Nonlocal stochastic mixing-length theory and the velocity profile in the turbulent boundary layer

Abstract

Nonlocal stochastic mixing-length theory and the velocity profile in the turbulent boundary layer Dekker, H.; de Leeuw, G.; Maassen van den Brin, A. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. AbstractTurbulence mixing by finite size eddies will be treated by means of a novel formulation of nonlocal K-theory, involving sample paths and a stochastic closure hypothesis, which implies a well defined recipe for the calculation of sampling and transition rates. The connection with the general theory of stochastic processes will be established. The relation with other nonlocal turbulence models (e.g. transilience and spectral diffusivity theory) is also discussed. Using an analytical sampling rate model (satisfying exchange) the theory is applied to the boundary layer (using a scaling hypothesis), which maps boundary layer turbulence mixing of scalar densities onto a nondiffusive (Kubo-Anderson or kangaroo) type stochastic process. The resulting transport equation for longitudinal momentum Px = ptJ is solved for a unified description of both the inertial and the viscous sublayer including the crossover. With a scaling exponent ~ 0.58 (while local turbulence would amount to s ~ oo) the velocity profile U+ =f(y+) is found to be in excellent agreement with the experimental data. Inter alia (i) the significance of e as a turbulence Cantor set dimension, (ii) the value of the integration constant in the logarithmic region (i.e. if y+ --* oo), (iii) linear timescaling, and (iv) finite Reynolds number effects will be investigated. The (analytical) predictions of the theory for near-wall behaviour (i.e. if y+ ~ 0) of fluctuating quantities also perfectly agree with recent direct numerical simulations. Introduction The velocity profileDimensional analysis of simple shear flow along an infinitely extended smooth surface immediately yields that the nondimensional mean velocity U+ = t J/u, (in the x-direction) must be a function of y+ = u,y/v, where u, is a reference (friction) velocity, y is the distance from the surface, and v is the kinematic viscosity. The friction * Corresponding author. Dekker et al./ Physica A 218 (1995) velocity is defined by means of the total stress z = pu2, in the boundary layer (p being the fluid mass density). In fully developed turbulent fl...

Enhancement of finite Reynolds number effects: Inner-outer sublayer interaction in the turbulent boundary layer

“…First, one recovers the result for the infinite boundary layer (inertial sublayer, buffer layer and viscous sublayer) of Dekker et al [5][6][7][8] by letting 7r --+ oo for finite y and a, viz. The analytic solution of (2) with (5) has been given in [7]:…”

“…In addition, OlC/Ou = 0 also at y = g (as it is symmetric in y -"R). The details of this symmetry are not relevant to the results (pertaining to the inertial sublayer 1 << y << 7-r for a << Tr The viscous correlation length has the value a ~ 6 [6][7][8]. The parameter r controls the flow type (pipes or channels are modeled by r ~ 0.3-0.8 while boundary layers per se are represented by r ~ 0; see, e.g., [12,18]).…”

“…Combining a wake model (where E(y) --+ y ify --+ 0; see [3]) and viscous sublayer model (where E(y) --+ y ify --+ ec; see [6][7][8]) leads to …”

“…In previous work (e.g., [6][7][8]) we inter alia re-examined the established data for the mean velocity profile (e.g., [10,11,13]) for turbulent flow over a smooth surface and found that its Reynolds number dependence can be described by the formula Aff+ ~ c0Re -1 + O(Re-2),…”

“…In the present paper we clarify Equation (1) by deriving it from eddy-viscosity theory, thereby showing that the remarkably large value of co (typically significant in the inertial sublayer) arises from the interplay between the outer part of the boundary layer (i.e., the wake) and its inner part (i.e., the viscous sublayer). The analysis proceeds in Section 2 by combining two (on their own rigorously solvable) eddy-viscosity models: one valid in the viscous sublayer (taken from Dekker et al [6][7][8]), the other pertaining to the wake (taken from Dekker [3]), and both asymptotically containing the inertial sublayer (for y+ -+ c~ and y/5 -4 O, respectively). The wake model covers both channel (or pipe) flow and boundary layers per se.…”