Macroscopic effects of the structure of the fluctuations in turbulent flows
Tuan Tran$^1$
Pinaki Chakraborty$^2$
Nicholas Guttenberg$^3$
Alisia Prescott$^4$
Hamid Kellay$^5$
Walter Goldburg$^4$
Nigel Goldenfeld$^{3\star}$ and Gustavo Gioia$^{1\star}$
% Two aspects of turbulent flows have been extensively % researched: Two aspects of turbulent flows\cite{FRIS95,SREE99,pope2000tf,hof2006flt} have been the subject of extensive, split research efforts: macroscopic properties, such as the frictional drag\cite{schlichting2000blt} experienced by a flow past a wall, and the turbulent spectrum.\cite{taylor1938spectrum,FRIS95,SREE97} % The turbulent spectrum may be said to represent % the fabric of a turbulent state; % in practice it is a function---a power law % of exponent $\alpha$, the ``spectral exponent''---that % gives the revolving velocity of a % turbulent fluctuation (or ``eddy'') % of any given size in a flow.\cite{FRIS95} The turbulent spectrum may be said to represent the fabric of a turbulent state; in practice it is a power law of exponent $\boldsymbol{ \alpha}$ (the ``spectral exponent'') that gives the revolving velocity of a turbulent fluctuation (or ``eddy'') of size $\boldsymbol{s}$ as a function of $\boldsymbol{s}$.\cite{FRIS95} % The turbulent spectrum may be said to represent % the fabric of a turbulent state; % in practice it is a power law of % exponent $\alpha$---the ``spectral exponent''---that % gives the revolving velocity of any % turbulent fluctuation (or ``eddy'') as % a function of the size of the eddy.\cite{FRIS95} The link, if any, between macroscopic properties and the turbulent spectrum remains missing. Might it be found by contrasting the frictional drag in flows with differing types of spectra? Here we perform unprecedented measurements of the frictional drag in soap-film flows,\cite{KELL02} where the spectral exponent $\boldsymbol{\alpha=3}$,\cite{KRAI67,batchelor1982tht} and compare the results with the frictional drag in pipe flows,\cite{schlichting2000blt} where the spectral exponent $\boldsymbol{\alpha=5/3}$.\cite{richardson26,KOLM41} For moderate values of the Reynolds number $\Re$ (a measure of the strength of the turbulence), we find that in soap-film flows the frictional drag scales as $\boldsymbol{\Re^{-1/2}}$, whereas in pipe flows the frictional drag scales\cite{mckeon2005ffs} as $\boldsymbol{\Re^{-1/4}}$. Each of these scalings may be predicted from the attendant value of %the spectral exponent $\boldsymbol{\alpha}$ by using a new theory,\cite{GIOI06, GUTT09} in which the frictional drag is explicitly linked to the turbulent spectrum. Our work indicates that in turbulence, as in continuous phase transitions, macroscopic properties are governed by the spectral structure of the fluctuations.\cite{GOLD06,MEHR08} %ALTERNATIVE % % Two aspects of turbulent flows\cite{FRIS95,SREE99,pope2000tf} % have been the subject % of extensive, split research efforts: % global properties, such as the % frictional drag experienced by a flow % past a wall,\cite{SREE99,pope2000tf,schlichting2000blt} % and the turbulent spectrum.\cite{taylor1938spectrum,FRIS95,SREE97} % The turbulent spectrum may be said to represent % the fabric of the turbulence; in practice it is % a function---a power law of exponent $\alpha$---that % gives the revolving velocity of a % turbulent fluctuation (or ``eddy'') % of any given size in the flow.\cite{FRIS95} % The link, if any, % between global properties and the turbulent spectrum % remains missing. Might it be found by contrasting % flows with differing types of spectra? % Here we perform unprecedented measurements of the %frictional drag in soap-film flows,\cite{KELL02} where the % spectrum is of a type known % as the ``enstrophy cascade''\cite{KRAI67,batchelor1982tht} % (for which $\alpha=3$), % and compare the results with the frictional drag % in pipe flows,\cite{schlichting2000blt} % where the spectrum is of a type known % as the ``energy cascade''\cite{richardson26,KOLM41} % (for which $\alpha=5/3$). % For moderate values of the Reynolds % number $\Re$ (a measure of the strength of the turbulence), % we find that in soap-film flows % the frictional drag scales as $\Re^{-1/2}$, whereas % in pipe flows the frictional drag scales\cite{mckeon2004ffs} % as $\Re^{-1/4}$. Each of these scalings may be % predicted from the attendant % value of $\alpha$ by %using a new theory\cite{GIOI06, GUTT09} %in which the frictional drag is explicitly linked % to the turbulent spectrum. %Our work indicates that in turbulence, %as in continuous phase transitions, %global properties are governed by the spectral structure of % the fluctuations.\cite{GOLD06,MEHR08} % Two aspects of turbulent flows have been %%the subject of most reserach: % researched the most: global properties, such as the % frictional drag experienced by a flow % past a wall, and the turbulent spectrum. % The turbulent spectrum may be said to represent % the internal structure of the turbulence; % in practice it is a function, % usually a power law with a characteristic spectral % exponent $\alpha$, that gives the revolving velocity of a % turbulent fluctuation (or ``eddy'') % of any given size in the flow. %% in practice, it is an account of the distribution of %% kinetic energy among the turbulent fluctuations %% (or ``eddies'') of different sizes in a flow. %% in pratice, it is an account of the distribution of %% kinetic energy among the turbulent fluctuations %% (or ``eddies'') of different sizes in a flow. %% in practice, it is a mathematical function that describes %% the form in which kinetic energy is apportioned among %% the turbulent fluctuations (or ``eddies'') of different sizes in %% a flow. % The link, if any, % between global properties and the turbulent spectrum % is presently unknown. %% This state of affairs brings to mind %% continuous phase transitions in the early 60's, %% before a link was found between the %% global properties, such as the heat capacity, %% and the fluctuations of thermodynamic systems. % Here we perform unprecedented measurements of the %frictional drag in soap-film flows, where the %turbulent spectrum is of a type known % as the ``enstrophy cascade'' (for which $\alpha=3$), % and compare the results with the frictional drag % in pipe flows, % where the turbulent spectrum is of a type known % as the ``energy cascade'' (for which $\alpha=5/3$). % For moderate values of the Reynolds % number $\Re$ (a measure of the strength of the turbulence), % we find that in soap-film flows % the frictional drag scales as $\Re^{-1/2}$, whereas % in pipe flows the frictional drag scales % as $\Re^{-1/4}$. These experimental scalings may be predicted %using a new theory\cite{GIOI06, GUTT09} %in which the frictional drag is explicitly linked % to the turbulent spectrum, so that the functional % dependence of the frictional drag on % Re is set by the value of the spectral exponent $\alpha$. %Our work indicates that in turbulence, %as in continuous phase transitions\cite{GOLD06,MEHR08}, %global properties are governed by the spectral structure of % the fluctuations. %and underscores the value of using %two-dimensional turbulent flows to test %and extend our understanding of %turbulence. % are due to %we show that the frictional drag is % governed by the turbulent spectrum. To that end, % we compare the frictional drag in %flows with distinct turbulent spectra: %three-dimensional pipe flows and two-dimensional %soap-film flows, in %which the turbulent spectrum is of the type known respectively as % the ``energy cascade'' \cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97} % and the ``enstrophy cascade'' \cite{batchelor1982tht,KRAI67,KELL02}. %We perform %Despite much progress in understanding the turbulent %spectrum\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}---the %form in which kinetic energy is apportioned among %fluctuations (or ``eddies'') of different sizes in a % turbulent flow---the link between % the turbulent spectrum and the global % properties of a %flow, such as the frictional drag, remains unknown. %%---the problem of predicting the global %% properties of a %%flow, such as the frictional drag, from a knowledge of the %%turbulent spectrum remains unsolved. %Here we show that the frictional drag is % governed by the turbulent spectrum. To that end, % we compare the frictional drag in %flows with distinct turbulent spectra: %three-dimensional pipe flows and two-dimensional %soap-film flows, in %which the turbulent spectrum is of the type known respectively as % the ``energy cascade'' \cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97} % and the ``enstrophy cascade'' \cite{batchelor1982tht,KRAI67,KELL02}. %We perform unprecedenced measurements of the %frictional drag in two-dimensional % soap-film flows and find that %for moderate values of the Reynolds number $\Re$ %(a measure of the strength of the turbulence) % the frictional drag scales as $\Re^{-1/2}$, %whereas in three-dimensional pipe flows % the frictional drag scales as %$\Re^{-1/4}$ (also for moderate values of Re). %These experimental scalings may be predicted %using a new theory\cite{GIOI06, GUTT09} %in which the frictional drag is derived directly %from the spectrum. %Our work indicates that in turbulence, %as in continuous phase transitions\cite{GOLD06,MEHR08}, %global properties are governed by the spectral structure of % the fluctuations, %and underscores the value of using %two-dimensional turbulent flows to test %and extend our understanding of %turbulence. %Turbulent flows past a wall experience frictional drag\cite{NIKU33}, %the global property of a flow that sets the cost %of pumping oil through a pipeline, %the draining capacity of a river in flood, %and other quantities of %interest\cite{SREE99,pope2000tf,schlichting2000blt,JIME04}. %Despite much progress in understanding the turbulent %spectrum\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}, %a function that describes the form in which turbulent % energy is apportioned amongst %fluctuations (or ``eddies'') of different sizes in a flow, %the problem of predicting global properties of a %flow---such as the frictional drag---from a knowledge of the %turbulent spectrum remains unsolved. %To acertain the relation between the frictional drag %and the turbulent spectrum, a comparison can be made %between %flows with distinct turbulent spectra: %three-dimensional, turbulent pipe flows in %which the spectrum is of the type known as % ``energy cascade''\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}, and %two-dimensional, %turbulent soap-film flows in which %the spectrum is of the type known as %``enstrophy cascade'' \cite{batchelor1982tht,KRAI67,KELL02}. %Here we perform unprecedenced measurements of the %frictional drag in two-dimensional %turbulent soap-film flows. %For moderate values of the Reynolds number $\Re$ %(a measure of the strength of the turbulence), %we find that in these flows the frictional drag %scales as $\Re^{-1/2}$, %whereas, in three-dimensional, %turbulent pipe flows, the frictional drag scales as %$\Re^{-1/4}$. %%We find that in these flows the frictional drag %%scales as $\Re^{-1/2}$ (where $\Re$ is the Reynolds number, %%a measure of the strength of the turbulence), whereas, in three-dimensional, %%turbulent pipe flows, the frictional drag scales as %%$\Re^{-1/4}$. %These experimental scalings may be predicted %using a new theory\cite{GIOI06, GUTT09} %in which the frictional drag is derived directly %from the spectrum. %%in which any specific type of spectrum is shown to correspond to %%a specific scaling. %Our work indicates that in turbulence, as in continuous phase transitions\cite{GOLD06,MEHR08}, %global properties are governed by the spectrum of the fluctuations, %and underscores the value of using %two-dimensional turbulent flows to test and extend our understanding of %turbulence. %% Turbulent flows past a wall experience a frictional drag\cite{NIKU33} %% whose prediction has posed a challenge to a fundamental understanding %% of turbulence for over two hundred years, and which influences many %% practical applications ranging from oil pipeline delivery to the flow %% of rivers\cite{SREE99,pope2000tf,schlichting2000blt,JIME04}. Despite %% much progress in understanding the statistical properties of %% small-scale velocity %% fluctuations\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}, the central %% problem of predicting large-scale flow phenomena---such as the %% frictional drag---remains famously unsolved. Here we report %% experiments that systematically measure the fluctuations and frictional %% drag in a two-dimensional turbulent soap film\cite{KELL02}, a flow %% which is qualitatively different from that in three-dimensional %% turbulence\cite{batchelor1982tht,KRAI67,KELL02}. Our data show a %% novel dependence on flow velocity that is classically unexpected and %% markedly different from experimental results in three-dimensional %% pipes\cite{NIKU33}. Nevertheless, these results are predicted, with no %% adjustable parameters, by the two-dimensional extension \cite{GUTT09} %% of a recent theory for the momentum transfer between the wall and the %% flow\cite{GIOI06}. Our demonstration of a deep connection between the %% spectrum of small-scale velocity fluctuations and the properties of %% large-scale flow is analogous to the relationship between order %% parameter fluctuations and thermodynamics in continuous phase %% transitions\cite{GOLD06,MEHR08}, and underscores the value of using %% two-dimensional turbulent flows to test and extend our understanding of %% turbulent phenomena. (209 words approximately)
friction2d.tex
%% This paper uses a Template for a preprint Letter or Article for submission
%% to the journal Nature.
%% Written by Peter Czoschke, 26 February 2004
%%

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%\title{Macroscopic effects of the structure of the fluctuations in turbulent flows}
\title{Macroscopic effects of the spectral structure in turbulent flows}


\author{Tuan Tran$^1$,  Pinaki Chakraborty$^2$, Nicholas Guttenberg$^3$, 
Alisia Prescott$^4$, Hamid Kellay$^5$,  Walter Goldburg$^4$,
  Nigel Goldenfeld$^{3\star}$ and Gustavo Gioia$^{1\star}$
}

\begin{document}
\def\Re{\textrm{Re}}

\maketitle

\spacing{1}\begin{affiliations}
\item Department of Mechanical Science and Engineering, University of
Illinois, Urbana, IL 61801, USA
\item Department of Geology, University of Illinois, Urbana, IL 61801, USA
\item Department of Physics, University of Illinois,
% at Urbana-Champaign, 1110 W.\ Green St., 
 Urbana, IL 61801, USA
\item Department of Physics and Astronomy, University of Pittsburgh,
Pittsburgh, PA 15260, USA
\item  {Centre de Physique Mol\'eculaire Optique et Hertzienne,
  Universit\'e Bordeaux I, %351 Cours de la Liberation,
 33405 Talence, France \\ $^\star$  These authors contributed equally to this work } 
\end{affiliations}

\spacing{1}\begin{abstract}

%  Two aspects of turbulent flows have been extensively
% researched: 
  Two aspects of turbulent flows\cite{FRIS95,SREE99,pope2000tf,hof2006flt}
 have been the subject
 of extensive, split research efforts:
  macroscopic properties, such as the 
  frictional drag\cite{schlichting2000blt}
  experienced by a flow past a wall,
  and the turbulent spectrum.\cite{taylor1938spectrum,FRIS95,SREE97} 
%  The turbulent spectrum may be said to represent 
%  the fabric of a turbulent state; 
%  in practice it is a function---a power law 
% of exponent $\alpha$, the ``spectral exponent''---that 
% gives the revolving velocity of a 
%  turbulent fluctuation (or ``eddy'') 
%  of any given size in a flow.\cite{FRIS95} 
  The turbulent spectrum may be said to represent 
  the fabric of a turbulent state; 
  in practice it is a power law of 
 exponent $\boldsymbol{ \alpha}$ (the ``spectral exponent'') that 
 gives the revolving velocity of a
  turbulent fluctuation (or ``eddy'') of 
 size $\boldsymbol{s}$ as
 a function of $\boldsymbol{s}$.\cite{FRIS95}  
%  The turbulent spectrum may be said to represent 
%  the fabric of a turbulent state; 
%  in practice it is a power law of 
% exponent $\alpha$---the ``spectral exponent''---that 
% gives the revolving velocity of any 
%  turbulent fluctuation (or ``eddy'') as
% a function of the size of the eddy.\cite{FRIS95} 
 The link, if any,
 between macroscopic properties and the turbulent spectrum
 remains missing. Might it be found by contrasting
 the frictional drag in
 flows with differing types of spectra?
  Here we perform unprecedented measurements of the
frictional drag in soap-film flows,\cite{KELL02} where 
 the spectral exponent
  $\boldsymbol{\alpha=3}$,\cite{KRAI67,batchelor1982tht}
 and compare the results with the frictional drag 
 in pipe flows,\cite{schlichting2000blt} 
 where the spectral exponent 
 $\boldsymbol{\alpha=5/3}$.\cite{richardson26,KOLM41}
   For moderate values of the Reynolds 
 number $\Re$ (a measure of the strength of the turbulence),
 we find that in soap-film flows
  the frictional drag scales as $\boldsymbol{\Re^{-1/2}}$, whereas 
  in pipe flows the frictional drag scales\cite{mckeon2005ffs}
  as $\boldsymbol{\Re^{-1/4}}$. Each of these scalings may be 
 predicted from the attendant 
 value of %the spectral exponent
 $\boldsymbol{\alpha}$ by 
using a new theory,\cite{GIOI06, GUTT09} 
in which the frictional drag is explicitly linked
 to the turbulent spectrum.
Our work indicates that in turbulence, 
as in continuous phase transitions,
 macroscopic properties are governed by the spectral structure of
 the fluctuations.\cite{GOLD06,MEHR08}


%ALTERNATIVE
%
%  Two aspects of turbulent flows\cite{FRIS95,SREE99,pope2000tf}
% have been the subject
% of extensive, split research efforts:
%  global properties, such as the 
%  frictional drag experienced by a flow 
%  past a wall,\cite{SREE99,pope2000tf,schlichting2000blt}
%  and the turbulent spectrum.\cite{taylor1938spectrum,FRIS95,SREE97} 
%  The turbulent spectrum may be said to represent 
%  the fabric of the turbulence; in practice it is 
% a function---a power law of exponent $\alpha$---that 
% gives the revolving velocity of a 
%  turbulent fluctuation (or ``eddy'') 
%  of any given size in the flow.\cite{FRIS95} 
% The link, if any,
% between global properties and the turbulent spectrum
% remains missing. Might it be found by contrasting
% flows with differing types of spectra?
%  Here we perform unprecedented measurements of the
%frictional drag in soap-film flows,\cite{KELL02} where the 
% spectrum is of a type known 
% as the ``enstrophy cascade''\cite{KRAI67,batchelor1982tht}
% (for which $\alpha=3$),
% and compare the results with the frictional drag 
% in pipe flows,\cite{schlichting2000blt} 
% where the spectrum is of a type known 
% as the ``energy cascade''\cite{richardson26,KOLM41}
% (for which $\alpha=5/3$).
%   For moderate values of the Reynolds 
% number $\Re$ (a measure of the strength of the turbulence),
% we find that in soap-film flows
%  the frictional drag scales as $\Re^{-1/2}$, whereas 
%  in pipe flows the frictional drag scales\cite{mckeon2004ffs}
%  as $\Re^{-1/4}$. Each of these scalings may be 
% predicted from the attendant 
% value of $\alpha$ by 
%using a new theory\cite{GIOI06, GUTT09} 
%in which the frictional drag is explicitly linked
% to the turbulent spectrum.
%Our work indicates that in turbulence, 
%as in continuous phase transitions,
%global properties are governed by the spectral structure of
% the fluctuations.\cite{GOLD06,MEHR08}



%  Two aspects of turbulent flows have been 
%%the  subject of most reserach:
% researched the most: global properties, such as the 
%  frictional drag experienced by a flow 
%  past a wall, and the turbulent spectrum. 
%  The turbulent spectrum may be said to represent 
%  the internal structure of the turbulence; 
%  in practice it is a function, 
% usually a power law with a characteristic spectral  
%  exponent $\alpha$, that gives the revolving velocity of a 
%  turbulent fluctuation (or ``eddy'') 
%  of any given size in the flow.
%%  in practice, it is an account of the distribution of
%%  kinetic energy among the turbulent fluctuations 
%%  (or ``eddies'') of different sizes in a flow.
%%  in pratice, it is an account of the distribution of
%%  kinetic energy among the turbulent fluctuations 
%%  (or ``eddies'') of different sizes in a flow.
%% in practice, it is a mathematical function that describes
%% the form in which kinetic energy is apportioned among
%% the turbulent fluctuations (or ``eddies'') of different sizes in 
%%  a flow.
% The link, if any,
% between global properties and the turbulent spectrum
% is presently unknown.
%% This state of affairs brings to mind 
%% continuous phase transitions in the early 60's, 
%% before a link was found between the 
%% global properties, such as the heat capacity,
%%  and the fluctuations of thermodynamic systems.
%  Here we perform unprecedented measurements of the
%frictional drag in soap-film flows, where the 
%turbulent spectrum is of a type known 
% as the ``enstrophy cascade'' (for which $\alpha=3$),
% and compare the results with the frictional drag 
% in pipe flows, 
% where the turbulent spectrum is of a type known 
% as the ``energy cascade'' (for which $\alpha=5/3$).
%   For moderate values of the Reynolds 
% number $\Re$ (a measure of the strength of the turbulence),
% we find that in soap-film flows
%  the frictional drag scales as $\Re^{-1/2}$, whereas 
%  in pipe flows the frictional drag scales
%  as $\Re^{-1/4}$. These experimental scalings may be predicted
%using a new theory\cite{GIOI06, GUTT09} 
%in which the frictional drag is explicitly linked
% to the turbulent spectrum, so that the functional
% dependence of the frictional drag on
%  Re is set by the value of the spectral exponent $\alpha$. 
%Our work indicates that in turbulence, 
%as in continuous phase transitions\cite{GOLD06,MEHR08},
%global properties are governed by the spectral structure of
% the fluctuations.

%and underscores the value of using
%two-dimensional turbulent flows to test 
%and extend our understanding of
%turbulence.


% are due to  
%we show that the frictional drag is 
% governed by the turbulent spectrum.  To that end, 
% we compare the frictional drag in 
%flows with distinct turbulent spectra:
%three-dimensional pipe flows and  two-dimensional 
%soap-film flows, in
%which the turbulent spectrum is of the type known respectively as 
% the ``energy cascade'' \cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}
% and the ``enstrophy cascade'' \cite{batchelor1982tht,KRAI67,KELL02}.
%We perform

%Despite much progress in understanding the turbulent 
%spectrum\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}---the 
%form in which kinetic energy is apportioned among
%fluctuations (or ``eddies'') of different sizes in a 
%  turbulent flow---the link between 
% the turbulent spectrum and  the global
% properties of a 
%flow, such as the frictional drag, remains unknown.
%%---the problem of predicting the global
%% properties of a 
%%flow, such as the frictional drag, from a knowledge of the 
%%turbulent spectrum remains unsolved.
%Here we show that the frictional drag is 
% governed by the turbulent spectrum.  To that end, 
% we compare the frictional drag in 
%flows with distinct turbulent spectra:
%three-dimensional pipe flows and  two-dimensional 
%soap-film flows, in
%which the turbulent spectrum is of the type known respectively as 
% the ``energy cascade'' \cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}
% and the ``enstrophy cascade'' \cite{batchelor1982tht,KRAI67,KELL02}.
%We perform unprecedenced measurements of the
%frictional drag in two-dimensional
% soap-film flows and find that 
%for moderate values of the Reynolds number $\Re$
%(a measure of the strength of the turbulence)
%  the frictional drag scales as $\Re^{-1/2}$, 
%whereas in three-dimensional pipe flows
% the frictional drag scales as 
%$\Re^{-1/4}$ (also for moderate values of Re). 
%These experimental scalings may be predicted
%using a new theory\cite{GIOI06, GUTT09} 
%in which the frictional drag is derived directly
%from the spectrum. 
%Our work indicates that in turbulence, 
%as in continuous phase transitions\cite{GOLD06,MEHR08},
%global properties are governed by the spectral structure of
% the fluctuations,
%and underscores the value of using
%two-dimensional turbulent flows to test 
%and extend our understanding of
%turbulence.

%Turbulent flows past a wall experience frictional drag\cite{NIKU33},
%the global property of a flow that sets the cost 
%of pumping oil through a pipeline,
%the draining capacity of a river in flood,
%and other quantities of 
%interest\cite{SREE99,pope2000tf,schlichting2000blt,JIME04}.
%Despite much progress in understanding the turbulent 
%spectrum\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97},
%a function that describes the form in which turbulent
% energy is apportioned amongst
%fluctuations (or ``eddies'') of different sizes in a flow,
%the problem of predicting global properties of a 
%flow---such as the frictional drag---from a knowledge of the 
%turbulent spectrum remains unsolved.
%To acertain the relation between the frictional drag
%and the turbulent spectrum, a comparison can be made
%between
%flows with distinct turbulent spectra:
%three-dimensional, turbulent pipe flows in
%which the spectrum is of the type known as 
% ``energy cascade''\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}, and
%two-dimensional,
%turbulent soap-film flows in which
%the spectrum is of the type known as
%``enstrophy cascade'' \cite{batchelor1982tht,KRAI67,KELL02}.
%Here we perform unprecedenced measurements of the
%frictional drag in two-dimensional
%turbulent soap-film flows.
%For moderate values of the Reynolds number $\Re$
%(a measure of the strength of the turbulence), 
%we find that in these flows the frictional drag
%scales as $\Re^{-1/2}$, 
%whereas, in three-dimensional,
%turbulent pipe flows, the frictional drag scales as 
%$\Re^{-1/4}$. 
%%We find that in these flows the frictional drag
%%scales as $\Re^{-1/2}$ (where $\Re$ is the Reynolds number,
%%a measure of the strength of the turbulence), whereas, in three-dimensional,
%%turbulent pipe flows, the frictional drag scales as 
%%$\Re^{-1/4}$. 
%These experimental scalings may be predicted
%using a new theory\cite{GIOI06, GUTT09} 
%in which the frictional drag is derived directly
%from the spectrum. 
%%in which any specific type of spectrum is shown to correspond to
%%a specific scaling.
%Our work indicates that in turbulence, as in continuous phase transitions\cite{GOLD06,MEHR08},
%global properties are governed by the spectrum of the fluctuations,
%and underscores the value of using
%two-dimensional turbulent flows to test and extend our understanding of
%turbulence.


%% Turbulent flows past a wall experience a frictional drag\cite{NIKU33}
%% whose prediction has posed a challenge to a fundamental understanding
%% of turbulence for over two hundred years, and which influences many
%% practical applications ranging from oil pipeline delivery to the flow
%% of rivers\cite{SREE99,pope2000tf,schlichting2000blt,JIME04}.  Despite
%% much progress in understanding the statistical properties of
%% small-scale velocity
%% fluctuations\cite{KOLM41,OBUK41,FRIS95,SREE97,CHOR97}, the central
%% problem of predicting large-scale flow phenomena---such as the
%% frictional drag---remains famously unsolved.  Here we report
%% experiments that systematically measure the fluctuations and frictional
%% drag in a two-dimensional turbulent soap film\cite{KELL02}, a flow
%% which is qualitatively different from that in three-dimensional
%% turbulence\cite{batchelor1982tht,KRAI67,KELL02}.  Our data show a
%% novel dependence on flow velocity that is classically unexpected and
%% markedly different from experimental results in three-dimensional
%% pipes\cite{NIKU33}. Nevertheless, these results are predicted, with no
%% adjustable parameters, by the two-dimensional extension \cite{GUTT09}
%% of a recent theory for the momentum transfer between the wall and the
%% flow\cite{GIOI06}. Our demonstration of a deep connection between the
%% spectrum of small-scale velocity fluctuations and the properties of
%% large-scale flow is analogous to the relationship between order
%% parameter fluctuations and thermodynamics in continuous phase
%% transitions\cite{GOLD06,MEHR08}, and underscores the value of using
%% two-dimensional turbulent flows to test and extend our understanding of
%% turbulent phenomena. (209 words approximately)

\end{abstract}

Turbulent flows past a wall experience frictional 
 drag, the macroscopic property of a flow that sets the cost 
of pumping oil through a pipeline,
the draining capacity of a river in flood,
and other quantities of engineering
 interest\cite{NIKU33,SREE99,pope2000tf,schlichting2000blt,JIME04}.
The frictional drag is 
defined as the dimensionless ratio $f=\tau/\rho U^2$,
 where $\tau$ is the shear stress or force per unit area that
develops between the flow and the wall,
 $\rho$ is the density of the fluid,
 and $U$ is the mean velocity of the flow. 
Already in XVIII Century France, $f$
%the frictional drag in open-channel flows
  was the subject of large-scale experiments
 performed in connection with the design of 
 a waterworks for the city of Paris.\cite{CHEZY1775, DOOGE92} 
%Our concern here is 
%a functional dependence of $f$ on the dimensionless Reynolds
%number $Re \equiv U D/\nu$, 
%where $D$ is the diameter of the pipe
%and $\nu$ is the kinematic viscosity of the fluid.
Modern experiments with  pipes %performed over two-hundred years 
have shown that $f$ depends on the Reynolds
number ${\rm Re}= U d/\nu$, 
where $d$ is the diameter of the pipe
and $\nu$ is the kinematic viscosity of the fluid.
For pipe flows of moderate turbulent strength 
 ($\rm  Re$ up to 98,000) %$f$ is well represented by
the experimental results are well described
(within 1.4\% error\cite{mckeon2005ffs})
 by the Blasius empirical scaling,
  $f \propto {\rm Re}^{-1/4}$. 
 (Throughout this paper, the symbol ``$\propto$''
 may be changed to the symbol ``$=$'' by 
  introducing a dimensionless proportionality factor,
 e.g., $f=C\, {\rm Re}^{-1/4}$.)
A celebrated theory\cite{schlichting2000blt,pope2000tf}
 of the frictional drag % that gives $f$ as a function of $\Re$
  was formulated
eighty years ago by Ludwig Prandtl, the founder of 
 turbulent hydraulics, and
 numerous variants\cite{schlichting2000blt,mckeon2005ffs,ALLE07}
 and alternatives\cite{BARE93,BARE04} of Prandtl's theory have since
 been proposed.
%\cite{mckeon2004ffs,shockling2006ret,ALLE07,kunkel2007further,langelandsvik2008flow,bailey2008azimuthal,BARE93,porporato2001ist,BARE04}
%To calculate $f$ as a function of $\Re$, engineers
%continue to use a celebrated eighty-old theory due
%to Prandtl, the founder of turbulent hydraulics. 
%Numerous variants of Prandtl's theory have since
% been proposed,\cite{mckeon2004ffs,shockling2006ret,ALLE07,kunkel2007further,langelandsvik2008flow,bailey2008azimuthal}
% and recently a theory that
%departs from the basic premises of Prandtl's theory
%has been formulated by Barenblatt.\cite{BARE93,porporato2001ist,BARE04}
Although Prandtl's theory and its variants and alternatives
 yield disparate mathematical
expressions for $f$ as a function of $\Re$,
  for moderate values of $\Re$ they all
give predictions in good numerical accord with the
Blasius empirical scaling.
Yet these theories have been predicated on
dimensional analysis and similarity assumptions,
without reference to the spectral
  structure of the turbulent fluctuations.
%as represented by the turbulent spectrum.
%(Prandtl's and Baranblatt's theories differ in the similarity assumptions.) 
As a result, these theories cannot be used to 
reveal the missing link between the frictional drag and 
the turbulent spectrum.
%Our aim here is to find this missing link.
%relate the frictional drag to the turbulent spectrum.
%Our aim here is to find the missing link between the
%frictional drag and the turbulent spectrum.
%find the missing link between the
%frictional drag and the turbulent spectrum.


%To establish that the measurements are indeed made in the enstrophy
%range, it is necessary to measure the power spectrum of the velocity
%fluctuations to assure that over a significant range of wavenumbers,
%$E(k) \propto k^{- \beta}$ with $\beta$  close to the well-established
%value of 3\cite{KRAI67,kraichnan1980tdt,FRIS95}.  Here $E(k)$ is the
%energy per unit wave number $k$, which is related to the total
%turbulent kinetic energy by the relation $E=(1/2) \langle ((U-u_x)^2 +
%u_y^2) \rangle =\int_0^\infty dk E(k)$.  It is assumed that the
%small-scale velocity fluctuations are isotropic.  To relate the

The turbulent spectrum is a function of the wavenumber $k$, $E(k)$,
 whose physical significance may be grasped from the expression 
 $u_s\propto (\int_{1/s}^\infty E(k) dk)^{1/2}$, which gives the 
 revolving velocity $u_s$ of a turbulent eddy 
 of size $s$ in the flow. In general, we can write\cite{GUTT09} 
 $E(k) \propto U^2 L^{(1-\alpha)} k^{-\alpha}$, and therefore 
\begin{equation} \label{theq}
u_s\propto U (s/L)^{(\alpha-1)/2},
\end{equation} 
 where $\alpha$ is the spectral exponent,
  $U$ is the mean velocity
 of the flow, and $L$ is a characteristic length. 
 A single  type of spectrum is possible 
  in three-dimensional (3D) flows: 
 the ``energy cascade,'' for 
 which $\alpha=5/3$.\cite{KOLM41,OBUK41} 
  Thus, e.g., in turbulent pipe flows
 the spectral exponent is $5/3$ and $L=d$, the diameter
 of the pipe. Two-dimensional (2D) turbulence 
 (a type of turbulence that may be realized in a soap film)
 differs from 3D turbulence in a
 number of crucial respects, most notably 
 in that in two dimensions there is no vortex
  stretching.
% in 2D there is no vortex stretching and two
% quadratic quantities are conserved, enstrophy and
% energy\cite{batchelor1982tht,KRAI67,KELL02}.  
As a result, a different type of 
spectrum is possible in 2D flows: the ``enstrophy cascade,'' for which 
 $\alpha=3$.\cite{batchelor1982tht,KRAI67} Thus, e.g., in
  turbulent soap-film flows the spectral exponent is $3$ and 
 $L=w$, the width of the soap film. 

 To study soap-film flows we hang a soap film 
 between two long, vertical, mutually parallel
  wires a few centimeters apart from one another
   (Fig.~\ref{setup}a). %\cite{rutgers2001} 
%The film drains into a reservoir at the bottom of the 
% setup and draws a soapy solution from a reservoir 
% at the top.
  Driven by gravity, a steady vertical flow soon 
 becomes established within the film.
 Then, the thickness $h$ of the film is roughly
 uniform on any %horizontal
 cross section of the film, % \cite{georgiev2002}
 typically  $h\approx 10\,\mu$m, much smaller
 than the width $w$  and the length 
 of the film  (Fig.~\ref{setup}a). As a result,
 the velocity of the flow 
 lies on the plane of the film, and 
 the flow is 2D. 
%There is, of course,
%  frictional energy loss to the air, coupling the
%film to a third dimension, but, being velocity dependent, this
%primarily influences the profile in the center of the film, and does
%not affect significantly the velocity gradient near the wall.  

We make the flow turbulent by piercing
 the film with a comb, as indicated in 
 Fig.~\ref{setup}a, so that the flow is stirred as it
 moves past the teeth of the comb.
% (The comb is the 
%2D counterpart of the grid used to generate 
% turbulence in 3D flows.) 
% The results were robust to variation of these
% parameters. 
To visualize the flow, 
 we cast monochromatic light
 on a face of a film and observe the 
 interference fringes that form there. 
%In Fig.~\ref{setup}b we show 
% a photograph of the interference  fringes.
 These fringes (Fig.~\ref{setup}b) 
 reflect small changes 
 in the local thickness of the film. 
%(the thickness
% of the film on any given fringe 
% differs by one wavelength of the
%  light, or about ????MICRONS, from the thickness
% of the film on a successive fringe). 
(The thickness is constant along
 a fringe; it differs by one-half
  wavelength of 
 the light, or a fraction of a $\mu$m,
 between any two successive fringes.)
The small changes in thickness in turn reflect 
 small changes in the absolute value of the 
 instantaneous velocity of the flow.\cite{KELL02}    
Thus Fig.~\ref{setup}b
 may be interpreted 
 as a map of the instantaneous spatial distribution 
 of turbulent fluctuations downstream of the comb.
% The turbulence is apparent 
% downstream from the comb in the photograph of 
% Fig.~\ref{setup}b. 

%To verify the type of turbulent spectrum 
% of the soap-film flows,
 We compute the  spectrum $E(k)$ at numerous
  points  on the film from measurements performed 
 with a Laser Doppler Velocimeter (LDV; see Methods).  
 In Fig.~\ref{setup}c we show 
 a few typical log-log plots of $E$ vs.\ $k$.
 The slope of these plots %a log-log plot of $E(k)$ vs.\ $k$
 represents the spectral exponent $\alpha$; 
 in our experiments the slope 
 is slightly larger than 3, consistent with
  prior experiments with
 soap-film flows \cite{kellay1995experiments,KELL02},
 and close to the theoretical value of $\alpha$ 
 for the enstrophy cascade ($\alpha=3$). 

 By using the same Laser Doppler Velocimeter
  we measure the mean (time-averaged) 
 velocity $u$ at any point on the film (see Methods).
 Successive measurements of $u$ along a
 cross section of the film 
 gives the ``mean velocity profile''
  $u(y)$ of that cross section.
   In Fig.~\ref{velpro}a we show a few typical
 plots of $u(y)$ over the entire width of the film
  (i.e., from wire to wire, or for $0\le y\le w$;
  see Fig.~\ref{setup}a).
  From a mean velocity profile
  we compute the mean velocity of the flow as
 $U=(1/w)\,\int_0^w u(y)dy$, and the 
 Reynolds number as ${\rm Re}=U w/\nu$.
 
 In Fig.~\ref{velpro}b we show a few typical 
 plots of $u(y)$ close to one of the wires,
 where the mean velocity profile is linear
 on a narrow  (about 0.2$\,$mm) viscous layer.
 From the slope $G$ 
 of the mean velocity profile in the viscous layer, 
  we compute the shear stress between the flow
 and the wire as $\tau=\rho \nu G$, and 
 the frictional drag as $f=\tau/\rho U^2=\nu G/U^2$.
 % we observed our
%films in yellow light (wavelength of 579nm) and counted at most two
%fringes, showing that the film thickness varies by no more than one
%wavelength, i.e. at most by 10\%.
%This is important because the viscosity of soap films will in principle 
%depend on its thickness \cite{Vorobieff99}
% thus there is no significant change of
%film viscosity over the measurement region.  
 In Fig.~\ref{friction_factor} we show
 a log-log plot of $f$ vs.\ Re.
% for films of varying width $0.5 < w < 2.6$cm.  
 The plot consists of five sets
of data points from numerous turbulent soap-film flows; 
 four sets were taken at Pittsburgh, and
one at Bordeaux in an independent experimental setup.
The cloud of data points is consistent 
 with the scaling, $f\propto {\rm Re}^{-1/2}$,
 and inconsistent with the Blasius empirical scaling, 
 $f\propto {\rm Re}^{-1/4}$, which is known
 to prevail %obtain
  in turbulent pipe flows.
 
% The data cloud shows little deviation from
%the theoretical prediction, and moreover, strongly deviates from the
%corresponding result in 3D flows\cite{NIKU33}, which show a slope of
%$-1/4$ up to Reynolds numbers of order $10^5$.
% The solid line has a slope of $-1/2$, which is the analytical
%prediction\cite{GUTT09} based on the momentum-transfer
%theory\cite{GIOI06}, and close to the results of direct numerical
%simulations\cite{GUTT09}.  The data cloud shows little deviation from
%the theoretical prediction, and moreover, strongly deviates from the
%corresponding result in 3D flows\cite{NIKU33}, which show a slope of
%$-1/4$ up to Reynolds numbers of order $10^5$.


%Our experimental results may be explained 
% using a recently proposed
%  theory of the frictional drag.\cite{GIOI06,GUTT09}  
%% in which the functional dependence 
%% of the friction drag on the Reynolds number 
%% is set by the type of turbulent spectrum.
%% The theory has been formulated for the general case
%% of a turbulent flow that moves past a rough wall, 
%% but
% Here we will sketch this new theory for the special
% case in which the turbulent flow
% moves past a smooth wall (as in our experiments). 
%% We start by ascertainig the size $\eta$ of the smallest 
%% eddies in a turbulent flow. 
%To a generic eddy of size 
% $s$ we ascribe an eddy Reynolds number 
% ${\rm Re}_s=u_s s/\nu$,
% where $u_s$ is the revolving velocity of the eddy.
% The Reynolds number of the smallest eddies in the flow
% is of order 1 (because these eddies 
% are viscous\cite{KOLM41,FRIS95}), and we can write
%  $u_\eta \eta/\nu\approx 1$, where $\eta$ is the 
% size of the smallest eddies in the flow. 
% A second expression for $u_\eta$ may be obtained by
% setting $s=\eta$ in (\ref{theq});
% by combining these two expressions
%  and recalling that ${\rm Re}=U L/\nu$, we conclude
% that $\eta \propto L\, {\rm Re}^{(1-\alpha)/2}$, and therefore that 
%  $u_\eta \propto U\, {\rm Re}^{(1-\alpha)/(1+\alpha)}$.

Our experimental results may be explained 
 using a recently proposed
  theory of the frictional drag.\cite{GIOI06,GUTT09} 
  In this new theory, the frictional drag 
 is produced by turbulent eddies that transfer momentum
  between the wall or wire (where the fluid carries a negligible 
 momentum per unit mass) and the turbulent flow (where the
 fluid carries a sizable momentum 
 per unit mass). The theory is generally applicable
 to flows that move past rough walls, but for 
 the special case in which the wall is smooth
  (as in our experiments), the theory predicts that
  $f\propto u_\eta/U$,\cite{GIOI06,GUTT09,GB02,GCB06,calzetta2009friction,MEHR08}  where $\eta$ is the size of the smallest eddies in the flow and
 $u_\eta$ is the revolving velocity of those eddies.
 Note that in this scaling for $f$,
  $u_\eta$ provides a connection to the 
 turbulent spectrum, as shown next.

%The theory is generally applicable
% to flows that move past rough walls, but for 
% the special case in which the wall is smooth
%  (as in our experiments), the theory predicts that the
%eddies that are most effective
% at transferring momentum close to the wall are shown to be the smallest 
% eddies in the flow. Then, 
%%the frictional drag   scales in the form
% $f\propto u_\eta/U$,\cite{GIOI06,GUTT09,GB02,GCB06,calzetta2009friction,MEHR08}  where $\eta$ is the size of the smallest eddies in the flow and
% $u_\eta$ is the revolving velocity of those eddies.
% Note that in this scaling for $f$,
%  $u_\eta$ provides a connection to the 
% turbulent spectrum, as shown next.

To a generic eddy of size  $s$ 
  we ascribe an eddy Reynolds number ${\rm Re}_s=u_s s/\nu$,
 where $u_s$ is the revolving velocity of the eddy.
 The Reynolds number of the smallest eddies in the flow
 is of order 1 (because these eddies 
 are viscous\cite{KOLM41,FRIS95}), and we can write
  ${\rm Re}_\eta =u_\eta \eta/\nu\approx 1$, or 
 $u_\eta\propto \nu/\eta$.
%, where $\eta$ is the  size of the smallest eddies in the flow. 
 A second expression for $u_\eta$ may be obtained 
 from the spectrum by 
 setting $s=\eta$ in (\ref{theq}); if we combine
  these two expressions so as to 
 eliminate $\eta$,   we can 
 write $u_\eta \propto U\, {\rm Re}^{(1-\alpha)/(1+\alpha)}$,
 where we have used the definition ${\rm Re}=U L/\nu$.
 As $f\propto u_\eta/U$, it follows that 
 \begin{equation} \label{fofalpha}
f\propto {\rm Re}^{(1-\alpha)/(1+\alpha)},
\end{equation}
 and the functional dependence 
 of the frictional drag on the Reynolds number 
 is set by the spectral exponent $\alpha$.
 For pipe 
 flows $\alpha=5/3$, and (\ref{fofalpha}) yields 
 the prediction\cite{GIOI06}
 $f\propto {\rm Re}^{-1/4}$, consistent
  with the Blasius scaling. In contrast, 
for soap-film flows $\alpha=3$, and 
 (\ref{fofalpha}) yields the prediction\cite{GUTT09} 
 $f\propto {\rm Re}^{-1/2}$, consistent
  with our experimental results 
 (Fig.~\ref{friction_factor}).

 
%% in which the functional dependence 
%% of the friction drag on the Reynolds number 
%% is set by the type of turbulent spectrum.
%% The theory has been formulated for the general case
%% of a turbulent flow that moves past a rough wall, 
%% but
% Here we will sketch this new theory 
%% We start by ascertainig the size $\eta$ of the smallest 
%% eddies in a turbulent flow. 
%To a generic eddy of size 
% $s$ we ascribe an eddy Reynolds number 
% ${\rm Re}_s=u_s s/\nu$,
% where $u_s$ is the revolving velocity of the eddy.
% The Reynolds number of the smallest eddies in the flow
% is of order 1 (because these eddies 
% are viscous\cite{KOLM41,FRIS95}), and we can write
%  $u_\eta \eta/\nu\approx 1$, where $\eta$ is the 
% size of the smallest eddies in the flow. 
% A second expression for $u_\eta$ may be obtained by
% setting $s=\eta$ in (\ref{theq});
% by combining these two expressions
%  and recalling that ${\rm Re}=U L/\nu$, we conclude
% that $\eta \propto L\, {\rm Re}^{(1-\alpha)/2}$, and therefore that 
%  $u_\eta \propto U\, {\rm Re}^{(1-\alpha)/(1+\alpha)}$.
%Consider now the shear stress $\tau$ 
% that acts on the wetted surface $S$ parallel 
% to the wall of a pipe (or to one of 
% the wires of a soap-film), where $S$ is
%  at a distance $\eta$ above the wall,
% which we take to be horizontal. Then, 
% $\tau$ is effected by momentum transfer across $S$.
% Above $S$, the velocity of the flow scales
% with $U$, and the fluid carries
% a momentum per unit volume $\approx\rho U$.
% Below $S$, the velocity of the 
% flow is negligible, and the fluid carries
% a momentum per unit volume 
%$\approx 0$. An eddy that straddles the wetted
% surface $S$ transfers 
% high-momentum fluid downwards across $S$, 
%  and low-momentum fluid
% upwards across $S$.
% The net rate of momentum transfer across $S$ 
% is set by the velocity normal to $S$,
% which velocity is provided by the eddy. 
% Eddies much larger than $\eta$
%  can provide only a negligible velocity
% normal to $S$.
% (This observation is purely a matter of
%  geometry.) It follows that 
%  the momentum transfer across S is 
% dominated by eddies of size $\approx \eta$,
% and $\tau$ can be estimated as the product 
% of the rate of momentum transfer of these
% eddies ($u_\eta$) multiplied the momentum contrast 
% spanned by the eddies ($\rho U$), or
% $\tau\propto u_\eta\,\rho\,U$.
% By recalling that $f=\tau/\rho U^2$ and 
%$u_\eta\propto U {\rm Re}^{(1-\alpha)/(1+\alpha)}$,
% we conclude that 
%\begin{equation} \label{fofalpha}
%f\propto {\rm Re}^{(1-\alpha)/(1+\alpha)}.
%\end{equation}
% Thus the functional dependence 
% of the frictional drag $f$ on the Reynolds number Re 
% is set by the spectral exponent $\alpha$.
% For pipe 
% flows $\alpha=5/3$, and (\ref{fofalpha}) yields 
% the prediction\cite{GIOI06}
% $f\propto {\rm Re}^{-1/4}$, consistent
%  with the Blasius scaling. In contrast, 
%for soap-film flows $\alpha=3$, and 
% (\ref{fofalpha}) yields the prediction\cite{GUTT09} 
% $f\propto {\rm Re}^{-1/2}$, consistent
%  with our experimental results 
% (Fig.~\ref{friction_factor}).


 From our experiments with two-dimensional soap-film flows
 we infer that the long-standing and
widely-accepted theory\cite{schlichting2000blt}
  of the frictional drag between a turbulent
flow and a wall is incomplete. This classical
theory does not take into account the structure of the 
 turbulent fluctuations, and cannot distinguish 
 between two-dimensional and three-dimensional 
 turbulent flows. Our  data on soap-film flows,
 as well as the available data on  
 pipe flows, are, however, consistent with the predictions 
of a recently proposed theory of the frictional
  drag.\cite{GIOI06,GUTT09} This new theory
  perforce relates the frictional drag 
  to the turbulent spectrum, and is sensitive 
 to the dimensionality of the flow via the dependence
 of the turbulent spectrum on the dimensionality.
%  We can illustrate the possible applications of our work
% by means of an example. 
%A river in flood entrains much sediment,
% which alters the turbulent spectrum
% in a way that may be measured and 
% characterized with relative ease. 
% Our findings will allow engineers to 
%  predict the concomitant 
% effect on the frictional drag,
% and thereby to estimate the draining 
%capacity of the river in flood.
Our findings lead us to conclude
  that the macroscopic properties of both three-dimensional
 and two-dimensional turbulent flows are linked to 
 the turbulent fluctuations, in a manner analogous to
 the way in which order-parameter fluctuations are linked to
 the macroscopic properties of thermodynamic
  systems close to a critical point.\cite{GOLD06,MEHR08}
%in equilibrium
%  continuous phase transitions\cite{GOLD06,MEHR08}.
 In addition, our findings serve to underscore the value of using
 two-dimensional soap-film
 flows to test and extend our understanding of
 turbulent phenomena. 

% From our experiments with two-dimensional soap-film flows
% we infer that the long-standing and
%widely-accepted theory\cite{schlichting2000blt}
%  of the frictional drag between a turbulent
%flow and a wall is incomplete. This classical
%theory does not take into account the structure of the 
% turbulent fluctuations, and cannot distinguish 
% between two-dimensional and three-dimensional 
% turbulent flows. Our  data on soap-film flows,
% as well as the available data on  
% pipe flows, are, however, consistent with the predictions 
%of a recently proposed theory of the frictional
%  drag.\cite{GIOI06,GUTT09} This new theory
%  perforce relates the frictional drag 
%  to the turbulent spectrum, and is sensitive 
% to the dimensionality of the flow via the dependence
% of the turbulent spectrum on the dimensionality.
%Our findings lead us to conclude
%  that the macroscopic properties of both three-dimensional
% and two-dimensional turbulent flows
%  are linked to
%  the turbulent fluctuations, in a manner analogous to
% the way in which order-parameter fluctuations are linked to
% the macroscopic properties of thermodynamic
%  systems close to a critical point.\cite{GOLD06,MEHR08}
%%in equilibrium
%%  continuous phase transitions\cite{GOLD06,MEHR08}.
% In addition, our findings serve to underscore the value of using
% two-dimensional soap-film
% flows to test and extend our understanding of
% turbulent phenomena. 


\begin{methods}
We measure the vertical component $u(t)$ of the 
 instantaneous velocity at a point on the film
 using a Laser Doppler Velocimeter with
 a sampling rate of 5 kHz.
%The LDV generates 
% a pair of laser beams which converge on the film 
% and scatter light from a spot 10$\,\mu$m
% in diameter. The scattering is from sub-micron 
% particles which we seed in the soapy solution. The LDV
 By performing measurements over a time period 
  of about 10$\,$s, we collect a time series $u(t_i)$.
 From the time series we  
 compute the local mean %(time-averaged) 
 velocity $u$ as the time average 
 of $u(t_i)$, $u=\langle u(t_i)\rangle$. 
 From the same 
 time series $u(t_i)$ we compute the local
 turbulent spectrum (more precisely, the longitudinal 
 turbulent spectrum). To that end, we invoke 
 Taylor's frozen-turbulence hypothesis\cite{taylor1938spectrum}
 to perform a space-for-time substitution
 $t\rightarrow x/u$ on the time series
   $(u(t_i)-u)$ to obtain a space series
  $v(x_i)=(u(x_i/u)-u)$, where $x_i=u t_i$. 
  (The frozen-turbulence 
 hypothesis is justified because in all our experiments the root 
 mean square of the velocity fluctuations is less than 20\% 
 of  $u$.\cite{belmonte2000experimental}) 
The spectrum $E(k)$ is the square of the magnitude of the
% coefficients of the
 discrete  Fourier transform of $v(x_i)$. 
\end{methods}



\bibliography{friction2d.bib}


\begin{addendum}
\item[Author Information] Reprints and permissions information is available
at npg.nature.com/reprints and permissions. The authors declare that they
have no competing financial interests. 
\item We thank discussions with J.\ M.\ Larkin.  This work was 
 funded by the US National Science Foundation through  
 NSF/DMR grant 06-04477 and NSF/DMR grant 06-04435 
 (W.\ Fuller--Mora, Programme Director).
 T.\ Tuan acknowledges support from the Vietnam Education Foundation.
\item[Author Contributions] Experiments were 
 performed primarily in Pittsburgh by T.\ Tuan,
  and in Bordeaux by H.\ Kellay.
 Analysis of data was carried out by all authors.  
 Research was designed by G.\ Gioia, N.\ Goldenfeld, and W.\ Goldburg.
  G.\ Gioia wrote the paper with assistance 
 from N.\ Goldenfeld and P.\ Chakraborty 
 (who also wrote the Supplementary Information). 
%\item[Supplementary Information] is linked to the online version of the
%paper at www.nature.com/nature.
\end{addendum}

\begin{figure}[hpdp]
\begin{center}
\includegraphics[width=9cm]{nature_fig1.pdf}
\spacing{1} \caption{\small{Experimental setup used to study 
   steady, gravity-driven, 2D soap-film flows. {\bf (a)} 
% Axis $x$  
% runs vertically along the centerline of the flow.
 Wires WL and WR are thin nylon wires
 (diameter $=0.5\,$mm) kept taut by  
 weight W. The film hangs from the wires;
 its width increases from 0 to $w$ over
 an expansion section,
  then remains constant and equal to $w$ over
 a section of length $\approx 1\,$m.
 Reservoir RT contains a soapy solution 
 ($2.5\,$\% Dawn Nonultra in water; $\nu=0.01\,$cm$^2$/s),
 which flows through valve V and into the film.
 After flowing through the film, 
 the soapy solution drains into reservoir RB 
  and returns to reservoir RT via pump P.
  Turbulence is generated by comb C
 of tooth diameter $\approx 0.5\,$mm and
  tooth spacing $\approx 3\,$mm.
 We perform all measurements at a distance 
 of at least 10$\,$cm downstream of the comb.
 Axis $y$ ($0\le y\le w$) has its origin at wire WL, 
 as indicated.
 {\bf (b)} Interference 
 fringes in yellow light (wavelength $=0.6\,\mu$m)
 make it possible to visualize the generation of 2D turbulence
from the comb. {\bf (c)}  Typical log-log plots 
 of the spectrum $E(k)$, from LDV measurements 
 performed on points of the film 
 close to one of the wires and (inset)
  along the centerline of the flow
  }} \label{setup}
\end{center}
\end{figure}
%works!

\begin{figure}[hpdp]
\begin{center}
\includegraphics[width=9cm]{nature_fig2.pdf}
\spacing{1}
\caption{\small{ The mean velocity profile $u(y)$ in 
turbulent soap-film flows, from LDV measurements.
 {\bf (a)} Typical plots of $u(y)$ in a film of width 
 $w=12\,$mm. {\bf (b)} Typical 
  plots of $u(y)$ close to one of the wires,
  in the viscous layer where $du(y)/dy=G$ and the thickness 
 of the film is uniform and $\approx 10\,\mu$m
  (Supplementary Information). 
The velocity profiles correspond to $\Re = 7893, 17648, 25912$.
Points on the film
 closer than $\approx 20\mu$m (the diameter 
  of the beam of the LDV) from the edge of the wire cannot
 be probed with the LDV; thus the first data point, 
 which we position at $y=0$, is at a distance of 
 $\approx 20\mu$m from the edge of the wire.
% (Supplementary Information). 
%Extrapolation of $u(y)$ to the edge of the wire gives 
 The apparent slip velocity is likely to 
 represent 3D and surface-tension effects associated with 
 the complex flow at the contact 
 between the film and the wire.
}}
\label{velpro}
\end{center}
\end{figure}

\begin{figure}[hpdp]
\begin{center}
\includegraphics[width=9cm]{nature_fig3.pdf}
\spacing{1}
\caption{\small{ 
Log-log plot of the frictional drag vs.\ the
 Reynolds number in 2D
 turbulent soap-film flows of Reynolds number
  $1300\le{\rm Re}\le 25000$, from independent
 experiments performed in Pittsburgh and Bordeaux.
   The cloud of data points may be represented as
  a straight line of slope $1/2$, consistent 
 with the scaling $f \propto {\rm Re}^{-1/2}$. The straight 
 dashed line of slope $1/4$ corresponds to the 
 Blasius empirical scaling, $f \propto {\rm Re}^{-1/4}$. }}
\label{friction_factor}
\end{center}
\end{figure}



\end{document}
  


%The strength of turbulence is characterized by the Reynolds number 
%$Re = U {\cal L}/\nu$, where 
%$U$ is a characteristic velocity, ${\cal L}$ is a characteristic length,
%and $\nu$ is the kinematic viscosity of the fluid.
%%We parameterize turbulent flows by the Reynolds number $\Re\equiv
%%UL/\nu $, expressed in terms of the mean flow speed $U$, characteristic
%%length scale $L$ and kinematic
%%viscosity of the fluid $\nu$.  
%A viscous fluid flowing through a  pipe
%exerts a stress (force per unit area) $\tau_w$ on the pipe wall, which
%is conventionally represented in dimensionless form by the friction
%factor $f\equiv\tau_w/\rho U^2$. When the flow is turbulent, this
%friction factor is found to be a non-monotonic function of the Reynolds
%number, for which a satisfactory and quantitative explanation is still
%lacking\cite{NIKU33}. The widely accepted theory of turbulent flows
%near a boundary\cite{schlichting2000blt,pope2000tf} attempts to
%describe how $f$ depends on $\Re$,
%based on general similarity arguments; a more general approach that
%does not make such assumptions has also been
%formulated\cite{BARE93,porporato2001ist,BARE04}, as well as recent
%phenomenological
%proposals\cite{AFZA06,LVOV08,calzetta2009friction,yang2009virtual}.
%That body of work is concerned with experiments on three-dimensional
%flows, primarily pipe
%flows\cite{mckeon2004ffs,shockling2006ret,ALLE07,kunkel2007further,langelandsvik2008flow,bailey2008azimuthal}
%but also channel flows (for recent developments, see e.g.
%\cite{schultz2009turbulent,schultz:015104}).  Of essential importance
%is that the conventional theory for the friction factor is formulated
%without explicit reference to the spatial dimensionality $D$, and has
%no fundamental way to represent the nature of the turbulent state: the
%character of turbulence is represented by statistical assumptions about
%the velocity fluctuations.  However, it is known that the nature of the
%turbulent state in two dimensions (2D) is different from that in three
%dimensions (3D): there is no vortex stretching and there are two
%quadratic quantities that are conserved, enstrophy and
%energy\cite{batchelor1982tht,KRAI67,KELL02}.  
%In 3D there is only one forward energy cascade\cite{KOLM41,OBUK41} 
%having an energy spectrum $E(k)$ 
%that depends on the wave number $k$ and the mean energy transfer rate $\bar{\epsilon}$ 
%in the form $E(k) = \bar{\epsilon}^{2/3}k^{-5/3}$ in the so-called inertial range.
%%In 3D there is only a
%%forward energy cascade\cite{KOLM41,OBUK41}, in which the dependence of
%%the fluctuation energy spectrum $E(k)$ on wavenumber and mean energy
%%transfer rate $\bar\epsilon$ is independent of the kinematic viscosity
%%$\nu$ and has the form $E(k)= \bar\epsilon^{2/3} k^{-5/3}$ for values
%%of wavenumber in the so-called inertial range. 
%In 2D, however, there
%are two cascades: an energy inverse cascade that runs from small to
%large scales\cite{batchelor1982tht, KRAI67}, and a forward
%cascade\cite{KRAI67} in the enstrophy $\Omega \equiv |\bf{\nabla}
%\times \bf{v}|^2$, where $\bf{v}$ is the fluid velocity field.  This
%enstrophy cascade yields an energy spectrum of the form
%$E(k)=\beta^{2/3} k^{-3}$, where $\beta$ is the rate of transfer of
%enstrophy and logarithmic corrections have been omitted.  The rich
%structure of 2D turbulence permits, in principle, a way to explore the
%dependence, if any, of macroscopic flow properties on dimension,
%cascade and the underlying nature of the turbulent state\cite{GUTT09}.%

%The standard theory of wall-bounded turbulent shear flows due to
%Prandtl and others is
%independent of the dimension $D$ of the system, but has only been tested
%in 3D systems until now.  Our 2D experiments, carried out in a
%vertically flowing soap film,  yield a very different a result, showing
%the Prandtl theory to be fundamentally flawed. However, our findings
%are in accord with a recent theory that has a more fundamental starting
%point. \cite{GUTT09}.

%We performed experiments on quasi-2D turbulent soap films, measuring
%how the friction factor $f$ depends on $U$ and also the energy spectrum
%$E(k)$, so that we could ascertain the nature of the turbulent state in
%our experiment. 

 
%, possibly indicative of the logarithmic corrections to the pure power law of 3\cite{KRAI67}. 

%We conclude that our experiments are consistent with the enstrophy
%cascade, although on the basis of just these measurements, we cannot
%determine that the direction of energy transfer is indeed to small
%scales; for the purposes of testing the momentum-transfer theory,
%however, only the spectrum's functional form serves to represent the
%nature of the turbulent state, not the direction of the energy
%transfer. Our friction factor measurements presented below were all
%taken in flows with this enstrophy cascade.  We also attempted to
%generate flows which exhibited the inverse cascade range, as has been
%reported previously\cite{RUTG98}, where $\beta$ = 5/3
%\cite{batchelor1982tht,KRAI67}, 
%but were unsucessful to generate such flows over a range of $\Re$
%that would provide a
%meaningful test of the momentum-transfer theory prediction for the
%friction factor in a 2D flow with an inverse energy cascade.
%but there was no configuration of tooth
%size, tooth spacing and comb orientation that would generate a flow
%with a sufficiently extensive inverse energy spectrum to provide a
%meaningful test of the momentum-transfer theory prediction for the
%friction factor in a 2D flow with an inverse energy cascade.
 

% illustrated in Fig. (\ref{setup}).  We
%used a soap solution (2 \% detergent + water) that flows between wires
%having various diameters in the 0.5 mm range. By
%definition,$f=\tau_w/\rho U^2$, or equivalently, $f=G_w \nu/U^2$, where
%$G_w$ is the velocity gradient at the wall.  The mean flow is in the
%$x$ direction and $y$ is the distance from either of the support wires.
%The wire separation is adjustable in the range $0.5$cm $< w <2.6$cm.
%The spent soap solution at the bottom of the film is pumped back into
%the top reservoir, so the system can be maintained in a steady state
%for hours at a time. This solution is seeded with small particles

%. Because the film thickness is in the micron
%range which is much less than $w$, the system is quasi-two dimensional.
%There is, of course, frictional energy loss to the air, coupling the
%film to a third dimension, but, being velocity dependent, this
%primarily influences the profile in the center of the film, and does
%not affect significantly the velocity gradient near the wall.  

%The measurements were made at various distances $X$ below a horizontal
%comb having a tooth diameter and tooth spacing of the order of 0.5 mm and
%3 mm respectively. 
% The results were robust to variation of these
%parameters. 
%The velocity $U$ was varied by changing the observation point $X$ and by 
%changing the soap solution flux through the valve. 
%(Typically $X \simeq $ 50 cm.)  
%The velocity was varied by adjusting X and by changing the soap solution flux through the valve. 
%The observation point $X$ was varied to permit the widest
%possible range of $U$.   Typically $X \simeq $ 50 cm.  The velocity $U$
%was varied primarily by adjusting $X$, but other means that we used
%included adjusting a valve in the line that feeds the soap solution
%from the top reservoir into the film, and even changing the film length
%$L$, which is typically 1 m. 
%The measurements were made in a region
%where the turbulence was fully-developed and completely filled the
%channel, as verified by measuring the velocity profile in the
%$y$-direction and verifying that the flow did not show maxima away from
%the centerline.
% which would be indicative of jet formation coexisting with a laminar state.  
%To extract $f(\Re)$, where $\Re=Uw/\nu$, it is
%necessary to measure the vertical velocity  close enough to a wire that
%the time averaged vertical velocity $\langle u(y) \rangle$ is
%proportional to $y$.  The outer limit $y=y_{max}$ was determined by
%moving the laser head with a computer-controlled micrometer.  These
%measurements established that $y_{max}$ is 0.2 mm for the film widths
%explored in this experiment.

% TUAN please check carefully the paragraph above for accuracy.

%To establish that the measurements are indeed made in the enstrophy
%range, it is necessary to measure the power spectrum of the velocity
%fluctuations to assure that over a significant range of wavenumbers,
%$E(k) \propto k^{- \beta}$ with $\beta$  close to the well-established
%value of 3\cite{KRAI67,kraichnan1980tdt,FRIS95}.  Here $E(k)$ is the
%energy per unit wave number $k$, which is related to the total
%turbulent kinetic energy by the relation $E=(1/2) \langle ((U-u_x)^2 +
%u_y^2) \rangle =\int_0^\infty dk E(k)$.  It is assumed that the
%small-scale velocity fluctuations are isotropic.  To relate the
%temporal to spatial fluctuations, Taylor frozen turbulence
%hypothesis\cite{taylor1938spectrum} has been invoked. This is
%justifiable if the rms velocity fluctuations are relatively small
%compared to $U$ \cite{belmonte2000experimental,FRIS95}.  In our
%experiments, this ratio was less than 20 \%.

% TUAN correct the numbers in the paragraph above if needed.  I got them from Walter's first draft.
% Also are you really measuring E(k) which is the isotropic spectrum or the longitudinal spectrum
% We think that what you are showing in the data is actually E_{11}
% as defined in Pope.  So we should either have a different formula, which would be confusing to
% the reader, or show the actual E(k) and not E_{11}.
% In any case the figure 2 needs to have the y-axis properly labelled, and the Re indicated.
% Is the 20 percent number correct?  It seems alarmingly high.

%Figure \ref{power-spectrum} shows $E(k)$ (arbitrary units) vs $k$ at
%three values of the mean velocity $U$. The slopes of these lines is close to 3.3, as in
%prior experiments \cite{kellay1995experiments}.
%%, possibly indicative of the logarithmic corrections to the pure power law of 3\cite{KRAI67}. 
%We conclude that our experiments are consistent with the enstrophy
%cascade, although on the basis of just these measurements, we cannot
%determine that the direction of energy transfer is indeed to small
%scales; for the purposes of testing the momentum-transfer theory,
%however, only the spectrum's functional form serves to represent the
%nature of the turbulent state, not the direction of the energy
%transfer. Our friction factor measurements presented below were all
%taken in flows with this enstrophy cascade.  We also attempted to
%generate flows which exhibited the inverse cascade range, as has been
%reported previously\cite{RUTG98}, where $\beta$ = 5/3
%\cite{batchelor1982tht,KRAI67}, 
%but were unsucessful to generate such flows over a range of $\Re$
%that would provide a
%meaningful test of the momentum-transfer theory prediction for the
%friction factor in a 2D flow with an inverse energy cascade.
%but there was no configuration of tooth
%size, tooth spacing and comb orientation that would generate a flow
%with a sufficiently extensive inverse energy spectrum to provide a
%meaningful test of the momentum-transfer theory prediction for the
%friction factor in a 2D flow with an inverse energy cascade.

%We deduced the friction factor from measurements of the slope of the
%linear part of the velocity profile near the wall of the soap
%film.  In this region, extending over about 0.2mm, we observed our
%films in yellow light (wavelength of 579nm) and counted at most two
%fringes, showing that the film thickness varies by no more than one
%wavelength, i.e. at most by 10\%.
%This is important because the viscosity of soap films will in principle 
%depend on its thickness \cite{Vorobieff99}.
%%; thus there is no significant change of
%%film viscosity over the measurement region.  
%Figure \ref{friction_factor} shows a log-log plot of $f(\Re)$ for films of
%varying width $0.5 < w < 2.6$cm.  Shown here are five independent sets
%of data, of which four were taken at the University of Pittsburgh, and
%one at Bordeaux in a completely independent experimental setup. The
%solid line has a slope of $-1/2$, which is the analytical
%prediction\cite{GUTT09} based on the momentum-transfer
%theory\cite{GIOI06}, and close to the results of direct numerical
%simulations\cite{GUTT09}.  The data cloud shows little deviation from
%the theoretical prediction, and moreover, strongly deviates from the
%corresponding result in 3D flows\cite{NIKU33}, which show a slope of
%$-1/4$ up to Reynolds numbers of order $10^5$.


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