Turbulent Marginal Separation and the Turbulent Goldstein Problem

Abstract

Symbols α Slenderness parameter β Control parameter, Eq. (13) χ Coupling parameter, Eq. (93) χ b Upper bound of χ, Eq. (95) δ BL thickness ℓ Mixing length, Eq. (8) ǫ Bifurcation parameter (redefined), Eq. (42) ε Notion for gauge function, Eqs. (18), (31) η,η Similarity variables based on s, Eq. (19), andŝ, Eq. (77) η,η,η Similarity variables, Eqs. (30), (35), (55) Γ Gamma function γ Bifurcation parameter, Eq. (15) κ V. Kármán constant, Eqs. (11), (135) λ, µ Invariance parameters, Eq. (120) ω Exponent, Eqs. (31), (36) φ Local scaling function, Eqs. (35), (37) ψ Stream function ρ, ϑ Polar coordinates, Eq. (71) ρ, ϑ Polar coordinates (UD), Eq. (110) σ TD length scale, Eq. (94) τ Surface friction, Eq. (135) θ Heaviside functioñ ν Kinematic viscosity ∆ BL thickness (BL solution), Eq. (6) Γ Upstream limit ofÛ s , Eqs. (94), (118), (124) Λ Strength of induced pressure, Eqs. (96), (101), (115) Ω Eigenvalue Ψ Stream function (BL solution), Eq. (6) Υ Shear stress gradient evaluated at surface, Eq. (126) Ξ, Φ Coefficients in asymptotic series, Eq. (127) ξ BL coordinate related to ∆, Eq. (12) ζ Auxiliary variable Superscripts * Onset of separation

Break-away separation for high turbulence intensity and large Reynolds number

“…On the other hand, it is a widely accepted fact that a free turbulent shear layer is a fully developed and 'thick' one insofar as its thickness is essentially independent of Re and measures the turbulence intensity concentrated in it. However, from an empirical point of view, such a shear layer can still be regarded as relatively slender though, as was first put in a formal asymptotic concept successfully by Schneider (1991); for boundary layers see Melnik (1989) and Scheichl & Kluwick (2007b). Hence, the existence of separated flow here does not restrain us from regarding the Reynolds stresses as negligibly small entirely within an extent of a typical body dimension from the body surface, so that (2.2a) and (2.2b) reduce to the Euler equations in the present regime.…”

Turbulent flow on a planar moving belt and a rotating disk: modelling and comparisons

“…The resolution of this problem requires an enhanced boundary layer where the thickness must be taken to be significantly larger than for flows driven by an outer stream. This phenomenon has been noted independently in a different context by Scheichl (2001) (see also Scheichl & Kluwick 2007a, b). Analytical and numerical solutions are then found for the velocity components in both layers, and these results are used to determine the displacement thickness.…”

Turbulent interactions for rotating blades and wakes

“…The simpler case of the turbulent boundary layer on a rotating disk was examined in [10] where it was found that the boundary layer is relatively massive, a result also reported in [11] concerning turbulent jets and [12] concerning marginal separation. In this paper we extend the work of [10] to include non-axisymmetry by considering the fully three-dimensional turbulent boundary layer (cf.…”