scholarly journals A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 750-786
Author(s):  
Samer Israwi ◽  
Henrik Kalisch ◽  
Theodoros Katsaounis ◽  
Dimitrios Mitsotakis
Keyword(s):  
Boundary Conditions ◽  
Water Waves ◽  
Mathematical Proof ◽  
Boussinesq System ◽  
Dispersive Waves ◽  
Wall Boundary ◽  
Wall Boundary Conditions ◽  
Boussinesq Systems ◽  
Well Posed ◽  
New System

Abstract The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

scholarly journals A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis

2020 ◽  
Author(s):  
Samer Israwi ◽  
Henrik Kalisch ◽  
Theodoros Katsaounis ◽  
Dimitrios Mitsotakis
Keyword(s):  
Boundary Conditions ◽  
Water Waves ◽  
Mathematical Proof ◽  
Boussinesq System ◽  
Dispersive Waves ◽  
Wall Boundary ◽  
Wall Boundary Conditions ◽  
Boussinesq Systems ◽  
Well Posed ◽  
New System

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche's method. Convergence of the numerical method is analyzed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

scholarly journals A regularized shallow-water waves system with slip-wall boundary conditions in a basin: Theory and numerical analysis

2020 ◽  
Author(s):  
Samer Israwi ◽  
Henrik Kalisch ◽  
Theodoros Katsaounis ◽  
Dimitrios Mitsotakis
Keyword(s):  
Boundary Conditions ◽  
Water Waves ◽  
Mathematical Proof ◽  
Boussinesq System ◽  
Dispersive Waves ◽  
Wall Boundary ◽  
Wall Boundary Conditions ◽  
Boussinesq Systems ◽  
Well Posed ◽  
New System

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche's method. Convergence of the numerical method is analyzed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

Extension of classical stability theory to viscous planar wall-bounded shear flows

2019 ◽  
Vol 877 ◽  
pp. 1134-1162 ◽  
Author(s):  
Harry Lee ◽  
Shixiao Wang
Keyword(s):  
Boundary Conditions ◽  
Shear Flows ◽  
Early Stage ◽  
Linear Instability ◽  
Slip Condition ◽  
Translation Invariant ◽  
Alternative Derivation ◽  
Wall Boundary ◽  
Wall Boundary Conditions ◽  
Parallel Shear Flows

A viscous extension of Arnold’s inviscid theory for planar parallel non-inflectional shear flows is developed and a viscous Arnold’s identity is obtained. Special forms of the viscous Arnold’s identity have been revealed that are closely related to the perturbation’s enstrophy identity derived by Synge (Proceedings of the Fifth International Congress for Applied Mechanics, 1938, pp. 326–332, John Wiley) (see also Fraternale et al., Phys. Rev. E, vol. 97, 2018, 063102). Firstly, an alternative derivation of the perturbation’s enstrophy identity for strictly parallel shear flows is acquired based on the viscous Arnold’s identity. The alternative derivation induces a weight function. Thereby, a novel weighted perturbation’s enstrophy identity is established, which extends the previously known enstrophy identity to include general streamwise translation-invariant shear flows. Finally, the validity of the enstrophy identity for parallel shear flows is rigorously examined and established under global nonlinear dynamics imposed with two classes of wall boundary conditions. As an application of the enstrophy identity, we quantitatively investigate the mechanism of linear instability/stability within the normal modal framework. The investigation reveals a subtle interaction between a critical layer and its adjacent boundary layer, which determines the stability nature of the disturbance. As an implementation of the relaxed wall boundary conditions imposed for the enstrophy identity, a control scheme is proposed that transitions the wall settings from the no-slip condition to the free-slip condition, through which a flow is stabilized quickly in an early stage of the transition.

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