Navier-Stokes Equations

In its most general form, including the chemical reactions at high temperatures, Eq. 2.49 was introduced as the set of equations for external flows. Global continuity equation and the conservation of momentum equations deal with the average values of flow parameters, therefore they are of mechanical nature, whereas the energy equation deals with the effect of heating as well the enthalpy increase caused by the diffusion of species. If we do not consider the chemical reactions, then there will not be diffusion terms present and the related specie conservation terms disappear. Therefore, Eq. 2.49 reduces to the well known Navier-Stokes Equations (Schlichting 1968). Since the Navier – Stokes equations can model all laminar and turbulent flows, they have a wide range of their implementation in aerodynamical applications. For the case of turbulent flows, we have to include the effective viscosity iT into the consti­tutive relations to model the Reynolds stresses. Now, we can re-write the constitutive relation 2.39 and the heat flux term 2.48 with the turbulent Prandtl number PrT as follows

The non dimensional similarity parameters appearing in the equations are well known Reynolds, Mach and Prandtl numbers which are defined with their physical meanings attached as follows

Reynolds number: Re = Vic/^, (inertia forces/viscous forces)

Mach number: Мж = , (kinetic energy of the flow/internal energy)

Prandtl number: Pr = ср1^0/к1; (energy dissipation/heat conduction).

From the perfect gas assumption: й = (y — 1)pe and T = yM^p/p relations among the non dimensional parameters are obtained.

In most of the aerodynamics applications there is high free stream speed involved. For the classical applications usually unseparated flows are considered.

Regardless of flow being attached or separated, for the flows with high free stream speeds we can apply some approximations to Eq. 2.53 to obtain simpler solutions. Let us now, see this approximations and conditions for their applicability.