Vortical Flows

Vortex Cavitation

Cavitation in vortex flow is an important class of problems in hydraulic and marine engineering. Vortical motions of all types are common, e.g. tip vortices in propellers. Unfortunately the mechanism of cavitation under these circumstances is not fully understood. In spite of extensive research, details about the structure of tip vortices are still unknown.

Figure 1
Figure 1. Cavitation induced tip vortices on a propeller.

Scaling Laws

The most significant paramter is the cavitation index, defined as


where pv is the vapor pressure, p∞, ρ, and U are the static pressure, density, and velocity in the free stream, respectively. The occurence of cavitation is related to the pressure distribution in the liquid. The relationship is as follows:


where σi is the value of cavitation index at inception, Cpmin is the minimum pressure in normalized form:

In the case of the tip vortex, Cpmin is a function of angle of attack, vortex roll-up process, core size, boundry layer on the foil, and other parameters. Δσ is a complex function of dissolved gas and nuclei distribution.

Figure 2
Figure 2. The experimental set up for the velocity profile measurements.

Four facilities were utilized in the experimental setup:
  1. 7.5" variable pressure, high speed water tunnel: cavitation inception and noise study.
  2. 7" variable pressure, high speed water tunnel: velocity measurement with laser Doppler velocimetry.
  3. 12" water channel for velocity profile measurement at low Reynolds numbers with laser Doppler velocimetry.
  4. 12x17" wind tunnel: a surface flow visualization study.

The experimental set up for the velocity profile measurement was comprised of two types of laser Doppler velocimeters (LDV's). One was a two-component system with an argon laser and Bragg cells. The other, shown below, is a relatively simple one-component system.

Figure 3 demonstrates the complex interaction between the effects of gas content and viscosity. At high gas content, the variation with Reynolds number is small. On the other hand, there is a bifurcation in the inception data for Reynolds numbers greater than or less than 600,000 which occurs at higher angles of attack.

Figure 3
Figure 3. Effects of gas content and viscosity. ENLARGE

The inception index varies with the level of pressurization at the start of the test (expressed in the nondimensional form of σinit). The difference between σi with high values of σinit and σi determined with low values of σinit is conjected to be due to tension in the liquid, as a result of a dearth of nuclei in the approaching flow.

Figure 4
Figure 4. σi vs. σinit ENLARGE

The tip vortex roll-up is of practical importance in many areas in addition to the cavitation problem. In the present investigation, tip vortex cavitation provides an excellant method for visualizing the mechanism of vortex roll-up. The center of the vortex moves inward, approaching the theoretical asymtotic position as depicted in Figure 5.

Figure 5
Figure 5. Tip vortex roll-up ENLARGE

The velocity profile of the tip vortex was measured under various test conditions. The tangential velocity profiles at different downstream locations are shown Figure 6. The detail of the circulation distribution from the core region to the outer flow can be seen on the second figure on the far bottom. The results of a semi-empirical model of the vortex structure is also shown in Figure 7.

Figure 6
Figure 6. Tip vortex roll-up ENLARGE
Figure 7
Figure 7. Tip vortex roll-up ENLARGE

Surface cavitation develops at lower pressure in the region where the trailing edge boundry layer separation and reattachment were predicted and observed. The altered spanwise lift distribution produces secondary vortices which become entrained into the primary vortex. Figure 8 shows the predicted viscous flow pattern on the suction side at α - α0 = 5.5 degrees, as well as the observed behavior.

Figure 8
Figure 8. Viscous flow pattern on the suction side. ENLARGE

At certain angles of attack, boundry layer characteristics were found to be very sensitive to the Reynolds number as shown in Figure 9. This qualitatively explains the complex interrelationships between the gas content and the Reynolds number.

Figure 9
Figure 9. Boundry layer for different Reynolds numbers. ENLARGE
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