Temporal discretization

The flow solver uses the Backward Euler and Crank-Nicolson schemes, and the local time step method as time discretization methods.

The flow solver uses the following flow time integration methods:

Backward Euler scheme, which is the fully implicit first orders scheme.
Crank-Nicolson scheme, which is the semi-implicit second-order scheme.

The semi-discrete form of the integral transport equation for a conserved quantity ϕ, written for the control volume centered at the node P depicted in the figure on the shape function section:

where:

S^m is the local volumetric rate of generation of ϕ.
F is the total flux vector of ϕ.
n is the outward unit normal vector.

Backward Euler scheme

In the fully implicit first order scheme, the integral transport equation is integrated from the time t_n to the time t_{n + 1} = t_n - Δt_n, yielding:

Crank-Nicolson scheme

In the semi-implicit second-order scheme, the integral transport equation is integrated by applying a second-order quadrature rule in the approximation of the time integral of the fluxes and sources, yielding:

In comparison with the fully-implicit time integration option, which results in an O(Δt²_n) truncation error, the use of the second-order accurate temporal discretization of the equation allows the accurate resolution of transient flow features with a significantly larger timestep.

However, the semi-implicit time integration method is susceptible to spurious, oscillatory temporal variation of the solution field if too large a timestep is employed. This method should not be selected when a timestep is large in comparison with the bulk time scale of the problem.

Note:

When you use the AUTO TURN-OFF FLUIDS EQUATION SOLVE advanced parameter in a transient analysis, the flow solver reactivates a frozen solution at the beginning of each time step and iterates at least 3 times, even if the solution converged in prior iterations. If the solution did not converge, the solver continues iterating within the time step until it reaches the convergence.

Steady state - Local time step method

The flow solver computes the local time step method using the quantity, ϕ, between two consecutive iterations:

where:

n is the number of the iteration.
Δϕ is the change in the quantity ϕ.

Using the control volume centered at the node P, which is depicted in the figure on the Shape function section, the discrete form of the transport equation for the conserved quantity is given:

where b is the source term.

A combination of the two previous equations gives:

where

The flow solver computes the local time step at each control volume, using the local velocity and element length scale. It solves the following equation using the local time step method and relaxation factor, α:

All the diagonal terms of the coefficient matrix, A_p, are scaled by , in which the lower value of the relaxation factor provides more relaxation to the convergence process but usually increases the computational cost.