4.1.1. Surface-wave eigenmodes and their impacts on energy growth

Figure 10(a) shows the eigenvalues in the free-surface open channel at $Re_\tau =180$, $Fr_\tau =0.142$, $\alpha h =0$ and $\beta h=1$. The eigenvalues in the corresponding rigid-lid open channel at the same wavenumber and Reynolds number are also plotted in figure 10(a) for comparison. There are two additional eigenvalues in the free-surface open channel case compared with the rigid-lid open channel case. These two additional eigenvalues and the corresponding eigenmodes induced by the free surface are referred to as the ‘surface-wave eigenvalues’ and ‘surface-wave eigenmodes’, respectively, while the other eigenvalues and eigenmodes are called the ‘shear eigenvalues’ and ‘shear eigenmodes’, respectively (Yih Reference Yih1972). In contrast to the zero values of the real parts of the eigenvalues of the shear eigenmodes, $\lambda _r=0$, the two surface-wave eigenmodes have the same imaginary parts $\lambda _i$ but opposite non-zero real parts $\lambda _r$, indicating that the two surface-wave eigenmodes propagate in opposite spanwise directions with identical phase speeds $|\lambda _r/\beta |$ and the same damping rate $\lambda _i$. Here, $|\lambda _r|$ represents the frequency of the surface-wave eigenmodes. The velocity moduli of the two surface-wave eigenmodes have the same vertical profiles. Figure 10(b) shows the profiles of the velocity moduli $|\bar {u}|$, $|\bar {v}|$ and $|\bar {w}|$ of the surface-wave eigenmodes. The streamwise velocity modulus $|\bar {u}|$ is non-zero due to the presence of the non-uniform base flow $U(y)$, and a viscous Stokes layer appears in the profiles of $|\bar {u}|$ and $|\bar {w}|$ near the bottom. The modulus of the vertical velocity $|\bar {v}|$ increases monotonically with $y$, reaching its maximum value at the surface and inducing surface deformation, which is another distinct feature of surface-wave eigenmodes compared with shear eigenmodes (Yih Reference Yih1972).

Figure 10.

(a) Comparison of eigenvalues $\lambda$ between the free-surface open channel with $Fr_\tau =0.142$ and the rigid-lid open channel. The points represent the eigenvalues of the shear eigenmodes in the free-surface open channel, the diamonds represent the eigenvalues of the surface-wave eigenmodes in the free-surface open channel and the hollow circles represent the eigenvalues in the rigid-lid open channel. (b) Vertical profiles of the velocity moduli of the surface-wave eigenmodes. The results are obtained at $Re_\tau =180$, $\alpha h =0$ and $\beta h=1$.

The surface-wave eigenmodes induced by the free surface have significant effects on the transient growth of the total energy $E$ and kinetic energy $E_{K}$. Figure 11 shows the evolution of $E(t)/E^{in}$ and $E_K(t)/E_K^{in}$ with time. The results are compared among the rigid-lid open channel, free-surface open channel ($Fr_\tau =0.142$) with surface-wave eigenmodes, and free-surface open channel ($Fr_\tau =0.142$) without surface-wave eigenmodes at $Re_\tau =180$, $\alpha h=0$ and $\beta h =1$. In each case, the initial input perturbations are those that lead to the homologous maximum transient growth rates. As shown in figure 11(a), in the rigid-lid case, $E(t)/E^{in}$ initially increases monotonically with time due to the non-orthogonality of the eigenmodes; then the maximum value $G_{max}$ is reached at $tu_\tau / h \approx 1.5$; and finally, $E(t)/E^{in}$ decreases because the eigenmodes are stable. For the free-surface open channel case, when the surface-wave eigenmodes are excluded, $E(t)/E^{in}$ is the same as in the rigid-lid case, indicating that the surface elevation induced by the shear eigenmodes is negligible. In contrast, when the surface-wave eigenmodes are considered, $E(t)/E^{in}$ increases with oscillations for $0< tu_\tau / h <1.5$. The oscillation period is close to the period of the surface-wave eigenmodes $T=2{\rm \pi} h/(|\lambda _r|u_\tau )=1.05 h/u_\tau$ (see $|\lambda _r|$ in figure 10a). Accompanied by the oscillation, $E(t)/E^{in}$ is larger in the free-surface open channel case with surface-wave eigenmodes than in the other two cases for most of the duration $0< tu_\tau /h<1.5$. The above comparison of $E(t)/E^{in}$ among the three cases indicates that the surface-wave eigenmodes are responsible for the larger $G_{max}$ in the free-surface open channel than in the rigid-lid open channel.

Figure 11.

Comparisons of (a) $E(t)/E^{in}$ and (b) $E_K(t)/E_K^{in}$ among the rigid-lid open channel flow, free-surface open channel flow ($Fr_\tau =0.142$) with surface-wave eigenmodes and free-surface open channel flow ($Fr_\tau =0.142$) without surface-wave eigenmodes. Note that the curves in the first case and the third case overlap. The results are obtained at $Re_\tau =180$, $\alpha h =0$ and $\beta h=1$, with the initial perturbations leading to $G_{max}$.

The temporal development of the kinetic energy $E_K(t)/E_K^{in}$ is also considerably influenced by the surface-wave eigenmodes. As shown in figure 11(b), when the surface-wave eigenmodes are excluded, $E_K(t)/E_K^{in}$ in the free-surface open channel is equal to $E_K(t)/E_K^{in}$ (which is equal to $E(t)/E^{in}$) in the rigid-lid open channel (figure 11a). However, when the surface-wave eigenmodes in the free-surface open channel are included, $E_K(t)/E_K^{in}$ oscillates with a value larger than $E(t)/E^{in}$ shown in figure 11(a). The maximum value of $E_K(t)/E_K^{in}$ in the free-surface open channel case is approximately 27 % larger than that in the rigid-lid open channel case. Similar to the analyses of $G_K$ (figure 5) based on the relationship in (3.1), the ratio of the kinetic energy to the total energy satisfies $R_K^{in}< R_K(t)\leq 1$, implying that the larger value of $E_K(t)/E_K^{in}$ than $E(t)/E^{in}$ is correlated with the initial potential energy induced by the surface-wave eigenmodes. This result is confirmed by the analyses in the next section.

4.1.2. Impact of surface-wave eigenmodes on the energy budget

Next, the governing equations of the total perturbation energy and kinetic energy are derived and analysed to elucidate how the surface-wave eigenmodes affect the maximum transient growth rate. By applying the dot product of $\boldsymbol {{u}}$ on both sides of (2.2), integrating over the whole open channel and over time, and replacing the surface pressure with the surface boundary condition (2.7), we derive that

(4.1)\begin{equation} \frac{E(t)}{E^{in}} = 1+ \frac{P(t)}{E^{in}} + \frac{S(t)}{E^{in}}+ \frac{V(t)}{E^{in}} +\frac{\epsilon(t)}{E^{in}}. \end{equation}

In (4.1), $P$, $S$, $V$ and $\epsilon$ refer to the integrals of the energy production term, the surface energy transfer term, the non-constant eddy viscosity term and the viscous dissipation term, respectively. They are defined as

(4.2)\begin{gather} P(t)={-}\int_0^t\int_0^h {\overline{uv}} U^\prime\,{\rm d} y\,{\rm d}t, \end{gather}

(4.3)\begin{gather} S(t)=\int_0^t\left[-\nu{\frac{\partial \overline{vv}}{\partial y}}+ \frac{1}{2}\nu_T \frac{\partial \overline{u_iu_i}}{\partial y} \right]_{y=h}\,{\rm d}t, \end{gather}

(4.4)\begin{gather} V(t)= \int_0^t\int_0^h {\frac{\partial \nu_T}{\partial y}}\left({\frac{\partial \overline{vv}}{\partial y}+\frac{1}{2} \frac{\partial \overline{u_iu_i}}{\partial y}}\right)\,{\rm d} y\,{\rm d}t, \end{gather}

(4.5)\begin{gather} \epsilon(t)={-} \int_0^t\int_0^h {\nu_T \overline{\frac{\partial {u_i}}{\partial x_j} \frac{\partial {u_i}}{\partial x_j}}}\,{\rm d} y\,{\rm d}t. \end{gather}

The overline denotes averaging over the horizontal plane. The term $S$ is zero in the rigid-lid open channel flow. For a given wavenumber, $(u_i, \eta ) = (\hat {u}_i,\hat {\eta }) \exp ({{\rm i}(\alpha x +\beta z-\lambda t)})+{\rm c.c.}$

Figure 12(a) shows the time evolution of the budget terms normalised by $E^{in}$ in (4.1) at $Re_\tau =180$, $\alpha h =0$, $\beta h =1$ and $Fr_\tau =0.142$. The vertical dashed line indicates the time at which the maximum transient growth rate is reached. In contrast to common expectations, the contribution of the surface transfer term $S$ to the total energy growth is negligible in the free-surface open channel case, although the surface is deformable. The non-uniform eddy viscosity term $V$ is also negligible in both the rigid-lid and free-surface open channel cases, because it works as a turbulent diffusion term in the transport of turbulent kinetic energy. Therefore, the transient growth rates in the two cases are governed by the net effect of the other two terms, the production term $P$ and the dissipation term $\epsilon$.

Figure 12.

Time evolution of the budget terms (4.1) and their components (4.7) and (4.9) normalised by $E^{in}$ in the rigid-lid open channel (red dashed lines with symbols) and free-surface open channel (black solid lines with symbols) cases. The results are obtained at $Re_\tau =180$, $\alpha h =0$ and $\beta h=1$, with the initial input perturbations leading to $G_{max}$. For the free-surface open channel, $Fr_\tau =0.142$.

The effect of the free surface on the balance between the energy production term $P$ and the dissipation term $\epsilon$ is investigated next. Assume that the perturbation consists of a surface-wave part and a shear part, i.e.

(4.6)\begin{equation} ({{u_i}}, {\eta})=({{u_i}}_w+{{u_i}}_s, {\eta}_w+{\eta}_s). \end{equation}

The subscript ‘$w$’ denotes the surface-wave part, which is a linear combination of the surface-wave eigenmodes, and the subscript ‘$s$’ refers to the shear part, which is a linear combination of the shear eigenmodes. Correspondingly, $P$ and $\epsilon$ can also be decomposed into a surface-wave part, a shear part and an interaction part, i.e.

(4.7)\begin{equation} P={-}\int_0^t\int_0^h {\left(\overline{{u_w}{v_w}} +\overline{{u_s}{v_s}} +\overline{{u_s}{v_w}+{u_w}{v_s}}\right) U^\prime}\,{\rm d} y\,{\rm d}t=P_w + P_s + P_{inter}, \end{equation}

and

(4.8)\begin{align} \epsilon&={-} \int_0^t\int_0^h {\nu_T \left(\overline{\frac{\partial {u_i}_w}{\partial x_j} \frac{\partial {u_i}_w}{\partial x_j}}+ \overline{\frac{\partial {u_i}_s}{\partial x_j} \frac{\partial {u_i}_s}{\partial x_j}}+\overline{\frac{\partial {u_i}_w}{\partial x_j} \frac{\partial {u_i}_s}{\partial x_j}+\frac{\partial {u_i}_s}{\partial x_j} \frac{\partial {u_i}_w}{\partial x_j}}\right)}\,{\rm d} y\,{\rm d}t \end{align}

(4.9)\begin{align} &=\epsilon_w + \epsilon_s + \epsilon_{inter}. \end{align}

The interaction parts, denoted by the subscript ‘inter’, exist because the eigenmodes are non-normal. For the rigid-lid case, the surface-wave part and the interaction part do not exist, and thus, $P=P_s$ and $\epsilon =\epsilon _s$.

For the shear part, figure 12(b) shows that the magnitudes of $P_s$ and $\epsilon _s$ normalised by $E^{in}$ are slightly smaller in the free-surface open channel than in the rigid-lid open channel, and $P_s+\epsilon _s$ is comparable between these two cases. Therefore, the free surface has an insignificant effect on the contribution of the shear part to the energy growth rate $E(t)/E^{in}$.

For the surface-wave part, as illustrated in figure 12(c), the surface wave gains little energy from the base flow because the surface-wave part of the energy production term $P_w$ is nearly zero. The dissipation $\epsilon _w$ is negative, consistent with the negative imaginary parts of the eigenvalues $\lambda _i$ in figure 10(a). Consequently, the combined surface-wave part $P_w+\epsilon _w$ has a negative contribution to the energy growth rate $E(t)/E^{in}$. However, when the surface-wave part is present, figure 12(d) shows that the net contribution of the interaction between the surface-wave part and the shear part, $P_{inter}+\epsilon _{inter}$, oscillates with mainly positive values, and the magnitude of this term is greater than $P_w+\epsilon _w$. Therefore, the presence of the surface-wave part increases the energy growth rate $E(t)/E^{in}$ through the interaction part.

Due to the small magnitude of $\epsilon _{inter}$, the quasi-positive contribution of the interaction part to $E(t)/E^{in}$ occurs due to $P_{inter}$ (figure 12d). The interaction production term $P_{inter}$ can be decomposed as follows:

(4.10)\begin{equation} P_{inter} = P_{inter1} + P_{inter2} ={-}\int_0^t\overline{{u}_s{v}_w}U^\prime \,{\rm d} y\,{\rm d}t -\int_0^t\overline{{u}_w{v}_s}U^\prime \,{\rm d} y\,{\rm d}t. \end{equation}

As shown in figure 13, $P_{inter}$ is dominated by $P_{inter1}$, i.e.

(4.11)\begin{equation} P_{inter} \approx P_{inter1} ={-}\int_0^t\overline{{u}_s{v}_w}U^\prime \,{\rm d} y\,{\rm d}t . \end{equation}

This is because among the four velocity components in (4.10), only $u_s$ increases with time (shown below). In conclusion, the free surface leads to the amplification and oscillation of $E(t)/E^{in}$ due to the energy production, which is related to the interaction between the vertical velocity of the surface-wave part $v_w$ and the streamwise velocity of the shear part $u_s$. The effect of the interaction production term is also confirmed at $Re_\tau =6000$ in Appendix B.

Figure 13.

Decomposition of $P_{inter}$ (4.10) obtained at $Re_\tau =180$, $Fr_\tau =0.142$, $\alpha h =0$ and $\beta h=1$.

The governing equation of the kinetic energy $E_K(t)/E_K^{in}$ can be derived from (4.1) as

(4.12)\begin{equation} \frac{E_K(t)}{E_K^{in}} = 1+\frac{P+S+V+\epsilon}{E_K^{in}}+\frac{E_\eta^{in}-E_\eta(t)}{E_K^{in}}. \end{equation}

The second term on the right-hand side of (4.12),

(4.13)\begin{equation} \frac{P+S+V+\epsilon}{E_K^{in}} =\frac{P+S+V+\epsilon}{E^{in}R_K^{in}}, \end{equation}

represents the net contribution of the energy production, surface energy transfer, and the effects of non-uniform eddy viscosity and dissipation. The third term, $(E_\eta ^{in}-E_\eta (t))/{E_K^{in}}$, represents the transformation of potential energy to kinetic energy. Figure 14 shows the time evolution of the budget terms in (4.12), with the term $(P+S+V+\epsilon )/E^{in}$ included for comparison. The results demonstrate that $(P+S+V+\epsilon )/E_K^{in}$ is larger than $(P+S+V+\epsilon )/E^{in}$, indicating the large initial potential energy $E_\eta ^{in}$. In addition, $(E_\eta ^{in}-E_\eta (t))/{E_K^{in}}$ is positive and oscillates at a period equal to half of the period of the surface-wave eigenmodes $T=1.05h/u_\tau$, suggesting that the initial potential energy is associated with the surface-wave part.

Figure 14.

Time evolution of the budget terms in (4.12). The results are obtained in a free-surface open channel at $Re_\tau =180$, $\alpha h =0$, $\beta h=1$ and $Fr_\tau =0.142$.

The above analyses of the energy budget equation show that the initial surface-wave part contains considerable potential energy $E_\eta ^{in}$ and interacts with the shear part, leading to a maximum in the quasi-positive energy production term. This in turn leads to greater values of $G_{max}$ and $G_K$ in the free-surface case than in the rigid-lid case, as shown in figures 3–7.

4.1.3. Initial fields that enhance the energy growth rate of streamwise-uniform motions

Next, we study the state of the initial flow, especially the phase relationship between the surface-wave part and the shear part, which can maximise the energy production term $P_{inter1}/E^{in}$ in (4.11).

Figure 15 displays the flow fields, surface deformation, and their shear and surface-wave parts at $t=iT/4$, with $i=0,1,\ldots,6$. Here, $T$ is the period of the two surface-wave eigenmodes corresponding to the frequency $|\lambda _r|$ in figure 10(a), which equals $1.05h/u_\tau$. The initial flow field consists of a pair of strong streamwise vortices denoted by the vector $(v,w)$ and relatively weak streaks of streamwise velocity $u$, with an apparent surface elevation $\eta$ (figure 15a i). The initial $(v,w)$ originates from the shear part $(v_s,w_s)$ (figure 15b i), because the initial vertical and spanwise velocities contributed by the surface-wave part are approximately zero, i.e. $v_w\approx 0$ and $w_w\approx 0$ (figure 15c i). The initial streamwise velocity $u$ (figure 15a i) is contributed by the shear part $u_s$ (figure 15b i) and the relatively weak surface-wave part $u_w$, which have opposite signs (figure 15c i). The initial surface deformation $\eta$ is determined by the surface-wave part $\eta _w$ (figure 15a i,c i). The zero initial vertical and spanwise velocities under the obvious spanwise wave (figure 15c i) indicate that the surface wave is not a progressive wave but rather a standing wave composed of two surface-wave eigenmodes with the same frequency propagating in opposite directions (figure 10a). This finding is confirmed by the time evolution of the surface deformation of the surface-wave part (figure 15c i–c vii). The surface elevation of the initial standing wave is in phase with the streamwise velocity generated by the lifting-up of the mean shear by streamwise vortices.

Figure 15.

Velocity field and surface elevation, as well as their shear and surface-wave parts, at (a i–c i) $t=0$, (a ii–c ii) $t=T/4$, (a iii–c iii) $t=T/2$, (a iv–c iv) $t=3T/4$, (a v–c v) $t=T$, (a vi–c vi) $t=5T/4$ and (a vii–c vii) $t=3T/2$. Panels (a i–a vi) show the total, panels (b i–b vii) depict the shear part and panels (c i–c vii) display the surface-wave part. Here, $T$ is the period of the two surface-wave eigenmodes, and the maximum transient growth is reached at $t=3T/2$. The vector is composed of the vertical and spanwise velocities, and the contours are the streamwise velocity.

Over time, the streamwise vortices (figure 15b i–b vii) in the shear part attenuate to amplify the streamwise velocity $u_s$, similar to the generation of streaks by the lifting-up effect in closed channel flows (Pujals et al. Reference Pujals, García-Villalba, Cossu and Depardon2009). Moreover, due to the small negative value of $\lambda _i$ for the two surface-wave eigenmodes (figure 10a), the magnitudes of the surface elevation and the velocity fields induced by the standing wave attenuate slowly and oscillate (figure 15c i–c vii), which is consistent with the oscillation of the positive $(E_\eta ^{in}-E_\eta (t))/{E_K^{in}}$ term in figure 14. Combining the two parts, the amplified flow field at the time when $G_{max}$ is reached is featured by strong streaks (figure 15a vii). Therefore, the streaks are amplified at the expense of the initial streamwise vortices and the standing wave in the free-surface open channel.

It can be proven that only the initial state of the surface-wave part shown in figure 15(c i) can maximise the energy production term $P_{inter1}/E^{in}$, which is correlated with the interaction between the surface-wave part and the shear part (4.11). For the shear part, the shear eigenmodes have $\lambda _r$ values of zero (figure 10a), i.e. zero frequency. The distribution of the shear part in the $y\unicode{x2013}z$ plane is similar at different times, with the magnitude varying with time (figure 15b i–b vii). In particular, the locations of the streamwise vortices and streaks do not change with time. However, the magnitudes of the streamwise vortices decrease with time, and the intensity of the streaks increases with time. Therefore, the velocities of the shear part can be assumed to be

(4.14)\begin{gather} u_s=f_{us}(y)g(t)\cos(\beta z + \theta_s), \end{gather}

(4.15)\begin{gather} v_s={-}f_{vs}(y)m(t)\cos(\beta z + \theta_s), \end{gather}

(4.16)\begin{gather} w_s={-}f_{ws}(y)n(t)\sin(\beta z + \theta_s). \end{gather}

Here, $f_{us}$ and $f_{vs}$ are positive, and the sign of $f_{ws}$ changes near the channel centre. The time functions satisfy $g(t)\geq 1$, $m(t)\leq 1$ and $n(t)\leq 1$, with $g(0)= m(0)= n(0)= 1$. The angle $\theta _s$ indicates the phase of the shear part in the spanwise direction.

The surface-wave part is a linear combination of the two surface-wave eigenmodes, i.e.

(4.17)\begin{gather} u_w = f_{uw}(y)\left[{-}a_1 \sin(\beta z +|\lambda_r| t+\theta_{w1} ) + a_2 \sin(\beta z - |\lambda_r| t+ \theta_{w2})\right] {\rm e}^{\lambda_i t}, \end{gather}

(4.18)\begin{gather} v_w = f_{vw}(y)\left[a_1 \cos(\beta z +|\lambda_r| t+\theta_{w1} ) + a_2 \cos(\beta z - |\lambda_r| t+ \theta_{w2})\right] {\rm e}^{\lambda_i t}, \end{gather}

(4.19)\begin{gather} w_w = f_{ww}(y)\left[{-}a_1 \sin(\beta z +|\lambda_r| t+\theta_{w1} ) - a_2 \sin(\beta z - |\lambda_r| t+ \theta_{w2})\right] {\rm e}^{\lambda_i t}, \end{gather}

(4.20)\begin{gather} \eta_w = f_{\eta}\left[a_1 \sin(\beta z +|\lambda_r| t+\theta_{w1} ) - a_2 \sin(\beta z - |\lambda_r| t+ \theta_{w2})\right] {\rm e}^{\lambda_i t}, \end{gather}

where $f_{uw}(y)$, $f_{vw}(y)$ and $f_{ww}(y)$ are the moduli of the surface-wave eigenmodes (figure 10b) and $\theta _{w1}$ and $\theta _{w2}$ are the phases of the two surface-wave eigenmodes. Note that $\lambda _i$ and $\lambda _r$ are the attenuation rate and frequency of the surface-wave eigenmodes, respectively (figure 10a), and $f_{\eta }$ is the surface elevation in the eigenmodes. The parameters $a_1$ and $a_2$ are positive. We assume that for the surface-wave part, the small phase change in the vertical velocity of the surface-wave eigenmodes can be ignored. The phase relationships $\theta _s-\theta _{w1}$ and $\theta _s-\theta _{w2}$, and the relationship between $a_1$ and $a_2$ needed to maximise $P_{inter1}/E^{in}$ are derived below.

Here, $P_{inter1}$ (4.11) can be approximated as

(4.21)\begin{align} P_{inter1}&\approx{-}\frac{1}{2} F \sum_{n=0}^N \frac{\varPhi^{(n)}(0)}{|\lambda_r| n! } t^{n}(a_1\sin(|\lambda_r| t+\theta_{w1}-\theta_{s})+a_2\sin(|\lambda_r| t+\theta_{s}-\theta_{w2})\nonumber\\ &\quad + \frac{a_1F\sin(\theta_{w1}-\theta_s) +a_2F\sin(\theta_{s}-\theta_{w2})}{2|\lambda_r|}, \end{align}

with

(4.22)\begin{equation} F=\int_0^hf_{us}(y)f_{vw}(y) U^\prime \,{\rm d} y \end{equation}

and

(4.23)\begin{equation} \varPhi (t)=g(t){\rm e}^{\lambda_i t}. \end{equation}

Here, $\varPhi ^{(n)}(0)$ is the $n$th derivative of $\varPhi$ at $t=0$. Equation (4.21) is obtained based on the Taylor expansion of $\varPhi$ at $t=0$. The first term on the right-hand side of (4.21) is responsible for the oscillation of $P_{inter1}$, and the second term is a constant. When $\theta _{w1}-\theta _s={\rm \pi} /2$ and $\theta _{s}-\theta _{w2}={\rm \pi} /2$, the second term reaches its positive maximum value, which leads to a quasi-positive $P_{inter}$. Under such conditions,

(4.24)\begin{equation} P_{inter1}={-}\frac{(a_1+a_2)F}{2} \sum_{n=0}^N \frac{\varPhi^{(n)}(0)}{|\lambda_r| n! } t^{n}\sin(|\lambda_r| t+{\rm \pi}/2)+\frac{(a_1+a_2)F}{2|\lambda_r|}. \end{equation}

For specified $Re_\tau$, $Fr_\tau$, $\alpha h$ and $\beta h$, the values of $F$, $\lambda _r$ and $\varPhi ^{(n)}(0)$ are constant. The maximum value of $P_{inter1}$ increases with $a_1+a_2$.

To obtain the maximal $P_{inter}/E^{in}$, the relationship between $a_1$ and $a_2$ that minimises $E^{in}$ must be determined. Keeping the shear part constant, $a_1$ and $a_2$ should be chosen to minimise the sum of the surface-wave part of the kinetic energy $(E_K)_w^{in}$, the interaction part of the kinetic energy $(E_K)_{inter}^{in}$ and the surface-wave part of the potential energy $(E_\eta )_w^{in}$. The interaction part of the potential energy is ignored because the surface deformation of the shear part is negligible. With $\theta _{w1}-\theta _s={\rm \pi} /2$ and $\theta _{s}-\theta _{w2}={\rm \pi} /2$, the energy $(E_K)_w^{in} + (E_K)_{inter}^{in}+(E_\eta )_w^{in}$ contributed by the surface-wave eigenmodes is

(4.25)\begin{align} & (E_K)_w^{in} + (E_K)_{inter}^{in} + (E_\eta)_w^{in} =\frac{(a_1+a_2)^2}{2} \left(\int_0^{h} f_{uw}^2(y)\,{\rm d} y + f_{\eta}^2\right)\nonumber\\ &\qquad -2(a_1+a_2)\int_0^hf_{uw}(y)f_{us}(y)\,{\rm d} y\nonumber\\ &\qquad + \frac{(a_1-a_2)^2}{2} \int_0^{h} f_{vw}^2(y)+f_{ww}^2(y)\,{\rm d} y. \end{align}

Because $a_1$, $a_2$ and $f_{uw}(y)f_{us}(y)$ are positive and $f_{uw}< f_{ww}$, $a_1=a_2$ minimises $(E_K)_w^{in} + (E_K)_{inter}^{in} + (E_\eta )_w^{in}$ as

(4.26)\begin{equation} (E_K)_w^{in} + (E_K)_{inter}^{in} + (E_\eta)_w^{in}=2a_1^2 \left(\int_0^{h} f_{uw}^2(y)\,{\rm d} y + f_{\eta}^2\right)-4a_1\int_0^hf_{uw}(y)f_{us}(y)\,{\rm d} y, \end{equation}

which is finally determined by $a_1$.

Therefore, under the conditions of $a_1=a_2$, $\theta _{w1}-\theta _s={\rm \pi} /2$ and $\theta _{s}-\theta _{w2}={\rm \pi} /2$, the final analytical result of $P_{inter1}/E^{in}$ is

(4.27)\begin{equation} \frac{P_{inter1}}{E^{in}}=\frac{-\dfrac{a_1F}{|\lambda_r|} \left(\sum_{n=0}^N \dfrac{\varPhi^{(n)}(0)}{ n! } t^{n}\sin(|\lambda_r| t+{\rm \pi}/2)-1\right)}{2a_1^2 \left(\int_0^{h} f_{uw}^2(y)\,{\rm d} y + f_{\eta}^2\right)-4a_1\int_0^hf_{uw}(y)f_{us}(y)\,{\rm d} y + (E_K)_s^{in}}, \end{equation}

which is a function of $a_1$ and $t$ for specified Reynolds number, Froude number and wavenumber. For the maximum $P_{inter1}/E^{in}$, the value of $a_1$ can be obtained from $\partial (P_{inter1}/E^{in})/\partial a_1=0$.

Note that when $a_1=a_2$, $\theta _{w1}-\theta _s={\rm \pi} /2$ and $\theta _{s}-\theta _{w2}={\rm \pi} /2$, the velocities and surface elevation perturbations of the surface-wave part satisfy

(4.28)\begin{gather} u_w ={-}2 a_1 f_{uw}(y) \cos(\beta z +\theta_s) \cos(|\lambda_r| t) {\rm e}^{\lambda_i t}, \end{gather}

(4.29)\begin{gather} v_w ={-}2 a_1 f_{vw}(y) \cos(\beta z +\theta_s) \sin(|\lambda_r| t) {\rm e}^{\lambda_i t}, \end{gather}

(4.30)\begin{gather} w_w = 2 a_1 f_{ww}(y) \sin(\beta z +\theta_s) \sin(|\lambda_r| t){\rm e}^{\lambda_i t}, \end{gather}

(4.31)\begin{gather} \eta = 2 a_1 f_{\eta} \cos(\beta z +\theta_s) \cos(|\lambda_r| t){\rm e}^{\lambda_i t}, \end{gather}

which represent a standing wave, as shown in figures 15(c i)–15(c vii).

4.1.4. Effects of the spanwise wavenumber and Froude number at different Reynolds numbers

The above analyses indicate that the free surface enhances the transient growth rate through the interaction production term $P_{inter}$ (4.11). Next, we confirm this mechanism at different spanwise wavenumbers, Froude numbers and Reynolds numbers, and explain the variation in the free-surface effect on the transient growth rate of streamwise-uniform motions with different spanwise wavenumbers and Froude numbers.

Figure 16 shows a comparison of the time histories of the interaction production term $P_{inter}$ at different Froude numbers and spanwise wavenumbers for $Re_\tau =180$, $360$ and $1000$. The term $P_{inter}$ increases with increasing $Fr_\tau$ but decreases sharply with increasing spanwise wavenumber $\beta h$. For the same Froude number and spanwise wavenumber, the change in $P_{inter}$ with the Reynolds number is negligible. Recall that at three different Reynolds numbers $Re_\tau =180$, $360$ and $1000$, the free-surface effects on the transient growth rate are similar (figures 5 and 7), increasing with increasing Froude number but attenuating obviously with increasing spanwise wavenumber. Thus, the variation in $P_{inter}$ with the Froude number and spanwise wavenumber is consistent with the variation in the transient growth rate with these parameters, confirming that the interaction production term is an important mechanism underlying that the free surface influences the transient growth of motions.

Figure 16.

Time evolution of the interaction production term $P_{inter}$ (a i–c i) at different Froude numbers $Fr_\tau$ for $\alpha h = 0$ and $\beta h=1$ and (a ii–c ii) at different spanwise wavenumbers $\beta h$ for $Fr_\tau =0.142$ and $\alpha h = 0$. The results are obtained at (a i,a ii) $Re_\tau =180$, (b i,b ii) $Re_\tau =360$ and (c i,c ii) $Re_\tau =1000$.

The variations in $P_{inter}$ and the transient growth rate with the Froude number and spanwise wavenumber can be explained using the analytical expression of the production term (4.27). The interaction production term $P_{inter}$ decreases as the frequency of the surface-wave eigenmodes $|\lambda _r|$ increases. Additionally, $P_{inter}$ in (4.27) is proportional to the integral term $F$ (4.22) associated with the vertical velocity modulus of the surface-wave eigenmodes. The variations in the frequency and vertical velocity modulus of the surface-wave eigenmodes with the Froude number and spanwise wavenumber are shown in figure 17. The frequency of the surface-wave eigenmodes decreases with increasing Froude number, whereas the vertical velocity modulus does not change with the Froude number. Therefore, an increase in the Froude number leads to an increase in $P_{inter}$ (figure 16a i–c i), and this escalation can be attributed solely to the decrease in the frequency of the surface-wave eigenmodes $|\lambda _r|$. Moreover, an increase in the spanwise wavenumber has dual effects on the surface-wave eigenmodes, increasing their frequency while also reducing the integral term $F$. The integral term $F$ is determined by the product of the streamwise velocity, which is composed of shear eigenmodes, and the vertical velocity, which is composed of surface-wave eigenmodes. As the spanwise wavenumber increases, the moduli of the vertical velocity of the surface-wave eigenmode becomes more concentrated near the free surface and rapidly diminishes away from it. Simultaneously, in the present result (figure 15b i–b vii) and other canonical wall-bounded turbulence (Cossu et al. Reference Cossu, Pujals and Depardon2009), the streamwise velocity of the perturbation composed of shear eigenmodes is more concentrated near the bottom wall. Consequently, the vertical locations where both components are significant start to separate, leading to a reduction in their product and thus a decrease in $F$. Combining with the increase of $|\lambda _r|$, $P_{inter}$ decreases with increasing spanwise wavenumber (figures 16a ii–c ii). Thus, the effect of the free surface on the transient growth rate is significant only for small spanwise wavenumbers at relatively high Froude numbers.

Figure 17.

Variations in (a,c) the eigenvalues and (b,d) the vertical velocity modulus of the surface-wave eigenmodes (a,b) with the Froude number $Fr_\tau$ ($Fr_\tau = 0.071, 0.1, 0.142$ and $0.212$) for $\alpha h = 0$ and $\beta h=1$ and (c,d) with the spanwise wavenumber $\beta h$ ($\beta h = 1, 2, 3$ and $4$) for $Fr_\tau =0.142$ and $\alpha h = 0$. The results are obtained at $Re_\tau =180$.

In summary, the free surface affects the transient growth process of streamwise-uniform motions through the interaction between spanwise standing waves consisting of surface-wave eigenmodes and streamwise vortices consisting of shear eigenmodes. In particular, the growth rates of the total energy and kinetic energy are enhanced by the energy extracted from the mean velocity gradient through the correlation between the streamwise velocity induced by the lifting-up effect of the streamwise vortices and the vertical velocity induced by the spanwise standing wave. This occurs if the initial surface elevation of the standing wave is at its maximum and the phase is the same as that of the streamwise velocity induced by the lifting-up effect of the streamwise vortices. The maximal transient growth rate is inversely proportional to the frequency of the standing wave, which decreases with increasing Froude number, but increases with increasing spanwise wavenumber. In the transient growth process, only the energy of the streamwise velocity is amplified, which is at the expense of the attenuation of the streamwise vortices and standing waves. Because the initial flow field tends to trigger the transient growth, the amplified flow field represents commonly observed flow states in turbulence. The amplified flow field is featured by energetic very-large-scale $u$ and relatively weak $v$ and $w$, which is consistent with the results of direct numerical simulations and laboratory experiments of free-surface open channel turbulent flows (Borue et al. Reference Borue, Orszag and Staroselsky1995; Kumar, Gupta & Banerjee Reference Kumar, Gupta and Banerjee1998; Duan et al. Reference Duan, Chen, Li and Zhong2020; Yoshimura & Fujita Reference Yoshimura and Fujita2020).

4.2.1. Effect of the free surface on the energy budgets for the streamwise-varying case

Similar to the streamwise-uniform case analysed in § 4.1, the energy growth rate $E(t)/E^{in}$ in the streamwise-varying case is dominated by the difference between the production term and the dissipation term. According to (4.7) and (4.9), the budget terms can be decomposed into a shear part, a wave part and an interaction part, with the latter two associated with the free surface. To examine the net contribution of these three parts to $E(t)/E^{in}$, figure 18(a) illustrates the time evolution of the combination of the shear parts of the production term and dissipation term $(P_{s}+\epsilon _{s})/E^{in}$, the combination of the wave parts of the production term and dissipation term $(P_{w}+\epsilon _{w})/E^{in}$, and the combination of the interaction parts of the production term and dissipation term $(P_{inter}+\epsilon _{inter})/E^{in}$ in the streamwise-uniform case (dashed line with symbols) and the streamwise-varying case (solid line with symbols). As shown, for both the streamwise-varying and streamwise-uniform cases, the net contribution of the wave part is negligible, while the interaction part $(P_{inter}+\epsilon _{inter})/E^{in}$ has a significant positive contribution. As shown in figure 18(b), the net contribution of $(P_{inter}+\epsilon _{inter})/E^{in}$ is due mainly to $P_{inter}$. Further decomposing $P_{inter}$ into $P_{inter1}$ and $P_{inter2}$, as shown in (4.10), we find that $P_{inter}$ is governed by $P_{inter1}$. Therefore, similar to the streamwise-uniform case, the free surface enhances the energy growth rate of the streamwise-varying motions through the energy production term that arises due to the correlation between the streamwise velocity of the shear part and the vertical velocity of the surface-wave part (4.11).

Figure 18.

(a) Comparison of the time evolution of the budget terms $(P_{s}+\epsilon _{s})/E^{in}$, $(P_{w}+\epsilon _{w})/E^{in}$, $(P_{inter}+\epsilon _{inter})/E^{in}$ and $E(t)/E^{in}$ (definitions provided in § 4.1.2) between the streamwise-uniform case ($\alpha h = 0$, dashed lines with symbols) and the streamwise-varying case ($\alpha h = 0.2$, solid lines with symbols). (b) Time evolution of the budget terms $P_{inter}$, $P_{inter1}$, $P_{inter2}$ and $\epsilon _{inter}$ in the streamwise-varying case ($\alpha h = 0.2$). The results are obtained at $Re_\tau =180$, $\beta h=1$ and $Fr_\tau =0.142$.

Because the values of $(P_{inter}+\epsilon _{inter})/E^{in}$ in the streamwise-varying and streamwise-uniform cases in figure 18(a) are comparable, the energy growth rate $E(t)/E^{in}$ in the streamwise-varying case is smaller than that in the streamwise-uniform case, mainly due to the smaller net contribution from the shear part $(P_{s}+\epsilon _{s})/E^{in}$. In closed channel flows, the energy growth rate, which is governed by the difference between the shear parts of the production and dissipation terms, also decreases with increasing $\alpha h$ (Cossu et al. Reference Cossu, Pujals and Depardon2009).

4.2.2. Initial conditions for the transient growth of streamwise-varying motions

Next, the features of the surface-wave part and shear part in the streamwise-varying case are analysed to study the initial state of the surface-wave part, which can lead to the maximum production term $P_{inter1}/E^{in}$ that is responsible for the enhancement of the energy growth rate by the free surface.

Figure 19 shows the eigenvalues of the surface-wave and shear eigenmodes, and the velocity moduli of the two surface-wave eigenmodes. As shown in figure 19(a), there are two surface-wave eigenmodes with different $\lambda _r$, indicating different propagation speeds $c=\lambda _r/k$ along the wavenumber direction $(\alpha /k, \beta /k)$. The propagation speeds of the two surface-wave eigenmodes are $c_1=c_s+c_w$ and $c_2=c_s-c_w$, with $c_s=(\lambda _{r1}+\lambda _{r2})/(2k)=\lambda _r^s/k$ and $c_w=|\lambda _{r1}-\lambda _{r2}|/(2k)=\lambda _r^w/k$. The velocity $c_s$ is approximately equal to the projection of the surface base flow speed in the wavenumber direction $U(h)\alpha /k$. In addition, the $\lambda _r$ values of the shear eigenmodes are all close to $\lambda _r^s$, implying that the shear part consisting of the shear eigenmodes propagates in the direction of $(\alpha /k, \beta /k)$ at a speed of $c_s=\lambda _r^s/k$. The velocity $c_w$ represents the propagation speed of the surface-wave eigenmodes relative to the shear eigenmodes.

Figure 19.

(a) Eigenvalues of the surface-wave eigenmodes (diamonds) and the shear eigenmodes (circles). (b,c) Velocity moduli of the two surface-wave eigenmodes. The surface-wave eigenmode with the larger $\lambda _r$ is shown by lines without symbols, and the other eigenmode is shown by lines with symbols. The velocity moduli are shown in the coordinate system (b) $(x,y,z)$ and (c) $(\tilde {x},\tilde {y},\tilde {z})$. The results are obtained at $Re_\tau =180$, $\alpha h=0.2$, $\beta h=1$ and $Fr_\tau =0.142$.

Based on the propogation directions of the surface-wave eigenmodes and shear eigenmodes, a new coordinate system is defined using the transformation

(4.32)\begin{equation} \tilde{x}= x\frac{\beta}{k} -z \frac{\alpha}{k},\quad \tilde{y}= y,\quad \tilde{z}= z\frac{\beta}{k} +x \frac{\alpha}{k}. \end{equation}

Here, the $\tilde {z}$-axis is parallel to the wavenumber vector. Correspondingly, the velocity components $\tilde {u}_i$ in the new coordinate system $(\tilde {x},\tilde {y},\tilde {z})$ are

(4.33)\begin{equation} \tilde{u}= u\frac{\beta}{k} -w \frac{\alpha}{k},\quad \tilde{v}= v,\quad \tilde{w}= w\frac{\beta}{k} +u \frac{\alpha}{k}. \end{equation}

As shown in figure 19(b), in the coordinate system $(x,y,z)$, the vertical velocity moduli of the two surface-wave eigenmodes are similar, while the streamwise velocity moduli of the two surface-wave eigenmodes are noticeably different. However, in the new coordinate system $(\tilde {x},\tilde {y},\tilde {z})$, the streamwise velocity moduli $|\tilde {u}|$ of the two eigenmodes in the $\tilde {x}$-direction are much closer (figure 19c).

The difference between the exponential damping rates of the two surface-wave eigenmodes, i.e. ${\rm e}^{\lambda _{i1}t}$ and ${\rm e}^{\lambda _{i2}t}$ (figure 19a), is small within the time needed to reach the maximum transient growth rate ($t_{max}\approx 1.5 h/u_\tau$). Therefore, the velocity of the surface-wave part can be approximated as

(4.34)$$\begin{gather} \tilde{u}_w=a_1 f_{\tilde{u}w} \sin(k\tilde{\tilde{z}}+\lambda^w_rt + \theta_{w1}) {\rm e}^{\lambda_{i} t} - a_2 f_{\tilde{u}w} \sin (k\tilde{\tilde{z}}-\lambda^w_rt + \theta_{w2}) {\rm e}^{\lambda_{i} t}, \end{gather}$$

(4.35)$$\begin{gather}\tilde{v}_w=a_1 f_{\tilde{v}w} \cos(k\tilde{\tilde{z}}+\lambda^w_rt+\theta_{w1}) {\rm e}^{\lambda_{i} t} +a_2 f_{\tilde{v}w} \cos (k\tilde{\tilde{z}}-\lambda^w_rt + \theta_{w2}) {\rm e}^{\lambda_{i} t}, \end{gather}$$

(4.36)$$\begin{gather}\tilde{w}_w={-}a_1 f_{\tilde{w}w} \sin(k\tilde{\tilde{z}}+\lambda^w_rt + \theta_{w1}) {\rm e}^{\lambda_{i} t} - a_2 f_{\tilde{w}w} \sin (k\tilde{\tilde{z}}-\lambda^w_rt + \theta_{w2}) {\rm e}^{\lambda_{i} t}, \end{gather}$$

(4.37)$$\begin{gather}\tilde{\eta}_w=a_1 f_{\eta} \sin(k\tilde{\tilde{z}}+\lambda^w_rt + \theta_{w1}) {\rm e}^{\lambda_{i} t} - a_2 f_{\eta} \sin (k\tilde{\tilde{z}}-\lambda^w_rt + \theta_{w2}) {\rm e}^{\lambda_{i} t}. \end{gather}$$

Here, $f_{\tilde {u}w}$, $f_{\tilde {v}w}$, $f_{\tilde {w}w}$, $f_{\eta }$ and $\lambda _{i}$ are approximations of the moduli of $\tilde {u}$, $\tilde {v}$, $\tilde {w}$, $\eta$ and the damping rate of the two surface-wave eigenmodes, respectively, and

(4.38)\begin{equation} \tilde{\tilde{z}}=\tilde{z}-c_st. \end{equation}

The two surface-wave eigenmodes in the streamwise-varying case propagate in opposite $\tilde {\tilde {z}}$-directions at the same phase speed; hence, the surface-wave part has a similar expression to that in the streamwise-uniform case (4.17)–(4.20).

Figure 20 displays the flow field and surface deformation in the transient growth process in the plane $(\tilde {y}=y,\tilde {\tilde {z}}=\tilde {z}-c_st)$, as well as the corresponding shear and surface-wave parts, at $t=iT/4$, with $i=0,1,\ldots,6$. Here, the period $T$ is equal to $2{\rm \pi} /(kc_w)$, and the velocity $c_w$ is the propagation speed of the surface-wave eigenmode relative to the speed of the shear eigenmodes (figure 19a). As shown in figure 20(b i–b vii), the shear part consists of initially strong vortices in the $\tilde {x}$-direction, as evidenced by $(\tilde {v},\tilde {w})$ and the streaks of $\tilde {u}$. As time increases, the vortices decay and the streaks are enhanced. Because the phase speed of the shear eigenmodes in the wavenumber vector direction is approximately $c_s$ (figure 19a), the spanwise locations of the vortices are essentially unchanged in the plane $(\tilde {y}=y,\tilde {\tilde {z}}=\tilde {z}-c_st)$. As a result, the velocities of the shear part can be approximated as

(4.39)\begin{gather} \tilde{u}_s=f_{\tilde{u}s}(\tilde{y})\cos(k\tilde{\tilde{z}} + \theta_s)\tilde{g}(t), \end{gather}

(4.40)\begin{gather} \tilde{v}_s={-}f_{\tilde{v}s}(\tilde{y})\cos(k\tilde{\tilde{z}}+ \theta_s) \tilde{m}(t), \end{gather}

(4.41)\begin{gather} \tilde{w}_s={-}f_{\tilde{w}s}(\tilde{y})\sin(k\tilde{\tilde{z}}+ \theta_s) \tilde{n}(t), \end{gather}

with $\tilde {g}(t)\geq 1$, $\tilde {m}(t)\leq 1$ and $\tilde {n}(t)\leq 1$. The moduli $f_{\tilde {u}s}$ and $f_{\tilde {v}s}$ are positive, while the sign of $f_{\tilde {w}s}(y)$ changes near the channel centre. The shear part in the streamwise-varying case, expressed using $\tilde {y}$ and $\tilde {\tilde {z}}$, is similar to that in the streamwise-uniform case (4.14)–(4.16).

Figure 20.

Velocity field ($\tilde {u}$, $\tilde {v}$, $\tilde {w}$) and the surface elevation $\eta$, as well as their shear and surface-wave parts, in the plane $(\tilde {y}=y,\tilde {\tilde {z}}=\tilde {z}-c_st)$ at (a i–c i) $t=0$, (a ii–c ii) $t=T/4$, (a iii–c iii) $t=T/2$, (a iv–c iv) $t=3T/4$, (a v–c v) $t=T$, (a vi–c vi) $t=5T/4$ and (a vii–c vii) $t=3T/2$. Panels (a i–a vii) show the total, panels (b i–b vii) depict the shear part and panels (c i–c vii) display the surface-wave part. Here, $T$ is equal to $2{\rm \pi} /(kc_w)$, where $c_w$ is the propagation speed of the surface-wave eigenmodes relative to that of the shear eigenmodes, and the maximum transient growth rate is reached at $t=3T/2$. The vector is composed of the vertical and spanwise velocities ($\tilde {v}$, $\tilde {w}$), and the contours are the streamwise velocity $\tilde {u}$. The results are obtained at $Re_\tau =180$, $\alpha h=0.2$, $\beta h=1$ and $Fr_\tau =0.142$.

Here, $P_{inter1}$ can be expressed as

(4.42)$$\begin{gather} P_{inter1}= \int_0^h\int_0^t\overline{{u}_s{v}_{w}}U^\prime \,{\rm d}t\,{\rm d} y=\int_0^h\int_0^t \left(\frac{\beta}{k}\overline{{\tilde{u}}_s{\tilde{v}}_{w}}+ \frac{\alpha}{k}\overline{{\tilde{w}}_s{\tilde{v}}_{w}}\right)U^\prime \,{\rm d} t \,{\rm d} y\nonumber\\ \approx\int_0^h\int_0^t \frac{\beta}{k}\overline{{\tilde{u}}_s{\tilde{v}}_{w}}U^\prime \,{\rm d}t\,{\rm d} y. \end{gather}$$

Because the sign of $\tilde {w}_s$ changes in the $y$ (or $\tilde {y}$) direction and the sign of $\tilde {v}_{w}$ remains the same in the $y$ direction, the integral ${\alpha }\overline {{\tilde {w}}_s{\tilde {v}}_{w}}/k$ in the $y$ direction is negligible. Additionally, $\alpha$ is much smaller than $\beta$ for streamwise-elongated motions, which we focus on in the present study. Therefore, the ${\alpha }\overline {{\tilde {w}}_s{\tilde {v}}_{w}}/{k}$ part is ignored. Because ${\tilde {u}}_s$ and ${\tilde {v}}_{w}$ in (4.39) and (4.34) have the same forms as those in the streamwise-uniform case, we obtain following (4.21) that

(4.43) $$\begin{gather} P_{inter1}\approx \frac{\beta}{k}\left(-\frac{1}{2} \tilde{F} \sum_{n=0}^N \frac{\tilde{\varPhi}^n(0)}{|\lambda_r^w| n! } t^{n}(a_1\sin(\lambda_r^w t+\theta_{w1}-\theta_{s})+a_2\sin(\lambda_r^w t+\theta_{s}-\theta_{w2}))\right)\nonumber\\ +\frac{\beta}{k}\frac{a_1\tilde{F}\sin(\theta_{w1}-\theta_s) +a_2\tilde{F}\sin(\theta_{s}-\theta_{w2})}{2|\lambda_r^w|}, \end{gather}$$

with

(4.44)\begin{equation} \tilde{F}=\int_0^hf_{\tilde{u}s}(y)f_{\tilde{v}w}(y) U^\prime \,{\rm d} y \end{equation}

and

(4.45)\begin{equation} \tilde{\varPhi} (t)=\tilde{g}(t){\rm e}^{\lambda_i t}. \end{equation}

Similar to the streamwise-uniform case, the maximum value of $P_{inter1}$ is obtained when $\theta _{w1}-\theta _s={\rm \pi} /2$ and $\theta _{s}-\theta _{w2}={\rm \pi} /2$ are satisfied. Under this condition, the portion of the initial input energy associated with the surface-wave part is

(4.46)$$\begin{gather} (E_K)_w^{in} + (E_K)_{inter}^{in} + (E_\eta)_w^{in} =\frac{(a_1+a_2)^2}{2} \left(\int_0^{h} f_{\tilde{u}w}^2(y)\,{\rm d} y + f_{\eta}^2\right)\nonumber\\ -2(a_1+a_2)\int_0^hf_{\tilde{u}w}(y)f_{\tilde{u}s}(y)\,{\rm d} y + \frac{(a_1-a_2)^2}{2} \int_0^{h}\left( f_{\tilde{v}w}^2(y)+f_{\tilde{w}w}^2(y)\right)\,{\rm d} y. \end{gather}$$

The minimum value is obtained when $a_1\approx a_2$. Because the moduli of the two surface-wave eigenmodes are only approximately the same, i.e. not identical, $a_1$ is not exactly equal to $a_2$.

The above analyses show that in the streamwise-varying case, the maximum $P_{inter1}/E^{in}$ is obtained when $a_1\approx a_2$, $\theta _{w1}-\theta _s={\rm \pi} /2$ and $\theta _{s}-\theta _{w2}={\rm \pi} /2$. Under these conditions, the velocities and surface elevation of the surface-wave part are

(4.47)\begin{gather} \tilde{u}_w \approx{-}2 a_1 f_{\tilde{u}w}(y) \cos(k \tilde{\tilde{z}} +\theta_s) \cos(|\lambda^w_r| t) {\rm e}^{\lambda_i t}, \end{gather}

(4.48)\begin{gather} \tilde{v}_w \approx{-}2 a_1 f_{\tilde{v}w}(y) \cos(k \tilde{\tilde{z}} +\theta_s) \sin(|\lambda^w_r| t) {\rm e}^{\lambda_i t}, \end{gather}

(4.49)\begin{gather} \tilde{w}_w \approx 2 a_1 f_{\tilde{w}w}(y) \sin(k \tilde{\tilde{z}} +\theta_s) \sin(|\lambda^w_r| t) {\rm e}^{\lambda_i t} \end{gather}

and

(4.50)\begin{equation} \eta \approx 2 a_1 f_{\eta} \cos(k \tilde{\tilde{z}} +\theta_s) \cos(|\lambda^w_r| t){\rm e}^{\lambda_i t}. \end{equation}

This solution represents a standing wave in the plane $(\tilde {y},\tilde {\tilde {z}})$ parallel to the wavenumber vector that moves with speed $c_s$ in the wavenumber-vector direction. Initially, the surface elevation of the standing wave is at its maximum value, and its phase is the same as that of $\tilde {u}_s$. The standing wave can be confirmed by the variation in the surface elevation at different times, as shown in figure 20(c i–c vii).

In summary, the above analyses show that for the streamwise-varying case, in a plane parallel to the wavenumber vector moving at a speed equal to the surface mean velocity projected in the wavenumber-vector direction, the two surface-wave eigenmodes propagate in opposite directions at the same speed, and the vortices consisting of the shear eigenmodes are approximately fixed in space. If a standing wave reaches its maximum surface elevation and its phase is identical to that of the streamwise velocity induced by the lifting-up effect of the vortices at the initial time, the transient growth rate can be enhanced by the free surface to a maximal level. In other words, the transient growth rate is enhanced by a standing wave riding on the vortices moving at a speed equal to the projection of the surface mean velocity in the wavenumber direction. While the interaction between the surface-wave part and the shear part can enhance the transient growth rate of the streamwise-varying motions in a way similar to that of streamwise-uniform motions, the growth of the total perturbation energy of the streamwise-varying motions is slower because of the decrease in the transient growth rate of the shear part with increasing streamwise wavenumber.