Mathematical and numerical study of a model for navigation in stratified waters

TL;DR

The paper develops a linear, two-layer fluid model for ship navigation in stratified waters, incorporating a possibly variable ship speed and Rayleigh damping. Through a spectral (Fourier) formulation, it proves global well-posedness of the Cauchy problem and analyzes both subcritical and supercritical regimes, yielding a dispersion relation and wake predictions in 1D and 2D. A Fourier-based spatial discretization combined with an exponential time integrator provides a numerically stable scheme with a proven first-order time accuracy and exact preservation of dispersion/dissipation properties; this is validated by comprehensive 1D and 2D simulations showing Wake patterns and the role of damping. The work bridges mathematical analysis and numerical methods to illuminate dead-water phenomena and wake formation in stratified fluids, offering a foundation for more realistic towing-force or traction studies.

Abstract

We derive a linear model of navigation in a two-layer fluid with a variable velocity of the ship. A spectral version of the model including a Rayleigh damping term is analyzed. We prove that the Cauchy problem has a unique solution if the velocity and if the initial data are sufficiently regular. The case of a constant speed is thoroughly investigated and the importance of a critical speed which separates two types of regimes is pointed out. We propose a numerical scheme based on the discrete Fourier transform for the space discretization and on an exponential integrator for the time discretization. We prove an error estimate for the exponential integrator. Numerical experiments in one and two space dimensions complete the theoretical results.

Figures (12)

• Figure 1: Illustration of the dead water phenomenon. $s$ represents the surface on which a boat sails at velocity $\bm{U}(t)$ over stratified water. $X(t)$ is the boat position at time $t$ and $\eta$ corresponds to the interface between two layers of fluid.
• Figure 2: Illustration of two dimensional wave propagation. The lines correspond to straight lines along which simple waves are constant. Blue ones form an angle $\varphi$ with the $x-$axis and correspond to divergent waves. Transverse waves (red lines) occur only in a subcritical configuration. In the supercritical case, divergent waves are contained within a cone with angle $\varphi^{\star}$ (green line).
• Figure 3: Supercritical case $U_x>U_c$ using $\rho_1 = 999~kg \cdot m^{-3}$, $\rho_2 = 1022.3~kg \cdot m^{-3}$, $h_1 = 1~m$, $h_2 = 6~m$ and $g = 9.81~m \cdot s^{-2}$. Left: Domain $\mathcal{D}$, the line of singularities is the blue curve while the green dashed lines define the limit angle $\varphi^{\star}$. Right : curve $r \mapsto \varphi(r)$ is the blue line which is bounded by $0$ and $\varphi^{\star} \approx 0.5470$.
• Figure 4: Subcritical case $U_x<U_c$ using $\rho_1 = 999~kg \cdot m^{-3}$, $\rho_2 = 1022.3~kg \cdot m^{-3}$, $h_1 = 1~m$, $h_2 = 6~m$ and $g = 9.81~m \cdot s^{-2}$. Left: Domain $\mathcal{D}$ is plotted with the blue curve, the red cross corresponds to the transverse wave. Right : the curve $r \mapsto \varphi(r)$ corresponds to the blue line and the red cross to the transverse wave with $r^{\star} \approx 2.8589 \cdot 10^{-2}$ that depends on the chosen physical parameters.
• Figure 5: Example \ref{['ex:converence']}. Relative $\ell^2$ error $\tfrac{\| \eta_{\varepsilon}(t^N) - \eta_{\varepsilon}^N \|_{\ell^2}}{\| \eta_{\varepsilon}(t^N) \|_{\ell^2}}$ between the numerical and analytical solutions for various values of $\varepsilon \in \{ 10^{-12}, 10^{-4}, 10^{-1} \}$ in the subcritical regime.

Theorems & Definitions (31)

• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 3
• proof
• Definition 4
• Remark 5
• Remark 6
• Remark 7