2. Flow configuration and numerical simulations

We consider the motion of an incompressible viscous fluid through an infinitely long duct with a rectangular cross-section of half-height $h$ and half-span $s$ ( ${\geqslant}h$ ). The fluid motion is driven by a constant flow rate $F$ , and the bulk mean velocity $u_{b}=F/(4sh)$ is temporally invariant. A right-handed Cartesian coordinate system $(x,y,z)$ is introduced with its origin on the centre axis along which the coordinate $x$ is defined. The coordinates $y$ and $z$ are taken in the height direction and the spanwise direction, respectively. The velocity components are denoted by $u$ , $v$ and $w$ along the $x$ -, $y$ - and $z$ -directions, respectively. No-slip and impermeable conditions $u=v=w=0$ are imposed on the duct walls, and the flow field is assumed to be periodic of length $L_{x}$ in the $x$ -direction. The dimensionless control parameters are the Reynolds number $\mathit{Re}=u_{b}h/{\it\nu}$ and the aspect ratio $A=s/h$ ( ${\geqslant}1$ ), where ${\it\nu}$ is the kinematic viscosity of the fluid.

We solve numerically the incompressible Navier–Stokes equations with a fractional-step method (see Pinelli et al. (Reference Pinelli, Uhlmann, Sekimoto and Kawahara2010) for the details of the numerical method and its validation). The time integration is based on the Crank–Nicolson scheme for the viscous terms and a three-step Runge–Kutta method for the nonlinear terms including the pressure term. The flow variables are expanded in terms of truncated Fourier series in the $x$ -direction and Chebyshev polynomials in the $y$ - and $z$ -directions. We use a collocated grid arrangement in physical space constructed from an equidistant spacing ${\rm\Delta}x$ in the $x$ -direction and the Chebyshev–Gauss–Lobatto points in the $y$ - and $z$ -directions. The nonlinear terms are evaluated in physical space whereas the explicit contribution of the linear terms is evaluated in spectral space. The fields are transformed back and forth by means of the fast Fourier transform and fast cosine transform. De-aliasing according to the $2/3$ rule is performed in the $x$ -direction. In numerical simulations the aspect ratio is set at $A=1$ , $2$ , $3$ , $4$ , $5$ , $9$ and the Reynolds number $\mathit{Re}$ is varied in the range $650\leqslant \mathit{Re}\leqslant 1500$ . In all the simulations the streamwise period is fixed at $L_{x}=40{\rm\pi}h$ . This is larger than the streamwise period ( $32{\rm\pi}$ times the pipe radius) used by Avila et al. (Reference Avila, Willis and Hof2010) in pipe-flow simulations to sufficiently avoid the interaction between leading and trailing edges of a turbulent puff that would arise in short domains.

3. Identifications of the lowest Reynolds number for transient turbulence

Although the onset Reynolds number of sustaining turbulence in pipes has been determined as a consequence of counteractions between the decay and splitting of turbulent puffs (Avila et al. Reference Avila, Moxey, Lozar, Avila, Barkley and Hof2011), the onset of sustaining turbulence in plane channels or rectangular ducts is difficult to identify. In this study, therefore, we consider the lowest Reynolds number of localized transient turbulence to investigate the dependence of turbulence structures at the lowest Reynolds number on the aspect ratio of a rectangular duct.

We first identify the lowest Reynolds numbers $\mathit{Re}_{T}$ that allow transiently sustaining turbulence for $A=1$ – $9$ . For each value of $A$ the Reynolds number $\mathit{Re}$ of the flow is gradually reduced by $5$ units from $\mathit{Re}=1500$ , at which a fully developed turbulent state can be observed. After each reduction of $\mathit{Re}$ the computations are run until either the flow relaminarizes or a minimum time $900{\it\nu}/u_{{\it\tau}}^{2}$ has elapsed, where $u_{{\it\tau}}=\sqrt{{\it\tau}/{\it\rho}}$ is the friction velocity, ${\it\tau}$ and ${\it\rho}$ being the wall shear stress averaged over the four walls and time, and the mass density of the fluid, respectively. The sustainment of turbulence is verified at every time step by monitoring the maximum value of the (positive) second invariant $Q$ of the velocity gradient tensor (which represents vortex structures) over the whole flow field. If the maximum value of $Q({\it\nu}/u_{{\it\tau}}^{2})^{2}$ is greater than $2\times 10^{-3}$ , then the turbulence is judged to survive at that time. We measure the lowest value, $\mathit{Re}_{\mathit{L}}$ , of the Reynolds number for the survival. For each value of the aspect ratio simulations are run with five distinct initial data, among which the minimum of $\mathit{Re}_{\mathit{L}}$ is identified as the lowest Reynolds number henceforth denoted as $\mathit{Re}_{T}$ . It has been checked that changes in the threshold value for $Q({\it\nu}/u_{{\it\tau}}^{2})^{2}$ do not lead to a significant difference in $\mathit{Re}_{T}$ . The simulation parameters in the typical runs where $\mathit{Re}_{T}$ has been identified are shown in table 1. Therein $\mathit{Re}_{{\it\tau}}=u_{{\it\tau}}h/{\it\nu}$ denotes the friction Reynolds number, and ${\rm\Delta}y$ and ${\rm\Delta}z$ are the grid intervals in the $y$ - and $z$ -directions, respectively.

Table 1.

Simulation parameters in typical runs.

In the present identification a relatively short duration time $900{\it\nu}/u_{{\it\tau}}^{2}$ has been taken, because localized turbulence will always decay after its finite lifetime, the probability distribution of which is exponential (Hof et al. Reference Hof, Westerweel, Schneider and Eckhardt2006). If the duration time were longer, therefore, much higher values for $\mathit{Re}_{T}$ would be obtained. Near-wall turbulence is known to survive by the regeneration cycle consisting of streamwise vortices and streaks (Hamilton, Kim & Waleffe Reference Hamilton, Kim and Waleffe1995), and the period of the cycle is about $282{\it\nu}/u_{{\it\tau}}^{2}$ at the Reynolds number $\mathit{Re}=1333$ in the minimal flow unit ( ${\rm\pi}h\times 2h\times 0.35{\rm\pi}h$ ) of plane channel turbulence (Jiménez & Moin Reference Jiménez and Moin1991). In this study, accordingly, the duration time has been set to be short in bulk time units (say, $238h/u_{b}$ – $292h/u_{b}$ ) but longer than the period of the near-wall regeneration cycle. A change of the duration time to $900{\it\nu}/u_{{\it\tau}}^{2}-400{\it\nu}/u_{{\it\tau}}^{2}$ (or $+400{\it\nu}/u_{{\it\tau}}^{2}$ ) was found to lead to a variation of $\mathit{Re}_{T}$ by $-2.5$ (or $+10.8$ ) on average. The variation of $\mathit{Re}_{T}$ with decreasing duration time is much smaller than that with increasing duration time, implying that a duration time comparable to the near-wall regeneration period would bring about a roughly consistent value of $\mathit{Re}_{T}$ . This might be because the near-wall regeneration cycle would play a key role in the transient sustainment of turbulence. It should be noted that the values of $\mathit{Re}_{T}$ have been estimated in a specific way, and thus they are neither universal nor strict approximations to universal values. However, we would like to stress that localized turbulence itself can be strictly defined as a non-attracting chaotic invariant set, i.e. a chaotic saddle (Kantz & Grassberger Reference Kantz and Grassberger1985; Lai & Winslow Reference Lai and Winslow1995), in phase space and that the purpose of this work is to demonstrate the dependence of localized turbulence structures on the aspect ratio rather than to strictly determine their onset Reynolds number.

Figure 1.

The lowest Reynolds number $\mathit{Re}_{T}$ (the minimal $\mathit{Re}_{\mathit{L}}$ ) of transient turbulence as a function of the aspect ratio $A$ . The values of $\mathit{Re}_{\mathit{L}}$ measured in the other runs are also shown by the small symbols +. The dashed horizontal line denotes the value of $\mathit{Re}_{T}=667$ .

Figure 1 shows the dependence of the lowest Reynolds number $\mathit{Re}_{T}$ of localized turbulence on the aspect ratio $A$ . The lowest Reynolds number for a square duct measures $\mathit{Re}_{T}=730$ , which is lower than the recently reported marginal Reynolds number, $\mathit{Re}\approx 1100$ , in a square duct with minimal streamwise extension (Uhlmann et al. Reference Uhlmann, Pinelli, Kawahara and Sekimoto2007). It is also lower than the onset Reynolds number (based on a pipe radius), $\mathit{Re}=860$ , of turbulent puffs in pipe flow (Avila et al. Reference Avila, Willis and Hof2010). Let us define an equivalent radius of a square duct with half-height $h$ in the sense of a consistent cross-sectional area as $r=(2/\sqrt{{\rm\pi}})h$ ; the lowest Reynolds number corresponding to pipe flow can then be expressed as $(r/h)\mathit{Re}_{T}=824$ , which is much closer to the reported value $860$ . It can be seen from figure 1 that $\mathit{Re}_{T}$ decreases monotonically with increasing $A$ . When based on the maximum velocity of a laminar profile for a fixed bulk velocity the reduction of the lowest Reynolds number (not shown) is even more striking than that of $\mathit{Re}_{T}$ shown in figure 1. The monotonic decrease of $\mathit{Re}_{T}$ is thought intuitively to be a consequence of the weaker constraint that the sidewalls in the wider duct impose on the flow. The stabilization of infinitesimal disturbances by the sidewalls has also been found in laminar rectangular-duct flow (Tatsumi & Yoshimura Reference Tatsumi and Yoshimura1990). For $A=5$ , however, the lowest Reynolds number nearly attains the minimal value of $\mathit{Re}_{T}\approx 670$ , which is comparable to the onset Reynolds number for turbulent spots in a plane channel (Carlson et al. Reference Carlson, Widnall and Peeters1982; Hashimoto et al. Reference Hashimoto, Hasobe, Tsukahara, Kawaguchi and Kawamura2009). Dean (Reference Dean1978) reported experimentally that in a fully developed turbulent state in rectangular-duct flows a skin friction coefficient is independent of the aspect ratio $A$ if $A\geqslant 7$ . Although there should be a significant difference between transitional and fully turbulent flows, the effects of the sidewalls on the lowest Reynolds number of turbulence as well as the turbulent skin friction almost disappear for comparably small values of the aspect ratio $A=5$ –7.

4. Structures of localized turbulence

We next discuss the turbulence structures at the lowest Reynolds numbers $\mathit{Re}=\mathit{Re}_{T}$ for each geometrical aspect ratio. In figure 2 are shown vortex structures visualized by positive isosurfaces of the second invariant of the velocity gradient tensor. It is observed that turbulence structures exhibit spatial localization for all the values of $A$ . Localized turbulence can be seen to be composed of a large number of streamwise tubular vortices. For aspect ratios close to unity ( $A=1$ – $3$ ) there appear streamwise-localized clusters of vortical structures that are strikingly similar to turbulent puffs in pipe flow (Wygnanski & Champagne Reference Wygnanski and Champagne1973). Similar to equilibrium puffs in pipe flow the localized clusters for $A=1$ – $3$ do not extend or shrink in time, and their streamwise extent is about $30h$ – $40h$ , which is comparable with the length of approximately $20$ diameters of the equilibrium puffs (Mullin Reference Mullin2011). For the larger aspect ratio ( $A=4$ , $5$ , $9$ ), on the other hand, the vortical structures are found to disappear from the vicinity of the sidewalls ( $A=4$ ), and the vortical clusters are localized not only in the streamwise direction but also in the spanwise direction ( $A=5$ , $9$ ). The vortical clusters for $A=5$ , $9$ will change their shape but will not fully extend in the spanwise direction. The roundish clusters for $A=5,9$ in figure 2 are reminiscent of turbulent spots in plane channel flow.

In the wider duct a spanwise localization of turbulence has been found in figure 2; however, the shapes of the visualized turbulence clusters depend on the threshold of isosurfaces. In order to supplement the demonstration of this crucial structural change with $A$ , we shall compare turbulence intensity on the sidewalls $z/h=\pm A$ with that on the longer two walls $y/h=\pm 1$ . As the typical turbulence intensity on the walls, let us take the average, $\langle {\it\omega}_{xs}^{2}\rangle$ or $\langle {\it\omega}_{xl}^{2}\rangle$ , of the squared streamwise vorticity on the sidewalls or the longer walls. Here the average is performed over time and over the ‘vigorous streamwise extent’, the latter being defined as the streamwise interval in which $\langle {\it\omega}_{x}^{2}\rangle _{yz}\geqslant \max (\langle {\it\omega}_{x}^{2}\rangle _{yz})/2$ , $\langle \cdot \rangle _{yz}$ denoting integration over the cross-plane. The turbulence intensity on the sidewalls compared with that on the other walls, $\langle {\it\omega}_{xs}^{2}\rangle /\langle {\it\omega}_{xl}^{2}\rangle$ , is shown as a function of $A$ in figure 3. It can be seen that the relative turbulence intensity on the sidewalls attenuates with increasing $A$ . It is confirmed quantitatively that the turbulence on the sidewalls significantly decays for $A=4,5,9$ . In the ducts of $A=5,9$ , for which turbulent spots have been observed, the turbulence on the sidewalls has disappeared almost completely.

Figure 2.

Instantaneous vortical structures visualized by isosurfaces of the second invariant of the velocity gradient tensor, $Q\,({\it\nu}/u_{{\it\tau}}^{2})^{2}=2\times 10^{-3}$ at the lowest Reynolds numbers $\mathit{Re}=\mathit{Re}_{T}$ (see table 1 for the values of $\mathit{Re}_{T}$ for each $A$ ). (a) $A=1$ , (b) $A=2$ , (c) $A=3$ , (d) $A=4$ , (e) $A=5$ , (f) $A=9$ . All the localized structures have been centred in the $x$ -direction.

Figure 3.

The relative turbulence intensity $\langle {\it\omega}_{xs}^{2}\rangle /\langle {\it\omega}_{xl}^{2}\rangle$ on the sidewalls as a function of  $A$ .

The expulsion of turbulence from the vicinity of the sidewalls should be a consequence of relaminarization of turbulent flow over the sidewalls rather than the other two walls. In order to inspect this aspect we consider the behaviour of the main contributor to near-wall turbulence energy production $P=-\langle u^{\prime }w^{\prime }\rangle \partial \langle u\rangle /\partial z|_{y/h=0}$ (with respect to the sidewall $z/h=-A$ ) along the sidewall bisector, where the average has been taken over time and the vigorous streamwise extent, and the prime denotes a fluctuation around the average. As shown in figure 4, the turbulence production for $A=2$ possesses a remarkable peak of about $0.22u_{{\it\tau}s}^{4}/{\it\nu}$ around $(z+s)u_{{\it\tau}s}/{\it\nu}=12$ , which corresponds to that in a plane channel, where $u_{{\it\tau}s}$ is the friction velocity averaged on the sidewall over time and the vigorous streamwise extent. This outstanding production stems from buffer-layer streamwise vortices associated with streamwise streaks on the sidewall as in a plane channel. The turbulence production, however, decreases with increasing $A$ , and for $A\gtrsim 4$ there is no longer any significant production.

Figure 4.

The turbulence production $P$ on the sidewall bisector $y/h=0$ as a function of the distance from the sidewall. ——, $A=2$ ; $\cdots \cdots$ , $A=3$ ; – - – - –, $A=4$ ; - - - -, $A=5$ .

Figure 5.

The full duct width $2h$ along the sidewall normalized with the friction length on the sidewall as a function of $A$ . The dashed horizontal line corresponds to the size $100{\it\nu}/u_{{\it\tau}s}$ of a minimal flow unit in plane channel flow.

Figure 6.

Same as figure 2 but for the higher Reynolds number $\mathit{Re}=800$ than $\mathit{Re}_{T}$ . (a)  $A=2$ , (b) $A=3$ , (c) $A=9$ .

Figure 7.

The skewness indicator $I$ of turbulence structures as a function of $A$ at $\mathit{Re}=800$ . The standard deviation of the skewness indicator is also shown by the vertical bars.

In their numerical simulations of plane channel turbulence Jiménez & Moin (Reference Jiménez and Moin1991) minimized the size of the periodic computational domain while sustaining turbulence activity, obtaining a minimal flow unit with a width of approximately $100$ friction lengths, which accommodates a pair of counter-rotating staggered streamwise vortices and one low-velocity streak. If the width of the flow is less than this minimal value, turbulence dies out. Figure 5 shows the full width of the duct along the sidewall normalized with the friction length ${\it\nu}/u_{{\it\tau}s}$ as a function of $A$ . It can be seen that the normalized width diminishes with increasing $A$ , measuring approximately $100$ for $A=2$ – $3$ . This is a direct consequence of the lower wall shear rate on the sidewalls and the greater kinematic viscosity (i.e. the increased friction length on the sidewalls) for turbulent flow in ducts of larger aspect ratio for the lower values of $\mathit{Re}_{T}$ (see figure 1). For $A\gtrsim 4$ the duct width $2h$ is smaller than $100$ wall units, i.e. it cannot accommodate a minimal flow unit of near-wall turbulence. This is the reason why the flow in the vicinity of the sidewalls relaminarizes, so that the spanwise-localized spots arise for $A\gtrsim 4$ . The remarkable decrease of $2h\,u_{{\it\tau}s}/{\it\nu}$ from $100$ at $A=4$ in figure 5 is due to the relaminarization.

5. Localized turbulence structures at higher Reynolds number

In the following we briefly discuss the modifications to the localized turbulence when the Reynolds number is increased. Instantaneous vortical structures at $\mathit{Re}=800$ (well above $\mathit{Re}_{T}$ ) are shown in figure 6. No essential difference from the turbulent puff at $\mathit{Re}=\mathit{Re}_{T}$ can be seen in the duct with aspect ratio close to unity ( $A=2$ ); however, the turbulent puff in the wider duct ( $A=3$ ) is skewed in the spanwise direction. The meandering turbulence structures in the much wider duct for $A=9$ are remarkably reminiscent of turbulent stripes found in plane channel flow by Hashimoto et al. (Reference Hashimoto, Hasobe, Tsukahara, Kawaguchi and Kawamura2009), who observed that the stripes appear in the Reynolds number range $\mathit{Re}=850$ – $1000$ . The oblique bands in figure 6(c) could arise from growing spots at the higher Reynolds number, as reported in Aida et al. (Reference Aida, Tsukahara and Kawaguchi2010).

In order to examine quantitatively the dependence on the aspect ratio of the skewness of the localized turbulence at the higher Reynolds number, a skewness indicator is introduced as follows. At a given time $t$ let us define $f(x,z,t)=\langle {\it\omega}_{x}^{2}\rangle _{y}/\langle \langle {\it\omega}_{x}^{2}\rangle _{y}\rangle _{xz}$ as a two-dimensional ‘distribution function’ of a vortical cluster, where $\langle \cdot \rangle _{xz}$ denotes the $(x,z)$ -integration in the vigorous streamwise extent and $\langle \cdot \rangle _{y}$ integration over the $y$ -direction. The relative coordinates $(x^{\prime },z^{\prime })$ with respect to the centre of the vortical cluster are then defined as $x^{\prime }=x-\langle xf\rangle _{xz}$ and $z^{\prime }=z-\langle zf\rangle _{xz}$ . Now we compute the averaged modulus of the ‘cross-correlation’ of $(x^{\prime },z^{\prime })$ given by $I=\overline{\,|\langle x^{\prime }z^{\prime }f\rangle _{x^{\prime }z^{\prime }}/\sqrt{\langle x^{\prime 2}f\rangle _{x^{\prime }z^{\prime }}\langle z^{\prime 2}f\rangle _{x^{\prime }z^{\prime }}}|\,}$ which defines the skewness indicator $I$ , where $\overline{(\cdot )}$ represents an ensemble average over $25$ – $70$ realizations for each value of $A$ . In the computations the vigorous streamwise extent of a meandering cluster as in figure 6(c) is split into individual pieces with distinct skewness indicators. As shown in figure 7 the computed skewness of localized turbulence patches gradually increases with increasing $A$ (except for a kink at $A=4$ which is presumably due to a statistical error), attaining a plateau for $A\gtrsim 4$ .

6. Concluding remarks

Direct numerical simulations have been performed for transitional flow in rectangular ducts with various aspect ratios $A$ . The lowest Reynolds number $\mathit{Re}_{T}$ for spatially localized (transiently sustaining) turbulence decreases monotonically from $\mathit{Re}_{T}=730$ for a square duct with increasing $A$ due to the weakening constraint by the sidewalls in the wider duct, and for $A=5$ it approaches the minimal value $\mathit{Re}_{T}\approx 670$ , comparable to the onset Reynolds number of turbulent spots in a plane channel. Turbulent states are characterized by localized structures: there appear streamwise-localized turbulent puffs for $A=1$ – $3$ , while streamwise- and spanwise-localized turbulent spots emerge for $A=5$ – $9$ . A puff–spot transition in localized turbulence at the lowest Reynolds numbers has been found for $A\approx 4$ . The disappearance of turbulence from the vicinity of the sidewalls is responsible for this critical change of the localized structures. In the wider duct flow with $A\gtrsim 4$ the friction length on the sidewalls is so long that the span (approximately $100$ friction lengths) of a self-sustaining minimal flow unit of streamwise vortices and low-velocity streaks is larger than the duct width and, therefore, it cannot be accommodated.

The phenomenological interpretation proposed in this paper of the spanwise localization of turbulence leading to turbulent spots for rectangular-duct flow should be followed by a theoretical demonstration. One candidate for a theoretical approach would be the characterization of localized structures in terms of simple invariant solutions to the Navier–Stokes equation (see Kawahara, Uhlmann & van Veen Reference Kawahara, Uhlmann and van Veen2012). In phase space localized turbulence could be represented by a chaotic saddle (Kantz & Grassberger Reference Kantz and Grassberger1985; Lai & Winslow Reference Lai and Winslow1995). When there exists a chaotic saddle in phase space, trajectories originating from random initial conditions usually wander in the close vicinity of the chaotic saddle for a finite amount of time before escaping the chaotic saddle and settling into the laminar attractor. Simple invariant solutions, such as fixed points, periodic orbits and relative periodic orbits, are embedded in the chaotic saddle. The coherence of localized structures might be expressed by the embedded invariant solutions in turbulence. Since a state point in the chaotic saddle occasionally approaches the simple solutions, a good initial guess of Newton iteration for the localized invariant solutions might be extracted from localized turbulence.

Recently, a streamwise-localized relative periodic solution has been discovered in pipe flow (Avila et al. Reference Avila, Mellbovsky, Roland and Hof2013) while a streamwise- and spanwise-localized relative periodic solution has been found in plane channel flow (Zammert & Eckhardt Reference Zammert and Eckhardt2014). For a shear-driven system Brand & Gibson (Reference Brand and Gibson2014) have discovered a streamwise- and spanwise-localized equilibrium solution in plane Couette flow. In rectangular-duct flow, to our knowledge, there are no reports of spatially localized invariant solutions; however, our own preliminary study suggests that a travelling-wave solution for $A=1$ recently obtained by Uhlmann, Kawahara & Pinelli (Reference Uhlmann, Kawahara and Pinelli2010) would exhibit the spanwise localization when tracked to $A>1$ . More recent inspection done by S. Okino (2014, private communication) implies that there would also exist a streamwise-localized travelling wave in rectangular-duct flow. These latest findings might be good starting points for a future theoretical demonstration of turbulence localization in transitional rectangular-duct flow.