Function es::reynolds_stress_13

Function Documentation

void es::reynolds_stress_13(Eigen::ArrayXd &r13_a, Eigen::ArrayXd &r13_b, const double beta, const Eigen::ArrayXd &eta, const double kappa, const double pi_coles, const double shear_ratio, const double zeta)

Compute Horizontal-Vertical Reynolds Stress R13 profile.

Gets Reynolds Stress profiles due to Type A and B eddies.

Can use either the Lewkowicz formulation for velocity deficit (per Perry and Marusic 1995) or

TODO - presently, the integrations here are sensitive to the chosen eta distribution. Refactor according to issue #38

TODO - Determine whether it will be an advantage to template this function 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 eta values.

References

[1] Perry AE and Marusic I (1995) A wall-wake model for turbulent boundary layers. Part 1. Extension of the attached eddy hypothesis J Fluid Mech vol 298 pp 361-388

Future Improvements

[1] Optional different mean profile formulations including account for the free surface

[2] Support directional variation with height; i.e. compatible with mean profile formulations using U(y)

[3] Added formulation for contribution of smaller Type C eddies

[4] For the given wall formulation, can f be calculated more efficiently? See schlichting and gersten p. 593

Parameters
  • beta: Clauser parameter \( \Beta \), representing acceleration/decelaration of the boundary layer
  • eta: Nondimensional vertical coordinates at which you want to get stress \( R_{13} \). Values must ascend but not necessarily be monotonic.
  • kappa: von Karman constant.
  • pi_coles: Coles wake parameter \( \Pi \)
  • shear_ratio: Ratio between free-stream and skin friction velocities \( S = U_{inf}/U_{\tau} \)
  • zeta: Scaled streamwise derivative \( \zeta \) of the Coles wake parameter \( \Pi \)