4.1. Definition of wave-following streamline coordinates

To facilitate the definition of the streamline coordinates, the flow is translated to the wave-following frame such that the mean flow, i.e. the wave motions, becomes a quasi-steady, two-dimensional flow in the $x$–$y$ plane. The translated mean velocity field $(\langle {u}\rangle -c, \langle {v}\rangle )$ is plotted in figure 4(a) using vector arrows, which show that the flow direction is opposite to the direction of wave propagation. Following Finnigan (Reference Finnigan1983), for the two-dimensional steady mean flow that depends on $x$ and $y$, the transformation from the Cartesian coordinates $(x, y, z)$ to the streamline coordinates $(\xi _1,\xi _2,\xi _3)$ is performed only for the first two dimensions and does not affect the transverse direction $z$ or $\xi _3$, i.e.  $z=\xi _3$. The coordinate lines of $\xi _1$ and $\xi _2$ are defined as the streamlines and the lines that are orthogonal to the streamlines, respectively. The streamlines are given by the lines of $\psi (x,y)=\text {constant}$, where $\psi (x,y)$ is the streamfunction defined by the relations

(4.1a,b)\begin{equation} \frac{\partial{\psi}}{\partial{y}} = \langle{u}\rangle - c, \quad \frac{\partial{\psi}}{\partial{x}} ={-}\langle{v}\rangle. \end{equation}

The other group of coordinate lines is given by the function $\phi (x, y)$, defined as

(4.2a,b)\begin{equation} \frac{\partial{\phi}}{\partial{x}} = \zeta (\langle{u}\rangle - c), \quad \frac{\partial{\phi}}{\partial{y}} = \zeta \langle{v}\rangle. \end{equation}

Note that the lines of $\phi (x,y)=\text {constant}$ are orthogonal to the streamlines. The parameter $\zeta$ in the above definition is calculated using

(4.3)\begin{equation} \zeta = \exp\left(\int \frac{\varOmega}{U^{2}} \,\mathrm{d}\psi\right),\end{equation}

where $U=|(\langle {u}\rangle -c,\langle {v}\rangle )|$ is the mean velocity magnitude and $\varOmega = \partial \langle {v}\rangle /\partial x - \partial \langle {u}\rangle /\partial y$ is the vorticity of the mean flow. We remark that when the flow is irrotational, the parameter $\zeta$ calculated from (4.3) is simply $\zeta =1$, and the function $\phi (x,y)$, (4.2a,b), reduces to the velocity potential, i.e.  $\boldsymbol {\nabla } \phi = (\langle {u}\rangle -c, \langle {v}\rangle )$. Although $\phi$ and $\psi$ can be directly used as the coordinates of an orthogonal coordinate system, the magnitudes of the base vectors of the $(\phi,\psi )$ system vary from place to place with respect to the Cartesian frame, in addition to the difference in dimensions, which makes it difficult to measure quantities using these coordinates and compare them with those measured under Cartesian coordinates. Therefore, we further normalize the basis vectors by their local magnitudes and the resulting coordinates $\xi _1$ and $\xi _2$ satisfy

(4.4a,b)\begin{equation} \mathrm{d}\xi_1= {(\zeta U )}^{{-}1} \mathrm{d}\phi, \quad \mathrm{d}\xi_2=U^{{-}1} \mathrm{d}\psi. \end{equation}

In this way, the quantities in the streamline coordinates $(\xi _1,\xi _2,\xi _3)$ defined above are properly scaled and have the same physical dimensions as the Cartesian coordinates.

Figure 4.

(a) The mean velocity vectors $(\langle {u}\rangle -c, \langle {v}\rangle )$ and contours of the mean velocity magnitude ${U}=|(\langle {u}\rangle -c, \langle {v}\rangle )|$. Note that the along-stream component of the mean velocity in streamline coordinates satisfies $\hat {U}=U$. (b) The $\xi _1$ and $\xi _2$ coordinate lines of the streamline coordinates.

A $\xi _1$–$\xi _2$ plane of the wave-following streamline coordinates is plotted schematically in figure 4(b). The wave surface, as a material boundary, is a streamline in the wave-following frame. The other streamlines ($\xi _2=\text {constant}$) are roughly parallel to the wave surface, with decreasing curvature as the distance from the surface increases. This is expected because the wave motions decay with the depth and the streamlines should become straight in deep waters with the influence of the wave diminishing. We note that the directions of the axes of the above streamline coordinates are different from those of Cartesian coordinates. The along-stream coordinate $\xi _1$ increases in the $-x$-direction, consistent with the mean flow direction shown in figure 4(a). The positive direction of the cross-stream coordinate $\xi _2$ roughly points downward to the interior of the flow and the right-hand system is maintained.

4.2. Mean flow properties

In this section we discuss properties of the wave using the wave-following streamline coordinate system for the subsequent analyses of the wave–turbulence interaction. For more intuitive interpretation of the figures, hereafter, the results are plotted in the physical domain using Cartesian coordinates $(x, y, z)$ but the dependent variables, such as the velocity, are defined based on the streamline coordinates. The values evaluated in streamline coordinates are denoted by $\widehat {(\cdot )}$. For example, the velocity components in the $\xi _1, \xi _2$ and $\xi _3$ directions of the streamline coordinates are denoted by $\hat {u}, \hat {v}$ and $\hat {w}$, respectively.

In the streamline coordinates, the only non-zero component of the mean velocity $\hat {\boldsymbol {U}}$ is the along-stream component $\hat {U}$, which is equal to the velocity magnitude $U$. The cross-stream mean velocity is zero by definition. The dimensionless mean momentum equation in the $\xi _1$-direction is written as (Finnigan Reference Finnigan1983)

(4.5)\begin{align} \frac{\partial \hat{U}}{\partial t} + \hat{U}\frac{\partial \hat{U}}{\partial \xi_1} &={-}\frac{\partial \langle{p}\rangle}{\partial \xi_1} - \frac{\partial \langle{\hat{u}'^{2}}\rangle}{\partial \xi_1} + \frac{\langle{\hat{u}'^{2}}\rangle - \langle{\hat{v}'^{2}}\rangle}{\gamma} - \frac{\partial \langle{\hat{u}'\hat{v}'}\rangle}{\partial \xi_2} + \frac{2 \langle{\hat{u}'\hat{v}'}\rangle}{R} \nonumber\\ &\quad + \frac{1}{Re} \left[\frac{\partial \hat{U}}{\partial \xi_1^{2}} + \frac{\partial \hat{U}}{\partial \xi_2^{2}} - \frac{2}{\gamma}\frac{\partial \hat{U}}{\partial \xi_1} - \frac{1}{R}\frac{\partial \hat{U}}{\partial \xi_2} - \frac{\hat{U}}{R^{2}}\right]. \end{align}

In the above equation, $\gamma$ and $R$ are two length scales arising naturally from the definitions of the wave-following streamline coordinates and their physical meanings are detailed below.

Figure 4(a) plots the contours of $\hat {U}$ for case I with $ak=0.1$ and $S=1$ as an example. We can see that the maximum and minimum velocity occurs under the wave trough and crest, respectively. Under the forward slope, the flow slows down along the streamlines, accompanied by the elevation of the water surface. The reverse process, i.e. the flow acceleration, occurs under the backward slope as the surface elevation decreases. Such periodic acceleration and deceleration along the streamlines can be quantified using a length scale $\gamma$ defined as

(4.6)\begin{equation} \gamma^{{-}1} = \frac{1}{\hat{U}}\frac{\partial{\hat{U}}}{\partial{\xi_1}} = \frac{\partial{\ln \hat{U}}}{\partial{\xi_1}}.\end{equation}

According to the above definition, $\gamma$ is the ‘e-folding’ length of the streamwise velocity $\hat {U}$. A larger $\gamma ^{-1}$ indicates that the local relative acceleration rate of the flow is larger. The contours of $\gamma ^{-1}$ are shown in figure 5(a). The negative and positive $\gamma ^{-1}$ under the wave forward and backward slopes correspond to the flow deceleration and acceleration, respectively. This parameter is also associated with the streamwise stretching and compression imposed on the fluid elements by the wave motions. As the flow accelerates, the fluid elements are stretched along the streamline. On the contrary, the deceleration of the flow leads to the effect of streamwise compression. It should be noted that for steady progressive waves, the mean flow does not detach from the wave surface even with the deceleration. In a more violent scenario, such as in a spilling breaker (Dabiri & Gharib Reference Dabiri and Gharib1997) or hydraulic jump (Misra et al. Reference Misra, Kirby, Brocchini, Veron, Thomas and Kambhamettu2008), a sharp deceleration of the mean flow can occur and the associated flow separation can induce intense generation of turbulence vorticity.

Figure 5.

Contours of (a) $\gamma ^{-1}$ and (b) $R^{-1}$ for case I. The values are normalized by $ak^{2}$. The yellow line with arrows represents a streamline and the direction of the mean flow.

Another important parameter introduced by the streamline coordinate system is the local curvature of the streamlines, $R^{-1}$, defined using the mean vorticity $\varOmega$ and streamwise velocity $\hat {U}$ as

(4.7)\begin{equation} R^{{-}1} = \frac{1}{\hat{U}}\left(\varOmega + \frac{\partial{\hat{U}}}{\partial{\xi_2}}\right).\end{equation}

It can be shown that the definition above is equivalent to the geometric curvature of the streamline (or the coordinate lines of $\xi _1$), for which the details are given in Appendix A. The contours of the streamline curvature for case I are plotted in figure 5(b). The figure shows that the maximum and minimum curvatures occur under the wave crest and trough, respectively. In our definition (4.7), $R^{-1}$ is positive when the centre of the curvature lies in the $+\xi _2$-direction. Therefore, the concave streamlines under the wave crest have positive curvature values whereas the convex flow under the trough has negative curvatures. We note that the flow curvature is associated with the centripetal acceleration of the flow and can be used to quantify the straining effect on the fluid elements caused by the rotation of the mean flow. This is important to the wave-phase variation of the Reynolds shear stress as discussed in the subsequent sections.

Both the acceleration length scale $\gamma$ and the curvature $R^{-1}$ can be estimated using the linear irrotational wave theory as detailed in Appendix B. The results (B10) and (B11) are written below,

(4.8)$$\begin{gather} \gamma^{{-}1} = a k^{2} \, \textrm{e}^{ky} \cos {kx} + O(a^{2} k^{2}), \end{gather}$$

(4.9)$$\begin{gather}R^{{-}1} = a k^{2} \, \textrm{e}^{ky} \sin {kx} + O(a^{2} k^{2}). \end{gather}$$

As discussed below, the viscous wave considered in the present study is not perfectly irrotational but the rotational region is confined to a thin viscous layer below the surface. Therefore, the use of the irrotational wave theory can be justified for most of the region away from the surface. Nonetheless, we find that $\gamma ^{-1}$ and $R^{-1}$ can still be accurately estimated even in the viscous layer using the irrotational wave velocity because the viscous effect does not significantly affect the along-stream acceleration nor the geometry of the streamlines. The numerical results of $\gamma ^{-1}$ and $R^{-1}$ (figure 5) are consistent with the above theoretical estimates (not plotted). Both quantities vary sinusoidally with the wave phase and the magnitudes of their variations scales with $a k^{2}$ and decays exponentially with the depth. This indicates that these two quantities are correlated for the wave orbital motions. However, it should be noted that they are associated with different mechanisms, i.e.  streamwise acceleration for $\gamma$ and centripetal acceleration for $R$ but with a phase difference of ${\rm \pi} /2$. As a result, they appear in different budget terms in the transport equation of the Reynolds shear stress, as shown below in § 4.4. If we assume that the characteristic length scale of the energy-containing eddies is the integral length scale $L_\infty ^{cf}$, we can define a relative curvature as $L_\infty ^{cf}/R$. The relative curvature, representative of the curvature effect on the turbulence (Moser & Moin Reference Moser and Moin1987; Baskaran et al. Reference Baskaran, Smits and Joubert1991), is $0.11$ for case I and $0.16$ for cases II and III. These correspond to comparatively large curvature that can induce significant modifications in a curved channel (Moser & Moin Reference Moser and Moin1987, Reference Moser and Moin1984). As a result, we may expect an appreciable effect of the wave motions on the turbulence.

Now that, in the wave-following frame, the wave motions are interpreted as nearly parallel flows with alternating concave and convex curvatures under the wave crest and trough, respectively, it should be noted that a crucial difference of the wave flow from the flow over a curved no-slip wall is that the wave is nearly irrotational. This means that, based on (4.7), in most of the region below the surface, the cross-stream gradient of the mean velocity, $\partial \hat {U}/\partial \xi _2$, is approximately equal to the straining associated with the curvature, $\hat {U}/R$. By comparison, the shear flow over a curved no-slip boundary has strong mean vorticity and the rotational shearing is dominant. For viscous waves, there still exists a viscous layer at the surface where the wave deviates from irrotational motions (Longuet-Higgins Reference Longuet-Higgins1953; Guo & Shen Reference Guo and Shen2013). It has been shown that, for a curved surface with vanishing tangential shear stress, the vorticity generated at the surface is $2\hat {U}/R$ (Longuet-Higgins Reference Longuet-Higgins1992), which is $O(2ak\sigma )$ to the leading order, as illustrated in figure 6(a). Consequently, based on (4.7), the cross-stream gradient in the Stokes layer satisfies $\partial \hat {U}/\partial \xi _2\approx -\hat {U}/R$. Therefore, $\partial \hat {U}/\partial \xi _2$ changes rapidly as the surface is approached and becomes negative under the wave crest and positive under the wave trough (figure 6b). As the vorticity arises from the oscillating wave surface, the thickness of this viscous layer is of the same order of magnitude of the Stokes layer, $\delta _S=\sqrt {2\nu /\sigma }$. For example, for case I, the dimensionless thickness of the Stokes layer is $k\delta _S = 8.9\times 10^{-3}$ and that of the vorticity layer is approximately $2k\delta _S$ (figure 6a). Therefore, the region where the rotational shearing effect of the wave takes place is quite limited. Another difference of the wave motions from the curved wall-bounded flows is that the free-surface boundary condition allows surface-tangential velocity fluctuations. This can lead to differences in the mechanisms responsible for the production of the Reynolds shear stress as discussed below.

Figure 6.

Contours of (a) mean vorticity $\varOmega$ and (b) cross-stream gradient $\partial \hat {U}/\partial \xi _2$, which show the Stokes layer at the surface. The values are normalized by $ak\sigma$. Case I is plotted.

At last, we shall point out some differences between the definition of the streamline curvature in the present paper and those used in the literature. The present definition (4.7), which represents the geometric curvature of the streamlines ( Appendix A), uses the cross-stream gradient of the mean velocity $\partial \hat {U}/\partial \xi _2$ and the mean vorticity $\varOmega$. On the other hand, many studies of the curved shear layer define the streamline curvature and the associated strain rates using $\partial \langle {v}\rangle /\partial x$ or $\partial \langle {v}\rangle /\partial \xi _1$ (see e.g. Bradshaw Reference Bradshaw1973; Lucarelli et al. Reference Lucarelli, Lugni, Falchi, Felli and Brocchini2018). These definitions use the rate of change of the velocity component normal to the coordinate lines to approximate the rotation of the mean flow and can only be calculated for Cartesian coordinates and non-streamline coordinates such as the $(s,n)$ coordinates. As shown by (B6) in Appendix B, $\partial \langle {v}\rangle /\partial x$ only contributes partially to the geometric curvature of the streamlines $R^{-1}$. However, for the wave flow considered here, the contribution from $\partial \langle {v}\rangle /\partial x$ is dominant, indicating that the streamline curvature defined using $\partial \langle {v}\rangle /\partial x$ is a good approximation for the streamline curvature in the present study. Because of the difference in the definitions of the streamline curvature, the curvature-induced strain rates in Bradshaw (Reference Bradshaw1973) and Lucarelli et al. (Reference Lucarelli, Lugni, Falchi, Felli and Brocchini2018) cannot be directly compared with the curvature straining in the present study. In Bradshaw (Reference Bradshaw1973) and Lucarelli et al. (Reference Lucarelli, Lugni, Falchi, Felli and Brocchini2018) there are two extra strain rates that are related to different curvatures. The first one, $U/R^{-1}$, is considered to be the strain rate due to geometric curvature, which refers to the curvature of the coordinate lines. The second strain rate is related to $\partial \hat {v}/\partial \xi _1$, which is considered to be the strain rate caused by the streamline curvature. The distinction of the geometric and streamline curvatures in Bradshaw (Reference Bradshaw1973) and Lucarelli et al. (Reference Lucarelli, Lugni, Falchi, Felli and Brocchini2018) is due to the use of a general curvilinear coordinate system where the coordinate lines and the flow direction are not aligned. As a result, the velocity vector have both the components tangential and normal to the coordinate lines, which are both rotating and lead to a straining effect on turbulence. By contrast, in the streamline coordinates, the rotation of the coordinate line is the same as the rotation of the mean flow direction and, thus, there is only one unified curvature-induced strain rate $\hat {U}/R$, which simplifies the interpretation of the strain rates of the mean flow.

4.3. Reynolds shear stress in streamline coordinates

By projecting the instantaneous velocity onto the wave-following streamline coordinates defined in § 4.1 and subtracting the mean velocity $\hat {\boldsymbol {U}}$ from it, we can obtain the along-stream and cross-stream velocity fluctuations, $\hat {u}'$ and $\hat {v}'$, respectively. Using this definition, $\langle {-\hat {u}'\hat {v}'}\rangle$, the correlations between the along-stream and cross-stream velocity fluctuations is indicative of the momentum transport across the curved streamlines, i.e.  the momentum exchange between the free-surface boundary and the interior in the wave-following frame.

The contours of Reynolds shear stress $\langle {-\hat {u}'\hat {v}'}\rangle$ defined above are plotted in figure 7. We can see that the variation of $\langle {-\hat {u}'\hat {v}'}\rangle$ defined in the streamline coordinates is drastically different from that defined in Cartesian coordinates (figure 2). This confirms that the Reynolds stress defined in Cartesian coordinates can be misleading for the analyses of the transport of momentum near the wave surface due to the strong influence of the surface kinematic blockage effect. To be specific, immediately below the surface, $\langle {-\hat {u}'\hat {v}'}\rangle$ is negative under the forward slope of the wave (figure 7), in contrast to the positive Reynolds shear stress under Cartesian coordinates (figure 2). The disappearance of the positive–negative regions around the wave trough as observed in Cartesian coordinates indicates that the kinematic effect is removed by the use of the wave-following streamline coordinates.

Figure 7.

Contours of the normalized Reynolds shear stress, $\langle {-\hat {u}'\hat {v}'}\rangle /{(u^{rms,cf})}^{2}$, defined in the streamline coordinates for (a) case I ($ak=0.1, Fr=0.1$), (b) case II ($ak=0.15, Fr=0.15$) and (c) case III ($ak=0.15, Fr=0.1$). The increment of contours is 0.015. The yellow lines represent the streamlines of $k\xi =0.01$ (– – –) and $k\xi =0.2$ (— $\cdot$ —) that are used in figure 8. The arrows on the streamlines indicate the direction of the mean flow in the wave-following frame.

Near the surface, $\langle {-\hat {u}'\hat {v}'}\rangle$ increases sharply along the mean flow direction under the wave crest and then gradually decreases under the wave trough. The values of $\langle {-\hat {u}'\hat {v}'}\rangle$ are slightly negative under the forward slope of the wave, which seems to be a counter-gradient momentum flux with $\partial \hat {U}/\partial \xi _2>0$ (figure 6b). This may be attributed to the deceleration of the mean flow from a kinematic point of view. Considering a flow particle with $\hat {v}'>0$ that moves in the $+\xi _2$ direction (away from the surface), despite the fact that $\partial \hat {U}/\partial \xi _2>0$ (figure 6b), the particle also moves in the $\xi _1$ direction with $\hat {U}$ decreasing (figure 5a). When the velocity difference between the origin and the end location of the particle is mostly due to the flow deceleration, the particle carries high momentum into the low momentum region with $\hat {u}'>0$, leading to $-\hat {u}'\hat {v}'<0$. Under the wave trough, the negative $\langle {-\hat {u}'\hat {v}'}\rangle$ near the surface can be due to the negative $\partial \hat {U}/\partial \xi _2$. The results above indicate that the physics of the Reynolds shear stress in the wave flow is complex involving processes other than the simple shear straining.

Away from the surface, the distribution of $\langle {-\hat {u}'\hat {v}'}\rangle$ becomes similar to that under Cartesian coordinates, which is expected because the effect of the wave surface weakens as the depth increases. The values of $\langle {-\hat {u}'\hat {v}'}\rangle$ become positive at all wave phases because the shear straining of the Stokes drift of the wave cumulatively tilts turbulence vortices towards the streamwise direction (Teixeira & Belcher Reference Teixeira and Belcher2002; Guo & Shen Reference Guo and Shen2014). Meanwhile, the wave-phase dependency of the Reynolds shear stress is still obvious. Along the mean flow direction, the values of $\langle {-\hat {u}'\hat {v}'}\rangle$ increase rapidly under the wave crest and reach maximum under the backward slope of the wave. This indicates that the momentum exchange is more active under the backward slope. This result is similar to the phenomenon that the concave curvature has a destabilizing effect on turbulence and can intensify the Reynolds shear stress (Hoffmann et al. Reference Hoffmann, Muck and Bradshaw1985; Moser & Moin Reference Moser and Moin1987).

We note that in the momentum equation (4.5), in addition to the cross-stream stress flux term $\partial \langle {-\hat {u}'\hat {v}'}\rangle /\partial \xi _2$, the Reynolds shear stress has an extra contribution to the momentum balance in the $\xi _1$-direction owing to curvature, $2\langle {\hat {u}'\hat {v}'}\rangle /R$. However, we find that the magnitude of this term is at least one order of magnitude smaller compared with $\partial \langle {-\hat {u}'\hat {v}'}\rangle /\partial \xi _2$, which is consistent with the analysis of the boundary layer formulation for shear layers in the literature where this curvature-related Reynolds shear stress term is often neglected by the order-of-magnitude analysis (Bradshaw Reference Bradshaw1973; Yousefi & Veron Reference Yousefi and Veron2020). With $\partial \langle {-\hat {u}'\hat {v}'}\rangle /\partial \xi _2$ being the dominant term through which $\langle {-\hat {u}'\hat {v}'}\rangle$ contributes to the momentum balance, the Reynolds shear stress defined under the streamline coordinates can still be considered as the momentum flux across the streamlines in the present study.

Although the choice of the Cartesian or streamline coordinate system can significantly change the values of the Reynolds shear stress, we find that the values of Reynolds normal stress are only weakly affected by the change of the coordinate frame. The reason why the Reynolds shear stress is more sensitive to the choice of the coordinate frame than the Reynolds normal stresses for the cases considered in the present study is discussed in detail in Appendix C. Considering that the wave-phase variation of the Reynolds normal stress has been discussed extensively in Guo & Shen (Reference Guo and Shen2014), we continue focusing on the behaviour of the Reynolds shear stress.

To illustrate the wave-phase variation of Reynolds shear stress more clearly, we plot in figure 8 the fluctuations of $\langle {-\hat {u}'\hat {v}'}\rangle$ along streamlines, defined as $\widetilde {\langle {-\hat {u}'\hat {v}'}\rangle } = \langle {-\hat {u}'\hat {v}'}\rangle - \overline {\langle {-\hat {u}'\hat {v}'}\rangle }$ with $\overline {\langle {-\hat {u}'\hat {v}'}\rangle }$ being the mean $\langle {-\hat {u}'\hat {v}'}\rangle$ averaged along the streamline. We can see that, after subtracting the mean shear stress, the wave-phase variation of the Reynolds shear stress is qualitatively similar in both the near-surface viscous layer (figure 8a) and away from the surface (figure 8b). The variation is not a simple sinusoidal shape. Under the wave crest, the Reynolds shear stress increases sharply. By contrast, under the wave trough, its variation is milder. This suggests that the physical processes governing the wave-phase variation of the Reynolds shear stress are different under the wave crest and trough, which are investigated in § 4.4.

Figure 8.

Normalized variation of the Reynolds shear stress, $\widetilde {\langle {-\hat {u}'\hat {v}'}\rangle }/ak\langle {\hat {u}'^{2}}\rangle$, along the streamlines of (a) $k\xi _2 = 0.01$ (the yellow dashed line in figure 7) and (b) $k\xi _2=0.2$ (the yellow dash–dotted line in figure 7) for case I (——–), case II (– – –) and case III (— $\cdot$ —). The arrow indicates that the mean flow direction points from right to left.

We have also found that, despite the notable difference in the magnitudes of $\langle {-\hat {u}'\hat {v}'}\rangle$ among cases I–III, as shown in figure 7, the variation of $\langle {-\hat {u}'\hat {v}'}\rangle$ under the wave crest can be normalized by $ak\langle {\hat {u}'^{2}}\rangle$. Figure 8 shows that the normalized shear stress collapses well for different cases. In other words, the variation of the Reynolds shear stress is proportional to the streamwise fluctuations and the wave steepness. Under the wave trough, this normalization is slightly less than ideal. However, since the variation of $\langle {-\hat {u}'\hat {v}'}\rangle$ is relatively small under the trough, especially when compared with under the crest, we can still consider the above normalization as acceptable for the description of the wave-phase variation of the Reynolds shear stress. The mechanism underlying the variation and scaling is further analysed below in § 4.4.

4.4. Reynolds stress budget

Next, we examine the budget of the Reynolds shear stress to investigate the mechanism responsible for the wave-phase variation of the stress. The budget equation in the streamline coordinates is Finnigan (Reference Finnigan1983),

(4.10)\begin{equation} \frac{\mathrm{D} \langle{-\hat{u}'\hat{v}'}\rangle}{\mathrm{D} t} = \frac{\partial{\langle{-\hat{u}'\hat{v}'}\rangle}}{\partial{t}} + \hat{U}\frac{\partial{\langle{-\hat{u}'\hat{v}'}\rangle}}{\partial{\xi_1}} = \mathcal{P} + \varPi + \mathcal{T} + \mathcal{T}^{p} + \mathcal{V}. \end{equation}

In the above equation, the right-hand side terms are production $\mathcal {P}$, pressure–strain correlation $\varPi$, turbulent diffusion $\mathcal {T}$, pressure diffusion $\mathcal {T}^{p}$ and viscous effect $\mathcal {V}$, respectively. These terms are defined as

(4.11)$$\begin{gather} \mathcal{P} = 2\langle{\hat{u}'^{2}}\rangle\frac{\hat{U}}{R} - \langle{\hat{v}'^{2}}\rangle\varOmega, \end{gather}$$

(4.12)$$\begin{gather}\varPi ={-}\left\langle{p'\frac{\partial{\hat{v}'}}{\partial{\xi_1}}+p'\frac{\partial{\hat{u}'}}{\partial{\xi_2}}} \right\rangle, \end{gather}$$

(4.13)$$\begin{gather}\mathcal{T} = \left(\frac{\partial{\langle{\hat{u}'^{2} \hat{v}'}\rangle}}{\partial{\xi_1}} + \frac{\partial{\langle{\hat{u}'\hat{v}'^{2}}\rangle}}{\partial{\xi_2}}\right) - \frac{1}{\gamma} (2\langle{\hat{u}'^{2} \hat{v}'}\rangle - \langle{\hat{v}'^{3}}\rangle) - \frac{1}{R} (2\langle{\hat{u}'\hat{v}'^{2}}\rangle - \langle{\hat{u}'^{3}}\rangle), \end{gather}$$

(4.14)$$\begin{gather}\mathcal{T}^{p} = \left\langle{\frac{\partial{p'\hat{v}'}}{\partial{\xi_1}}+\frac{\partial{p'\hat{u}'}}{\partial{\xi_2}}}\right\rangle, \end{gather}$$

(4.15)\begin{gather} \mathcal{V} ={-}\frac{1}{Re}\left[\langle{\hat{u}' \nabla^{2} \hat{v}'}\rangle + \langle{\hat{w}' \nabla^{2} \hat{u}'}\rangle + \frac{\partial}{\partial \xi_1}\left(\frac{\langle{\hat{u}'^{2}}\rangle - \langle{\hat{v}'^{2}}\rangle}{R}\right) - \frac{\partial}{\partial \xi_2}\left(\frac{\langle{\hat{u}'^{2}}\rangle - \langle{\hat{v}'^{2}}\rangle}{\gamma}\right)\right. \nonumber\\ \hspace{-6pc}- \left.\frac{1}{\gamma}\frac{\partial{\langle{\hat{u}'\hat{v}'}\rangle}}{\partial{\xi_1}} -\frac{1}{R}\frac{\partial{\langle{\hat{u}'\hat{v}'}\rangle}}{\partial{\xi_2}}- 2\langle{\hat{u}'\hat{v}'}\rangle\left(\frac{1}{\gamma^{2}}+\frac{1}{R^{2}}\right) \right]. \end{gather}

Because of the statistically steady state in the wave-following frame, the unsteady term, $\partial \langle {-\hat {u}'\hat {v}'}\rangle /\partial t$, is approximately zero. The convection term, $\hat {U} \partial \langle {-\hat {u}'\hat {v}'}\rangle /\partial \xi$, represents the rate of change of the Reynolds shear stress along the streamline. The positive–negative variation of $\hat {U} \partial \langle {-\hat {u}'\hat {v}'}\rangle /\partial \xi$ with the wave phase, which corresponds to the increase and decease of the Reynolds shear stress, is contributed by the various effects on the right-hand side of (4.10). We also note that most of the above terms contain components that are related with $\gamma$ and $R$, the two length scales associated with the streamline coordinates defined by (4.6) and (4.7), respectively. These components are not present in the corresponding budget equation in Cartesian coordinates and appear as the result of the curved coordinate lines in the streamline coordinates defined based on the wave flow. These $R$- and $\gamma$-containing terms directly quantify the impacts of the flow curvature and acceleration caused by the wave on the budget of the Reynolds shear stress.

From the simulation data, we found that the dominant effects that contribute to the variation of $\langle {-\hat {u}'\hat {v}'}\rangle$ with the wave phase are the production $\mathcal {P}$, the pressure diffusion $\mathcal {T}^{p}$ and the pressure–strain correlation $\varPi$. Because the balance of the budget is qualitatively similar for all the cases, here we only discuss these terms for case I, which are plotted in figure 9. Compared with other terms, these three terms have much larger magnitudes and all show substantial variations with the wave phase and, thus, lead to the wave-phase variations of $\langle {-\hat {u}'\hat {v}'}\rangle$.

Figure 9.

Contours of the dominant budget terms of $\langle {-\hat {u}'\hat {v}'}\rangle$ (4.10): (a) production $\mathcal {P}$, (b) pressure diffusion $\mathcal {T}^{p}$ and (c) pressure–strain correlation $\varPi$. The results of case I are shown and are normalized by ${(u^{rms,cf})}^{2} ck$. The yellow lines represent the streamlines of $k\xi =0.2$ (— $\cdot$ —) and $k\xi =0.01$ (– – –) that are used in figure 10. The arrows on the streamlines indicate the direction of the mean flow.

As shown in figure 9(a), the production term, $\mathcal {P}$, is positive and negative under the wave crest and trough, respectively. This indicates that the production by the interaction of turbulence with the mean flow increases the Reynolds shear stress $\langle {-\hat {u}'\hat {v}'}\rangle$ when a wave crest passes by and decreases it as a trough passes by. The magnitude of the variation of the production increases towards the surface, leading to a larger rate of change of $\langle {-\hat {u}'\hat {v}'}\rangle$ near the surface. This behaviour of the production is closely related to the curvature effect, which is discussed in detail later in this section and § 4.5.1.

The effects associated with the turbulence pressure fluctuations are quantified by the pressure diffusion and pressure–strain correlation terms (figures 9b and 9c, respectively). The pressure–strain correlation, $\varPi$, is negative under the wave crest and positive under the wave trough. The vertical variation of $\varPi$ is not monotonic. Towards the surface, the magnitude of $\varPi$ first increases and then decreases approximately within the region $k\xi _2<0.05$. The decrease is due to the reduction of the strain rate, $\partial \hat {v}'/\partial \xi _1+\partial \hat {u}'/\partial \xi _2$, within the surface layer to satisfy the shear-free boundary condition (Shen et al. Reference Shen, Zhang, Yue and Triantafyllou1999; Guo & Shen Reference Guo and Shen2013).

The behaviour of the pressure diffusion, $\mathcal {T}^{p}$, is more complex. Under the wave crest, the pressure diffusion is positive. We find that the positive $\mathcal {T}^{p}$ is mostly contributed by $\partial (p'\hat {u}')/\partial \xi _2$ (not plotted), indicating that $\langle {-\hat {u}'\hat {v}'}\rangle$ is diffused from the deep region towards the surface by the pressure effect. Under the wave trough, the pressure diffusion exhibits a two-layer structure. This term is negative except in a thin layer in the immediate vicinity of the surface. Within this thin layer, $\mathcal {T}^{p}$ changes drastically to positive values. Therefore, the pressure diffusion of $\langle {-\hat {u}'\hat {v}'}\rangle$ under the wave trough occurs much closer to the surface, from the near-surface region into the thin surface layer. We find that this thin layer has a thickness of the same order of magnitude as the Stokes layer (figure 6) and, therefore, the appearance of this positive region is likely related to the viscous effect.

To show the balance of the budget of the Reynolds shear stress more clearly, we plot in figure 10 the variations of the above dominant budget terms along the streamlines to discuss the mechanisms of the wave-phase variation. Considering the two-layer structures of the pressure diffusion, two representative streamlines, one outside the Stokes layer and one inside the Stokes layer, are chosen.

Figure 10.

Along-stream variation of the dominant budget terms, the production $\mathcal {P}$ (——–), pressure diffusion $\mathcal {T}^{p}$ (– – –), pressure–strain correlation $\varPi$ (— $\cdot$ —) and the net effect (..........) at (a) $k\xi _2=0.2$ (the dash–dotted yellow line in figure 9) and (b) $k\xi _2=k\delta _S/2$ (the dashed yellow line in figure 9). The results of case I are shown and are normalized by ${(u^{rms,cf})}^{2} ck$.

The three dominant budget terms and their net effect along the streamline of $k\xi _2=0.2$ outside of the Stokes layer are plotted in figure 10(a). We can see that, under the wave crest, the pressure diffusion is approximately balanced by the pressure–strain correlation. As a result, the net effect is dominated by the production, which is positive and increases the values of $\langle {-\hat {u}'\hat {v}'}\rangle$ under the crest. Under the wave trough, the two pressure effects are no longer balanced. We can see that the pressure–strain correlation overweights the pressure diffusion. As a result, the combined pressure effects are positive and offset the negative production under the wave trough. The combined effect of the three terms is still negative and decreases $\langle {-\hat {u}'\hat {v}'}\rangle$ under the trough. This leads to the maxima and minima of the Reynolds shear stress under the backward and forward slope, respectively (figures 7 and 8). However, due to the offset effect of the pressure terms, the rate of change of $\langle {-\hat {u}'\hat {v}'}\rangle$ under the wave trough does not vary as much as under the crest, which is consistent with the observations in § 4.3.

The processes occurring inside the Stokes layer, represented by the streamline of $k\xi _2=k\delta _S/2$ ($k\delta _S=k\sqrt {2\nu /\sigma }=8.9\times 10^{-3}$ for case I as shown in § 4.2), are plotted in figure 10(b). Under the wave crest, the two pressure terms are small compared with those outside of the Stokes layer (figure 10a) and, therefore, the production term dominates. Under the wave trough, the production term is still dominant but is partially offset by the positive pressure diffusion and pressure–strain correlation. Note that the offset effect of the two pressure terms is mainly from the positive pressure diffusion in the Stokes layer (figure 9b) whereas the pressure–strain correlation is more important outside of the Stokes layer. Overall, the net effect of the three major budget terms inside the Stokes layer is qualitatively similar to that outside of the Stokes layer, i.e.  the net effect increases the Reynolds shear stress $\langle {-\hat {u}'\hat {v}'}\rangle$ rapidly under the wave crest and decreases it with a mild rate under the wave trough.

Based on the analyses above, we can see that the production by the wave straining plays a crucial role in determining the wave-phase variation of the Reynolds shear stress, especially under the wave crest. Under the wave trough, even though the dynamics of the Reynolds shear stress are complicated by the pressure effects, the main source of its variation is still the production. We can use this conclusion to explain the scaling of the Reynolds shear stress proposed in § 4.3 (figure 8). Under the wave crest, because the net effect is dominated by the production, we have

(4.16)\begin{equation} \hat{U}\frac{\partial \langle{-\hat{u}'\hat{v}'}\rangle}{\partial \xi_1} \approx \mathcal{P}. \end{equation}

Because the wave is almost irrotational ($\varOmega =0$), the production (4.11) can be estimated as

(4.17)\begin{equation} \mathcal{P} \approx 2 \langle{\hat{u}'^{2}}\rangle \frac{\hat{U}}{R}. \end{equation}

Even though the assumption of the irrotationality is invalid inside the Stokes layer, the cross-stream fluctuations are significantly suppressed at the surface and, thus, the second part in (4.11) is still negligibly small compared with the first part. Using the theoretical estimation of the curvature (4.9), the variation of $\langle {-\hat {u}'\hat {v}'}\rangle$ under the wave crest is approximately

(4.18)\begin{equation} \frac{\partial \langle{-\hat{u}'\hat{v}'}\rangle}{\partial \xi_1} \approx \frac{2\langle{\hat{u}'^{2}}\rangle}{R} \approx 2(ak) \, \textrm{e}^{ky} k \sin (kx)\langle{\hat{u}'^{2}}\rangle. \end{equation}

Utilizing the approximation that $\mathrm {d}\xi _1 \approx -\mathrm {d} x$ to the leading order in the wave slope (B 14), we can integrate the above equation with respect to $\xi _1$ and obtain that the magnitude of the along-stream variation of $\langle {-\hat {u}'\hat {v}'}\rangle$ is proportional to the wave steepness $ak$ and the streamwise fluctuations $\langle {\hat {u}'^{2}}\rangle$, which supports the scaling we find in § 4.3.