Template Function es::marusic_jones_speed¶
- Defined in File velocity.h
Function Documentation¶
-
template <typename T_z, typename T_pi_j>
T_zes
::
marusic_jones_speed
(T_z const &z, T_pi_j const pi_j, const double kappa, const double z_0, const double delta, const double u_inf, const double u_tau)¶ Compute speed profile according to Marusic and Jones’ relations.
Templated so that it can be called with active scalars (allows use of autodiff), doubles/floats, Eigen::Arrays (directly) or Eigen::VectorXds (via template specialisation) of z values.
Speed is computed as:
\[\begin{split} \frac{\overline{U}}{U_{\tau}} & = & \frac{1}{\kappa} \ln \left( \frac{z+z_0}{k_s} \right) + Br + \frac{\Pi_j}{\kappa} W_c[\eta, \Pi_j] \\ W_c[\eta, \Pi_j] & = & 2 \eta^2 \left( 3 - 2\eta \right) - \frac{1}{3\Pi_j}\eta^3 \\ \eta & = & \frac{z+z_0}{\delta + z_0} \end{split}\]which reduces to the defecit relation:
\[ U_{D}^{*} = \frac{U_\infty-\overline{U}}{U_{\tau}} = -\frac{1}{\kappa} \ln \left( \eta \right) + \frac{1}{3\kappa} \left(\eta^3 - 1 \right) + 2 \frac{\Pi_j}{\kappa} \left(1 - 3\eta^2 + 2\eta^3 \right) \]- Parameters
z
: Height(s) in m at which you want to get speed.pi_j
: Jones’ modification of the Coles wake factor. Double or AutoDiffScalar type acccepted.kappa
: von Karman constantz_0
: Roughness length - represents distance of hypothetical smooth wall from actual rough wall z0 = 0.25k_s (m)delta
: Boundary layer thickness (m)u_inf
: Speed of flow at z = delta (m/s)u_tau
: Shear / skin friction velocity (governed by ratio parameter S = u_inf / u_tau)