8,804 research outputs found
A least-squares finite element method for incompressible Navier-Stokes problems
A least-squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady incompressible Navier-Stokes problems. This method leads to a minimization problem rather than to a saddle-point problem by the classic mixed method, and can thus accommodate equal-order interpolations. This method has no parameter to tune. The associated algebraic system is symmetric, and positive definite. Numerical results for the cavity flow at Reynolds number up to 10,000 and the backward-facing step flow at Reynolds number up to 900 are presented
Numerical Solutions of 2-D Steady Incompressible Driven Cavity Flow at High Reynolds Numbers
Numerical calculations of the 2-D steady incompressible driven cavity flow
are presented. The Navier-Stokes equations in streamfunction and vorticity
formulation are solved numerically using a fine uniform grid mesh of 601x601.
The steady driven cavity solutions are computed for Re<21,000 with a maximum
absolute residuals of the governing equations that were less than 10-10. A new
quaternary vortex at the bottom left corner and a new tertiary vortex at the
top left corner of the cavity are observed in the flow field as the Reynolds
number increases. Detailed results are presented and comparisons are made with
benchmark solutions found in the literature
Least-squares solution of incompressible Navier-Stokes equations with the p-version of finite elements
A p-version of the least squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady state incompressible viscous flow problems. The resulting system of symmetric and positive definite linear equations can be solved satisfactorily with the conjugate gradient method. In conjunction with the use of rapid operator application which avoids the formation of either element of global matrices, it is possible to achieve a highly compact and efficient solution scheme for the incompressible Navier-Stokes equations. Numerical results are presented for two-dimensional flow over a backward facing step. The effectiveness of simple outflow boundary conditions is also demonstrated
Vector potential methods
Vector potential and related methods, for the simulation of both inviscid and viscous flows over aerodynamic configurations, are briefly reviewed. The advantages and disadvantages of several formulations are discussed and alternate strategies are recommended. Scalar potential, modified potential, alternate formulations of Euler equations, least-squares formulation, variational principles, iterative techniques and related methods, and viscous flow simulation are discussed
Least-squares finite element method for fluid dynamics
An overview is given of new developments of the least squares finite element method (LSFEM) in fluid dynamics. Special emphasis is placed on the universality of LSFEM; the symmetry and positiveness of the algebraic systems obtained from LSFEM; the accommodation of LSFEM to equal order interpolations for incompressible viscous flows; and the natural numerical dissipation of LSFEM for convective transport problems and high speed compressible flows. The performance of LSFEM is illustrated by numerical examples
Numerical Solutions of 2-D Steady Incompressible Flow in a Driven Skewed Cavity
The benchmark test case for non-orthogonal grid mesh, the "driven skewed
cavity flow", first introduced by Demirdzic et al. (1992, IJNMF, 15, 329) for
skew angles of alpha=30 and alpha=45, is reintroduced with a more variety of
skew angles. The benchmark problem has non-orthogonal, skewed grid mesh with
skew angle (alpha). The governing 2-D steady incompressible Navier-Stokes
equations in general curvilinear coordinates are solved for the solution of
driven skewed cavity flow with non-orthogonal grid mesh using a numerical
method which is efficient and stable even at extreme skew angles. Highly
accurate numerical solutions of the driven skewed cavity flow, solved using a
fine grid (512x512) mesh, are presented for Reynolds number of 100 and 1000 for
skew angles ranging between 15<alpha<165
Incorporating variable viscosity in vorticity-based formulations for Brinkman equations
In this brief note, we introduce a non-symmetric mixed finite element
formulation for Brinkman equations written in terms of velocity, vorticity and
pressure with non-constant viscosity. The analysis is performed by the
classical Babu\v{s}ka-Brezzi theory, and we state that any inf-sup stable
finite element pair for Stokes approximating velocity and pressure can be
coupled with a generic discrete space of arbitrary order for the vorticity. We
establish optimal a priori error estimates which are further confirmed through
computational example
Finite element techniques for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation
Finite element procedures for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation have been implemented. For both formulations, streamline-upwind/Petrov-Galerkin techniques are used for the discretization of the transport equations. The main problem associated with the vorticity stream-function formulation is the lack of boundary conditions for vorticity at solid surfaces. Here an implicit treatment of the vorticity at no-slip boundaries is incorporated in a predictor-multicorrector time integration scheme. For the primitive variable formulation, mixed finite-element approximations are used. A nine-node element and a four-node + bubble element have been implemented. The latter is shown to exhibit a checkerboard pressure mode and a numerical treatment for this spurious pressure mode is proposed. The two methods are compared from the points of view of simulating internal and external flows and the possibilities of extensions to three dimensions
Optimal Reconstruction of Inviscid Vortices
We address the question of constructing simple inviscid vortex models which
optimally approximate realistic flows as solutions of an inverse problem.
Assuming the model to be incompressible, inviscid and stationary in the frame
of reference moving with the vortex, the "structure" of the vortex is uniquely
characterized by the functional relation between the streamfunction and
vorticity. It is demonstrated how the inverse problem of reconstructing this
functional relation from data can be framed as an optimization problem which
can be efficiently solved using variational techniques. In contrast to earlier
studies, the vorticity function defining the streamfunction-vorticity relation
is reconstructed in the continuous setting subject to a minimum number of
assumptions. To focus attention, we consider flows in 3D axisymmetric geometry
with vortex rings. To validate our approach, a test case involving Hill's
vortex is presented in which a very good reconstruction is obtained. In the
second example we construct an optimal inviscid vortex model for a realistic
flow in which a more accurate vorticity function is obtained than produced
through an empirical fit. When compared to available theoretical vortex-ring
models, our approach has the advantage of offering a good representation of
both the vortex structure and its integral characteristics.Comment: 33 pages, 10 figure
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