# 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 an LES this statistical character is not a nuisance to be averaged away, it is the answer the analysis is after.
The unsteady loads on an immersed body are carried by the fluctuations, not by the mean flow alone; in computational wind engineering (CWE), for instance, the peak suction that lifts a roof, the along-wind force that sways a tower and the pedestrian-level gust that determines comfort are all properties of the fluctuating field.
A single instantaneous velocity field from a turbulent simulation is essentially irreproducible: two runs started from slightly different conditions diverge within a few eddy turnover times {footcite:p}`pope2000turbulent`.
What *is* reproducible, and what you should report and validate, are the statistical moments of the signal: its mean, its variance, its spectrum, and its extremes.
This page builds those statistics from the ground up, because interpreting a Nassu result means reading it as a sample of a random field rather than as a single picture of the flow.

```{admonition} Key idea
Almost every flow of engineering interest is turbulent, and in a turbulent flow the fluctuations *are* the result, not noise on top of it. A single instantaneous field is irreproducible; treat a turbulent simulation as a sample of a random field and report its statistics, the mean, variance, spectrum and extremes, rather than any one snapshot.
```

## Reynolds decomposition and stationarity

The starting point of the statistical description is the Reynolds decomposition, which splits every instantaneous field into a mean part and a zero-mean fluctuation {footcite:p}`pope2000turbulent,davidson2015turbulence`:

$$
u_{\alpha}\left(\mathbf{x},t\right) = \bar{u}_{\alpha}\left(\mathbf{x}\right) + u'_{\alpha}\left(\mathbf{x},t\right)
$$ (reynolds_decomposition)

By construction the fluctuation averages to zero, $\overline{u'_{\alpha}}=0$, so the mean carries the bulk transport and the fluctuation carries the turbulent energy and the gust loading.

```{figure} /_static/img/theory/reynolds_decomposition.svg
---
align: center
width: 70%
---
Reynolds decomposition. The instantaneous velocity $u(t)$ splits into a steady mean $\bar{u}$ and a zero-mean fluctuation $u'(t)$; in a statistically stationary flow the mean is constant and the fluctuation carries the unsteady loading.
```

Separating the two is only meaningful when the statistics themselves do not drift in time.
A flow is *statistically stationary* once its averages stop depending on when you start measuring: the mean, variance and higher moments computed over a window are the same whether that window opens now or one turnover time later.
This is exactly the "statistically developed" state reached after the turbulence-generation transient has flushed through the domain and the body wake has settled into a self-sustaining unsteady regime.
Collecting statistics before stationarity contaminates them with the start-up transient and is the most common cause of non-physical coefficients, so the averaging windows below assume the flow has reached this state first.

For an instant $t_{1}$ at which the flow is statistically developed, the average velocity field is given by:

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

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

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

where the $\Delta t$ has to 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 ($\mathrm{ETT}=l/u'$), where $u'$ is a characteristic large-eddy velocity scale (not the friction velocity $u^{*}$ defined below).

The $\mathrm{ETT}$ is the typical time scale for an eddy of length scale $l$ to undergo significant distortion.

These two requirements, a small $\Delta t$ and a large $N$, answer different questions.
The sampling interval $\Delta t$ must resolve the fastest fluctuation you care about (see the spectral discussion in the turbulence spectra page), while the total averaging window $T = N\,\Delta t$ must be long enough for the slow, energy-containing eddies to repeat many times so that the average sees a representative population of them.
The natural yardstick for "long enough" is the eddy turnover time: a turbulent average converges only after the flow has cycled through many turnovers, because successive turnovers are the statistically independent samples the average is built from {footcite:p}`pope2000turbulent`.
A practical consequence is that the statistical error on a mean decays only as $1/\sqrt{T_{\mathrm{eff}}}$, where $T_{\mathrm{eff}}$ is the averaging time measured in turnover times, so halving the uncertainty on a reported coefficient requires roughly four times the averaging window.
Higher moments converge more slowly still: the variance needs a longer window than the mean, and the peak and kurtosis (introduced on the {ref}`surface-coefficients page <surface_coefficients>`) the longest of all, because they are dominated by rare events that appear only a few times per window.

```{warning}
A subtle trap: sampling before convergence

Statistics collected before the flow is statistically stationary are contaminated by the start-up transient, and this is the most common cause of non-physical coefficients. A statistic that still drifts as you extend the run signals too short a window, not a wrong model, and no post-processing recovers a moment from a record that never converged. Discard the transient first, then average over many eddy turnover times.
```

The friction velocity is representative of the mean wall shear stress $\tau_{\mathrm{w}}$, and is calculated with:

$$
u^{*}=\sqrt{\frac{\tau_{\mathrm{w}}}{\rho}}
$$ (friction_velocity)

Where $\tau_{\mathrm{w}}$ is the wall shear-stress.

## Reynolds stresses and turbulence intensity

The second moments of the fluctuation field carry the turbulent energy and momentum exchange.
Averaging the products of fluctuation components defines the Reynolds stress tensor {footcite:p}`pope2000turbulent`:

$$
R_{\alpha\beta}\left(\mathbf{x}\right) = \overline{u'_{\alpha}\,u'_{\beta}}
$$ (reynolds_stress)

Each diagonal term $R_{\alpha\alpha}$ (no summation over $\alpha$) is the variance of that single velocity component, so $\bar{u}_{\mathrm{rms}}=\sqrt{R_{\alpha\alpha}}$ recovers the root-mean-square fluctuation of Eq. {math:numref}`vc_rms_velocity`.
The off-diagonal terms such as $R_{xz}=\overline{u'_{x}u'_{z}}$ are the turbulent shear stresses: they measure how vertical gusts correlate with along-wind gusts and thereby transport streamwise momentum toward the wall, the mechanism that produces the log-law profile discussed below.
The tensor is symmetric, so in three dimensions it has six independent components, and it is positive semi-definite because its diagonal entries are variances.

For reporting and for prescribing inflow it is convenient to normalize the fluctuations by a reference velocity, giving the turbulence intensity:

$$
I_{\alpha} = \frac{\sqrt{\overline{u'_{\alpha}u'_{\alpha}}}}{U_{\mathrm{ref}}}
$$ (turbulence_intensity)

where the index $\alpha$ is not summed and $U_{\mathrm{ref}}$ is typically the mean speed at the reference height.
The longitudinal intensity $I_{x}$ is the quantity wind codes and wind-tunnel reports tabulate for each terrain category, and it is one of the profiles Nassu validates against the target atmospheric boundary layer {footcite:p}`OLIVEIRA2026106465`.

```{admonition} Why this matters for Nassu
The Reynolds stress tensor $R_{\alpha\beta}$ and the integral length scales are precisely the inputs the synthetic eddy method (SEM) consumes to build a turbulent inlet: it Cholesky-decomposes $R_{\alpha\beta}$ to imprint the correct stresses on the synthetic fluctuations, while the length scales set the eddy size. Defining these statistics here is what makes the inlet specification well posed.
```

## 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 $u^{+} = u/u^{*}$.

Such units are 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 into the near-wall regions shown 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 $y^{+}$.

```{figure} /_static/img/theory/wall_layers.svg
---
align: center
width: 75%
---
The regions of a wall-bounded turbulent flow in wall coordinates. The viscous sublayer ($y^{+} < 5$) and buffer layer ($5 < y^{+} < 30$) govern the near-wall dynamics; from $y^{+} \approx 30$ the velocity follows the logarithmic law, which is where a wall model samples the flow.
```

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

$$
\Delta y^{+} = \Delta y u^{*}/\nu
$$ (delta_y)

The smallest length scale in a turbulent flow is the Kolmogorov microscale, which in wall units $\eta^{+}$ is of order unity in the near-wall region of a Newtonian turbulent channel flow (approximately 1.5 in the buffer layer, with the exact value being Reynolds-number dependent) {footcite:p}`KimMoinMoiser1987-177`.
Resolving the near-wall turbulence down to these scales therefore requires a very fine mesh: the common DNS guideline is a first wall-normal spacing of $\Delta y^{+} \approx 1$. Since meeting such a resolution everywhere would require an overwhelming amount of elements for high Reynolds simulations, workarounds such as the use of Large-eddy simulations and wall models are adopted to overcome these numerical limitations.

```{admonition} Worked example
A uniform-mesh channel with 144 nodes in the wall-normal direction at $Re_{\mathrm{b}} \approx 54000$ gives $\Delta y^{+} \approx 30$, putting the first node at the start of the log-law region rather than inside the viscous sublayer. This is the regime a wall model is built to handle.
```

## 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.

```{eval-rst}
.. footbibliography::
```

```{toctree}
---
hidden: true
maxdepth: '-1'
---
Turbulence Spectra <turbulence_spectra.md>
```