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Blevins Wake Model

[1] Blevins R.D., Saint-Marcoux J-F, Influence of length of exposure on riser interference in deepwater from comparison between experimental and analytical work, Proceeding ISOPE 2007.

[2] Huse, Current force on individual elements of riser arrays, Proceeding ISOPE 1992.

The model proposed by Blevins is similar to the model proposed by Huse [2] for the drag coefficient but add a lift coefficient to push the downstream riser in the mid wake of the upstream riser as outlined in the next figure:

If U0 is the freestream velocity, the velocity behind the cylinder is expressed as U0-Ud, where is the deficit velocity. The formulation proposed by Blevins gives the following total velocity behind the riser: $$ U(x,y)=U_{0}\left[1-a_{1} \sqrt{\frac{C_{D10}D_{1}}{x}}exp\left(-\frac{a_{2}y^2}{C_{D20}D_{2}x} \right) \right] $$

  • \(a_{1}\), \(a_{2}\) and \(a_{3}\) are derived from experimental results,

  • \(C_{D10}\) and \(D_{1}\) are the upstream riser drag coefficient and diameter.

Introducing \(C_{D20}\) and \(D_{2}\) as the downstream riser drag coefficient and diameter, the resulting drag force on the second cylinder is:

\[ F_{D2}=\frac{1}{2}{\rho}D_{2}C_{D20}U^2(x,y) \]

The drag force on the first cylinder is the usual formulation:

\[ F_{D1}=\frac{1}{2}{\rho}D_{1}C_{D10}U^2_{0} \]

Blevins proposed to use a formulation for the lift coefficient based on the derivative of the drag coefficient. The upstream cylinder does not experience drag and the downstream cylinder will experience a lift towards the center wake of the first cylinder given by:

\[ F_{L2}=\frac{1}{2}{\rho}D_{2}C_{L2}U^2_{0} \]

where

\[ C_{L2}=\frac{a_{3}}{a_{1}a_{2}}D_{2}\frac{\partial{C_{D2}(x,y)}}{\partial{y}} ~~ \text{and} ~~ C_{D2}(x,y)=C_{D20} \left[1-a_{1} \sqrt{\frac{C_{D1}D_{1}}{x}}exp\left(-\frac{a_{2}y^2}{C_{D1}D_{1}x} \right) \right]^2 \]

This can also be rewritten as

\[ F_{L2}=\frac{1}{2}{\rho}D_{2}C_{D20}\frac{2a_{3}}{a_{1}a_{2}}U(x,y)\frac{\partial{U}}{\partial{y}}(x,y) \]

This formula is generalized to take into account more than one cylinder, based on the previous work done in DeepLines with the Huse formulation. The cylinder are sorted in the current direction so that a cylinder sees the wake of all cylinders located upstream.