Asymptotic multi-layer analysis of wind over unsteady monochromatic surface waves

Abstract

Asymptotic multi-layer (AML) analyses and computation of solutions for turbulent flows over steady and unsteady monochromatic surface waves are reviewed in the limits of low-turbulence stresses and small wave amplitude. The structure of the flow is defined in terms of asymptotically matched thin layers, namely the surface layer and a critical layer (CL), whether it is ‘elevated’ or ‘immersed’, corresponding to its location above or within the surface layer. The results particularly demonstrate the physical importance of the singular flow features and physical implications of the elevated CL in the limit of the unsteadiness tending to zero. These agree with the variational mathematical solution of Miles (J Fluid Mech, 3:185–204, 1957) for a small but finite growth rate, but they are not consistent physically or mathematically with his analysis in the limit of a growth rate tending to zero. As this and other studies conclude, in the limit of zero growth rate, the effect of the elevated CL is eliminated by finite turbulent diffusivity, so that the perturbed flow and the drag force are determined by the asymmetric or sheltering flow in the surface shear layer and its matched interaction with the upper region. But for groups of waves, in which the individual waves grow and decay, there is a net contribution of the elevated CL to the wave growth. CLs, whether elevated or immersed, affect this asymmetric sheltering mechanism, but in quite a different way from their effect on growing waves. These AML methods lead to physical insights and suggest approximate methods for analysing higher-amplitude and more complex flows, such as flow over wave groups.

Appendix: Effect of the inertial critical layer

In a frame of reference moving with waves, the vertical perturbation to the air flow, $\Delta w=\fancyscript{W}(z)\hbox {e}^{\mathrm{i}k(x-c_\mathrm{r}t)+kc_\mathrm{i}t}$, satisfies the Orr–Sommerfeld-like equation [30, 31]³

$$\begin{aligned} {\fancyscript{T}}''\equiv (\nu _\mathrm{e}\fancyscript{W}'')''=\hbox {i}k[(\fancyscript{U}-\hbox {i}c_\mathrm{i}) (\fancyscript{W}''-k^2\fancyscript{W})-U''\fancyscript{W}], \end{aligned}$$

(9)

where $\nu _\mathrm{e}$ is the eddy viscosity.

In the outer region, turbulence is negligible, and thus the left-hand side of (9) can be neglected compared to the right-hand side, and hence we obtain the Rayleigh equation

$$\begin{aligned} (\fancyscript{U}-\hbox {i}c_\mathrm{i})(\fancyscript{W}''-k^2\fancyscript{W})-U''\fancyscript{W}=0. \end{aligned}$$

(10)

As shown by Sajjadi [32], the leading-order solution to (10) is

$$\begin{aligned} \fancyscript{W}=(\fancyscript{U}-\hbox {i}c_\mathrm{i})\hbox {e}^{-kz}\left[ A+\fancyscript{W}_\mathrm{c}U_\mathrm{c}'\hbox {e}^{kz_\mathrm{c}} \int \limits _{0}^\infty \Big \{\displaystyle \frac{1}{(\fancyscript{U}-\hbox {i}c_\mathrm{i})^2}-1\Big \}\,\hbox {d}z\right] , \end{aligned}$$

(11)

where $A$ is constant, which can be determined by matching the solutions to the outer and inner regions.

For slowly growing waves $c_\mathrm{i}>0$, the CL lies within the inner region close to the surface wave and the integral in (11) is regular since $\fancyscript{U}>0$ there. Let us now suppose that

$$\begin{aligned} c_\mathrm{i}\ll U_\mathrm{c}^{\prime 2}/U_\mathrm{c}''; \end{aligned}$$

then the integral in (11) can be evaluated approximately. Hence, indenting the path of integration in (11) under the singularity $z=z_\mathrm{c}$ we obtain

$$\begin{aligned} \fancyscript{W}=(\fancyscript{U}-\hbox {i}c_\mathrm{i})\hbox {e}^{-kz}\left[ A+\fancyscript{W}_\mathrm{c}U_\mathrm{c}'\hbox {e}^{kz_\mathrm{c}} \left( {\int \limits _0^{\infty }\!\!\!\!\!\!-\quad } \Big \{\displaystyle \frac{1}{(\fancyscript{U}-\hbox {i}c_\mathrm{i})^2}-1\Big \} \,\hbox {d}z-I\right) \right] , \end{aligned}$$

(12)

where

$$\begin{aligned} I=\lim _{\varpi \rightarrow 0}\int \limits _{\eta _\mathrm{c}-\varpi }^{\eta _\mathrm{c}+\varpi } \left\{ \displaystyle \frac{1}{(\fancyscript{U}-\hbox {i}c_\mathrm{i})^2}-1\right\} \hbox {d}z. \end{aligned}$$

(13)

If we expand $\fancyscript{U}\!(z)$ as a Taylor expansion in the vicinity of the critical point, i.e.

$$\begin{aligned} \fancyscript{U}(z)\sim \eta U_\mathrm{c}'+\textstyle \frac{1}{2}\eta ^2U_\mathrm{c}''+O(\eta ^3),\quad \eta \equiv z-z_\mathrm{c}, \end{aligned}$$

and set $z=z_\mathrm{c}\varpi \hbox {e}^{\mathrm{i}\theta }$, where $\varpi \equiv c_\mathrm{i}/U_*\ll 1$, and $\tan \theta =-c_\mathrm{i}/U_\mathrm{c}'\eta$, then (13) becomes

$$\begin{aligned} I&\sim \displaystyle \frac{1}{U_\mathrm{c}^{\prime 2}}\left\{ \lim _{\varpi \rightarrow 0} \int \limits _{z_\mathrm{c}-\varpi }^{z_\mathrm{c}+\varpi }\displaystyle \frac{\hbox {d}z}{(z-z_\mathrm{c})^2} +\hbox {i}\pi \displaystyle \frac{U_\mathrm{c}''}{U_\mathrm{c}'}\right\} = \displaystyle \frac{\hbox {i}\pi U_\mathrm{c}''}{U_\mathrm{c}^{\prime 3}}, \end{aligned}$$

(14)

which is in agreement with the result obtained by Belcher et al. [21].

As also pointed out by Belcher et al. [21], for a logarithmic mean velocity profile $\tan \theta =\varpi z_\mathrm{c}/(z-z_\mathrm{c})$. Hence $\theta$ varies between $0$ and $\pi$ as $(z-z_\mathrm{c})/l_\mathrm{c}$ tends to $\pm \infty$, respectively. Note that the transition between these limiting values occurs across the layer of thickness $l_\mathrm{c}=\varpi z_\mathrm{c}$. Note also that the significance of the term $\hbox {i}U_\mathrm{c}''/U_\mathrm{c}^{\prime 3}$ in the solution for $I$ is that it leads to an out-of-phase contribution to the wave-induced vertical velocity. This gives rise to the same wave growth rate as that of Miles’ [15] CL model.

The result of the present analysis confirms the earlier finding [21] in that Miles’ [15] solution is only valid when waves grow significantly slowly such that

$$\begin{aligned} c_\mathrm{i}\ll U_\mathrm{c}'z_\mathrm{c}\sim U_* . \end{aligned}$$

(15)

Our analysis also shows that when inertial effects control the behaviour around the CL, there is a smooth behaviour around the CL of thickness [21]

$$\begin{aligned} l_\mathrm{c}\sim c_\mathrm{i}/U_\mathrm{c}'\sim z_\mathrm{c}c_\mathrm{i}/U_* . \end{aligned}$$

(16)

Hence, this proves the effects of a CL [15] are only valid in the limit $c_\mathrm{i}/U_*\downarrow 0$.

To calculate the energy-transfer parameter due to the CL, $\beta _\mathrm{c}$, we let $\fancyscript{W}=-\fancyscript{VM}$, where $\fancyscript{V}=\fancyscript{U}-\hbox {i}c_\mathrm{i}$. Thus, (9) becomes

$$\begin{aligned}{}[\nu _\mathrm{e}({\fancyscript{V}}{\fancyscript{M}}''+2U'{\fancyscript{M}}'+U''{\fancyscript{M}})]''=\hbox {i}k[({\fancyscript{V}}^2{\fancyscript{M}}')'- k^2{\fancyscript{V}}^2{\fancyscript{M}}]. \end{aligned}$$

(17)

In the quasi-laminar limit the left-hand side of (17) is negligible, and thus we have

$$\begin{aligned} ({\fancyscript{V}}^2{\fancyscript{M}}')'-k^2{\fancyscript{V}}^2{\fancyscript{M}}=0. \end{aligned}$$

(18)

Multiplying (18) by ${\fancyscript{M}}$, integrating by parts over $0<z<\infty$, and invoking the inner limits ${\fancyscript{M}}\rightarrow a$ and ${\fancyscript{V}}^2{\fancyscript{M}}'\rightarrow {\fancyscript{P}}_0$ (the complex amplitude of the surface pressure) and a null condition at $z=\infty$, we obtain

$$\begin{aligned} a{\fancyscript{P}}_0=-\int \limits _0^\infty {\fancyscript{V}}^2({\fancyscript{M}}^{\prime 2}+k^2{\fancyscript{M}}^2)\,\hbox {d}z. \end{aligned}$$

(19)

The simplest admissible trial function for the variational integral (19), may be taken as

$$\begin{aligned} {\fancyscript{M}}=a\hbox {e}^{-kz/\varsigma }, \end{aligned}$$

(20)

where $\varsigma$ is a free parameter. Substituting (20) into (19) together with the approximation ${\fancyscript{V}}\approx U_1\ln (z/z_\mathrm{c})-\hbox {i}c_\mathrm{i}$ we obtain

$$\begin{aligned} \hat{{\fancyscript{P}}}_0\equiv \displaystyle \frac{\fancyscript{P}}{kaU_1^2}&= -k(\varsigma ^{-2}+1)\int \limits _0^\infty \hbox {e}^{-2kz/\varsigma } {\fancyscript{F}}(z)\,\hbox {d}z,\nonumber \end{aligned}$$

where

$$\begin{aligned} {\fancyscript{F}}(z)=\ln ^2(z/z_\mathrm{c})-2\hbox {i}\hat{c}_\mathrm{i}\ln (z/z_\mathrm{c})-\hat{c}_\mathrm{i}^2 \end{aligned}$$

and $\hat{c}_\mathrm{i}=c_\mathrm{i}/U_1$. Evaluating the integral we obtain

$$\begin{aligned} \hat{{\fancyscript{P}}}_0=-\displaystyle \frac{\varsigma +\varsigma ^{-1}}{2} \left\{ \displaystyle \frac{\pi ^2}{6}+\ln ^2\left( \displaystyle \frac{2\gamma \xi _\mathrm{c}}{\varsigma }\right) -2\hbox {i}\hat{c}_\mathrm{i}\ln \left( \displaystyle \frac{2\gamma \xi _\mathrm{c}}{\varsigma }\right) +\hat{c}_\mathrm{i}^2\right\} , \end{aligned}$$

(21)

where $\xi _\mathrm{c}\equiv kz_\mathrm{c}$ (cf. [15]; see also the caption of Fig. 1), $\gamma =0.5772$ is Euler’s constant, $U_1=U_*/\kappa$, and $\kappa =0.41$ is von Kármán’s constant. It then follows from the variational condition $\partial \hat{{\fancyscript{P}}}_0/\partial \varsigma =0$ that

$$\begin{aligned} \varsigma ^2=\displaystyle \frac{L_\varsigma ^2-2(1+\hbox {i}\hat{c}_\mathrm{i})L_\varsigma +(\hat{c}_\mathrm{i}^2+2\hbox {i}\hat{c}_\mathrm{i}+\pi ^2/6)}{L_\varsigma ^2+2(1-\hbox {i}\hat{c}_\mathrm{i})L_\varsigma +(\hat{c}_\mathrm{i}^2-2\hbox {i}\hat{c}_\mathrm{i}+\pi ^2/6)}, \end{aligned}$$

(22)

where $L_\varsigma =-(L_0+\ln \varsigma )$ and $L_0=\gamma -\ln (2\xi _\mathrm{c})=\Lambda ^{-1}$.

The corresponding CL approximation to the energy-transfer parameter $\beta$ may then be calculated from (12), which implies $\fancyscript{W}_\mathrm{c}={\fancyscript{P}}_\mathrm{c}/U_\mathrm{c}'\approx {\fancyscript{P}}_0/U_\mathrm{c}'$, and (14), which yields

$$\begin{aligned} \beta _\mathrm{c}&= \pi \xi _\mathrm{c}|\fancyscript{W}_\mathrm{c}/U_1a|^2=\pi \xi ^3_\mathrm{c}|\hat{\fancyscript{P}}_0|^2\nonumber \\&= \textstyle \frac{1}{4}\pi (\varsigma +\varsigma ^{-1})^2 \left| \left( L_\varsigma ^2-2\hbox {i}\hat{c}_\mathrm{i}L_\varsigma +\hat{c}_\mathrm{i}^2+\textstyle \frac{1}{6}\pi ^2\right) \right| ^2\nonumber \\&= \pi \xi _\mathrm{c}^3L_0^4\left[ 1+\left( 4-\textstyle \frac{1}{3}\pi ^2+10\hat{c}_\mathrm{i}^2\right) \Lambda ^2+\mathcal{O}(\Lambda ^3)\right] . \end{aligned}$$

(23)

To obtain the corresponding expression for the component of the energy-transfer parameter, $\beta _T$, due to turbulence, we multiply (9) by $-{\fancyscript{M}}$, integrating over $0<z<\infty$, invoking the conditions

$$\begin{aligned} {\fancyscript{M}}=a,\quad {\fancyscript{M}}'=ka,\quad {\fancyscript{T}}'=\hbox {i}k[{\fancyscript{P}}_0-kac^2] \end{aligned}$$

on $z=0$ and the null condition for $z\rightarrow 0$, and we obtain

$$\begin{aligned} \int \limits _0^\infty {\fancyscript{M}}{\fancyscript{T}}''\,\hbox {d}z&= ka [{\fancyscript{T}}_0-\hbox {i}{\fancyscript{P}}_0]+\hbox {i}(kac)^2+ \int \limits _0^\infty {\fancyscript{M}}''{\fancyscript{T}}\,\hbox {d}z = \hbox {i}(kac)^2+\hbox {i}k\int \limits _0^\infty {\fancyscript{V}}^2\left( {\fancyscript{M}}^{\prime ^2}+k^2{\fancyscript{M}}^2\right) \hbox {d}z, \end{aligned}$$

(24)

where ${\fancyscript{T}}_0$ is the complex amplitude of the surface shear stress and $c=c_\mathrm{r}+\hbox {i}c_\mathrm{i}$. In the limit as $s\equiv \rho _\mathrm{a}/\rho _\mathrm{w}\downarrow 0$, where $\rho _\mathrm{a}$ and $\rho _\mathrm{w}$ are densities of the air and water, respectively, we obtain

$$\begin{aligned} \alpha +\hbox {i}\beta \equiv (c^2-c_\mathrm{w}^2)/sU_1^2= (\fancyscript{P}_0+\mathrm{i}\fancyscript{T}_0)/kaU_1^2\equiv (\hat{\fancyscript{P}}_0 +\hbox {i}\hat{\fancyscript{T}}_0), \end{aligned}$$

(25)

In (25) $c$ is the complex wave speed,

$$\begin{aligned} c_\mathrm{w}=\sqrt{g/k}-2\mathrm{ik}\nu _\mathrm{w},\quad |k\nu _\mathrm{w}/c|\ll 1, \end{aligned}$$

is the speed of water waves in the absence of the airflow above it, $\nu _\mathrm{w}$ is the kinematic viscosity of water, and the suffix zero denotes evaluation at $z=0$. Then it follows from (24) and (21) that

$$\begin{aligned} \alpha _T+\hbox {i}\beta _T&= (kaU_1)^{-2}\int \limits _0^\infty \left\{ \hbox {i}\nu _\mathrm{e}\left[ {\fancyscript{V}\fancyscript{M}}^{\prime \prime ^2}+ 2U'{\fancyscript{M}}{\fancyscript{M}}''+U''{\fancyscript{M}}{\fancyscript{M}}''\right] - k{\fancyscript{V}}^2\left( {\fancyscript{M}}^{\prime ^2}+k^2{\fancyscript{M}}^2\right) \right\} \hbox {d}z. \end{aligned}$$

The preceding integral can be evaluated asymptotically,⁴ and its imaginary part yields

$$\begin{aligned} \beta _T=5\kappa ^2L_0+\mathcal{O}(\Lambda ). \end{aligned}$$

(26)

In Fig. 6, we show a comparison of the energy-transfer rate, $\beta$, between the present result for a monochromatic unsteady (growing) wave, both analytically and numerically, and those calculated by Miles [15] and Janssen [18] for the steady wave counterpart. Miles and Janssen both assumed that the drag $C_D$, and thence $\beta$, was dominated by the limiting inviscid wave growth mechanism, and thus their formulation is independent of $c_\mathrm{i}$. In contrast, the present calculation is for a viscous unsteady (growing) wave, where $c_\mathrm{i}/U_*=0.01$ and $kz_0=10^{-4}$.

Fig. 6

Total energy-transfer parameter, $\beta$, due to combined effect of sheltering and inertial critical layer for growing waves (where $c_\mathrm{i}\ll U_*$) as a function of wave age $c_\mathrm{r}/U_1$. Plus symbol Miles’ [15] calculation ($c_\mathrm{i}=0, \nu _\mathrm{e}=0$) from his formula $\beta =\pi \xi _\mathrm{c}\big \{\textstyle \frac{1}{6}\pi ^2+\log ^2(\gamma \xi _\mathrm{c})+2\sum _{n=1}^{\infty }\textstyle \frac{(-1)^n\xi _\mathrm{c}^n}{n!n^2}\big \}^2$, where $\xi _\mathrm{c}=kz_\mathrm{c}$ is the critical height $\xi _\mathrm{c}=\Omega (U_1/c_\mathrm{r})^2\hbox {e}^{c_\mathrm{r}/U_1}$ and $\Omega =gz_0/U_1^2$ is Charnock’s constant [33]. Thick solid line: parameterisation of Miles’ formula [18] for $c_\mathrm{i}=0, \nu _\mathrm{e}=0$: $\beta =1.2\kappa ^{-2}\xi _\mathrm{c}\log ^4\xi _\mathrm{c}$, where $\xi _\mathrm{c}=\min \left\{ 1,kz_0\hbox {e}^{[\kappa /(U_*/c+0.011)]}\right\}$. Thin solid line: present formulation: ($\beta _\mathrm{T}+\beta _\mathrm{c}$) for $c_\mathrm{i}\ne 0, \nu _\mathrm{e}\ne 0$. $\circ$, Numerical simulation using Reynolds-stress closure model [34] for $c_\mathrm{i}\ne 0, \nu _\mathrm{e}\ne 0$. Note that $\beta$ given in [15, 18] is equivalent to $\beta _\mathrm{c}$ in our notation

We emphasize that the various models [10, 12, 35] all generally agree with our numerical simulations performed using the Reynolds-stress closure scheme [34] for the energy transfer parameter, $\beta$, shown in Fig. 6. This shows the consistency between these models and the unimportance of very small $c_\mathrm{i}$ for which viscous processes are significant.

These parameterisations have been incorporated into and tested in spectral the wave models WaveWatch and WindWave, which yields superior results compared with field data [36, 37].

Figure 7 shows a comparison of $\beta _\mathrm{c}$ as a function of wave age $c_\mathrm{r}/U_1$, calculated from the numerical solution of the inviscid Orr–Sommerfeld equation [38], against the numerical solution of Eq. (1) for $c_\mathrm{i}/U_*=0.01, kz_0=10^{-4}$ and $\nu _\mathrm{e}\ne 0$. Increasing $c_\mathrm{i}/U_*$ from 0.01 to 0.1 (not shown here) makes no significant difference in the magnitude of $\beta _\mathrm{c}$. We conclude, therefore, for a finite value of $\nu _\mathrm{e}$, the right-hand side of Eq. (1) is dominant, and therefore the magnitude of $\beta _\mathrm{c}$, calculated from the solution of (1), is practically zero over a wide range of the wave age, in particular for a ‘young’ wave, where $c_\mathrm{r}/U_1<2.$ We thus conclude that the CL mechanism plays an insignificant role for $c_\mathrm{r}/U_1<9$ and has very little effect for $9\le c_\mathrm{r}/U_1\le 10.5$.

Fig. 7

Component of energy-transfer parameter, $\beta _\mathrm{c}$, due to inertial CL for growing waves (where $c_\mathrm{i}\ll U_*$) as a function of wave age $c_\mathrm{r}/U_1$. Filled circle: numerical solution of inviscid Orr–Sommerfeld equation [38] for $c_\mathrm{i}=0$ and $\nu _\mathrm{e}=0$ using singular CL approach; open circle: numerical solution of Eq. (1) for $c_\mathrm{i}\ne 0$ and $\nu _\mathrm{e}\ne 0$