Turbulent flow

A turbulent flow is characterized by chaotic fluctuations of pressure and velocity such that it can only be assessed statistically. The presence of small perturbations in a high Reynolds number flow can result in a laminar to turbulent transition.

For a instant \(t_{1}\) to which the flow is statistically developed, the average velocity field is given by:

(1)\[\bar{u}\left(\mathbf{x}\right) = \frac{1}{N}\sum_{n=0}^{N-1}u\left(\mathbf{x},t_{1}+n\Delta t\right)\]

While the averaged root mean square-fluctuation velocity is given by:

(2)\[\bar{u}_{\mathrm{rms}}\left(\mathbf{x}\right) = \sqrt{\frac{1}{N}\sum_{n=0}^{N-1}\left[u\left(\mathbf{x},t_{1}+n\Delta t\right) - \bar{u}\left(\mathbf{x}\right)\right]^{2}}\]

where the \(\Delta t\) has be sufficiently small, and the amount of time steps \(N\) must be enough to correctly represent the flow statistics. The interval between each velocity measurement is usually estimated from the flow’s eddy turnover time (\(\mathrm{ETT}=l/u^{*}\)).

The \(ETT\) is the typical time scale for an eddy of length scale \(l\) to undergo significant distortion. The friction velocity is representative of the mean wall shear stress \(\sigma_{\mathrm{w}}\), and is calculated with:

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

Where \(\sigma_{\mathrm{w}}\) is the wall shear-stress.

Wall coordinates

It is common practice to work with wall normalized scales in turbulent flows, those are identified by the superscript \(^{+}\). The normalization is performed according to flow’s friction (\(u^{*}\)) velocity and fluid’s viscosity (\(\nu\)). Hence, the length \(y^{+} = y u^{*}/\nu\) is the normalized distance from wall (\(y^{+} = 0\) at the wall). The velocity is normalized with the wall velocity \(u^{+} = u/u^{*}\).

The use of such units is mostly used to compare a numerical solution with the logarithmic law of the wall, a self similar solution for the mean velocity parallel to the wall, valid for flows at high Reynolds numbers. For a turbulent channel flow, the average flow profile can be generalized as below

The viscous sublayer is the first near wall region in a turbulent flow, in which flow retains aspects of a laminar flow, with viscous shear-stresses predominant over turbulent shear-stresses. It has the highest gradients of time averaged velocity. The buffer layer is also often called turbulent-generation layer, since it stands between both viscous and inertial dominated regions of the flow, it also presents a strong gradient of the time averaged velocity. Beyond \(\Delta y^{+} \approx 30\) there is the log-law of the wall region, in which the flow is characterized by small average velocity gradient and in which the average velocity of a turbulent flow can be approximated by a logarithmic law written in terms of \(y^{+}\).

When dealing of numerical models, the normalization of length measurement with \(\nu/u^{*}\) can be applied to determine the resolution of the numerical simulation

(4)\[\Delta y^{+} = \Delta y u^{*}/\nu\]

The smallest length scale in a turbulent flow is given by the Kolmogorov microscale \(k^{+}\), which for a Newtonian turbulent channel flow is around 1.5. Which means that for a \(\Delta y^{+}>1.5\), the numerical simulation will not be able to reproduce all near a wall eddies. Since this would require a overwhelming amount of elements for high Reynolds simulations, workarounds such as the use of Large-eddy simulations and wall models are adopted to overcome this numerical limitations.

Note

A uniform mesh channel with 144 nodes at wall direction performing a simulation with \(Re_{\mathrm{b}} \approx 54000\) would present a \(\Delta y^{+} \approx 30\).

How to generate turbulent flow

Computationally, a stable turbulent flow can be generated by setting a high Reynolds number and using a temporary body to generate the perturbations. In this case, the body stays in the computational domain for a limited amount of time, and is removed after sufficient turbulence is produced.

After the removal of the solid body, additional time-steps must be conducted before a statistically developed turbulent flow is obtained. The average and standard deviation velocities and average density are calculated from the subsequent time-steps.