# how to creat contraction cone of a wind tunnel based on higher order polynomial equation

i want to creat the contraction cone of wind tunnel based on higher order polynomial curve. can anyone help me?

### 1 Answer

(1) y = ax + bx + cx + dx + ex + fx + g

The chosen profile has 7 parameters (a-g). Five of these are specified by the inlet and outlet height, zero slope at the inlet and outlet and zero curvature at outlet. This leaves two parameters available for optimisation. These are specified by the inlet curvature and the axial position of the point of inflection relative to the contraction length. The 7 conditions defining the profile are thus:

(2) y(x=0)=h y"(x=i)=0 y'(x=0)=0 y(x=L)=0

y"(x=L)=0 y"(x=0)α y'(x=L)=0

where:

h = inlet half height – exit half height

α = inlet curvature

i = axial location of inflection point

L = length of contraction

The conditions specified by (2) directly provide the following constants for the polynomial (1):

g = h; f= 0 ; e= α/2

The other constants are defined by the equation:

(3) Aw = B

where, for α = 0 for the standard case

(with no inlet curvature):

(4) A=[30i^4, 20i^3, 12i^2, 6i, L^6, L^5, L^4, L^3, 6L^5,

5L^4, 4L^3, 3L^2, 30L^4, 20L^3, 12L^2, 6L],

B=[0, -h, 0, 0],

w=[a, b, c, d]

The range of the variable, i, distance to the point of inflection, which gives a sensible, monotonically decreasing curve is 0.4-0.6 L. In order to optimise the shape, the optimal position of the point of inflection was determined first, and the degree of curvature at inlet was varied for this optimal design.