2.1 Asymptotic suction boundary layer

The asymptotic suction boundary layer (ASBL) is a boundary-layer flow above a porous flat plate subject to constant homogeneous suction. ‘Asymptotic’ refers to the regime once the flow has developed spatially and reached a constant boundary-layer thickness. In the laminar case the incompressible Navier–Stokes equations admit a steady analytical solution

(2.1a-c ) $$\begin{eqnarray}u_{L}(y)=U_{\infty }(1-\text{e}^{-yV_{S}/{\it\nu}}),\quad v_{L}=-V_{S},\quad w_{L}=0,\end{eqnarray}$$

which is independent of the streamwise direction (see figure 1). Here $(u_{L},v_{L},w_{L})$ is the dimensional laminar velocity field in the streamwise $x$ , wall-normal $y$ and spanwise $z$ directions; $U_{\infty }$ and $V_{S}>0$ are the free-stream and suction velocities, respectively; and ${\it\nu}$ is the kinematic viscosity. The Reynolds number

(2.2) $$\begin{eqnarray}Re=\frac{U_{\infty }{\it\delta}^{\ast }}{{\it\nu}}\end{eqnarray}$$

is defined using the displacement boundary-layer thickness ${\it\delta}^{\ast }$ of the laminar solution,

(2.3) $$\begin{eqnarray}{\it\delta}^{\ast }=\int _{0}^{\infty }\left(1-\frac{u_{L}(y)}{U_{\infty }}\right)\text{d}y.\end{eqnarray}$$

Using expression (2.1) for the laminar solution, equation (2.3) becomes ${\it\delta}^{\ast }={\it\nu}/V_{S}$ . Hence $Re=U_{\infty }/V_{S}$ is the ratio between the free-stream and suction velocities. The base flow is linearly stable for $Re$ up to $Re_{c}=54\,370$ (Hocking Reference Hocking1975). However, turbulence can be observed experimentally for values of $Re$ much below $Re_{c}$ (Antonia et al. Reference Antonia, Fulachier, Krishnamoorthy, Benabid and Anselmet1988). Sustained turbulence has been reported in recent numerical experiments for $Re$ as low as $300$ (Schlatter & Örlü Reference Schlatter and Örlü2011; Khapko et al. Reference Khapko, Duguet, Kreilos, Schlatter, Eckhardt and Henningson2014).

One particular property of the ASBL follows directly from the integral momentum conservation. In both the laminar and turbulent cases the skin friction $c_{f}=2(u_{{\it\tau}}/U_{\infty })^{2}$ is fixed once the asymptotic state is reached. This leads to the equilibrium relation

(2.4) $$\begin{eqnarray}\left(\frac{u_{{\it\tau}}}{U_{\infty }}\right)^{2}=\frac{1}{Re},\end{eqnarray}$$

where $u_{{\it\tau}}$ is the friction velocity. Accordingly, the inner length scale $\ell _{\star }={\it\nu}/u_{{\it\tau}}$ used for the viscous scaling reads $\ell _{\star }={\it\delta}^{\ast }/\sqrt{Re}$ . The friction Reynolds number is defined as $Re_{{\it\tau}}={\it\delta}_{99}u_{{\it\tau}}/{\it\nu}$ or equivalently $Re_{{\it\tau}}={\it\delta}_{99}^{+}$ , where ${\it\delta}_{99}$ is the $99\,\%$ boundary-layer thickness $u({\it\delta}_{99})=0.99U_{\infty }$ , and $(\cdot )^{+}$ indicates inner scaling.

The laminar displacement thickness ${\it\delta}^{\ast }$ together with the free-stream velocity $U_{\infty }$ are used as characteristic units for the non-dimensionalisation. From the next section onwards, all quantities are non-dimensional.

2.2 Numerical method

The ASBL is studied numerically using the fully spectral code SIMSON developed at KTH (Chevalier et al. Reference Chevalier, Schlatter, Lundbladh and Henningson2007). The streamwise and spanwise velocity components are expanded on $N_{x}$ and $N_{z}$ Fourier modes, respectively, whereas the wall-normal component is expanded using $N_{y}$ Chebyshev modes. Dealiasing with the $3/2$ rule is performed in the wall-parallel directions. Time stepping is carried out using a third-order Runge–Kutta method for the nonlinear terms and a second-order Crank–Nicolson method for the linear terms. The same numerical method was used in Khapko et al. (Reference Khapko, Kreilos, Schlatter, Duguet, Eckhardt and Henningson2013, Reference Khapko, Duguet, Kreilos, Schlatter, Eckhardt and Henningson2014) to study the dynamics on the laminar–turbulent boundary in ASBL, and by Schlatter & Örlü (Reference Schlatter and Örlü2011) for turbulent ASBL.

The simulations were performed in a cuboid geometry ${\it\Omega}=[0,L_{x}]\times [0,L_{y}]\times [0,L_{z}]$ (see figure 1). Periodic boundary conditions are imposed in the wall-parallel directions as a consequence of using a Fourier decomposition. At the lower wall a no-slip condition is imposed. Owing to the finite wall-normal extent of the domain, free-stream boundary conditions are used at $y=L_{y}$ . Typically Neumann-type boundary conditions have been used for this purpose in boundary-layer simulations. However, in the case of ASBL, they lead to a slow temporal growth of the boundary layer in the laminar regime (Khapko Reference Khapko2014). Instead, Dirichlet boundary conditions are used here at both lower and upper walls of the domain,

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle (u,v,w)_{y=0}=(0,-1/Re,0), & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle (u,v,w)_{y=L_{y}}=(1,-1/Re,0), & \displaystyle\end{eqnarray}$$

fixing the suction velocity at the upper wall and allowing the correct steady laminar ASBL solution to be captured.

The code is parallelised using both shared and distributed memory parallelisation with the Open Multi-Processing (OpenMP) and Message Passing Interface (MPI), respectively. Two different versions of the MPI parallelisation were used, with domain decomposition in one ( $z$ ) and two ( $xz$ ) directions.

2.3 Study protocol

There are two practical approaches to estimate $Re_{g}$ , defined as the largest value of $Re$ below which turbulence is never sustained. The first one is to trigger turbulence locally and see whether it persists or decays (Bottin et al. Reference Bottin, Daviaud, Manneville and Dauchot1998; Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). However, the method is known to overestimate the value of $Re_{g}$ , since the results are dependent on the initial condition. Furthermore, it requires statistical analysis based on a large number of realisations. The second method is the adiabatic approach: the control parameter (here $Re$ ) is decreased slowly, either continuously or in very small steps, starting from a value where the flow is fully turbulent (Prigent Reference Prigent2001; Barkley & Tuckerman Reference Barkley and Tuckerman2005; Moxey & Barkley Reference Moxey and Barkley2010; Philip & Manneville Reference Philip and Manneville2011). This not only allows one to study the turbulent regime on the verge of laminarisation, but also yields a better quantitative estimate for $Re_{g}$ . In our study we have chosen the adiabatic protocol, and proceeded by lowering $Re$ from a fully turbulent regime in discrete steps ( $1.5\,\%$ of the starting value $Re=333$ ). At each step of the lowering procedure, we wait for the statistically steady regime to be reached (by considering the fluctuating ${\it\delta}_{99}$ ). Statistics of the flow are then gathered over at least $1.6\times 10^{5}$ convective time units, after which the Reynolds number is decreased further. Technically, $Re$ is decreased by increasing the suction velocity $V_{S}$ , while the free-stream velocity $U_{\infty }$ and viscosity ${\it\nu}$ are kept constant. This would also be the natural way to proceed in a laboratory experiment, enabling opportunities for future comparison. As $Re$ is varied, the value of the laminar displacement thickness ${\it\delta}^{\ast }$ changes as well. Therefore, the displacement thickness ${\it\delta}_{s}^{\ast }$ at the start value $Re=333$ is used as the reference length in the system.

We rely on former numerical studies for an estimation of the initial $Re$ to start the adiabatic lowering procedure. Mariani et al. (Reference Mariani, Spalart, Kollmann, So, Speziale and Launder1993) reported sustained turbulence at a value of $Re$ as low as $278$ (note that they use a different definition for the Reynolds number in their paper). Schlatter & Örlü (Reference Schlatter and Örlü2011) observed sustained turbulence at $Re=333$ and laminarisation at $Re=280$ starting from a localised perturbation inside the boundary layer. We choose to continue from the latter study, by first obtaining a fully turbulent field at $Re=333$ before starting the adiabatic lowering from there.

As discussed in the introduction, the size of the numerical domain plays an important role in correctly capturing the spatio-temporal dynamics of the flow. Moreover, the statistically steady turbulent ASBL becomes independent of the dimensions of the system only for large enough numerical domains (Schlatter & Örlü Reference Schlatter and Örlü2011), with the final $99\,\%$ boundary-layer thickness saturating at comparably high values for even moderate $Re$ . Using the results of Schlatter & Örlü (Reference Schlatter and Örlü2011) as guidelines, we have chosen the box dimensions to be $L_{x}=800$ , $L_{z}=400$ and $L_{y}=300$ in terms of ${\it\delta}_{s}^{\ast }$ ( ${\it\delta}^{\ast }$ at $Re=333$ ), with the domain becoming even larger in units of ${\it\delta}^{\ast }(Re)$ as $Re$ is lowered. Since in most planar shear flows turbulent regions at the onset tend to organise in the form of oblique bands, this computational box was estimated sufficient to fit at least one band, assuming that the band has the same wavelength in inner units as for plane Couette flow. As $Re$ is gradually reduced, the $99\,\%$ boundary-layer thickness also decreases. Therefore the domain height $L_{y}$ was safely reduced for the lowest values of $Re$ . The change of grid is performed technically by using spectral interpolation with the Clenshaw recursive algorithm.

Since the value of $Re_{g}$ and the observation times required to reach statistically steady regimes at each value of $Re$ are not known in advance, preliminary runs using relatively low resolution have been performed first. We used ${\rm\Delta}x^{+}\approx 60$ and ${\rm\Delta}z^{+}\approx 30$ , which amounts to $N_{x}\times N_{y}\times N_{z}=256\times 201\times 256$ spectral modes (case R1 in table 1). Other related under-resolved studies (Willis & Kerswell Reference Willis and Kerswell2009; Manneville & Rolland Reference Manneville and Rolland2011) have pointed out how lower resolution affects the flow quantitatively, mainly by shifting the value of $Re_{g}$ , whereas the qualitative comparison with experiments and higher-accuracy numerics remains good. Note that the numerical simulations performed here do not involve any subgrid-scale modelling. These simulations are used in an exploratory manner in order to obtain a rough estimate of $Re_{g}$ and, more importantly, to generate turbulent fields close to $Re_{g}$ . The lowest values of $Re$ are re-examined using higher resolution, with ${\rm\Delta}x^{+}\approx 15$ and ${\rm\Delta}z^{+}\approx 7.5$ , and a lower value for $L_{y}$ (case R2 in table 1).

3.1 Exploratory runs

As mentioned in the previous section, the literature gives only a rough estimate of the value of the critical Reynolds number $Re_{g}$ . Initial explorations were performed using the low R1 resolution.

Figure 2.

The adiabatic protocol results for the exploratory R1 case. (a) Time evolution of the $99\,\%$ boundary-layer thickness ${\it\delta}_{99}$ in the stationary regime. Time $t$ is scaled with ${\it\delta}_{s}^{\ast }/U_{\infty }$ . The Reynolds number was first lowered from $Re=333$ down to $Re=315$ and from there on in steps of  $5$ . (b) The corresponding time-averaged values of ${\it\delta}_{99}$ (filled circles) scaled with the laminar ${\it\delta}_{99L}$ at each $Re$ value and $Re_{{\it\tau}}={\it\delta}_{99}^{+}$ (open squares, axis to the right) versus $Re$ . For the values of $Re<Re_{g}$ where the flow is always laminar ${\it\delta}_{99}/{\it\delta}_{99L}\equiv 1$ , the value marked with the dashed line.

Table 1.

Simulations performed using different resolutions and domain heights.

We begin the protocol by focusing on the turbulent regime at a fixed value of $Re$ , which is then used for the adiabatic lowering. The fully stationary turbulent state is obtained by perturbing the laminar base flow with a localised finite-amplitude disturbance at $Re=333$ . In the asymptotic turbulent regime, the $99\,\%$ boundary-layer thickness ${\it\delta}_{99}$ equilibrates around the value of $155$ (see figure 2 a). This value is relatively high compared to the one in the laminar state, ${\it\delta}_{99L}\approx 4.6$ , as mentioned in Schlatter & Örlü (Reference Schlatter and Örlü2011) and Bobke et al. (Reference Bobke, Örlü and Schlatter2015), and corresponds to the friction Reynolds number $Re_{{\it\tau}}\approx 2800$ (see figure 2 b), reflecting the scale separation in the asymptotic turbulent flow. Recall that $Re_{{\it\tau}}$ , sometimes called the Kármán number, expresses the ratio between the largest scales of the boundary layer and the inner scale $\ell _{\star }$ . From figure 3(a) it is evident that several scales coexist within the same flow, with the large-scale structures extending far beyond the near-wall region. This figure is consistent with the attached-eddy hypothesis of Townsend (Reference Townsend1956). The large-scale structures have an influence on the near-wall region, leaving an energy footprint near the wall and modulating the small-scale fluctuations (Bobke et al. Reference Bobke, Örlü and Schlatter2015), a phenomenon now known as amplitude modulation. This term was coined in Mathis, Hutchins & Marusic (Reference Mathis, Hutchins and Marusic2009), who have investigated the large-scale flow influence in the case of a spatially growing turbulent boundary layer. In their study $Re_{{\it\tau}}=2800$ was the lower limit, with the modulation effect getting stronger with increasing $Re_{{\it\tau}}$ , as the superstructures also increase in both size and energy.

Figure 3.

Two-dimensional $yz$ plane visualisations of the streamwise velocity fluctuations $u^{\prime }$ at (a) $Re=333$ and (b) $Re=260$ computed using R1 resolution. Only part of the wall-normal extent of the domain is shown. The black dashed line marks the height $y^{+}=100$ . The values of the time-averaged 99 % boundary-layer thickness are ${\it\delta}_{99}\approx 155$ and $27$ for $Re=333$ and $260$ , respectively.

The lowering procedure is now described in detail. At the first step, the Reynolds number is decreased from $Re=333$ down to $Re=315$ , and from there on in intervals of  $5$ in $Re$ . For each value of $Re$ , the total simulation time is at least $2\times 10^{5}$ convective time units ${\it\delta}_{s}^{\ast }/U_{\infty }$ in order to ensure stationarity, with the transients not exceeding $0.4\times 10^{5}$ convective units. In figure 2(a) the time evolution of ${\it\delta}_{99}$ is shown for the statistically steady part without the transients. The equilibrium value of ${\it\delta}_{99}$ decreases monotonically with $Re$ . During the step from $Re=260$ to $Re=255$ , the flow eventually laminarises completely. This gives the guiding value of $Re_{g}\approx 260$ that can be used for the high-resolution runs R2, with smaller steps in $Re$ deemed unnecessary for the R1 case.

Note that, close to the threshold, the obtained value of $Re_{{\it\tau}}=500$ is relatively large. For comparison, $Re_{{\it\tau}}$ close to the onset is approximately 60–80 (Tsukahara et al. Reference Tsukahara, Seki, Kawamura and Tochio2005) for plane Poiseuille flow and approximately 30–50 (Brethouwer et al. Reference Brethouwer, Duguet and Schlatter2012) for plane Couette flow (based on the half-height between the walls $h$ ). The latter values are consistent with the estimate for the minimal value of $Re_{{\it\tau}}\approx 50$ required for the self-regeneration of turbulent structures in planar shear flows (Waleffe Reference Waleffe1997; Alfredsson & Matsubara Reference Alfredsson and Matsubara2000). In the case of a spatially developing turbulent boundary layer, a value of $Re_{{\it\tau}}=140$ was reported close to the inflow by Schlatter & Örlü (Reference Schlatter and Örlü2010a ) (based on the 99 % boundary-layer thickness ${\it\delta}_{99}$ ). In ASBL the large-scale structures diminish in size with decreasing $Re$ ; however, the multi-scale nature of the flow remains down to $Re_{g}$ , as attested by the large value of the Kármán number $Re_{{\it\tau}}$ . This is also evident from figure 3(b), where structures of different scales can be seen, especially comparing the regions below and above the $y^{+}=100$ line.

3.2 Identification of partial collapse

As for the laminar–turbulent coexistence, unlike in other wall-bounded shear flows, no laminar regions have been identified visually close to the onset. For instance, at $Re=260$ , which is only slightly above $Re_{g}$ for the R1 case, turbulence occupies the complete spanwise extent, as evident from figure 3(b). For a more quantitative check, the local wall-shear stress in the spanwise direction ${\it\tau}_{z}=|\partial w/\partial y|_{y=0}$ is analysed. This quantity, monitored at an unambiguous wall-normal location, has shown a good indicator of turbulent activity (Kreilos et al. Reference Kreilos, Khapko, Schlatter, Duguet, Henningson and Eckhardt2016). For the sake of laminar–turbulent distinction, we need to define the smallest elementary cell that can unambiguously be in one of the two states, either laminar or turbulent. Following the ideas of Manneville (Reference Manneville2009), such an elementary cell should have a size comparable with the minimal flow unit (Jiménez & Moin Reference Jiménez and Moin1991), and we have chosen here $\text{d}x^{+}\times \text{d}z^{+}\approx 120\times 120$ (that is, $\text{d}x\times \text{d}z\approx 6.5\times 6.5$ ). The average of ${\it\tau}_{z}$ computed over one such cell $[{\it\tau}_{z}]\equiv \langle {\it\tau}_{z}\rangle _{cell}$ enables a clear distinction between laminar/quasi-laminar flow and turbulence. While $[{\it\tau}_{z}]\approx 0$ in the laminar or quasi-laminar case, $[{\it\tau}_{z}]>0$ applies for turbulent flow. Thus, the probability density function (p.d.f.) of $[{\it\tau}_{z}]$ computed over the minimal cells in the case of laminar–turbulent coexistence is expected to feature two clear peaks: one close to zero for laminar/quasi-laminar flow, and another one away from zero for turbulence. The local minimum between the two would define the value of the threshold $[{\it\tau}_{z}]^{c}$ that would best separate laminar and turbulent values (for more details, see Kreilos et al. (Reference Kreilos, Khapko, Schlatter, Duguet, Henningson and Eckhardt2016)).

Figure 4.

The p.d.f.s of (a) the local wall-shear stress in the spanwise direction ${\it\tau}_{z}$ and (b)  ${\it\tau}_{z}$ averaged over elementary cells $[{\it\tau}_{z}]$ capturing independent laminar/turbulent activity for $Re=333$ (solid), $Re=295$ (dot-dashed) and $Re=260$ (dashed), scaled in inner units. The elementary cell size is $\text{d}x^{+}\approx 120$ and $\text{d}z^{+}\approx 120$ .

In figure 4(a) the p.d.f. of ${\it\tau}_{z}$ scaled in inner units is shown. Since the p.d.f. of $\partial w/\partial y|_{y=0}$ is symmetric with the mean at zero, ${\it\tau}_{z}=|\partial w/\partial y|_{y=0}$ also features a clear peak at zero for all $Re$ . But, as mentioned in the previous paragraph, for the purpose of laminar/turbulent discrimination, the data need to be coarse-grained at the level of the minimal cells capturing independent laminar/turbulent activity. The p.d.f. of the coarsened data $[{\it\tau}_{z}]$ is shown in figure 4(b). The peak away from zero for all $Re$ corresponds to the most probable turbulent value. The probability for a cell to be laminar ( $[{\it\tau}_{z}]$ being close to 0) increases with decreasing $Re$ , but remains low for all values of $Re$ . Hence, the flow is fully turbulent even close to the onset, without featuring any extended laminar holes. This behaviour persists when the size of the coarse-graining is varied. Moreover, as the p.d.f.s for different values of $Re$ look only self-similar without being superimposable, the increase of the zero-value probability might be attributed to the inner scaling becoming not fully appropriate at low $Re$ . The inner scaling is known not to perform well near the onset of sustained turbulence (Ching, Djenidi & Antonia Reference Ching, Djenidi, Antonia, Adrian, Durão, Durst, Heitor, Maeda and Whitelaw1996). In other words, the data at hand do not yield any clear value of $[{\it\tau}_{z}]^{c}$ to discriminate between laminar and turbulent, irrespective of the scaling used. The interpretation is that there exists no proper laminar subregion in this flow, i.e. no local turbulence collapse. Note however that the global collapse is easily detectable by monitoring the total kinetic energy of the fluctuations.

3.3 High-resolution reassessment

Turbulent simulations using the R1 resolution suggest no laminar subdomain. This intriguing result has been later verified using high-resolution computations. From here onwards, all the results presented are obtained using the finer R2 resolution.

The velocity fields obtained using the R1 resolution serve as the initial conditions for the R2 case at the same value of $Re$ . The resolution is then increased by steps in all directions until the desired R2 accuracy is reached. This approach avoids repeating the costly lowering procedure and to immediately focus on the most interesting range where $Re\gtrsim Re_{g}$ . With this method the critical Reynolds number is found to be $Re_{g}=270$ . Turbulence collapses globally every time $Re$ is decreased from 270 to 269 or lower, independent of the initial condition. Thus, the resolution improvement from R1 to R2 corresponds to a shift of approximately 10 in $Re_{g}$ , i.e. less than 4 %. This agrees well with the results of Manneville & Rolland (Reference Manneville and Rolland2011), who documented a similar trend – under-resolved simulations lead to a down-shift of the transitional range in terms of  $Re$ .

A snapshot of the turbulent regime at $Re=270$ is shown in figure 5. The $xz$ plane close to the wall at $y^{+}\approx 12$ (figure 5 a) reveals that the whole near-wall region is filled with streaks. Some of these streaks are in the bursting phase, recognised in figure 5(a) by small-scale fluctuations distorting the elongated streaky structures, while others are in a calm phase at the considered time instant, but bursting at another time. Figure 5(b) shows isolevels of the vortex criterion ${\it\lambda}_{2}$ (Jeong & Hussain Reference Jeong and Hussain1995). A uniformly dense forest of vortices is present not only near the wall but also extending away from it. From figure 5 it is evident there are no clear laminar holes close to $Re_{g}$ . This confirms the conclusions drawn from the wall-shear stress (figure 4 b). High-accuracy computations improve the estimation of $Re_{g}$ , but qualitatively display the same features as in the R1 case. In particular, neither STI nor patterning has been detected near the onset.

Figure 5.

Three-dimensional visualisation of the well-resolved flow field (resolution R2) at $Re=270$ . (a) The $xz$ plane at $y^{+}\approx 12$ with positive (red) and negative (blue) streamwise velocity fluctuations (values range between $u^{\prime +}=\pm 2.46$ ). (b) Isosurfaces of the streamwise velocity fluctuations $u^{\prime +}=\pm 1.15$ (red and blue) and ${\it\lambda}_{2}$ criterion ${\it\lambda}_{2}^{+}=-5\times 10^{-5}$ (green). Flow is from lower left to upper right.

Figure 6.

Statistical quantities for the well-resolved case. (a) Mean streamwise velocity profiles for $Re=270$ , $Re=280$ and $Re=290$ , with the lighter colours corresponding to increasing $Re$ . The green dot-dashed line corresponds to the modified linear law (Simpson Reference Simpson1967). The green dotted line represents the empirical log law $u^{+}=(1/{\it\kappa})\ln y^{+}+B$ for $Re=270$ with ${\it\kappa}=1.1$ and $B=10.7$ . The free-stream velocity in inner scaling $U_{\infty }^{+}$ is marked with the green dashed lines. (b) Reynolds stresses: $u_{\mathit{rms}}$ (marked with a square), $v_{\mathit{rms}}$ (star), $w_{\mathit{rms}}$ (triangle) and $\overline{u^{\prime }v^{\prime }}$ (circle) at $Re=270$ , and lighter colours for increasing $Re$ , i.e.  $Re=280$ and $Re=290$ . (c) Two-dimensional premultiplied energy spectrum ${\it\phi}_{uu}^{+}({\it\lambda}_{x}^{+},{\it\lambda}_{z}^{+})$ at position $y^{+}=12$ for $Re=270$ (red solid line), $Re=280$  (green solid line) and $Re=290$ (black solid line). Contour lines correspond to $0.08$ and $0.6$ of the maximum value at $Re=290$ and pseudocolours of the spectrum at $Re=290$ are shown in the background. (d) One-dimensional premultiplied energy spectrum ${\it\phi}_{uu}^{+}({\it\lambda}_{z}^{+})$ at $y^{+}=12$ with the same colour scheme as in panel (c).

After confirming the space-filling nature of the turbulent regime near the onset, the usual turbulence statistics can be computed in order to shed further light on the structure of the flow. Selected results are reported in figure 6: the mean streamwise velocity profiles (figure 6 a) feature a distinct log layer, with the overall shape being comparable with higher- $Re$ statistics (Schlatter & Örlü Reference Schlatter and Örlü2011). A feature of the asymptotic turbulent regime is the absence of a clear wake region, as opposed to the untranspired zero-pressure-gradient turbulent boundary layer. This confirms that the appropriate size requirements are satisfied. The modified linear law, derived for the cases with transpiration (Simpson Reference Simpson1967), provides good agreement very close to the wall for all suction rates. Inside the logarithmic region the empirical log law $u^{+}=(1/{\it\kappa})\ln y^{+}+B$ is fitted to the $Re=270$ profile with ${\it\kappa}=1.1$ and $B=10.7$ , i.e. different from the canonical boundary layers. The parameters ${\it\kappa}$ and $B$ are $Re$ -dependent and increase here with decreasing $Re$ . The same trend was observed at much higher $Re$ by Bobke et al. (Reference Bobke, Örlü and Schlatter2015). In figure 6(b) the Reynolds stresses are shown. Applying suction decreases all stresses compared to the spatially developing boundary layer. The reduction gets more pronounced with decreasing $Re$ . This trend is seen both in figure 6(b) and in the works of Mariani et al. (Reference Mariani, Spalart, Kollmann, So, Speziale and Launder1993) and Schlatter & Örlü (Reference Schlatter and Örlü2011).

Furthermore, the two-dimensional premultiplied spectrum of the streamwise velocity fluctuations, defined as ${\it\phi}_{uu}^{+}({\it\lambda}_{x},{\it\lambda}_{z})=k_{x}k_{z}{\it\phi}_{uu}({\it\lambda}_{x},{\it\lambda}_{z})/u_{{\it\tau}}^{2}$ , is considered inside the buffer layer (see figure 6 c). The peak near ${\it\lambda}_{x}^{+}=10^{3}$ and ${\it\lambda}_{z}^{+}=10^{2}$ captures the near-wall streamwise streaks. This peak shifts slightly towards wider scales with decreasing $Re$ . The one-dimensional spanwise premultiplied spectrum ${\it\phi}_{uu}^{+}({\it\lambda}_{z})=k_{z}{\it\phi}_{uu}({\it\lambda}_{z})/u_{{\it\tau}}^{2}$ is shown in figure 6(d). The peak is located at ${\it\lambda}_{z}^{+}\approx 150$ and varies only weakly with $Re$ . As noted before by Schlatter & Örlü (Reference Schlatter and Örlü2011) and Bobke et al. (Reference Bobke, Örlü and Schlatter2015), the streaks in turbulent ASBL are slightly wider compared to the case without suction. These data confirm that turbulence is sustained without intermittency down to the lowest $Re$ before global collapse.

Figure 7.

Two-dimensional $xz$ plane visualisations demonstrating the flow on its way to laminarisation when the Reynolds number is decreased from $Re=270$ down to $Re=269$ . The streamwise velocity $u$ at $y^{+}\approx 12$ is shown at times (a) $t=0$ , (b)  $t\approx 1.38\times 10^{4}$ , (c)  $t\approx 1.7\times 10^{4}$ and (d) $t\approx 3\times 10^{4}$ .

3.4 Global collapse of turbulence

Up to this point, no steady or even transient laminar–turbulent coexistence has been observed in ASBL. Therefore, it is only when the turbulence collapses globally that clear laminar regions emerge. By definition, when the flow is adiabatically lowered below $Re_{g}$ , global collapse of turbulence eventually occurs with unit probability. Beyond the quantitative value of $Re_{g}$ , the analysis of the laminarisation process itself is rich in information.

The current focus is on cases where $Re$ is reduced from $Re>Re_{g}$ to $Re\lesssim Re_{g}$ . A robust sequence of events leading to laminarisation of the flow is observed in all cases; hence it is enough to scrutinise one such case to identify the generic features of the laminarisation process in ASBL. A global collapse event occurring when lowering $Re$ from $270$ to $269$ is chosen for this purpose.

The whole process is shown using a sequence of snapshots of $u(x,y^{+}=12,z)$ displayed in figure 7. In addition, the evolution of the p.d.f. of $[{\it\tau}_{z}]$ is shown in figure 8. When the flow is fully laminar, the p.d.f. peaks at zero. Thus, in order to be able to compare the p.d.f.s in turbulent and laminar flow, normalisation with the local maximum is used. Unlike figure 4(b), the p.d.f. in figure 8 is obtained by averaging over only a few consecutive snapshots. Consequently, the data in figure 8 appear more noisy, but clearly highlight the difference between the laminar and turbulent regimes.

Figure 8.

Time evolution of the probability density function of $[{\it\tau}_{z}]$ computed using the same size of elementary cells as in figure 4(b). The Reynolds number is lowered from $Re=270$ to $Re=269$ . The values of the p.d.f. are rescaled with the maximum at each time instant.

The initial time ( $t=0$ ) corresponds to the fully turbulent regime at $Re=270$ (figure 7 a). The p.d.f. of $[{\it\tau}_{z}]$ features a clear zero probability for $[{\it\tau}_{z}]=0$ , confirming again the absence of laminar holes. As the flow adapts to the new value of $Re$ , the most probable values of $[{\it\tau}_{z}]$ shift towards zero. At the same time, spatial regions with reduced fluctuations appear at the top and bottom in figure 7(b). This weakening of the turbulent activity ultimately leads to partial laminarisation of the flow. At $t\approx 1.7\times 10^{4}$ , the first laminar holes form (see figure 7 c). At this point in figure 8 a non-zero probability is attained at $[{\it\tau}_{z}]=0$ , which then grows with the expansion of the laminar holes. As the weaker turbulence regions from figure 7(b) laminarise they also spread fast in the streamwise direction until their length reaches $L_{x}$ . This seems to be a general feature of ASBL, as the formation of laminar holes predominantly happens through elongation in the streamwise direction. This leads to a state featuring turbulent and laminar bands, both aligned with the streamwise direction. The fluctuations in the turbulent zones progressively weaken, until these further break up into turbulent patches (see figure 7 d). These patches do not spread at this value of $Re$ and all eventually decay between $t\approx 3\times 10^{4}$ and $t\approx 4\times 10^{4}$ .

In summary, the whole collapse process can be split into two phases. During the first one, the laminar holes appear and grow to form streamwise laminar avenues. Next, the turbulence activity in the streamwise bands diminishes, and they laminarise through the (dis)appearance of transient turbulent spots.

This phenomenon differs from the collapse process reported, for example, in plane Couette and plane Poiseuille flows. There, laminar–turbulent bands are oblique with respect to the $x$ direction and persist for a long time. Once a hole spontaneously forms inside one turbulent band, turbulence retracts in the direction of the band (Manneville Reference Manneville2011). In ASBL no robust oblique interface separating laminar from turbulent regions has been detected; instead, only streamwise-elongated interfaces. Since in the other planar shear flows the large-scale flow is held responsible for the obliqueness of interfaces, any explanation for the peculiarity of ASBL has to incorporate the notion of large-scale flow as well.

3.5 Hypothesis about the role of the large-scale flow

A distinctive feature of turbulent ASBL is the clear scale separation that remains down to the lowest $Re$ with sustained turbulence. The explanation lies in the fact that the large-scale fluctuations in the wall-normal direction, i.e. ejections and sweeps, may appear at relatively low $Re$ . These events lead to the creation of a large-scale flow (LSF) away from the wall. In order to understand the difference between the LSF at the onset in ASBL and in the wall-bounded internal shear flows, we elaborate on the ideas of Duguet & Schlatter (Reference Duguet and Schlatter2013). Consider the continuity equation valid at any location

(3.1) $$\begin{eqnarray}\partial _{x}u+\partial _{y}v+\partial _{z}w=0.\end{eqnarray}$$

Applying large-scale spatial filtering to (3.1) leads to

(3.2) $$\begin{eqnarray}\partial _{x}U+\partial _{y}V+\partial _{z}W=0,\end{eqnarray}$$

where large-scale components are denoted with the upper case. The cutoff wavelength should be chosen larger than the average size of the near-wall structures.

Integrating (3.2) from the wall, here located at $y=0$ , to some arbitrary value $y=l$ yields

(3.3) $$\begin{eqnarray}\partial _{x}\bar{U}+\overline{\partial _{y}V}+\partial _{z}\bar{W}=0,\end{eqnarray}$$

where the overbar denotes wall-normal integration, i.e. $\bar{(\cdot )}=\int _{0}^{l}(\cdot )\,\text{d}y$ . Performing the integration of the wall-normal term leads to

(3.4) $$\begin{eqnarray}\partial _{x}\bar{U}+\partial _{z}\bar{W}=V(0)-V(l).\end{eqnarray}$$

For plane Couette and plane Poiseuille flow close to $Re_{g}$ , the near-wall structures actually scale with the gap width $2h$ . In that case, the walls are impermeable, and $y$ -averaging performed between the two walls in (3.4) leads to

(3.5) $$\begin{eqnarray}\partial _{x}\bar{U}+\partial _{z}\bar{W}=0.\end{eqnarray}$$

The $y$ -integrated LSF in such a case is hence two-dimensional and divergence-free in the plane. Duguet & Schlatter (Reference Duguet and Schlatter2013) have suggested that the transport of weak fluctuations by this flow close to a laminar–turbulent interface should result in an oblique stripe in the $xz$ plane, precisely because $(\bar{U},\bar{W})$ displays obliqueness due to the streamwise deceleration at the laminar–turbulent interfaces.

In ASBL, wall-normal integration between 0 and any finite $l$ leads to

(3.6) $$\begin{eqnarray}\partial _{x}\bar{U}+\partial _{z}\bar{W}=-V_{S}-V(l).\end{eqnarray}$$

For $l$ inside the boundary layer, there are large eddies, typically large-scale sweeps and ejections; hence the term $V(l)$ is not equal to $-V_{S}$ , resulting in a non-vanishing right-hand side in (3.6). Thus, even if $\partial _{x}\bar{U}$ is different from zero (for instance, near a laminar–turbulent interface), (3.5) does not hold. However, the obliqueness of laminar–turbulent interfaces is closely connected to (3.5). This suggests that fully oblique laminar–turbulent interfaces are not generically expected in ASBL, even in the case of growing turbulent spots. This hypothesis can actually be tested numerically by manipulating the LSF of ASBL so that it becomes nearly two-dimensional, as in plane Couette or plane Poiseuille flow. In the next section we will show the effects of suppressing the flow in the outer region of the boundary layer.