Capturing vertical information in radially symmetric flow using hyperbolic shallow water moment equations

TL;DR

This work addresses the lack of global hyperbolicity in axisymmetric shallow water moment models by deriving axisymmetric SWME (ASWME) from cylindrical Navier–Stokes and exposing hyperbolicity breakdown through eigenvalue analysis. Building on the 1D hyperbolic regularization approach, the authors construct Hyperbolic Axisymmetric Shallow Water Moment Equations (HASWME) with a block-structured system matrix that preserves real eigenvalues, enabling stable, finite-velocity propagation. Analytical results provide explicit eigenvalue formulations, while a tailored finite-volume scheme on cylindrical grids demonstrates that HASWME reduces oscillations in dam-break scenarios and achieves convergence with increasing moment order for smooth data. These findings offer a robust intermediate step toward fully two-dimensional hyperbolic moment models, with future work focusing on equilibrium analysis and 2D extensions to broaden applicability to geophysical flows in cylindrical geometries.

Abstract

Models for shallow water flow often assume that the lateral velocity is constant over the water height. The recently derived shallow water moment equations are an extension of these standard shallow water equations. The extended models allow for vertical changes in the lateral velocity, resulting in a system that is more accurate in situations where the horizontal velocity varies considerably over the height of the fluid. Unfortunately, already the one-dimensional models lack global hyperbolicity, an important property of partial differential equations that ensures that disturbances have a finite propagation speed. In this paper we show that the loss of hyperbolicity also occurs in two-dimensional axisymmetric systems. First, a cylindrical moment model is obtained by starting from the cylindrical incompressible Navier-Stokes equations. We derive two-dimensional axisymmetric Shallow Water Moment Equations by imposing axisymmetry in the cylindrical model. The loss of hyperbolicity is observed by directly evaluating the propagation speeds and plotting the hyperbolicity region. A hyperbolic axisymmetric moment model is then obtained by modifying the system matrix in analogy to the one-dimensional case, for which the hyperbolicity problem has already been observed and overcome. The new model is written in analytical form and we prove that hyperbolicity can be guaranteed. To test the new model, numerical simulations with both discontinuous and continuous initial data are performed. We show that the hyperbolic model leads to less oscillations than the non-hyperbolic model in the case of a discontinuous initial height profile. The hyperbolic model is preferred over the non-hyperbolic model in this specific scenario for shorter times. Further, we observe that the error decreases when the order of the model is increased in a test case with smooth initial data.

Figures (7)

• Figure 1: Axisymmetric flow.
• Figure 2: Hyperbolicity region of the second order axisymmetric system (a) and the third order axisymmetric system (b). Red regions indicate loss of hyperbolicity.
• Figure 3: Graphical representation of a grid cell (a) and the boundary conditions (b).
• Figure 4: 3D visualization of radial dam break simulation using the third order HASWME at time $t=0.5$
• Figure 5: Radial dam break test case for the third order ASWME and HASWME at times $t=0.1$ and $t=0.3$. HASWME gives comparable accuracy for $h$ and $v_{r,m}$ but improves the prediction of $\alpha_1$ at $t=0.1$.

Theorems & Definitions (14)

• Remark 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof
• Theorem 3.6
• proof