# 低速風洞收縮段計算
###### tags: `Sponsorship`

For low-speed wind tunnels to be used in aerospace industries, the best area ratio $(AR)$ is typically between 7 to 12. Here we choose 12 to be our design point. The area of the test section is:

$$
A_{test} = h_{test} w_{test} = (4.0\text{ m})(2.5\text{ m}) = 10\text{ m}^2
\\
A_{inlet} = (AR) A_{test} = 120\text{ m}^2
$$
With a wind speed of $40 \text{ m/s}$, the volume flow rate through the test section is:
$$
Q = A_{test} V_{test} = (10 \text{ m}^2) (40 \text{ m/s}) = 400 \text{ m}^3\text{/s}
$$
The contraction inlet length should be approximately as the height of the rectangular inlet:
$$
L_{contraction} = h_{inlet} = h_{test} \sqrt{AR} = (4.0 \text{ m}) (3.464) = 13.86\text{ m}
$$
The contraction angle of height and width would be:
$$
tan(\frac{\alpha}{2}) = \frac{h_{inlet}-h_{test}}{2L_{contraction}} = 0.36
\\
tan(\frac{\beta}{2}) = \frac{w_{inlet}-w_{test}}{2L_{contraction}} = 0.22
\\
\frac{\alpha}{2} = 19.80 \deg
\\
\frac{\beta}{2} = 12.41 \deg
$$

Where:
$$
w_{inlet} = w_{test} \sqrt{AR} = (2.5\text{ m}) (3.464) = 8.66\text{ m}
$$

There are several options including parabolic, elliptical, and ogive curves, just like the fairings of the rocket. However, the best curvature of contraction is a 5th degree polynomial. Here we follow the recommendations of J. Bell and R. Metha (1988), the polynomials are:
$$
h = [−10Ϛ^3 + 15Ϛ^4 − 6Ϛ^5](𝐻_𝑖 − 𝐻_0 ) + 𝐻_i
\\
w = [−10Ϛ^3 + 15Ϛ^4 − 6Ϛ^5](W_𝑖 − W_0 ) + W_i
$$

Where, $Ϛ = x/L$ ; $H_i$ and $H_o$ is the height of contraction inlet and outlet.

The flow velocity is calculated by:
$$
V(x) = \frac{Q}{A(x)}
$$
By Bernoulli’s principle, we can calculate the pressure of each cross section by:
$$
p(x) = \frac{1}2 \rho \left(V_{inlet}^2-V(x)^2\right) + p_0
$$
where ambient air density, pressure and the inlet flow velocity are:
$$
\rho = 1.225 \text{ kg/m}^3
\\
p_0 = 101325 \text{ N/m}^2
\\
V_{inlet} = \frac{Q}{A_{inlet}} = 3.33 \text{ m/s}
$$
From the equations above, we can get the result below:

| x (cm) | h (m) | w (m) | A (m^2) | AR | V (m/s) | P (N/m^2) |
|-|-|-|-|-|-|-|
| 0 | 4.000 | 2.500 | 10.000 | 1.000 | 40.000 | 100351.79 |
| 10 | 4.000 | 2.500 | 10.000 | 1.000 | 39.999 | 100351.83 |
| 20 | 4.000 | 2.500 | 10.001 | 1.000 | 39.994 | 100352.08 |
| 30 | 4.001 | 2.501 | 10.005 | 1.000 | 39.981 | 100352.74 |
| 40 | 4.002 | 2.501 | 10.011 | 1.001 | 39.955 | 100354.01 |
| 50 | 4.004 | 2.503 | 10.022 | 1.002 | 39.912 | 100356.07 |
| 60 | 4.007 | 2.505 | 10.037 | 1.004 | 39.851 | 100359.10 |
| 70 | 4.012 | 2.507 | 10.059 | 1.006 | 39.766 | 100363.23 |
| 80 | 4.017 | 2.511 | 10.087 | 1.009 | 39.655 | 100368.62 |
| 90 | 4.024 | 2.515 | 10.123 | 1.012 | 39.516 | 100375.38 |
| 100 | 4.033 | 2.521 | 10.166 | 1.017 | 39.345 | 100383.61 |
| 110 | 4.044 | 2.527 | 10.219 | 1.022 | 39.142 | 100393.39 |
| 120 | 4.056 | 2.535 | 10.282 | 1.028 | 38.904 | 100404.78 |
| 130 | 4.070 | 2.544 | 10.355 | 1.035 | 38.629 | 100417.80 |
| 140 | 4.087 | 2.554 | 10.439 | 1.044 | 38.318 | 100432.47 |
| 150 | 4.106 | 2.566 | 10.535 | 1.053 | 37.969 | 100448.77 |
| 160 | 4.127 | 2.579 | 10.643 | 1.064 | 37.583 | 100466.66 |
| 170 | 4.150 | 2.594 | 10.765 | 1.076 | 37.159 | 100486.07 |
| 180 | 4.176 | 2.610 | 10.900 | 1.090 | 36.698 | 100506.92 |
| 190 | 4.205 | 2.628 | 11.049 | 1.105 | 36.201 | 100529.10 |
| 200 | 4.236 | 2.647 | 11.214 | 1.121 | 35.670 | 100552.48 |
| 210 | 4.270 | 2.669 | 11.394 | 1.139 | 35.106 | 100576.94 |
| 220 | 4.306 | 2.691 | 11.591 | 1.159 | 34.511 | 100602.32 |
| 230 | 4.346 | 2.716 | 11.804 | 1.180 | 33.887 | 100628.46 |
| 240 | 4.388 | 2.743 | 12.035 | 1.204 | 33.236 | 100655.20 |
| 250 | 4.433 | 2.771 | 12.284 | 1.228 | 32.562 | 100682.37 |
| 260 | 4.481 | 2.801 | 12.552 | 1.255 | 31.867 | 100709.81 |
| 270 | 4.533 | 2.833 | 12.840 | 1.284 | 31.153 | 100737.35 |
| 280 | 4.587 | 2.867 | 13.148 | 1.315 | 30.424 | 100764.85 |
| 290 | 4.643 | 2.902 | 13.476 | 1.348 | 29.682 | 100792.15 |
| 300 | 4.703 | 2.940 | 13.826 | 1.383 | 28.931 | 100819.13 |
| 310 | 4.766 | 2.979 | 14.198 | 1.420 | 28.173 | 100845.64 |
| 320 | 4.832 | 3.020 | 14.593 | 1.459 | 27.411 | 100871.60 |
| 330 | 4.901 | 3.063 | 15.011 | 1.501 | 26.647 | 100896.89 |
| 340 | 4.973 | 3.108 | 15.454 | 1.545 | 25.884 | 100921.43 |
| 350 | 5.047 | 3.154 | 15.921 | 1.592 | 25.124 | 100945.16 |
| 360 | 5.125 | 3.203 | 16.413 | 1.641 | 24.370 | 100968.02 |
| 370 | 5.205 | 3.253 | 16.932 | 1.693 | 23.624 | 100989.95 |
| 380 | 5.288 | 3.305 | 17.477 | 1.748 | 22.887 | 101010.95 |
| 390 | 5.374 | 3.359 | 18.049 | 1.805 | 22.161 | 101030.97 |
| 400 | 5.463 | 3.414 | 18.650 | 1.865 | 21.448 | 101050.02 |
| 410 | 5.554 | 3.471 | 19.278 | 1.928 | 20.749 | 101068.10 |
| 420 | 5.648 | 3.530 | 19.936 | 1.994 | 20.065 | 101085.20 |
| 430 | 5.744 | 3.590 | 20.622 | 2.062 | 19.396 | 101101.36 |
| 440 | 5.843 | 3.652 | 21.339 | 2.134 | 18.745 | 101116.58 |
| 450 | 5.945 | 3.715 | 22.086 | 2.209 | 18.111 | 101130.89 |
| 460 | 6.048 | 3.780 | 22.864 | 2.286 | 17.495 | 101144.33 |
| 470 | 6.154 | 3.847 | 23.673 | 2.367 | 16.897 | 101156.92 |
| 480 | 6.263 | 3.914 | 24.513 | 2.451 | 16.318 | 101168.71 |
| 490 | 6.373 | 3.983 | 25.386 | 2.539 | 15.757 | 101179.72 |
| 500 | 6.486 | 4.054 | 26.290 | 2.629 | 15.215 | 101190.00 |
| 510 | 6.600 | 4.125 | 27.226 | 2.723 | 14.692 | 101199.58 |
| 520 | 6.716 | 4.198 | 28.194 | 2.819 | 14.187 | 101208.51 |
| 530 | 6.835 | 4.272 | 29.195 | 2.920 | 13.701 | 101216.82 |
| 540 | 6.955 | 4.347 | 30.228 | 3.023 | 13.233 | 101224.54 |
| 550 | 7.076 | 4.423 | 31.294 | 3.129 | 12.782 | 101231.72 |
| 560 | 7.199 | 4.499 | 32.391 | 3.239 | 12.349 | 101238.39 |
| 570 | 7.323 | 4.577 | 33.521 | 3.352 | 11.933 | 101244.58 |
| 580 | 7.449 | 4.656 | 34.682 | 3.468 | 11.533 | 101250.32 |
| 590 | 7.576 | 4.735 | 35.874 | 3.587 | 11.150 | 101255.64 |
| 600 | 7.704 | 4.815 | 37.098 | 3.710 | 10.782 | 101260.58 |
| 610 | 7.833 | 4.896 | 38.352 | 3.835 | 10.430 | 101265.16 |
| 620 | 7.963 | 4.977 | 39.635 | 3.964 | 10.092 | 101269.41 |
| 630 | 8.094 | 5.059 | 40.948 | 4.095 | 9.768 | 101273.35 |
| 640 | 8.226 | 5.141 | 42.290 | 4.229 | 9.459 | 101277.00 |
| 650 | 8.358 | 5.224 | 43.659 | 4.366 | 9.162 | 101280.38 |
| 660 | 8.490 | 5.307 | 45.055 | 4.506 | 8.878 | 101283.52 |
| 670 | 8.623 | 5.390 | 46.477 | 4.648 | 8.606 | 101286.42 |
| 680 | 8.757 | 5.473 | 47.924 | 4.792 | 8.347 | 101289.12 |
| 690 | 8.890 | 5.556 | 49.395 | 4.939 | 8.098 | 101291.63 |
| 700 | 9.023 | 5.640 | 50.888 | 5.089 | 7.860 | 101293.95 |
| 710 | 9.157 | 5.723 | 52.403 | 5.240 | 7.633 | 101296.10 |
| 720 | 9.290 | 5.806 | 53.938 | 5.394 | 7.416 | 101298.11 |
| 730 | 9.423 | 5.889 | 55.491 | 5.549 | 7.208 | 101299.97 |
| 740 | 9.555 | 5.972 | 57.061 | 5.706 | 7.010 | 101301.69 |
| 750 | 9.687 | 6.054 | 58.647 | 5.865 | 6.820 | 101303.30 |
| 760 | 9.818 | 6.136 | 60.247 | 6.025 | 6.639 | 101304.79 |
| 770 | 9.949 | 6.218 | 61.860 | 6.186 | 6.466 | 101306.18 |
| 780 | 10.078 | 6.299 | 63.483 | 6.348 | 6.301 | 101307.47 |
| 790 | 10.207 | 6.379 | 65.115 | 6.512 | 6.143 | 101308.68 |
| 800 | 10.335 | 6.459 | 66.754 | 6.675 | 5.992 | 101309.80 |
| 810 | 10.461 | 6.538 | 68.398 | 6.840 | 5.848 | 101310.84 |
| 820 | 10.586 | 6.617 | 70.046 | 7.005 | 5.711 | 101311.82 |
| 830 | 10.710 | 6.694 | 71.695 | 7.169 | 5.579 | 101312.73 |
| 840 | 10.833 | 6.770 | 73.343 | 7.334 | 5.454 | 101313.57 |
| 850 | 10.954 | 6.846 | 74.989 | 7.499 | 5.334 | 101314.36 |
| 860 | 11.073 | 6.921 | 76.630 | 7.663 | 5.220 | 101315.10 |
| 870 | 11.190 | 6.994 | 78.264 | 7.826 | 5.111 | 101315.79 |
| 880 | 11.306 | 7.066 | 79.890 | 7.989 | 5.007 | 101316.44 |
| 890 | 11.420 | 7.137 | 81.505 | 8.150 | 4.908 | 101317.04 |
| 900 | 11.531 | 7.207 | 83.107 | 8.311 | 4.813 | 101317.60 |
| 910 | 11.641 | 7.276 | 84.694 | 8.469 | 4.723 | 101318.13 |
| 920 | 11.748 | 7.343 | 86.264 | 8.626 | 4.637 | 101318.62 |
| 930 | 11.853 | 7.408 | 87.815 | 8.782 | 4.555 | 101319.08 |
| 940 | 11.956 | 7.473 | 89.345 | 8.935 | 4.477 | 101319.52 |
| 950 | 12.057 | 7.535 | 90.853 | 9.085 | 4.403 | 101319.92 |
| 960 | 12.155 | 7.597 | 92.335 | 9.234 | 4.332 | 101320.30 |
| 970 | 12.250 | 7.656 | 93.791 | 9.379 | 4.265 | 101320.65 |
| 980 | 12.343 | 7.714 | 95.219 | 9.522 | 4.201 | 101320.98 |
| 990 | 12.433 | 7.771 | 96.616 | 9.662 | 4.140 | 101321.29 |
| 1000 | 12.521 | 7.825 | 97.981 | 9.798 | 4.082 | 101321.58 |
| 1010 | 12.606 | 7.878 | 99.313 | 9.931 | 4.028 | 101321.86 |
| 1020 | 12.688 | 7.930 | 100.609 | 10.061 | 3.976 | 101322.11 |
| 1030 | 12.767 | 7.979 | 101.869 | 10.187 | 3.927 | 101322.35 |
| 1040 | 12.843 | 8.027 | 103.091 | 10.309 | 3.880 | 101322.57 |
| 1050 | 12.917 | 8.073 | 104.273 | 10.427 | 3.836 | 101322.78 |
| 1060 | 12.987 | 8.117 | 105.415 | 10.541 | 3.795 | 101322.97 |
| 1070 | 13.055 | 8.159 | 106.515 | 10.651 | 3.755 | 101323.15 |
| 1080 | 13.119 | 8.200 | 107.572 | 10.757 | 3.718 | 101323.32 |
| 1090 | 13.181 | 8.238 | 108.586 | 10.859 | 3.684 | 101323.48 |
| 1100 | 13.240 | 8.275 | 109.556 | 10.956 | 3.651 | 101323.63 |
| 1110 | 13.295 | 8.310 | 110.481 | 11.048 | 3.621 | 101323.76 |
| 1120 | 13.348 | 8.343 | 111.360 | 11.136 | 3.592 | 101323.89 |
| 1130 | 13.398 | 8.374 | 112.193 | 11.219 | 3.565 | 101324.01 |
| 1140 | 13.445 | 8.403 | 112.981 | 11.298 | 3.540 | 101324.11 |
| 1150 | 13.489 | 8.431 | 113.722 | 11.372 | 3.517 | 101324.21 |
| 1160 | 13.530 | 8.456 | 114.417 | 11.442 | 3.496 | 101324.31 |
| 1170 | 13.569 | 8.480 | 115.067 | 11.507 | 3.476 | 101324.39 |
| 1180 | 13.604 | 8.503 | 115.671 | 11.567 | 3.458 | 101324.47 |
| 1190 | 13.637 | 8.523 | 116.229 | 11.623 | 3.441 | 101324.54 |
| 1200 | 13.667 | 8.542 | 116.743 | 11.674 | 3.426 | 101324.60 |
| 1210 | 13.695 | 8.559 | 117.214 | 11.721 | 3.413 | 101324.66 |
| 1220 | 13.720 | 8.575 | 117.642 | 11.764 | 3.400 | 101324.71 |
| 1230 | 13.742 | 8.589 | 118.028 | 11.803 | 3.389 | 101324.76 |
| 1240 | 13.762 | 8.601 | 118.374 | 11.837 | 3.379 | 101324.80 |
| 1250 | 13.780 | 8.613 | 118.681 | 11.868 | 3.370 | 101324.83 |
| 1260 | 13.796 | 8.622 | 118.950 | 11.895 | 3.363 | 101324.87 |
| 1270 | 13.809 | 8.631 | 119.184 | 11.918 | 3.356 | 101324.89 |
| 1280 | 13.821 | 8.638 | 119.384 | 11.938 | 3.351 | 101324.92 |
| 1290 | 13.831 | 8.644 | 119.552 | 11.955 | 3.346 | 101324.94 |
| 1300 | 13.839 | 8.649 | 119.691 | 11.969 | 3.342 | 101324.95 |
| 1310 | 13.845 | 8.653 | 119.803 | 11.980 | 3.339 | 101324.96 |
| 1320 | 13.850 | 8.656 | 119.891 | 11.989 | 3.336 | 101324.97 |
| 1330 | 13.854 | 8.659 | 119.956 | 11.996 | 3.335 | 101324.98 |
| 1340 | 13.857 | 8.660 | 120.003 | 12.000 | 3.333 | 101324.99 |
| 1350 | 13.858 | 8.661 | 120.033 | 12.003 | 3.332 | 101324.99 |
| 1360 | 13.859 | 8.662 | 120.051 | 12.005 | 3.332 | 101324.99 |
| 1370 | 13.860 | 8.662 | 120.060 | 12.006 | 3.332 | 101324.99 |
| 1380 | 13.860 | 8.662 | 120.062 | 12.006 | 3.332 | 101324.99 |
| 1386 | 13.860 | 8.663 | 120.062 | 12.006 | 3.332 | 101324.99 |

## Reference
1. [Design Methodology for a Quick and Low-Cost Wind Tunnel](www.intechopen.com/chapters/42327)
2. [Analysis of transformed fifth order polynomial curve for the contraction of wind tunnel by using OpenFOAM](https://iopscience.iop.org/article/10.1088/1757-899X/377/1/012048/pdf)
3. [Contraction Design for Small Low-Speed Wind Tunnels](https://www.semanticscholar.org/paper/Contraction-Design-for-Small-Low-Speed-Wind-Tunnels-Bell-Mehta/d78fabf4192488181c6642994e868834cfe501e7)