VOL. 25, NO. 11, NOVEMBER 1988 J. AIRCRAFT 993 Downstream Vorticity Measurements from Ultrasonic Pulses Wolfgang Send . DFVLR—Institute of Aeroelasticity, Gottingen, Federal Republic of Germany Abstract The vorticity distribution behind a profile is obtained from the measurement of the running-time shift, which an ultra- sonic pulse experiences in wind tunnel flow. The spatial gradient of the running-time shift is related to the vorticity vector. The method is limited by the assumptions of incom- pressible flow and a high Reynolds number. Introduction T he investigation of flow patterns by means of ultrasonic pulses and some initial applications were published in 1982. 1 Running-time measurements were part of an investiga- tion concerning various blade tip shapes of rotor blades. 2 The velocity field induced by the downstream vorticity determines the typical shape of the running-time signals. The mathemati- cal model applied to describe the three-dimensional flow around a profile of finite thickness is a higher-order panel method. 3 The physical assumptions for the fluid are infinitely high Reynolds number and incompressible flow, which lead to a simplified vorticity transport equation. 4 The method covers the wide range from fixed wings to rotating systems; it is not bound by the need for the mathematical modeling of a given configuration. The predic- tion of lift is based theoretically on a relation between the running-time shift of an ultrasonic pulse passing the wake and the vorticity contained in it. Mathematical models of selected flowfields allow the theoretical simulation of wind tunnel tests and give some insight into the applicability * accurace and limitations of the method. 5 Preferred applications are compar-. ative measurements of tip shapes or the investigation of rotor wakes. 6 Compared with point-to-point measurements in a flowfield, e.g., by laser anemometry, the method provides integral values in thin and sensitive regions (such as vorticity in the wake). A previous paper presented by the author extends the method of ultrasonic pulse measurements to measure- ments of vorticity. 7 Running-Time Shift The running time is mathematically a well-defined function in the flowfield, and its gradient may be related to the vorticity density along the path of the signal. An ultrasonic pulse passing the flow around a wing experiences a typical shift of its running time. For any point r in the flowfield and any time t the relative velocity V rQl (r,t) with respect to the wing may be described by the kinematic motion with respect to a frame at rest and the perturbation velocity induced by the moving obstacle. A pulse transmitted at a point r T into a direction n(r T ,t) is initially propagated at a velocity c s n(r T ,t) (1) where c s /is the local velocity of sound. Assuming a constant velocity of sound throughout the fluid is equivalent to an Presented as Paper 86-1.9.1 at the 15th Congress of the Interna- tional Council of Aeronautical Sciences (1CAS), London, England, Sept. 7-12, 1986; received Dec. 11, 1986; revision received Oct. 26, 1987. Full paper available from AIAA Library, 555 W. 57th St., New York, NY, 10019. Price: microfiche, $4.00; hard copy, $9.00. Remit- tance must accompany order. Copyright © 1986 by ICAS and AIAA. All rights reserved. * Scientist, Department of Unsteady Aerodynamics. almost contant temperature in a wind tunnel, the drift of temperature during a long operating time of the tunnel has to be considered as a time-dependent correction of c s . The signal is converted by the fluid; its position after a given time interval results from an integration of the velocity in Eq. (1) and, therefore, is unknown in advance. However, as long as the relative velocity is small compared to the velocity of sound, the final position may be estimated quite accurately. In addition, the pulse widens during its passage through the fluid and finally covers an area large enough to be detected. We approximate the path of the signal by a straight line and prescribe the point r R , where the signal is supposed to be received by a microphone. The given spatial distance D and direction n D = \r R -r T \, n = l/D(r R -r T ) (2) between transmitter and receiver defines the running time t D by integration along r($) = r T + sn, which leads to c s n (3) (4) The arc length s parameterizes the p^th of the signal from the transmitter point r T = r(0) to the receiver point r R = r(D). The constant time t = t 0 in Eq. (3) indicates that the fluid is assumed to be frozen. The assumption is justified as long as changes in the flow patter are small during the interval t D , in which the signal travels through the fluid. With D/c s as the running-time shift At is defined by At: = t D - (5) At is exactly the physical variable, which originates from experiments with ultrasonic pulses, as well as from the corresponding theoretical predictions, and reflects basic phys- 6.0 At [jas] U Theory 0.0 (p [deg.l ——> 360. Fig. 1 Running-time shift in unsteady flow. 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