Dean number

The Dean number 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).

Definition

The Dean number is typically denoted by the symbol De. For flow in a pipe or tube it is defined as: \[\text{De} = \frac{\rho V D}{\mu} \left( \frac{D}{2 R} \right)^{1/2}\] where

• \(\rho\) is the density of the fluid
• \(\mu\) is the dynamic viscosity
• \(V\) is the axial velocity scale
• \(D\) is the diameter (other shapes are represented by an equivalent diameter, see Reynolds number)
• \(R\) is the radius of curvature of the path of the channel.

The Dean number is therefore the product of the Reynolds number (based on axial flow \(V\) through a pipe of diameter \(D\)) and the square root of the curvature ratio.

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 \(a/r \ll 1\)).

We use an orthogonal coordinates \((x,y,z)\) with corresponding unit vectors \((\hat{\boldsymbol{x}},\hat{\boldsymbol{y}},\hat{\boldsymbol{z}})\) aligned with the centre-line of the pipe at each point. The axial direction is \(\hat{\boldsymbol{z}}\), with \(\hat{\boldsymbol{x}}\) being the normal in the plane of the centre-line, and \(\hat{\boldsymbol{y}}\) the binormal. For an axial flow driven by a pressure gradient \(G\), the axial velocity \(u_z\) is scaled with \(U=Ga^2/\mu\). The cross-stream velocities \(u_x, u_y\) are scaled with \((a/R)^{1/2} U\), and cross-stream pressures with \(\rho a U^2/L\). Lengths are scaled with the tube radius \(a\).

In terms of these non-dimensional variables and coordinates, the Dean equations are then \[D \left( \frac{\mathrm{D} u_x}{\mathrm{D} t} + u_z^2 \right) = -D\frac{\partial p}{\partial x} + \nabla^2 u_x \] \[D \frac{\mathrm{D} u_y}{\mathrm{D} t} = -D\frac{\partial p}{\partial y} + \nabla^2 u_y\] \[D \frac{\mathrm{D} u_z}{\mathrm{D} t} = 1 + \nabla^2 u_z \] \[\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0\] where \[\frac{\mathrm{D}}{\mathrm{D} t} = u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y}\] 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 outrside of the bend across the centre and back around the edges. This solution is stable up to a critical Dean number \(D_c \approx 956\) (Dennis & Ng 1982). For larger D, there are multiple solutions, many of which are unstable.

References

• Mestel, J. Flow in curved pipes: The Dean equations, Lecture Handout for Course M4A33, Imperial College.

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