Dean number

Physical key figure
Surname	Dean number
Formula symbol	${\displaystyle {\mathit {De}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {De}}={\frac {u}{\nu }}{\sqrt {\frac {s^{3}}{2r}}}}$
${\displaystyle u}$ Flow velocity ${\displaystyle \nu }$ kinematic viscosity ${\displaystyle s}$ Dimension of the pipe ${\displaystyle r}$ Radius of curvature
Named after	William Reginald Dean
scope of application	Flow in elbows

The Dean number is a dimensionless number from fluid mechanics that is used to describe the flow, e.g. B. the pressure loss , is used in a circularly curved pipe or channel . It was named after William Reginald Dean (1896–1973), who published results in 1928 on his work on currents in curved crevices. ${\displaystyle {\mathit {De}}}$

The Dean number is defined as:

${\displaystyle {\mathit {De}}={\frac {u}{\nu }}\cdot {\sqrt {\frac {s^{3}}{2r}}}}$

With

• the flow velocity in the pipe ${\displaystyle u}$
• the kinematic viscosity of the fluid ${\displaystyle \nu }$
• the geometric dimensions of the curved flow channel:
  • the radius of curvature of the inner boundary surface ${\displaystyle r}$
  • the distance between the curved surfaces, possibly also the pipe diameter . ${\displaystyle s}$

One can formulate with the Reynolds number ${\displaystyle Re}$

${\displaystyle \Leftrightarrow {\mathit {De}}={\mathit {Re}}\cdot {\sqrt {\frac {s}{2r}}}}$

The Dean number is a criterion for whether the deflection of the fluid flow creates eddies in curved channels:

• According to Dean's investigation, no secondary eddies are formed in a curved gap with a fully developed profile of the inflow , the flow is stable in this area. ${\displaystyle {\mathit {De}}<54}$
• For larger Dean numbers, if there are minor disturbances, the faster core flow is pressed against the outer pipe wall by centrifugal force and displaces the slower wall flow, so that typical counter-rotating eddies (Dean eddies) form on the outer of the curved surfaces.

Individual evidence

1. ^ ^{A b} W. R. Dean: Fluid Motion in a Curved Channel. Proc. Roy. Soc., Series A, 121, 402-420 (1928).
2. ↑ Günther Hämmerlin: The stability of the flow in a curved channel. Archive for Rational Mechanics and Analysis, 1957, Vol.1 (1), 212-224.