2.1 Representativity of the partial slip model

As in the pioneering work of Min & Kim (Reference Min and Kim2004, Reference Min and Kim2005), in this paper the presence of a submerged superhydrophobic surface has been modelled using a spatially homogeneous partial slip boundary condition. As will be discussed in the following, when the spanwise length of the grooves and the shear-free fraction are sufficiently small with respect to the channel thickness, so to retain the lubricating layer within the surface micro-roughnesses even in transitional/turbulent conditions (Seo & Mani Reference Seo and Mani2016), this simple partial slip condition allows us to model accurately enough the macroscopic flow behaviour observed in experimental essays.

In this work, slip length values of $L_{s}=0.005,0.01,0.02$ have been considered in order to match the values employed by Min & Kim (Reference Min and Kim2005), enabling a direct comparison of our results with those in the literature. However, we should verify that these slip length values can be realistically obtained by microtextured SHS in wetting-stable conditions. Equation (15) of the recent work by Seo & Mani (Reference Seo and Mani2016) allows us to determine the equivalent slip length $L_{s}$ provided by a microtextured superhydrophobic surface, given the geometrical pattern and the length scale $L$ of the micro roughness. Combining equations (3) and (15) from Seo & Mani (Reference Seo and Mani2016) we obtain

(2.4) $$\begin{eqnarray}L^{+}=\frac{L_{s}^{+}\sqrt{\unicode[STIX]{x1D719}_{s}}}{0.325-0.44\sqrt{\unicode[STIX]{x1D719}_{s}}}+0.328(L_{s}^{+}\sqrt{\unicode[STIX]{x1D719}_{s}})^{3},\end{eqnarray}$$

where $L^{+}=LRe_{\unicode[STIX]{x1D70F}}$ and $L_{s}^{+}=L_{s}Re_{\unicode[STIX]{x1D70F}}$ , respectively, $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}H/\unicode[STIX]{x1D708}$ being the friction Reynolds number, and $\unicode[STIX]{x1D719}_{s}$ is the solid fraction, quantifying the fraction of solid contact within the SHS roughness pattern, typically spanning from $0.1$ to $0.3$ (Bidkar et al. Reference Bidkar, Leblanc, Kulkarni, Bahadur, Ceccio and Perlin2014; Zhang et al. Reference Zhang, Yao and Hao2016). For the purposes of this section, we have set $\unicode[STIX]{x1D719}_{s}=0.25$ . Using this equation we evaluate the equivalent roughness size required to obtain the largest slip length considered here, $L_{s}=0.02$ , for which an undesired depletion of the plastron layer might be more likely to occur (Zhang et al. Reference Zhang, Yao and Hao2016). Setting $\unicode[STIX]{x1D719}_{s}=0.25$ and $Re_{\unicode[STIX]{x1D70F}}\approx 200$ (as obtained by Min & Kim (Reference Min and Kim2005) for the same flow configuration used in the present work), the largest equivalent texture size attains $L^{+}\approx 19$ . This value complies with the conditions discussed by Seo et al. (Reference Seo, García-Mayoral and Mani2017) to avoid wetting transition, either triggered by capillarity or stagnation pressure. Moreover, the equivalent texture size $L^{+}$ is small enough to not interact with the overlying coherent structures characteristic of a turbulent flow (Fairhall et al. Reference Fairhall, Abderrahaman-Elena and García-Mayoral2018), which have a length scale which is typically of $O(100)$ (Lee, Eckelman & Hanratty Reference Lee, Eckelman and Hanratty1974). These conditions ensure that the simple spatially homogeneous partial slip employed in the present study would provide virtually the same result that a heterogeneous alternation of slip/no-slip patches, both in laminar (Ybert et al. Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2007) and in fully turbulent flows (Seo & Mani Reference Seo and Mani2016). Finally, imposing an $L^{+}$ of $O(10)$ leads to a micro-roughness size comparable with those characterising superhydrophobic surfaces successfully employed in experimental campaigns (Daniello et al. Reference Daniello, Waterhouse and Rothstein2009; Zhang et al. Reference Zhang, Yao and Hao2016; Gose et al. Reference Gose, Golovin, Boban, Mabry, Tuteja, Perlin and Ceccio2018). Using slip lengths larger than $L_{s}=0.02$ would enhance the effect of surface slip, but lead to an increase of the equivalent roughness size that must not exceed that of superhydrophobic surfaces characterised by contact angle hysteresis in the $5^{\circ }{-}10^{\circ }$ range (Zhang et al. Reference Zhang, Yao and Hao2016; Gose et al. Reference Gose, Golovin, Boban, Mabry, Tuteja, Perlin and Ceccio2018), capable of successfully sustaining a fully developed turbulent flow.

2.2 Linear stability analysis (LSA)

Linearising the governing equations around the base flow we obtain the linearised Navier–Stokes equations, which, once projected onto a divergence-free vector space, can be compactly written as

(2.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}=\boldsymbol{L}\boldsymbol{u},\end{eqnarray}$$

together with the linearised wall boundary conditions:

(2.6a-c ) $$\begin{eqnarray}u=L_{s}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y},\quad v=0,\quad w=L_{s}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}y},\end{eqnarray}$$

$u,v,w$ being the streamwise, wall-normal and spanwise components of the velocity perturbation, respectively. With this system being autonomous in time and the base flow being periodic in both the streamwise and spanwise directions, one can apply a Fourier–Laplace transform to any field $\boldsymbol{q}$ so that $\boldsymbol{q}(\boldsymbol{x},t)=\tilde{\boldsymbol{q}}(y)\exp [\text{i}(\unicode[STIX]{x1D6FC}x+\unicode[STIX]{x1D6FD}z)+\unicode[STIX]{x1D706}t]+\text{c.c.}$ , where $\text{c.c.}$ is the complex conjugate, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the real streamwise and spanwise wavenumbers, respectively, $\unicode[STIX]{x1D706}$ is the complex circular frequency and $\tilde{\boldsymbol{q}}\in \mathbb{C}^{4}$ is the associated eigenfunction.

In this framework, the behaviour of a generic solenoidal perturbation $\boldsymbol{u}$ is linked to the eigenpairs of $\boldsymbol{L}$ : $\{\unicode[STIX]{x1D706}^{eig},\tilde{\boldsymbol{u}}^{eig}\}$ (Orszag Reference Orszag1971), since they can be written as

(2.7) $$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t)=\mathop{\sum }_{l}\unicode[STIX]{x1D705}_{l}\tilde{\boldsymbol{u}}(y)_{l}^{eig}\exp (\text{i}(\unicode[STIX]{x1D6FC}x+\unicode[STIX]{x1D6FD}z)+\unicode[STIX]{x1D706}_{l}^{eig}t),\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{l}$ is a scalar weight. For a given couple $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$ the temporal behaviour of each eigenvector of the linearised operator $\boldsymbol{L}$ is then described by its associated eigenvalue $\unicode[STIX]{x1D706}_{i}^{eig}=\unicode[STIX]{x1D70E}_{i}^{eig}+\text{i}\unicode[STIX]{x1D714}_{i}^{eig}$ , where $\unicode[STIX]{x1D70E}_{i}^{eig}$ represents its asymptotic growth/decay rate and $\unicode[STIX]{x1D714}_{i}^{eig}$ its circular frequency.

Solving the one-dimensional local stability problem, which is to say, seeking the eigenvalues of $\boldsymbol{L}$ which govern the behaviour of small perturbations in the time-asymptotic limit, by means of a spectral code (Schmid & Henningson Reference Schmid and Henningson2001), we retrieve the same results found by Min & Kim (Reference Min and Kim2005). The most unstable modes of the operator $\boldsymbol{L}$ , i.e. the Tollmien–Schlichting (TS) waves, are strongly affected by the value of the slip length imposed at the wall. This is not surprising considering that TS waves are near-wall perturbations and that SHS reduce the velocity gradient near the wall. Figure 3(a) shows, for each considered value of $L_{s}$ , the relative neutral curve in the $Re{-}\unicode[STIX]{x1D6FC}$ plane, which separates the unstable region of the plane (right side of the curve) from the stable one (left side). Increasing the value of $L_{s}$ the neutral curve moves towards higher values of $Re$ , suggesting that SHS might delay transition to turbulence, at least in supercritical conditions (i.e. when the Reynolds number overtakes the critical value for linear instability).

In subcritical conditions, small-amplitude perturbations can transiently grow in time due to non-modal mechanisms, linked to the non-normality of the Navier–Stokes equations. To investigate how the non-modal dynamics of small perturbations is affected by SHS in a finite-time framework, we seek the ‘optimal’ perturbation at initial time, $\boldsymbol{u}_{0}^{Opt}(\boldsymbol{x})$ , capable of inducing the maximum energy growth at a target time $T$ (Butler & Farrell Reference Butler and Farrell1992; Luchini Reference Luchini2000). The energy gain to be optimised reads

(2.8) $$\begin{eqnarray}G(T)=\max _{\boldsymbol{u}_{0}}\frac{E(\boldsymbol{u}_{T})}{E(\boldsymbol{u}_{0})},\end{eqnarray}$$

where the integral energy in the computational domain $V$ is defined as

(2.9) $$\begin{eqnarray}E(\boldsymbol{u}(t))=\frac{1}{2L_{x}L_{z}}\int _{V}\boldsymbol{u}^{\text{T}}(t)\boldsymbol{u}(t)\,\text{d}V.\end{eqnarray}$$

To obtain the optimal energy gain and the corresponding initial and final optimal perturbations $\boldsymbol{u}_{0}^{Opt}(\boldsymbol{x})$ , $\boldsymbol{u}_{T}^{Opt}(\boldsymbol{x})$ , a singular value decomposition of the linearised governing equation (2.5) has been used, following the approach of Schmid & Brandt (Reference Schmid and Brandt2014). As shown in figure 3(b), a variation of $L_{s}$ induces but a slight modification of the maximum energy gain and of the optimal perturbation shape, consisting of modulations in the spanwise direction of the streamwise velocity, i.e. streaks (Farrell Reference Farrell1988). In fact, since these optimal structures essentially lie in the flow bulk far from the walls, they are much less affected by the surface boundary condition than the near-wall TS waves. The fact that velocity perturbations are slightly or strongly affected by SHS depending on their distance from the wall suggests that some specific transition scenarios can be better controlled than others through the introduction of SHS.

Figure 3.

Linear stability and transient growth analyses for different values of $L_{s}$ : neutral curve indicating the $Re-\unicode[STIX]{x1D6FC}$ values for which the growth rate of the most unstable mode with $\unicode[STIX]{x1D6FD}=0$ is zero (a) and optimal energy gain contours in the $\unicode[STIX]{x1D6FC}{-}\unicode[STIX]{x1D6FD}$ plane for $Re=1000$ (b), both in perfect match with Min & Kim (Reference Min and Kim2005). The slip length strongly affects the time asymptotic behaviour, while being ineffective in controlling finite-time dynamics, as shown by the unmodified contours of the optimal transient energy growth. Solid, dashed and dashed-dotted contours in (b) are related to the same values of $L_{s}$ reported in (a). The values of the isocontours in (b), from outer to inner, correspond to $G_{max}=[110,120,130,\ldots ,180,190,200]$ .

2.3 Direct numerical simulations

In order to investigate the influence of SHS on laminar–turbulent transition, we superpose different perturbations built on the basis of the previously introduced modal and non-modal stability analyses to the laminar base flow and follow their evolution using DNS. The flow in a streamwise-periodic channel with constant flow rate is simulated using the incompressible solver Nek5000 (Fischer, Lottes & Kerkemeier Reference Fischer, Lottes and Kerkemeier2008), based on the spectral element method. The code, which has been modified in order to account for the Robin boundary condition (1.1) (see the supplementary material for some details of the implementation) is based on a $\mathbb{P}_{N}-\mathbb{P}_{N-2}$ spatial discretisation. Viscous terms are advanced in time using a backward differentiation of order 3, whereas for convective terms an extrapolation of order 3 is used, resulting in the time-advancement scheme labelled BDF3/EXT3. Throughout the present work, the domain size is set to $[L_{x},L_{y},L_{z}]=[2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC},2H,2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}]$ , where, unless otherwise stated, we have imposed $\unicode[STIX]{x1D6FC}=1.12,\unicode[STIX]{x1D6FD}=2.10$ to match the domain size typically employed to study K-type transition in a channel flow (Zang & Krist Reference Zang and Krist1989; Sandham & Kleiser Reference Sandham and Kleiser1992; Min & Kim Reference Min and Kim2005; Schlatter Reference Schlatter2005). In most of the computations the Reynolds number is set to $Re=5000$ in order to match literature results about K-type transition in channel flows (Zang & Krist Reference Zang and Krist1989). Using 24 spectral elements in each direction with a polynomial order equal to 8, the resulting spatial resolution is $[N_{x},N_{y},N_{z}]=[192,192,192]$ . Under these conditions and considering a no-slip wall, the resulting fully turbulent state is characterised by a friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}=210$ (Schlatter, Stolz & Kleiser Reference Schlatter, Stolz and Kleiser2006) (albeit we will show that imposing a constant flow rate will result in a lower $Re_{\unicode[STIX]{x1D70F}}$ for a turbulent flow over SHS). Therefore, the grid size expressed in friction units is sufficiently small to accurately simulate a turbulent channel flow by means of DNS (Seo & Mani Reference Seo and Mani2018). In particular we have $\unicode[STIX]{x0394}x^{+}\approx 6.2,\unicode[STIX]{x0394}z^{+}\approx 3.0$ , $\unicode[STIX]{x0394}y_{min}^{+}\approx 0.16$ at the wall and $\unicode[STIX]{x0394}y_{max}^{+}\approx 11.5$ at the centreline, which are lower than the values used by Min & Kim (Reference Min and Kim2004, Reference Min and Kim2005).

3.1 Supercritical case: TS waves

First, we study how slippery boundaries can affect the laminar–turbulent transition process when $Re\geqslant Re_{critical}$ , namely in the supercritical regime. Thus, for the computations presented in this paragraph we set $Re=10\,000$ , $\unicode[STIX]{x1D6FC}=1.0$ and $\unicode[STIX]{x1D6FD}=2.0$ , to ensure a linearly unstable flow even for a large slip length, see figure 3(a). In these flow conditions, the most unstable eigenvalue of $\boldsymbol{L}$ is the two-dimensional TS wave with $\unicode[STIX]{x1D6FC}=1.00,\unicode[STIX]{x1D6FD}=0.0$ , which we will refer to as $\boldsymbol{u}_{TS_{2D}}$ . Supercritical transition is triggered using as initial condition this unstable mode superposed on the base flow as $\boldsymbol{U}(\boldsymbol{x},t=0)=\boldsymbol{U}_{b}(\boldsymbol{x})+\unicode[STIX]{x1D716}\boldsymbol{u}_{TS_{2D}}(\boldsymbol{x})$ first for a no-slip wall and then for a slip value compatible with the experiments of Gose et al. (Reference Gose, Golovin, Boban, Mabry, Tuteja, Perlin and Ceccio2018), namely $L_{s}=0.005$ . The amplitude $\unicode[STIX]{x1D716}$ is set to ${\approx}3\times 10^{-3}$ to guarantee an initial phase of exponential growth before the onset of nonlinear effects, provided that $E(t=0)=10^{-5}$ .

In order to detect transition we track the evolution of the kinetic energy density of the velocity perturbation defined in (2.9). The result is plotted in figure 4 for $L_{s}=0$ and $L_{s}=0.005$ . In both cases, after an initial phase characterised by exponential growth, as predicted by the LSA, secondary instability sets in, leading to a rapid increment of $E$ towards saturation to a fully turbulent state. For both the considered values of $L_{s}$ the onset of secondary instability is observed for $E\approx 5\times 10^{-5}$ , and the energy saturates towards the same value characterising the turbulent regime. The initial phase of linear amplification of the perturbation energy density matches the predictions of the linear stability theory, as one can notice comparing the DNS results (solid lines) with the dashed lines representing an exponential energy growth with exponent $2\unicode[STIX]{x1D70E}$ , $\unicode[STIX]{x1D70E}$ being the growth rate of the most unstable eigenmode of $\boldsymbol{L}$ . As predicted by LSA, in the presence of SHS, the initial exponential phase is characterised by a lower growth rate, thus the threshold energy for secondary instability is reached in more time, leading to a time delay of the consequent transition to turbulence. For the chosen amplitude of the unstable mode, most of the transition time is spent during the initial exponential growth and the physical mechanisms leading to transition are essentially the same in both the considered cases. Therefore, the transition time might be estimated using the $e^{N}$ method (van Ingen Reference van Ingen2008) also for the SHS case.

Figure 4.

Influence of SHS on the supercritical transition path at $Re=10\,000$ . Simulations have been initialised with the most unstable eigenvector of $\boldsymbol{L}$ , with initial perturbation kinetic energy density equal to $E(t)=10^{-5}$ , for different slip lengths $L_{s}$ . The initial linear amplification phase matches the prediction of the linear stability theory as shown by the dashed lines with slopes equal to $2\unicode[STIX]{x1D70E}$ , with $\unicode[STIX]{x1D70E}(L_{s}=0.000)\approx 0.00374$ and $\unicode[STIX]{x1D70E}(L_{s}=0.005)\approx 0.002978$ , where $\unicode[STIX]{x1D70E}$ is the growth rate of the most unstable eigenmode of $\boldsymbol{L}$ , for $\unicode[STIX]{x1D6FC}=1.0,\unicode[STIX]{x1D6FD}=0.0$ and $Re=10\,000>Re_{critical}$ .

3.2 Subcritical case: K-type transition

In this section we study in detail how the K-type transition scenario is influenced by a slippery wall at $Re=5000$ , pursuing the study made by Min & Kim (Reference Min and Kim2005) in order to unveil the physical mechanisms leading to transition delay in subcritical conditions. Following their work, the K-type transition scenario is triggered setting as initial disturbances a sum of the two-dimensional (2-D) TS wave $\boldsymbol{u}_{TS_{2D}}$ with $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})=(1.12,0.0)$ , and a sum of equal and opposite oblique three-dimensional (3-D) fundamental Tollmien–Schlichting waves $\boldsymbol{u}_{TS_{\pm 3D}}$ characterised by spatial wavenumbers $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})=(1.12,\pm 2.10)$ . All the considered TS waves are the most unstable eigenmodes of $\boldsymbol{L}$ obtained for the imposed $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},Re)$ values using a one-dimensional (1-D) local stability analysis code implementing a Chebyshev spectral collocation approach on a 200 points grid, as indicated in § 2.2. From now on, the domain size is set to $[L_{x},L_{y},L_{z}]=[2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC},2H,2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}]$ , with $\unicode[STIX]{x1D6FC}=1.12,\unicode[STIX]{x1D6FD}=2.10$ .

Figure 5.

Time evolution of the instantaneous wall shear normalised by its laminar value in a standard K-type transition scenario as in (3.2). Compared to the benchmark no-slip case ( $L_{s}=0.00$ ), using SHS modelled superhydrophobic surfaces transition to turbulence can be retarded ( $L_{s}=0.01$ ) or even avoided ( $L_{s}=0.02$ ). The symbols $L_{s}^{V}$ are validations with respect to the case 3 from Min & Kim (Reference Min and Kim2005).

The initial condition for K-type transition (Kleiser & Zang Reference Kleiser and Zang1991; Schlatter Reference Schlatter2005) then reads

(3.2) $$\begin{eqnarray}\boldsymbol{U}(t=0)=\boldsymbol{U}_{BF}+0.03\boldsymbol{u}_{TS_{2D}}\pm 0.001\boldsymbol{u}_{TS_{\pm 3D}},\end{eqnarray}$$

where the shape of the modes depends on the imposed value of $L_{s}$ and their phase is set so that the maximum amplitude of the disturbances occurs at $z=L_{z}/2$ (Zang & Krist Reference Zang and Krist1989; Sandham & Kleiser Reference Sandham and Kleiser1992). For both waves, the maximum value of the streamwise velocity component is set to the desired amplitude and the other components are scaled accordingly. Figure 5 shows the temporal evolution of the instantaneous wall shear normalised by its laminar value for $L_{s}=0.00,0.01,0.02$ . The first two curves accurately reproduce those in figure 3(a) of the work by Min & Kim (Reference Min and Kim2005) (where only $L_{s}=0.00$ and $L_{s}=0.01$ are considered), validating our numerical approach. The highest value of $L_{s}=0.02$ , already used by the same authors in a previous study (Min & Kim Reference Min and Kim2004), would result from a SHS having a roughness texture period of $L^{+}\approx 15$ in the turbulent regime (Seo & Mani Reference Seo and Mani2016). This value of $L^{+}$ is still sufficiently low to ensure the suitability of the spatially homogeneous numerical approach (Seo & Mani Reference Seo and Mani2016) and to avoid wetting transition (Zhang et al. Reference Zhang, Yao and Hao2016) and is comparable to values of the texture period in turbulent conditions reported in the literature, namely $L^{+}\approx 0.5{-}10$ (Daniello et al. Reference Daniello, Waterhouse and Rothstein2009; Woolford et al. Reference Woolford, Prince, Maynes and Webb2009; Park, Sun & Chang-Jin Kim Reference Park, Sun and Chang-Jin Kim2014; Gose et al. Reference Gose, Golovin, Boban, Mabry, Tuteja, Perlin and Ceccio2018; Rowin et al. Reference Rowin, Hou and Ghaemi2018). While for $L_{s}=0.01$ transition is only delayed in time (see also Min & Kim (Reference Min and Kim2005)), for $L_{s}=0.02$ it is completely avoided, although the wall shear increases to a value larger than the laminar one (but still much lower than the turbulent one). In fact, for $L_{s}=0.02$ , whilst initial perturbations are not capable of triggering laminar–turbulent transition, their intensity is high enough to induce the development and successive saturation of streamwise streaks, which translate into a slight increase of the shear stress at $t\approx 250$ . Since the amplitude of this saturated flow field is not sufficiently large to induce transition, streaky perturbations are slowly dissipated through viscous effects, inducing an asymptotic decrease of the wall shear stress which eventually regains its laminar, unperturbed value. This behaviour will be more deeply discussed in § 3.2.2 for slightly different flow conditions.

Figure 6.

Time evolution of the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ , for the modified K-type transition scenario (3.3), using increasingly larger slip lengths. Differently from the case in figure 5 transition occurs even for $L_{s}=0.02$ .

In order to investigate the physical mechanisms affecting transition in the presence of a slippery boundary, we place ourselves in a threshold condition, slightly increasing the initial perturbation intensity up to a value for which transition is triggered even in the $L_{s}=0.02$ flow case. This is achieved for a $10\,\%$ increase of the perturbation amplitude, namely

(3.3) $$\begin{eqnarray}\boldsymbol{U}(t=0)=\boldsymbol{U}_{BF}+1.10(0.03\boldsymbol{u}_{TS_{2D}}\pm 0.001\boldsymbol{u}_{TS_{\pm 3D}}).\end{eqnarray}$$

In the remainder of the paper we will focus on this latter case (3.3), referring to it simply as K-type transition for the sake of simplicity, after having verified that initialising the simulation using (3.2) or (3.3) provides essentially the same results for $L_{s}=[0.00,0.01]$ . To detect whether transition has taken place, in figure 6 we provide the evolution of the friction Reynolds number, defined as

(3.4) $$\begin{eqnarray}Re_{\unicode[STIX]{x1D70F}}=\sqrt{Re\left|\frac{\unicode[STIX]{x2202}\langle U(\boldsymbol{x},t)\rangle }{\unicode[STIX]{x2202}y}\right|_{wall}},\end{eqnarray}$$

where $\langle \cdot \rangle$ represents the spatial average computed on the wall-parallel planes $x{-}z$ at a given time $t$ and the laminar Reynolds number is set to the constant value $Re=5000$ (Zang & Krist Reference Zang and Krist1989). Both the laminar ( $t<100$ ) and turbulent ( $t>500$ ) mean values of $Re_{\unicode[STIX]{x1D70F}}$ change with $L_{s}$ . The former can be easily derived injecting (2.3) within (3.4)

(3.5) $$\begin{eqnarray}Re_{\unicode[STIX]{x1D70F}}=\sqrt{Re\left|\frac{2y}{3L_{s}+1}\right|_{wall}}=\sqrt{\frac{2Re}{(3L_{s}+1)}},\end{eqnarray}$$

while the latter is reduced due to the decreased friction drag. This is a consequence of the fact that we have imposed a constant flow rate. Therefore, we cannot directly compare our results with literature studies of turbulent flows over SHS where $Re_{\unicode[STIX]{x1D70F}}$ is kept constant (Min & Kim Reference Min and Kim2004; Seo & Mani Reference Seo and Mani2016).

Looking at the transitional phase, we observe that while $Re_{\unicode[STIX]{x1D70F}}$ increases monotonically for $L_{s}=[0.00,0.01]$ , this is not the case for $L_{s}=0.02$ , where a transient growth of the friction Reynolds number is observed, followed by a saturation and a rapid increase towards the turbulent value. In the following sections we will analyse the coherent structures and associated Fourier modes arising in the different stages of transition with the aim of identifying the instability mechanisms inhibited by the presence of SHS.

Figure 7.

Overview of the K-type transition over superhydrophobic surfaces with different slip lengths. For each slip length, snapshots extracted at different times are placed next to each other in order to provide an overall view of the laminar–turbulent transition process. The time values are reported on the black lines separating the snapshots. The isosurfaces show the $\unicode[STIX]{x1D706}_{2}$ criterion, coloured by its distance from the wall, and the isocontours represent the streamwise velocity measured at the lower wall, shifted in the spanwise direction for visualisation purposes. Only the lower half of the channel is shown for the sake of visualisation.

3.2.2 Fourier harmonics

With the aim of providing a more quantitative description of how the K-type transition scenario is affected by the use of SHS, we track in time the amplitude of selected Fourier harmonics (Zang & Krist Reference Zang and Krist1989),

(3.6) $$\begin{eqnarray}E_{k}(k_{x},k_{z})=\frac{1}{2E_{0}}\int _{-1}^{+1}\left|\hat{\boldsymbol{u}}_{k_{x},k_{z}}(y,t)\right|^{2}\,\text{d}y,\end{eqnarray}$$

where $\hat{\boldsymbol{u}}_{k_{x},k_{z}}(y,t)$ is a single component of the Fourier transform of the perturbation velocity field in the streamwise and spanwise directions with wavenumbers $k_{x},k_{z}$ , respectively, while $E_{0}$ is the kinetic energy of the laminar flow. Following the literature, the wavenumbers of the different Fourier modes will be indicated as multiples of the fundamental wavenumber, the couple $(i_{x},i_{z})$ representing the $i\text{th}$ multiple of the fundamental streamwise and spanwise wavenumbers, $2\unicode[STIX]{x03C0}/L_{x},2\unicode[STIX]{x03C0}/L_{z}$ , respectively. We recall that in this framework the presence of the TS waves $u_{TS_{2D}},u_{TS_{\pm 3D}}$ is translated into spikes in the $(1,0)$ and $(1,\pm 1)$ Fourier modes, respectively. The time evolution of the most energetic Fourier modes during the different phases of the K-type transition is provided in figure 8 for the considered values of the slip length. In all cases, despite the fact that energy is initially provided only to the $(1,0)$ and $(1,\pm 1)$ modes, a streamwise invariant, subharmonic mode $(0,2)$ is immediately generated, rapidly increasing in energy. This is a clear sign that, even if the modes initial amplitude is relatively small, nonlinear interactions are already taking place, indicating that predictions based on linear amplification mechanisms are not sufficient to model the transition scenario (Min & Kim Reference Min and Kim2005). For $L_{s}=0.00$ and $L_{s}=0.01$ the time evolution of the Fourier modes is substantially similar. On the other hand, for the $L_{s}=0.02$ case we observe a decrease of the amplitude of the $(1,0)$ mode and a much slower increase of $(1,1)$ and $(0,2)$ modes. At $t\approx 200$ , the streamwise-dependent modes begin to decrease in energy whereas mode $(0,2)$ saturates for a rather long time range. Thus, for $t>200$ the flow is mostly characterised by streamwise-invariant velocity modulations, i.e. streaky structures.

Figure 8.

Time evolution of the energy associated with (1, 0), (1, 1) and (0, 2) Fourier modes during K-type transition for different values of $L_{s}$ . While the 2-D and 3-D TS waves, respectively identified by modes (1, 0) and (1, 1), evolve in a similar way in all cases, for the $L_{s}=0.02$ case streaks ((0, 2) modes) dominate the flow evolution for a large time range, leading to a considerable delay in transition.

3.2.3 Late stages of transition

In order to follow the development and eventual disruption of these streaky structures, in figure 9 we provide the time evolution of the streamwise and spanwise vorticities $(\unicode[STIX]{x1D714}_{x}=(\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}y)-(\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}z);\unicode[STIX]{x1D714}_{z}=(\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}z)-(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y))$ within the channel section, as well as the friction Reynolds number (bottom) coloured by the amplitude of the spanwise modulation, which is defined as

(3.7) $$\begin{eqnarray}A_{s}(t)=[\max _{y,z}(\boldsymbol{U}(\boldsymbol{x},t)-\boldsymbol{U}_{b}(\boldsymbol{x},t))-\min _{y,z}(\boldsymbol{U}(\boldsymbol{x},t)-\boldsymbol{U}_{b}(\boldsymbol{x},t))]\end{eqnarray}$$

as proposed by Andersson, Berggren & Henningson (Reference Andersson, Berggren and Henningson1999), Brandt et al. (Reference Brandt, Cossu, Chomaz, Huerre and Henningson2003). Monitoring these quantities in time allows us to identify the hairpin legs and heads, characterised by high values of $\unicode[STIX]{x1D714}_{x},\unicode[STIX]{x1D714}_{z}$ , respectively (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999), as well as the onset of spanwise vortices, associated with peaks of $\unicode[STIX]{x1D714}_{z}$ and $A_{s}$ (Brandt et al. Reference Brandt, Cossu, Chomaz, Huerre and Henningson2003). The time evolution of these three quantities is qualitatively and quantitatively similar for the $L_{s}=[0.00,0.01]$ cases, as reported in figure 9(a,b,d,e,g,h). In both cases, the formation of $\unicode[STIX]{x1D706}$ vortices (localised peak in $\unicode[STIX]{x1D714}_{x}$ for $t<100$ in figure 9(a,b)) is followed by their evolution into hairpin vortices, as suggested by the $\unicode[STIX]{x1D714}_{z}$ spot at $t\approx 125$ indicating the rapid development of hairpin heads (see figure 9 d,e). Although this $\unicode[STIX]{x1D714}_{z}$ peak is weaker for the $L_{s}=0.01$ case, in both cases it is followed by a wall-normal spreading of both vorticity components together with a rapid increase of $Re_{\unicode[STIX]{x1D70F}}$ , indicating the breakdown of hairpin vortices and consequent transition to turbulence (Sandham & Kleiser Reference Sandham and Kleiser1992). Figure 9(g,h) is also very similar, although for the slip case the transitional phase is longer and $Re_{\unicode[STIX]{x1D70F}}$ is characterised by a smaller value at large time.

This process is radically modified for the largest slip length considered here, $L_{s}=0.02$ . Whilst the initial condition has been designed to promote the formation of $\unicode[STIX]{x1D706}$ vortices (Sandham & Kleiser Reference Sandham and Kleiser1992), $\unicode[STIX]{x1D714}_{x}$ increases only transiently and then drops to very low values, as depicted in figure 9(c). The reduced wall shear restrains the formation of $\unicode[STIX]{x1D706}$ vortices, therefore inhibiting the consequent creation of hairpin vortices. Right after the disappearance of the $\unicode[STIX]{x1D714}_{x}$ transient peak, at $t\approx 200$ , $\unicode[STIX]{x1D714}_{z}$ begins to increase, initially close to the wall and then migrating through the channel bulk up to $t\approx 400$ (see figure 9 f). In this time range the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ increases to a value larger than the laminar but lower than the turbulent mean one and the spanwise modulation amplitude saturates, as shown in figure 9(i). In fact, in this phase streaky structures are created as a response of the flow to the residual perturbations present in the computational domain (Luchini Reference Luchini2000). For $180<t<200$ the streaks linearly increase their amplitude ( $A_{s}$ ) due to flow receptivity. Once $A_{s}$ reaches a threshold amplitude, nonlinear effects set in (according to Brandt et al. (Reference Brandt, Cossu, Chomaz, Huerre and Henningson2003), $A_{s}\approx 26\,\%$ of the free-stream velocity for sinuous instability), and lead to a saturation of both $Re_{\unicode[STIX]{x1D70F}}$ and $A_{s}$ , while the region of high amplitude $\unicode[STIX]{x1D714}_{z}$ departs from the wall. This indicates that the initial ‘linear’ streaks are deformed in the wall-normal direction due to nonlinear effects (Mao et al. Reference Mao, Zaki, Sherwin and Blackburn2017). Finally, at $t>400$ , abrupt secondary instability of these nonlinearly saturated flows occurs and breakdown to turbulence is finally reached.

Figure 9.

Streamwise and spanwise vorticities averaged along the homogeneous directions $x$ – $z$ , $\langle \unicode[STIX]{x1D714}_{x}\rangle ,\langle \unicode[STIX]{x1D714}_{z}\rangle$ in (a–c) and (d–f), respectively. Panels (g–i) show the time evolution of the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ , coloured by the spanwise deformation amplitude $A_{s}$ . Plots are arranged in columns according to the $L_{s}$ to which they belong.

The development of nonlinearly saturated streaks and the consequent creation of hairpin vortices can be further analysed by following the time evolution of the streamwise perturbation and of the $uv$ product averaged in the wall-parallel planes as plotted in figure 10 together with the time variation of $A_{s}$ coloured by the skin friction coefficient $C_{f}$ . The first quantity, $\langle u\rangle$ allows us to distinguish linear from nonlinear streaks. In fact, linear high- and low-speed streaks are placed at the same wall-normal positions (Schmid & Henningson Reference Schmid and Henningson2001), thus spanwise averaging cancels out their contribution in the $\langle u\rangle$ term. When nonlinearity sets in, since low- (high-) speed streaks migrate upwards (downwards), the quantity $\langle u\rangle$ departs from zero. The second quantity, $\langle uv\rangle$ , allows us to identify the presence of sweep and ejection events, characterised by large values of $\langle uv\rangle$ with $u$ and $v$ anticorrelated in sign, thus belonging to the Q4 and Q2 quadrants of the $u{-}v$ plane, as in Adrian (Reference Adrian2007) and Farano et al. (Reference Farano, Cherubini, Robinet and De Palma2017), although in the present work $u,v$ represent fluctuations relative to the base flow (see also Farano et al. (Reference Farano, Cherubini, Robinet and De Palma2015)) and not to the mean flow as in the previously mentioned work. The presence of these events has been indicated in the figure using the Q4 and Q2 nomenclature, as a reminder of the $\langle uv\rangle$ velocity quadrant to which they belong. Comparing figures 10(a), 10(b) and 10(c), one can see that, when $L_{s}$ increases from $0$ to $0.02$ , in the time range $t=[0,120]$ the quantity $\langle u\rangle$ decreases to zero close to the wall, indicating that near-wall linear streaks almost disappear. In the same time range, sweeps and ejections are completely absent in the $L_{s}=0.02$ case, whereas for $L_{s}=0,0.01$ they occur in the narrow $t=[75,100]$ window (compare figures 10 d, e and f). In this time range Q4 (Q2) events occur close to (far from) the wall, as typically observed in the presence of hairpin vortices. On the contrary, at $200<t<400$ for the $L_{s}=0.02$ case, ejections are observed close to the wall whereas sweeps lay in the outer region (see figure 10 f). This feature can be associated with the onset of nonlinearities saturating the streaks, where the lift-up of the low-speed streaks initially placed in the near-wall region induces Q2 events close to the wall and the downward motion of high-speed streaks produces Q4 events far from the wall. The same wall-normal arrangement of Q2 and Q4 events is observed in all the three cases in the turbulent regime, namely at $t>150$ for $L_{s}=0,0.01$ and at $t>400$ for $L_{s}=0.02$ (as indicated by the rapid increase of $C_{f}$ at these times), due to the presence of nonlinear streaks in the buffer and viscous layer in the fully turbulent flow (Jiménez Reference Jiménez2013).

Figure 10.

(a–c) Contours of the streamwise velocity disturbances averaged onto the $x{-}z$ plane, $\langle u\rangle$ . (d–f) Contours of the $uv$ product measuring Q2 and Q4 events averaged on wall-parallel planes, $\langle uv\rangle$ . (g–i) Time evolution of the spanwise amplitude deformation $A_{s}$ , coloured by the friction coefficient $C_{f}$ . Each column is associated with a given slip length, $L_{s}=0.00,0.01,0.02$ from (a–i).

The effect of the slip boundary condition on sweep and ejection events in the nonlinear phase of the transition scenario can be further analysed inspecting the distribution of the $u{-}v$ probability density function averaged over the wall-parallel planes in the near-wall region ( $0<y<1$ ), which is provided in figure 11(a–c), together with the wall-normal distribution of the positive/negative streamwise velocity disturbances averaged on $x{-}z$ planes ( $u^{+},u^{-}$ ) provided in figure 11(d–f), for different values of $L_{s}$ . Sweep and ejection events are attenuated when the slip length is increased from $L_{s}=0.0$ to $L_{s}=0.01$ (compare figure 11 a with 11 b), whereas the $u^{+}$ and $u^{-}$ velocity profiles in figure 11(d,e), representing the wake of hairpin legs and heads respectively, remain qualitatively similar. In the case $L_{s}=0.02$ , Q4 events are strongly inhibited and the probability density function is mostly dominated by Q2–Q3 events (see figure 11 c), indicating that the flow is characterised by stronger low-speed streaks whose peak value moves upwards in the wall-normal direction (see figure 11 c, bottom), further evidence of the development of nonlinear streaks (Mao et al. Reference Mao, Zaki, Sherwin and Blackburn2017). Very similar $u{-}v$ distribution is found further from the wall (not shown), confirming that the reduced shear succeeds in inhibiting Q4 events everywhere in the flow. These results clearly indicate that for a sufficiently large slip length ( $L_{s}=0.02$ in the present configuration) the SHS considerably alters the mechanisms leading to the creation of $\unicode[STIX]{x1D706}$ and hairpin vortices, inhibiting Q4 events and consequent transition to turbulence. To sum up, K-type transition can be delayed or even completely avoided in the presence of SHS of sufficiently large slip length, for a given amplitude of the imposed initial perturbations.

Figure 11.

(a–c) Contours of the logarithm of the probability density function (p.d.f.) of the wall-normal and streamwise velocity disturbances in a $u{-}v$ plane ( $w=0$ ) averaged in the wall-parallel planes in the region $0<y<1$ . Values of the logarithm of the p.d.f. have been normalised with respect to the total number of points of the computational domain. (d–f) Profiles of the positive/negative ( $u^{+}/u^{-}$ ) streamwise velocity disturbances, averaged on $x{-}z$ planes. For each computed $L_{s}$ , arranged in columns, we plot the data at time $T$ so that the $Re_{\unicode[STIX]{x1D70F}}(T)=1.1\cdot Re_{\unicode[STIX]{x1D70F}}(t=0)$ , at the beginning of the nonlinear phase.

Figure 12.

Wall-normal integral of the averaged spanwise and streamwise components of vorticity for different slip lengths. A scaled time $t^{\ast }=t/t_{R}$ is used for sake of visualisation, with reference time $t_{R}$ conventionally set to the time at which $Re_{\unicode[STIX]{x1D70F}}$ reaches $1.1Re_{\unicode[STIX]{x1D70F}}(t=0)$ . Straight lines highlight the exponential growth of $\unicode[STIX]{x1D714}_{x}$ indicating the vorticity stretching phase (Malm, Schlatter & Sandham Reference Malm, Schlatter and Sandham2011).

3.2.4 How slippery surfaces damp the growth of $\unicode[STIX]{x1D6EC}$ vortices

We have shown that a slippery boundary is capable of delaying K-type transition by considerably modifying the coherent structures occurring in the process (Bake, Meyer & Rist Reference Bake, Meyer and Rist2002; Sayadi, Hamman & Moin Reference Sayadi, Hamman and Moin2013). In particular, figure 7 clearly shows that introducing a slip length inhibits the development of hairpin vortices by strongly damping the growth of $\unicode[STIX]{x1D6EC}$ vortices. Since during K-type transition $\unicode[STIX]{x1D6EC}$ vortices result from the vortex tilting of the spanwise vorticity associated with the initially imposed TS waves (Malm et al. Reference Malm, Schlatter and Sandham2011), we focus on the effect of the slippery walls on the streamwise and spanwise vorticities. Figure 12 provides the time evolution of the wall-normal averaged values of $\langle \unicode[STIX]{x1D714}_{x}\rangle ,\langle \unicode[STIX]{x1D714}_{z}\rangle$ for the three considered slip lengths. For $L_{s}=0.00$ we observe an initial increase of $\unicode[STIX]{x1D714}_{z}$ due to the presence of TS waves near the walls. After the spanwise vorticity has reached a plateau value, $\unicode[STIX]{x1D714}_{x}$ begins to increase exponentially due to vortex tilting (Malm et al. Reference Malm, Schlatter and Sandham2011), producing $\unicode[STIX]{x1D6EC}$ vortices. The vortex tilting phase leading to the creation of $\unicode[STIX]{x1D6EC}$ and subsequent hairpin vortices can be visualised in figure 13. Increasing $L_{s}$ results in a reduction of $\unicode[STIX]{x1D714}_{z}$ right during the vortex tilting phase, with a consequent weaker growth of $\unicode[STIX]{x1D714}_{x}$ (see figure 14). This effect can be analysed by evaluating the different terms of the vorticity transport equation, which for an incompressible flow reads

(3.8) $$\begin{eqnarray}\frac{\text{D}\unicode[STIX]{x1D74E}}{\text{D}t}=(\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}+\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D74E},\end{eqnarray}$$

where $\unicode[STIX]{x1D74E}$ is the three-dimensional vorticity vector field. Assuming $Re$ to be large enough so that the diffusive term is small (Ye, Schrijer & Scarano Reference Ye, Schrijer and Scarano2018), the temporal evolution of the spanwise vorticity reduces to

(3.9) $$\begin{eqnarray}\frac{\text{D}\unicode[STIX]{x1D714}_{z}}{\text{D}t}\approx \underbrace{-\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}z}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x}}_{T_{zx}}\underbrace{+\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}y}}_{T_{zy}}+\underbrace{\unicode[STIX]{x1D714}_{z}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}}_{S_{zz}},\end{eqnarray}$$

where $T_{zx},T_{zy}$ and $S_{zz}$ represent the vorticity tilting and stretching terms for the spanwise component, respectively. Further development of the latter relation provides the equation which describes the temporal evolution of the spanwise vorticity

(3.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}}{\unicode[STIX]{x2202}t}\approx -\underbrace{\left(u\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}}{\unicode[STIX]{x2202}y}+w\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}}{\unicode[STIX]{x2202}z}\right)}_{A_{z}}+\,T_{zx}+T_{zy}+S_{zz}.\end{eqnarray}$$

The volume integral of each term of (3.10) for different values of $L_{s}$ , normalised by their value at $L_{s}=0$ , is represented in figure 15 for $t=23$ . At this time, for $L_{s}=0.00$ , $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}/\unicode[STIX]{x2202}t\approx 0$ results from a balance of all terms in (3.10) (although the balance is not exact due to the terms which have been neglected from (3.8)) so that $\unicode[STIX]{x1D714}_{z}$ remains constant in time for a large time range. Introducing a slippery boundary, the wall-normal gradients $\unicode[STIX]{x2202}(\bullet )/\unicode[STIX]{x2202}y$ (Min & Kim Reference Min and Kim2004) are reduced, inducing a strong decrease of the vortex tilting term $T_{zy}=\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z\cdot \unicode[STIX]{x2202}w/\unicode[STIX]{x2202}y$ . It is important to remark that all terms in (3.10) are affected by the slip length. However, the reduction of the stretching term is indirectly due to the decrease of $\unicode[STIX]{x1D714}_{z}$ , rather than to the reduction of the wall-normal shear directly induced by the slippery boundary. Thus, as the slip length is increases, $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}/\unicode[STIX]{x2202}t$ departs from zero due to the imbalance of the advection, tilting and stretching terms in (3.10). This leads to a decrease of the spanwise vorticity and consequently of the vortex-tilting-induced streamwise one, inhibiting the growth of $\unicode[STIX]{x1D6EC}$ vortices. Thus, introducing a slippery boundary mildly influences the evolution of the TS spanwise vortices, while strongly affecting the subsequent onset of $\unicode[STIX]{x1D6EC}$ vortices by reducing $\unicode[STIX]{x1D714}_{z}$ , as clearly shown in figure 14.

Figure 13.

Isosurfaces of $\unicode[STIX]{x1D706}_{2}$ (red, on the left) and $\unicode[STIX]{x1D714}_{z},\unicode[STIX]{x1D714}_{x}$ (blue, green, on the right) during K-type transition over a no-slip ( $L_{s}=0.00$ ) wall at different times indicated in the plot: $\unicode[STIX]{x1D706}_{2}=-0.001,\unicode[STIX]{x1D714}_{z}=2.0$ for $t=90$ ; $\unicode[STIX]{x1D706}_{2}=-0.02,\unicode[STIX]{x1D714}_{x}=0.8,\unicode[STIX]{x1D714}_{z}=3.0$ for the other snapshots. Isosurfaces are coloured with their wall-normal location (darker is closer to the wall). PPF, plane Poiseuille flow.

Figure 14.

Isosurfaces of $\unicode[STIX]{x1D706}_{2}=-0.001$ (red, on the left) and $\unicode[STIX]{x1D714}_{z}=1.475,\unicode[STIX]{x1D714}_{x}=1.0$ (blue, green, on the right) at $t=90$ during K-type transition over a no-slip wall and for increasing slip lengths. Whilst spanwise vortices arising from the weakly nonlinear interaction of TS waves appear to remain substantially unchanged, $\unicode[STIX]{x1D714}_{z}$ is strongly reduced by the introduction of a slippery wall.

Figure 15.

Volume integral of the different terms in the spanwise vorticity equation (3.10) for $t=23$ and different slip lengths $L_{s}$ , normalised with respect to their values measured for $L_{s}=0.00$ . On the right-hand side, the $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}/\unicode[STIX]{x2202}t$ term, scaled with respect to its value at $L_{s}=0.02$ is shown. The scaling chosen enhances how the evolution of each term of (3.10) changes when wall slip is increased.

4.1 Optimal perturbations

In subcritical conditions in a noisy environment, streaky perturbations are usually observed prior to transition (Kendall Reference Kendall1998; Saric, Reed & Kerschen Reference Saric, Reed and Kerschen2002; Manneville Reference Manneville2015), since small-amplitude environmental perturbations project in time onto the largest singular value of the linearised Navier–Stokes operator $\boldsymbol{L}$ (Luchini Reference Luchini2000), whose associated response corresponds to the optimal perturbation for the considered flow (Farrell Reference Farrell1988). Optimal perturbations are able to induce, by means of non-normal yet linear interaction of eigenmodes of $\boldsymbol{L}$ , the maximum possible transient energy growth in a finite time range (Schmid & Henningson Reference Schmid and Henningson2001). This energy growth allows for the development of nonlinear interactions, leading the flow to transition. To study the effect of the SHS on this optimal transition scenario, we impose as initial condition $\boldsymbol{U}(\boldsymbol{x},t=0)=\boldsymbol{U}_{b}(\boldsymbol{x})+\unicode[STIX]{x1D716}\boldsymbol{u}_{0}^{Opt}(\boldsymbol{x})$ , where $\unicode[STIX]{x1D716}\boldsymbol{u}_{0}^{Opt}(\boldsymbol{x})$ is the initial optimal perturbation computed in the chosen flow conditions and $\unicode[STIX]{x1D716}$ is a given amplitude, which has to be set sufficiently high to trigger nonlinear effects and consequently lead to transition (Reshotko Reference Reshotko2001). Being derived from linearised equations, the optimal perturbation can be freely scaled so to match an arbitrarily imposed energy level. If the initial perturbation energy is set to a very low value, the initial perturbation, composed of streamwise vortices, linearly evolves in time, creating optimal streaks which asymptotically fade away, as predicted by the linear theory. Increasing the initial energy level to $E(0)=10^{-6}$ , as shown in figure 16 for the no-slip case, after an initial linear growth phase ( $0<t<200$ ), the streaks saturate nonlinearly (see the inset at the panel c) to an amplitude $A_{s}$ . As soon as this amplitude overcomes a critical threshold (Brandt et al. Reference Brandt, Cossu, Chomaz, Huerre and Henningson2003) secondary instabilities of the nonlinearly saturated optimal streaks set in ( $t\approx 400$ ), immediately followed by breakdown to turbulence. Further increasing the initial energy to $E(0)=10^{-5}$ leads to a more rapid departure from the linearly predicted energy growth curve due to the onset of nonlinearities already at very small times, resulting in a different saturation energy threshold. Nonlinearly saturated streaks, strongly deformed with respect to the linear optimal solution, as shown in the top right inset, are formed already at $t\approx 150$ and due to secondary instability, breakdown to turbulence is reached already at $t\approx 200$ .

Figure 16.

(a) Time evolution of the kinetic energy density of the velocity perturbation during laminar–turbulent transitions triggered by linear optimal perturbations, for a no-slip wall ( $L_{s}=0.00$ ) and for two prescribed values of the initial energy, $E(t=0)=10^{-6}$ and $E(t=0)=10^{-5}$ (thick solid lines coloured by the amplitude of spanwise modulation $A_{s}$ , as defined in equation 3.7). The dashed lines indicate the energy growth as predicted by the linear theory for the two imposed initial energy levels. (b) Isocontours of the streamwise velocity perturbation in a $z{-}y$ plane for the case with $E(t=0)=10^{-6}$ at $t=150$ showing the highly deformed nonlinearly saturated streaks. (c) Isocontours of the streamwise velocity perturbation in a $z{-}y$ plane for the case with $E(t=0)=10^{-5}$ at $t=380$ . In both (b) and (c), the velocity field is invariant in the streamwise direction, therefore any $x$ -normal plane will provide the same data.

Introducing a slippery boundary does not modify this scenario, as depicted in figure 17. For the smallest considered energy level, the growth rate curves for the cases $L_{s}=0.00,0.02$ are very close to each other, although a slightly larger energy gain is reached in the slippery case. The same effect is observed for the largest energy level considered here, where the case at $L_{s}=0.02$ appears to follow more closely the linear evolution of $G(t)$ with respect to the no-slip one. This is probably due to the fact that in the no-slip case a slightly larger energy growth is predicted by the linear theory at small times (compare the solid with the dashed black lines in the left frame), resulting in a more rapid onset of nonlinear effects which translates into an earlier and more pronounced deviation from the linear energy growth curve. However, in both cases, secondary instability is reached practically at the same time, since the energy level reached in the saturation phase (see the right frame of figure 17) and the shape of the nonlinearly saturated streaks (see figure 18) appear to be almost independent on the imposed slip length. The fact that, differently from the K-type one, the optimal transition scenario is virtually unaffected by the introduction of a slip length can be explained by the fact that the optimal perturbations are localised in the bulk of the channel, as opposed to TS waves which are placed close to the wall. Since SHS act by modifying the flow shear near the wall, it appears clear why optimal streaks are unaffected by this boundary modification even when nonlinearities kick in, as depicted in figure 18.

Figure 17.

Influence of SHS on subcritical transition at $Re=5000$ triggered by linear optimal perturbations. Both panels represent the same dataset, but with different scales. In (a) the energy gain evolution is plotted to highlight the initial linear phases, whereas (b) provides the energy, for two initial energy levels and two slip lengths, as indicated in the legend. The dashed lines indicate the energy growth as predicted by the linear theory for the two imposed initial energy levels.

Figure 18.

Streamwise velocity perturbation measured at $t=150$ in a DNS initialised with the optimal perturbation having initial energy $E(t)=10^{-5}$ for different slip lengths. The nonlinear streaks observed in the no-slip case ( $L_{s}=0.0$ , figure 16 c) match those developing on a slippery surface with $L_{s}=0.02$ .

4.2 Uncontrolled transition

In this section we aim to study how uncontrolled transition, namely that naturally occurring in a noisy environment in subcritical conditions, is affected by the presence of superhydrophobic walls. Uncontrolled transition in a channel flow can be triggered using random noise, a random superposition of Stokes modes or, as recently proposed in Picella et al. (Reference Picella, Bucci, Cherubini and Robinet2019), using an ad hoc volume forcing, constructed by a superposition of optimal forcing functions, able to trigger a large-amplitude response in the flow as a consequence of receptivity. The latter technique, which we refer to as F-type transition, has been chosen here to study the effect of SHS on laminar–turbulent uncontrolled transition. Following the procedure described by Picella et al. (Reference Picella, Bucci, Cherubini and Robinet2019), we have constructed a noisy volume forcing using a basis of harmonic disturbances that maximise (within a linear framework) the flow response for different given frequencies (Schmid & Henningson Reference Schmid and Henningson2001). The synthetic noisy volume forcing is then constructed as a linear composition of these optimal volume forcings issued from resolvent analysis, suitably weighted in order to respect a chosen energy spectrum. In the present case the noisy forcing has been constructed using a basis of $N_{\unicode[STIX]{x1D714}}\times N_{\unicode[STIX]{x1D6FC}}\times N_{\unicode[STIX]{x1D6FD}}=64\times 4\times 4$ optimal forcing functions (referring to the number of temporal, streamwise and spanwise wavenumbers taken into account for constructing it), with an initial turbulence intensity set to $0.35\,\%$ of the reference velocity $U_{r}$ . In figure 19 we show the impact of SHS on uncontrolled transition. Increasing $L_{s}$ is totally ineffective in delaying the onset of turbulence, as shown by the evolution of the friction Reynolds number during F-type transition (figure 19 a). As explained by Picella et al. (Reference Picella, Bucci, Cherubini and Robinet2019) in the framework of a channel flow with no-slip walls, the imposed volume forcing acts similarly to a noisy environmental disturbance, promoting the development of streaks as a response of the flow to external disturbances (Jacobs & Durbin Reference Jacobs and Durbin2001; Schmid & Henningson Reference Schmid and Henningson2001; Brandt, Schlatter & Henningson Reference Brandt, Schlatter and Henningson2004). As previously shown in figure 17, streaks are mildly affected by a slippery boundary condition, being localised far from the walls. This is confirmed by figure 19(b), where the time evolution of the Fourier modes $(0,1)$ , $(1,1)$ and $(0,2)$ is provided, clearly showing the onset of streaks (i.e. $(0,1)$ ) and $(0,2)$ modes) from a noisy disturbance due to receptivity, followed by their nonlinear saturation as well as their final breakdown. Practically no difference can be noticed between the modes’ evolution for the considered three slip lengths. The time evolution of the coherent structures associated with uncontrolled transition is provided in figure 20. The flow evolution appears very similar for the three slip lengths considered, except for the non-zero velocities measured at the wall in the SHS cases, which have practically no influence on the mechanisms leading to transition. Thus, we can conclude that uncontrolled transition cannot be effectively controlled by SHS, at least for the considered values of the slip length. In fact, similarly to the optimal transition scenario, uncontrolled transition mostly relies on the onset and instability of streamwise streaks, which are both intrinsically inviscid mechanisms not strongly affected by the presence of a slippery wall.

Figure 19.

Time evolution of the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ and of the energy of three selected Fourier modes for channel flows with different slip lengths undergoing F-type transition. The process is completely unaffected by the increase of the slip length $L_{s}$ .

Figure 20.

Overview of F-type transition over superhydrophobic surfaces. As in figure 7, snapshots depict the isosurfaces of the $\unicode[STIX]{x1D706}_{2}$ criterion and streamwise velocity at the wall for different values of the slip length $L_{s}$ . This uncontrolled scenario, characterised by the onset of streaky velocity perturbations occurring in the flow bulk, appears to be completely unaffected by the introduction of a slippery boundary.