PS7_solution - AAE 334 PS-7 SOLUTIONS Note In general the...

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AAE 334 PS-7 SOLUTIONS Note: In general, the final values of flow variables are presented with 5 significant digits .The numbers used in the calculations are of 7 significant digits. The symbol * represents arithmetic multiplication. PROBLEM 1a For linearized supersonic theory, we have And Where is expressed in radians. The deviation of the lift and drag coefficient to the exact values as given by shock expansion theory can be expressed as |( ) ( ) | ( ) |( ) ( ) | ( ) The results for three angles of attack are given in table-1 Table-1 (deg) (rad) ( ) ( ) ( ) ( ) 5 15 30
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It is quite evident from this table that as the angle of attack is increased, the estimated values from the linear theory for both the lift coefficient and the wave drag coefficient become increasingly inaccurate. This can be explained by the fact that the linear theory assumes small angles of attack and this is a fairly good assumption up to an angle of attack of 15 degrees, as far as the accuracy of the lift and drag coefficient for a flat plate are concerned. PROBLEM 1b We have by definition the expression for the pressure coefficient to be In class it was proved that ( ) Solving for the pressure ratio, For the lower surface, from linear theory And for the upper surface From equations E1.1 and E1.2, we have ( ) With ( )
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From equations E1.1 and E1.3, we have ( ) With ( ) The deviation of the pressure ratio from the exact values as given by shock expansion theory can be expressed as ( ) |(( ) ) (( ) ) | (( ) ) ( ) |(( ) ) (( ) ) | (( ) ) The results for the three angles of attack are given in table-2a and table-2b. Table-2a (deg) (rad) .( ) / .( ) / ( ) 5 15 30 We see that as the angle of attack increases, the values for the pressure ratio as given by the linear theory become more inaccurate. As far as the pressure ratio is concerned, an angle of attack greater
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than 5 degrees yields unacceptable values for the pressure ratio as given by the linear theory, with a percentage deviation of more than 5 percent. Table-2b (deg) (rad) .( ) / .( ) / ( ) 5 15 30 Beyond an angle of attack of five degrees, the linear theory gives unphysical values for the pressure coefficient over the upper surface of the airfoil. Up to an angle of attack of about five degrees, the percentage deviation is reasonable.
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