Determination of Statistical Characteristics of Isotropic Turbulence

Ultrasonic flow measurement technology has the potential to significantly improve flow measurement accuracy beyond that presently achieved. However, current applications fail to achieve theoretical accuracies because of turbulence effects on ultrasonic wave transit time. Herein, the ultrasonic flowmeter equation is reconsidered, where the effects of turbulent velocity and sound speed fluctuations are included. The result is an integral equation for the corresponding correlation functions. In this paper experimental velocity data are used to solve this integral equation analytically. As a result, some statistical characteristics of the turbulent flow are developed and can explain the limitations of measurement accuracy observed in applications. NOMENCLATURE c Mean speed of sound c ' Speed of sound fluctuations u Mean velocity component in X ­ direction u ' Fluctuations of velocity component in X direction S Sound wave travel path *Address all correspondence to this author. w.w. Durgin Mechanical Engineering Department Worcester Polytechnic Institute Worcester, Massachusetts 01609 Email: wwdurgin@wpLedu H. Johari Mechanical Engineering Department Worcester Polytechnic Institute Worcester, Massachusetts 01609 Email: [email protected] 1)..1 Sound wave travel time N Number of experiments KtJ (S, S') Spatial correlation function oftravel time t1..t K ' ( s, s ') Spatial correlation function of u X component velocity fluctuations KC' (s, s ') Spatial correlation function of sound speed fluctuations 'Pi (X), If/j (X ') Shape functions INTRODUCTION Technical advances in ultrasonics for the past twenty years have resulted in the development of electronic instrumentation capable of measuring very small time differences associated with changes in the ultrasound wave propagation time (Lynnworth, 1989). These capabilities are used effectively in oblique path flowmeters. The ability of ultrasound to measure noninvasively, nondestructively and rapidly is clearly desirable (Yeh and Mattingly, 1998). Nevertheless, the accuracy of these Requestors must com I ." COPyright law (Title 17 uPl We'th .. ~ . Ode) Copyright© 2001 by ASME 571 devices has not improved very ~uch, certainly not proportional to improvements in the measurement technology. We believe the explanation lies in the effect of turbulence on the ultrasonic wave propagation. Our objective in this research is an attempt to find analytical solution of the ultrasonic flowmeter equation; namely an integral equation for the corresponding correlation functions derived from general ultrasonic flowmeter equation. The paper focuses on construction of statistical characteristics of the turbulent flow based on experimental data. Our approach in this paper is: 1. Develop statistical characteristics of the ultrasonic wave travel time using an ensemble of experiments to construct a spatial correlation function of the pulse propagation time period. 2. To solve an inverse problem of finding analytical spatial correlation functions of turbulent velocity and sound speed variations based on the experimentally obtained spatial correlation function of time. METHOD OF APPROACH The method utilizes ultrasonic pulses that travel in straight paths. Figure 1 illustrates the ultrasonic flowmeter in a duct. The upper and lower transducer locations and signal path are shown. In the limiting case, the mean flow should not affect the transit time of the signal.