Session D29: Turbulent Boundary Layers II

Turbulence: The view from the wall

For many practical applications, from naive observation to active control strategies, it is desirable to know the internal state of wall-bounded turbulence. However, practical measurements are limited to the wall, where everything vanishes except for the pressure and the wall-parallel vorticities. We focus on the reconstruction of the second invariant of the velocity gradient tensor, $\Pi$, away from the wall from these three measurements, using its linear relation with the inertial component of the pressure, $p$; $\Nabla^2p= 2\Pi$. Statistically optimal approximations are computed for the velocity fields, $\mathbf{u}$, recovering similar results to those of $\Pi$. Their relative error close to the wall ($y^+\approx 10$) is less than $10\%$ of the turbulent kinetic energy. While both the reconstructions of $\Pi$ and $\mathbf{u}$ capture most of the flow features at $y^+ < 30$, they degrade rapidly, holding reasonable errors only for scales comparable or larger than the distance to the wall, or ``attached''.   

Multiscale analysis of non-equilibrium boundary layers at moderate Reynolds numbers

Turbulent boundary layers with mean-flow three dimensionality occur frequently in complex flow configurations. In these cases, the mean flow direction changes continuously across the boundary layer thickness due to the cross-stream pressure gradient induced by the geometry of the immersed body. This results in counter intuitive reduction of turbulent stresses, with important implications for turbulence modeling. We investigate temporally developing three
dimensional turbulent boundary layers at friction Reynolds numbers up to 1000 using direct numerical simulation of planar channels subjected to a sudden spanwise pressure gradient.  It is shown that the scaling properties and regimes of the flow across different Reynolds numbers can be explained in terms of a hierarchy of self-similar wall-attached eddies with characteristic time-scales proportional to their distance the wall. These momentum-carrying eddies are shown to undergo Reynolds stress depletion caused by the formation of a spanwise boundary layer which inhibits the redistribution of energy via pressure-strain correlation. Our findings answer fundamental questions regarding the structural changes of equilibrium wall-turbulence under the imposition of additional strain.   

Modelling inner--outer interactions of average wall-shear-stress in rough-wall turbulent channel flows using amplitude modulation

DNS of turbulent channel flows over hemispherical roughness were performed with friction Reynolds numbers $Re_\tau=200$, $400$ and $600$. The inner scaled roughness height $k^+=20$ was maintained for all Reynolds numbers while the spacing between hemispheres was varied from $d/k=2$--$4$. Wall-shear-stresses on both walls were extracted by integrating the stress tensor over the rough surfaces and averaged within arrays of cells of various sizes. Analysis of amplitude modulation was applied to the cell-averaged wall-shear-stress to investigate the inner--outer interactions between the outer large-scale structures and the wall-shear-stress. Pre-multiplied streamwise spectra of the universal wall-shear-stress signals (based on AM model) was investigated for all roughness and a scaling relation between the spectra and cell sizes was found, which collapses all spectra at large wavelengths between various roughness parameters. However, roughness shedding and spectral aliasing modify the scaling at small wavelengths. Spherical harmonics was used to extract the detailed power-spectrum of the wall-shear-stress to help quantify the observed behavior.   

Characterizing Energy Transfer in Restricted Nonlinear Wall-Bounded Turbulence

The computational expense of resolving all scales of turbulence in wall-bounded flows is impractical for engineering applications, prompting the development of reduced-order models. Numerical and experimental evidence showing the existence of structures elongated in the streamwise direction of turbulent shear flows motivates the exploration of a streamwise coherent modeling framework. The restricted nonlinear (RNL) modeling paradigm takes such an approach by omitting the streamwise varying part of the perturbation-perturbation nonlinearity in the Navier-Stokes equations, thereby reducing the number of active streamwise Fourier modes in order to reduce computational cost. This model has shown promise in reproducing first- and second-order statistics with as few as one nonzero streamwise mode. Despite the promise of this approach, certain aspects of the dynamics have yet to be fully understood.  Here we focus on the production, dissipation, and transport of energy on these restricted modes. Spectra and energy budgets from the RNL system supported by its natural set and selected set of streamwise modes are compared to DNS data in the low-to-moderate Reynolds number range.   

Evidence of an inverse cascade in a plane wall jet through large-scale forcing.

The present work focuses on understanding the scale interactions within a turbulent boundary layer through large-scale, large-perturbation forcing. A plane wall jet (PWJ), the model flow field, was subject to acoustic forcing. The streamwise wavelengths of the forcing $\lambda_{ex}$, were larger than the local integral length scale $\delta(x)$, where $x$ is the streamwise direction. When the forcing wavelength $\lambda_{ex}$ was larger than the wavelengths of the naturally occurring outer large-scales of the unperturbed jet $\lambda_{n}$, the forcing energy was transferred to scales smaller than $\lambda_{ex}$ in a forward cascade. As the PWJ develops downstream, the local integral length scale $\delta(x)$ gets larger due to the spreading of the PWJ. Hence, as the PWJ develops at downstream locations where $\lambda_{ex}<\lambda_n$, the forcing energy was transferred to scales larger than $\lambda_{ex}$ indicating an inverse cascade. This transfer was accompanied by a transfer of momentum away from the near-wall region to the outer region of the plane wall jet and reduction of the friction velocity for all $\lambda_{ex}$ considered.   

Influence of wall-attached structures on the interface of the quiescent core region in turbulent pipe flow

The quiescent core region in pipe flow is explored using the direct numerical simulation data of turbulent pipe flow at Reτ = 926. The quiescent core region is one of the uniform momentum zones and contains the highest convection velocity in turbulent pipe flow. The interface of the quiescent core region has a similar physical behavior to the turbulent/non-turbulent interface and underlying internal layer. The turbulent structures in pipe flow are detected by using the percolation theory. The wall-attached structure is closely associated with geometrical characteristics of the interface and entrainment, which is a transport phenomenon at the interface. The wall-attached structure has a huge influence on dynamics in the vicinity of the interface. The wall-attached u-structure acquired in the present work is a structure that is subject to the attached-eddy hypothesis, and the influence of the wall-attached structures near the interface varies according to height change of the wall-attached structures. Finally, dynamics of the interface is reconstructed using conditional statistics according to the growth of the wall-attached structure.