An attached-eddy elliptic approximation for two-dimensional structure functions in turbulent channel flow

We investigate the equal-height, two-dimensional second-order structure function of the streamwise velocity fluctuation u u u in turbulent channel flow. Specifically, at a given wall-normal location z z z we consider upper S 2 left parenthesis r Subscript x Baseline comma r Subscript y Baseline semicolon z right parenthesis equals left angle bracket left bracket u left parenthesis x plus r Subscript x Baseline comma y plus r Subscript y Baseline comma z right parenthesis minus u left parenthesis x comma y comma z right parenthesis right bracket squared right angle bracket S 2 ( r x , r y ; z ) = ⟨ [ u ( x + r x , y + r y , z ) − u ( x , y , z ) ] 2 ⟩ S₂(rₓ,ry;z)= [u(x+rₓ,y+ry,z)-u(x,y,z)]² , where left parenthesis r Subscript x Baseline comma r Subscript y Baseline right parenthesis ( r x , r y ) (rₓ,ry) are the streamwise and spanwise separations in a wall-parallel plane and left angle bracket bold dot right angle bracket ⟨ ⋅ ⟩ · denotes wall-parallel spatial averaging. Focusing on separations within the logarithmic region, we build on Townsend’s attached-eddy hypothesis and its hierarchical random additive-process formulation to introduce an attached-eddy elliptic approximation, in which the two-point geometry is represented by an elliptic in-plane footprint. The model yields a compact prediction for upper S 2 left parenthesis r Subscript x Baseline comma r Subscript y Baseline semicolon z right parenthesis <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.