Prediction of Turbulent Boundary-Layer Separation Influenced by Blowing

Abstract

presented previously. 1 ' 2 For some applications, the method of Ref. 2 might be preferred because with it the velocity vector of the moving object can be found without knowing the coordinates of the listening stations. However, at least three such stations are needed. The present Note gives an alternate method of computation in which only two observation posts are required. This can be advantageous in some cases.Let t be the time at which the shock wave arrives at a particular point. Finite difference approximations for dt/dx, dt/dy and dt/dz can be obtained by having two or more microphones along the x, y and z axes through the point, respectively. This gives Vt, the gradient of t. But the shock wave travels with the speed of sound, C, in the direction perpendicular to the wave front. Hence Vt is in the direction of the vector C, and its magnitude is 1/C. That is,(2) If the speed of sound is known, then only two of the partial derivatives of t need be measured. The third one can be calculated, except for sign, by means of Eq. (2). Thus a cluster of microphones can serve as an observation post to determine Vt, and therefore C. Fig. 1 The object at B 9 moving with velocity V greater than C, creates a shock cone as shown.Refer now to Fig. 1. It shows a portion of the shock cone generated by an object moving along path A 1 A 2 B with velocity V. Observers are at P 1 and P 2 . The shock wave has just arrived at P i and Q 2 . The time will be designated as t 19 and the arrival time at P 2 will be called t 2 . Hence Q 2 = P 2 -C 2 (; 2 -gThis gives the point Q 2 . Let i\ be the time required for the shock wave to travel from A l to P v and i 2 the time it spends going from A 2 to Q 2 . Then V(T 1 -T 2 ) = C 1 T 1 + Q 2 -P 1 -C 2 i 2 (4) But the cosine of the angle between V and C is equal to C/V. Therefore V C = C 2 . So dotting Eq. (4) with C 2 and then C 1 gives (C'-CVC^^C^-PJ (5) (G 2 -C 1 -C 2 )i 2 = C 1 -(P 1 -Q 2 ) (6) These equations give r 1 and i 2 unless C t = C 2 . If (C 1 + C 2 )-(Q 2 -P x ) is not zero, then T I and T 2 will be different, and V can be obtained from Eq. (4). Also, A^P^C^ (7) Thus the point A 1 on the path of the moving object, and the time t 1 -i 1 when it was there, are also known.