## Main

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 interest2,3,5,17,18. The frictional drag is defined as the dimensionless ratio f=τ/ρ U2, where τ is the shear stress or force per unit area that develops between the flow and the wall, ρ is the density of the fluid and U is the mean velocity of the flow. Already in eighteenth-century France, f was the subject of large-scale experiments carried out in connection with the design of a waterworks for the city of Paris19,20. In 1883 it was predicted21, and subsequently confirmed by numerous experiments, that in pipe flows f depends on the Reynolds number Re=U d/ν, where d is the diameter of the pipe and ν is the kinematic viscosity of the fluid. For pipe flows of moderate turbulent strength (starting from Re≈2,500 and up to Re≈100,000) the experimental results are well described13 by the Blasius empirical scaling, fRe−1/4. (Throughout this letter, the symbol ‘’ may be changed to the symbol ‘=’ by introducing a dimensionless proportionality factor, for example, f=CRe−1/4.) The celebrated theory3,5 of the frictional drag was formulated 80 years ago by Ludwig Prandtl, the founder of turbulent hydraulics, and numerous variants5,13,22 and alternatives23,24 of Prandtl’s theory have since been proposed. 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 a result, these theories cannot be used to reveal the missing link between the frictional drag and the turbulent spectrum.

The turbulent spectrum is a function of the wavenumber k, E(k). The physical significance of E(k) can be grasped from the expression3 , which gives the characteristic velocity us of a turbulent eddy of size s in the flow. For the spectrum of the spectral exponent α we can write16 E(k)U2L(1−α)kα, and therefore

where 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 α=5/3 (ref. 12). Thus, for example, 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 several crucial respects, most notably in that in two dimensions there is no vortex stretching. As a result, a different type of spectrum is possible in 2D flows: the ‘enstrophy cascade’, for which α=3 (refs 9, 10). Thus, for example, 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 centimetres apart from one another (Fig. 1a). Driven by gravity, a steady vertical flow soon becomes established within the film. In this case, the thickness h of the film is roughly uniform on any cross-section of the film, typically h≈10 μm, much smaller than the width w and the length of the film (Fig. 1a). As a result, the velocity of the flow lies on the plane of the film, and the flow is 2D.

We make the flow turbulent by piercing the film with a comb, as indicated in Fig. 1a, so that the flow is stirred as it moves past the teeth of the comb. To visualize the flow, we cast monochromatic light on a face of a film and observe the interference fringes that form there. These fringes (Fig. 1b) reflect small changes in the local thickness of the film. (The thickness is constant along a fringe; it differs by a fraction of the wavelength of the light, or a fraction of a micrometre, 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 flow8. Thus, Fig. 1b may be interpreted as a map of the instantaneous spatial distribution of turbulent fluctuations downstream of the comb.

We compute the spectrum E(k) at numerous points on the film from measurements carried out with a laser Doppler velocimeter (LDV; see the Methods section). In Fig. 1c we show a few typical log–log plots of E versus k. The slope of these plots represents the spectral exponent α; in our experiments the slope is slightly larger than 3, consistent with previous experiments with soap-film flows8,25 and close to the theoretical value of α for the enstrophy cascade (α=3).

By using the same LDV we measure the mean (time-averaged) velocity u at any point on the film (see the Methods section). Successive measurements of u along a cross-section of the film give the ‘mean velocity profile’ u(y) of that cross-section. In Fig. 2a we show a few typical plots of u(y) over the entire width of the film (that is, from wire to wire, or for 0 ≤ yw). From a mean velocity profile we compute the mean velocity of the flow as , and the Reynolds number of the flow as Re=U w/ν.

In Fig. 2b we show a few typical plots of u(y) close to one of the wires, where u depends linearly on y on a narrow (about 0.2 mm) viscous layer. We have verified that the Reynolds shear stress vanishes in the viscous layer (Fig. 2c,d). We have also verified that the thickness of the film is nearly uniform in the viscous layer (Fig. 2e,f).

From the slope G of a mean velocity profile in the viscous layer (for example, Fig. 2b), we compute the shear stress between the flow and the wire as τ=ρ ν G. The frictional drag follows from the definition, f=τ/ρ U2, as f=ν G/U2.

An apparent slip velocity US is conspicuous in the plots of Fig. 2b, and is likely to represent 3D and surface-tension effects associated with the complex flow at the contact between a film and a wire. The value of US tends to lessen where we use thinner wires or brand new wires. By using a variety of wires, we have been able to realize several flows with the same value of Re but widely differing values of US. We have verified that the frictional drag of these flows is the same within experimental error (for example, Fig. 3), in spite of the widely differing values of US. We conclude that the frictional drag does not depend on the apparent slip velocity (except perhaps through the Reynolds number).

In Fig. 4 we show a log–log plot of f versus Re. 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 set-up. The cloud of data points is consistent with the scaling, fRe−1/2, and inconsistent with the Blasius empirical scaling, fRe−1/4, which is known to prevail in turbulent pipe flows.

Our experimental results may be explained using a recently proposed theory of the frictional drag14,15,16. In this 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 applicable to flows on rough walls, but where the walls are smooth (as in our experiments), the theory predicts that fuη/U (refs 14, 15, 16, 22, 26, 27, 28, 29). Here η is the size, and uη the characteristic velocity, of those eddies with an intrinsic Reynolds number of 1, so that uηη/ν=1. By setting s=η in (1) and combining the result with uηη/ν=1 and Re=U L/ν, we conclude that if the energy spectrum of the flow has the spectral exponent α, then the scaling, uηURe(1−α)/(1+α), must hold. It follows that

and the functional dependence of the frictional drag on the Reynolds number is set by the spectral exponent α. For pipe flows α=5/3, and (2) yields15 fRe−1/4, consistent with the Blasius scaling. In contrast, for soap-film flows α=3, and (2) yields16 fRe−1/2, consistent with our experimental results (Fig. 4).

From our experiments with 2D soap-film flows we infer that the long-standing and widely accepted theory5 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 2D and 3D 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 drag14,15,16. This new theory perforce relates the frictional drag to the turbulent spectrum, and is sensitive to the dimensionality of the flow through the dependence of the turbulent spectrum on the dimensionality. Our findings lead us to conclude that the macroscopic properties of both 3D and 2D turbulent flows are closely linked to the turbulent fluctuations. In addition, our findings serve to underscore the value of using 2D soap-film flows to test and extend our understanding of turbulence.

## Methods

To measure the components u(t) (along the mean flow) and v(t) (transverse to the mean flow) of the instantaneous velocity at a point on the film, we use an LDV with a sampling rate of 5 kHz. By carrying out measurements over a time period of about 200 s, we collect time series u(ti) and v(ti). From these time series we calculate the local mean velocities u and v as the time averages, u≡〈u(ti)〉 and v≡〈v(ti)〉. To compute the local turbulent spectrum (more precisely, the longitudinal turbulent spectrum), we invoke Taylor’s frozen-turbulence hypothesis6 to carry out a space-for-time substitution tx/u on the time series (u(ti)−u). This space-for-time substitution gives the space series, u′(xi)≡(u(xi/u)−u), where xi=u ti. (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 (ref. 30).) The spectrum E(k) is the square of the magnitude of the discrete Fourier transform of u′(xi). We compute the local Reynolds shear stress as τRe=ρ〈(u(ti)−u)(v(ti)−v)〉. To measure the thickness h of the film, we mix the soapy solution with a fluorescent dye and focus a blue laser beam of diameter 20 μm on a spot of the film. The spot becomes fluorescent, and we monitor the intensity of the fluorescence by means of a photodetector with a counting rate proportional to h.