For turbulent flows, physical quantities $f(x_i,t)$ (for example a velocity component, pressure, temperature, or their products) fluctuate irregularly in time and space, so different statistical averages are used under different circumstances. If the turbulence is statistically steady, a time average is commonly used. At spatial position $x_i$, the time-average is defined as $$ \overline{f(x_i)_{\,t}} = \lim_{T \to \infty} \frac{1}{T}\int_{0}^{T} f(x_i, t)\, dt $$ Here $T$ is the averaging time; $T\to\infty$ means that $T$ is much larger than the time scale of the turbulent fluctuations

If the turbulence is statistically homogeneous in one or several spatial directions, a spatial average in that direction may be used. At time $t$ and coordinates $x_k\ (k\ne j)$, the average in the $x_j$ direction is $$ \overline{f(x_k, t)_{\,x_j}} = \lim_{X_j \to \infty} \frac{1}{X_j}\int_{0}^{X_j} f(x_k, t, x_j)\, dx_j $$ Here $X_j$ denotes the extent of the averaging region: a length ($j=1$), area ($j=1,2$), or volume ($j=1,2,3$). The notation $X_j\to\infty$ means that $X_j$ is much larger than the length scale of the turbulent fluctuations. In these cases, the spatial average represents a line average, a surface average, or a volume average respectively

In any situation, one can also use an ensemble average. The ensemble average of $f$ is defined by $$ \overline{f(x_i, t)_{\,n}} = \lim_{N \to \infty} \frac{1}{N}\sum_{n=1}^{N} f_n(x_i, t) $$ where $n$ indexes the realizations (or the $n$-th measurement result) of statistically identical flows