Session R20: Turbulence: Theory: Wall-Bounded Flows

Streamwise mean flow and turbulent intensity profiles in turbulent pipe flow

The Townsend-Perry attached eddy spectral model predicts that theintegral length-scale varies very slowly with distance to the wall inthe intermediate layer. The only way for the integral length scale'svariation to be more realistic while keeping with the Townsend-Perryattached eddy spectrum is to add a new wavenumber range to the modelat wavenumbers smaller than that spectrum. This necessary additionalso accounts for the high Reynolds number outer peak of the turbulentkinetic energy in the intermediate layer. An analytic expression isobtained for this outer peak in agreement with extremely high Reynoldsnumber data by Hultmark, Vallikivi, Bailey {\&} Smits (2012,2013). Townsend's (1976) production-dissipation balance and thefinding of Dallas, Vassilicos {\&} Hewitt (2009) that, in theintermediate layer, the eddy turnover time scales with skin frictionvelocity and distance to the wall implies that the mean flow gradienthas an outer peak at the same location as the turbulent kineticenergy. This is seen in the data of Hultmark, Vallikivi, Bailey Smits (2012, 2013). The same approach also predicts that the mean flowgradient has a logarithmic decay at distances to the wall larger thanthe position of the outer peak. This qualitative prediction is alsosupported by the aforementioned data.   

Modeling height-dependent spatio-temporal spectra in wall-bounded turbulence

Spatio-temporal spectra of wall-bounded turbulence show a non-trivial structure in the wavenumber-frequency domain. We here study spectra of streamwise velocity fluctuations by means of large-eddy simulations. Such spectra, for instance, indicate wavenumber-dependent frequency shifts induced by mean flow advection as well as frequency broadening related to large-scale velocity perturbations. In previous work, we introduced an advection model combining Taylor's frozen eddy hypothesis and the Kraichnan-Tennekes random sweeping hypothesis to capture these observations. For the logarithmic layer of the flow, we furthermore introduced analytical model parameterizations for the height-dependent wavenumber part of the spectrum as well as the frequency shift and broadening. After summarizing these results, we will present further comparisons of the model with simulation data. In particular, we will validate the model for a range of heights across the logarithmic layer by focusing on comparisons of wavenumber-frequency spectra in the streamwise wavenumber-frequency plane as well as in the spanwise wavenumber-frequency plane.   

Scaling laws of turbulent Couette flow with wall-normal transpiration

An extensive DNS study of turbulent plane Couette flows with permeable boundary conditions, i.e. wall-normal transpiration, was conducted at $Re_\tau=250,500,1000$ and varying transpiration velocities $v_0$. The discretization employed is speudo-spectral in wall-parallel and compact finite differences in wall-normal direction (see Hoyas et al., Phys. Fluids 2006). We derived a global stress relation for the flow, balancing total shear stresses, with very different friction velocities at lower and upper wall. This, in turn, was used to validate convergence of DNS statistics. Most important, we derived a viscous sublayer velocity scaling for the suction wall employing asymptotic methods. Moreover, using Lie group symmetry analysis applied to the multi-point correlation equation we derived scaling laws for the near-wall region on the blowing wall and the channel center, predicting mean velocity $\left< U_1\right>$ and the Reynolds-stress components $\left< u_i u_j\right>$, (see Oberlack et al., JSME Mech. Eng. Rew., 2015), which were nicely validated against DNS data.   

Properties of the total kinetic energy balance in wall-bounded turbulent flows

The properties of the total kinetic energy balance in turbulent boundary layer and channel flows are explored empirically. The total kinetic energy transport equation, which is the combination of mean and turbulent kinetic energy transport equations, is appropriately simplified for fully developed turbulent channel flow and the two-dimensional flat plate boundary layer. Different from the turbulence kinetic energy equation, a suitable grouping of terms is found that cleanly segregates the leading balances in the total energy equation. Available high-quality data reveal a four-layer structure for the energetics that is qualitatively different from the four-layer description of the mean dynamics [Wei \textit{et al}. 2005,\textit{ J. Fluid Mech}. \textbf{522}, 303]. The wall-normal widths of the layers exhibit significant Reynolds number dependencies, and these are empirically quantified. Present findings indicate that each of the four layers is characterized by a predominance of some of the terms in the governing equations. Particular significance is attached to the ratio of the sum of viscous diffusion and dissipation terms to the production/turbulent diffusion term, since these groupings allow the characterization of the layer widths. The third layer exhibits a complex leading order balance exchange that is described in detail.   

High Reynolds number decay of turbulent Taylor-Couette flow

We study the decay of high-Reynolds number turbulence in a Taylor-Couette facility for pure inner cylinder rotation. The rotation of the inner cylinder ($\mathrm{Re}_i=2 \times 10^6$) is suddenly decelerated as fast as possible, thus removing the energy input within seconds. Local velocity measurements show that the decay in this wall-bounded inhomogeneous flow is faster than observed for homogeneous isotropic turbulent flows, due to the strong viscous drag applied by the inner and outer cylinder surfaces. We found that the decay over time can be described with the differential equation $\dot{\mathrm{Re}}(t)=c_f(\mathrm{Re})\mathrm{Re}^2$, where the effects of the walls are included through the friction coefficient. A self-similar behavior of the azimuthal velocity is found: its normalized velocity profile as a function of the radius collapses over time during the decay process.   

Large eddy simulation study of spanwise spacing effects on secondary flows in turbulent channel flow

The structure of turbulent flow over a complex topography composed of streamwise-aligned rows of cones with varying spanwise spacing, $s$ is studied with large-eddy simulation (LES). Similar to the experimental study of Vanderwel and Ganapathisubramani, 2015: J. Fluid Mech., we investigate the relationship between secondary flow and $s$, for $0.25 \leq s/\delta \leq 5$. For cases with $s/\delta > 2$, domain-scale rollers freely exist. These had previously been called ``turbulent secondary flows'' (Willingham et al., 2014: Phys. Fluids; Barros and Christensen, 2014: J. Fluid Mech.; Anderson et al., 2015: J. Fluid Mech.), but closer inspection of the statistics indicates these are a turbulent tertiary flow: they only remain ``anchored'' to the conical roughness elements for $s/\delta > 2$. For $s/\delta < 2$, turbulent tertiary flows are prevented from occupying the domain by virtue of proximity to adjacent, counter-rotating tertiary flows. Turbulent secondary flows are associated with the conical roughness elements. These turbulent secondary flows emanate from individual conical topographic elements and set the roughness sublayer depth. The turbulent secondary flows remain intact for large and small spacing. For $s/\delta < 1$, a mean tertiary flow is not present.   

Finite Reynolds number properties of a turbulent channel flow similarity solution

Finite Reynolds number behaviors of the asymptotically logarithmic mean velocity profile in fully developed turbulent channel flow are investigated. This is accomplished by exploiting invariance properties admitted by the appropriately simplified form of the mean momentum equation. These properties underlie the existence of a similarity solution over an interior inertial domain. This similarity solution, which was originally demonstrated by numerically integrating the relevant nonlinear equation, is consistent with the emergence of a logarithmic mean velocity profile as the Reynolds number becomes large. It is now shown that the governing nonlinear equation has an analytical solution that contains both linear and logarithmic terms, but with the coefficient on the linear term decaying to zero with Reynolds number. Existing DNS are used to elucidate Reynolds number dependent properties of this finite Reynolds number form of the similarity solution. Correspondences between these properties and those indicated by finite Reynolds number corrections to the classical overlap layer formulation for the mean velocity profile are described and discussed.   

Transport of heat and momentum in oscillatory wall-bounded flow

The balance of the leading order terms in the mean momentum and energy equations and their thrice integrated forms are investigated in oscillatory wall-bounded flow using both DNS and experimental data. The integrated forms of the equations are used to investigate the dynamical contributions to the phase-averaged wall shear stress and wall heat flux. Preliminary results indicate that phases corresponding to flow acceleration are dynamically similar to oscillatory laminar flow and phases corresponding to flow deceleration are dynamically similar to fully developed turbulent flow. Moreover, the flow becomes more turbulent-like with increasing period of oscillation.   

Isotropic boundary adapted wavelets for coherent vorticity extraction in turbulent channel flows

We present a construction of isotropic boundary adapted wavelets, which are orthogonal and yield a multi-resolution analysis. We analyze DNS data of turbulent channel flow computed at a friction-velocity based Reynolds number of 395 and investigate the role of coherent vorticity. Thresholding of the wavelet coefficients allows to split the flow into two parts, coherent and incoherent vorticity. The statistics of the former, i.e., energy and enstrophy spectra, are close to the ones of the total flow, and moreover the nonlinear energy budgets are well preserved. The remaining incoherent part, represented by the large majority of the weak wavelet coefficients, corresponds to a structureless, i.e., noise-like, background flow and exhibits an almost equi-distribution of energy.   

A Reduced Nonlinear Model of Wall-Bounded Shear Flow Turbulence

The roll/streak is the dominant structure in the dynamics of wall-bounded shear flow turbulence. It appears that this structure arises from a nonlinear instability, the various proposed mechanisms for which are referred to as self-sustaining processes. However, even once the nonlinear instability is identified there remains the problem of understanding how this instability is regulated to maintain the observed turbulent state. Here both of these questions will be addressed by adopting the perspective of statistical state dynamics (SSD), specifically its reduced nonlinear (RNL) implementation. RNL comprises the joint evolution of the streamwise constant mean flow (first cumulant) and second order perturbation statistics (second cumulant). This restriction greatly reduces the complexity of the dynamics while retaining a realistic SSP. The perturbations supporting the SSP in RNL arise from parametric instability of the time-dependence streak the statistical stability of these perturbations being enforced by a feedback mediated control process operating between the mean flow and the perturbations. In this talk it will be shown how the maintenance and regulation of RNL turbulence allows insight into the mechanism of turbulence in wall-bounded shear flow.   

Non-unique frictional drag in turbulent plane Couette flows

There is a long standing mystery concerning frictional drag in fully developed turbulent plane Couette flows. In manifest defiance to the predictions from dimensional analysis, experiments have consistently shown that the frictional drag, f, is not a unique function of the Reynolds number, Re. In fact, the f vs. Re data fall on two distinct curves. The origin of these two curves dates back to the 1950s when Reichardt and Robertson independently performed their classical experiments. Subsequent works have found f vs. Re data to be in accord with the Reichardt curve or with the Robertson curve. Here we examine this problem from the perspective of the spectral link, the link between macroscopic properties (like f and the mean velocity profile, MVP) and the turbulent energy spectrum. We argue that since the flow is driven by moving boundaries, the boundaries affect the large length scales of the spectrum differently in the different setups. Using the spectral link we predict that the Reichardt and Robertson curves correspond to disparate features in the MVP: the presence or absence of an overshooting wake, respectively. Whilst the different experiments and simulations did not report the spectrum, we verify our predictions by comparing the f vs. Re curves with the attendant MVPs.   

Mean dynamics of a turbulent plane wall jet

Experimental and large-eddy simulation data are used to investigate the balances between viscous and inertial forces in plane turbulent wall jets. In recent years, analysis of the mean momentum balance in its unintegrated form has been shown to provide a mathematically and physically useful means for clarifying the leading order mean dynamics as a function of the transverse coordinate. Distinct from its laminar counterpart, each of the terms in the appropriately simplified form of the mean dynamical equation for the planar turbulent wall jet is leading order somewhere, but not everywhere, across the flow domain. Similar to what is observed in the canonical turbulent wall-flows, there is a wall region where the mean viscous force retains leading order. The wall jet, however, contains two peaks of opposite sign in its Reynolds stress profile. With distance from the wall, the first peak is associated with the loss of a leading order viscous force, while the outer peak is akin to the wholly inertial balance exchange that occurs in shear-wake flows. The physics of these balances exchanges are described, the scaling behaviors of the leading order balance layers are estimated, and the present findings are compared with previous models of planar wall jet structure.