.4). If a base flow is employed, that is not a Navier-Stokes solution, one inherently introduces a residual R i fulfilling and therefore one solves for instead of Eq. (A.1). In other words, R i is incorporated as an implicit forcing, which is not explicitly calculated but relaxes the base flow towards the prescribed form. This is similar to the procedure of manually resetting the base flow in every timestep, or explicitly calculating Reynolds stresses and including them in the equations.