Derivation of dynamic wake model

Hello,everybody.
Question 1: after using the DWM model of fast. Farm, I want to deduce the Wake-Deficit model. According to the user guide, I know that the momentum equation used is eq. (1), but I get eq. (2) by deriving the thin shear layer (TL) approximation. I find that the Wake-Deficit model ignores the influence of the second term on the right. Why?
Question 2: we know the eddy visibility formulation expressions the influence of the ambient vapor (first term on the right-hand side) and wake shear layer (second term) on the vapor stresses in the wake, formula 3. According to the mixed length theory, I know that the eddy visibility is equal to the second term, but what theory is the first term on the right hand side based on? I have read all the articles on DWM development and quoted the three articles of Ainslie [1-3], but there are no resources on the Internet. Does anyone know how this is derived? Or have anyone share these three articles?

[1] Ainslie, J. F., 1986, “Wake Modelling and the Prediction of Turbulence Properties,” Proceedings of the Eighth British Wind energy Association Conference, Cambridge, Mar. 19–21, pp. 115–120.
[2] Ainslie, J. F., 1988, “Calculating the Flow Field in the Wake of Wind Turbines,” J. Wind. Eng. Ind. Aerodyn., 27, pp. 213–224.
[3] Ainslie, J. F., 1985, “Development of an Eddy Viscosity Model for Wind Turbine Wakes,” Proceedings of the BWEA Conference, pp. 61–66.

Dear YuMing.Yuan,

Regarding (1), I’m not sure what your second term represents, but it appears to be a time-average of turbulent fluctuations. However, the thin shear-layer approximation applied in FAST.Farm assumes quasi-steady conditions, so, there are no such fluctuations.

Regarding (2), the first term in FAST.Farm’s eddy viscosity formulation is similar to that given by the following paper:

H. A. Madsen and et al. Calibration and validation of the dynamic wake meandering model for implementation in an aeroelastic code. Journal of Solar Energy Engineering, November 2010. doi:https://doi.org/10.1115/1.4002555.

Best regards,