SPH Adaptive Time-Step Providers Reference

The SPH method is not fully implicit. Therefore, the time step is restricted by stability conditions. Adaptive time-stepping is used to determine a time-step size $\mathrm{\Delta }t$ such that the various Courant number (CFL) conditions are met within the computational domain.

Convective CFL

The Convective CFL time-step provider sets the time-step based on the CFL number, the convective time scale. The time-step is adjusted such that the maximum CFL number of all particles is lower than the specified Target Convective CFL Number. The convective CFL number is defined as:

$${CFL}_{convective,i}=\mathrm{\Delta }t\frac{v_i}{\mathrm{\Delta }{x}_i}<{CFL}_{convective,tar}$$

(102)

where $v_i$ is the magnitude of the velocity of particle i and $\mathrm{\Delta }{x}_i=\sqrt[3]{V_i}$ is the size of the cube with the same volume $V$ as the particle i.

Gravity CFL

The Gravity CFL time-step provider sets the time-step based on a CFL number that takes into account the resultant of body forces $f_{b,i}$ . The time-step is adjusted such that the maximum gravity CFL number of all particles is lower than the specified Target Gravity CFL Number. The CFL number based on body forces is defined as:

$${CFL}_{gravity,i}=\sqrt{\frac{f_{b,i}}{{\rho }_0\mathrm{\Delta }{x}_i}}<{CFL}_{gravity,tar}$$

(103)

where ${\rho }_0$ is the fluid density. This time-step provider is available only when gravity is selected in the simulation.

Viscous CFL

In viscous flow simulations, the Viscous CFL time-step provider sets the time-step based on a CFL number that takes into account the fluid viscosity. The time-step is adjusted such that the maximum viscous CFL number is lower than the specified Target Viscous CFL Number. The CFL number based on fluid viscosity is defined as:

$${CFL}_{visocus,i}=\mathrm{\Delta }t\frac{{\mu }_i}{{\rho }_0{\mathrm{\Delta }{x}_i}^2}<{CFL}_{visocus,tar}$$

(104)

where ${\mu }_i$ is the dynamic viscosity of the particle i.

For SPH simulations with the explicit velocity solver, you are advised to use all the SPH compatible time-step providers (Convective CFL, Gravity CFL, and Viscous CFL). The Viscous CFL time-step provider is not required for simulations that use the implicit velocity solver.

Convective CFL Properties

Target Convective CFL Number

Specifies ${CFL}_{convective,tar}$ in Eqn. (102). The default value is 0.72.

Include Boundaries

When activated, includes the boundary velocities in the calculation of the time-step for the convective CFL condition.

Gravity CFL Properties

Target Gravity CFL Number

Specifies ${CFL}_{gravity,tar}$ in Eqn. (103). The default value is 0.073.

Viscous CFL Properties

Target Viscous CFL Number

Specifies ${CFL}_{viscous,tar}$ in Eqn. (104). The default value is 0.01.