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Register a new userNumerical Algorithms for Water Waves with Background Flow over Obstacles and Topography
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We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface. We prove that the resulting second-kind Fredholm integral equations are invertible. In the velocity potential formulation, invertibility is achieved after a physically motivated finite-rank correction. The integral equations for the two methods are closely related, one being the adjoint of the other after modifying it to evaluate the layer potentials on the opposite side of each interface. In addition to a background flow, both formulations allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. The proposed boundary integral methods are compatible with graph-based or angle-arclength parameterizations of the free surface. In the latter case, we show how to avoid curve reconstruction errors in interior Runge-Kutta stages due to incompatibility of the angle-arclength representation with spatial periodicity. The proposed methods are used to study gravity-capillary waves generated by flow over three elliptical obstacles with different choices of the circulation parameters. In each case, the free surface forms a structure resembling a Crapper wave that narrows and eventually self intersects in a splash singularity.
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We conduct depth-resolved three-dimensional Direct Numerical Simulations (DNS) of bi-disperse turbidity currents interacting with complex bottom topography in the form of a Gaussian bump. Several flow characteristics such as suspended particle mass, instantaneous wall shear stress, transient deposit height are shown via videos. Furthermore, we investigate the influence of the obstacle on the vortical structure and sedimentation of particles by comparing the results against the same setup and but with a flat bottom surface. We observe that the obstacle influences the deposition of coarse particles mainly in the vicinity of the obstacle due to lateral deflection, whereas for the sedimentation of fine particles the effects of topographical features are felt further downstream. The results shown in this fluid dynamics video help us develop a fundamental understanding of the dynamics of turbidity currents interacting with complex seafloor topography.
We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose-Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum $n(omega)$ at the zero frequency $omega$. The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at $omega=0$ and a power-law asymptotic $n(omega) to omega^{-x}$ at $omega to infty ;; xin mathbb{R}^+$. Finding it amounts to solving a nonlinear eigenvalue problem, i.e. finding the value $x^*$ of the exponent $x$ for which these two boundary conditions can be satisfied simultaneously. To solve this problem we develop a new high-precision algorithm based on Chebyshev approximations and double exponential formulas for evaluating the collision integral, as well as the iterative techniques for solving the integro-differential equation for the self-similar shape function. This procedures allow to achieve a solution with accuracy $approx 4.7 %$ which is realized for $x^* approx 1.22$.
We investigate the behaviour of a system where a single phase fluid domain is coupled to a biphasic poroelastic domain. The fluid domain consists of an incompressible Newtonian viscous fluid while the poroelastic domain consists of a linear elastic solid filled with the same viscous fluid. The properties of the poroelastic domain, i.e. permeability and elastic parameters, depend on the inhomogeneous initial porosity field. The theoretical framework highlights how the heterogeneous material properties enter the linearised governing equations for the poroelastic domain. To couple flows through this domain with a surrounding Stokes flow, we show case a numerical implementation based on a new mixed formulation where the equations in the poroelastic domain are rewritten in terms of three fields: displacement, fluid pressure and total pressure. Coupling single phase and multiphase flow problems are ubiquitous in many industrial and biological applications, and here we consider an example from in-vitro tissue engineering. We consider a perfusion system, where a flow is forced to pass from the single phase fluid to the biphasic poroelastic domain. We focus on a simplified two dimensional geometry with small aspect ratio, and perform an asymptotic analysis to derive analytical solutions for the displacement, the pressure and the velocity fields. Our analysis advances the quantitative understanding of the role of heterogeneous material properties of a poroelastic domain on its mechanics when coupled with a fluid domain. Specifically, (i) the analytical analysis gives closed form relations that can be directly used in the design of slender perfusion systems; (ii) the numerical method is validated by comparing its result against selected theoretical solutions, opening towards the possibility to investigate more complex geometrical configurations.
The regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations. In a recent work, the first two authors proposed a so-called Hamiltonian regularisation for nonlinear shallow water and isentropic Euler equations. The characteristic property of this method is that the regularisation of solutions is achieved without adding any artificial dissipation or ispersion. The regularised system possesses a Hamiltonian structure and, thus, formally preserves the corresponding energy functional. In the present article we generalise this approach to shallow water waves over general, possibly time-dependent, bottoms. The proposed system is solved numerically with continuous Galerkin method and its solutions are compared with the analogous solutions of the classical shallow water and dispersive Serre-Green-Naghdi equations. The numerical results confirm the absence of dispersive and dissipative effects in presence of bathymetry variations.
An electrohydrodynamic (EHD) flow in a point-to-ring corona configuration is investigated experimentally, analytically and via a multiphysics numerical model. The interaction between the accelerated ions and the neutral gas molecules is modeled as an external body force in the Navier-Stokes equation (NSE). The gas flow characteristics are solved from conservation principles with spectral methods. The analytical and numerical simulation results are compared against experimental measurements of the cathode voltage, ion concentration, and velocity profiles. A nondimensional parameter, X, is formulated as the ratio of the local electric force to the inertial term in the NSE. In the region of X > 1, the electric force dominates the flow dynamics, while in the X << 1 region, the balance of viscous and inertial terms yields traditional pipe flow characteristics.
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