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} (x) = \frac{1}{N} \sum_{n = 0}^{N - 1} u (x, t_{1} + n Δ t)

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

(2)#

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

where the $Δ t$ 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 ( $ETT = l / u^{*}$ ).

The $E T T$ 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 $σ_{w}$ , and is calculated with:

(3)#

u^{*} = \sqrt{\frac{σ_{w}}{ρ}}

Where $σ_{w}$ 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 ( $ν$ ). Hence, the length $y^{+} = y u^{*} / ν$ 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 $Δ y^{+} \approx 30$ 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 $ν / u^{*}$ can be applied to determine the resolution of the numerical simulation

(4)#

Δ y^{+} = Δ y u^{*} / ν

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 $Δ y^{+} > 1.5$ , 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 $R e_{b} \approx 54000$ would present a $Δ y^{+} \approx 30$ .

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.

Turbulent flow

Contents

Turbulent flow#

Wall coordinates#

How to generate turbulent flow#