quiz-solution2

# quiz-solution2 - 650:460 Aerodynamics Quiz No 2 Solution...

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Unformatted text preview: 650:460 Aerodynamics Quiz No. 2 - Solution Prof. Doyle Knight Tel: 732 445 4464, Email: [email protected] Oﬃce hours: Tues and Thur, 4:30 pm - 6:00 pm and by appointment Fall 2009 1 Quiz No. 2 Consider uniform ﬂow along the x −axis with magnitude U∞ past a line source of strength K located at (x , y ) = (0, h). The x −axis is a solid boundary. The ﬂuid is incompressible and irrotational. The freestream pressure is p∞ . Find an expression for the pressure coeﬃcient cp (x ) on the solid boundary. y . .... .. . .. .. K.......... .. . ... .... .... ... h ....... . .... .. .. .. . ....... . ... x 2 U∞ ... p∞ . ....... Quiz No. 2 Solution A second source of equal magnitude must be added at y = −h to obtain a streamline coincident with the x −axis. The velocity potential is φ = U∞ x + uniform ﬂow upper source lower source K log 2π x 2 + (y − h)2 + K log 2π x 2 + (y + h)2 The x −velocity on the x −axis is K K x x x K ∂φ + = U∞ + = U∞ + 2 + h2 2 + h2 2 + h2 ∂x 2π x 2π x πx upper source lower source 3 Quiz No. 2 The pressure coeﬃcient is cp = From Bernoulli’s equation 1 1 2 p∞ + ρ∞ U∞ = p + ρ∞ (u 2 + v 2 ) 2 2 and therefore on the x − axis where v = 0, cp = 1 − u U∞ 2 (p − p∞ ) 1 2 2 ρ∞ U∞ x 2K − =− π U∞ x 2 + h2 K π U∞ 2 x2 (x 2 + h2 )2 4 ...
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