Pressure beneath a periodic travelling water-wave in constant-vorticity flow over a flat bed

Abstract

We investigate within the framework of linear theory the behaviour of the total (hydrodynamic) pressure and of the dynamic pressure in a regular wave train which propagates at the surface of water with a flat bed in a flow with constant vorticity. We show that nonzero vorticity, the hallmark of a non-uniform underlying current, may strongly alter the behaviour with respect to the case of irrotational flows, for which the maximum and minimum of the dynamic pressure always occur at the wave crest and at the wave trough, respectively (the extrema of the dynamic pressure may occur along the flat bed or along the critical level, depending on the vorticity strength). While vorticity does not modify the increase of the hydrodynamic pressure with depth, it can significantly alter the location of the extrema of the hydrodynamic pressure at a fixed depth level.

Figures (8)

• Figure 1: Depiction of a wave propagating to the right with velocity $c>0$ and an underlying current with velocity $\Omega(Y+d)$ vanishing at the flat bed $Y=-d$. Left: $\Omega>0$. Right: $\Omega<0$.
• Figure 2: Representation of $\hat{c}-\omega(1+y)$, $y \in [-1,0]$. Left: unidirectional flow for $\hat{c} > \omega>0$. Middle: unidirectional flow for $\hat{c} >0 >\omega$. Right: flow-reversal for $0< \hat{c} < \omega$.
• Figure 3: Illustration of the wave speed $\hat{c}(\omega)$ (in blue) and of the depth $(\hat{c}(\omega)-\omega)/\omega$ of the critical layer (in red) obtained from the dispersion relation \ref{['eqfc2']} for $2\pi \delta=1$ and $\sqrt{gd}=1$. Left: sign $-$. Right: sign $+$.
• Figure 4: In a non-dimensional reference frame moving at the wave speed $\hat{c}$, the linear waves $h(x)=A\sin(2\pi x)$ with principal period $1$ are steady sinusoidal oscillations of the flat free surface $y=0$, with the wave crest/trough at $x=\pm\tfrac{1}{4}$. The necessary and sufficient condition for the existence of a critical line $y=\frac{\hat{c}-\omega}{\omega}$, where the reversal of the underlying mean flow occurs, is \ref{['frl']}.
• Figure 5: The monotonicity of the dynamic pressure in a periodicity box beneath an irrotational wave (linear theory), in accordance to the relations in \ref{['derpzx']}-\ref{['derpzyl']}. The maximum/minimum of the dynamic pressure $\hat{p}$ are attained at the wave crest/trough.