The Dean Eccentricity Parameter and Data Condition

Abstract

The ability to identify the empirical force coefficients used in the Morison wave force equation depends on the condition of the wave kinematic data and the measured force data. Herein the condition of these data is evaluated by three methods: two geometric and one numerical. The Dean error ellipse method is shown to depend explicitly on the Keulegan-Carpenter parameter, K, for simple harmonic kinematics via the Dean eccentricity parameter (E =3K/2π2 ). For simple harmonic data, the alignment of the semi-minor axis of the Dean error ellipse is shown to be parallel to the Cd –Cm -axes and to depend on K. When E <1.0, then K<11.40 and the semi-minor axis of the error ellipse is parallel to the Cm -axis. When E >1.0, then K>11.40 and the semi-minor axis is parallel to the Cd -axis. When E =1.0, then K=11.40 and the error ellipse is a circle with zero eccentricity. The Dean error ellipse method is compared geometrically with the amplitude/phase error analysis in which the condition of the data is demonstrated geometrically by the slopes of contours of dimensionless coefficient ratios passing through a zero phase error. These two geometric methods are shown to be related by the Dean eccentricity parameter, E . The matrix A condition number is also shown to depend on E . The matrix for simple harmonic kinematics is Hermitian, unitary and becomes a unit matrix with a unit condition number when K=13.16 and E =1.15. There is only a 15-percent difference between K=11.4 and K=13.16, which is usually well within the data scatter band for wave force tests on vertical smooth cylinders near these values of K. Three sets of physical data show that at K values where the data are equally well conditioned to determine both Cd and Cm , a stable condition of periodic, repetitive vortices are formed. For such stable vortex shedding in periodic flow, the force on the cylinder is stable periodic, yielding repetitive values for Cd and Cm from cycle to cycle.