The Attached Eddy Hypothesis and von Kármán’s Constant

Townsend’s attached eddy hypothesis states that the flow in the logarithmic region of wall-bounded turbulent flows will be dominated at the energy-containing scales by a hierarchy of eddies, whose corresponding velocity fields extend to the wall [20]. These eddies are assumed to be geometrically self-similar, differing from each other only in their size, which scales with their distance from the wall. The hypothesis has subsequently gained significant support from high Reynolds number experiments and from numerical simulations [17]. Recently, a more rigorous physical and mathematical basis for the attached eddy hypothesis has been put forward by Marusic and Woodcock [12]. In this present work, we utilise this analysis to investigate the predicted nature of von Kármán’s constant (κ), which has been a source of controversy, particularly since Townsend [20] argued that κ should change at very high Reynolds numbers. We show that strictly applying the attached eddy hypothesis results in von Kármán’s constant rapidly approaching a constant value as the Reynolds number increases. Introduction The great complexity of turbulent flows has always been a huge barrier to the development of practical physical models of the phenomenon. Furthermore, the direct numerical simulation of turbulent flows is limited by Reynolds number due to the increasing multitude of scales that need to be resolved. One prominent model for wall-bounded flows stems from the so-called attached eddy hypothesis of A. A. Townsend [19]. Townsend’s hypothesis, states that the flow in the log-region consists of a series of geometrically self-similar eddies, which scale with their distance from the wall, and whose corresponding velocity fields extend to the wall. In this way, the study of a highly complicated phenomenon involving many scales of motion is effectively reduced to the study of a single representative eddying motion. In order to produce statistical predictions from the attached eddy hypothesis, Townsend adopted a distribution of eddy sizes specifically in order to obtain a constant Reynolds shear stress [20]. Using this model, Townsend was able to derive the second-order moments of the velocity as a function of the distance from the wall. If u, v and w represent the velocity fluctuations in the streamwise, spanwise and wall-normal directions respectively, he obtained 〈 u2 〉+ = B1−A1 log(z/δ) , (1) 〈 v2 〉+ = B1,v−A1,v log(z/δ) , (2) 〈 w2 〉+ = B1,w, (3) −〈uw〉 = 1, (4) where the angled brackets represent ensemble averages. Here, δ denotes the maximum distance from the wall at which the flow is dominated by the presence of the attached eddies (i.e. the boundary layer thickness), and the superscript + indicates that the quantities have been scaled according to wall variables, that is with Uτ, the friction velocity or ν/Uτ, the viscous length scale. All of the As and Bs above are constants. This result only applies where the flow is sufficiently close to the wall to be affected by its presence and yet sufficiently far from the wall that the effect of viscosity is negligible. The above equations have subsequently been vindicated by high Reynolds number experiments [3, 4, 9, 11, 13, 21] and direct numerical simulations [17]. Various authors have reviewed the nature of the log-region in recent years [1, 5, 6, 10, 18]. While there remain differing interpretations of the causal relationships between the coherent structures in the log-region, a consensus has emerged that the region contains a hierarchy of eddies, whose behaviours and distribution concur with Townsend’s hypothesis. It has been shown that such a self-similar hierarchical structure is consistent with invariant solutions associated with the leading order dynamics [7, 8]. Townsend’s result was extended by Perry and coworkers [14, 15, 16], who also used the attached eddy hypothesis along with a prescribed distribution of eddy sizes to obtain the classical logarithmic law of the wall: 〈U〉 = 1 κ log ( z ) +C, (5) where κ is von Kármán’s constant, and C depends on the roughness of the surface, but is otherwise constant. It is noted that the both of these derivations (for equations 1-5) were predicated on the adoption of a prescribed distribution of eddy sizes, and also on the assumption that there are no correlations between eddies of different sizes. Recently, Woodcock & Marusic [12, 22] formulated a new derivation of the attached eddy model, which minimised the number of assumptions. They avoided specifying either a prescribed distribution of eddy sizes or a constant Reynolds shear stress. In order to do this, they presented an extended form of Campbell’s theorem (originally a method used to account for the random arrival of electrons at an anode, and now applied to the random placement of eddies on a wall). Using this, they were able to derive all of the moments of the velocity fluctuations. In this work, we look at the implications of this new derivation of the attached eddy model for von Kármán’s constant. Previously, Townsend [20] and Davidson [2] have predicted that the attached eddy hypothesis should result in a von Kármán’s constant that continually changes with the Reynolds number. Townsend argued that von Kármán’s constant will increase as the ratio of the energy present in the fluctuations to that present within the mean flow increases. He therefore concluded that any such variations would be unlikely to be detectable under ordinary circumstances, but would become important at extremely large Reynolds numbers. Conversely however, we find that von Kármán’s constant initially increases with the Reynolds number, but rapidly converges to a constant. Mathematical Formulation Following the attached eddy hypothesis, the velocity distribution is modelled as the superposition of the velocity fields corresponding to each of the eddies present. The eddies are all of identical shape and relative dimensions, and differ only in their heights. Each individual eddy can therefore be seen as a separate system. Its defining characteristics are its height, h, and its location on the wall, xe. The length scale of the eddy will therefore be h, while the friction velocity will be its velocity scale. Therefore, if Q is the velocity field at x corresponding to an individual eddy, then its spatial and height dependence will be Q = Q ( x−xe h ) . (6) The total velocity, U(x), is then simply the superposition of the velocity fields corresponding to each of the individual eddies. However, we could never postulate the locations and sizes of all eddies present, and so we must instead consider only the statistical properties of the entire flow. (We apply the method of images at the wall, z = 0, in order to determine the boundary conditions. This will become important subsequently.) The distribution of eddy sizes follows from the observation that h is the system’s only natural length scale. From dimensional analysis, for eddies that are space-filling we can see that if ρh denotes the density of eddies of size h, then