Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows

Published versio

Stabilized mixed approximation of axisymmetric Brinkman flows

This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order k ≥ 0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k, respectively. The well-posedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuška–Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions

Institute for Computational Mechanics in Propulsion (ICOMP)

The Institute for Computational Mechanics in Propulsion (ICOMP) is a combined activity of Case Western Reserve University, Ohio Aerospace Institute (OAI) and NASA Lewis. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1991 are described

Validation of a Discontinuous Galerkin Implementation of the Time-Domain Linearized Navier–Stokes Equations for Aeroacoustics

The propagation of small perturbations in complex geometries can involve hydrodynamic-acoustic interactions, coupling acoustic waves and vortical modes. A propagation model, based on the linearized Navier–Stokes equations, is proposed. It includes the mechanism responsible for the generation of vorticity associated with the hydrodynamic modes. The linearized Navier–Stokes equations are discretized in space using a discontinuous Galerkin formulation for unstructured grids. Explicit time integration and non-reflecting boundary conditions are described. The linearized Navier–Stokes (LNS) model is applied to two test cases. The first one is the time-harmonic source line in an incompressible inviscid two-dimensional mean shear flow in an infinite domain. It is shown that the proposed model is able to capture the trailing vorticity field developing behind the mass source and to represent the redistribution of the vorticity. The second test case deals with the analysis of the acoustic propagation of an incoming perturbation inside a circular duct with a sudden area expansion in the presence of a mean flow and the evaluation of its scattering matrix. The computed coefficients of the scattering matrix are compared to experimental data for three different Mach numbers of the mean flow, M0 = 0.08, 0.19 and 0.29. The good agreement with the experimental data shows that the proposed method is suitable for characterizing the acoustic behavior of this kind of network

Fourier spectral computation of geometrically confined two-dimensional flows

Large-scale flow phenomena in the atmosphere and the oceans are predominantly two-dimensional (2D) due to the large aspect ratio of the typical horizontal and vertical length scales in the flow. The 2D nature of large-scale geophysical flows motivates the use of a conceptual approach known as "2D turbulence". It usually involves the (forced/damped) Navier-Stokes equations on a square domain with periodic boundaries or on a spherical surface. This setup may be useful for numerical studies of atmospheric flow. For the oceans, on the other hand, geometrical confinement due to the continental shelves is of crucial importance. The physically most relevant boundary condition for oceanographic flow is probably the no-slip condition. Previous numerical and experimental studies have shown that confinement by no-slip boundaries dramatically affects the dynamics of (quasi-)2D turbulence due to its role as vorticity source. An important process is the detachment of high-amplitude vorticity filaments from the no-slip sidewalls that subsequently affect the internal flow. The first part of the thesis concerns the development and extensive testing of a Fourier spectral scheme for 2D Navier-Stokes flow in domains bounded by rigid noslip walls. An advantage of Fourier methods is that higher-order accuracy can, in principle, be achieved. Moreover, these methods are fast, relatively easy to implement even for performing parallel computations. The no-slip boundary condition is enforced by using an immersed boundary technique called "volume-penalization". In this method an obstacle with no-slip boundaries is modelled as a porous medium with a small permeability. It has recently been shown that in the limit of infinitely small permeability the solution of the penalized Navier-Stokes equations converges towards the solution of the Navier-Stokes equations with no-slip boundaries. Therefore the penalization error can be controlled with an arbitrary parameter. A possible drawback is that the sharp transition between the fluid and the porous medium can trigger Gibbs oscillations that might deteriorate the stability and accuracy of the scheme. Using a very challenging dipole-wall collision as a benchmark problem, it is, however, shown that higher-order accuracy is retrieved by using a novel 159 post-processing procedure to remove the Gibbs effect. The second topic of the thesis is the dynamics of geometrically confined 2D turbulent flows. The role of the geometry on the flow development has been studied extensively. For this purpose high resolution Fourier spectral simulations have been conducted where different geometries are implemented by using the volumepenalization method. A quantity that is of particular importance on a bounded domain is the angular momentum. On a circular domain production of angular momentum is virtually absent. Therefore the amount of angular momentum carried by the initial flow has important consequences for the evolution of the flow. The results of the simulations are consistent with previous numerical and experimental work on this topic performed in a lower Reynolds number regime. The typical vortex structures of the late time evolution of the flow are explained by means of a minimum enstrophy principle and the presence of weak viscous dissipation. For an elliptic geometry it is shown that strong spin-up events of the flow occur even for small eccentricities. The spin-up phenomenon can be related to the role of the pressure along the boundary of the domain. It is found that the magnitude of the torque exerted on the internal fluid can be scaled with the eccentricity. Furthermore, it is observed that angular momentum production in a non circular geometry is not restricted to moderate Reynolds numbers. Significantly higher Reynolds number flow computations in a square geometry clearly reveal strong and rapid spin-up of the flow. Finally the scale-dependence of the vorticity and velocity statistics in forced 2D turbulence on a bounded domain has been studied. A challenging aspect is that a statistically steady state can be achieved by a balance between the injection of kinetic energy by the external forcing and energy dissipation at the no-slip sidewalls. It is important to note that on a double periodic domain a steady state is usually achieved by introducing volumetric drag forces. Several studies reported that this strongly affects the spatial scaling behaviour of the flow. Therefore it is very interesting to quantify the small-scale statistics in the bulk of statistically steady flow on a domain with no-slip boundaries in the absence of bottom drag. It is observed that the internal flow shows extended self-similar, locally homogeneous and isotropic scaling behaviour at small scales. It is further demonstrated that a direct enstrophy cascade develops in the interior of the flow domain. Some deviations from the classical scaling theory of 2D turbulence developed independently by Kraichnan, Batchelor and Leith may be associated to the presence of coherent structures in the flow. It is, however, anticipated that higher-resolution simulations are required in order to draw more decisive conclusions. The parallel Fourier spectral scheme with volume-penalization is very suitable for pursuing such simulations on high performance machines in the near future. In summary the thesis contributes to both the development of numerical techniques and understanding of wall-bounded two-dimensional flows. The Fourier spectral scheme with volume-penalization is found very suitable for pursuing direct numerical simulations in complex geometries. The high-resolution simulations considered in the thesis clearly reveal that spontaneous production of angular momentum due to interaction with non-circular domain boundaries is present for significantly higher Reynolds numbers than considered previously

Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations

Whether the 3D incompressible Euler and Navier–Stokes equations can develop a finite-time singularity from smooth initial data with finite energy has been one of the most long-standing open questions. We review some recent theoretical and computational studies which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. We also investigate the dynamic stability of the 3D Navier–Stokes equations and the stabilizing effect of convection. A unique feature of our approach is the interplay between computation and analysis. Guided by our local non-blow-up theory, we have performed large-scale computations of the 3D Euler equations using a novel pseudo-spectral method on some of the most promising blow-up candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. Moreover, we observe that the support of maximum vorticity becomes severely flattened as the maximum vorticity increases and the direction of the vortex filaments near the support of maximum vorticity is very regular. Our numerical observations in turn provide valuable insight, which leads to further theoretical breakthrough. Finally, we present a new class of solutions for the 3D Euler and Navier–Stokes equations, which exhibit very interesting dynamic growth properties. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions

Transitional cylindrical swirling flow in presence of a flat free surface

This article is devoted to the study of an incompressible viscous flow of a fluid partly enclosed in a cylindrical container with an open top surface and driven by the constant rotation of the bottom wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry and of the prescribed boundary conditions of Dirichlet type -- no-slip on the cavity walls. The top surface of the cylindrical cavity is left open with an imposed stress-free boundary condition, while a no-slip condition with a prescribed rotational velocity is imposed on the bottom wall. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new flow states that were investigated based on a fully three-dimensional solution of the Navier--Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a Legendre spectral element method.Comment: Computers and Fluids, In Pres