The Dean number (D) is a dimensionless group in fluid mechanics, which occurs in the study of flow in curved pipes and channels. It is named after the British scientist W. R. Dean, who studied such flows in the 1920s (Dean, 1927, 1928).
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Definition
The Dean number is typically denoted by De (or Dn). For a flow in a pipe or tube it is defined as:
where
The Dean number is therefore the product of the Reynolds number (based on axial flow
The Dean equations
The Dean number appears in the so-called Dean equations. These are an approximation to the full Navier–Stokes equations for the steady axially uniform flow of a Newtonian fluid in a toroidal pipe, obtained by retaining just the leading order curvature effects (i.e. the leading-order equations for
We use orthogonal coordinates
In terms of these non-dimensional variables and coordinates, the Dean equations are then
where
is the convective derivative.
The Dean number D is the only parameter left in the system, and encapsulates the leading order curvature effects. Higher-order approximations will involve additional parameters.
For weak curvature effects (small D), the Dean equations can be solved as a series expansion in D. The first correction to the leading-order axial Poiseuille flow is a pair of vortices in the cross-section carrying flow form the inside to the outside of the bend across the centre and back around the edges. This solution is stable up to a critical Dean number