4.1 Phase-averaged scalar field

To quantify the effect of surface waves on the turbulent flow and scalar fields, a triple decomposition is applied to analyze the instantaneous quantities. Specifically, an instantaneous quantity $f(x,y,z,t)$ obtained from the DNS can be decomposed as (e.g. Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Kihara et al. Reference Kihara, Hanazaki, Mizuya and Ueda2007; Yang & Shen Reference Yang and Shen2010)

where $\langle f\rangle$ denotes the phase average, $\overline{f}$ denotes the time and $\unicode[STIX]{x1D701}$ -plane average over all $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D713},\unicode[STIX]{x1D70F})$ in the computational space (hereinafter simply referred to as the time and plane average), $\widetilde{f}$ denotes the wave-induced variation of $f$ , and $f^{\prime }$ denotes the instantaneous fluctuation. In particular, the phase average $\langle f\rangle$ is calculated by shifting the instantaneous quantity $f$ to the same phase with respect to the surface wave, then averaging in time and along the spanwise direction, and lastly further averaging among the four streamwise waves (Yang & Shen Reference Yang and Shen2010). The wave-induced variation is then obtained by subtracting from the phase average the time and plane average at the corresponding $\unicode[STIX]{x1D701}$ level in the computational space, i.e.  $\widetilde{f}(x,z)=\langle f\rangle (x,z)-\overline{f}(\unicode[STIX]{x1D701})$ . As a result of the averaging among the four different waves, only one wavelength is necessary when presenting $\langle f\rangle$ and $\widetilde{f}$ . Moreover, when presenting the vertical profile of the time and plane average $\overline{f}$ , the $\unicode[STIX]{x1D701}$ coordinate is converted to the physical coordinate above the wave surface as $\overline{z}=\overline{H}\unicode[STIX]{x1D701}$ (for a flat-surface case, $\overline{z}=z$ ). Hereinafter, for simplification $\overline{f}$ is also referred to as the mean of $f$ .

Note that to calculate the mean, one could also choose to first interpolate the physical quantities obtained from the boundary-fitted DNS to a regular Cartesian grid and then average on planes of constant $z$ . However, by doing so the flow region below the height of the wave crest would be excluded (e.g. Rutgersson & Sullivan Reference Rutgersson and Sullivan2005). As shown in the following subsections, because the current DNS uses boundary-fitted grid with the $\unicode[STIX]{x1D701}$ -plane nearly parallel to the wave surface near the lower boundary (see figure 29 a in appendix B), calculating $\overline{f}$ by averaging on planes of constant $\unicode[STIX]{x1D701}$ allows us to capture the linear profiles of mean velocity and scalar in the thin viscous sublayer very close to the wave surface. Meanwhile, in the log-law region the difference of the mean profiles obtained by averaging on $z$ planes and $\unicode[STIX]{x1D701}$ planes is found to be small because the $\unicode[STIX]{x1D701}$ plane becomes flatter as $\unicode[STIX]{x1D701}$ increases (see more information in appendix B). Moreover, defining and calculating $\overline{f}$ based on $\unicode[STIX]{x1D701}$ allows us to have meaningful values for the wave-correlated variation $\widetilde{f}$ in the entire phase-averaged domain, including the region above the trough but below the elevation of the crest. Similar triple decomposition approaches as (4.1) have been used in previous studies of turbulent flows over waves (e.g. Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Sullivan & McWilliams Reference Sullivan and McWilliams2002; Kihara et al. Reference Kihara, Hanazaki, Mizuya and Ueda2007; Hara & Sullivan Reference Hara and Sullivan2015).

Figure 3.

Phase-averaged (a,c,e,g) streamlines and (b,d,f,h) scalar concentration $\langle \unicode[STIX]{x1D703}\rangle$ (for $Sc=1.0$ ) for various wave cases: (a,b) S25; (c,d) W25C2; (e,f) W25C14; and (g,h) W25C25. The streamlines are based on $(\langle u\rangle -c,\langle w\rangle )$ , i.e. in the reference frame moving with the wave phase speed $c$ . The red dash-dot line indicates the location of the critical layer where $\langle u\rangle -c=0$ (it is located higher than $z/\unicode[STIX]{x1D706}=0.3$ in case W25C25 and thus is not shown in (g)). The contour interval for $\langle \unicode[STIX]{x1D703}\rangle$ is $0.1$ .

Figure 4.

Comparison of phase-averaged scalar concentration $\langle \unicode[STIX]{x1D703}\rangle$ with $Sc=0.7$ over a stationary wavy wall with steepness $ak=0.314$ . Profiles of $\langle \unicode[STIX]{x1D703}\rangle$ at $x/\unicode[STIX]{x1D706}=0.1$ , $0.3$ , $0.5$ , and $0.7$ are plotted: $\circ$ , DNS of channel flow by Rossi (Reference Rossi2010); $-\!\!\!-\!\!\!-$ , current DNS with $Re_{\ast }=283$ . The reference value $\unicode[STIX]{x1D703}_{ref}$ for the normalization is based on the mean scalar concentration at $z=0.3\unicode[STIX]{x1D706}$ .

Figure 3 shows the phase-averaged streamlines and scalar concentration $\langle \unicode[STIX]{x1D703}\rangle$ for various wave cases. The streamlines are plotted based on the velocity vector $(\langle u\rangle -c,\langle w\rangle )$ , representing the mean flow pattern in the wave-following reference frame. In case S25 (figure 3 a), due to the large amplitude of the stationary wavy surface, the phase-averaged flow separates on the leeward face of the crest and reattaches on the windward face of the succeeding crest. The streamlines in the separation zone exhibit a closed-circle pattern, which does not extend vertically beyond the elevation of the wave crest. Above the separation zone, the streamlines are nearly horizontal, with a slightly concave curvature above the reattachment region. The distribution of the phase-averaged scalar concentration $\langle \unicode[STIX]{x1D703}\rangle$ exhibits correlation with the streamline patterns (figure 3 b). Near the wavy surface, the low- $\langle \unicode[STIX]{x1D703}\rangle$ region is lifted up in the separation zone between the crest and the succeeding trough (i.e. roughly between $x/\unicode[STIX]{x1D706}=0.1$ and $0.5$ ) due to the mean upward flow motion associated with the separation, resulting in a reduced vertical gradient for $\langle \unicode[STIX]{x1D703}\rangle$ ; near the reattachment zone (i.e. around $x/\unicode[STIX]{x1D706}=0.7$ ), the mean flow sweeps towards the windward face of the succeeding crest, resulting in a strong near-surface shear layer (from $x/\unicode[STIX]{x1D706}=0.6$ to $1.0$ ) with a high vertical gradient for both $\langle u\rangle$ and $\langle \unicode[STIX]{x1D703}\rangle$ . These wave-phase-dependent modulations of the near-surface distribution of $\langle \unicode[STIX]{x1D703}\rangle$ are revisited when the Sherwood number $Sh$ is discussed in the next section (§ 4.2). Above the elevation of the crest, the contour lines of $\langle \unicode[STIX]{x1D703}\rangle$ become nearly horizontal, consistent with the horizontal streamlines at these high elevations. Similar near-surface modulation of $\langle \unicode[STIX]{x1D703}\rangle$ has also been reported in the DNS study by Rossi (Reference Rossi2010) for transport of a scalar with $Sc=0.7$ in a turbulent channel flow over a stationary wavy surface with steepness $ak=0.314$ . Figure 4 shows a corresponding case obtained from our DNS (with $ak=0.314$ and $Re_{\ast }=283$ ), which yields reasonably good agreement with the DNS results from Rossi (Reference Rossi2010).

In case W25C2 (figure 3 c,d), the wave surface has non-zero orbital velocities $(u_{s},w_{s})$ that induce additional distortion to the flow field above it (in addition to the boundary curvature effect as in case S25). In the reference frame that moves with the wave at its phase speed $c$ , near the surface the mean streamwise velocity $\langle u\rangle -c$ is in the $-x$ -direction. As $\langle u\rangle$ increases with height, $\langle u\rangle -c$ becomes zero at the height indicated by the dash-dot line in figure 3(c), and becomes positive further above. The layer where $\langle u\rangle -c=0$ , known as the ‘critical layer’, was found to play a crucial role in the momentum transfer between the turbulence and the surface waves (e.g. Miles Reference Miles1957; Lighthill Reference Lighthill1962; Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Hristov et al. Reference Hristov, Miller and Friehe2003). Around the critical layer, the streamlines form a closed-circle pattern, named as the ‘cat’s eye’ by Lighthill (Reference Lighthill1962). For case W25C2, the cat’s-eye streamlines are located right above the wave trough. Note that the cat’s eyes pattern in figure 3(c) does not imply the occurrence of mean flow separation above the wave trough (see Banner & Melville Reference Banner and Melville1976; Gent & Taylor Reference Gent and Taylor1977). Unlike in the flow over a stationary wave, in which a negative surface stress (i.e. reversed near-surface flow) would indicate a flow separation, in flows over progressive waves the near-surface flow is in the reversed direction in the wave-following reference frame everywhere above the wave surface. Yang & Shen (Reference Yang and Shen2010) used the local relative velocity (with respect to the local surface orbital velocity) and found occasional instantaneous flow separations occur on the leeward face of slow progressive waves, while the mean flow was found to remain unseparated. Similarly, in case W25C2 of the current DNS the mean flow is found to be unseparated. Compared with the mean flow separation bubble in case S25, in case W25C2 the cat’s-eye streamline pattern extends further upwards and upstream, As a result, the low- $\langle \unicode[STIX]{x1D703}\rangle$ region above the wave trough is lifted further upwards, and this effect also extends further upstream when compared with case S25.

For cases W25C14 (figure 3 e,f) and W25C25 (figure 3 g,h), the wave phase speed $c$ is large and the critical layer is located far from the wave surface. Below the critical layer, the streamlines are nearly parallel to the wave surface with the mean streamwise velocity $\langle u\rangle -c$ in the $-x$ -direction. Associated with this smooth streamline pattern, the contour lines for $\langle \unicode[STIX]{x1D703}\rangle$ are also nearly parallel to the wave surface. In cases W25C14 and W25C25 the scalar concentration $\langle \unicode[STIX]{x1D703}\rangle$ near the wave surface is in general higher than that in cases S25 and W25C2, as indicated by the lower elevations of the $\langle \unicode[STIX]{x1D703}\rangle =0.6$ and $0.8$ contour lines, respectively, in figures 3(f) and 3(h) compared with those in figures 3(b) and 3(d).

Figure 5.

Distribution of wave-induced variation of scalar concentration, $\widetilde{\unicode[STIX]{x1D703}}$ , for various wave cases: (a,e) S25; (b,f) W25C2; (c,g) W25C14; and (d,h) W25C25. (a–d) Show the contours of $\widetilde{\unicode[STIX]{x1D703}}$ for $Sc=1.0$ . The contour intervals are (a,e) $0.04$ , (b,f) $0.04$ , (c,g) $0.01$ and (d,h) $0.01$ . Dashed contour lines indicate negative values. (e–h) Show the vertical profiles of $\overline{|\widetilde{\unicode[STIX]{x1D703}}|}$ (i.e. the average of $|\widetilde{\unicode[STIX]{x1D703}}|$ over all $\unicode[STIX]{x1D709}$ ) for various values of $Sc$ : ——, $Sc=0.5$ ; – – –, $Sc=1.0$ ; — ⋅ —, $Sc=2.0$ ; and — ⋅ ⋅ —, $Sc=4.0$ .

The effect of surface waves on the scalar field can be directly visualized from the distribution of the wave-induced variation of the scalar concentration, $\widetilde{\unicode[STIX]{x1D703}}$ , in which the time and plane average $\overline{\unicode[STIX]{x1D703}}$ has been removed. Figure 5 shows the two-dimensional contours of $\widetilde{\unicode[STIX]{x1D703}}$ and the vertical profile of the $\unicode[STIX]{x1D709}$ -average of its magnitude, $\overline{|\widetilde{\unicode[STIX]{x1D703}}|}$ . For cases S25 (figure 5 a) and W25C2 (figure 5 b), the distribution of $\widetilde{\unicode[STIX]{x1D703}}$ is asymmetric about the wave crest, with negative values above the leeward face and trough, and positive values above the windward face and crest. For cases W25C14 (figure 5 c) and W25C25 (figure 5 d), the distribution of $\widetilde{\unicode[STIX]{x1D703}}$ becomes nearly symmetric with respect to the wave trough, with positive values above the trough and negative values above the crest. For all the four wave cases, the averaged magnitude $\overline{|\widetilde{\unicode[STIX]{x1D703}}|}$ has its maximum near the wave surface (at $\unicode[STIX]{x1D701}/\unicode[STIX]{x1D706}\lesssim 0.04$ ), and reduces to a small value at higher height. The peak values of $\overline{|\widetilde{\unicode[STIX]{x1D703}}|}$ for cases S25 and W25C2 are significantly higher than those for cases W25C14 and W25C25. Figure 5 also shows the effect of $Sc$ on $\overline{|\widetilde{\unicode[STIX]{x1D703}}|}$ . For all of the cases, the increase of $Sc$ causes an increase of the peak value and a reduction in the height of the peak.

4.2 Sherwood number

The wave-induced variation of the near-surface scalar field causes considerable wave-correlated variation in the Sherwood number. The Sherwood number is usually used to measure the relative importance of convection and molecular diffusion in convective mass transfer processes (corresponding to the Nusselt number in heat transfer). The local Sherwood number along the wave surface based on the phase-averaged scalar field can be calculated as

where $h$ is the convective mass transfer coefficient, $D$ is the molecular diffusivity as in (2.3), $\overline{\unicode[STIX]{x1D703}}_{1/2}$ is the averaged $\unicode[STIX]{x1D703}$ at the centre height of the domain ( $\overline{z}=\overline{H}/2$ ), and $\unicode[STIX]{x1D703}_{0}=0$ is the imposed bottom surface value of $\unicode[STIX]{x1D703}$ as the boundary condition. Figure 6 shows the wave-correlated variation of $Sh$ for the steep stationary wave case (i.e. the case with $ak=0.314$ and $Re_{\ast }=283$ as shown in figure 4), in which the current DNS result is compared with the results from several previous numerical studies (Park et al. Reference Park, Choi and Suzuki2004; Dellil et al. Reference Dellil, Azzi and Jubran2004; Choi & Suzuki Reference Choi and Suzuki2005). For the steep stationary wave case, the profile of $Sh$ has a nearly sinusoidal shape. On the stationary wavy surface, the value of $Sh$ is small above the leeward face and large above the windward face, as a result of the smaller and larger vertical gradients of $\langle \unicode[STIX]{x1D703}\rangle$ on the leeward and windward faces, respectively, as shown in figures 3(a) and 4.

Figure 6.

(a) Profiles of phase-averaged local Sherwood number $Sh$ over the stationary wavy wall: ○, RANS result of Park et al. (Reference Park, Choi and Suzuki2004) for $(ak,Sc)=(0.1\unicode[STIX]{x03C0},0.7)$ ; ▫, RANS result of Dellil et al. (Reference Dellil, Azzi and Jubran2004) for $(ak,Sc)=(0.08\unicode[STIX]{x03C0},0.9)$ ; ▵, RANS result of Dellil et al. (Reference Dellil, Azzi and Jubran2004) for $(ak,Sc)=(0.12\unicode[STIX]{x03C0},0.9)$ ; ▿, LES result of Choi & Suzuki (Reference Choi and Suzuki2005) for $(ak,Sc)=(0.1\unicode[STIX]{x03C0},0.7)$ ; and ——, current DNS result for $(ak,Sc)=(0.1\unicode[STIX]{x03C0},0.7)$ and $Re_{\ast }=283$ (see figure 4). The corresponding wave surface phase is indicated by the wave elevation $\unicode[STIX]{x1D702}$ of $ak=0.1\unicode[STIX]{x03C0}$ in panel (b).

Figure 7.

(a) Profiles of phase-averaged local Sherwood number $Sh$ for $Sc=1.0$ and for various wave cases: ○, S25 ( $c/u_{\ast }=0$ ); ▫, W25C2 ( $c/u_{\ast }=2$ ); ▵, W25C14 ( $c/u_{\ast }=14$ ); and ▿, W25C25 ( $c/u_{\ast }=25$ ). The corresponding wave surface phase is indicated by the wave elevation $\unicode[STIX]{x1D702}$ in panel (b).

Figure 7 shows the effect of wave motion on $Sh$ . The plotted curves are for $Sc=1.0$ . When the wave is slow (case W25C2), similar to the stationary wave condition, $Sh$ is also small above the leeward face and large above the windward face. However, the shape of the $Sh$ curve apparently differs from the sinusoidal shape, with its minimum point closer to the wave crest (near $x/\unicode[STIX]{x1D706}=0.12$ ). For cases with intermediate (case W25C14) and fast (W25C25) waves, the distribution of $Sh$ become symmetric with respect to the wave form, with the maxima above the wave trough and the minima above the crest, corresponding to the regions of positive and negative $\widetilde{\unicode[STIX]{x1D703}}$ in figure 5, respectively.

Figure 8.

Dependence of mean Sherwood number $\overline{Sh}$ on $Sc$ for various wave cases: ○, S25 ( $c/u_{\ast }=0$ ); ▫, W25C2 ( $c/u_{\ast }=2$ ); ▵, W25C14 ( $c/u_{\ast }=14$ ); and ▿, W25C25 ( $c/u_{\ast }=25$ ). The lines are the curve fittings: – – –, $\overline{Sh}=27.09Sc^{0.579}$ for S25; — ⋅ —, $\overline{Sh}=26.54Sc^{0.609}$ for W25C2; $\cdots \,$ , $\overline{Sh}=23.68Sc^{0.514}$ for W25C14; and — ⋅ ⋅ —, $\overline{Sh}=21.13Sc^{0.518}$ for W25C25. The two solid lines show the averaged slope of the four surface wave conditions, i.e.  $\overline{Sh}\propto Sc^{0.555}$ .

For a given wave condition, varying the $Sc$ value mainly causes changes in the magnitudes of the mean and variation of $Sh$ , while the shapes of the curves are similar to those of the corresponding curves shown in figure 7. Thus, these additional $Sh$ curves with different $Sc$ are not plotted here, and only the effect of $Sc$ on the averaged Sherwood number $\overline{Sh}$ ( $=(1/\unicode[STIX]{x1D706})\int _{0}^{\unicode[STIX]{x1D706}}Sh(x)\text{d}x$ ) is discussed. Figure 8 shows the dependence of $\overline{Sh}$ on $Sc$ . Previous studies on scalar transport in wall-bounded or free-surface-bounded turbulent flows have shown that the scaling of $\overline{Sh}$ with $Sc$ follows a power law (e.g. Shaw & Hanratty Reference Shaw and Hanratty1977; Jähne & Haußecker Reference Jähne and Haußecker1998; Na, Papavassiliou & Hanratty Reference Na, Papavassiliou and Hanratty1999; Hasegawa & Kasagi Reference Hasegawa and Kasagi2008), i.e.

The current DNS results also exhibit this power-law trend for each wave condition as shown in figure 8. For the four different surface wave conditions considered in this study, the power-law coefficient $\unicode[STIX]{x1D6FD}$ varies slightly around an averaged value of $0.555$ , while $\unicode[STIX]{x1D6FC}$ varies between $21.13$ and $27.09$ . The pronounced variation of $\unicode[STIX]{x1D6FC}$ with the surface wave condition indicates the importance of wave motion on the magnitude of scalar transport. To further elucidate the effect of wave motion on scalar transport, the averaged Sherwood number $\overline{Sh}$ is plotted in figure 9 as a function of the wave age  $c/u_{\ast }$ . When $Sc$ is fixed, in general $\overline{Sh}$ decreases as $c/u_{\ast }$ increases (figure 9 a). When normalized by $Sc^{0.555}$ (as suggested by the curve fitting in figure 8), the $\overline{Sh}$ curves for all the different Schmidt numbers collapse reasonably well.

Figure 9.

Dependence of mean Sherwood number $\overline{Sh}$ on wave age $c/u_{\ast }$ for various Schmidt numbers:

, $Sc=0.5$ ;

, $Sc=1.0$ ;

, $Sc=2.0$ ; and

, $Sc=4.0$ . In panel (b), $\overline{Sh}$ is rescaled by $Sc^{0.555}$ .

4.3 Profiles of time- and plane-averaged quantities

The phase-averaged statistics in § 4.1 have shown significant wave-induced variation in the scalar field, which is highly dependent on the wave phase. In many practical applications, the surface wave effect is usually not directly captured in a phase-resolved context. Instead, the averaged wave effect is parameterized as the effective wave roughness that appears in the one-dimensional profiles of the time- and plane-averaged velocity and scalar concentration, which can be obtained based on field and laboratory measurement data (e.g. Charnock Reference Charnock1955; Smith Reference Smith1988; Fairall et al. Reference Fairall, Bradley, Rogers, Edson and Young1996; Johnson et al. Reference Johnson, Høstrup, Vested and Larsen1998; Toba et al. Reference Toba, Smith, Ebuchi, Jones and Toba2001). Although limited to low Reynolds number conditions, DNS has been shown to provide valuable insights for understanding the dependence of the effective wave roughness on the characteristics of the waves (Sullivan et al. Reference Sullivan, McWilliams and Moeng2000). In this subsection, time and plane averaging is performed for the instantaneous snapshots of scalar and velocity fields obtained from the current DNS to help quantify the averaged wave effect. Note that the plane average is performed on the $\unicode[STIX]{x1D701}$ -plane in the computational space, as defined in § 4.1.

Figure 10.

Profiles of mean streamwise velocity $\overline{u}^{+}$ for various wave cases: ○, S25 ( $c/u_{\ast }=0$ ); ▫, W25C2 ( $c/u_{\ast }=2$ ); ▵, W25C14 ( $c/u_{\ast }=14$ ); and ▿, W25C25 ( $c/u_{\ast }=25$ ). The semi-logarithmic plot in panel (a) highlights the log-law region, with the thin solid lines indicating the logarithmic law $\overline{u}^{+}=(1/\unicode[STIX]{x1D705})\ln (\overline{z}^{+}/z_{0}^{+})$ and the corresponding values of $\unicode[STIX]{x1D705}$ and $z_{0}^{+}$ for each simulation case given in table 1. The logarithmic plot in panel (b) highlights the near-wall viscous layer, with the thin solid lines indicating the linear law $\overline{u}^{+}=\overline{z}^{+}$ . In the figure, the DNS result of flat-boundary channel flow from Kim, Moin & Moser (Reference Kim, Moin and Moser1987) is indicated by the thick line, and the experimental data from Eckelmann (Reference Eckelmann1974) (corrected by Kim et al. Reference Kim, Moin and Moser1987) are indicated by the filled circles.

Figure 10 shows the profiles of the mean streamwise velocity $\overline{u}$ (for the definition, see (4.1) and the associated discussion). The slow wave case (W25C2, $c/u_{\ast }=2$ ) shows a mean velocity profile close to that of the stationary wave case (S25). As $c/u_{\ast }$ increases, the profile of $\overline{u}^{+}$ shifts monotonically towards higher values, which is expected since the wave form drag decreases as $c/u_{\ast }$ increases (Belcher & Hunt Reference Belcher and Hunt1998). For the current DNS, the normalized pressure-induced form drag $F_{p}$ and surface viscous drag $F_{\unicode[STIX]{x1D708}}$ per unit projected horizontal area can be calculated based on the phase-averaged pressure and velocity as

and the total surface drag per unit area equals to their sum, $F_{total}=F_{p}+F_{\unicode[STIX]{x1D708}}$ . As shown in figure 11, in cases S25 and W25C2, the form drag makes a significant contribution to the total surface drag. In case W25C14, the form drag is very small and the total surface drag is mostly due to the viscous effect, indicating that the mean flow and the surface wave are nearly in equilibrium. In case W25C25, the wave phase speed is fast and the surface wave pushes the mean flow forwards, resulting in negative form drag (i.e. a thrust). A similar dependence of the surface drag on the wave age has been reported in various previous studies (e.g. Li et al. Reference Li, Xu and Taylor2000; Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Kihara et al. Reference Kihara, Hanazaki, Mizuya and Ueda2007; Yang & Shen Reference Yang and Shen2010).

Figure 11.

Budget of surface drag as a function of wave age:

, total surface drag $f_{total}$ ;

, pressure-induced form drag $f_{p}$ ; and

, viscous drag $f_{\unicode[STIX]{x1D708}}$ .

For all the four wave conditions considered in this study, the mean velocity profiles exhibit a distinct logarithmic region, as indicated by the thin solid lines in the semi-logarithmic plot in figure 10(a). These lines are determined by fitting the mean velocity profiles in the range of $\overline{z}^{+}\in [50,150]$ (see appendix B) with the logarithmic profile

where the superscript ‘ $+$ ’ denotes the quantities normalized by the wall units (i.e. velocity is normalized by $u_{\ast }$ and length is normalized by $\unicode[STIX]{x1D708}/u_{\ast }$ ), $\unicode[STIX]{x1D705}$ is the von Kármán constant, $z_{0}^{+}$ is the effective wave surface roughness, and $B$ is the profile offset. For each case, the corresponding log-law profile parameters $\unicode[STIX]{x1D705}$ , $z_{0}^{+}$ and $B$ are listed in table 1. Similar to the DNS results reported in Sullivan et al. (Reference Sullivan, McWilliams and Moeng2000), the evaluated $z_{0}^{+}$ values in most of the DNS cases considered in this study fall in the transitional regime ( $0.1<z_{0}^{+}<2.2$ ) in terms of the sea-surface roughness, which was categorized to be a common sea state by Kitaigorodskii & Donelan (Reference Kitaigorodskii, Donelan, Brutsaert and Jirka1984). The slow wave case W25C2 exhibits a roughness of $z_{0}^{+}=2.31$ that is slightly higher than the upper bound of the transitional regime. For cases S25, W25C2 and W25C14, the values for $\unicode[STIX]{x1D705}$ are close to the standard value of $\unicode[STIX]{x1D705}=0.41$ for the flat-wall case. The fast wave case W25C25 exhibits a smaller value of $\unicode[STIX]{x1D705}=0.27$ , which is likely due to the strong surface motion of the fast propagating surface waves when $c/u_{\ast }=25$ . Similarly, Sullivan et al. (Reference Sullivan, McWilliams and Moeng2000) reported a $10\,\%$ reduction of $\unicode[STIX]{x1D705}$ in DNS of turbulence over progressive waves with $ak=0.1$ (i.e. the waves are less steep than the ones studied here). Note that due to this smaller $\unicode[STIX]{x1D705}$ in case W25C25, its evaluated $z_{0}^{+}$ is larger than that in case W25C14. Therefore, the larger $z_{0}^{+}$ in case W25C25 does not imply that the turbulence sees the fast wave ‘rougher’ than the intermediate wave. It should be pointed out that caution should be taken when interpreting the mean velocity profile for the present fast wave case due to the limited Reynolds number used in this study. Under real oceanic conditions, the fast wave case usually corresponds to the wind over storm-generated ocean swells, for which the Reynolds number is much higher than what a DNS can afford. The current DNS of $c/u_{\ast }=25$ are more relevant to the condition of light wind over fast mechanically generated waves in laboratory-scale wind–wave tanks (e.g. Buckley & Veron Reference Buckley and Veron2016). Further study is required in the future in order to understand the effect of surface waves on the von Kármán constant and the wave surface roughness.

The near-surface behaviour of the mean velocity profile is shown by the logarithmic plot in figure 10(b). The mean velocity profiles exhibit a nearly linear behaviour in the viscous sublayer (i.e.  $\overline{z}^{+}<5$ ), with some non-negligible difference from the linear law of the flat-wall turbulence. In particular, the viscous sublayer profiles of the intermediate (W25C14) and fast (W25C25) wave cases are close to the linear profile of the flat-wall case (e.g. Kim et al. Reference Kim, Moin and Moser1987; Lee & Moser Reference Lee and Moser2015), i.e.  $\overline{u}^{+}=\overline{z}^{+}$ , but with larger value towards the wave surface due to the positive mean streamwise surface orbital velocity of the nonlinear Stokes wave. On the other hand, the stationary (S25) and slow (W25C2) wave cases also show a linear profile in the viscous sublayer, but with a velocity deficit compared with the flat-wall linear law. The viscous sublayer profile of the stationary wave case (S25) is nearly parallel to that of the flat wall, while the one for the slow wave case (W25C2) is tilted slightly towards higher values when approaching the surface due to the positive mean surface orbital velocity of the Stokes waves.

Figure 12.

Profiles of rescaled mean scalar concentration $\overline{\unicode[STIX]{x1D703}}\,^{+}/Sc$ . Panel (a) shows various wave cases with a fixed Schmidt number $Sc=1.0$ : ○, S25 ( $c/u_{\ast }=0$ ); ▫, W25C2 ( $c/u_{\ast }=2$ ); ▵, W25C14 ( $c/u_{\ast }=14$ ); and ▿, W25C25 ( $c/u_{\ast }=25$ ). Panel (b) shows case W25C14 ( $c/u_{\ast }=14$ ) with various Schmidt numbers: ●, $Sc=0.5$ ; ▪, $Sc=1.0$ ; ▴, $Sc=2.0$ ; and ▾, $Sc=4.0$ . The profiles are shown on a logarithmic scale to highlight the near-wall region, with the solid line indicating the linear law $\overline{\unicode[STIX]{x1D703}}\,^{+}/Sc=\overline{z}^{+}$ . In panel (a), the DNS result from Kawamura et al. (Reference Kawamura, Ohsaka, Abe and Yamamoto1998) is also plotted (the thick line).

Figure 13.

Profiles of mean scalar concentration $\overline{\unicode[STIX]{x1D703}}\,^{+}$ . The definitions of the symbols are the same as those in figure 12. The profiles are shown on a semi-logarithmic scale to highlight the log-law region, with the thin solid lines indicating the logarithmic law $\overline{\unicode[STIX]{x1D703}}\,^{+}=(1/\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D703}})\ln (\overline{z}^{+}/z_{0,\unicode[STIX]{x1D703}}^{+})$ and the corresponding values of $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D703}}$ and $z_{0,\unicode[STIX]{x1D703}}^{+}$ for each simulation case given in table 1. In panel (a), the DNS result from Kawamura et al. (Reference Kawamura, Ohsaka, Abe and Yamamoto1998) is also plotted (the thick line).

A similar analysis can be done for the mean scalar concentration. By compiling a number of experimental datasets, Kader (Reference Kader1981) showed that the mean temperature and scalar concentration in fully developed turbulent boundary layers follow a linear profile in the viscous sublayer and a logarithmic profile in the log-law region. In the near-surface region, the Kader correlation suggests that

where $c_{4}\approx 1.5\times 10^{-4}$ as estimated based on the experimental data (Kader Reference Kader1981). Therefore, for the flat-wall case it is expected that

to the leading order near the surface, with the deviation from this linear profile increasing towards larger $\overline{z}\,^{+}$ and with higher $Sc$ . Similar near-surface linear behaviour is found in the current DNS results over wave surfaces. Here $\overline{\unicode[STIX]{x1D703}}\,^{+}=\overline{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D703}_{\ast }$ , where

is a characteristic scalar scale (in the case of heat transfer, $\unicode[STIX]{x1D703}_{\ast }$ is also referred to as the friction temperature) (Kim & Moin Reference Kim, Moin, André, Cousteix, Durst, Launder, Schmidt and Whitelaw1989). Figure 12 shows the profiles of $\overline{\unicode[STIX]{x1D703}}\,^{+}/Sc$ (in a logarithmic plot) for some representative cases obtained by the current DNS. In particular, figure 12(a) shows the profiles for various wave cases with $Sc=1.0$ , and figure 12(b) shows the profiles for case W25C14 with various Schmidt numbers (other cases also follow consistent trends for the wave and $Sc$ dependences, and are thus not shown here due to space limitation). In the viscous sublayer ( $\overline{z}^{+}<5$ ), the profiles for different wave conditions and Schmidt numbers collapse well and agree with the linear law found in the flat-wall condition.

The logarithmic law behaviour of $\overline{\unicode[STIX]{x1D703}}\,^{+}$ can be visualized in a semi-logarithmic plot as shown in figure 13. Similar to the profile of $\overline{u}^{+}$ , for a fixed $Sc$ the $\overline{\unicode[STIX]{x1D703}}\,^{+}$ profile shifts towards higher values when $c/u_{\ast }$ increases, with the profile for $c/u_{\ast }=2$ close to that for the stationary wave case (figure 13 a). For each wave condition, the $\overline{\unicode[STIX]{x1D703}}\,^{+}$ profile shifts monotonically to higher values when $Sc$ increases (see for example the case W25C14 shown in figure 13 b). Similar to (4.6), the logarithmic law for scalar concentration can be written as (e.g. Kader Reference Kader1981)

where $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D703}}$ is the von Kármán constant of scalar, $z_{0,\unicode[STIX]{x1D703}}^{+}$ is the effective wave surface roughness for scalar, and $B_{\unicode[STIX]{x1D703}}$ is the offset of the mean scalar profile. For each wave case, the corresponding parameters $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D703}}$ , $z_{0,\unicode[STIX]{x1D703}}^{+}$ , and $B_{\unicode[STIX]{x1D703}}$ can be obtained by fitting the DNS data in the log-law region (defined as $50<z^{+}<150$ in this study as explained in appendix B) with the logarithmic profile (4.10). The estimated values of these parameters are listed in table 1. For each wave case, the evaluated value of the scalar von Kármán constant $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D703}}$ exhibits a small variation when the Schmidt number is changed, while the profile offset $B_{\unicode[STIX]{x1D703}}$ (or the roughness $z_{0,\unicode[STIX]{x1D703}}^{+}$ ) increases significantly as $Sc$ increases.

Figure 14.

Dependence of $B_{\unicode[STIX]{x1D703}}$ on the Schmidt number for various wave cases: ○, S25 ( $c/u_{\ast }=0$ ); ▫, W25C2 ( $c/u_{\ast }=2$ ); ▵, W25C14 ( $c/u_{\ast }=14$ ); and ▿, W25C25 ( $c/u_{\ast }=25$ ). Equation (4.14) suggests that $B_{\unicode[STIX]{x1D703}}+L_{1/2}=Sc^{1-\unicode[STIX]{x1D6FD}}\overline{H}\,^{+}/(2\unicode[STIX]{x1D6FC})$ , as shown by the lines: – – –, S25; — ⋅ —, W25C2; $\cdots \,$ , W25C14; and — ⋅ ⋅ —, W25C25. For the current DNS, $\unicode[STIX]{x1D6FD}\approx 0.555$ when averaged among the four wave conditions, and the $\unicode[STIX]{x1D6FC}$ value for each wave condition can be found in the caption of figure 8.

The dependence of $B_{\unicode[STIX]{x1D703}}$ on $Sc$ can be further quantified. Recall that the averaged Sherwood number $\overline{Sh}$ follows a power-law dependence on $Sc$ , as shown in (4.3). Applying the surface average to (4.2) and combining it with (4.3) yields

The near-surface linear profile (4.8) gives

and the logarithmic profile (4.10) gives

Substituting (4.12) and (4.13) into (4.11) gives

where

As shown in table 1 and figure 9, for a given wave condition the parameters $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D703}}$ estimated based on the current DNS data are nearly constant. Therefore, equation (4.14) suggests that $B_{\unicode[STIX]{x1D703}}\sim Sc^{1-\unicode[STIX]{x1D6FD}}$ , where $\unicode[STIX]{x1D6FD}\approx 0.555$ as shown in § 4.2. Figure 14 shows the estimated $B_{\unicode[STIX]{x1D703}}+L_{1/2}$ as a function of the Schmidt number. With $\unicode[STIX]{x1D6FD}=0.555$ and $\unicode[STIX]{x1D6FC}$ given by the power-law fitting in figure 8, equation (4.14) is found to provide a reasonable parameterization for $B_{\unicode[STIX]{x1D703}}$ , which agrees well with the $B_{\unicode[STIX]{x1D703}}$ values obtained directly from the log-law profile fitting (the values are listed in table 1).

Figure 15.

Profiles of (a) $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ , (b) $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ , and (c) $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ on a linear scale; and profiles of (d) $\unicode[STIX]{x1D703}_{rms}^{\prime +}/Sc$ , (e) $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}/Sc$ , and (f) $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}/Sc$ on a logarithmic scale. The plotted DNS results are for $Sc=1.0$ with various wave cases: ○, S25 ( $c/u_{\ast }=0$ ); ▫, W25C2 ( $c/u_{\ast }=2$ ); ▵, W25C14 ( $c/u_{\ast }=14$ ); and ▿, W25C25 ( $c/u_{\ast }=25$ ). The solid lines in (d–f) indicate the expressions proposed by Kawamura et al. (Reference Kawamura, Ohsaka, Abe and Yamamoto1998).

4.4 Time- and plane-averaged statistics of scalar fluctuations

The presence of surface waves can induce a strong effect on the air turbulence. Statistics of velocity fluctuations above plane progressive waves of various wave ages and steepnesses have been studied in detail in previous studies (e.g. Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Kihara et al. Reference Kihara, Hanazaki, Mizuya and Ueda2007; Sullivan et al. Reference Sullivan, Edson, Hristov and McWilliams2008; Yang & Shen Reference Yang and Shen2010; Hara & Sullivan Reference Hara and Sullivan2015). On the other hand, the statistics of scalar fluctuations over waves have rarely been studied, with the exception of some studies for stationary wavy walls (e.g. Dellil et al. Reference Dellil, Azzi and Jubran2004; Park et al. Reference Park, Choi and Suzuki2004; Rossi Reference Rossi2010) and for stratified turbulence over water waves (e.g. Sullivan & McWilliams Reference Sullivan and McWilliams2002; Druzhinin et al. Reference Druzhinin, Troitskayaa and Zilitinkevich2016).

In this subsection, the time- and plane-averaged statistics of scalar fluctuations are studied based on the DNS data. Figure 15(a–c) shows the profiles of $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ , $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ , and $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ for various wave cases with $Sc=1.0$ . The profiles of $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ (figure 15 a) and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ (figure 15 b) have a similar trend. As $\overline{z}^{+}$ increases from $0$ , the value first increases to a peak and then decreases to approach a constant value. In particular, the maximum is at $\overline{z}^{+}\approx 35$ for case S25, $\overline{z}^{+}\approx 47$ for case W25C2, and $\overline{z}^{+}\approx 15$ for cases W25C14 and W25C25. Among the three progressive wave cases, the peak values of both $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ increase with $c/u_{\ast }$ , indicating the increase of wave effect as the wave propagates faster. The peak values of $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ in case S25 are slightly higher than the corresponding ones in case W25C2, because the presence of flow separation in case S25 (figure 3 a) enhances the turbulence intensity above the wave trough. The wave-induced effect on $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ appears to be mainly in the viscous wall region (i.e.  $\overline{z}^{+}<50$ ), with only slight differences at higher elevation.

Figure 15(c) shows the profiles of $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ . Assuming a statistically steady state and integrating the scalar transport equation over a control volume, the total vertical scalar flux at any height $z$ ,

is a constant in the stress-driven Couette flow considered in the present study. In the viscous wall region ( $\overline{z}^{+}<50$ ), the contribution from the molecular flux $D\text{d}\overline{\unicode[STIX]{x1D703}}/\text{d}\overline{z}$ is significant, therefore the value of $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ ( $=\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}/\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ ) is less than $1$ . In this region, the values of $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ in cases S25 and W25C2 are larger than those in cases W25C14 and W25C25. In the outer region ( $\overline{z}^{+}>50$ ), the molecular diffusion effect is negligible and thus $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}\approx 1.0$ for all of the cases. Similar to $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ , the profile of $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ is also affected by the surface waves mainly in the viscous wall region. In particular, as $\overline{z}^{+}$ increases, $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ approaches $1$ more rapidly in cases S25 and W25C2 than in cases W25C14 and W25C25, indicating more intensive turbulent mixing in cases S25 and W25C2.

Previous studies on scalar transport over flat boundary have shown that in the viscous sublayer (i.e.  $\overline{z}^{+}<5$ ) the scalar fluctuation profiles can be expressed (to the leading order) as (e.g. Antonia & Kim Reference Antonia and Kim1991; Kawamura et al. Reference Kawamura, Ohsaka, Abe and Yamamoto1998)

To look closely at the viscous sublayer, we plot in figure 15(d–f) the profiles of $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}/Sc$ , $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}/Sc$ , and $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}/Sc$ on a logarithmic scale. For $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}/Sc$ and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}/Sc$ , the DNS results for various wave cases are found to be close to the expressions given by (4.17) and (4.18), respectively. Differently, for $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}/Sc$ the presence of waves causes an obvious discrepancy from the flat-wall expression given by (4.19). For better understanding of the effect of surface waves on the vertical scalar flux $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }$ , more turbulence statistics are quantified and discussed later in §§ 4.5 and 4.6.

Figure 16.

Profiles of (a) $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ , (b) $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ , and (c) $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ on a linear scale; and profiles of (d) $\unicode[STIX]{x1D703}_{rms}^{\prime +}/Sc$ , (e) $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}/Sc$ , and (f) $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}/Sc$ on a logarithmic scale. The plotted DNS results are for the wave case W25C14 with various Schmidt numbers:

, $Sc=0.5$ ; 

,  $Sc=1.0$ ; 

,  $Sc=2.0$ ; and

, $Sc=4.0$ . The solid lines in (d–f) indicate the expressions proposed by Kawamura et al. (Reference Kawamura, Ohsaka, Abe and Yamamoto1998).

Figure 16 shows the effect of $Sc$ on the time- and plane-averaged statistics of scalar fluctuation and fluxes. In particular, figure 16(a–c) shows the profiles of $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ , $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ , and $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ for the wave case W25C14 with various $Sc$ . For $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ (figure 16 a) and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ (figure 16 b), when $Sc$ increases, their peaks in the viscous wall region (i.e. $\overline{z}^{+}<50$ ) increase significantly, while in the outer layer (i.e.  $\overline{z}^{+}>50$ ) the profiles only slightly shift towards larger values. When rescaled by $Sc$ and plotted on a logarithmic scale, in the viscous sublayer (i.e.  $\overline{z}^{+}<5$ ) the profiles of $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}/Sc$ (figure 16 d) and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}/Sc$ (figure 16 e) for all of the four $Sc$ values are close to the expressions given by (4.17) and (4.18), respectively. Differently, the change of $Sc$ has much less effect on the profile of $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ (figure 16 c). When plotted on a logarithmic scale (figure 16 f), the profiles of $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}/Sc$ differ significantly from the flat-wall expression given by (4.19), similar to the results shown in figure 15(f).

In figures 15 and 16, only a number of selected cases are shown to illustrate the effect of surface waves and Schmidt number on the statistics of scalar fluctuations. Other cases exhibit similar trends for the surface wave and $Sc$ effects. Note that in the logarithmic plots shown in figures 15(f) and 16(f), for most of the cases the viscous sublayer portion of the $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}/Sc$ profile cannot be displayed because $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}/Sc<0$ there, which is caused by the strong wave-induced modulation to the vertical scalar flux $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }$ . About this point, more discussions are given in §§ 4.5 and 4.6.

4.5 Wave-correlated distribution of scalar variance and fluxes

To help understand the effect of surface waves on the scalar fluctuation statistics, in this subsection the phase averaging method (see § 4.1 for the mathematical definition) is used to quantify the wave-correlated distribution of the scalar variance and fluxes. In particular, for each wave case the DNS results for $Sc=1.0$ are presented. The results for other Schmidt numbers exhibit similar wave-correlated behaviours, and are thus not shown here.

Figure 17.

Contours of $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle /\unicode[STIX]{x1D703}_{\ast }^{2}$ for various wave cases: (a) S25; (b) W25C2; (c) W25C14; and (d) W25C25. Contours of $\langle {u^{\prime }}^{2}\rangle /u_{\ast }^{2}$ for various wave cases: (e) S25; (f) W25C2; (g) W25C14; and (h) W25C25. $Sc=1.0$ .

Figure 17 shows the contours of the phase-averaged scalar variance, $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ , and the phase-averaged streamwise velocity variance, $\langle {u^{\prime }}^{2}\rangle$ . The scalar variance $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ shows considerable correlation with $\langle {u^{\prime }}^{2}\rangle$ , and exhibits strong phase-dependent variation induced by the surface waves. In particular, for the stationary wave case S25 (figure 17 a,e) there exists a strong shear layer with high intensity for $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ and $\langle {u^{\prime }}^{2}\rangle$ , which originates from the crest and extends downstream to above the succeeding trough. In case W25C2 (figure 17 b), the high- $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ region is also located above the trough, but the maximum is located more upstream than that in case S25. The high- $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ regions in cases S25 and W25C2 are consistent with the high vertical gradient regions for $\langle \unicode[STIX]{x1D703}\rangle$ shown in figures 3(b) and 3(d), respectively.

Different from cases S25 and W25C2, in cases W25C14 and W25C25 the high- $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ regions are located above the windward face of the wave crest, and are much closer to the surface as a result of the smooth streamlines and $\langle \unicode[STIX]{x1D703}\rangle$ contour lines that are parallel to the wave surface in these two faster wave cases (figure 3 e–h). The streamwise velocity variances $\langle {u^{\prime }}^{2}\rangle$ for cases W25C14 (figure 17 g) and W25C25 (figure 17 h) show similar wave-correlated distributions as the corresponding $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ .

Figure 18.

Contours of $\langle \unicode[STIX]{x1D703}^{\prime }u^{\prime }\rangle /\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ for $Sc=1.0$ and for various wave cases: (a) S25; (b) W25C2; (c) W25C14; and (d) W25C25.

Consistent with the similarity between the $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ and $\langle {u^{\prime }}^{2}\rangle$ distributions, the horizontal scalar flux $\langle \unicode[STIX]{x1D703}^{\prime }u^{\prime }\rangle$ shown in figure 18 exhibits a similar wave-correlated distribution and wave condition dependence as in $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ (cf. figure 17). Moreover, the wave-induced variations of $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ and $\langle \unicode[STIX]{x1D703}^{\prime }u^{\prime }\rangle$ are mainly within the viscous wall region $\overline{z}^{+}<50$ (corresponding to $(z-\unicode[STIX]{x1D702})/\unicode[STIX]{x1D706}<0.12$ ), consistent with the wave-affected region of the time- and plane-averaged profiles $\overline{\unicode[STIX]{x1D703}^{\prime }}\,\!_{rms}^{+}$ and $\overline{\unicode[STIX]{x1D703}^{\prime }u^{\prime }}\,^{+}$ shown in figure 15.

Figure 19 shows the contours of the vertical scalar flux $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ . Similar to $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ and $\langle {u^{\prime }}^{2}\rangle$ , the presence of surface waves also induces significant wave-correlated variation in $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ . In cases S25 and W25C2, the maximum $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ is located above the trough, in a region close to the region of maximum $\langle {u^{\prime }}^{2}\rangle$ (cf. figure 17 e,f). In cases W25C14 and W25C25, the maximum $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ is located above the leeward face of the crest. Moreover, there exists a distinct negative $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ region above the windward face of the crest, which does not exist in flat-wall boundary layer turbulence. The negative $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ region is insignificant in cases S25 and W25C2, but is prominent in cases W25C14 and W25C25. The distinct wave-induced negative $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ region near the windward face causes the time and plane average of $\overline{-\unicode[STIX]{x1D703}^{\prime }w^{\prime }}\,^{+}$ in the viscous sublayer to be significantly lower than that above a flat surface, as shown in figures 15(f) and 16(f).

A similar distinct negative vertical flux region was found for the Reynolds stress $\langle -u^{\prime }w^{\prime }\rangle$ in turbulence over intermediate and fast waves in the DNS studies by Yang & Shen (Reference Yang and Shen2009, Reference Yang and Shen2010). Recently, the existence of the negative $\langle -u^{\prime }w^{\prime }\rangle$ region has also been observed in the laboratory experiment of Buckley & Veron (Reference Buckley and Veron2016) using the particle image velocimetry technique. Yang & Shen (Reference Yang and Shen2009) studied the characteristics of coherent vortical structures in turbulence over progressive surface waves, and suggested that the significant negative Reynolds stress above the windward face is caused by the vertically bent quasi-streamwise vorticies. A similar mechanism is expected for generating the negative $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ , which is studied in more detail in § 4.6.

Figure 19.

Contours of $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle /\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ for $Sc=1.0$ and for various wave cases: (a) S25; (b) W25C2; (c) W25C14; and (d) W25C25.

4.6 Correlation between scalar and vortical structures

Coherent vortical structures in turbulent boundary layer play an important role in the mixing and transport of momentum as well as scalars (e.g. Robinson Reference Robinson1991; Kawamura et al. Reference Kawamura, Ohsaka, Abe and Yamamoto1998; Zedler & Street Reference Zedler and Street2001; Choi & Suzuki Reference Choi and Suzuki2005; Wallace Reference Wallace2016). In this subsection, the correlation between vortical structures and scalar fluctuations and fluxes is studied by direct observation of instantaneous vortices and the scalar field, and by conditional averages.

Figure 20.

Instantaneous flow and scalar fields in turbulence over the slow wave (case W25C2, $Sc=1.0$ ). Here only a small portion of the simulation domain is shown for illustration purposes. In (a), five characteristic vortices are marked, i.e. the reversed horseshoe vortices A and B and the quasi-streamwise vortices C–E. Two $(y,z)$ -planes at the leeward face (plane i) and crest (plane ii) are plotted, with the contours of $\unicode[STIX]{x1D703}^{\prime }/\unicode[STIX]{x1D703}_{\ast }$ and vectors of $(v^{\prime },w^{\prime })$ shown simultaneously. Zoom-in views of planes (i) and (ii) are plotted with the contours of $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }/\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ shown in (b) and (c), respectively. The instantaneous vortices are identified by the iso-surface of $\unicode[STIX]{x1D706}_{2}=-0.02$ , with white colour for $\unicode[STIX]{x1D714}_{x}>0$ and black colour for $\unicode[STIX]{x1D714}_{x}<0$ . In (b), the leg of vortex A with $\unicode[STIX]{x1D714}_{x}>0$ is labelled as $\text{A}_{p}$ , and the one with $\unicode[STIX]{x1D714}_{x}<0$ is labelled as $\text{A}_{n}$ . Similar notations are used to mark the two legs of vortex B.

Figure 20 shows the instantaneous vortical structures and scalar fluctuations in the slow wave case W25C2. In particular, the vortices are visualized using the $\unicode[STIX]{x1D706}_{2}$ method (Jeong & Hussain Reference Jeong and Hussain1995). Based on the DNS data, the strain-rate tensor $\unicode[STIX]{x1D64E}$ and the rotation tensor $\unicode[STIX]{x1D734}$ are calculated, and $\unicode[STIX]{x1D706}_{2}$ is the second largest eigenvalue of the tensor $\unicode[STIX]{x1D64E}^{2}+\unicode[STIX]{x1D734}^{2}$ . In the flow field, the regions with $\unicode[STIX]{x1D706}_{2}<0$ correspond to the interior of vortices (Jeong & Hussain Reference Jeong and Hussain1995). Moreover, in figure 20(a) the instantaneous vortices are coloured based on the streamwise vorticity, i.e. white if $\unicode[STIX]{x1D714}_{x}>0$ and black if $\unicode[STIX]{x1D714}_{x}<0$ . In figure 20(a), several characteristic instantaneous vortices can be observed: reversed horseshoe vortices A and B, and quasi-streamwise vortices C–E. In particular, these reversed horseshoe vortices have their heads upstream and the two legs extended downstream, with $\unicode[STIX]{x1D714}_{x}>0$ for the left leg and $\unicode[STIX]{x1D714}_{x}<0$ for the right leg (observed when facing the $+x$ -direction).

The correlation between these characteristic vortices and the scalar fluctuations can be clearly seen in figure 20(b,c). The reversed horseshoe vortices A and B are located above the wave trough. Taking vortex B as an example, the rotating motions of its head and two legs induce strong sweep motion ( $w^{\prime }<0$ ) towards the wave trough and bring high $\unicode[STIX]{x1D703}$ downwards, resulting in $\unicode[STIX]{x1D703}^{\prime }>0$ and $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }>0$ (figure 20 b). The effect of the reversed horseshoe vortex A on the scalar transport is similar to that of vortex B. Vortices C and D appear as a counter-rotating pair, and induce an upwelling motion (i.e.  $w^{\prime }>0$ ) between them. Because on average $\unicode[STIX]{x1D703}$ increases with height, the upwelling flow ejects low $\unicode[STIX]{x1D703}$ upwards and generates a region of $\unicode[STIX]{x1D703}^{\prime }<0$ ; the combination of $w^{\prime }>0$ and $\unicode[STIX]{x1D703}^{\prime }<0$ results in a region with $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }>0$ between vortices C and D (figure 20 c). The single quasi-streamwise vortex E generates a downwelling event ( $w^{\prime }<0$ ) on its right side that brings high $\unicode[STIX]{x1D703}$ downwards and results in $\unicode[STIX]{x1D703}^{\prime }>0$ and $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }>0$ there.

The above correlations between the vortices and scalar fluctuations in the slow wave case are similar to those in turbulence over flat boundaries (see the reviews by Robinson Reference Robinson1991; Wallace Reference Wallace2016). As shown by the statistical analysis in Yang & Shen (Reference Yang and Shen2009, Reference Yang and Shen2010), in turbulence over progressive surface waves, different types of characteristic vortices are present for different wave conditions, and these different characteristic vortices also have different preferential locations relative to the surface waves. In the case of slow waves, reversed horseshoe vortices are usually generated above the wave trough, and then break up into quasi-streamwise vortices as their legs are stretched downstream (Yang & Shen Reference Yang and Shen2009). Then the quasi-streamwise vortices are convected downstream and lifted upwards over the succeeding wave crests and troughs. The combination of the wave-phase-dependent distribution of the coherent vortices and their strong correlations with the scalar fluctuations provides a mechanism for generating the wave-correlated distributions of the scalar variance $\langle {\unicode[STIX]{x1D703}^{\prime }}^{2}\rangle$ (figure 17) and the scalar fluxes $\langle \unicode[STIX]{x1D703}^{\prime }u^{\prime }\rangle$ (figure 18) and $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ (figure 19).

Figure 21.

Educed flow and scalar fields for case W25C2 with $Sc=1.0$ by conditional average of the Q2 events above the wave trough. The horseshoe vortex is visualized by the iso-surfaces of $\unicode[STIX]{x1D706}_{2}=-0.002$ , with white colour on the leg with $\unicode[STIX]{x1D714}_{x}>0$ and black colour on the leg with $\unicode[STIX]{x1D714}_{x}<0$ . Contours of $\unicode[STIX]{x1D703}^{\prime }/\unicode[STIX]{x1D703}_{\ast }$ are shown on the vertical plane across the two legs. Contours of $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }/\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ and vectors of $(v^{\prime },w^{\prime })$ are shown in the zoom-in view of the vertical plane, on which the thick white and black lines are the iso-contour lines of $\unicode[STIX]{x1D706}_{2}=-0.002$ .

Figure 22.

Educed flow and scalar fields for case W25C2 with $Sc=1.0$ by conditional average of the Q4 events above the wave trough. The reversed horseshoe vortex is visualized by the iso-surface of $\unicode[STIX]{x1D706}_{2}=-0.0008$ , with white colour on the leg with $\unicode[STIX]{x1D714}_{x}>0$ and black colour on the leg with $\unicode[STIX]{x1D714}_{x}<0$ . Contours of $\unicode[STIX]{x1D703}^{\prime }/\unicode[STIX]{x1D703}_{\ast }$ are shown on the vertical plane across the two legs. Contours of $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }/\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ and vectors of $(v^{\prime },w^{\prime })$ are shown in the zoom-in view of the vertical plane, on which the thick white and black lines are the iso-contour lines of $\unicode[STIX]{x1D706}_{2}=-0.0008$ .

The correlation between vortices and scalar fluctuations can be further investigated by a conditional average based on the quadrants of the vertical turbulent flux (Wallace Reference Wallace2016). Similar to the quadrant analysis for Reynolds stress, the vertical scalar flux $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ can be divided into four quadrants based on the signs of $\unicode[STIX]{x1D703}^{\prime }$ and $w^{\prime }$ : Q1 $(\unicode[STIX]{x1D703}^{\prime }>0,w^{\prime }>0)$ , Q2 $(\unicode[STIX]{x1D703}^{\prime }<0,w^{\prime }>0)$ , Q3 $(\unicode[STIX]{x1D703}^{\prime }<0,w^{\prime }<0)$ and Q4 $(\unicode[STIX]{x1D703}^{\prime }>0,w^{\prime }<0)$ . The coherent flow structure related to a specific quadrant can be educed by conditionally sampling and averaging the instantaneous flow events of the chosen quadrant. A brief introduction to the quadrant conditional average scheme used in the current analysis is given in appendix C.

For case W25C2, quadrant analysis indicates that the strong vertical scalar flux above the wave trough shown in figure 19(b) is mainly due to the Q2 and Q4 events (see figure 30). By applying the quadrant-based conditional average above the wave trough, the coherent flow structures associated with the Q2 and Q4 events can be educed. Figure 21 shows that a horseshoe vortex (with its head on the downstream side) is educed by a conditional average of Q2 events above the wave trough. Such a vortex induces ejection of lower $\unicode[STIX]{x1D703}$ upwards (since $\unicode[STIX]{x1D703}$ has lower value near the bottom boundary), which results in $\unicode[STIX]{x1D703}^{\prime }<0$ and $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }>0$ between the two legs. It should be pointed out that direct observation of the instantaneous snapshots suggests the presence of quasi-streamwise vortices and horseshoe vortices (with the heads located downstream relative to the legs) in this flow region, both of which contribute to the Q2 events. Because the quadrant-based conditional average does not distinguish these two types of vortices, the educed flow field in figure 21 should account for the contribution of both types of vortices.

Figure 22 shows the conditionally averaged flow and scalar fields for Q4 events in case W25C2. The educed vortical structure exhibits a clear reversed horseshoe shape located slightly above the wave trough, which represents well the instantaneous reversed horseshoe vortices observed from the instantaneous snapshots (figure 20). The reversed horseshoe vortex induces downwelling motion that sweeps high $\unicode[STIX]{x1D703}$ towards the wave trough, which results in $\unicode[STIX]{x1D703}^{\prime }>0$ and $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }>0$ between the two legs. The reversed horseshoe vortex educed from Q4 events (figure 22) is located lower than the vortex pair educed from Q2 events (figure 21).

Figure 23.

Instantaneous flow and scalar fields in turbulence over the intermediate wave (case W25C14, $Sc=1.0$ ). Only a small portion of the simulation domain is shown for illustration purposes. Four characteristic vertically bent quasi-streamwise vortices (A–D) are marked. In (a), the instantaneous vortices are identified by the iso-surface of $\unicode[STIX]{x1D706}_{2}=-0.02$ , with white colour for $\unicode[STIX]{x1D714}_{x}>0$ and black colour for $\unicode[STIX]{x1D714}_{x}<0$ . The $(y,z)$ -plane above the leeward face (plane i) and the $(x,z)$ -plane above the windward face (plane ii) are plotted, with the contours of $\unicode[STIX]{x1D703}^{\prime }/\unicode[STIX]{x1D703}_{\ast }$ and vectors of velocity fluctuation (the components parallel to each plane) shown simultaneously. Panels (b) and (c) show the contours of the scalar flux $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }/\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ on planes (i) and (ii), respectively. In (b) and (c), the locations of the vortices are indicated by the iso-contours of $\unicode[STIX]{x1D706}_{2}=-0.02$ , with white colour for $\unicode[STIX]{x1D714}_{x}>0$ and black colour for $\unicode[STIX]{x1D714}_{x}<0$ .

Different from the slow wave case discussed above, in the intermediate (W25C14) and fast (W25C25) wave cases the dominant flow structures are the vertically bent quasi-streamwise vortices, e.g. the vortices A–D shown in figure 23(a). Yang & Shen (Reference Yang and Shen2009) showed that the characteristics of coherent vortices in turbulence over intermediate and fast waves are very similar, with the difference mainly in the population and intensity of the vortices. Therefore, the analysis here focuses on the intermediate wave case W25C14 as a representative case. Figure 23 shows the instantaneous vortex and scalar fields in case W25C14, and figure 24 shows a sketch that illustrates the correlation between a vortex and the vertical scalar flux. In a reference frame moving at the wave phase speed, the near-surface vortices travel backwards in the $-x$ -direction (see the mean streamlines shown in figure 3). They appear to be straight and horizontally oriented above the leeward face and the crest, and then bend downwards as they travel over the windward face. The unique shape and wave-phase correlation of the vortices result in the wave-correlated scalar flux shown in figure 19(c,d).

Taking the vortex C in figure 23 as an example, its horizontal part above the leeward face induces an upwelling motion (i.e.  $w^{\prime }>0$ ) on its left side (observed when facing the $+x$ -direction), which ejects low $\unicode[STIX]{x1D703}$ upwards and results in $\unicode[STIX]{x1D703}^{\prime }<0$ (see plane-i in figure 23 a) and leads to a Q2 flux $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }>0$ (figure 23 b). This mechanisms is similar to the Q2 flux generation mechanism in case W25C2. On the other hand, the vertical part of the vortex C above the windward face induces a horizontal velocity fluctuation $u^{\prime }<0$ on its left side (observed when facing the $+x$ -direction), which induces a vertical velocity fluctuation $w^{\prime }<0$ because $\unicode[STIX]{x2202}\langle w\rangle /\unicode[STIX]{x2202}x<0$ there (Yang & Shen Reference Yang and Shen2009) (also see figure 3 e and the sketch in figure 24 a). Similarly, the fluctuation $u^{\prime }<0$ induces a scalar fluctuation of $\unicode[STIX]{x1D703}^{\prime }<0$ because $\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D703}\rangle /\unicode[STIX]{x2202}x<0$ there (figure 3 f and the sketch in figure 24 a). These correlations result in $\unicode[STIX]{x1D703}^{\prime }u^{\prime }>0$ and $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }<0$ above the windward face of the wave crest, as shown in figures 18(c) and 19(c), respectively.

Figure 24.

Sketch of the characteristic vortical structure (i.e. the vertically bent quasi-streamwise vortex) in turbulence over intermediate and fast waves. Panel (a) illustrates the correlation between the vortex and the turbulent fluctuations of velocity and scalar. The arrows near the vortex indicate the velocity fluctuation vectors associated with the vortex, with solid arrows in front of the vortex and dashed arrows behind the vortex. Panels (b–d) show three representative samples of the instantaneous vortices extracted from the DNS data of case W25C14.

Figure 25.

Educed flow and scalar fields for case W25C14 with $Sc=1.0$ by conditional average of the Q2 events above the leeward face of the wave crest. The pair of vortices are visualized by the iso-surfaces of $|\unicode[STIX]{x1D714}_{x}/(u_{\ast }^{2}/\unicode[STIX]{x1D708})|=0.05$ , with white colour for $\unicode[STIX]{x1D714}_{x}>0$ and black colour $\unicode[STIX]{x1D714}_{x}<0$ . Contours of $\unicode[STIX]{x1D703}^{\prime }/\unicode[STIX]{x1D703}_{\ast }$ are shown on the vertical plane across the vortex pair. Contours of $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }/\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ and vectors of $(v^{\prime },w^{\prime })$ are shown in the zoom-in view of the vertical plane, on which the thick white and black lines are the iso-contour lines of $\unicode[STIX]{x1D714}_{x}/(u_{\ast }^{2}/\unicode[STIX]{x1D708})=0.05$ and $-0.05$ , respectively.

Similar to case W25C2, a quadrant-based conditional average can be done for case W25C14 to educe a statistical description of the correlation between vortical structures and scalar fluxes. As shown in figure 31 in the appendix C, in case W25C14 the positive vertical scalar flux $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }$ above the leeward face is mainly due to the Q2 and Q4 events, while the negative vertical scalar flux above the windward face is mainly due to the Q1 and Q3 events. This result is consistent with the direct observation of the instantaneous flow and scalar fields shown in figure 23. Based on these wave-phase preferences, Q2 and Q4 events are conditionally sampled above the leeward face, and Q1 and Q3 events are conditionally sampled above the windward face. The specific locations where quadrant events detectors are applied are marked in figure 31.

Figure 25 shows the conditionally averaged Q2 event above the leeward face of case W25C14. Note that unlike in figures 21 and 22 where the vortices are visualized by the $\unicode[STIX]{x1D706}_{2}$ method, in figure 25 the vortices are visualized using the iso-surfaces of $|\unicode[STIX]{x1D714}_{x}|$ . This is because in turbulence over intermediate and fast propagating surface waves (as well as in oscillating flow over stationary waves), the strong relative oscillation between the mean flow and the surface waves generates thin vortex sheets with significant spanwise vorticity in the vicinity of the wave crests and troughs (e.g. Tseng & Ferziger Reference Tseng and Ferziger2003; Yang & Balaras Reference Yang and Balaras2006; Yang & Shen Reference Yang and Shen2009). In figure 23, those thin vortex sheets have been removed to provide a clear visualization of other coherent flow structures. However, because of their persistent presences at fixed wave phases (i.e. above crests and troughs), these vortex sheets also appear in the conditionally averaged field, with magnitudes larger than those of the vortices associated with the quadrant events. As a result, the desired vortices would not be clearly seen when the $\unicode[STIX]{x1D706}_{2}$ method is used for visualization. Instead, the iso-surfaces of $|\unicode[STIX]{x1D714}_{x}|$ are good alternative for visualization because the horizontal part of a quasi-streamwise vortex above the leeward face in case W25C14 has a significant streamwise vorticity $\unicode[STIX]{x1D714}_{x}$ . Similarly, the iso-surfaces of $|\unicode[STIX]{x1D714}_{z}|$ are used for visualizing the vertical part of the vortex above the windward face. As shown in figure 25, the conditional average of Q2 events exhibits a pair of counter-rotating streamwise vortices above the leeward face, which induce upwelling motion ( $w^{\prime }>0$ ) between them and result in $\unicode[STIX]{x1D703}^{\prime }<0$ and $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }>0$ . Note that in figure 25 the vortices appear in pairs because the Q2 events generated by vortices with $\unicode[STIX]{x1D714}_{x}>0$ and $\unicode[STIX]{x1D714}_{x}<0$ are both sampled and included in the averaging process. In the instantaneous snapshots, these vortices usually appear individually.

Figure 26.

Side view of the educed flow and scalar fields for case W25C14 with $Sc=1.0$ by conditional average based on the four different quadrants of $\langle -\unicode[STIX]{x1D703}^{\prime }w^{\prime }\rangle$ : (a) Q1 $(\unicode[STIX]{x1D703}^{\prime }>0,w^{\prime }>0)$ events above the windward face; (b) Q2 $(\unicode[STIX]{x1D703}^{\prime }<0,w^{\prime }>0)$ events above the leeward face; (c) Q3 $(\unicode[STIX]{x1D703}^{\prime }<0,w^{\prime }<0)$ events above the windward face; and (d) Q4 $(\unicode[STIX]{x1D703}^{\prime }>0,w^{\prime }<0)$ events above the leeward face. In the figure, the colour contours are for $-\unicode[STIX]{x1D703}^{\prime }w^{\prime }/\unicode[STIX]{x1D703}_{\ast }u_{\ast }$ on the $(x,z)$ -plane across the centre of the educed field. In each panel, the vortical structures are represented by the iso-surfaces of (a) $\unicode[STIX]{x1D714}_{z}/(u_{\ast }^{2}/\unicode[STIX]{x1D708})=-0.05$ , (b) $\unicode[STIX]{x1D714}_{x}/(u_{\ast }^{2}/\unicode[STIX]{x1D708})=0.05$ , (c) $\unicode[STIX]{x1D714}_{z}/(u_{\ast }^{2}/\unicode[STIX]{x1D708})=0.05$ , and (d) $\unicode[STIX]{x1D714}_{x}/(u_{\ast }^{2}/\unicode[STIX]{x1D708})=-0.05$ , respectively. In particular, light colour iso-surfaces correspond to positive vorticities (i.e.  $\unicode[STIX]{x1D714}_{x}>0$ or $\unicode[STIX]{x1D714}_{z}>0$ ) and dark colour ones correspond to negative vorticities (i.e.  $\unicode[STIX]{x1D714}_{x}<0$ or $\unicode[STIX]{x1D714}_{z}<0$ ).

A similar quadrant-based conditional average can be applied for all the four quadrants. Figure 26 shows the results for the four quadrants. Note that, as discussed above, in figure 26(a,c) the vortices are visualized using the iso-surfaces of $|\unicode[STIX]{x1D714}_{z}|$ , while in figure 26(b,d) the vortices are visualized based on $|\unicode[STIX]{x1D714}_{x}|$ . For all the four quadrants the educed vortices appear in pair, but only the ones in front of the central $(x,z)$ -plane of the averaged field are shown in figure 26. Both the conditionally averaged vortices and the scalar flux fields agree with the observation of the instantaneous snapshot in figure 23 and the phase-averaged scalar flux contours in figures 19(c) and 31.

It should be noted that the characteristics of the vortical structures may vary with the Reynolds number. The results presented in § 4.6 are based on DNS with a Reynolds number lower than that in a real marine wind condition. Therefore, caution should be taken when interpreting the vortical structures reported in this DNS study and linking them to those in the more realistic field condition. Nevertheless, the vortical structures found in the current DNS as well as their correlations with the distribution of scalar flux still provide useful insights for explaining the flow structures over progressive waves on a laboratory scale such as the recent experiment by Buckley & Veron (Reference Buckley and Veron2016), in which they observed negative Reynolds stress above the windward face of air flow over fast propagating surface waves similar to the results obtained from the current DNS (see figures 19, 26 and 31).