2,910 research outputs found
An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary
Common efficient schemes for the incompressible Navier-Stokes equations, such
as projection or fractional step methods, have limited temporal accuracy as a
result of matrix splitting errors, or introduce errors near the domain
boundaries (which destroy uniform convergence to the solution). In this paper
we recast the incompressible (constant density) Navier-Stokes equations (with
the velocity prescribed at the boundary) as an equivalent system, for the
primary variables velocity and pressure. We do this in the usual way away from
the boundaries, by replacing the incompressibility condition on the velocity by
a Poisson equation for the pressure. The key difference from the usual
approaches occurs at the boundaries, where we use boundary conditions that
unequivocally allow the pressure to be recovered from knowledge of the velocity
at any fixed time. This avoids the common difficulty of an, apparently,
over-determined Poisson problem. Since in this alternative formulation the
pressure can be accurately and efficiently recovered from the velocity, the
recast equations are ideal for numerical marching methods. The new system can
be discretized using a variety of methods, in principle to any desired order of
accuracy. In this work we illustrate the approach with a 2-D second order
finite difference scheme on a Cartesian grid, and devise an algorithm to solve
the equations on domains with curved (non-conforming) boundaries, including a
case with a non-trivial topology (a circular obstruction inside the domain).
This algorithm achieves second order accuracy (in L-infinity), for both the
velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure
A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations
A new projection method based on radial basis functions (RBFs) is presented
for discretizing the incompressible unsteady Stokes equations in irregular
geometries. The novelty of the method comes from the application of a new
technique for computing the Leray-Helmholtz projection of a vector field using
generalized interpolation with divergence-free and curl-free RBFs. Unlike
traditional projection methods, this new method enables matching both
tangential and normal components of divergence-free vector fields on the domain
boundary. This allows incompressibility of the velocity field to be enforced
without any time-splitting or pressure boundary conditions. Spatial derivatives
are approximated using collocation with global RBFs so that the method only
requires samples of the field at (possibly scattered) nodes over the domain.
Numerical results are presented demonstrating high-order convergence in both
space (between 5th and 6th order) and time (up to 4th order) for some model
problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure
A Simulation Method to Resolve Hydrodynamic Interactions in Colloidal Dispersions
A new computational method is presented to resolve hydrodynamic interactions
acting on solid particles immersed in incompressible host fluids. In this
method, boundaries between solid particles and host fluids are replaced with a
continuous interface by assuming a smoothed profile. This enabled us to
calculate hydrodynamic interactions both efficiently and accurately, without
neglecting many-body interactions. The validity of the method was tested by
calculating the drag force acting on a single cylindrical rod moving in an
incompressible Newtonian fluid. This method was then applied in order to
simulate sedimentation process of colloidal dispersions.Comment: 7pages, 7 figure
Meshfree finite differences for vector Poisson and pressure Poisson equations with electric boundary conditions
We demonstrate how meshfree finite difference methods can be applied to solve
vector Poisson problems with electric boundary conditions. In these, the
tangential velocity and the incompressibility of the vector field are
prescribed at the boundary. Even on irregular domains with only convex corners,
canonical nodal-based finite elements may converge to the wrong solution due to
a version of the Babuska paradox. In turn, straightforward meshfree finite
differences converge to the true solution, and even high-order accuracy can be
achieved in a simple fashion. The methodology is then extended to a specific
pressure Poisson equation reformulation of the Navier-Stokes equations that
possesses the same type of boundary conditions. The resulting numerical
approach is second order accurate and allows for a simple switching between an
explicit and implicit treatment of the viscosity terms.Comment: 19 pages, 7 figure
Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet it only achieves first-order
spatial accuracy near embedded boundaries. In this paper, we introduce a new
high-order numerical method which we call the Immersed Boundary Smooth
Extension (IBSE) method. The IBSE method achieves high-order accuracy by
smoothly extending the unknown solution of the PDE from a given smooth domain
to a larger computational domain, enabling the use of simple Cartesian-grid
discretizations (e.g. Fourier spectral methods). The method preserves much of
the flexibility and robustness of the original IB method. In particular, it
requires minimal geometric information to describe the boundary and relies only
on convolution with regularized delta-functions to communicate information
between the computational grid and the boundary. We present a fast algorithm
for solving elliptic equations, which forms the basis for simple, high-order
implicit-time methods for parabolic PDE and implicit-explicit methods for
related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat,
Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise
convergence for Dirichlet problems and third-order pointwise convergence for
Neumann problems
An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies
We present an immersed interface method for the vorticity-velocity form of
the 2D Navier Stokes equations that directly addresses challenges posed by
multiply connected domains, nonconvex obstacles, and the calculation of force
distributions on immersed surfaces. The immersed interface method is
re-interpreted as a polynomial extrapolation of flow quantities and boundary
conditions into the obstacle, reducing its computational and implementation
complexity. In the flow, the vorticity transport equation is discretized using
a conservative finite difference scheme and explicit Runge-Kutta time
integration. The velocity reconstruction problem is transformed to a scalar
Poisson equation that is discretized with conservative finite differences, and
solved using an FFT-accelerated iterative algorithm. The use of conservative
differencing throughout leads to exact enforcement of a discrete Kelvin's
theorem, which provides the key to simulations with multiply connected domains
and outflow boundaries. The method achieves second order spatial accuracy and
third order temporal accuracy, and is validated on a variety of 2D flows in
internal and free-space domains
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