Hydrodynamics on an Oscillating Cylinder by Morison’s Equation

The hydro loads on a circular cylinder in an unsteady viscous flow can be determined by Morison’s Equation, a combination of an inertial term and a drag term.

\[ F(t)=\rho C_m V \dot{v}+\frac{1}{2} \rho C_d A v|v|,

where  $\rho,\ C_m,\ C_d,\ V, \ A,\ v$ are the density, mass coefficient, drag coefficient, displacement, projected area (to vertical plane), and fluid speed, respectively. For linear waves, the range of drag coefficient is 1.0-1.4 and mass coefficient is 2.0. These values are for rough estimates, and in real situation these coefficients vary widely with the various flow parameters and with time. The drag coefficients should account for cylinder roughness and Reynolds number effects. For smooth cylinders at Reynolds numbers around  $10^5$ , laminar flow transitions to turbulent flow, and there is a dip in  $C_d$ .

Moving cylinder in unsteady flow.

For an oscillating cylinder in an unsteady viscous flow, the hydrodynamics can be calculated by extending the Morison’s equation as

\[ F_n=\rho C_m V (\dot{U}_n-\dot{V}_n)-\rho(C_m-1)V\dot{U}_n+\frac{1}{2} \rho C_d A (U_n-V_n)|U_n-V_n|,

     \[ F_t=\frac{1}{2}\rho C_f S(U_t-V_t)|U_t-V_t|\]

where  $U_n$ and  $V_n$ are the decomposition of  $U$ and  $V$ , respectively. It should be noted that the hydrodynamics at the top and bottom of the cylinder can usually not be neglected.

An oscillating cylinder under regular waves (implemented by Simulink) using S-function, as shown in following figure, can be download from morisonMovingCylinder.zip. The cylinder is divided into 10 elements. Note that the source files are compiled as mexw64 file.

morison equation

The source files can be found here. The codes for calculating the hydrodynamic force in irregular waves is here.

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