skip to main content

Title: A scaling law for the shear-production range of second-order structure functions
Dimensional analysis suggests that the dissipation length scale ( $\ell _{{\it\epsilon}}=u_{\star }^{3}/{\it\epsilon}$ ) is the appropriate scale for the shear-production range of the second-order streamwise structure function in neutrally stratified turbulent shear flows near solid boundaries, including smooth- and rough-wall boundary layers and shear layers above canopies (e.g. crops, forests and cities). These flows have two major characteristics in common: (i) a single velocity scale, i.e. the friction velocity ( $u_{\star }$ ) and (ii) the presence of large eddies that scale with an external length scale much larger than the local integral length scale. No assumptions are made about the local integral scale, which is shown to be proportional to $\ell _{{\it\epsilon}}$ for the scaling analysis to be consistent with Kolmogorov’s result for the inertial subrange. Here ${\it\epsilon}$ is the rate of dissipation of turbulent kinetic energy (TKE) that represents the rate of energy cascade in the inertial subrange. The scaling yields a log-law dependence of the second-order streamwise structure function on ( $r/\ell _{{\it\epsilon}}$ ), where $r$ is the streamwise spatial separation. This scaling law is confirmed by large-eddy simulation (LES) results in the roughness sublayer above a model canopy, where the imbalance between local production and dissipation of TKE more » is much greater than in the inertial layer of wall turbulence and the local integral scale is affected by two external length scales. Parameters estimated for the log-law dependence on ( $r/\ell _{{\it\epsilon}}$ ) are in reasonable agreement with those reported for the inertial layer of wall turbulence. This leads to two important conclusions. Firstly, the validity of the $\ell _{{\it\epsilon}}$ -scaling is extended to shear flows with a much greater imbalance between production and dissipation, indicating possible universality of the shear-production range in flows near solid boundaries. Secondly, from a modelling perspective, $\ell _{{\it\epsilon}}$ is the appropriate scale to characterize turbulence in shear flows with multiple externally imposed length scales. « less
Award ID(s):
Publication Date:
Journal Name:
Journal of Fluid Mechanics
Page Range or eLocation-ID:
459 to 474
Sponsoring Org:
National Science Foundation
More Like this
  1. We discuss the Onsager theory of wall-bounded turbulence, analysing the momentum dissipation anomaly hypothesized by Taylor. Turbulent drag laws observed with both smooth and rough walls imply ultraviolet divergences of velocity gradients. These are eliminated by a coarse-graining operation, filtering out small-scale eddies and windowing out near-wall eddies, thus introducing two arbitrary regularization length-scales. The regularized equations for resolved eddies correspond to the weak formulation of the Navier–Stokes equation and contain, in addition to the usual turbulent stress, also an inertial drag force modelling momentum exchange with unresolved near-wall eddies. Using an Onsager-type argument based on the principle of renormalization group invariance, we derive an upper bound on wall friction by a function of Reynolds number determined by the modulus of continuity of the velocity at the wall. Our main result is a deterministic version of Prandtl’s relation between the Blasius − 1 / 4 drag law and the 1/7 power-law profile of the mean streamwise velocity. At higher Reynolds, the von Kármán–Prandtl drag law requires instead a slow logarithmic approach of velocity to zero at the wall. We discuss briefly also the large-eddy simulation of wall-bounded flows and use of iterative renormalization group methods to establish universal statistics inmore »the inertial sublayer. This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.« less
  2. We employ numerically implicit subgrid-scale modeling provided by the well-known streamlined upwind/Petrov–Galerkin stabilization for the finite element discretization of advection–diffusion problems in a Large Eddy Simulation (LES) approach. Whereas its original purpose was to provide sufficient algorithmic dissipation for a stable and convergent numerical method, more recently, it has been utilized as a subgrid-scale (SGS) model to account for the effect of small scales, unresolvable by the discretization. The freestream Mach number is 2.5, and direct comparison with a DNS database from our research group, as well as with experiments from the literature of adiabatic supersonic spatially turbulent boundary layers, is performed. Turbulent inflow conditions are generated via our dynamic rescaling–recycling approach, recently extended to high-speed flows. Focus is given to the assessment of the resolved Reynolds stresses. In addition, flow visualization is performed to obtain a much better insight into the physics of the flow. A weak compressibility effect is observed on thermal turbulent structures based on two-point correlations (IC vs. supersonic). The Reynolds analogy (u′ vs. t′) approximately holds for the supersonic regime, but to a lesser extent than previously observed in incompressible (IC) turbulent boundary layers, where temperature was assumed as a passive scalar. A much longermore »power law behavior of the mean streamwise velocity is computed in the outer region when compared to the log law at Mach 2.5. Implicit LES has shown very good performance in Mach 2.5 adiabatic flat plates in terms of the mean flow (i.e., Cf and UVD+). iLES significantly overpredicts the peak values of u′, and consequently Reynolds shear stress peaks, in the buffer layer. However, excellent agreement between the turbulence intensities and Reynolds shear stresses is accomplished in the outer region by the present iLES with respect to the external DNS database at similar Reynolds numbers.« less
  3. The inertial subrange of turbulent scales is commonly reflected by a power law signature in ensemble statistics such as the energy spectrum and structure functions – both in theory and from observations. Despite promising findings on the topic of fractal geometries in turbulence, there is no accepted image for the physical flow features corresponding to this statistical signature in the inertial subrange. The present study uses boundary layer turbulence measurements to evaluate the self-similar geometric properties of velocity isosurfaces and investigate their influence on statistics for the velocity signal. The fractal dimension of streamwise velocity isosurfaces, indicating statistical self-similarity in the size of ‘wrinkles’ along each isosurface, is shown to be constant only within the inertial subrange of scales. For the transition between the inertial subrange and production range, it is inferred that the largest wrinkles become increasingly confined by the overall size of large-scale coherent velocity regions such as uniform momentum zones. The self-similarity of isosurfaces yields power-law trends in subsequent one-dimensional statistics. For instance, the theoretical 2/3 power-law exponent for the structure function can be recovered by considering the collective behaviour of numerous isosurface level sets. The results suggest that the physical presence of inertial subrange eddies ismore »manifested in the self-similar wrinkles of isosurfaces.« less
  4. Turbulent processes in the ocean surface boundary layer (OSBL) play a key role in weather and climate systems. This study explores a Lagrangian analysis of wave-driven OSBL turbulence, based on a large-eddy simulation (LES) model coupled to a Lagrangian stochastic model (LSM). Langmuir turbulence (LT) is captured by Craik–Leibovich wave forcing that generates LT through the Craik–Leibovich type 2 (CL2) mechanism. Breaking wave (BW) effects are modeled by a surface turbulent kinetic energy flux that is constrained by wind energy input to surface waves. Unresolved LES subgrid-scale (SGS) motions are simulated with the LSM to be energetically consistent with the SGS model of the LES. With LT, Lagrangian autocorrelations of velocities reveal three distinct turbulent time scales: an integral, a dispersive mixing, and a coherent structure time. Coherent structures due to LT result in relatively narrow peaks of Lagrangian frequency velocity spectra. With and without waves, the high-frequency spectral tail is consistent with expectations for the inertial subrange, but BWs substantially increase spectral levels at high frequencies. Consistently, over short times, particle-pair dispersion results agree with the Richardson–Obukhov law, and near-surface dispersion is significantly enhanced because of BWs. Over longer times, our dispersion results are consistent with Taylor dispersion. Inmore »this case, turbulent diffusivities are substantially larger with LT in the crosswind direction, but reduced in the along-wind direction because of enhanced turbulent transport by LT that reduces mean Eulerian shear. Our results indicate that the Lagrangian analysis framework is effective and physically intuitive to characterize OSBL turbulence.

    « less
  5. Abstract Direct numerical simulations are performed to compare the evolution of turbulent stratified shear layers with different density gradient profiles at a high Reynolds number. The density profiles include uniform stratification, two-layer hyperbolic tangent profile and a composite of these two profiles. All profiles have the same initial bulk Richardson number ( $$Ri_{b,0}$$ R i b , 0 ); however, the minimum gradient Richardson number and the distribution of density gradient across the shear layer are varied among the cases. The objective of the study is to provide a comparative analysis of the evolution of the shear layers in term of shear layer growth, turbulent kinetic energy as well as the mixing efficiency and its parameterization. The evolution of the shear layers in all cases shows the development of Kelvin–Helmholtz billows, the transition to turbulence by secondary instabilities followed by the decay of turbulence. Comparison among the cases reveals that the amount of turbulent mixing varies with the density gradient distribution inside the shear layer. The minimum gradient Richardson number and the initial bulk Richardson number do not correlate well with the integrated TKE production, dissipation and buoyancy flux. The bulk mixing efficiency for fixed $$Ri_{b,0}$$ R i b ,more »0 is found to be highest in the case with two-layer density profile and lowest in the case with uniform stratification. However, the parameterizations of the flux coefficient based on buoyancy Reynolds number and the ratio of Ozmidov and Ellison scales show similar scaling in all cases.« less