Question

The aerodynamic behavior of a flying insect is to investigate in a water tunnel using a 10X scale model. The lift force F is believed to be a function of wing size L, flight speed u, flapping frequency w, fluid density p, and fluid viscosity u. Assume that air has a density 1/1000th and a dynamic viscosity 1/100that of water. List all parameters, n. List all primary dimensions, r. Find all dimensionless parameters. (Hint: Buckingham 1 Theorem) Assuming the insect flap its wings 50 times per second when flying at 1 m/s in air. Find the water tunnel speed and wing flapping frequency for dynamic similarity. What is the ratio between the drag measured in the water tunnel and that of the flying insect in air, under dynamically similar conditions

Answer 1

Varaiable SI Unit MIT dimension lift force F MLT 1 wing site L flight speed a mis LT-1 1 flapping Frequency w sec T? -3 fluid density s kghm? ME viscosity a lasec -1 -1 MLI Here, Total variables n = 6 and all primary dimension, (mass, length, timelt Consider L, u and 3 45 hepeating vaziable а b с T = 1 ust . MOLT = (6² CLT") (ML-3° CM LT2) (+1 atb-36+1 molt a - b-2 T

... (+420 -b-2-0 b=-2 atb-3c+1=0 a-2+3+1=0 >fa a= -2 -2 -2 Tg = Louber L и E Ty = 2 8u²L² a b c ha Lu's w molt = (La CLT 2 b Cm 2-33 (+7) atb-3c : moloto= mu -6-1 T C =0 ab-1=0 (b=-1 atb-30=0= a - 1 - 00 a = 1 с b T= Illus W = L u pºw T₂= WL e T3 = Loubs en Mo Leto = (La CLT CML CM[²7)

MOLT = mcts, atb-6-1 -b-1 T ctito (c = -1 -b-1=0 (b= -1 atb-3e-1=0 & a-1 +3-1=0 [a=-1 us u Lu S ll T = ll Sul given, Prototype Model L L water dannel 1 Se Im le Jooo 1 Joo -7 Wp- 50 sec = 1 mis Lox Scale & Lm 1 10 Up - Le using Tg = M SUL elp Sp uphp Ilm Sm Um Lm i u 'm = Upx 3p Le Lm Um Im le Anower um = I x 1oo Jooo X 10x - Un = I mis Water tunnel speed J

Using Ty = WL u Wple - Wm lm Up 50x wm - Wp x Lp Lm Um Um Up Wm= 50x 1 10 1 I -1 cm = 5 sei in sec & Brower wing flapping frequency for water tunnel Now Use Tg = Fm 3u²12 Fe Sp up LP 2 Sm2 Lm 2 um Lm Fm F CU Sp Up 4 Fm 1000 x plan Fe Fm F - 1o Answer Ratio between the drag measured in the londer tunnel and that of the flying insect in arz