2.1 The Navier-Stokes equation and the Reynolds number

Equations which describe the dynamics of real incompressible fluid flows have been introduced by C. Navier in 1823 and improved by G. Stokes. They are nothing but the momentum equation based on Newton’s second law, which relates the acceleration of a fluid particle² to the resulting volume and body forces acting on it. These equations have been introduced by L. Euler, however, the main contribution by C. Navier was to add a friction forcing term due to the interactions between fluid layers which move with different speed. This term results to be proportional to the viscosity coefficients $j$ and $q$ and to the variation of speed. By defining the velocity field $u(r, t)$ the kinetic pressure $p$ and the density $r$ , the equations describing a fluid flow are the continuity equation to describe the conservation of mass

$@r --- + (u . \~/ ) r = - r \~/ .u, (1) @t$

the equation for the conservation of momentum

$[ ] @u ( j ) r ---+ (u .\ ~/ ) u = - \~/ p + j \~/ 2u + q + -- \~/ ( \~/ .u) , (2) @t 3$

and an equation for the conservation of energy

$[ ] ( )2 rT @s-+ (u . \~/ )s = \~/ .(x\ ~/ T ) + j @ui-+ @uk- - 2-dik \~/ .u + q( \~/ .u)2, (3) @t 2 @xk @xi 3$

where $s$ is the entropy per mass unit, $T$ is the temperature, and $x$ is the coefficient of thermoconduction. An equation of state closes the system of fluid equations. The above equations considerably simplify if we consider the tractable incompressible fluid, where $r = const.$ so that we obtain the Navier-Stokes (NS) equation

$( ) @u \~/ p 2 ---+ (u . \~/ ) u = - ---- + n \~/ u, (4) @t r$

where the coefficient $n = j/r$ is the kinematic viscosity. The incompressibility of the flow translates in a condition on the velocity field, namely the field is divergence-free, i.e., $\~/ .u = 0$ . The non-linear term in equations represents the convective (or substantial) derivative. Of course, we can add on the right hand side of this equation all external forces, which eventually act on the fluid parcel.

We use the velocity scale $U$ and the length scale $L$ to define dimensionless independent variables, namely $r = r'L$ (from which $\~/ = \~/ '/L$ ) and $t = t'(L/U )$ , and dependent variables $u = u'U$ and $' 2 p = p U r$ . Then, using these variables in Equation (4 ), we obtain

$@u' ' ' ' '' - 1 '2 ' @t'-+ (u . \~/ )u = - \~/ p + Re \~/ u . (5)$

The Reynolds number $Re = UL/n$ is evidently the only parameter of the fluid flow. This defines a Reynolds number similarity for fluid flows, namely fluids with the same value of the Reynolds number behaves in the same way. Looking at Equation (5 ) it can be realized that the Reynolds number represents a measure of the relative strength between the non-linear convective term and the viscous term in Equation (4 ). The higher $Re$ , the more important the non-linear term is in the dynamics of the flow. Turbulence is a genuine result of the non-linear dynamics of fluid flows.