Capturing Vertical Information in Radially Symmetric Flow Using Hyperbolic Shallow Water Moment Equations

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 a vertically changing velocity profile, resulting in more accuracy when the 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, cylindrical shallow water moment equations are formulated by starting from the cylindrical incompressible Navier-Stokes equations. We formulate two-dimensional axisymmetric Shallow Water Moment Equations by imposing axisymmetry in the cylindrical model. The loss of hyperbolicity is analyzed and a hyperbolic axisymmetric moment model is then derived by modifying the system matrix in analogy to the one-dimensional case, for which the hyperbolicity problem has already been observed and overcome. Numerical simulations with both discontinuous and continuous initial data in a cylindrical domain are performed using a finite volume scheme tailored to the cylindrical mesh. The newly derived hyperbolic model is clearly beneficial as it gives more stable solutions and still converges to the reference solution when increasing the number of moments.