Moment-enhanced shallow water equations for non-slip boundary conditions

Abstract

The shallow water equations often assume a constant velocity profile along the vertical axis. However, this assumption does not hold in many practical applications. To better approximate the vertical velocity distribution, models such as the shallow water moment expansion models have been proposed. Nevertheless, under non-slip bottom boundary conditions, both the standard shallow water equation and its moment-enhanced models struggle to accurately capture the vertical velocity profile due to the stiff source terms. In this work, we propose modified shallow water equations and corresponding moment-enhanced models that perform well under both non-slip and slip boundary conditions. The primary difference between the modified and original models lies in the treatment of the source term, which allows our modified moment expansion models to be readily generalized, while maintaining compatibility with our previous analysis on the hyperbolicity of the model. To assess the performance of both the standard and modified moment expansion models, we conduct a comprehensive numerical comparison with the incompressible Navier--Stokes equations -- a comparison that is absent from existing literature.

Figures (17)

• Figure 1: Simulation of the 2D dam-break problem with OpenFOAM: Comparison of water height $h$ for different vertical resolutions $N_z=160, 320,$ and 640.
• Figure 2: Simulation of the 2D dam-break problem with OpenFOAM: Comparison of depth-averaged velocity $u_m$ along $x$-axis for different vertical resolutions $N_z=160, 320,$ and 640.
• Figure 3: Simulation of the 2D dam-break problem: Comparison of water height $h$. Panels (a)-(d) show the water height at times $t=0,1,2,$ and 3. (1) Black solid line denotes the incompressible Navier--Stokes (OpenFOAM); (2) Gray solid line denotes the SWE; (3) blue solid line denotes the HSWME with $N=1$; (4) purple solid line denotes the HSWME with $N=2$; dimensionless models we derived: (5) red dashed line denotes SWE; (6) Orange dash-dot line denotes the HSWME with $N=1$; (7) yellow-green dotted line denotes the HSWME with $N=2$. The results show that, over time, the OpenFOAM simulation produces a sharp water front that propagates to the right. In the region where the water level drops, the water surface exhibits oscillatory behavior. In contrast, the water height predicted by the SWE model changes very slowly over time, with no significant variation observed between $t=0$ and $t=3$. The remaining models provide smooth approximations of the water height in the region of water drops.
• Figure 4: Simulation of the 2D dam-break problem: Comparison of depth-averaged velocity $u_m$ along $x$-axis. Panels (a)-(d) show the mean velocity $u_m$ at times $t=0,1,2,$ and 3. (1) Black solid line denotes the incompressible Navier--Stokes (OpenFOAM); (2) Gray solid line denotes the SWE; (3) blue solid line denotes the HSWME with $N=1$; (4) purple solid line denotes the HSWME with $N=2$; dimensionless models we derived: (5) red dashed line denotes SWE; (6) Orange dash-dot line denotes the HSWME with $N=1$; (7) yellow-green dotted line denotes the HSWME with $N=2$. The OpenFOAM results show that during the water drop, the water front and the drop region exhibit the highest velocity, while the velocity in the intermediate region gradually oscillates and stabilizes. The averaged velocity produced by the SWE model remains nearly zero, whereas the other models provide good approximations of the averaged velocity.
• Figure 5: Simulation of the 2D dam-break problem: Comparison of vertical velocity at $x=55$. Panels (a)-(d) show the vertical velocity of $u$ at times $t=0,1,2$, and 3. (1) Black solid line denotes the incompressible Navier--Stokes (OpenFOAM); (2) Gray solid line denotes the SWE; (3) blue solid line denotes the HSWME with $N=1$; (4) purple solid line denotes the HSWME with $N=2$; dimensionless models we derived: (5) red dashed line denotes SWE; (6) Orange dash-dot line denotes the HSWME with $N=1$; (7) yellow-green dotted line denotes the HSWME with $N=2$. The OpenFOAM results show that the vertical velocity exhibits a linear profile between the bottom and the water surface. The SWE, HSWME with $N=1$ and $N=2$ provide constant, linear, and quadratic approximations of the vertical velocity, respectively, all starting at the point $(0,0)$. The MSWE offers a constant approximation centered around the mid-depth value of the OpenFOAM profile, while the MHSWME with $N=1$ and $N=2$ deliver accurate linear approximations of the vertical velocity.

Theorems & Definitions (5)

• Theorem 1: hyperbolicity of the HSWME
• Remark 1
• Example 4.1: 2D dam-break problem
• Example 4.2: 3D radial water collapse
• Example 4.3: 3D water collapse