The Einstein notation decribed in the theory section can be used to achieve notational brevity. This is particularly useful when applied to the Navier-Stokes equations! For example, the incompressible continuity equation The index variable j appears several times in a single term. Therefore, it represents a summation over all values of the index (1, 2 and 3), as can be seen in Eq.(1↑). Similarly, the convective term of the x₁-momentum equations It is important to make the distinction between the indexes that appear several times in a single term, and those that don’t. For example, within the full Navier-Stokes equations, the convective terms are typically written The j index is repeated, and therefore represents a summation. On the other hand, the index i is alone in this term. This is because Eq.(3↑) actually describes the three momentum equations, along x₁, x₂ and x₃. The index i therefore takes the value 1, 2 or 3, depending on the momentum equation considered (i = 1 in Eq.(2↑)). It is interesting to notice that the convective term as described in Eq.(3↑) is different from that described in the theory section. The latter is written in its conservative form, where all velocities are incorporated within the partial derivative. In the incompressible case, it is straightforward to show that these two expressions are equal. because ∂u_j ⁄ ∂x_j = 0 from the conservation of mass. Finally, the viscous term

As described in the theory section, each variable (eg. p) is decomposed into a mean (eg. P) and a fluctuating component (eg. p’). In this section, U_i and P are time averages. For example, U_i is defined by where t₀ is the starting instant of the averaging process. Eq.(7↑) is useful as long as the turbulence is stationary. If it is not, the average period T must be chosen more carefuly. It must be large enough to capture the maximum periods of the turbulent fluctuations T₁, but remains smaller than the time scale T₂ representative of the slow variations in the flow. Hence, the time average U_i for non-stationary turbulence is given by

Introducing the Reynolds decomposition into the continuity equation (Eq.(1↑)), one obtains Taking the average of Eq.(9↑),