Volume Average

The volume average of a scalar quantity is computed as:

Figure 1. EQUATION_DISPLAY

V o l u m e A v e r a g e \equiv \frac{1}{V} \int ϕ d V = \frac{\sum_{c}^{} ϕ_{c} V_{c}}{\sum_{c}^{} V_{c}}

(410)

where $ϕ_{c}$ is the cell value of the selected scalar and $V_{c}$ is the cell volume.

The volume average over parcels in a Lagrangian phase is computed as:

Figure 2. EQUATION_DISPLAY

V o l u m e A v e r a g e \equiv \frac{1}{V} \int ϕ d V = \frac{\sum_{π}^{} ϕ_{p} V_{p} n_{π}}{\sum_{π}^{} V_{p} n_{π}}

(411)

where $V_{p}$ is the volume of a single particle represented by a parcel.

To compute a generalized volume average, you can specify an additional field function $w_{c}$ that acts as a weighting function. The volume average report value over cells with weighting is:

Figure 3. EQUATION_DISPLAY

\frac{\int w ϕ d V}{\int w d V} = \frac{\sum_{c}^{} w_{c} ϕ_{c} V_{c}}{\sum_{c}^{} w_{c} V_{c}}

(412)

The volume average report value over a Lagrangian phase with weighting is:

Figure 4. EQUATION_DISPLAY

\frac{\int w ϕ d V}{\int w d V} = \frac{\sum_{π}^{} w_{p} ϕ_{p} V_{p} n_{π}}{{\sum w_{p}}_{π}^{} V_{p} n_{π}}

(413)

where $w_{p}$ is the weighting function.

Example Use Case: Mass-Averaged Enthalpy of O2

As an example of a weighted volume average, suppose that in a combustion simulation, you require the mass-averaged enthalpy of O2 only within a region. You would use the following procedure:

Create a volume average report.
Set the field function to ${Enthalpy of O2} (the enthalpy per mass of O2 in each cell).
Set the weighting function to a user-defined field function defined as ${Mass Fraction of O2}*$Density.

This user-defined field function gives the mass of O2 per volume in each cell. The ratio computed by the volume average report for the region is therefore:

Figure 5. EQUATION_DISPLAY

\frac{\sum_{c e l l s}^{} h_{O 2} Y_{O 2} ρ_{f} V_{c e l l}}{\sum_{c e l l s}^{} Y_{O 2} ρ_{f} V_{c e l l}}

(414)

Hence the ratio resulting from the volume average report is the average enthalpy per mass of O2 alone.

Volume Average Report Properties

Units: Units that are used for specifying the quantity.
Field Function: Scalar field function value $ϕ$ upon which the report is based.
Weighting Function: Optionally select a field function to be used as a weighting function in addition to volume for this report. The Weighting Function is always evaluated at cells independent of whether the Smooth Values option is activated.
Parts: A list of parts that is used as the source for this report. Opens the in-place part selector for editing.
Representation: Selects a representation. Additional selections appear as representations that are added to the simulation.
Smooth Values: When this property is activated, the report is calculated using node (interpolated) values of the chosen quantity rather than cell values.