Session LA: Turbulent Boundary Layers: Theory

Optimal linear amplification of partial energy norms in turbulent channels

When comparing ``optimally-amplified'' linearized porturbations in modeled turbulent channels to the observed structures in real multiscale flows, it is often useful to redefine the norm used in the comparison. For example, to study the kinetic energy at a given wall distance, it is useful to examine which initial perturbations amplify optimally the energy at that level, rather than the total one. The same is true of individual velocity components. Unfortunately, such norms are singular, vanishing for some nonzero perturbations, and require modifying the standard optimum-growth algorithm, We present such a modification, using a nonsingular norm to normalize the initial condition, and a possibly singular one to monitor growth, and apply it to model the energy spectra in a linearized eddy-viscosity approximation to turbulent channels. Using a norm that measures the buffer-layer energy leads to maximum amplification in the spectral region characteristic of near-wall streaks, while using energy windows farther from the wall isolates spectra similar to those in the log or outer layers. The results are used to investigate the likely origin of the different turbulent structures.   

Optimal energy amplification of plane turbulent channel flows with emphasis on different types of perturbation

Optimal perturbations in turbulent channel flow with mean velocity profile and its associated eddy viscosity are investigated, with emphasis on different types of perturbation. We look for linear amplification of both very large-scale outer structures and near-wall streaks arising from three types of perturbations: initial perturbation, harmonic forcing and stochastic excitation. Proper premultiplied energy amplification factors for optimal harmonic forcing and stochastic excitation are suggested to be identified with growth of the outer and inner structures. Finally, response of the turbulent flow to optimal perturbations is studied, based on direct numerical simulation to validate the linear analysis.   

Initial conditions and symmetry breaking for linear energy amplification in eddy-viscosity models of turbulent channels

We study maximally-growing linear perturbations in a turbulent channel, using linearized Navier Stokes equations and an eddy viscosity that is generally lower than the one required to maintain the full velocity profile. The new viscosity depends on the wavenumber, and can be rationalized from spectral considerations. Significantly, fully nonlinear simulations using it, are able to self-sustain. We find that it is important to consider not only the optimal perturbation for a given wavenumber, but also those associated to the next few singular values. In general, these come in pairs. For short-wavelength modes localized near the wall, or in the sublayer, symmetric and antisymmetric eigenfunctions have essentially the same growth properties, showing that the two walls are decoupled. For larger wavelengths, whose optimal perturbations span the whole flow thickness, symmetry is broken, and the solution with an antisymmetric streamwise velocity becomes dominant. It corresponds to a fast streak in one wall opposite to a slow one in the other, and agrees with the structure of global modes obtained from correlations in full-channel DNSes.   

Log law via first principles

The first-principles based theory of Fife et al. 2005 \textit{J. Fluid Mech.} \textbf{532}, Fife et al. 2009 \textit{J. Discrete \& Cont. Dyn. Sys.} \textbf{24} is tested relative to the properties of the logarithmic mean velocity profile. The theory demonstrates that the mean momentum balance (MMB) formally admits a hierarchy of scaling layers, with an associated length scale distribution that asymptotically scales with distance from the wall. DNS data are shown to support these and other analytical findings. The mean velocity profile exhibits logarithmic dependence (exact or approximate) when the solution to the MMB exhibits (exact or approximate) self-similarity on the hierarchy. Exact self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. An independent equation for this coefficient (von K\'{a}rm\'{a}n coefficient, $\kappa$), and its various equivalent forms are shown support by DNS data. Physically, $\kappa$ exists owing to approximately scale invariant dynamics over an internal layer hierarchy. The theory clarifies how and why logarithmic dependence occurs and that logarithmic dependence is inherently approximate.   

On the nature of Karman coefficient variantion in wall-bounded turbulent flow

It becomes increasingly recognized that the so-called Karman constant in wall-bounded flow is not universal, but varies from channel to pipe, and hence depends on the type of the boundary layers. Recent studies show that Karman coefficient vary with pressure gradients and unsteadiness of the smooth wall. In this study, we use an ensemble decomposition technique from a so- called structural ensemble dynamics theory, to analyze turbulent fluctuation data in a smooth channel. It is shown that a proper decompostion is able to characterize the mean property associated a set of turbulent structures. In particular, Karman coefficents obtained for bursts/streaks show systematic variations, which underlies the mechanism of variations of Karman coefficient with Reynolds numbers and with the geometry of the boundary layers.   

Recurrent dynamics in turbulent boundary layers

Recurrences, in which structures break-up, re-form and advect, are an important aspect of the fine scale motions in boundary layers. These recurrences can be captured using certain exact solutions of the Navier-Stokes equation, which are periodic in time but may or may not advect in space. We will use these solutions to demonstrate that spanwise advection is essential for the bursting phenomenon recorded in hotwire measurements near the wall. The importance of spanwise variation in the velocity field is well known. The new element in our work has to do with spanwise advection, not spanwise variation.   

A critical layer analogy for the very large scale motions in wall turbulence

A concatenation of experimental results in wall turbulence has shown the importance of the very large scale motions (VLSMs) in the streamwise direction, of the order of ten outer lengthscales. While Sreenivasan (1988) proposed a critical layer description of the variation of the wall-normal location of the peak in Reynolds shear stress, namely $y^+_{pk} \sim R^ {+ 1/2}$, where $y^+ = y u_\tau/\nu$, in this presentation we extend the critical layer interpretation to explain the existence and scaling of the VLSMs. An analytical expression for the location of peak streamwise VLSM energy agrees well with experimental results, including the hot-wire results of Morrison et al (2004) from the Princeton/ONR Superpipe.   

The fluctuating topography and acceleration statistics in a turbulent channel flow

Dallas, Vassilicos \& Hewitt (PRE, 2009) characterised the turbulent channel flow topography of the fluctuating velocity field in terms of its stagnation points and quantified this topography by observing that, in the intermediate layer, the number density of stagnation points is inversely proportional to wall-distance. Our DNS of turbulent channel flow confirm this observation at skin friction Reynolds numbers 360 and 720. This spatial structure of the fluctuating velocity's topography partly determines the mean flow profile. We then study, in the intermediate layer, the motion of stagnation points which depends on the acceleration field. The mean streamwise acceleration equals the square of the skin friction velocity divided by the half width, the mean spanwise acceleration is zero and the mean wall-normal acceleration equals the vertical gradient of the mean square vertical fluctuating velocity. The local and convective acceleration terms tend to cancel each other and their rms values are equal only if the terms involving products of mean and fluctuating velocity terms are taken into account. When these terms are excluded, the rms of all three acceleration components are inversely proportional to wall-distance. Consequently,the rms of the streamwise stagnation point velocity is inversely proportional to wall-distance but the rms of this velocity's two other components are independent of it.   

Streamwise Constant Dynamics in Plane Couette Flow

We have previously shown that when forced by small-amplitude Gaussian noise, a streamwise constant projection of the Navier Stokes equations captures many of the salient features of fully developed turbulent plane Couette flow. In this work we develop further the relationship between the nonlinearity in the model and the mathematical mechanism that results in the characteristic shape of the turbulent velocity profile. We use periodic spanwise-wall normal stream functions to represent an idealized model of the streamwise streaks and vortices that are thought to play an important role in both transition and fully developed turbulence in wall bounded shear flows. We demonstrate that using this model, such stream functions produce mean flows consistent with both DNS and experimental observations. Analysis of the amplification properties of the model around flow solutions arising from such stream functions is also studied in an effort to develop a quantitative bound on their energy contribution.   

Integral form of the skin friction coefficient suitable for experimental data

An integral method to evaluate skin friction coefficient for turbulent boundary layer flow is presented. The method replaces streamwise gradients with total stress gradients in the wall-normal direction and is therefore useful in cases when velocity profiles at multiple locations are not available or feasible. It is also shown to be especially useful for experimental data with typical noisy shear stress profiles such as rough-wall boundary layer flows. This is significant, particularly in view of the limited ways by which skin friction can be determined for rough-wall flows.