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. %% 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 %% the inertial range, and the wake to the energetic range. %% The parameters of the spectrum set %% the constants of the log law of the wall and the thickness of the %% viscous layer. % We conclude that % in turbulence, as in phase transitions, global % variables are governed by the structure % of the fluctuations. % qualitative agreement with Nikuradse's data; % they unveil the existence of unsuspected % ties between two milestones of experimental and % theoretical turbulence; and they suggest that % the phenomenological theory may be relevant % to other turbulent flows of practical interest.
velpro.tex
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%Title of paper
\title{The turbulent mean-velocity profile: it is all in the spectrum 
}

% repeat the \author .. \affiliation  etc. as needed
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\author{Gustavo Gioia}
\affiliation{Department of Mechanical Science and Engineering, University of Illinois, Urbana, IL 61801}
\author{Nicholas Guttenberg}
\affiliation{Department of Physics, University of Illinois, Urbana, IL 61801}
\author{Nigel Goldenfeld}
\affiliation{Department of Physics,  University of Illinois, Urbana, IL 61801}
\author{Pinaki Chakraborty}
\affiliation{Department of Geology, University of Illinois, Urbana, IL 61801}
%\email[]{Your e-mail address}
%\homepage[]{Your web page}
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\date{\today}

\begin{abstract}

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. 

%% 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
%%  the inertial range, and the wake to the energetic range.
%%  The parameters of the spectrum set 
%%  the constants of the log law of the wall and the thickness of the 
%%  viscous layer.


% We conclude that
% in turbulence, as in phase transitions, global
%  variables are governed by the structure 
% of the fluctuations.

% qualitative agreement with Nikuradse's data; 
% they unveil the existence of unsuspected
% ties between two milestones of experimental and
%  theoretical turbulence; and they suggest that 
% the phenomenological theory may be relevant
% to other turbulent flows of practical interest.
\end{abstract}

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%\subsubsection{}

% Turbulence is the unrest that 
% spontaneously takes over a streamline
%  flow adjacent to a wall or obstacle
% when the flow is 
% made sufficiently fast.
 Although most flows 
   in nature and technology
  are turbulent flows over confining walls, 
  these flows have remained amongst 
 the least understood phenomena of 
 classical physics \cite{noncapisco}.
 Take the simplest example (an example
 whose applications are legion):
 the turbulent flow in a long cylindrical 
 pipe with a %circular
 cross section of radius $R$
 and a smooth  internal wall. If the flux is kept 
  steady, the velocity of the flow at a distance
 $y$ from the wall of the pipe 
 may be averaged over a long period 
 of time to obtain a  mean velocity $u$.
%  at the distance $y$ from the wall. 
 The function $u(y)$ is called the
 mean-velocity profile (MVP), and there
 is a MVP for each value of the Reynolds 
 number, ${\rm Re}\equiv UR/\nu$,  where $U$ is the 
 mean velocity of the flow (i.e., 
 the flux divided by the cross-sectional 
 area of the pipe) 
%%$R$ is the radius of the  pipe,
 and $\nu$ is the kinematic viscosity
of the fluid \cite{tl,prandtl}. (Re quantifies 
 the relative importance of inertia and 
  viscosity in the flow; the higher the Re, the more
 turbulent the flow.)
%(The Reynolds number
% is the ratio between the viscous 
% and inertial forces in the flow. It
%  can be defined as ${\rm Re}= UR/\nu$, 
% of the fluid.)
  MVPs were first measured %experimentally more than
 70 years ago \cite{niku} and
  have recently been the subject of %two %excruciatingly 
 exacting experiments \cite{exper} and numerous computational
  simulations \cite{computgen,comput}. Theory has meanwhile
 lagged well behind experiments
 and simulations. 

The first theory came soon after the
 earliest experiments. 
Ludwig Prandtl
 showed that when plotted in terms of the 
 dimensionless %so-called
 ``wall variables'' ${\tilde u}$ and ${\tilde y}$,
 the MVPs for different values of 
 Re collapse into a single MVP close to the wall
  (Fig.~\ref{expe}) \cite{prandtl}.
 The wall variables are %the dimensionless variables
  defined by
 ${\tilde u}\equiv u/\sqrt{\tau_0/\rho}$ and
  ${\tilde y}\equiv (\sqrt{\tau_0/\rho})/\nu$, 
 where $\rho$ is the density of the fluid and 
 $\tau_0$ is the shear stress---or shear force per unit 
 area---that develops between the flow and the 
  wall of the pipe.
% Ludwig Prandtl
% showed that when plotted in the form 
% ${\tilde u}({\tilde y})$,
% the MVPs for different 
% Re collapse into a single MVP close to the wall.
% Here ${\tilde u}$ and ${\tilde y}$ are 
% the so-called ``plus variables,'' i.e., 
% the dimensionless variables  defined by
% ${\tilde u}\equiv u/\sqrt{\tau_0/\rho}$ and
%  ${\tilde y}\equiv (\sqrt{\tau_0/\rho})/\nu$, 
% where $\rho$ is the density of the fluid, and 
% $\tau_0$ is the shear stress---or shear force per unit 
% area---that develops between the flow and the 
% internal wall of the pipe (Fig.~1). 
 Prandlt also showed that over most of its domain
 ${\tilde u}({\tilde y})$
 follows the ``log law of the wall,''
 \begin{equation} \label{llw}
{\tilde u}({\tilde y})=
{{1}\over{\kappa}} \ln {\tilde y} +B,  
\end{equation}
 where $\kappa$ (the ``K\'arm\'an constant'') and $B$ are %universal 
 dimensionless constants which can be estimated
 by fitting experimental data \cite{prandtl}.
\begin{figure} 
\centering
%\resizebox{2.4in}{!}{\includegraphics{vpexpefigure.eps}}
\resizebox{3in}{!}{\includegraphics{vpexpefigure.eps}}
\caption{\label{expe} Log-linear plots of the 
 MVPs in the ``wall variables'' ${\tilde u}$ and ${\tilde y}$
  for four values of Re. The symbols are from experiments 
 \cite{exper} and the solid lines from simulations \cite{comput}. Each  MVP
 extends from the wall (which corresponds always to ${\tilde y}=0$)
 to the centerline of the pipe (which 
 corresponds to a value of ${\tilde y}$ that depends on 
 Re); the MVPs collapse on a single MVP close to 
 the wall. Focusing on any one MVP, we can parse the horizontal
 axis from left to right to find
  the ``viscous layer'' (where the MVP has a positive curvature),
 the ``buffer layer'' (where the MVP has a noticeable
  negative curvature),
 %that reminds us of an integral sign ``$\int$''), the 
 the ``log layer'' (where the MVP follows the log law of 
 the wall (\ref{llw})), and a ``wake'' (where the
  MVP overshoots the log law of the wall
 close to the centerline of the pipe) \cite{layers}. }
\end{figure}
 
 Numerous variants of the log law of the wall have
 been proposed since the time of Prandtl, and recently it has 
 been argued that the MVPs for different values of Re might not
 be logarithmic after all. Instead, they might be 
 power laws with Re-dependent exponents and a 
 logarithmic envelope \cite{baren}.
 Yet the original theory of Prandtl and 
 this recent theory of Barenblatt have %both
  been predicated on dimensional analysis and 
 similarity assumptions (they differ in the 
 similarity assumptions \cite{baren}), without reference
 to the turbulent spectrum. As a result, 
 these theories %contain little physics, and 
cannot be used to relate, for example, the
 log law of the wall or the constants thereof
 to the turbulent spectrum. 
(The turbulent 
spectrum may be said to embody the fabric of 
 a turbulent state; in practice it is an account 
 of the form in which the kinetic 
energy is apportioned among the eddies of different sizes 
in the flow.)  
  Our aim here is to find
  the  missing link between the MVP and 
 the turbulent spectrum.
% (The turbulent spectrum describes
%  the form in which the kinetic energy is 
%apportioned among the eddies of different sizes in the 
%flow.) 
%As we shall see, the spectrum suffices %can be used 
% to determine the MVP over the full extend of its domain.
 

% At the onset of turbulence \cite{hof}, 
% at a Re of about 3,000,
% with Blasius's empirical scaling \cite{prandtl},
% $f \sim {\rm Re}^{-1/4}$, as 
% exposed to the turbulent flow as Re increases \cite{chow}.


 We adopt the
 phenomenological imagery of 
 ``turbulent eddies'' \cite{rich,frisch}
 and use the 
 spectrum of turbulent energy \cite{tl,pope} at the wavenumber $k$,
 $E(k)$, to determine the velocity of 
 the eddies of size $s$, $v_s$, in the form
$v_s^2=\int_{1/s}^\infty E(k) dk$, where 
$E(k)={{2}\over{3}}\,(\kappa_\varepsilon \varepsilon)^{2/3}\, 
 k^{-5/3}\,c_d(\eta k)\,c_e(R k)$.
  Here $\kappa_\varepsilon$ is a dimensionless parameter,
   $\varepsilon$ is the turbulent power per unit mass \cite{taylor},
 $\eta=\nu^{3/4}\,\varepsilon^{-1/4}$ is the 
 viscous lengthscale \cite{tl},
  $R$ is the largest lengthscale in the flow, 
 ${{2}\over{3}}\,(\kappa_\varepsilon \varepsilon)^{2/3}\, k^{-5/3}$ is
 the Kolmog\'orov spectrum \cite{kolmo,aniso}, 
 %that prevails in the inertial range
%  ($\eta \ll k^{-1} \ll R$), 
 and $c_d$ and $c_e$ 
 are dimensionless correction functions---the 
 dissipative-range correction and the
 energetic-range correction, respectively.
% that correct the Kolmog\'orov spectrum
% outside of the inertial range. 
%  in the dissipative range ($k^{-1}\approx \eta$) 
% and the energetic range   ($k^{-1}\approx R$),  respectively. 
 For the dissipative-range correction 
   we adopt the usual exponential
 form, $c_d(\eta k)=\exp(-\beta_d\eta k)$, and for 
 the energetic-range correction % we adopt
 the form proposed by  K\'arm\'an, 
 $c_e(R k)=(1+ \beta_e(R k)^{-2})^{-17/6}$,
 where $\beta_d$ and $\beta_e$ are nonnegative 
  dimensionless parameters \cite{pope}.
% Note that $c_d\approx 1$ except in the dissipative range 
% ($k^{-1}\approx \eta$), where $c_d<1$, and  $c_e\approx 1$ 
%except in the energetic range ($k^{-1}\approx R$), 
% where $c_e<1$.

By introducing the 
  dimensionless variable $\xi\equiv s k$, we 
 can write $v_s= (\kappa_\varepsilon \varepsilon s)^{1/3}\sqrt{I}$,
 where  $I={\mathcal I}(\eta/s,s/R)\equiv {{2}\over{3}}\int_{1}^\infty 
 \xi^{-5/3} \exp(-\xi\,\beta_d\,\eta/s)\, 
 (1+ \beta_e (s/R)^2/\xi^2)^{-17/6}d\xi$.
 For $s$ in the inertial range ($\eta\ll s \ll R$), %we have 
 % $I=\lim_{\eta/s\rightarrow 0}\lim_{s/R\rightarrow 0} 
 %{\mathcal I}(\eta/s,s/R)=1$,
%  $I={\mathcal I}(0,0)=1$, and therefore 
  $I=1$, and therefore 
$v_s= (\kappa_\varepsilon \varepsilon s)^{1/3}$
 \cite{tl,frisch,pope}, with the implication that
   the velocity of an eddy of the inertial range 
   (i.e., an eddy of size $\eta\ll s\ll R$)
  increases with %is an increasing function of
 the size of the eddy.
 The same result, $v_s= (\kappa_\varepsilon \varepsilon s)^{1/3}$,
 may  be derived  from Kolmog\'orov's four-fifth law, which 
  also gives the estimate $\kappa_\varepsilon=4/5$ \cite{ff}.
  For $s$ outside of the inertial range, 
 %($s\approx \eta$ or $s\approx R$), %we have
  $I<1$. % (because either $c_d< 1$ or
 % $c_e<1$ over a part of the domain of integration).
  It follows that  %, for any given $\varepsilon$, 
 an eddy of the dissipative range or the energetic range
  (i.e., an eddy of size $s\approx \eta$ or $s\approx R$,
 respectively) has a velocity
 $v_s< (\kappa_\varepsilon \varepsilon s)^{1/3}$---i.e.,
   the eddy is {\it slower\/} than an imaginary eddy of the same 
 size in the inertial range.

% We adopt the
% phenomenological imagery of 
% ``turbulent eddies'' \cite{rich,frisch}
% and use the 
% spectrum of turbulent energy \cite{tl,pope} at the wavenumber $k$,
% $E(k)$, to determine the velocity of 
% the eddies of size $s$, $v_s$, in the form
%$v_s^2=\int_{1/s}^\infty E(k) dk$, where 
%$E(k)={{2}\over{3}}\,(\kappa_\varepsilon \varepsilon)^{2/3}\, 
% k^{-5/3}\,c_d(\eta k)\,c_e(R k)$.
%  Here $\kappa_\varepsilon$ is a dimensionless parameter,
%   $\varepsilon$ is the turbulent power per unit mass \cite{taylor},
% $\eta=\nu^{3/4}\,\varepsilon^{-1/4}$ is the 
% viscous lengthscale \cite{tl},
%  $R$ is the largest lengthscale in the flow, 
% ${{2}\over{3}}\,(\kappa_\varepsilon \varepsilon)^{2/3}\, k^{-5/3}$ is
% the Kolmog\'orov spectrum \cite{kolmo,aniso} 
% that prevails in the inertial range
%  ($\eta \ll k^{-1} \ll R$), and $c_d$ and $c_e$ 
% are dimensionless correction functions.
%% that correct the Kolmog\'orov spectrum
%% outside of the inertial range. 
%%  in the dissipative range ($k^{-1}\approx \eta$) 
%% and the energetic range   ($k^{-1}\approx R$),  respectively. 
% For the dissipative-range correction 
%   we adopt the usual exponential
% form, $c_d(\eta k)=\exp(-\beta_d\eta k)$, and for 
% the energetic-range correction % we adopt
% the form proposed by  K\'arm\'an, 
% $c_e(R k)=(1+ \beta_e(R k)^{-2})^{-17/6}$,
% where $\beta_d$ and $\beta_e$ are nonnegative 
%  dimensionless parameters \cite{pope}.
% Note that $c_d\approx 1$ except in the dissipative range 
% ($k^{-1}\approx \eta$), where $c_d<1$, and  $c_e\approx 1$ 
%except in the energetic range ($k^{-1}\approx R$), 
% where $c_e<1$.

% By introducing the 
%  dimensionless variable $\xi\equiv s k$, we 
% can write $v_s= (\kappa_\varepsilon \varepsilon s)^{1/3}\sqrt{I}$,
% where  $I={\mathcal I}(\eta/s,s/R)\equiv {{2}\over{3}}\int_{1}^\infty 
% \xi^{-5/3} \exp(-\xi\,\beta_d\,\eta/s)\, 
% (1+ \beta_e (s/R)^2/\xi^2)^{-17/6}d\xi$.
% For $s$ in the inertial range ($\eta\ll s \ll R$), %we have 
% % $I=\lim_{\eta/s\rightarrow 0}\lim_{s/R\rightarrow 0} 
% %{\mathcal I}(\eta/s,s/R)=1$,
%  $I={\mathcal I}(0,0)=1$, and therefore 
%$v_s= (\kappa_\varepsilon \varepsilon s)^{1/3}$
% \cite{tl,frisch,pope}, with the implication that
%   the velocity of an eddy of the inertial range 
%   (i.e., an eddy of size $\eta\ll s\ll R$)
%  increases with %is an increasing function of
% the size of the eddy.
% The same result, $v_s= (\kappa_\varepsilon \varepsilon s)^{1/3}$,
% may  be derived  from Kolmog\'orov's four-fifth law, which 
%  also gives the estimate $\kappa_\varepsilon=4/5$ \cite{ff}.
%  For $s$ outside of the inertial range, %we have
%  $I<1$. % (because either $c_d< 1$ or
% % $c_e<1$ over a part of the domain of integration).
%  It follows that  %, for any given $\varepsilon$, 
% an eddy of the dissipative range or the energetic range
%  (i.e., an eddy of size $s\approx \eta$ or $s\approx R$,
% respectively) has a velocity
% $v_s< (\kappa_\varepsilon \varepsilon s)^{1/3}$---i.e.,
%   the eddy is {\it slower\/} than an imaginary eddy of the same 
% size in the inertial range.
 
%   Suppose that we can compute the local value of 
%  $\varepsilon$ at a distance $y$ from the wall.
% This local
% value of $\varepsilon$ (and the attendant local value of 
% the viscous lengthscale, $\eta=\nu^{3/4}\varepsilon^{-1/4}$)
% will allow us to determine the local spectrum 
% (which depends only on the local values of $\varepsilon$ and 
% $\eta$) and therefore to 
% compute the velocity $v_s$ of the eddies of any size $s$
% at the distance $y$ from the wall.

 Consider now the flow in a smooth-walled pipe. 
  The energy equation gives and expression for $\varepsilon$
in the form $\varepsilon=\tau_t u'/\rho$, where 
 $\tau_t$ is the turbulent shear stress,
 $u'\equiv du/dy$, and $y$ is the distance to the wall
 \cite{pope}.%\cite{energy}. 
The sum of the turbulent 
 shear stress $\tau_t$ and the viscous shear stress 
 $\rho \nu u'$ is the total shear stress $\tau$, 
 which in a pipe of radius $R$ %circular cross section
  is given by 
 $\tau=\tau_0 (1-y/R)$, where $\tau_0$ is the 
  shear stress at the wall \cite{pope}. Using the definition of the 
 friction factor, $f\equiv \tau_0/\rho U^2$, we can write
 $\tau_0=\rho f U^2$ \cite{pope}; it follows that
 \begin{equation} \label{tau}
\tau_t+\rho \nu u' =\rho f U^2 (1-y/R)
\end{equation} 
and
\begin{equation} \label{varep}
\varepsilon= f U^2 (1-y/R) u'-\nu u'^2.
\end{equation}

We now seek to derive an expression for
 the turbulent shear stress $\tau_t$.
 Let us call $W_y$ the wetted surface
at a distance $y$ from the wall of the pipe
 (Fig.~\ref{mtrans}).
% Let us denote by $\tau_t(y)$ the turbulent shear 
% stress that acts on a wetted surface $W_y$ 
%at a distance $y$ from the wall of the pipe. 
The turbulent shear stress that acts on $W_y$
 is produced by eddies 
 that straddle $W_y$ and transfer momentum across $W_y$
 (Fig.~\ref{mtrans}).
 % For example, an eddy of size $s$ 
 %  takes fluid at a distance $s/2$ above
 %$W_y$, where the horizontal momentum per unit volume  
 %is high ($\rho u(y+s/2)$),
 % and releases this high-momentum
 %fluid at a distance  $s/2$ 
 %below $W_y$ (Fig.~\ref{mtrans}).
 %At the same time,  this eddy of size $s$ 
 %  takes fluid  at a distance $s/2$ below
 %$W_y$, where the horizontal momentum per unit volume  
 %is low ($\rho u(y-s/2)$), 
 %and releases this low-momentum
 % fluid at a distance
 %  $s/2$ above $W_y$ (Fig.~\ref{mtrans}). 
Thus an eddy 
 of size $s$
 carries fluid of high horizontal momentum 
 per unit volume (about $\rho u(y+s/2)$)
 downward across $W_y$ and fluid of low horizontal
 momentum per unit volume (about $\rho u(y-s/2)$) upwards
  across $W_y$, and the eddy
 may be said to span a momentum contrast 
  $\rho (u(y+s/2)-u(y+s/2))\approx \rho s u'(y)$ \cite{firsterm}.
 The rate of momentum transfer across $W_y$ 
 is set by the velocity of the eddy, $v_s$
 (i.e., the velocity normal to $W_y$). 
 Therefore, the turbulent shear 
 stress produced by an eddy of size $s$ 
  scales as the product of
  the momentum contrast times the rate of momentum transfer, 
 or $\rho s u'(y)\, v_s$.
\begin{figure} 
\centering
%\resizebox{1.8in}{!}{\includegraphics{mtrans.eps}}
\resizebox{2.2in}{!}{\includegraphics{mtrans.eps}}
\caption{\label{mtrans} Schematic for the derivation
 of the turbulent shear stress. $u(y+s/2)$ is the mean
 velocity at a distance $y+s/2$ from the wall; 
$u(y-s/2)$ is the mean
 velocity at a distance $y-s/2$ from the wall. 
   }
\end{figure}

 In order to  identify the {\sl dominant\/} eddies 
 that straddle $W_y$ 
(i.e., the eddies that dominate the momentum 
transfer across $W_y$), 
recall that $v_s$
 (and therefore $s v_s$) increases with $s$.  
 It follows that the larger the eddy the 
 larger the turbulent shear stress 
 produced by the eddy.
  Nonetheless, eddies much larger than $y$ 
 do not properly straddle $W_y$ and
  can provide only a negligible velocity
 normal to $W_y$, and therefore a negligible 
 turbulent shear stress.
 (This observation is purely a matter of
  geometry.) We conclude that the dominant
 eddies that straddle $W_y$ are the eddies of
 size $s=y$. The turbulent shear stress 
 at a distance $y$ from the wall is thus given by
  $\tau_t=\kappa_\tau  \rho y v_y u'(y)$,
 where $\kappa_\tau$ is a dimensionless 
 proportionality constant.

 We are now ready to obtain an equation
 for the MVP. We substitute (\ref{varep})
 in $v_y= (\kappa_\varepsilon \varepsilon y)^{1/3}\sqrt{I}$,
 substitute the resulting expression for $v_y$ 
  in $\tau_t=\kappa_\tau  \rho y v_y u'$, 
and substitute the resulting expression for $\tau_t$
  in (\ref{tau}). After some algebra, 
 we write the final result in terms
 of ${\rm Re}=UR/\nu$ and the dimensionless variables 
 ${\hat y}\equiv y/R$ and ${\hat u}\equiv u/U$:
\begin{equation} \label{lau}
 \kappa^2 I^3 {\hat y}^2 {\hat u}'^{2}
 +{\rm Re}^{-1} {\hat u}'-f (1-{\hat y})=0,
\end{equation}
where $\kappa \equiv(\kappa_\varepsilon \kappa_\tau^3)^{1/4}$,
${\hat u}'\equiv d{\hat u}/d{\hat y}$ and 
 $I={\mathcal I}(\eta/y,{\hat y})$. To obtain 
an equation for 
 $\eta/y$  we substitute (\ref{varep}) in
 $\eta=\nu^{3/4}\varepsilon^{-1/4}$ and change 
 variables to $\hat y$ and $\hat u$:
\begin{equation} \label{laeoy}
 \eta/y={\rm Re}^{-1/2}\,(f {\rm Re}\, {\hat u}' (1-{\hat y})- {\hat u}'^2)^{-1/4}\,{\hat y}^{-1}.
\end{equation}

%$(\kappa_\varepsilon \kappa_\tau^3)^{1/2} I^3 y^2 u'^2+\nu u'-\tau=0$.
%$\eta=(\nu^3/(u'-\nu u'^2))^{1/4}. 
 
 If for a fixed Re we let ${\hat y}\rightarrow 0$,
 then $\eta/y\rightarrow \infty$ (from (\ref{laeoy})),
  %$I=\lim_{\eta/y\rightarrow \infty} {\mathcal I}(\eta/y,{\hat y})=0$, 
  $I \rightarrow {\mathcal I}(\infty,{\hat y})=0$, 
 and (\ref{lau}) simplifies to 
  ${\hat u}'=f {\rm Re}$, which is the law of the viscous layer.
  If for a fixed ${\hat y}\ll 1$ we let ${\rm Re}\rightarrow \infty$,
 then $\eta/y \rightarrow 0$ (from (\ref{laeoy})),
  $y$ is in the inertial range (where $I=1$), and 
% Thus for any fixed ${\hat y}\ll 1$ and ${\rm Re}\rightarrow \infty$,
 (\ref{lau}) simplifies to
 ${\hat u}'= \sqrt{f} / \kappa {\hat y}$,
 which we recognize as the log-law of the wall with a K\'arm\'an 
 constant $\kappa=(\kappa_\varepsilon \kappa_\tau^3)^{1/4}$. 
Note that 
the log law of the wall prevails where $y$ is in the inertial 
range; it follows that the dominant eddies in the log layer 
are eddies of the inertial range. 
 The presence of $\kappa_\varepsilon$ and $\kappa_\tau$
 in the expression, $\kappa=(\kappa_\varepsilon \kappa_\tau^3)^{1/4}$,
 reminds us of the underpinnings of the theory: the spectrum and 
 the momentum transfer, respectively. 
As $\kappa_\varepsilon$ is fixed by 
Kolmog\'orov's four-fifth law, which gives $\kappa_\varepsilon$ = 4/5 
 \cite{ff,ff1},  the K\'arm\'an constant
$\kappa$ is but an alternative form of the momentum-transfer 
constant $\kappa_\tau$. 
 From  $\kappa$= 0.42 (the experimental value \cite{exper}), 
we have $\kappa_\tau$ = 0.34. 

% The substitutions ${\hat y}=\gamma_y {\tilde y}$ and 
% ${\hat u}=\gamma_u {\tilde u}$ in  ${\hat u}'=f {\rm Re}$
% and ${\hat u}'= \sqrt{f} / \kappa {\hat y}$ yield 
% $\gamma_u {\tilde u}'=\gamma_y f {\rm Re}$ and
% $\gamma_u {\tilde u}'=/ \kappa {\tilde y}$. 
% It follows that to make  the law of the viscous layer
%  and the the log law of the wall
%  invariant to changes is Re and $f$, in the form
%  ${\tilde u}'=1$ and ${\tilde u}'=1 / \kappa {\tilde y}$, respectively,
% we must choose  $\gamma_y=1/{\rm Re} \sqrt{f}$ and $\gamma_u=\sqrt{f}$,
% and therefore  ${\tilde y}\equiv {\rm Re} \sqrt{f}\, {\hat y}
%  = {\rm Re} \sqrt{f}\, y/R$ 
% and ${\tilde u}\equiv {\hat u}/\sqrt{f}=  u/U\sqrt{f}$, which we recognize
% as the plus variables.

% The substitutions ${\hat y}=\gamma_y {\tilde y}$ and 
% ${\hat u}=\gamma_u {\tilde u}$ in  ${\hat u}'=f {\rm Re}$
% and ${\hat u}'= \sqrt{f} / \kappa {\hat y}$ yield 
% $\gamma_u {\tilde u}'=\gamma_y f {\rm Re}$ and
% $\gamma_u {\tilde u}'=/ \kappa {\tilde y}$. 

%  There is a unique choice of variables 
% ${\tilde y}$ and $\tilde u$ that makes
% the law of the viscous layer
%  and the the log law of the wall
%  invariant to changes is Re and $f$, in the form
%  ${\tilde u}'=1$ and ${\tilde u}'=1 / \kappa {\tilde y}$, respectively:
%   ${\tilde y}\equiv {\rm Re} \sqrt{f}\, {\hat y}
%  = {\rm Re} \sqrt{f}\, y/R$ 
% and ${\tilde u}\equiv {\hat u}/\sqrt{f}=  u/U\sqrt{f}$, which we recognize
% as the plus variables.
% In the plus variables the equation for the MVP reads

  The law of the viscous layer
  and the the log law of the wall may be made 
  invariant to changes in Re and $f$, in the form
  ${\tilde u}'=1$ and ${\tilde u}'=1 / \kappa {\tilde y}$, 
 by choosing ${\tilde y}\equiv {\rm Re} \sqrt{f}\, {\hat y}
  = {\rm Re} \sqrt{f}\, y/R$ 
 and ${\tilde u}\equiv {\hat u}/\sqrt{f}=  u/U\sqrt{f}$, 
 which we recognize as the wall variables.
 In the wall variables (\ref{lau}) becomes
\begin{equation} \label{util}
\kappa^2 I^3 {\tilde y}^2 {\tilde u}'^{2}
 + {\tilde u}'- (1-{\tilde y}/ {\rm Re} \sqrt{f})=0,
\end{equation}
where $I\equiv {\mathcal I}(\eta/y,{\tilde y}/{\rm Re}\sqrt{f})$, 
 and (\ref{laeoy}) becomes
 \begin{equation} \label{etatil}
\eta/y= ( {\tilde u}' (1-{\tilde y}/{\rm Re}\sqrt{f})
 -{\tilde u}'^2)^{-1/4}\,{\tilde y}^{-1}.
\end{equation}
 From these equations it is apparent that in the wall
 variables there is a single MVP
 except close to the centerline of the pipe, %in the wake,
 where ${\tilde y}\approx {\rm Re}\sqrt{f}$ (or $y\approx R$).

  We now ascertain under what conditions 
 (\ref{util}) and (\ref{etatil}) are compatible with a
  nonvanishing turbulent shear stress close to the wall.
 Suppose that at a point ${\tilde y}\ll {\rm Re}\sqrt{f}$ (or $y\ll R$)
   we have $I={\mathcal I}(\eta/y,0)>0$,
  and therefore $\tau_t>0$.
Then, we can eliminate ${\tilde u}'$ from (\ref{util}) and (\ref{etatil}) 
to obtain %express ${\tilde y}$ as a function of $\eta/y$,
%\begin{equation}
%{\tilde y}=
%\left({ { \kappa^{2/3} I }\over{(\eta/y)^{8/3}}} +
% {{1}\over{ \kappa^{2/3} (\eta/y)^{4/3} I }}\right)^{1/2}.
%\end{equation}
\begin{equation} \label{ecuy}
{\tilde y}=\left({ {(\eta/y)^{4/3} + \kappa^{4/3}\, {\mathcal I}^2(\eta/y,0) }
\over{\kappa^{2/3}\, (\eta/y)^{8/3}\, {\mathcal I}(\eta/y,0) }} \right)^{1/2}.
\end{equation}
 Plots of ${\tilde y}$ vs.\ $\eta/y$ (Fig.~\ref{yvisfig}) 
reveal that for any given $\kappa$ and $\beta_d$ there exists a 
  minimum value of ${\tilde y}$, to be denoted ${\tilde y}_v$. 
  Therefore,  for  ${\tilde y}<{\tilde y}_v$ it must be 
 that $I=0$, $\tau_t=0$, and ${\tilde u}'=1$.
 As this latter equation is the law of the viscous layer,
  we identify ${\tilde y}_v$ with the thickness of 
the viscous layer \cite{layers}. Note that ${\tilde y}_v$ depends on 
 the dissipative-range parameter $\beta_d$ %and $\kappa$  
 (Fig.~\ref{yvisfig}b), and
  that for $\beta_d=0$ there is no viscous layer (${\tilde y}_v=0$).
% and that for $\beta_d=0$ 
% where 
% $\kappa=(\kappa_\varepsilon \kappa_\tau^3)^{1/4}$.
% Thus the thickness of the viscous layer is set
% by the momentum-transfer parameter $\kappa_\tau$ 
%and the spectral parameters $\kappa_\varepsilon$ and 
% $\beta_d$.
%For example, for $\kappa=0.42$ and $\beta_d=1$
% we have ${\tilde y}_v= 4$ (Fig.~\ref{yvisfig}b).
\begin{figure} 
\centering
\resizebox{3.5in}{!}{\includegraphics{vp_yviscousfig.eps}}
%\resizebox{4.2in}{!}{\includegraphics{vp_yviscousfig.eps}}
\caption{\label{yvisfig} (a) Plots of (\ref{ecuy}) for $\kappa=0.42$ and three
  values of $\beta_d$. (b) Plots of the thickness
 of the viscous layer ${\tilde y}_v$ as 
 a function of $\beta_d$ for three  values of $\kappa$.  }
\end{figure}

We are now ready to compute the MVP.
%As the friction factor $f$ does not affect
% the MVP except in the wake, 
For simplicity we use 
  the Blasius relation for the friction factor,
 $f=0.033\,{\rm Re}^{-1/4}$ \cite{exper}. 
% For %the spectral parameter 
% $\kappa_\varepsilon$
% we use the theoretical estimate 
% from Kolmog\'orov's four-fifth law \cite{ff}, 
% $\kappa_\varepsilon=4/5$.  
  Values of the spectral parameters
  $\beta_d$ and $\beta_e$ %determined 
  from the fitting of 
  spectra are of O(1) \cite{pope};
  here we use $\beta_d=1$ and $\beta_e=4$. 
% For the momentum-transfer parameter $\kappa_\tau$ 
%  we use the value $\kappa_\tau=0.34$, so that 
% $\kappa=(\kappa_\varepsilon \kappa_\tau^3)^{1/4}=0.42$ \cite{exper}.
 The thickness of the viscous layer is set by 
 $\beta_d$ and $\kappa$;
 for $\beta_d=1$ and $\kappa=0.42$, ${\tilde y}_v= 3.9$
 (Fig.~\ref{yvisfig}b).  Therefore, for
  ${\tilde y}<3.9$ we can write ${\tilde u}({\tilde y})={\tilde y}$, 
 and for   ${\tilde y} >3.9$ we must compute ${\tilde u}({\tilde y})$
 by integrating (\ref{util}) with
 boundary condition ${\tilde u}(3.9)=3.9$.
The results are shown in Fig.~\ref{ufig}a.
 %(From the figure, we estimate the constant $B$ 
 %of the log-law of the wall to be $B=7$ in this case.)
 %It is clear that, 
 Given the spectrum,
 the theory yields the entire MVP with all of its distinctive 
 features. The specific conection between each one of 
 these features and the spectrum will become apparent 
 in what follows.

To elucidate the effect of the 
 energetic-range correction, we recompute the MVP
  using $\beta_e=7$ (a larger value than before)
 and $\beta_e=0$ (the smallest possible value, 
 which corresponds to having no energetic-range 
 correction). 
 The results (Fig.~\ref{ufig}b)
  indicate that %the sole effect  of
 the energetic-range 
  correction steepens the MVP in the wake.
% The eddies that dominate the momentum transfer 
% in the wake must %therefore
%  be eddies of the energetic range. 
 The dominant eddies 
 in the wake must therefore
  be eddies of
 the energetic range. These eddies are slowed
 down by the energetic-range correction (and 
 made less adept at transfering momentum). 
 This effect explains the steepening
 of the MVP in the wake.
 
We have seen that the 
  dissipative-range correction sets
  the thickness of the viscous layer 
%(which depends on $\beta_d$ 
(Fig.~\ref{yvisfig}b)).
To understand further the effects of the 
 dissipative-range correction, we recompute the MVP
 using a few values of $\beta_d$, including
 $\beta_d=0$ (the smallest possible value, 
 which corresponds to having no dissipative-range 
 correction). 
 % (Note that for $\beta_d=0$, $y_v=0$ (Fig.~\ref{
 % so that we compute
 % ${\tilde u}({\tilde y})$
 %by integrating (\ref{util}) with
 %boundary condition ${\tilde u}(0)=0$.) 
 %The results, shown in Fig.~\ref{ufig}c, 
 %indicate that the sole effect 
 %of the dissipative-range 
 % correction is to give rise to
 %a buffer  a layer 
 %in which the MVP has a noticeble 
 %negative curvature \cite{layers}). For $\beta_d=0$,
%steepen the MVP in the wake.
 The results (Fig.~\ref{ufig}c) 
 indicate that the dissipative-range  
  correction causes the buffer layer to form, 
 so that for $\beta_d=0$ there is no buffer layer.
 (The buffer layer is the part of the MVP where 
 the MVP has a %noticeble 
 negative curvature \cite{layers}.) 
%  The buffer layer becomes less pronounced
%  as the value of $\beta_d$ is decreased, and 
%for $\beta_d=0$ there is no buffer layer.
% The eddies that dominate the momentum transfer 
% in the buffer layer must % therefore
% be eddies of the dissipative range. 
The dominant eddies in the buffer layer 
must therefore be eddies of the dissipative range. 
These eddies
  are slowed down by the dissipative-range 
 correction; the larger the eddies, the 
  less they are slowed down, and the more adept 
 they remain at transferring momentum. As the size of the 
 dominant eddies increases with the distance
 to the wall, the MVP becomes less steep
 as we traverse the buffer layer from the 
  outer edge of the viscous layer 
 (where the eddies are fully viscous) 
 to the inner edge of
  the log layer (where the eddies are
  fully inertial).
  This effect explains
 the negative curvature of the MVP in the buffer layer.

It is apparent from Fig.~\ref{ufig}c that the constant $B$ of the
log law of the wall is set by $\beta_d$. 
A plot of $B$ as a function 
of $\beta_d$ is shown in Fig.~\ref{ufig}d. 


\begin{figure} 
\centering
\resizebox{3.4in}{!}{\includegraphics{vpfigure.eps}}
%\resizebox{4in}{!}{\includegraphics{vpfigure.eps}}
\caption{\label{ufig} (a) The MVPs computed using 
  $\kappa=0.42$, $\beta_e=3$, and $\beta_d=1$.
  The thick dot %at $\ln {\tilde y}=\ln {\tilde y}_v=\ln 3.9$
 indicates the point of contact between the viscous layer and 
 the buffer layer. (b) The same as in (a) but using 
  $\beta_e=0$ (dashed lines) and $\beta_e=7$ (solid lines).
% For $\beta_e=0$ the energetic-range correction is null. 
Inset: detail of the wakes.
  (c) The same as in (a) but using 
   $\beta_d=0$ (bottom), 1, 2, and 3 (top).
 Inset: A plot of %the analytic solution of 
(\ref{ubeta0}) for $\kappa=0.42$.
% for the same value of $\kappa$ as in the other plots. 
%This is the 
% theoretical MVP where the dissipative-range correction 
%  is null and ${\rm Re}\to \infty$.
(d) A plot of $B$ %of the log law of the wall 
as a function $\beta_d$ for $\kappa=0.42$. }
\end{figure}

%we expect the MVP to be steeper close to the 
% viscous layer and less steep close to the log layer.
% as we traverse the buffer layer starting at the 
%  outer edge of the viscous layer (where 
% the MVP is the steepest) and ending at the inner edge of
%  the log layer
% (where the MVP is the least steep). This effect explains
% the negative curvature of the MVP in the buffer layer.

% At the boundary between
% the buffer layer and the log layer the dominant eddies
%  (whose size is proportional to the distance to the wall)
%  are large and virtually unaffected by the dissipative-range
% correction; they are therefore as efficient 
% at transferring momentum as the dominant eddies in the 
% the log layer, and the 
 %
 %As the size of
 %the dominant eddies is proportional to the distance
 %to the wall, the dominant eddies become 
 %more efficient at transferring momentum as 
 % the distance to the wall increases,  

%, and there is no buffer layer.
% at transfering momentum) than an imaginary eddy of the same
% size in the inertial range. This effect explains the steepening
% of the MVP in the buffer layer. Note also that the MVP
% is always steeper in the buffer than in the viscous layer:
% the dominat eddies are always 
% more efficient at producing shear stress than the viscosity,
% even when these eddies are slowed down by the dissipative-range
% correction.

 Interestingly,
  the MVP for $\beta_d=0$ (no dissipative-range correction)
 can be obtained  analytically everywhere save the wake. In fact,
  for $\beta_d=0$ and 
 ${\tilde y}\ll {\rm Re}\sqrt{f}$ (or $y\ll R$),
 $I=1$ and (\ref{util}) simplifies to 
 $\kappa^2 {\tilde y}^2 {\tilde u}'^{2}+ {\tilde u}'- 1=0$,
 which with  ${\tilde u}(0)=0$ yields
\begin{equation} \label{ubeta0}
{\tilde u}={{ {\rm arcsinh}(2 \kappa {\tilde y})}\over{\kappa}}+ 
 {{1-\sqrt{1+\kappa^2 {\tilde y}^2 }\over{2 \kappa^2 {\tilde y}}}}.
\end{equation}
 This is what the MVP would be away from the wake 
 if the Kolmog\'orov spectrum
 were valid even for vanishingly small eddies (inset of Fig.~\ref{ufig}c).
For ${0\ll \tilde y}\ll {\rm Re}\sqrt{f}$ (or $0\ll y\ll R$),
 (\ref{ubeta0}) simplifies asymptotically to  
%${\tilde u}\sim (1/\kappa) \ln {\tilde y} +(-1+\ln 4\kappa)/\kappa$.
% We conclude that for $\beta_d=0$ the constant $B$ of the 
% log law of the wall takes the value $(-1+\ln 4\kappa)/\kappa$. 
%For a general case with $\beta_d>0$, $B$ depends on 
% both $\kappa$ and $\beta_d$ (Fig.~\ref{ufig}d).
${\tilde u}\sim (1/\kappa) \ln {\tilde y} + B$, with 
$B = (-1+\ln 4\kappa)/\kappa$.
Thus, for 
$\beta_d=0$ and $\kappa=0.42$, $B=-1.15$ 
in accord with Fig.~\ref{ufig}d. 

%By using  the turbulent spectrum to compute the velocities
% of eddies of different sizes in the flow, we have been able to
%  identify the eddies that dominate the 
% production of shear stress, and
% to obtain an expression for the shear stress 
% produced by the dominant eddies. 
%%  We have formulated a theory that %provides This expression
%% % for the shear stress 
%% furnishes the long-surmised link \cite{tl} 
%%  between the mean velocity profile and
%%  the turbulent spectrum. 
%% To test the theory, we have shown
%%  that the usual model of the spectrum
%% %,consisting of %the Kolmog\'orov
%%  (a power-law
%%  inertial range with corrections for the dissipative range and 
%%  the energetic range) is in itself sufficient to compute
%%   with no additional assumptions 
%%  a mean velocity profile complete with viscous layer,
%%  buffer layer, log layer, and wake. 
%% The thickness of the viscous layer, the 
%%   two constants of the log law of the wall
%%   (including the K\'arm\'an constant), % $\kappa$),
%%   and the amplitude of the 
%%  wake are all set by a
%%   dimensionless momentum-transfer 
%%  parameter $\kappa_\tau$ and the usual spectral 
%%  parameters---the Kolmog\'orov
%%  exponent $-5/3$, the
%%  multiplicative constant $\kappa_\varepsilon=4/5$ \cite{ff}, 
%% % (which can be estimated from Kolmog\'orov's 
%% % four-fifths law to be $\kappa_\varepsilon=4/5$),
%%  and the parameters $\beta_d$ and $\beta_e$ of 
%%  the dissipative-range 
%%  and energetic-range corrections.
%% The relation between a specific feature of the MVP and
%%  the spectral parameters provides insight into 
%%  the underlying physics. Thus, for example,
%%  the K\'arm\'an constant is given by
%%   $\kappa=(\kappa_\varepsilon \kappa_\tau^3)^{1/4}$, and is 
%%  therefore independent of the energetic-range and
%%  dissipative-range corrections, with
%%  the implication that the eddies that dominate 
%%   the momentum transfer in the log layer are 
%%   eddies of the inertial range. 
%% % therefore does not depend on the 
%% %the exponent $-5/3$,
%%  More broadly, the close relation
%%   between the mean velocity profile 
%%   and the turbulent spectrum suggests that in turbulence, 
%%   as in phase
%%  transitions, global variables %(large-scale pehnomena?)
%%   are 
%%   governed by the statistics of the fluctuations. 

%$\tau_t=\rho \nu_t u'(y)$ where $\nu_t$ is 
% the turbulent kinematic viscosity of the dominant
% eddy at a distance $y$ from the wall
% $\nu_t=\kappa_t \kappa_\varepsilon^{-1/3} \nu (y/\eta)^{4/3} \sqrt{I(y/\eta,y/R)}$

We have established the long-surmised link \cite{tl} between
the mean velocity profile and the turbulent spectrum. To
test our results, we have shown that the usual model of
the spectrum (a power-law inertial range with corrections
for the dissipative range and the energetic range)
is in itself sufficient to compute with no additional assumptions
a mean velocity profile complete with viscous
layer, buffer layer, log layer, and wake. The thickness
of the viscous layer, the two constants of the log law of
the wall, and the amplitude of the wake are all set by
the dimensionless momentum-transfer constant $\kappa_\tau$ and
the usual spectral parameters---the parameter $\beta_e$ of the
energetic-range correction and the parameter $\beta_d$ of the
dissipative-range correction. The relation between a specific
feature of the MVP and the spectral parameters reminds
us of the underlying physics. Thus, for example,
the K\'arm\'an constant is independent of both $\beta_e$ and $\beta_d$,
and is therefore unaffected by the energetic-range and
dissipative-range corrections, with the implication that
the eddies that dominate the momentum transfer in the
log layer are eddies of the inertial range. More broadly,
the close relation between the mean velocity profile and
the turbulent spectrum indicates that in turbulence, as
in continuous phase transitions, global variables are governed by the
statistics of the fluctuations.

\begin{acknowledgments}
 NSF funded this work through
%  grant %DMR06--04477 (UP)
  grant DMR06--04435. 
\end{acknowledgments}

% Create the reference section using BibTeX:
%\bibliography{basename of .bib file}
\begin{thebibliography}{} 
\bibitem{noncapisco}B.\ Hof et al.,
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   Phys.\ Rev.\ Lett.\ {\bf 100}, 054504 (2008). 
%J.\ Jim\'enez, %Turbulent flows over rough walls.
% Annu.\ Rev.\ Fluid Mech.\ {\bf 36}, 173 %-196 (2004); 
%M.\ R.\ Raupach, R.\ A.\ Antonia,
%S.\ Rajagopalan, %Rough-wall turbulent boundary layers. 
% Appl.\ Mech.\ Rev.\ {\bf 44}, 1 %-26 (1991).} 
\bibitem{tl} H.\ Tennekes, J.\ L.\ Lumley,
{\it A First Course in Turbulence\/} (MIT Press, 1972).
\bibitem{prandtl}L.\ Prandtl, {\it Essentials of Fluid Dynamics\/}, 
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\bibitem{niku}Reprinted in English in 
   J.\ Nikuradse, %Laws of flow in rough pipes. 
   {\it NACA TM} {\bf 1292} (1950).
% For closely
%  related data on open-channel flows, see 
% O.\ Kirschmer,
%Pertes des charge dans les conduites forc\'ees et les canaux d\'ecouverts
% Revue g\'en\'erale de l'hydraulique (Paris)
% {\bf 15}, 115 %--138
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   buffer layer, etc.\ (caption of Fig.~\ref{expe}),
%  we follow \cite{tl}, in particular Chapter~5 and the figure on page 160.
as in ch.\ 5 of \cite{tl}, in particular Fig. 5.6.
\bibitem{baren} G.\ Barenblatt, 
%Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis. 
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G.\ Barenblatt, A.\ J.\ Chorin, 
%A mathematical model for the scaling of turbulence. 
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\bibitem{rich}L.\ F.\ Richardson,
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(Cambridge Univ. Press, Cambridge, 1995).
\bibitem{pope}S.\ B.\ Pope, {\it Turbulent Flows\/} 
(Cambridge Univ. Press, Cambridge, 2000). 
\bibitem{taylor}{G.\ I.\ Taylor, % The statistical theory  of turbulence. 
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 % Crossover from high to low Reynolds number turbulence.
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\bibitem{kolmo}Reprinted in English in A.\ N.\ Kolmog\'orov,
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  %for very large Reynolds numbers. 
    Proc.\ R.\ Soc.\ London A {\bf 434}, 9 %-13
     (1991). 
\bibitem{aniso} On the Kolmog\'orov spectrum and anisotropy, see
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    %Kolmogorov inertial range for inhomogeneous turbulent flows.
     Phys.\ Rev.\ Lett.\ {\bf 65}, 1356 %-1359
    (1990); T.\ S.\ Lundgren, 
    %Kolmogorov two-thirds law by matched asymptotic expansion. 
     Phys.\ Fluids {\bf 14}, 638 %-642 
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     % Kolmogorov turbulence by the method of matched asymptotic expansions. 
     ibid.\ {\bf 15}, 1024 %-1081 
    (2003).  The view of anisotropy  
 as a perturbation superposed on the isotropic
 flow  (see, e.g., L.\ Biferale, I.\ Procaccia, 
% {\it Anisotropy in turbulent flows and in turbulent transport\/},
 Phys.\ Rep.\ {\bf 414}, 43 (2005))  might
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themselves might change, albeit slightly (see 
 M.\ Casciola, P.\ Gualtieri, B.\ Jacob, R.\ Piva,
% Scaling Properties in the Production Range of Shear Dominated Flows.
 Phys.\ Rev.\ Lett.\ {\bf 95}, 024503 (2005)). 
% Nevertheless, note (i) that
%  the exponent of the second-order structure function 
%  appears to change only by %about 5\%, 
% little and (ii) that the Kolmog\'orov
%exponent does give the exact scalings of Blasius and Strickler, 
%whereas any modified exponent would not.}
\bibitem{ff}The four-fifth law reads
 $\overline{v_s^3} = -(4/5)\, \varepsilon \,s$,
 where the left-hand side
 is the third-order structure function.
 The estimate $|\overline{v_s^3}|=u_s^3$ leads to
 $\varepsilon=(5/4)\, u_s^3/s$, or $\kappa_\varepsilon=4/5$.
%\bibitem{energy} energy equation
\bibitem{firsterm} We retain only the 1st term of the Taylor
  series because (a) additional terms 
 would necessitate BCs other than 
 the no-slip BC and (b) in our final solution for $u(y)$  
the 2nd term $\le 11$\% of the 1st term for all $y$.
\bibitem{ff1}    Experimental and computational
 estimates of $\kappa_\varepsilon$ are invariably of
 O(1) but differ slightly for different types of flow; 
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K\'arm\'an constant may depend mildly on the type of flow.
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G.\ Gioia, F.\ A.\ Bombardelli, 
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\end{thebibliography}

\end{document}










0909.2714.extracted_bib
\begin{thebibliography}{} 
\bibitem{noncapisco}B.\ Hof et al.,
  Science {\bf 305}, 1594 (2004);
 V.\ S.\ L'vov et al., %Universal model of finite reynolds number turbulent flow in channels and pipes. 
   Phys.\ Rev.\ Lett.\ {\bf 100}, 054504 (2008). 
%J.\ Jim\'enez, %Turbulent flows over rough walls.
% Annu.\ Rev.\ Fluid Mech.\ {\bf 36}, 173 %-196 (2004); 
%M.\ R.\ Raupach, R.\ A.\ Antonia,
%S.\ Rajagopalan, %Rough-wall turbulent boundary layers. 
% Appl.\ Mech.\ Rev.\ {\bf 44}, 1 %-26 (1991).} 
\bibitem{tl} H.\ Tennekes, J.\ L.\ Lumley,
{\it A First Course in Turbulence\/} (MIT Press, 1972).
\bibitem{prandtl}L.\ Prandtl, {\it Essentials of Fluid Dynamics\/}, 
    (Blackie \& Son, London,  3rd ed., 1953), chap.\ III.
\bibitem{niku}Reprinted in English in 
   J.\ Nikuradse, %Laws of flow in rough pipes. 
   {\it NACA TM} {\bf 1292} (1950).
% For closely
%  related data on open-channel flows, see 
% O.\ Kirschmer,
%Pertes des charge dans les conduites forc\'ees et les canaux d\'ecouverts
% Revue g\'en\'erale de l'hydraulique (Paris)
% {\bf 15}, 115 %--138
% (1949) (in French). 
\bibitem{exper} M.\ V.\ Zagarola, A.\ J.\ Smits, A. J. 
%Mean-flow scaling of turbulent pipe flow. 
J.\ Fluid Mech.\ {\bf 373}, 33%--79. 
(1998); B.\ J.\ McKeon et al., 
%Further observations on the mean velocity distribution in fully developed pipe flow. J.\ Fluid Mech.\
ibid.\ {\bf 501} 135%--147
 (2004).
\bibitem{computgen} J.\ G.\ M.\  Eggels et al.,
J.\ Fluid Mech.\ {\bf 268}, 175%--210
 (1994);
J.\ M.\ J.\ den Toonder, F.\ T.\ M.\ Nieuwstadt,
Phys.\ Fluids {\bf 9} 3398 (1997).
\bibitem{comput}
S.\ Hoyas and J.\ Jimenez, Phys.\ Fluids {\bf 18} 011702 (2006);
del Alamo et al.,
J.\ Fluid Mech.\ {\bf 500} 135 (2004).
\bibitem{layers} We define the viscous layer, the 
   buffer layer, etc.\ (caption of Fig.~\ref{expe}),
%  we follow \cite{tl}, in particular Chapter~5 and the figure on page 160.
as in ch.\ 5 of \cite{tl}, in particular Fig. 5.6.
\bibitem{baren} G.\ Barenblatt, 
%Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis. 
J.\ Fluid Mech.\ {\bf 248} 513%--520 
 (1993);
G.\ Barenblatt, A.\ J.\ Chorin, 
%A mathematical model for the scaling of turbulence. 
Proc.\ Natl.\ Acad.\ Sci.\ {\bf 101}, 15023%--15026 
 (2004). 
\bibitem{rich}L.\ F.\ Richardson,
   Proc.\ Roy.\ Soc.\ London  A {\bf 110}, 709 %--737 
  (1926). 
\bibitem{frisch}U.\ Frisch, {\it Turbulence\/} 
(Cambridge Univ. Press, Cambridge, 1995).
\bibitem{pope}S.\ B.\ Pope, {\it Turbulent Flows\/} 
(Cambridge Univ. Press, Cambridge, 2000). 
\bibitem{taylor}{G.\ I.\ Taylor, % The statistical theory  of turbulence. 
 Proc.\ Roy.\ Soc.\ London  A {\bf 151}, 421 %-478 
  (1935); D.\ Lohse,
 % Crossover from high to low Reynolds number turbulence.
   Phys.\ Rev.\ Lett.\ {\bf 73}, 3223 %-3226 
  (1994).  The existence of an  upper bound on $\varepsilon$ 
 that is independent of the viscosity has been proved
  mathematically; see Doering, Ch.\ R.\ and P.\ Constantin, 
 Phys.\ Rev.\ Lett.\ {\bf 69}, 1648 (1992). } 
\bibitem{kolmo}Reprinted in English in A.\ N.\ Kolmog\'orov,
 %The local structure of turbulence in incompressible viscous fluid
  %for very large Reynolds numbers. 
    Proc.\ R.\ Soc.\ London A {\bf 434}, 9 %-13
     (1991). 
\bibitem{aniso} On the Kolmog\'orov spectrum and anisotropy, see
  B.\ Knight,  L.\ Sirovich, 
    %Kolmogorov inertial range for inhomogeneous turbulent flows.
     Phys.\ Rev.\ Lett.\ {\bf 65}, 1356 %-1359
    (1990); T.\ S.\ Lundgren, 
    %Kolmogorov two-thirds law by matched asymptotic expansion. 
     Phys.\ Fluids {\bf 14}, 638 %-642 
    (2002); T.\ S.\ Lundgren,
     % Kolmogorov turbulence by the method of matched asymptotic expansions. 
     ibid.\ {\bf 15}, 1024 %-1081 
    (2003).  The view of anisotropy  
 as a perturbation superposed on the isotropic
 flow  (see, e.g., L.\ Biferale, I.\ Procaccia, 
% {\it Anisotropy in turbulent flows and in turbulent transport\/},
 Phys.\ Rep.\ {\bf 414}, 43 (2005))  might
 break down close to a wall, where  the exponents 
themselves might change, albeit slightly (see 
 M.\ Casciola, P.\ Gualtieri, B.\ Jacob, R.\ Piva,
% Scaling Properties in the Production Range of Shear Dominated Flows.
 Phys.\ Rev.\ Lett.\ {\bf 95}, 024503 (2005)). 
% Nevertheless, note (i) that
%  the exponent of the second-order structure function 
%  appears to change only by %about 5\%, 
% little and (ii) that the Kolmog\'orov
%exponent does give the exact scalings of Blasius and Strickler, 
%whereas any modified exponent would not.}
\bibitem{ff}The four-fifth law reads
 $\overline{v_s^3} = -(4/5)\, \varepsilon \,s$,
 where the left-hand side
 is the third-order structure function.
 The estimate $|\overline{v_s^3}|=u_s^3$ leads to
 $\varepsilon=(5/4)\, u_s^3/s$, or $\kappa_\varepsilon=4/5$.
%\bibitem{energy} energy equation
\bibitem{firsterm} We retain only the 1st term of the Taylor
  series because (a) additional terms 
 would necessitate BCs other than 
 the no-slip BC and (b) in our final solution for $u(y)$  
the 2nd term $\le 11$\% of the 1st term for all $y$.
\bibitem{ff1}    Experimental and computational
 estimates of $\kappa_\varepsilon$ are invariably of
 O(1) but differ slightly for different types of flow; 
 K.\ R.\ Sreenivasan, Phys.\ Fluids {\bf 10}, 528 (1998).
 The implication here is that the 
K\'arm\'an constant may depend mildly on the type of flow.
\bibitem{previous} 
G.\ Gioia, F.\ A.\ Bombardelli, 
 %Scaling and similarity in rough channel flows.
   Phys.\ Rev.\ Lett.\ {\bf  88},  014501 (2002);
  G.\ Gioia, P.\ Chakraborty, 
    ibid.\ {\bf 96},  044502 (2006);
N.\ Goldenfeld,  ibid.\ {\bf 96},  044503 (2006);
G.\ Gioia, P.\ Chakraborty, F.\ A.\ Bombardelli, 
 Phys.\ Fluids {\bf 18}, 038107 (2006);
 M.\ Mehrafarin, N.\ Pourtolami, 
Phys.\ Rev.\ E {\bf 77}, 055304(R) (2008); 
 N.\ Guttenberg, N.\ Goldenfeld,  
 ibid.\ {\bf 79}, 065306(R) (2009).

\end{thebibliography}