A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3,700 are used to verify the accuracy and physical fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational Physic

Computation of Steady Incompressible Flows in Unbounded Domains

In this study we revisit the problem of computing steady Navier-Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction-vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background "skeleton" field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favourably against known theoretical predictions and the data available in the literature.Comment: 39 pages, 12 figures, accepted for publication in "Computers and Fluids

Spectral/hp elements in fluid structure interaction

summary:This work presents simulations of incompressible fluid flow interacting with a moving rigid body. A numerical algorithm for incompressible Navier-Stokes equations in a general coordinate system is applied to two types of body motion, prescribed and flow-induced. Discretization in spatial coordinates is based on the spectral/hp element method. Specific techniques of stabilisation, mesh design and approximation quality estimates are described and compared. Presented data show performance of the solver for various geometries in 2D from slowly moving cylinder to high speed flow around aerodynamic profiles

A fast lattice Green's function method for solving viscous incompressible flows on unbounded domains

A computationally efficient method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. The method formally discretizes the incompressible Navier–Stokes equations on an unbounded staggered Cartesian grid. Operations are limited to a finite computational domain through a lattice Green's function technique. This technique obtains solutions to inhomogeneous difference equations through the discrete convolution of source terms with the fundamental solutions of the discrete operators. The differential algebraic equations describing the temporal evolution of the discrete momentum equation and incompressibility constraint are numerically solved by combining an integrating factor technique for the viscous term and a half-explicit Runge–Kutta scheme for the convective term. A projection method that exploits the mimetic and commutativity properties of the discrete operators is used to efficiently solve the system of equations that arises in each stage of the time integration scheme. Linear complexity, fast computation rates, and parallel scalability are achieved using recently developed fast multipole methods for difference equations. The accuracy and physical fidelity of solutions are verified through numerical simulations of vortex rings

A Vortex Damping Outflow Forcing for Multiphase Flows with Sharp Interfacial Jumps

Outflow boundaries play an important role in multiphase fluid dynamics simulations involving low Weber numbers and large jump in physical variables. Inadequate treatment of these jumps at outflow generates undesirable fluid disturbances within the computational domain. We introduce a forcing term for incompressible Navier-Stokes equations that is coupled with a fixed pressure outflow boundary condition to enable stable exit of these disturbances from the domain boundary. The forcing term acts as a damping mechanism to control vortices that are generated by droplets/bubbles in multiphase flows, and is designed to be a general formulation that can be applied to a variety of fluid-flow simulations involving phase transition and sharp interfacial jumps. Validation cases are provided to demonstrate applicability of this formulation to pool and flow boiling problems, where bubble induced vortices during evaporation and condensation can lead to instabilities at outflow boundaries that eventually propagate downstream to corrupt numerical solution. Computational experiments are performed using \flashx, which is a composable open-source software instrument designed for multiscale fluid dynamics simulations on heterogenous architectures.Comment: Preprint Submitted to Journal of Computational Physic