Spatial Averaging

This section allows users to define the options for the the spatial averaging process, including the calculation of spatial gradients and spatial integrals.

Figure 67 - "Spatial Averaging" section of coarse-graining module.

Function

Users can select between different weight functions for the spatial averaging:

• Gaussian: particles' contibution varies according to the Gaussian function of the given width.
• Heaviside: particles' contibution will be equally-weighted while they lay in the averaging region being averaged over.

Width and Cutoff Factor

The value of the width determines the length scale for the spatial averaging function. For Gaussian function, this corresponds to the standard deviation of the funciton. For Heaviside, this determines the size of the representative volume.

The Cutoff Factor is used in combination with the Width to ignore particles whose contribution will be very small. Since the Gaussian function only becomes zero at infinity, all particles would contribute to the spatial average, most giving the tiniest of contributions.

The values of Width and Cutoff Factor are problem dependent, but for quasi-static problems or low speed flows a Width of 2.5 to 3.0 times the average particle radius can be considered a safe value. For high speed flows and high sampling frequencies, this value can be as low as half the average radius. A Cutoff Factor of 3.0 can be considered safe. In the case of high speed flows or sampling frequencies, this value can decrease to obtain a better resolution close to the boundaries.

Gradients

This features allows the user to activate the calculation of spatial gradients during the spatial averaging process. This creates the extra results in the Coarse-graining dataset that correspond to spatial gradients of the results: Density, Solid fraction, Momentum, and Velocity.

Spatial Integral

This feature allows the users to select if spatial intergral will be performed during the calculation of the coarse-graining results:

• None: spatial intergral will not be calculated.

• 1D: the coarse-grianing results are integrated along one of the main axis-directions: [X / Y / Z].

• 2D: the coarse-grianing results are integrated over a plane. Users can select among the following planes for the integration: [XY / YZ / XZ].

• 1D cylindrical: the coarse-grianing results are integrated along the tangential of cylindrical coordinate system defined by the user:

  • 1D Cylindrical Direction: defines the axial direction of the cylindrical coordinate system.

  • 1D Cylindrical Origin: defines the position of the origin of the cylindrical coordinate system.