First steps in Aerodynamics

with CFX TASCflow


III. Transonic flow

1. Mach 0.2

The fluids mechanics basements can be used in such a case because of the high Reynolds number property of the flow. The flow velocity is equal to 68 m.s-1, the reference length is about 5 meters, and the air viscosity is about 10-5. Thus, the Reynolds number is :

In such a way, the viscosity effects can be neglected in front of the pressure effects. By using the Euler equations, which represent an approximation of the Navier Stokes equations in that case, and considering a steady one-dimensional phenomenon, one can write :

Then, by integrating the first equation and by using this result in the integration of the second one, one obtains :

Thus, the quantity is also conserved along the stream lines. In addition, thanks to the perfect fluid theory and according to the profile design, it's possible to predict a speed increasing a speed increasing in the first part (upstream) of the extrados, followed by a speed decreasing in the other part (downstream) of the extrados. A zero velocity point must be observed too.

Residuals evolution

Mach number graph around the profile (initial Mach number = 0.2)

The results are relevant compared with the few theoretical points exposed in the top of this part.

2. Mach 0.5

The analysis used in the previous case is still adapted to this Mach number flow. The absolute speed variation on the extrados is more important because of the increasing of the initial flow velocity.

Residuals evolution

Mach numbers graph around the profile (initial Mach number = 0.5)

3. Mach 0.8

Residuals evolution

Mach numbers graph around the profile (initial Mach number = 0.8)

It can be noted that the number of iterations required to complete the calculations increases with the Mach number. In fact, it could be a consequence to the more dominant role of the fluid compressibility when the initial velocity get closer to the sound velocity.