Session MA: Turbulent Boundary Layers VII: Theory

Overlap region in turbulent boundary layer over a rough surface

The one term non-linear outer layer in George \& Castillo (1997, AMR 50, 689), based on their AIP argument, was matched with inner wall layer leading to power law velocity, which denied very existence of traditional log law, while Clauser (1956) patched same outer layer with inner wall log law. Jones, Nickles \& Marusic (2008, JFM 616, 195) proposal that free stream velocity (in GC97) and friction velocity (in Coles 1956) are potentially valid scalings according to their theoretical criterion in the outer layer, is misleading, being not correct. Further, in Nishioka (2010, FDR 42, 45502-5) and Prandtl (1935, AT) the additive constant in power law velocity is singular at large Reynolds numbers is also not correct, and this constant is shown to be zero. In the present work, two terms outer layer expansion is considered where leading term scales with free steam velocity and first order with friction velocity. The leading term turns out to be a non-linear wake type equation through application of Izakson-Millikan- Kolmogorov hypothesis. The first order terms lead to alternate functional equations, arising from ratios of two successive derivatives of the functional equations, each of which admits two functional solutions, the power law velocity profile in addition to log law velocity profile. The comparison with extensive data on rough \& smooth walls also provide strong support to present work.   

Behavior of local dissipation scales in turbulent pipe flow

Classically, dissipation of turbulence has been thought to occur around the Kolmogorov scales. However, the Kolmogorov scales are prescribed using mean dissipation rate, whereas dissipation is spatially intermittent. It therefore seems natural to instead describe dissipation using a continuum of local length scales rather than a single scale. By connecting a local dissipation scale $\eta$ to the velocity increment across this scale $\delta u_\eta$, it is possible to derive a probability density function (PDF) of $\eta$ which show how the dissipation is contained in scales larger and smaller than the Kolmogorov scale. Here we present a comparison between measured PDFs in turbulent pipe flow, the analytically derived PDF, and PDFs determined from direct numerical simulation of homogeneous isotropic turbulence. It was found that there is good general agreement between experiment, simulation and theory amongst both homogeneous and inhomogeneous turbulent flows, pointing to universality in the dissipation scales amongst different flows. It was also found that the PDFs are invariant with distance from the wall except for a region very near the wall ($y^+<80)$, where dissipation was found to occur at increasingly larger length scales as the wall is approached.   

Macroscopic effects of the spectral structure in turbulent flows

There is a missing link between macroscopic properties of turbulent flows, such as the frictional drag of a wall-bounded flow, and the turbulent spectrum. To seek the missing link we carry out unprecedented experimental measurements of the frictional drag in turbulent soap-film flows over smooth walls. These flows are effectively two-dimensional, and we are able to create soap-film flows with the two types of turbulent spectrum that are theoretically possible in two dimensions: the ``enstrophy cascade,'' for which the spectral exponent $\alpha = 3$, and the ``inverse energy cascade,'' for which the spectral exponent $\alpha = 5/3$. We find that the functional relation between the frictional drag $f $ and the Reynolds number $\rm{Re}$ depends on the spectral exponent: where $\alpha = 3$, $f \sim Re^{-1/2}$; where $\alpha = 5/3$, $f \sim Re^{-1/4}$. Each of these scalings may be predicted from the attendant value of $\alpha$ by using a recently proposed spectral theory of the frictional drag. In this theory the frictional drag of turbulent flows on smooth walls is predicted to be $f \sim Re^{(1-\alpha)/(1+\alpha)}$.   

Mean velocity of a free fully-developed turbulent boundary layer

A novel model equation for describing the profiles of the mean axial velocity of a free fully developed turbulent boundary layer over a smooth solid wall at zero pressure gradient is developed. The model uses the Reynolds-average equation for the mean axial speed and a modified Prandtl's mixing-length curve for the turbulent stress. This model is used to integrate the mean velocity profiles and compute the wall friction coefficient along the wall. The computed results of the velocity profiles and the friction coefficient show a remarkable agreement with much measured data and results from direct numerical computations for a wide range of $Re_{x}$ between 10 and 100 million, except for the transition region. Moreover, the present analysis demonstrates the four main regions that govern the flow and sheds new light on the structure of the boundary layer.   

The turbulent mean-velocity profile: it is all in the spectrum

It has long been surmised that the mean-velocity profile (MVP) of a pipe flow is closely related to the spectrum of turbulent energy. Here we perform a spectral analysis to identify the eddies that dominate the production of shear stress via momentum transfer. This analysis allows us to express the MVP as a functional of the spectrum. Each part of the MVP relates to a specific spectral range: the buffer layer to the dissipative range, the log layer to the inertial range, and the wake to the energetic range. The parameters of the spectrum set the thickness of the viscous layer, the amplitude of the buffer layer, and the amplitude of the wake.   

The viscous sublayer revisited

The viscous sublayer of wall bounded turbulent flows is a thin region, usually assumed to stretch out to about 5 viscous length units, where the mean velocity distribution is close to linear. Its thickness is typical of the order of one percent or less of the boundary layer thickness or channel height. Despite this fact its importance for the flow in the boundary layer cannot be overstated since the mean shear stress at the wall determines the velocity scale of the Reynolds stresses and hence the velocity scale of the turbulence itself. In this presentation we show how the variation of the flow statistics within the viscous sublayer can be understood from a simple analysis of the instantaneous velocity profile. Special emphasis is put on the near self-similarity of the probability density distribution (pdf) of the streamwise velocity in the viscous sublayer. We also describe how the pdf of the fluctuating streamwise velocity measured using hot-wire anemometry can be used to determine the wall position and the friction velocity despite the fact that such measurements are contaminated by interference effects close to the wall. We illustrate this analysis both with DNS results from turbulent boundary layers and channel flows as well as from experiments in turbulent boundary layers.   

Relevant length scales in wall-bounded turbulence

The structure of wall-bounded turbulence is different from the case of isotropic/homogeneous turbulence due to the presence of local mean shear rate $S$. $S$ produces kinetic energy that have been classically assumed to occur at length scales of the order of the integral length scale. We show that there are three independent length scales that are regulated by $S$ interacting with kinetic energy $k$, viscosity $\nu$ and dissipation rate $\epsilon$. The first two length scales: $L_{S,k}$ based on $S$ and $k$; and $L_{S,\nu}$ based on $S$ and $\nu$; signify the upper and lower bounds of the scales that represent turbulence production respectively. The third length scale, $L_{S,\epsilon}$ which is based on $S$ and $\epsilon$, is an intermediate scale that signifies the beginning of overlap between the energy cascade process and the production range set by the first two scales. We also illustrate the fundamental and independent nature of these three length scales in that they set two important classical length scales of motion in wall-bounded turbulence, namely the large eddy length scale $L_{k,\epsilon}$ based on $k$ and $\epsilon$; the Kolmogorov length scale $\eta$; and a third new smallest length scale $L_{k,\nu}$ based on $k$ and $\nu$. Analysis of the variation of all six length scales using a large high-resolution DNS database of turbulent channel flow is provided with fresh insights into the dynamic characteristics that define the viscous sublayer, buffer layer, and the inertial regime.   

New insights into adverse pressure gradient boundary layers

In a recent paper Shah et al.\ 2010 (Proc. of the WALLTURB Meeting, 2009), Lille, FR, Springer, in press) documented a number of adverse pressure gradient flows (APG's), with and without wall curvature, where the turbulence intensity peak moved quite sharply away from the wall with increasing distance. They further suggested that this peak was triggered by the adverse pressure gradient and had its origin in an instability hidden in the turbulent boundary layer, developing soon after the change of sign of the pressure gradient. They then offered that this may explain the difficulties encountered up to now in finding a universal scaling for turbulent boundary layers. We build on these observations, and show that in fact there is clear evidence in the literature (in most experiments, both old and new) for such a development downstream of the imposition of an adverse pressure gradient. The exact nature of the evolution and the distance over which it occurs depends on the upstream boundary layer and the manner in which the APG is imposed. But far enough downstream the mean velocity profile in all cases becomes an inflectional point profile with the location of the inflection point corresponding quite closely to the observed peak in the streamwise turbulence intensity. This does not seem to have been previously noticed.   

Turbulence structure in non-zero pressure gradient boundary layers

We present an extensive database of single- and multi--probe hot-wire measurements of streamwise velocity acquired in zero, adverse and favourable pressure gradient turbulent boundary layers. The primary aim of this investigation was to characterise the effects of pressure gradient on the structure of turbulence at high Reynolds numbers. Specifically, we examine the changes to turbulence intensity, energy spectra and two-point correlations of streamwise velocity. By systematically varying the pressure gradient (PG) at a fixed Reynolds number ($Re$) we were able to isolate PG effects from $Re$ effects. Results from the adverse pressure gradient case show a strong contribution to the energy spectra from length-scales of $\sim$3$\delta$ ($\delta$ is the boundary layer thickness). This contribution is observed throughout the flow, even near the wall. Whether `superstructures' (of length $\sim$6$\delta$) exist or are modified in strong pressure gradients is unclear since the energy spectra indicate the dominance of $\sim$3$\delta$--length structures in the logarithmic and outer regions. Two-point correlations indicate similar spanwise width-scales in the log region compared with the zero pressure gradient case, while the average structure becomes wider beyond the logarithmic region. Further results will be presented showing the effect of varying pressure gradient from favourable through zero to adverse at fixed Reynolds number.   

Analytical solutions of a quasilaminarized turbulent boundary layer

Analytical solutions to the characteristic equation, arising from similarity analysis as proposed by Cal and Castillo (2008),\footnote{R. B. Cal and L. Castillo (2008), Phys. Fluids. vol. 20, 105106.} describing a turbulent boundary layer subject to a strong favorable pressure gradient (FPG) approaching a quasilaminar state are found. By virtue of numerical analysis, solutions to this characteristic equation are obtained for several values of the pressure parameter, $\Lambda = -\frac{\delta}{U_\infty d\delta/dx}\frac{dU_\infty}{dx}$, in addition to the Pohlhausen parameter, $K_s=\frac{\delta^2}{\nu} \frac{dU_\infty}{dx}$. These solutions characterize the influence of the two parameters on a turbulent boundary layer subject to a strong FPG, and quantify these parameters for such flows with eventual quasilaminarization. Different cases are tested to observe the limits of these parameters. The analytical solutions obtained are compared to the experimental data obtained by Warnack and Fernholz (1998).\footnote{D. Warnack and H. H. Fernholz (1998), J. Fluid Mech. vol. 359, 357.} A confirmation of the validity of this method and understanding of the influence of the remnants of the turbulence in the quasilaminar flow is assessed.