Template Function es::deficit

Function Documentation

template <typename T_eta, typename T_param>
T_eta es::deficit(T_eta const &eta, const double kappa, T_param const &pi_coles, const double shear_ratio, const bool lewkowicz = false)

Get the velocity deficit integrand \( f \) used in computation of \( R_{13} \).

TODO move into velocity.h, reuse in lewkowicz_speed and jones_speed functions;. In their original derivation of the ADEM, Perry and Marusic 1995 eqn. 2 use the velocity distribution of Lewkowicz 1982 as a basis to determine the shear stress profile. This led to an integrand for the mean velocity deficit profile:

\[\begin{split} f & = & \frac{U_{1} - \overline{U}}{U_{\tau}} \\ & = & - \frac{1}{\kappa} \ln \left( \eta \right) + \frac{\Pi}{\kappa} W_c[1, \Pi] - \frac{\Pi}{\kappa} W_c[\eta, \Pi], \\ W_c[\eta, \Pi] & = & 2 \eta^2 \left( 3 - 2\eta \right) - \frac{1}{\Pi}\eta^2 \left( 1 - \eta \right) \left( 1 - 2\eta \right) \end{split}\]

Jones, Marusic and Perry (2001) eqn 1.9 uses an alternative formulation, thereby removing the non-physical dependency of the wake factor on . This leads to the velocity deficit function:

\[\begin{split} f & = & \frac{U_{1} - \overline{U}}{U_{\tau}} \\ & = & - \frac{1}{\kappa} \ln \left( \eta \right) + \frac{1}{3\kappa} \left(\eta^{3} - 1 \right) + 2 \frac{\Pi}{\kappa} \left( 1 - 3\eta^{2} + 2\eta^{3} \right) \end{split}\]

This function is templated so that it can be called with active scalars (allows use of autodiff), doubles/floats, Eigen::Arrays (directly) or Eigen::VectorXds (the latter via template specialisation) of z values.

Return
Parameters
  • eta: Nondimensional height values
  • kappa: von Karman constant
  • pi_coles: The coles wake parameter Pi
  • shear_ratio: Shear ratio S
  • lewkowicz: Boolean flag, default false. If true, use Lewkowicz (1982) velocity deficit funciton. If false, use Jones et at (2001).