# Aspect ratio dependence of charge transport in turbulent electroconvection

###### Abstract

We present measurements of the normalized charge transport or Nusselt number as a function of the aspect ratio for turbulent convection in an electrically driven film. In analogy with turbulent Rayleigh-Bénard convection, we develop the relevant theoretical framework in which we discuss the local-power-law-scaling of with a dimensionless electrical forcing parameter . For these experiments where we find that with either or , in excellent agreement with the theoretical predictions of and . Our measurements of the aspect ratio-dependence of for compares favorably with the function from the scaling theory.

###### pacs:

47.27.TeHeat transport has for many years been the cornerstone in the study of turbulent Rayleigh-Bénard convection (RBC) kadanoff_01 . Over the past three years there has been remarkable experimental ahlers_00 ; ahlers_side_00 ; niemela_00 ; ahlers_01 ; xia_lam_zhou_02 ; ahlers_03 ; niemela_03 and theoretical GL_00 ; GL_01 ; GL_sidewall_03 progress in characterizing the properties and mechanisms of heat transfer in fluids heated from below. So far, experiments and theory are relevant only to approximately unit aspect ratio systems i.e. geometries that have comparable lateral and vertical dimensions. Turbulent convection flows in nature, however, are laterally extended and the transport of heat across a layer of fluid has been inadequately studied as a function of the aspect ratio . In particular, experiments at have seldom been performed primarily because it is extremely difficult to achieve the strong forcing that is readily reached in containers.

In this Letter, we present a study of an analogous electrically driven convecting system which allows a broad range of to be explored at moderate levels of forcing. We show that measurements of the normalized charge transport varies with the aspect ratio consistent with a function given by the scaling theory. The strong dependence on especially for implies that experiment and theories relevant to this regime are restrictive. The data and the function become increasingly independent of for . This observation highlights the importance of developing experiments and theory for laterally extended systems where it appears that universal behavior, often lacking in systems, may be restored daya_ecke_01 . Our data also reveal that with and in agreement with a local-power-law-scaling theory developed in a manner identical to the Grossmann-Lohse (GL) model for turbulent RBC GL_00 ; GL_01 .

In previous work, it has been demonstrated that an electrically conducting, freely suspended liquid crystal film between parallel wires could be driven to convect when a sufficiently large potential drop was applied across its edges morris_90 ; mao_97 ; daya_97 ; dey_97 ; daya_98 ; daya_99 ; daya_thesis_99 ; daya_01 ; daya_02 . Experiments daya_98 ; daya_thesis_99 ; daya_01 ; daya_02 and theory daya_99 ; daya_thesis_99 were then extended to the naturally periodic geometry of an annular film shown schematically in Fig. 1. Analogous to RBC where an inverted mass density distribution is unstable to buoyant forcing, the thin film has an inverted surface charge density distribution that is unstable to electrical forcing. In this system, an initial bifurcation to convection rolls occurs when the applied voltage exceeds a critical voltage , corresponding to the critical temperature difference in RBC. Secondary bifurcations occur at higher forcing and result in changing the number of counter-rotating vortex pairs that are arranged around the annulus. At much higher forcing the fluid becomes turbulent but retains the large scale structure of the rolls or convection cells.

Electroconvecting smectic films have several advantages and disadvantages relative to traditional RBC experiments. In turbulent RBC experiments, the accurate measurement of heat transport requires detailed accounting of the heat conducted through the sidewalls ahlers_side_00 . The annular geometry of the smectic film (see Fig. 1a) is free of lateral boundaries and consequently without sidewall losses, which facilitates precise measurement of the charge transport between the inner and outer electrodes. The film is an annular disk of width mm and thickness m. Here are the radii of the inner (outer) electrodes that support the film (see Fig. 1b). These radii can be varied so as to achieve different values of . The tiny size of the film, which contains many orders of magnitude less working fluid than required for an RBC experiment, results in data acquisition timescales of minutes, rather than days. On the other hand, the smectic film is delicate and while many films survive vigorous forcing without thickness change, some do suffer sudden thickness variations. Since and other properties of the flows depend on thickness, these data must be discarded. The dc electrical forcing also results in conductivity drifts in the films, which result in systematic uncertainties. The experiments we describe here were carried out at atmospheric pressure, which increases the film stability. But air drag, which is known to produce a quantitative but not qualitative change to the dissipation, is not properly accounted for in the theory. Nevertheless, notwithstanding these caveats, we are able to efficiently explore the turbulent scaling regime over a broad range of , complementing RBC experiments.

Our experiment consists of a temperature controlled film of octylcyanobiphenyl (8CB), a smectic-A liquid crystal, suspended between two concentric gold-plated electrodes. In the smectic-A phase, the film flows as an isotropic, incompressible and Newtonian fluid in the plane of the film. The flow is strictly two-dimensional. The film is driven to electroconvect by a dc voltage applied to its inner edge while holding the outer edge at ground potential. The annular assembly is enclosed in a Faraday cage. An experiment consists of imposing a voltage to the inner electrode and measuring the current transported through the film with a sensitive electrometer. The applied voltage was varied between and volts in a sequence of small incremental and decremental steps resulting in a current-voltage characteristic. Further details of the apparatus and procedure can be found in Refs. daya_98 ; daya_99 ; daya_thesis_99 ; daya_01 ; daya_02 . Figure 2 shows a representative current-voltage (IV) characteristic. For the fluid is quiescent with the current sustained by ohmic conduction. When the fluid is organized in counter-rotating vortices and additional charge is transferred by convection; this is seen by the increase in slope of the IV-characteristic. At higher voltages, the fluid becomes turbulent and the transition is marked by a sudden increase in the rms fluctuations of the current, as shown in the inset of Fig. 2.

A film is a 2D annular sheet. The film has thickness/width . Geometrically, a film is described by its radius ratio and the aspect ratio , which is the ratio of the mid-radius circumference to the film width. So defined, . To make correspondence with the conventional aspect ratio for RBC, we define for the annulus. Two other dimensionless parameters describe the experimental system: and . Here is the control parameter and is a measure of the external electrical forcing and , the Prandtl-like parameter, is the ratio of the time scales of electrical and viscous dissipation processes in the film. In the above the fluid density, molecular viscosity, and conductivity are denoted by , , and , and is the permittivity of free space.

We normalize the measured total electric current by the portion due to conduction where is the ohmic conductance of the film obtained from the IV-characteristic when the film is not convecting, i.e. when . We define the normalized charge transport or Nusselt number . Figure 3 shows representative sets of vs data at and . We find that for with either or , depending on the size of . The error bar is obtained from the variation in the best-fit value of over data sets for all four values of .

We developed a theoretical model describing turbulent electroconvection by borrowing from turbulent RBC. The equations of motion that describe electroconvection consist of the incompressible Navier-Stokes equation supplemented with an electrical body force, the charge conservation equation that accounts for ohmic conduction and advection of charge and Maxwell’s equation that relates the surface charge density to the electric potential;

(1) | |||||

(2) | |||||

(3) |

These equations are constrained by the no-slip and applied electric potential boundary conditions. In the above equations , , and are the 2D gradient, velocity, pressure and kinematic viscosity respectively. The electric potential is three dimensional and extends outside the film, where . The Maxwell equation relates the perpendicular gradient of to the surface charge . The factor of 2 arises from the film’s two free surfaces. The relation between and is thus nonlocal and somewhat complex. It can be considerably simplified by making a local approximation, setting , where is a certain constant. This local approximation has been shown to be adequate in describing the onset of electroconvection. See Ref. daya_99 for a detailed discussion of the theoretical model.

The Eqns. 1, 2 are similar to the Boussinesq equations for turbulent RBC. In particular, in the local approximation, there is an almost precise correspondence between the temperature and the electric potential. The periodicity of the annular geometry and thereby of the velocity and electric potential allow for exact relations for the globally-averaged kinetic and electric dissipations. Denoting averages over the system volume by we find the following for the kinetic dissipation and for the electric dissipation :

(4) | |||||

(5) |

These relations are similar to those for the kinetic and thermal dissipation in RBC presented in Refs. GL_00 ; GL_01 ; shr_sig_90_94 . Following the GL theory, we decompose the kinetic and electric dissipations into bulk and boundary layer parts: . Considering the various combinations of the dominant contributors to the total dissipation (e.g. BL- and BULK- etc.) we can determine the corresponding local power-law scalings for each of the regimes. We assume that the turbulent flow is comprised of convection cells that tile the annulus as shown in Fig. 1a and that each cell is roughly square i.e. it has the same transverse and lateral dimension. The large scale circulation is then the vortex that defines a cell. Assuming laminar boundary layer scaling we can then show that

(6) | |||||

(7) |

As might be expected, we find the same set of exponents that appear in the GL theory. Here, is the Reynolds number of the large scale circulation.

In the relatively small regime, the total dissipation is dominated by the BL-contributions and the theory predicts and . In a neighboring regime where the dissipation is primarily and , the theory predicts . Our measured exponents, as shown in Figure 3 are in reasonable agreement with the theoretical predictions from the GL model, if we suppose that we traverse the appropriate regimes as varies. A detailed validation of the theory would depend crucially on either directly demonstrating the correctness of the assumptions about the dominant contributions to the dissipation or on systematically showing that the right local power-law scalings are indeed obtained as one varies and . Our experiments currently span only relatively low and sparsely cover the rather wide range . Many more experiments will be needed to adequately test the GL theory for turbulent electroconvection.

In the theoretical treatment described above, we have taken into account the dependence on the aspect ratio of the system, a consideration that was missing in the GL theory which treated only systems GL_00 ; GL_01 ; GL_sidewall_03 . We find that the charge transport is modified by a -dependent factor (). Since the large scale circulation is local to each convection vortex, is independent of . Unlike Ref. shr_sig_90_94 , we find the aspect-ratio dependence is not a power-law scaling but rather a function of the finite annular geometry. We can make a direct comparison with previous turbulent RBC experiments by making a correspondence between the different cell geometries, using as described above. We plot () vs. , with the constant . Following Ref. ahlers_00 , we choose this normalization so that . Then in the limit. The function decreases monotonically with with the greatest variation for , and is within of the limiting value for .

Our experimental data span the range . From power-law fits to vs. data we have extracted the exponents . To determine we then divide by . Because our data extend over a rather wide range of , we expect to cross regimes with differing and GL_00 ; GL_01 . To extract the aspect ratio dependence alone, we used the fitted value of , and where , we used the GL-theory prediction of . Using one free parameter for all the data, we again scaled these results so that . Our data for different are in reasonable quantitative agreement with the theoretical function , as shown in Fig. 4. Each data point is an average over runs. The error estimates are representative of the scatter in over these data. In particular, the largest contribution to the error at is systematic and arises from the ambiguity in the correct exponent for .

Data from several turbulent RBC experimentsahlers_00 ; ahlers_side_00 ; niemela_00 ; niemela_03 ; HKgroup_96 for values of are also broadly in agreement with the function , in spite of the difference in geometry and the higher range of Rayleigh numbers. Earlier RBC experimentsthrelfall_74 ; wu_libchaber_92 used gases as the working fluid and deviate significantly from .

Several important questions are raised by our results on the aspect ratio dependence of charge transport in turbulent electroconvection and, by extension, that of heat transport in turbulent RBC. The strong dependence of on for , suggests that the dynamics of turbulent convection is dependent on the system’s lateral extent. That the charge or heat transport, a global property, is sensitive to the aspect ratio further suggests that local properties such as temperature and charge fluctuations may be even more strongly dependent on . Experiments in turbulent RBC clearly show that while heat transport is insensitive to cell geometry at fixed , the fluctuations are strongly affected by the shape of the lateral boundary daya_ecke_01 . The relative independence of on for large aspect ratio, suggests that the normalized charge and heat transport approach a universal value in laterally extended systems. We conjecture that fluctuations in the interior may also become universal in large aspect ratio systems.

Our results emphasize the importance of extending convective turbulence experiments to larger aspect ratios. For turbulent smectic electroconvection, we would like to make a systematic study of the scaling of in the parameter space of and . can be varied in principle by changing the thickness and/or width of the film. We would also like to study long time series of in order to infer the characteristic size and duration of the fluctuations. It is a challenge to extend the forcing parameter to values much beyond , since this requires applying large voltages across the film that may result in dielectric breakdown. A first step would be to better control the electrochemistry of the liquid crystal film to reduce drifts and push down the critical voltage . It would also be interesting to extend our previous studies of electroconvection under shear daya_98 ; daya_99 ; daya_thesis_99 ; daya_01 ; daya_02 to the turbulent regime. Finally, we would like to develop local probes to study the fluctuations of the velocity and electric potential over the film.

We thank G. Ahlers, E. Ben-Naim, R. E. Ecke and E. Titi for helpful discussions and constructive comments. This research was supported by the Canadian NSERC and the U.S. DOE (W-7405-ENG-36).

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