For the velocity field of Prob. 4-23, calculate the fluid acceleration along the diffuser centerline as a function of x and the given parameters. For L = 1.56 m, uentrance = 24.3 m/s, and uexit=16.8 m/s, calculate the acceleration at x =0 and x = 1.0 m.

PROBLEM: Consider steady flow of air through the diffuser portion of a wind tunnel (Fig. P11–20). Along the centerline of the diffuser, the air speed decreases from uentrance to uexit as sketched. Measurements reveal that the centerline air speed decreases parabolically through the diffuser. Write an equation for centerline speed u(x), based on the parameters given here, from x = 0 to x = L.

FIGURE:

Solution 23P

Step 1</p>

In this problem, a fluid is flowing through a diffuser. So, we have to calculate the acceleration of the fluid through the center line of the diffuser.

So, we have the equation for velocity of the diffuser. It can be written as,

Where, - fluid velocity at entrance through the center line

- fluid velocity at exit through the center line

- Length of the diffuser

- Position at center line

Step 2</p>

Finding out the acceleration through center line

Standard expression for the acceleration in three dimensional space is,

But, in the problem, they have provided that, the flow is steady. So, the time derivative would be zero.

That is,

Similarly, the fluid flow is only towards the center line. Therefore, the terms and will be zero.

Therefore,

Therefore, the expression for the acceleration towards the center line is,