Fluid dynamic problems

Non-dimensional form

The characteristic equations of the CBS and SUPG/PSPG algorithms can be solved in non-dimensional form. To enable this, give scaleflag a nonzero value and supply the three scales:

[scaleflag (in)]

A nonzero value turns on the non-dimensional form.

lscale (rn), uscale (rn), dscale (rn)

Typical length, velocity and density scales. Required when scaleflag is nonzero.

Transient incompressible flow — CBS algorithm

Solves transient incompressible flow with an algorithm based on the Characteristics Based Split (CBS); for reference see O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, volume 3, Butterworth-Heinemann, 2000.

At present only the semi-implicit form of the algorithm is available, and the energy equation — and with it the temperature field — is not solved.

Record syntax

CBS nsteps (in) deltaT (rn) [theta1 (rn)] [theta2 (rn)] [cmflag (in)] [scaleflag (in)] [lscale (rn)] [uscale (rn)] [dscale (rn)] [lstype (in)] [smtype (in)]

Parameters

nsteps (in)

Number of solution steps.

deltaT (rn)

Time step length used for integration. It is adjusted automatically, if necessary, to satisfy the stability limits \(\Delta t \le h/\vert\boldsymbol{u}\vert\) and \(\Delta t \le h^2/(2\nu)\).

[theta1 (rn)], [theta2 (rn)]

Integration constants, \(\theta_1, \theta_2 \in \langle\tfrac12, 1\rangle\).

[cmflag (in)]

A nonzero value uses the consistent mass matrix instead of the default lumped one.

[scaleflag (in)], [lscale (rn)], [uscale (rn)], [dscale (rn)]

See Non-dimensional form.

[lstype (in)], [smtype (in)]

Linear solver type and sparse matrix storage scheme; the scheme must be compatible with the solver type. See Sparse linear solver parameters.

Transient incompressible flow — SUPG/PSPG algorithm

Solves transient incompressible flow using a stabilized formulation based on SUPG and PSPG stabilization terms. The stabilization provides stability and accuracy for advection-dominated problems and permits equal-order interpolation for velocity and pressure. It also improves the convergence rate substantially when large nonlinear systems are solved iteratively.

Varying \(\alpha\) selects different methods from the generalized midpoint family: forward Euler (\(\alpha=0\)), the midpoint rule (\(\alpha=0.5\)), Galerkin (\(\alpha=2/3\)) and backward Euler (\(\alpha=1\)). Except the first one, all are implicit and require a matrix inversion. Energy-method analysis suggests unconditional stability for \(\alpha\ge 0.5\); as far as accuracy goes, the midpoint rule is generally to be preferred.

Record syntax

SUPG nsteps (in) deltaT (rn) rtolv (rn) [atolv (rn)] [stopmaxiter (in)] [alpha (rn)] [cmflag (in)] [deltatltf (in)] [miflag (in)] [scaleflag (in)] [lscale (rn)] [uscale (rn)] [dscale (rn)] [lstype (in)] [smtype (in)]

Parameters

nsteps (in)

Number of solution steps.

deltaT (rn)

Time step length used for integration.

[deltatltf (in)]

Number of a time function used instead of deltaT to determine the time step length; its value evaluated for the solution step number yields the step length.

[alpha (rn)]

The \(\alpha\) parameter of the generalized midpoint family, as above.

rtolv (rn), [atolv (rn)]

Relative and absolute error norms for the residual vector. The equilibrium iteration stops when both limits are satisfied, or when the number of iterations exceeds stopmaxiter.

[stopmaxiter (in)]

Maximum number of equilibrium iterations.

[cmflag (in)]

A nonzero value uses the consistent mass matrix instead of the default lumped one.

[miflag (in)]

Enables multi-fluid analysis, letting the algorithm solve the flow of two immiscible fluids in a fixed spatial domain (currently in 2D only). This can also be used for free surface problems, with one of the fluids representing air. The supported values are described in Material interfaces. The initial distribution of the reference fluid volume must be provided, as well as constitutive models for both fluids.

[scaleflag (in)], [lscale (rn)], [uscale (rn)], [dscale (rn)]

See Non-dimensional form.

[lstype (in)], [smtype (in)]

Linear solver type and sparse matrix storage scheme. Note that this algorithm leads to a non-symmetric matrix, so the scheme must be compatible with the solver type. See Sparse linear solver parameters.

Transient incompressible flow — PFEM algorithm

Note

This problem type is part of the PFEM module and can be used only when that module is configured.

Solves transient incompressible flow using the particle finite element method, based on the Lagrangian formulation of the Navier-Stokes equations.

Mesh nodes are represented by PFEM particles (see PFEM particle), which can move freely and even separate from the main domain. A temporary mesh built from the particles is needed to integrate the governing equations in each solution step; it is rebuilt from scratch in every step to prevent large element distortion. The problem is therefore defined without any elements in the input file, and cs and material assign the cross section and material to the elements that are created.

The mesh is generated by Delaunay triangulation, with the alpha-shape technique identifying the free surface. On the free surface a zero-pressure boundary condition is enforced, which must be defined in the boundary condition record whose number is given by pressure.

Record syntax

PFEM nsteps (in) deltaT (rn) material (in) cs (in) pressure (in) [mindeltat (rn)] [maxiter (in)] [rtolv (rn)] [rtolp (rn)] [alphashapecoef (rn)] [removalratio (rn)] [scheme (in)] [lstype (in)] [smtype (in)]

Parameters

nsteps (in)

Number of solution steps.

deltaT (rn)

Time step length used for integration. To ensure numerical stability the step length is adapted to the mesh geometry and to the velocity of the individual nodes.

material (in), cs (in)

Numbers of the material and cross section records assigned to the elements created from the particles.

pressure (in)

Number of the boundary condition record enforcing zero pressure on the free surface.

[mindeltat (rn)]

Minimum time step length, preventing the step from becoming too short.

[removalratio (rn)]

Limit, relative to the element edge length, below which particles that are too close together are removed from the solution.

[maxiter (in)]

Maximum number of iterations. Default 50.

[rtolv (rn)], [rtolp (rn)]

Relative norms for the velocity and pressure difference between two successive iterations. Defaults 1.e-8.

[alphashapecoef (rn)]

Alpha-shape coefficient, which should reflect the initial distribution of the PFEM particles. A value of about 1.5 times the shortest distance between two neighbouring particles has been found to work well.

[scheme (in)]

Whether the equation system for the components of the auxiliary velocity is solved explicitly (0) or implicitly (1, the default).

[lstype (in)], [smtype (in)]

Linear solver type and sparse matrix storage scheme. Note that this algorithm leads to a non-symmetric matrix, so the scheme must be compatible with the solver type. See Sparse linear solver parameters.