Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

In this paper we prove the uniform-in-time $$L^p$$ L p convergence in the inviscid limit of a family $$\omega ^\nu $$ ω ν of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ ω of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $$\omega ^{\nu }$$ ω ν to $$\omega $$ ω in $$L^p$$ L p . Finally, we show that solutions of the Euler equations with $$L^p$$ L p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

Modeling Microbubble Microvessel Interaction for Sonoporation Application

Modeling the dynamics of microbubbles inside confined spaces has many potential applications in biomedicine, sonoporation being one classic example. Sonoporation is the permeabilization of a blood vessel’s endothelial cell membrane by acoustic waves in order to non-invasively deliver large-sized drug molecules into cells for therapeutic applications. By controlled activation of ultrasound contrast agents (UCA) in a microvessel, one can achieve better permeabilization without causing permanent damage associated with high intensity ultrasound. This paper considers numerically, the fluid-structure interactions (FSI) of UCA microbubbles with a microvessel accounting for large deformations. The modeling approach is based on a multi-material compressible flow solver that uses a Lagrangian treatment for numerical discretization of cells containing an interface between two phases and an Eulerian treatment for cells away from material interfaces. A re-mapping procedure is employed to map the Lagrangian solution back to the Eulerian grid. The model is first validated by simulating a microbubble oscillating due to an imposed ultrasound inside a microvessel and good agreement with experiments is obtained for both the bubble and vessel dynamics. The effect of vessel elasticity is then studied and it is shown that increasing the vessel elasticity damps the bubble oscillations. Then the effect of placing the bubble away from the axis of vessel is studied and it is shown that bubbles closer the vessel wall are capable of creating maximum deformation on the wall compared to those away from the wall.

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number in the carrier wave and in variation in the coefficient multiplying the nonlinear term. In the case of vorticity dependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient are constant. When the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. There are special cases (e.g., Gerstner waves) in which the vorticity is proportional to the squared wave amplitude and nonlinearity disappears, thus making the equations for wave packet dynamics linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution by simply changing the horizontal coordinates.

The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect

The nonlinear Schrödinger equation (NLS equation) describing weakly rotational wave packets in an infinity-depth fluid in the Lagrangian coordinates is derived. The vorticity is assumed to be an arbitrary function of the Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is proved that the modulation instability criteria of the low-vorticity waves and deep water potential waves coincide. All the known solutions of the NLS equation for rogue waves are applicable to the low-vorticity waves. The effect of vorticity is manifested in a shift of the wave number in the carrier wave. In case of vorticity dependence on the vertical Lagrangian coordinate only (the Gouyon waves) this shift is constant. In a more general case, where the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally heterogeneous. There is a special case with the Gerstner waves where the vorticity is proportional to the square of the wave amplitude, and the resulting non-linearity disappears, thus making the equations of the dynamics of the Gerstner wave packet linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables can be obtained from the Lagrangian solution by the ordinary change of the horizontal coordinates.

Toward a PV-Based Algorithm for the Dynamical Core of Hydrostatic Global Models

The diabatic contour-advective semi-Lagrangian (DCASL) algorithms previously constructed for the shallow-water and multilayer Boussinesq primitive equations are extended to multilayer non-Boussinesq equations on the sphere using a hybrid terrain-following–isentropic (σ–θ) vertical coordinate. It is shown that the DCASL algorithms face challenges beyond more conventional algorithms in that various types of damping, filtering, and regularization are required for computational stability, and the nonlinearity of the hydrostatic equation in the σ–θ coordinate causes convergence problems with setting up a semi-implicit time-stepping scheme to reduce computational cost. The prognostic variables are an approximation to the Rossby–Ertel potential vorticity Q, a scaled pressure thickness, the horizontal divergence, and the surface potential temperature. Results from the DCASL algorithm in two formulations of the σ–θ coordinate, differing only in the rate at which the vertical coordinate tends to θ with increasing height, are assessed using the baroclinic instability test case introduced by Jablonowski and Williamson in 2006. The assessment is based on comparisons with available reference solutions as well as results from two other algorithms derived from the DCASL algorithm: one with a semi-Lagrangian solution for Q and another with an Eulerian grid-based solution procedure with relative vorticity replacing Q as the prognostic variable. It is shown that at intermediate resolutions, results comparable to the reference solutions can be obtained.

Second-order Lagrangian description of tri-dimensional gravity wave interactions

We revisit and supplement the description of gravity waves based on perturbation expansions in Lagrangian coordinates. A general analytical framework is developed to derive a second-order Lagrangian solution to the motion of arbitrary surface gravity wave fields in a compact and vectorial form. The result is shown to be consistent with the classical second-order Eulerian expansion by Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480) and is used to improve the original derivation by Pierson (1961 Models of random seas based on the Lagrangian equations of motion. Tech. Rep. New York University) for long-crested waves. As demonstrated, the Lagrangian perturbation expansion captures nonlinearities to a higher degree than does the corresponding Eulerian expansion of the same order. At the second order, it can account for complex nonlinear phenomena such as wave-front deformation that we can relate to the initial stage of horseshoe-pattern formation and the Benjamin–Feir modulational instability to shed new light on the origins of these mechanisms.

Shared-Memory Parallelization for Two-Way Coupled Euler-Lagrange Modeling of Bubbly Flows

Cavitating bubbly flows are encountered in many engineering problems involving propellers, pumps, valves, ultrasonic biomedical applications, … etc. In this contribution an OpenMP parallelized Euler-Lagrange model of two-phase flow problems and cavitation is presented. The two-phase medium is treated as a continuum and solved on an Eulerian grid, while the discrete bubbles are tracked in a Lagrangian fashion with their dynamics computed. The intimate coupling between the two description levels is realized through the local void fraction, which is computed from the instantaneous bubble volumes and locations, and provides the continuum properties. Since, in practice, any such flows will involve large numbers of bubbles, schemes for significant speedup are needed to reduce computation times. We present here a shared-memory parallelization scheme combining domain decomposition for the continuum domain and number decomposition for the bubbles; both selected to realize maximum speed up and good load balance. The Eulerian computational domain is subdivided based on geometry into several subdomains, while for the Lagrangian computations, the bubbles are subdivided based on their indices into several subsets. The number of fluid subdomains and bubble subsets are matched with the number of CPU cores available in a share-memory system. Computation of the continuum solution and the bubble dynamics proceeds sequentially. During each computation time step, all selected OpenMP threads are first used to evolve the fluid solution, with each handling one subdomain. Upon completion, the OpenMP threads selected for the Lagrangian solution are then used to execute the bubble computations. All data exchanges are executed through the shared memory. Extra steps are taken to localize the memory access pattern to minimize non-local data fetch latency, since severe performance penalty may occur on a Non-Uniform Memory Architecture multiprocessing system where thread access to non-local memory is much slower than to local memory. This parallelization scheme is illustrated on a typical non-uniform bubbly flow problem, cloud bubble dynamics near a rigid wall driven by an imposed pressure function.