Viscous flows in corner regions: Singularities and hidden eigensolutions

Numerical issues arising in computations of viscous flows in corners formed by a liquid-fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the `moving contact-line problem' and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite-element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh-dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh-independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite element method in the context of the present problem are discussed.Comment: Submitted to the International Journal for Numerical Methods in Fluid

Variational multiscale error estimators for the adaptive mesh refinement of compressible flow simulations

This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. The proposed methodology employs the Variational Multi-Scale framework, and specifically, the idea is to use the variational subscales to estimate the error. These subscales are defined to be orthogonal to the finite element space, dynamic and non-linear, and both the subscales in the interior of the element and on the element boundaries are considered. Another particularity of the model is that we define some norms that lead to a dimensionally consistent measure of the compressible flow solution error inside each element; a scaled -norm, and the calculation of a physical entropy measure, are both studied in this work. The estimation of the error is used to drive the adaptive mesh refinement of several compressible flow simulations. Numerical results demonstrate good accuracy of the local error estimate and the ability to drive the adaptative mesh refinement to minimize the error through the computational domain.Peer ReviewedPostprint (author's final draft

GMRES physics‐based preconditioner for all Reynolds and Mach numbers: numerical examples

This paper presents several numerical results using a vectorized version of a 3D finite element compressible and nearly incompressible Euler and Navier–Stokes code. The assumptions were set on laminar flows and Newtonian fluids. The goal of this research is to show the capabilities of the present code to treat a wide range of problems appearing in laminar fluid dynamics towards the unification from incompressible to compressible and from inviscid to viscous flow codes. Several authors with different approaches have tried to attain this target in CFD with relative success. At the beginning the methods based on operator splitting and perturbation were preferred, but lately, with the wide usage of time‐marching algorithms, the preconditioning mass matrix (PMM) has become very popular. With this kind of relaxation scheme it is possible to accelerate the rate of convergence to steady state solutions with the modification of the mass matrix under certain restrictions. The selection of the mass matrix is not an easy task, but we have certain freedom to define it in order to improve the condition number of the system. In this paper we have used a physics‐based preconditioner for the GMRES implicit solver developed previously by us and an SUPG formulation for the semidiscretization of the spatial operator. In sections 2 and 3 we present some theoretical aspects related to the physical problem and the mathematical model, showing the inviscid and viscous flow equations to be solved and the variational formulation involved in the finite element analysis. Section 4 deals with the numerical solution of non‐linear systems of equations, with some emphasis on the preconditioned matrix‐free GMRES solver. Section 5 shows how boundary conditions were treated for both Euler and Navier–Stokes problems. Section 6 contains some aspects about vectorization on the Cray C90. The performance reached by this implementation is close to 1 Gflop using multitasking. Section 7 presents several numerical examples for both models covering a wide range of interesting problems, such as inviscid low subsonic, transonic and supersonic regimes and viscous problems with interaction between boundary layers and shock waves in either attached or separated flows

Boundary element solution of Poisson\u27s equations in axisymmetric laminar flows

The primitive variable Navier-Stokes equations may be replaced by two equations using the derived variable of vorticity. These equations model separately the kinematic and kinetic parts of the problem. Two boundary element solutions for the kinematic equations were developed for axisymmetric flow geometries. The first was based on the fluid mechanics analogy of the Biot and Savart formula for the magnetic effects of a current. The second was the solution of the vector Poisson\u27s velocity equation using the direct boundary element equation. Numerical integration algorithms were developed which were used for all integrals;Integral solutions for Poisson\u27s pressure equation and Poisson\u27s vector potential equation were derived using the direct boundary element equation. The equations were integrated using the algorithms developed for the velocity solutions;The axisymmetric laminar Navier-Stokes solution was completed by solving the kinetic vorticity transport equation with finite difference methods. Two finite difference methods developed for the complete 2 dimensional non-linear Burger\u27s equation were modified for use on the axisymmetric form of the vorticity transport equation;This complete Navier-Stokes solution was then used to verify the form of the six boundary element equations and the accuracy of the integration algorithm developed. This was done by solving three steady state flow problems and one time dependent flow problem which were designed to simulate flow in power hydraulic components;Flow problems were encountered which produced ill-conditioned kinematic systems with attendant unstable solutions and large errors. Solution algorithms were developed which stabilized the associated matrix operator and improved solution performance. The method is based on the theory and numerical methods of Tikhonov regularization as it applies to linear algebraic systems of equations

Summary of research in progress at ICASE

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period October 1, 1992 through March 31, 1993

Adaptive mesh simulations of compressible flows using stabilized formulations

This thesis investigates numerical methods that approximate the solution of compressible flow equations. The first part of the thesis is committed to studying the Variational Multi-Scale (VMS) finite element approximation of several compressible flow equations. In particular, the one-dimensional Burgers equation in the Fourier space, and the compressible Navier-Stokes equations written in both conservative and primitive variables are considered. The approximations made for the VMS formulation are extensively researched; the design of the matrix of stabilization parameters, the definition of the space where the subscales live, the inclusion of the temporal derivatives of the subscales, and the non-linear tracking of the subscales are formulated. Also, the addition of local artificial diffusion in the form of shock capturing techniques is included. The accuracy of the formulations is studied for several regimes of the compressible flow, from aeroacoustic flows at low Mach numbers to supersonic shocks. The second part of the thesis is devoted to make the solution of the smallest fluctuating scales of the compressible flow affordable. To this end, a novel algorithm for $h-$refinement of computational physics meshes in a distributed parallel setting, together with the solution of some refinement test cases in supercomputers are presented. The definition of an explicit a-posteriori error estimator that can be used in the adaptive mesh refinement simulations of compressible flows is also developed; the proposed methodology employs the variational subscales as a local error estimate that drives the mesh refinement. The numerical methods proposed in this thesis are capable to describe the high-frequency fluctuations of compressible flows, especially, the ones corresponding to complex aeroacoustic applications. Precisely, the direct simulation of the fricative [s] sound inside a realistic geometry of the human vocal tract is achieved at the end of the thesis.Esta tesis investiga métodos numéricos que aproximan la solución de las ecuaciones de flujo compresible. La primera parte de la tesis está dedicada al estudio de la aproximación numérica del flujo compresible por medio del método multiescala variacional (VMS) en elementos finitos. En particular, se consideran la ecuación de Burgers unidimensional descrita en el espacio de Fourier y las ecuaciones de Navier-Stokes de flujo compresible escritas en variables conservativas y primitivas. Las aproximaciones hechas para plantear la formulación VMS son ampliamente investigadas; el diseño de la matriz de parámetros de estabilización, la definición del espacio donde viven las subescalas, la inclusión de las derivadas temporales de las subescalas y el seguimiento no lineal de las subescalas son particularidades de la formulación que se analizan para cada una de las ecuaciones consideradas. Además, se incluye la adición de difusión artificial local en forma de técnicas de captura de choque. La precisión de las formulaciones se estudia para varios regímenes del flujo compresible, desde flujos aeroacústicos a bajos números de Mach hasta choques supersónicos. La segunda parte de la tesis está dedicada a hacer asequible la solución de las escalas fluctuantes más pequeñas del flujo compresible. Con este fin, se presenta un algoritmo novedoso para el refinamiento $h$ de las mallas de física computacional usadas en computación distribuida en paralelo. Además, se demuestra la solución en superordenadores de algunos casos de prueba del refinamiento de mallas. También se desarrolla la definición de un estimador de error explícito a posteriori que se puede usar en las simulaciones adaptativas de refinamiento de malla de flujos compresibles; la metodología propuesta emplea las subescalas variacionales como una estimación de error local que induce el refinamiento de la malla. Los métodos numéricos propuestos en esta tesis son capaces de describir las fluctuaciones de alta frecuencia de los flujos compresibles, especialmente los correspondientes a aplicaciones aeroacústicas complejas. Precisamente, la simulación directa del sonido consonántico fricativo [s] dentro de una geometría realista del tracto vocal humano se demuestra al final de la tesis

Innovative Approaches to the Numerical Approximation of PDEs

This workshop was about the numerical solution of PDEs for which classical approaches, such as the finite element method, are not well suited or need further (theoretical) underpinnings. A prominent example of PDEs for which classical methods are not well suited are PDEs posed in high space dimensions. New results on low rank tensor approximation for those problems were presented. Other presentations dealt with regularity of PDEs, the numerical solution of PDEs on surfaces, PDEs of fractional order, numerical solvers for PDEs that converge with exponential rates, and the application of deep neural networks for solving PDEs

Link-wise Artificial Compressibility Method

The Artificial Compressibility Method (ACM) for the incompressible Navier-Stokes equations is (link-wise) reformulated (referred to as LW-ACM) by a finite set of discrete directions (links) on a regular Cartesian mesh, in analogy with the Lattice Boltzmann Method (LBM). The main advantage is the possibility of exploiting well established technologies originally developed for LBM and classical computational fluid dynamics, with special emphasis on finite differences (at least in the present paper), at the cost of minor changes. For instance, wall boundaries not aligned with the background Cartesian mesh can be taken into account by tracing the intersections of each link with the wall (analogously to LBM technology). LW-ACM requires no high-order moments beyond hydrodynamics (often referred to as ghost moments) and no kinetic expansion. Like finite difference schemes, only standard Taylor expansion is needed for analyzing consistency. Preliminary efforts towards optimal implementations have shown that LW-ACM is capable of similar computational speed as optimized (BGK-) LBM. In addition, the memory demand is significantly smaller than (BGK-) LBM. Importantly, with an efficient implementation, this algorithm may be one of the few which is compute-bound and not memory-bound. Two- and three-dimensional benchmarks are investigated, and an extensive comparative study between the present approach and state of the art methods from the literature is carried out. Numerical evidences suggest that LW-ACM represents an excellent alternative in terms of simplicity, stability and accuracy.Comment: 62 pages, 20 figure