In tensor form, the momentum and continuity equations are $$ \underbrace{\frac{\partial \tilde{u}_i}{\partial t}}_{\text{local (unsteady) acceleration}} \;+\; \underbrace{\frac{\partial (\tilde{u}_i \tilde{u}_j)}{\partial x_j}}_{\text{advection}} = -\,\frac{1}{\rho}\, \underbrace{\frac{\partial \tilde{p}}{\partial x_i}}_{\text{pressure gradient}} \;+\; \underbrace{g_i}_{\text{body-force acceleration}} \;+\; \nu\, \underbrace{\frac{\partial^2 \tilde{u}_i}{\partial x_j \partial x_j}}_{\text{viscous diffusion}} (i,j=1,2,3) $$ $$ \underbrace{\frac{\partial \tilde{u}_j}{\partial x_j}}_{\text{incompressibility}} = 0 $$ The two equations above describe, at every instant, the instantaneous velocity and pressure at every point in the flow field. However closed-form analytical solutions of the full N–S equations are not available and direct numerical solutions for high-Reynolds-number turbulent flows are impractical. Therefore, it is necessary to develop averaged equations of motion that are capable of addressing practical problems