3.2.1. Wake height and lee waves

To form appropriate averaged measures in the near wake, the wake geometry and length scales must be specified and measured in terms that do not presume self-similarity of mean or turbulent velocity profiles. The most clear signal comes from $\overline{{\it\omega}}_{y}$ (shown in figure 3 e) and profiles of $\overline{{\it\omega}}_{y}(z)$ (shown for a range of $Nt$ in figure 5). The decay of the peak $\overline{{\it\omega}}_{y}$ can be observed by tracking the amplitude of crests, and their diffusion is noted in the increasing width of the crests. The peaks occur closer to the wake centre as $Nt$ increases up to $Nt=4$ and the trend is reversed for $Nt\geqslant 4$ . The crests roughly correspond to the locations of the maximum velocity gradient, as can be seen in figure 5(c,d). The redistribution of turbulent kinetic energy from wake edge to centre (see figure 3) is responsible for the changing shape of $\overline{u}$ , as the relatively irregular shape evolves into a more Gaussian-like profile at late time.

Figure 5.

Plots of $\overline{{\it\omega}}_{y}/N$ at specific $Nt$ , for $\mathit{Fr}=1.3$ and $\mathit{Re}=2700$ (R2F1, ♦).

The crests of $\overline{{\it\omega}}_{y}$ may be used to define the edge of the wake, and the distance between the top and bottom crests defines a wake height, $2h(Nt)$ . Though this procedure will eventually fail when the vortices dissipate due to viscosity, it is robust in the near-wake region. The dimensionless wake half-height is defined as $L_{z}=h/R$ , and is shown as a function of $Nt$ in figure 6.

Figure 6.

PLot of $L_{z}(Nt)$ , with least-squares sine series fit. Solid, dashed and dotted lines are for $\mathit{Re}=\{2700,5700,11\,000\}$ respectively; the various symbols correspond to the cases shown in table 1. The solid line with ♦ corresponds to R2F1 shown in figure 5. Data points are subsampled for clarity.

The minimum values, $L_{z,min}$ , for all curves lie between $3\leqslant Nt\leqslant 4$ independent of $\mathit{Fr}$ and $\mathit{Re}$ . If ${\it\lambda}$ is a wavelength in $x$ , it may be written as

(3.1) $$\begin{eqnarray}\frac{{\it\lambda}}{R}=\frac{U\,Nt_{{\it\lambda}}}{NR}=\mathit{Fr}\,Nt_{{\it\lambda}},\end{eqnarray}$$

where $Nt_{{\it\lambda}}$ is the wavelength in $Nt$ . There is a linear relationship between ${\it\lambda}/R$ and $\mathit{Fr}$ , with slope $Nt_{{\it\lambda}}$ . The wave pattern is more pronounced when plotted locally at the wake edge as in figure 7. By measuring the ${\rm\Delta}Nt$ between the first trough and first crest, ${\it\lambda}/R$ is calculated using (3.1) and shown in figure 8(a). The results agree well with the linear theory prediction, $Nt_{{\it\lambda}}=N\times 2{\rm\pi}/N=2{\rm\pi}$ . As $Nt_{{\it\lambda}}$ is associated with the wake contraction, then we may infer it to be a consequence of a steady lee wave behind the grid, with wavelength independent of $\mathit{Re}$ . Such a result is also consistent with experiments in Chomaz et al. (Reference Chomaz, Bonetton and Hopfinger1993) and the simulations in Hanazaki (Reference Hanazaki1988) for sphere wakes with $\mathit{Fr}>1$ .

Figure 7.

Plot of $\overline{w}/U$ at the wake edge.

Figure 8.

(a) Plot of ${\it\lambda}/R$ versus $\mathit{Fr}$ : $+$ and $\times$ , experimental results of Chomaz et al. (Reference Chomaz, Bonetton and Hopfinger1993), with $\mathit{Re}=2177$ and 4514, respectively; solid line, ${\it\lambda}/R=2{\rm\pi}\mathit{Fr}$ ; dashed line, numerical simulation of Hanazaki (Reference Hanazaki1988); other symbols, selected results from this study, see table 1 for $\mathit{Re}$ and $\mathit{Fr}$ . (b) Plot of ${\it\zeta}/R$ versus $\mathit{Fr}$ : solid line, $2{\it\zeta}/R=\mathit{Fr}$ ; other symbols are from the same data source as panel (a). (c) Lee-wave isophase curves (every ${\rm\pi}$ ) in the vertical centre plane, for R2F.6 ( $\bigstar$ ).

By inspection of figure 6, the wave amplitude, plotted in figure 8(b), appears to have no systematic dependence on either $\mathit{Re}$ or $\mathit{Fr}$ . This contrasts sharply with the sphere case shown by Chomaz et al. (Reference Chomaz, Bonetton and Hopfinger1993), where the scaled amplitude ${\it\zeta}/R$ increased linearly in the near wake with $\mathit{Fr}$ for $\mathit{Fr}\in [0.3,0.8]$ , had maximum amplitude ${\it\zeta}/R=1$ for $\mathit{Fr}\in [0.8,1.5]$ , and decreased slowly with $\mathit{Fr}$ after that. Here, the wave amplitude is disconnected from the outer length scale, $R$ , of the grid because fluid particles in the midline, for example, are never obliged to turn over the grid, instead passing straight through. This is a straightforward demonstration of how near-wake geometry can and must depend on the initial and boundary conditions.

The lee waves propagate into the exterior, and isophase lines can be drawn from the zero-crossings of the vertical velocity component. These lines are shown for the lower half-plane in figure 8(c). Similar isophase lines were also reported by Chomaz et al. (Reference Chomaz, Bonetton and Hopfinger1993) for the sphere wake.

The mix of grid permeability and vertical wave motions induced by the partial blockage leads to some interesting effects on the wake. When ${\it\lambda}$ is small, at low $\mathit{Fr}$ , even the inner wake region is influenced by the lee wave, as it reduces the effective wake area, somewhat akin to a nozzle in a jet. Figure 9(a) shows the mean streamwise centreline velocity, $\overline{u}_{0}$ , relative to the grid, for varying $\mathit{Fr}$ at constant $\mathit{Re}$ . (Since the grid is towed from right to left, both $\overline{u}_{0}$ and $U$ are negative.) For all $\mathit{Fr}$ , there is an initial decrease in the relative centreline velocity for $Nt\leqslant 2$ , most likely due to the blockage effect of the grid. For low $\mathit{Fr}=0.6$ and 1, the flow then accelerates to a local maximum at $Nt\sim 3.5$ , where the streamlines pinch inwards in figure 4(b) and at the first minimum in figure 6. With increasing Fr, the distance from the grid to the first trough, $x/R=\mathit{Fr}\,Nt$ , increases, and the local acceleration is less pronounced. The initial dip in $\overline{u}_{0}$ is less evident with higher Re, as shown in figure 9, probably because the grid blockage becomes less effective.

Figure 9.

Centreline streamwise velocity relative to the grid: (a) for $\mathit{Re}=2700$ and varying $\mathit{Fr}$ ; and (b) for $\mathit{Fr}=2.4$ and varying $\mathit{Re}$ .

This difference in the initial dip affects the local wake behaviour for cases with the same Fr but varying Re. When the nominal Re approximately doubles, the towing speed $U$ doubles, and so does $N$ to keep the nominal $\mathit{Fr}$ constant. However, the local flow speed $\overline{u}_{0}$ behind the grid increases by a factor less than 2 (as shown in figure 9 b). Therefore, the local $\mathit{Fr}$ right behind the grid is no longer the same, and, instead, cases with higher nominal $\mathit{Re}$ will have a lower local $\mathit{Fr}$ . Similarly, the local $\mathit{Re}$ is increased by a factor less than 2. An important and somewhat paradoxical consequence is that wakes with higher nominal $\mathit{Re}$ may be less turbulent because the local $\mathit{Fr}$ is lower. (Further evidence will be found later in $\mathit{Fr}_{H}$ in figure 18(a).)

These results suggest that a universal decay law may not exist for the centreline streamwise velocity in the early wake of a towed grid, at least for low $\mathit{Fr}$ . The lee wave is also dominant at low $\mathit{Fr}$ in the early wake of a towed sphere, where it is likely to affect the centreline velocity also. The universal decay law for the early wake suggested by Spedding (Reference Spedding1997) was entirely conjectural based on a presumed non-stratified initial wake, and may not be applied to low $\mathit{Fr}$ . If there are universal conditions, they can only exist at sufficiently high $\mathit{Fr}$ , at least $\mathit{Fr}>4$ .

3.2.2. Free shear layers or wake edge

The eddies at the wake edge may originate from KH instability of each layer. A Strouhal number can be estimated using $\mathit{St}=k{\it\theta}$ , where $k$ is a spectroscopic wavenumber and ${\it\theta}$ is computed from

(3.2) $$\begin{eqnarray}{\it\theta}=\frac{1}{2}\int _{-\infty }^{+\infty }\frac{u}{U}\left(1-\frac{u}{U}\right)\,\text{d}z,\end{eqnarray}$$

as for a stationary grid in a free stream, $U$ . The wavenumber $k$ is determined from the highest peak in the power spectrum of the wake edge fluctuating vertical velocity. The Strouhal number $\mathit{St}$ is shown in figure 10, and varies between 0.2 and 0.3, with an average $\mathit{St}\approx 0.24$ . If $\mathit{St}$ is computed by $\mathit{St}=kR$ , the resulting range $\mathit{St}\in [0.9,1.7]$ agrees well with the similar measure reported by Chomaz et al. (Reference Chomaz, Bonetton and Hopfinger1993) for the homogeneous sphere wake in this $\mathit{Re}$ range. However, there is no obvious dependence on either $\mathit{Re}$ or $\mathit{Fr}$ , in contrast with the $\mathit{Re}^{0.5}$ dependence for the sphere, perhaps because the grid shear layer does not start from a growing boundary layer, but always begins abruptly at the same corner point.

Figure 10.

Strouhal number of the wake-edge vortices. The symbols correspond to the cases listed in table 1.

Initially wake-edge vortices merge with neighbours (figure 3), growing in size but decaying in intensity. The early-time mean vorticity has two primary sources: the vertical shear due to the streamwise velocity gradient, and the horizontal shear due to the lee wave. The mean lateral vorticity $\overline{{\it\omega}}_{y}$ , the horizontal shear $\partial \overline{w}/\partial x$ and the vertical shear $\partial \overline{u}/\partial z$ are shown in figure 11 for $\mathit{Fr}=0.6$ and 9.0. In the early wake, though $\overline{w}$ is about the same magnitude as $\overline{u}$ (evident in figures 7 and 9), the horizontal shear is much smaller than the vertical shear. Even at low $\mathit{Fr}=0.6$ , when the early vertical motion is expected to be vigorous, the vertical shear $\partial \overline{u}/\partial z$ still predominates and accounts for most $\overline{w}_{y}$ at late times, when vertical motion is suppressed. For high $\mathit{Fr}$ , $\overline{w}$ is initially small, and $\partial \overline{u}/\partial z$ remains a close measure of $\overline{{\it\omega}}_{y}$ (figure 11 b). This implies that the decay of $\overline{{\it\omega}}_{y}$ at the wake edge is not strongly influenced by $\mathit{Fr}$ even at early times.

Figure 11.

Horizontal vorticity $\overline{{\it\omega}}_{y}$ (solid line), vertical shear $-\partial \overline{u}/\partial z$ (dashed line) and horizontal shear $\partial \overline{w}/\partial x$ (dotted line) for (a) R2F.6 ( $\bigstar$ ) and (b) R11F9 (▹).

To investigate the effect of $\mathit{Fr}$ and $\mathit{Re}$ on the decay of wake-edge vorticity, an appropriate scaling is sought. Because the major source of $\overline{{\it\omega}}_{y}$ is the vertical shear, it should be non-dimensionalized by $U/R$ . However, unlike the well-studied late wake of a sphere, where the velocity distribution is Gaussian, the early wake of the grid is not fully developed (see figure 5 c,d). Therefore the global length scale $R$ does not properly represent the length scale of the vortices themselves, which only occupy the edge of the wake, and the tow speed $U$ does not represent the characteristic shearing velocity. Instead, one can define a modified local vorticity thickness ${\it\delta}_{{\it\omega}}$ for this shear-layer problem, as the distance between the nearest two locations where $\overline{{\it\omega}}_{y}=0.2\overline{{\it\omega}}_{y,peak}$ , and one may also define a shearing velocity $\overline{u}_{s}$ as the difference between the streamwise velocities at those two locations. Taking the first available ${\it\delta}_{{\it\omega}}$ and $\overline{u}_{s}$ as the initial scales, $\overline{{\it\omega}}_{y}$ may be scaled by $\overline{u}_{s}/{\it\delta}_{{\it\omega}}$ .

The rescaled mean vorticity is plotted against $Nt$ and $x/R$ for $\mathit{Re}=11\,000$ and varying $\mathit{Fr}$ in figure 12. For low $\mathit{Fr}$ , more buoyancy periods are required for the rescaled vorticity to reach the same amplitude, compared with the high- $\mathit{Fr}$ case. This is counter-intuitive if one imagines buoyancy to play an important role in damping $\overline{{\it\omega}}_{y}$ by suppressing vertical motion. When $\overline{{\it\omega}}_{y}$ is plotted against $x/R$ , the curves collapse fairly well (figure 12 b), regardless of $\mathit{Fr}$ . The observation is on the mean vorticity, and $\mathit{Fr}$ could still affect the vertical shear through fluctuating quantities and through $\partial \overline{v}/\partial z$ . This possible influence on the flow instability will be discussed further in the following two sections.

Figure 12.

Rescaled vorticity decay at the wake edge for $\mathit{Re}=11\,000$ : (a) against $Nt$ ; and (b) against $x/R$ .

The effect of $\mathit{Re}$ is shown in figure 13(a) for $\mathit{Fr}=2.4$ . It can be seen that $\overline{{\it\omega}}_{y}$ decays more slowly at higher $\mathit{Re}$ because viscous forces weaken compared with inertia-related forces as $\mathit{Re}$ increases. A simple $\mathit{Re}^{-1}$ scaling as shown in figure 13(b) collapses the curves quite well.

Figure 13.

Rescaled vorticity decay at the wake edge for $\mathit{Fr}=2.4$ : (a) against $x/R$ ; and (b) against $(1/\mathit{Re})(x/R)$ .

Once a vertical structure is set, then the ensuing vertical gradients of the horizontal velocity slow down primarily by viscosity. The strong influence of viscosity and comparative unimportance of stratification parameters is quite consistent with the expected phenomenology in a strongly stratified regime with low buoyancy Reynolds number. It is this regime to which most previous laboratory experiments on strongly stratified turbulence gravitate (Fincham et al. Reference Fincham, Maxworthy and Spedding1996; Spedding Reference Spedding1997; Praud et al. Reference Praud, Fincham and Sommeria2005). The estimation of local buoyancy Reynolds number is considered later.

3.2.3. The magnitude and scaling of shear components

The importance of $\overline{(\partial u/\partial z)^{2}}$ in determining the kinetic energy dissipation is consistent with grid turbulence experiments by Fincham et al. (Reference Fincham, Maxworthy and Spedding1996) and Praud et al. (Reference Praud, Fincham and Sommeria2005). There is a closer resemblance to the sphere wakes of Spedding (Reference Spedding2002) because of the influence of the mean flow, which breaks the expected rough equivalence of $\overline{(\partial u/\partial z)^{2}}$ with $\overline{(\partial v/\partial z)^{2}}$ . Figure 14 shows selected shear components available in the experiments at $Nt=5$ and $Nt=10$ . Though $\overline{(\partial u/\partial z)^{2}}\gg \overline{(\partial v/\partial z)^{2}}$ at early times, the difference between them decreases.

Figure 14.

Shear components in vertical slices, for $\mathit{Fr}=1.3$ and $\mathit{Re}=2700$ (R2F1, ♦). The black and grey lines show $Nt=5$ and 10, respectively. Solid, dashed, dash-dotted and dotted lines show $\overline{(\partial u/\partial z)^{2}}$ , $\overline{(\partial w/\partial x)^{2}}$ , $\overline{(\partial v/\partial x)^{2}}$ and $\overline{(\partial v/\partial z)^{2}}$ , respectively.

It is expected that $\overline{(\partial w/\partial x)^{2}}$ be much smaller than $\overline{(\partial v/\partial x)^{2}}$ (Spedding Reference Spedding2002) since the stratification suppresses the horizontal gradient of vertical velocity, but, owing to the lee-wave-generated mean flow, $\overline{(\partial w/\partial x)^{2}}$ is similar in magnitude to $\overline{(\partial v/\partial x)^{2}}$ at early times, becoming smaller as the amplitude of the lee wave decreases in the far wake.

The strong vertical shear is thought to be an important mechanism for triggering secondary turbulence (Lilly Reference Lilly1983; Hebert & de Bruyn Kops Reference Hebert and de Bruyn Kops2006a ), even in strongly stratified flows (Riley & de Bruyn Kops Reference Riley and de Bruyn Kops2003; Diamessis et al. Reference Diamessis, Spedding and Domaradzki2011), and therefore has received significant attention in the literature. The averaged mean-squared vertical shear $S^{2}$ is

(3.3) $$\begin{eqnarray}S^{2}=\overline{\left(\frac{\partial u}{\partial z}\right)^{2}}+\overline{\left(\frac{\partial v}{\partial z}\right)^{2}}.\end{eqnarray}$$

As $\overline{(\partial u/\partial z)^{2}}$ dominates, non-negligible $S^{2}$ occurs primarily at the wake edge where the most vigorous KH modes occur, and here, as we have already seen, both the geometry and location of these layers are strongly influenced by the lee waves. The evolution of $\langle S^{2}\rangle _{z}/(U/R)^{2}$ ( $\langle \cdots \rangle _{z}$ is a temporal average and the value over the wake region between $[-L_{z},L_{z}]$ ) is shown in figure 15. Note that, since the mean vertical shear is taken over the entire wake region, the global length, $R$ , and velocity, $U$ , scales are used.

Figure 15.

Evolution of non-dimensional mean-square vertical shear integrated vertically over the wake region. Solid, dashed and dotted lines denote $\mathit{Re}=2700$ , 5700 and 11 000, respectively. Black, dark grey and light grey lines denote $\mathit{Fr}=\{2.4,4.7,9.1\}$ , respectively. A symbol is added to the beginning of each line for further clarity.

At fixed $\mathit{Re}$ , increasing $\mathit{Fr}$ leads to lower $\langle S^{2}\rangle _{z}/(U/R)^{2}$ , as expected. In Spedding (Reference Spedding2002) the $\mathit{Fr}$ dependence of $\overline{(\partial u/\partial z)^{2}}$ was removed when only fluctuating $u$ components were included. However, at fixed $\mathit{Fr}$ , higher $\mathit{Re}$ is also associated with a smaller $\langle S^{2}\rangle _{z}/(U/R)^{2}$ . For this particular experiment, this means that at the same $Nt$ (or, equivalently, same $x/R$ , since $\mathit{Fr}$ is fixed), though the towing speed is doubled from case to case, the average vertical shear is increased by a multiplier less than 2. The consequences of this will be discussed later.

Diamessis et al. (Reference Diamessis, Spedding and Domaradzki2011) found that a scaling of $[\langle S^{2}\rangle /(U/R)^{2}](\mathit{Fr}^{2}/\mathit{Re})$ , where $\langle \cdots \rangle$ denotes averaging over the volume of the wake, collapsed the late-wake data for $30<Nt<200$ . The $\mathit{Re}$ contribution comes from arguments in Riley & de Bruyn Kops (Reference Riley and de Bruyn Kops2003), who suggested that $[\langle S^{2}\rangle /(U/R)^{2}]/\mathit{Re}$ , which is a part of the full dissipation term, would be independent of $\mathit{Re}$ once truly turbulent-like behaviour is developed. Noting the differences above in how $\mathit{Re}$ and $\mathit{Fr}$ are varied, the same rescaling fails to collapse $\langle S^{2}\rangle _{z}$ (mean vertical shear over the vertical centreplane only) here. However, in the early wake of a grid, the velocity profile is not fully developed and not Gaussian-like, and the global scales of $R$ and $U$ may not be the best scaling parameter for $\langle S^{2}\rangle$ . Regardless of non-dimensionalization, a scaling of $\mathit{Re}^{-1}$ collapses the dimensional shear with moderate success in figure 16(a). If $\mathit{Fr}$ is varied systematically, then $\mathit{Fr}^{-2}$ overcompensates in early wakes. Arguing that a late-wake, $\mathit{Fr}$ -independent state has not yet been reached, one may empirically search for a different scaling for $\mathit{Fr}$ , as shown in figure 16(b), a result that is purely empirical, and may depend on this particular grid geometry. We return to this point later.

Figure 16.

Vertically averaged mean-square vertical shear: (a) rescaled with $1/\mathit{Re}$ , against $Nt$ , for $\mathit{Fr}=4.7$ , and (b) rescaled with $\mathit{Fr}^{0.7}/\mathit{Re}$ , against $Nt$ , for the same dataset as figure 15.

3.2.4. Flow instability and stratified turbulence

The strength of the vertical shear and its likely role in promoting local overturning instability can be measured by a minimum Richardson-number criterion. The local Richardson number, $\mathit{Ri}_{loc}$ , is

(3.4) $$\begin{eqnarray}\mathit{Ri}_{loc}=\frac{N^{2}}{S^{2}}.\end{eqnarray}$$

Richardson number $\mathit{Ri}_{loc}$ is more of a semi-local number because $S$ has a local value, varying across the wake, while $N$ is just the global $N$ . This is an enforced simplification because there are no data on local ${\it\rho}(x,z)$ , or its spatial gradients. In practice, one may expect quite strong initial variations in ${\it\rho}(x,z)$ in a turbulent stratified wake (Xu, Fernando & Boyer Reference Xu, Fernando and Boyer1995). With the resulting wide variation in local $N$ , it might be hard in any event to compute a meaningful $\mathit{Ri}_{loc}$ . Doing so in this experiment must await planned density profile measurements.

At any given $Nt$ , since $N$ is constant, $\mathit{Ri}_{loc,min}$ will occur at the wake edge. Its time evolution is given in figure 17. All cases begin at a value much smaller than the nominal critical value of $1/4$ . The initial values have a systematic dependence on both $\mathit{Fr}$ and $\mathit{Re}$ . For cases of the same $\mathit{Re}$ , the smaller the $\mathit{Fr}$ , the larger the initial $\mathit{Ri}_{loc,min}$ , meaning that vertical overturning is more suppressed in the beginning. This dependence is the same as reported by Diamessis et al. (Reference Diamessis, Spedding and Domaradzki2011). For $\mathit{Fr}$ fixed, larger $\mathit{Re}$ corresponds to larger initial $\mathit{Ri}_{loc,min}$ , perhaps counter-intuitively, and in contrast with the simulations.

Figure 17.

Local minimum Richardson number, against $Nt$ , for the same dataset as figure 15.

After the initial gradual increase, the further increase in $\mathit{Ri}_{loc,min}$ is $\mathit{Fr}$ -independent and scales as $(Nt)^{2}$ . For smaller $\mathit{Fr}$ , this late-time limit is reached at a relatively larger $Nt$ . The same $\mathit{Fr}$ independence was also observed by Spedding (Reference Spedding2002) and Diamessis et al. (Reference Diamessis, Spedding and Domaradzki2011) for the wake of a sphere. Though $\mathit{Re}$ independence is also observed here, the contrasting $\mathit{Re}$ dependence is observed in Diamessis et al. (Reference Diamessis, Spedding and Domaradzki2011) when ${\it\nu}$ is changed, and we can only claim $\mathit{Re}$ independence for ${\it\nu}$ fixed.

In figure 17, $\mathit{Ri}_{loc,min}$ crosses unity at $Nt\approx 10$ for all conditions. This value is the same as that reported by Diamessis et al. (Reference Diamessis, Spedding and Domaradzki2011) for the low $\mathit{Re}=5000$ runs. The criterion of $\mathit{Ri}_{loc}<1$ for development of shear instability, and therefore stratified turbulence, may be argued to be equivalent to $\mathit{Re}_{H}\mathit{Fr}_{H}^{2}>1$ (Riley & de Bruyn Kops Reference Riley and de Bruyn Kops2003; Hebert & de Bruyn Kops Reference Hebert and de Bruyn Kops2006a ,Reference Hebert and de Bruyn Kops b ), where $\mathit{Re}_{H}=u_{H}l_{H}/{\it\nu}$ is the horizontal Reynolds number and $\mathit{Fr}=u_{H}/(l_{H}N)$ is the horizontal Froude number.

If correctly defined on local fluctuating quantities, $\mathit{Re}_{H}\mathit{Fr}_{H}^{2}$ is also a close estimation of the buoyancy Reynolds number $\mathit{Re}_{b}$ . Here, $u_{H}$ is defined as the average of $u^{\prime }$ at the wake top edge and bottom edge, and $l_{H}$ represents the horizontal length scale of local overturning, and is therefore computed from the zero-crossing of the autocorrelation functions of fluctuating vertical velocity $w^{\prime }$ and vorticity ${\it\omega}_{y}^{\prime }$ at the wake edge. In each experiment run, for each wake edge location with the same $x/R$ (or equivalently $Nt$ ), $w^{\prime }$ and ${\it\omega}_{y}^{\prime }$ can be reconstructed in a time series. Then the time difference between the first positive zero-crossing and first negative zero-crossing is computed for the autocorrelation functions of the $w^{\prime }$ and ${\it\omega}_{y}^{\prime }$ sequences separately, and denoted as ${\rm\Delta}t_{l1}$ and ${\rm\Delta}t_{l2}$ , indicating the time it takes for an overturning to pass that location. The local time scale can be written as ${\rm\Delta}t_{lH}=({\rm\Delta}t_{l1}+{\rm\Delta}t_{l2})/2$ . Then $l_{H}$ is computed from $l_{H}={\rm\Delta}t_{lH}\,u_{loc}$ , where $u_{loc}$ is the local mean horizontal velocity with respect to the grid at the wake edge. Finally, results from all runs are averaged, and the average of $l_{H}$ at the top edge and bottom edge is taken as the final $l_{H}$ . This procedure only works for the early wake when the wake edge definition is valid and the vertical motion has not been completely suppressed.

Figure 18 shows the evolution of $\mathit{Fr}_{H}$ and $\mathit{Re}_{H}\mathit{Fr}_{H}^{2}$ for different cases. For constant $\mathit{Fr}$ , the smaller $\mathit{Re}$ leads to a relatively larger local $\mathit{Fr}_{H}$ at early times, which is consistent with the analysis in § 3.2.1. The $\mathit{Fr}_{H}$ value drops below unity at very small $Nt$ for all cases, and further decrease of $\mathit{Fr}_{H}$ at later time scales approximately with $(Nt)^{-1}$ , and is $\mathit{Fr}$ -independent (and given ${\it\nu}$ fixed, $\mathit{Re}$ -independent). Similar collapse of data is also observed in $\mathit{Re}_{H}\mathit{Fr}_{H}^{2}$ , but scales approximately with $(Nt)^{-2}$ .

Figure 18.

(a) Horizontal Froude number $\mathit{Fr}_{H}$ , and (b) $\mathit{Re}_{H}\mathit{Fr}_{H}^{2}$ , against $Nt$ , for the same dataset as figure 15.

3.2.5. Fluctuating quantities

The primary local source of turbulence is at the wake edge, where shear layers induce KH instability and eddies form. Then the eddies evolve by a pairing–merging mechanism and spread into the interior wake. In this section we examine the characteristics of the turbulent quantities.

Figure 19(a,b) shows the vertical profiles of $u^{\prime }/U$ and $w^{\prime }/U$ at different $Nt$ for $\mathit{Fr}=4.7$ and $\mathit{Re}=5700$ . Before $Nt=3$ , the shape and magnitude of $u^{\prime }/U$ and $w^{\prime }/U$ are closely matched, with peak values at $z/h=\pm 1$ . The magnitude and width of the peaks of both $u^{\prime }/U$ and $w^{\prime }/U$ increase with $Nt$ . Note that the latter peaks are slightly broader in $z$ as waves propagate to the exterior. After $Nt=3$ , the maxima in $u^{\prime }/U$ have grown in the midline, while $(w^{\prime }/U)_{max}$ remains around $z/h=\pm 1$ . Figure 19(c) shows the centreline $u_{0}^{\prime }/U$ and $w_{0}^{\prime }/U$ . The shapes of the two curves are perhaps similar, increasing at early times as the wake becomes more turbulent, and then dissipating due to viscosity, but the amplitudes are very different. Not only is the initial forcing not isotropic in $\{x,z\}$ , but wave modes propagate outwards, away from the centreline.

Figure 19.

Fluctuating quantities for $\mathit{Fr}=4.7$ and $\mathit{Re}=5700$ (R5F4, ▴): (a)  $u^{\prime }/U$ in vertical slices; (b)  $w^{\prime }/U$ in vertical slices; (c)  $u_{0}^{\prime }/U$ (solid line) and $w_{0}^{\prime }/U$ (dashed line) in wake centre; and (d) Reynolds stress in vertical slices. In panels (a), (b) and (d), $z$ is non-dimensionalized by the local wake half-height. Black solid line, black dashed line and black dotted line are for $Nt=\{1,2,3\}$ , respectively; grey solid line and grey dashed line are for $Nt=\{4,6\}$ .

The Reynolds stress in vertical slices is plotted in figure 19(d) for the same case. The peak values are located around $z/h=\pm 1$ , with magnitude increasing before $Nt=3$ , and decreasing after $Nt=4$ . The inner wake region becomes quite turbulent after $Nt=4$ as another opposite-sign peak occurs between the wake edge and centre, and grows even larger in magnitude than the wake edge peak at $Nt=6$ .

At early times, the fluctuating quantities reflect the simultaneous dynamics of KH modes with the lee wave at the wake edge. Figure 20(a) shows the maximum fluctuating vertical velocity against $Nt$ for $\mathit{Re}=2700$ but varying $\mathit{Fr}$ . As $\mathit{Fr}$ increases, so does $w_{max}^{\prime }$ and the peak occurs at smaller $Nt$ . When $\mathit{Fr}=(1/N)/(R/U)$ is high, $R/U$ is small compared with $1/N$ , and a maximum after a fixed number of overturns appears at smaller $Nt$ . If plotted against $x/R$ , the maxima occur at $x/R\in [3,5]$ as shown in figure 20(b).

Figure 20.

Plots of $w_{max}^{\prime }/|U|$ in the wake region for: (a) constant $\mathit{Re}=2700$ , varying $\mathit{Fr}$ , against $Nt$ ; (b) same data as (a), but against $x/R$ ; (c) constant $\mathit{Re}=11\,000$ , varying $\mathit{Fr}$ , against $Nt$ ; and (d) constant $\mathit{Fr}=2.4$ , varying $\mathit{Re}$ , against $Nt$ .

At higher $\mathit{Re}$ , the trend that $w_{max}^{\prime }$ peaks at smaller $Nt$ for larger $\mathit{Fr}$ persists but the amplitude does not increase systematically. Here $\mathit{Re}$ may be large enough so that initial turbulent dissipation has removed kinetic energy from the eddy motions before there is any significant interaction with the exterior. If that is so, then it presumably contrasts with low- $\mathit{Re}$ cases where this does not happen, and figure 20(a,c) has results from the same physical experiment, but in two rather different dynamical regimes, distinguished by $\mathit{Re}_{H}\,\mathit{Fr}_{H}^{2}=\mathit{Re}_{b}$ .

Because $\mathit{Re}_{H}\,\mathit{Fr}_{H}^{2}$ is a local parameter, to be consistent, $w_{max}^{\prime }$ is non-dimensionalized by the maximum $|\overline{u}|$ at the wake edge behind the grid, which occurs at $Nt$ roughly equal to that of the initial dip in figure 9. The results for similar cases are shown in figure 21. The peak locations and magnitudes match reasonably well with those shown in figure 18(b). For $\mathit{Re}=2700$ , $\mathit{Fr}=9.1$ and $\mathit{Fr}=4.7$ have $\mathit{Re}_{b,max}>200$ , $\mathit{Fr}=2.4$ has $\mathit{Re}_{b,max}\approx 30$ , and $\mathit{Fr}=0.6$ has $\mathit{Re}_{b,max}$ below unity, so the peak magnitudes of $w_{max}^{\prime }/|\overline{u}|_{max}$ differ considerably. The three cases with $\mathit{Re}=11\,000$ have $\mathit{Re}_{b,max}\in [20,100]$ , and the three cases with $\mathit{Fr}=2.4$ have $\mathit{Re}_{b,max}\in [10,30]$ , indicating that they are probably in the same dynamical regime, respectively, so peak magnitudes of $w_{max}^{\prime }/|\overline{u}|_{max}$ are similar.

Figure 21.

Plots of $w_{max}^{\prime }/|\overline{u}|_{max}$ in the wake region, where $|\overline{u}|_{max}$ is the maximum mean streamwise velocity at the wake edge behind the grid, for the same dataset as figure 20.