Incompressible Flow Navier-Stokes Equations

In a wide region of aerodynamical applications low subsonic speeds are encoun­tered. Since the free stream Mach number for these types of are very low, the flow is assumed incompressible. The continuity equation for the incompressible flow becomes

V • V = 0 (2.65)

Equation 2.65 implies that the flow is divergenless which in turn simplifies the constitutive relations, Eq. 2.51a, b. In addition, because of low speeds the temperature changes in the flow field will also be low which makes the viscosity remain constant. Since the viscosity is constant, the momentum equation is sim­plified also to take the following form

D~

PD =-Vp + lV2 ~ (2.66)

In case of turbulent flows, we use the effective viscosity: ie = i + iT in Eq. 2.66 which undergoes an averaging process after Reynolds decomposition which makes the final form of the equations to be called ‘Reynolds Averaged Navier-Stokes Equations’.

Another convenient form of incompressible Navier-Stokes equations is written in terms of a new variable called vorticity. The vorticity vector is derived from the velocity vector as

V = Vx~ (2.67)

The vorticity transport equation obtained from two dimensional version of Eq. 2.66 reads as

Here, x as the third component of the vorticity appears as a scalar quantity in Eq. 2.68, which does not have any pressure term involved. The integral form of Eqs. 2.65 and 2.67 reads as (Wu and Gulcat 1981),

1 X0x(r0 – r)

R

1 Z ~0 ■ r~) (~o – r) – (~0X~^)x(~o – r)

T 2 dB0

ro r

B

Here, R shows the region for vortical flow, B the boundaries, r and ro the position vectors and no the unit vector pointing outwards to the boundaries. The boundary B contains the airfoil surface and the far field boundary. While solving Eq. 2.68, we only consider the vertical region confined around the airfoil. Same is done for the evaluation of the velocity field via Eq. 2.69. The integro-differential formulation presented here, therefore, enables us to work with small computational domains. Another use of Eq. 2.69 comes into picture while determining the surface vortex sheet strength through the no-slip boundary condition.