Turbulent Pipe Flow#

The turbulent flow in a pipe is extensively studied in literature and can be used as validation case of turbulence models, wall functions, curved boundaries, among others. As illustrated below, this is a pressure driven flow which can be described periodically using a constant force density.

where \(u_{\mathrm{w}}\) is the friction velocity, which is representative of the mean wall shear stress \(\sigma_{\mathrm{W}}\), and is calculated with:

(1)#\[u^{*}=\sqrt{\frac{\sigma_{\mathrm{w}}}{\rho}}\]

As for the channel, the turbulent pipe flow does present a statistically developed state, to which average and standard deviation velocities are solely dependent on the flow Reynolds number.

The characterization of the flow is usually performed from its friction Reynolds number, given by:

(2)#\[\mathrm{Re}_{\tau}=\frac{u_{\mathrm{w}}\left(d/2\right)}{\nu}\]

For a turbulent pipe flow, the length scale for calculation of the eddy turnover time is \(l=d/2\).

There are many numerical studies of a turbulent pipe flow available. Peng et al.[1] presents the first DNS validation within LBM. It is shown that the D3Q19 velocity set lacks isotropy to represent a turbulent flow in a circular pipe. The validation is then performed with the D3Q27 velocity set for a friction Reynolds number \(\mathrm{Re}_{\tau}=180\). The average and rms velocity profiles were consistently validated against different DNS studies using spectral and finnite volume methods and are shown below:

For an accurate DNS, Peng et al.[1] suggests for a uniform grid a sufficient resolution as \(\Delta y^{+} \leq 2.5\). Such scale is suffices the criterion of three grid points inside the viscous sublayer.