The Direct Measurement of Circulation in Free Surface Vortices

Ultrasonic teclmiques have been used to directly and non-intrusively measure the circulation of free surface vortices. All experiments were performed in a vertical cylindrical tank with a central drain and a tangential inlet The circulation was measured on a closed triangular path by measuring the difference in upstream and downstream transit-times. Circulation was measured as a function of the Reynolds and Froude numbers and was found to increase as the Reynolds and Froude numbers increased. The circulation was also found to be proportional to the square of the ratio of the drain diameter to cylinder diameter while the ratio of fluid depth to cylinder diameter was held constant. Minimum surface elevations were measured at various conditions and attempts were made to correlate them with measured circulation. Background Nomenclature The rotating flow that is formed when a container C .. speed of sound is drained is an example of a free surface vortex. The d drain diameter size and strength of such vortices is an important D .. cylinder diameter consideration in many engineering applications such as Fr ... Froude Number'" Y/(gH)tf2 fuel and liquid transfers at various gravity levels, the g gravitational acceleration draining of storage tanks or fuel draw down in rockets, h ... vortex dimple (air core) depth at the inlets to pumps and piping systems, and in H .. fluid depth in the cylinder hydraulic structures and fluid machinery. 1·6 L length of ultrasonic beam path The free surface vortex is a complex, three­ Q volume flow rate through system dimensional, viscous flow field with unsteady axial, Re .. Reynolds Number" Yd/v radial, and tangential velocity components that vary Tlin .. downstream transit-time spatially. A relatively stable vortex can be formed in T~ upstream transit-time cylindrical tanks and many studies have been preformed Y Mean discharge (drain) velocity to characterize the flow field. -10 The flow in the outlet We Weber Number'" pyd/a may possess considerable axial velocity, stretching and .:\T '" transit-time difference .. T~ Tdft amplifying the vortex ftlaments. The radial velocity r .. circula tion component has been found to vary with both radius, r, p density and axial position, Z.7. I The tangential velocity a .. surface tension component of the flow has been found to be inversely I-L '" viscosity proportional to radius and independent of depth except v .. kinematic viscosity for a small region near the bottom of the tank where the r '" radial coordinate boundary layer affects the flow.'-9 Z .. axial coordinate Measurement of the three velocity components in free surface vortices is difficult because the vortex typically tends to move within the container. Measurements of the velocity components have been *Graduate Student, Member AIAA undertaken through various methods.7.9 A serious tProfcssor, Member AIAA problem in taking measurements in vortical flows is that lAssistant Professor, Member AIAA physical probes tend to affect the flow. Qualitative classification of vortex strength using visual methods Copyright @ 1994 by R. H. Smith, Jr., W. W. Durgin, was pursed by Durgin and Hecker. 1O In the present and H. Johari, Worcester Polytechnic Institute. effort, ultrasonic techniques were used to directly, and Published by the American Institute of Aeronautics non-intrusively measure the circulation of free surface and Astronautics, Inc. with pennission. vortices. Dimensional Arguments Figure I shows a general representation of a free surface vortex and the cylindrical tank. The following parameters were determined to be important to the problem Wlder study: gravitational acceleration g, drain diameter d, cylinder diameter D, dimple (air core) depth h, fluid depth H, circulation r, fluid density p, fluid viscosity ~, mean drain velocity V, and fluid surface tension o. Dimensional reasoning provides the following functional relationships for the circulation: Ultruon;c Beam V'OCOllS Cor. Path (l), A"Cor. r d H (I) Vd =F.(Re, Fr, We, D' D) and for the air core depth: h d H =F2 (Re Fr We -) (2) H '" D' D where: Vd V Re=Fr=-­ v J8H (3) Daggett & Keulegan' found the same parameters for similitude although their arrangement is slightly different. For the present experiments under consideration, the cylinder diameter D, the free surface depth H, and of course gravity g were held constant. The fluid properties p, ~, and 0 were assumed to remain constant since the experiments were preformed in an area with nearly constant temperature. No precautions were taken to control these parameters. The drain diameter d and the volume flow rate (Q ­ (ltVd)/4) were varied independently. The volume flow rate and the drain diameter are used to calculate the drain velocity which is used to calculate the Reynolds, Froude, and Weber numbers. The drain velocity is also used to non-dimensionalize the circulation. Previous measurements' have indicated that Weber number effects on the discharge coefficient and circulation are small. ~-~ --H t . \/ L_~ n.------J .J l. = d i;c= Figure 1. Schematic sketch of the cylindrical tank and relevant parameters Experimental Apparatus The experiment was designed to establish a stable nee surface vortex. The experimental apparatus consisted of a cylindrical tank, a pump, a needle valve, a 1/2H offset flow cell II, two Panametrics PT 868 Ultrasonic flowmeters, four 1 MHz ultrasonic transducers and a cathatometer. Distilled, deionized water was used as the working fluid. A block diagram of the apparatus is shown in Figure 2.