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r13_morrison

# r13_morrison - 13.42 Morrison's Equation SPRING 2004 A H...

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13.42 04/01/04: Morrison’s Equation SPRING 2004 ©A. H. TECHET 1. General form of Morrison’s Equation Flow past a circular cylinder is a canonical problem in ocean engineering. For a purely inviscid, steady flow we know that the force on any body is zero (D’Allembert’s paradox). For unsteady inviscid flow this is no longer the case and added mass effects must be considered. Of course in the “real” world, viscosity plays a large role and we must consider, in addition to added mass forces, viscous drag forces resulting from separation and boundary layer friction. Following on Tuesday’s lecture, the resulting force on a body in an unsteady viscous flow can be determined using Morrison’s Equation, which is a combination of an inertial term and a drag term. The force in the x-direction on a body in unsteady flow with velocity U(t) is 1 ( ) = D t F t ( ) = ρ C U ± + ρ C AU U (1.1) x m 2 d In order to obtain rough estimates of the magnitude of the force of a body, it is advantageous to use Morrison’s equation with constant coefficients. Supposing we want to find the estimates of the wave forces on a fixed structure, then the procedure would be as follows: 1.) Select an appropriate wave theory (linear waves, or other higher order if necessary). 2.) Select the appropriate C M and C D based on Reynolds number, and other factors (see table below). 3.) Apply Morrison’s Equation

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Wave Theory C d C m Comments Reference Linear Theory 1.0 0.95 Mean values for ocean wave data on 13-24in cylinders Wiegel, et al (1957) 1.0- 2.0 Recommended design values based Agerschou and 1.4 on statistical analysis of published Edens (1965) data Stokes 3 rd order 1.34 1.46 Mean Values for oscillatory flow Keulegan and for 2-3in cylinders Carpenter (1958) Stokes 5 th order 0.8- 2.0 Recommended values based on Agerschou and 1.0 statistical analysis of published data Edens (1965) We can see from the above table that for linear waves the recommended values for drag and mass coefficients are 1.0-1.4 and 2.0, respectively. The range of drag coefficients allows us to account for roughness and Reynolds number effects. These values are for rough estimates. In reality these coefficients vary widely with the various flow parameters and with time. Bretschneider showed that the values of C D and C M can even vary over one wave cycle. Even if we ignore the time dependence of these coefficients we must account for the influence of other parameters. Reynolds number and roughness effects: For smooth cylinders at Reynolds numbers around 10 5 , laminar flow transitions to turbulent flow, and there is a dip in C D as a function of Re. For larger Reynolds numbers the separation point remains essentially constant and thus so does the drag coefficient. In this range C D is Reynolds number independent. Roughness causes
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r13_morrison - 13.42 Morrison's Equation SPRING 2004 A H...

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