Session E20: Turbulent Boundary Layers III: Theory

Dynamics of Streak and Roll Structures in Turbulent Plane Couette Flow

The dominance of streamwise elongated features in transition and turbulence motivates the use of streamwise constant (so called 2D/3C) and statistical mean state dynamics models to investigate the dynamics of the well studied streamwise elongated roll and streak structures in plane Couette flow. Predictions resulting from two such theoretical models are compared to those of fully resolved DNS data. The 2D/3C model correctly captures the roll/streak structures, the momentum transfer mechanism and statistical features of turbulence such as the mean turbulent velocity profile. Furthermore, the statistical mean state dynamics model, which couples the streamwise constant mean flow and (streamwise varying) perturbation dynamics, adds the critical feedback mechanism arising from the important $k_x\neq0$ information to this streamwise constant framework. The resulting model not only predicts the roll/streak structures, mean profile and momentum transfer but also exhibits a bifurcation from the stable 2D/3C dynamics to a self-sustaining turbulent state that closely resembles that seen in DNS. \textbf{Acknowledgements:} This research was performed at the 2012 Stanford CTR Summer Program.   

Coherent structure, amplitude modulation and higher order statistics in wall turbulence

Coherent structure in wall turbulence is shown to be captured by the frequency-domain treatment of the Navier-Stokes equations as a directional amplifier proposed by McKeon \& Sharma (2010). Simple combinations of the predicted response modes (which take the form of radially-varying travelling waves), consistent with the nonlinear triadic interaction known for wavenumber interactions, are offered which minimally predict all types of structures including hairpin vortices and modulated hairpin packets. One such combination is understood to form a turbulence ``kernel,'' which, it is proposed, constitutes a self-exciting process analogous to the near-wall cycle. The phase relationships explain important skewness and correlation results known in the literature. It is shown that the local shear associated with very large scale motions acts to organize hairpin-like structures such that they co-locate with areas of low streamwise momentum. Compelling evidence for the theory is presented based on comparison to observations of structure and statistics reported in the experimental and numerical simulation literature and similarities with other analytical and empirical models are discussed   

Scaling of normal stresses in the turbulent boundary layer

Concentrating on the canonical zero pressure gradient (ZPG) turbulent boundary layer (TBL), different scalings of normal stresses, in particular of $$, have been proposed. In the range of Reynolds numbers where measurements are available, the best data collapse is obtained by scaling stream-wise fluctuations with the free stream velocity $U_{\infty}$. It is shown with the underlying RANS equations that this choice, together with the traditional Rotta outer scale $\delta^{\ast} U^+_{\infty}$ and the ``log law'' leads to a boundary layer thickness which decreases in the downstream direction. In other words, if one insists on both the traditional mean flow similarity and on scaling normal stresses with $U_{\infty}$, all (growing) TBLs seen so far are very far from their true asymptotic (shrinking) state. Alternative assumptions/scalings and their consequences will be discussed.   

A mean profile formulation for canonical wall-bounded turbulent flows consistent with the mean momentum equation

The mean velocity profile for wall bounded flows is formulated in a manner that is consistent with the magnitude ordering of terms and characteristic length scales associated with the mean momentum equation. Close to the wall, the viscous length characterizes the dynamics, and Prandtl's law-of-the-wall holds. In an outer inertial region where the dominant balance is between the Reynolds stress gradient and the pressure gradient (or mean advection), the mean flow is most closely approximated by a logarithmic function. The width of this region is (asymptotically) characterized by the outer length scale. As initially demonstrated by Wei et al (2005), for all canonical wall-flows the mean viscous force retains dominant order out to a wall-normal location that, in inner units, is $O(\sqrt{\delta^+})$, where $\delta^+$ is the Karman number. The present formulation respects these known properties. This formulation predicts that for low $\delta^+$ the log-law is approached from ``above'' the logarithmic line, while for high $\delta^+$ the log-law is attained from ``below.'' These subtle properties and the general functional form are shown to be in very good agreement with the mean velocity data available from boundary layer, pipe and channel flows.   

A new theory for the streamwise turbulent fluctuations in pipe flow

A new theory for the streamwise turbulent fluctuations, in fully developed pipe flow, is proposed. The new theory, which is based on the near asymptotic analysis introduced by George \& Castillo 1997, introduces a sensitivity function for the fluctuations, which is related to the increase of the non-dimensionalized fluctuations with Reynolds number. The theory predicts that the fluctuations will experience a logarithmic behavior if the sensitivity function is constant in a region in space and that the magnitude of the constant will correspond to the slope of the logarithm. The theory extends the similarities between the mean flow and the streamwise turbulence fluctuations, as observed in experimental high Reynolds number data, to also include the theoretical derivation. Experimental data show that a mesolayer, similar to that introduced by Wosnik et al. 2000, exists for the fluctuations for $300 > y^+ > 800$, which coincides with the mesolayer for the mean velocities. In the mesolayer, the flow is still affected by viscosity, which acts to decrease the fluctuations and to form an outer peak in the fluctuation profile.