The stagnation point von Kármán coefficient

الأصل · EN

On the basis of various DNS of turbulent channel flows the following picture is proposed. (i) At a height y from the y = 0 wall, the Taylor microscale λis proportional to the average distance lₛ between stagnation points of the fluctuating velocity field, i.e. λ(y) = B₁ lₛ(y) with B₁ constant, for δν<< y δ. (ii) The number density nₛ of stagnation points varies with height according to nₛ = Cₛ y+⁻¹ / δν³ where Cₛ is constant in the range δν<< y δ. (iii) In that same range, the kinetic energy dissipation rate per unit mass, ε= 2/3 E+ uτ³ / (κₛ y) where E+ is the total kinetic energy per unit mass normalised by uτ² and κₛ = B₁² / Cₛ is the stagnation point von Kármán coefficient. (iv) In the limit of exceedingly large Reτ, large enough for the production to balance dissipation locally and for -<uv> uτ² in the range δν<< y << δ, dU+/dy 2/3 E+/(κₛ y) in that same range. (v) The von Kármán coefficient κis a meaningful and well-defined coefficient and the log-law holds only if E+ is independent of y+ and Reτin that range, in which case κ κₛ. The universality of κₛ = B₁² / Cₛ depends on the universality of the stagnation point structure of the turbulence via B₁ and Cₛ, which are conceivably not universal. (vi) DNS data of turbulent channel flows which include the highest currently available values of Reτsuggest E+ 2/3 B₄ y+⁻²/¹⁵ and dU+/dy+ B₄/(κₛ) y+⁻¹ ⁻ ²/¹⁵ with B₄ independent of y in δν<< y << δif the significant departure from -<uv> uτ² is taken into account.