Theory

The uniformity index of Weltens et al. (1993) measures how evenly the flow passes through a cross-section, for example the inlet face of a catalyst, a filter, a fuel-cell stack or a liner: 1 when the normal velocity is the same everywhere, 0 in the worst case. Their definition averages over equal elements; the cells of a plane cut through a mesh are not equal, so every mean is weighted by the cell area, which is the same for equal elements. For example $\gamma = 0.5$ means that half the section carries twice the mean velocity and the other half none.

The plane is cut through the solved volume (the same conversion as the section plots); the velocity normal to it, $u_n$ , gives the index and the maximum-to-mean ratio, the tangential part $u_t$ the swirl angle, weighted by the flux through each cell. Cells where $u_n$ points against the mean flow make up the back-flow area. An inlet or outlet face uses the solution on that face. With "only inside a circle" just the cells whose centre lies within the circle count, e.g. a monolith face smaller than the duct.

Equations

\bar u_n = \frac{1}{A}\sum_i u_{n,i} A_i, \qquad \gamma = 1 - \frac{1}{2A}\sum_i \frac{|u_{n,i} - \bar u_n|}{\bar u_n} A_i

(1)

where	meaning	unit
$u_{n,i}$	Velocity normal to the plane in cell $i$	m/s
$A_i$	Area of cell $i$ on the plane	m²
$A$	Area of the plane (or of the circle's cells)	m²
$\gamma$	Uniformity index	-

\theta = \frac{\sum_i \arctan(|u_{t,i}|/|u_{n,i}|)\,|u_{n,i}| A_i}{\sum_i |u_{n,i}| A_i}

(2)

where	meaning	unit
$\theta$	Flux-weighted mean swirl angle	°
$u_{t,i}$	Velocity component in the plane	m/s

Default and rule

Default: normal to an axis
Rule: Normal to an axis.
Source: Weltens et al. (1993), eqs. 1-4 (area-weighted for unequal cells).

Valid range axis | inlet / outlet face | point + normal

Post-processing

No solve: the plane is cut through the stored solution.

References 1

Weltens, H., Bressler, H., Terres, F., Neumaier, H. and Rammoser, D. (1993). Optimisation of catalytic converter gas flow distribution by CFD prediction. SAE Technical Paper 930780. DOI 10.4271/930780

Theory

Equations

Default and rule

Valid range axis | inlet / outlet face | point + normal

Related 8

References 1