Integral Equations for Boundary Layers with Streamwise Vortices
G.L. De Oliveira Andrade (TU Delft - Wind Energy)
Nando Timmer (TU Delft - Wind Energy)
Bas W. van Oudheusden (TU Delft - Aerodynamics)
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Abstract
We explore integral boundary layer approximations for shear layer flows with vortex generators. The flow field is decomposed to highlight two phenomena: shear over the wall and vortex-driven mixing of the shear layer. The Navier-Stokes Equations are normalized to identify a new adimensional parameter: the vortex strength number (Vg). Usual boundary layer scales are valid when the
vortex strength number (Vg) is of order one or smaller. New Boundary Layer Equations comprising the effect of streamwise vortex filaments are obtained and integrated accross a periodic vortex cell. The new integral equations share their structure with the original Von Karmann Integral Equations but use different variables. The deduction concludes with an approximate interaction equation for
the construction of generalized closures from the classic set of Swafford turbulent closure relations. The new formulation is solved numerically and it is compatible with future integration in the Xfoil or Rfoil viscous-inviscid airfoil analysis codes.