Polynomial represents a C-D Nozzle!

[tareqkh]

Hi All,

I am working on a problem where I would hope that there is a solution or a general formula for it. I am trying to find a polynomial represents a C-D nozzle as in the following picture. The polynomial must except the length of the C-D nozzle, nozzle radius, diffuser radius, throat radius, as well as the number of points that lay on the polynomial.

Is there any formula for this shape like that for an airfoil shape or do I need to create it from scratch

? The polynomial must always fit by either change R1, R2, R, and L or by changing one variable?

Many thanks in advance,

 

[tkhunny]

Well, sure it CAN be done, but should it? Are you CERTAIN a polynomial is an appropriate model? Polynomials can exhibit unexpected behavior outside a reasonable region. Generally, you need a polynomial of degree n-1 if you are trying to fit n parameters. Show us a real problem and demonstrate your best work.

 

[tareqkh]

Thanks for the reply!

Many thanks for the reply. Is there any way to fit a quadratic polynomial to two points? I need to fit the nozzle, which contains only two points with a quadratic polynomial and the same for the divergent part with another quadratic polynomial.

Thanks

 

[mmm4444bot]

Is there any way to fit a quadratic polynomial to two points?

Yes, but an infinite number of parabolas pass through those two points. For a unique polynomial, you need three points.

I need to fit the nozzle, which contains only two points with a quadratic polynomial and the same for the divergent part with another quadratic polynomial.

So it's not a true C-D nozzle then, but, rather, some generalized representation?

The curves forming the image of the C-D nozzle below are not parabolas.

 

[tareqkh]

Thanks for the reply!

Well, that is the target. I need to fit an infinite number of quadratic polynomials.

 

[mmm4444bot]

Well, that is the target. I need to fit an infinite number of quadratic [polynomial curves].

I don't understand how to do that. If there were an infinite number of curves, it seems like each curve would need to be reduced to a single point (requiring you to know in advance the position of each point).

Is this a course assignment?

 

[tareqkh]

I don't understand how to do that. If there were an infinite number of curves, it seems like each curve would need to be reduced to a single point (requiring you to know in advance the position of each point).

Is this a course assignment?

No, it is not an assignment. I am working on a mesh for a CFD analysis and faced this issue. However, I have solved the problem by simply taking the derivative of the quartic polynomial to find the minimum and the then solve a 3X3 matrix. For instance, if the nozzle coordinates (Ln,RN) where Ln is the length of the nozzle and Rn is the radius and the throat is (Lt,Rt). Therefore,
Rn = a1(Ln)^2+a2(Ln)+a3,
Rt = a1(Lt)^2+a2(Lt)+a3,
dy/dx = 0=2a1x+a2.

Then solve the matrix for the coefficients. That's it. The same can be applied of the diffuser.

I hope that it is clear.

Thanks for hospitality anyway.

Regards,