ISSN 10637761, Journal of Experimental and Theoretical Physics, 2011, Vol. 113, No. 3, pp. 516–529. © Pleiades Publishing, Inc., 2011. Original Russian Text © A.E. Gledzer, E.B. Gledzer, A.A. Khapaev, O.G. Chkhetiani, 2011, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 140, No. 3, pp. 590–605.
STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Structure Functions of QuasiTwoDimensional Turbulence in a Laboratory Experiment A. E. Gledzera,*, E. B. Gledzera, A. A. Khapaeva, and O. G. Chkhetiania,b,** a
Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, 109017 Russia Space Research Institute, Russian Academy of Sciences, Profsoyuznaya ul. 84/32, Moscow, 117810 Russia *email:
[email protected] **email:
[email protected]
b
Received November 22, 2010
Abstract—The results of experiments for turbulent flows in a thin layer of conducting fluid above a solid sur face generated by the Ampere force when passing a current and under the action of a spatially periodic mag netic field are considered. The statistical characteristics of the flows are shown to exhibit threedimensional (3D) dynamics even on horizontal scales exceeding the layer thickness by an order of magnitude. In this case, the thirdorder longitudinal structure functions of the velocity field are approximately linear in spatial dis placement and negative, as in 3D turbulence, due to the dominant contribution of energy dissipation when the boundary condition for adhesion on the lower surface is met. The dissipation and basic energy production terms are estimated for the energy balance equation. DOI: 10.1134/S106377611108005X
1. INTRODUCTION Quasitwodimensional (2D) approximations are widely used in the description of hydrodynamic flows for models of geophysical hydrodynamics and for interpreting data from laboratory experiments simu lating atmospheric motions. In such approximations, the vertical structure of flows is generally neglected or it is taken into account via effective coefficients or additional terms modifying the 2D dynamics of flows [1–8] (the socalled bottom friction). Only the hori zontal velocity components referring to 2D motion can usually be measured in laboratory experiments using relatively thin fluid layers [9–11]. In this case, the vertical structure of the velocity field defies mea surement, although, given the development of a tech nique for measurements in thin fluid layers, works in this direction have been performed in recent years (see [12, 13]). The role of vertical motions can be assessed by comparing results with a theory incorporating the vertical motions in the layer. For flows with pro nounced random, chaotic variations in velocity com ponents (at least after the subtraction of timeaveraged values), the patterns of homogeneous turbulence in both 3D (given that the quasi2D velocity field actu ally has a 3D structure) and 2D (as an approximation) cases can act as such a theoretical basis. The second and thirdorder structure functions are among the most commonly used characteristics of turbulence. For these functions, there are Kolmog orov laws for the viscous and inertial scales in the limit of large Reynolds numbers confirmed in a large num ber of experimental, numerical, and theoretical stud
ies. However, for quasi2D flows, given the last two types of studies of these laws, it is rather difficult to obtain them in laboratory experiments. We can point out only a few papers that studied thin layers or films of various fluids where these structure functions were calculated. In a geometry similar to that considered here when an electric current interacted with a spa tially periodic external magnetic field acting on a con ducting incompressible fluid, the thirdorder structure functions were derived in [14–17], with the emphasis having been on detecting the patterns of the inverse (from small to large scales) energy cascade and the suppression of turbulence by a mean largescale orga nized motion. There are also other works in this direc tion using the same method for generating the small scale vortex structure of flows in the 2D approximation [18, 19]. A 2D flow in a soap film is used in another formulation [20], where it is possible to expand con siderably the spectrum of allowable scales in the direc tion of their decrease or increase. The thirdorder structure functions, along with other energy charac teristics of 2D dynamics, were also obtained here. In the experimental works [14–17], the influence of bottom friction and related 3D energy dissipation is suppressed when a 2D velocity field is generated above a heavy nonconducting fluid. This makes it possible to obtain experimentally the inverse energy cascade in a hydrodynamic flow. The goal of this paper is opposite and consists in studying the influence of 3D effects produced by friction of a thin fluid layer above a solid surface on the statistics when the adhesion condition is met. As has already been noted, these effects can man
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ifest themselves in direct measurements of the velocity field in thin layers [12, 13]. However, for a long time they had been considered mainly through the intro duction of effective parameters affecting the motion of a 2D film into the theory. This is linear or quadratic (in velocity) friction that appears when reducing the vis cous and nonlinear terms of the hydrodynamic equa tions containing the vertical velocities and derivatives with respect to the vertical coordinate (see [2–8, 21, 22]). The 3D energy dissipation is significant for atmo spheric processes if it is kept in mind that the idea of an inverse cascade formulated in 1989 by Lilly [23] can be used, as is pointed out in [16, 17], to explain some of the features in the spectral energy distribution when measuring the velocity field on scales of 10–100 km in the atmosphere. In this case, the friction is significant not only in the surface boundary layer and the adjacent part of the troposphere but also for the upper tropo sphere and lower stratosphere as well as for the consid erably higher mesosphere–lower thermosphere transi tion region, as is indicated by data from rocket launches and meteor trails [24–26]. The Ekman spi rals detected at these altitudes are direct evidence for a significant role of turbulent friction in their genera tion, which is taken into account for all layers in the atmosphere and when constructing prognostic numerical schemes. The energy dissipation is the main defining param eter for the turbulent energy balance equation that is written out in the next section by taking into account the possible nonstationarity and inhomogeneity of the process under the action of an external force. We pro vide formulas for the (longitudinal and transverse) structure functions where the vertical flow structure is taken into account via a nonzero horizontal diver gence of the velocity field. The results obtained subse quently from measurements in laboratory experiments are essentially determined by the 3D dissipation in the system and by the energy input. The latter not only takes place in the region of smallscale vortices but also, just as for the atmospheric processes due to the baroclinic flows into the barotropic component of motion, can contribute to the motions of other scales. In this case, the dissipation and basic energy produc tion terms were estimated for the energy balance equa tion. No largescale circulation flow (condensate, according to the terminology of [14–17]) emerges in these experiments or it is indistinct during its artificial generation. However, subtracting the averaged charac teristics of the velocity field, which result mainly from the action of external spatially periodic forces, turned out to be important for obtaining the statistical char acteristics. 2. STRUCTURE FUNCTIONS Below, we provide some formulas for the second and thirdorder structure functions for homogeneous,
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locally isotropic 3D and 2D turbulence, for 2D one under conditions when ∂u + ∂v div h u = ≠ 0, ∂x ∂y where u and v are the horizontal velocity components. The structure functions are defined by the projections of the spatial velocity difference v = (vx, vy, vz), v ( r, t ) ≡ v ( r, t x ) = u ( x + r, t ) – u ( x, t ) onto the direction of the vector r = (rx, ry, rz): vL(r, t) = (r/r)v(r, t) and vN(r, t) perpendicular to it, where r is the vector of differences between the coordinates of points x + r and x, x = (x, y, z). Under the conditions of stationarity, homogeneity, and local isotropy, only the following structure func tions are nonzero and depend on the magnitude r: 3
D LLL ( r ) = 〈 v L ( r, t )〉 , 2
D LNN ( r ) = 〈 v L ( r, t )v N ( r, t )〉 ,
(1)
where 〈 〉 denotes an ensemble averaging. Let us write the Navier–Stokes equations for the velocity differences v(r, t). In accordance with the pro cedure described, for example, in [27], we will obtain an equation for the energy E(r, t) = v2(r, t)/2: ∂E div + r ( vE ) + div x ( u ( x, t )E ) ∂t 1 ∂(p(x + r) – p(x)) = – v i ρ ∂x i
(2)
+ νΔ x + r u ⋅ u ( x + r ) – νΔ x u ⋅ u ( x ), where ri = (rx, ry, rz) and the condition for zero 3D divergence of the flow, divv = 0, is used. (An equation similar to (2) is written in [28].) When averaged, the righthand side gives –2ε in accordance with [27] and in the case of spatial homogeneity or in the absence of correlations between the velocity difference v(r, t) and the velocity u(x, t). Here, ε = ν
∂u i
⎞ ∑ ⎛⎝ ∂x ⎠ i, j
2
j
is the energy dissipation per unit mass. Therefore, for stationary and divergencefree 3D and 2D turbulence, we have div 〈 vE〉 = – 2ε. (3) If the 2D (horizontal) divergence of the velocity divhu is nonzero, then the terms –Edivrv and –Edivxu will appear on the lefthand side of Eq. (2) for 2D tur bulence. Instead of (3), this gives div r 〈 vE〉 + div x 〈 u ( x, t )E〉 – 〈 Ediv r v〉 (4) – 〈 Ediv x u〉 = – 2ε. Here, the second term on the lefthand side was retained to allow for the possible spatial inhomogene ity or the presence of correlations between the velocity difference v(r, t) and the velocity u(x, t), as in the
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experimentally realized case where the mean (spatially periodic) velocity field is nonzero. Using the work by Obukhov [29] (and [27]), we can obtain the following relation between DLLL and DLNN for a spatial dimension n under the conditions of local isotropy: 1 d D LNN = ⎛ r D LLL + ( n – 2 )D LLL⎞ . ⎠ 3 ( n – 1 ) ⎝ dr
(5)
Equation (4) is brought to the form 1 1 d n–1 r ( D LLL + ( n – 1 )D LNN ) 2 r n – 1 dr
(6) ∂ 〈 E〉 = –2ε + P v ( r ) + P u ( r ) – P d ( r ) – + Π ( r ), ∂t where Pv(r) = 0 and Pu(r) = 0 in the 2D and 3D cases of a divergencefree velocity field, 2 P v ( r ) = 1 〈 v ( r, t )div h v〉 , 2 1 2 P u ( r ) = 〈 v ( r, t )div h u〉 , 2
(7)
P d ( r ) = div x 〈 u ( x, t )E〉 for quasi2D flows. The righthand side of Eq. (6) also includes the terms attributable to the possible nonsta tionarity of the process, 1 〈 E〉 = ( D LL ( r, t ) + ( n – 1 )D NN ( r, t ) ), (8) 2 and the energy input under the action of an external force f(x) (the Ampere force for this paper), Π ( r ) = 〈 ( f ( x + r ) – f ( x ) ) ( u ( x + r ) – u ( x ) )〉 , (9) where the averaging over all directions of the vector r is assumed. The functions Pv(r), Pu(r), and Pd(r) mean the energy input (or output, depending on the sign) attrib utable to circulation in the vertical plane of the fluid layer or to the above correlation. Generally, the func tions have a dependence on z, which can be eliminated by assuming the averaging in (7) as well as in (1) to be also performed over the layer height. In this case, the first term on the righthand side of Eq. (6) is propor tional to the heightaveraged energy dissipation. This averaging should be performed for quasi2D flows, because formulas like (5) hold only for isotropy in the entire ndimensional space, in particular, the 2D one. From Eqs. (6) and (5) we obtain (at ∂/∂t = 0 and for the values of r at which there is no input of external energy, Π(r) = 0) D LLL + ( n – 1 )D LNN = – 4ε r + R ( r ), n r (10) n–1 1 r P ( r ) dr, R ( r ) = 2 n–1 r 0
∫
d D 12ε r + 3R ( r ), ( n + 1 )D LLL + r LLL = – dr n (11) P ( r ) = P v ( r ) + P u ( r ) – P d ( r ). In the absence of sources P(r), R(r) = 0, from (11) we have the solution (without any singularity at r = 0) 12 εr. D LLL = – (12) n(n + 2) At n = 3, this gives Kolmogorov’s wellknown ⎯4/5 law: DLLL = –(4/5)εr. Equation (12) is also available, for example, in [30], where a spectral representation is also used in part for its derivation. Note that the above formulas do not specify the direction of energy transfer over the spectrum of scales. The –4/5 law for 3D turbulence is based on the Richardson–Kolmogorov cascade mechanism of energy transfer toward small scales, where the energy dissipates at rate ε. For 2D turbulence, the –3/2 law, DLLL = –(3/2)εr, formally follows from (12), provided that the energy here is also transferred from large scales to small ones. However, for “pure” 2D turbu lence, the energy flux is directed from small scales to large ones (while the enstrophy flux is directed oppo sitely). Therefore, to use Eq. (12) at n = 2, it is neces sary to assume that the energy input with the opposite sign enters into it instead of ε: εinp = –ε. This implies that the term responsible for the energy production, +2εinp, appears in Eq. (6) instead of –2ε, while the energy is dissipated by the large scales that are disre garded in Eqs. (6)–(12) in the inertial range of r. For the quasi2D case, the formulas are invalid, because the term R(r), whose sign is, in general, inde terminate, is present in (11). However, under the assumption that R(r) is an alternating quantity, since Eq. (11) is linear, it is hoped that DLLL is negative, at least in the ranges of the difference r where the first term in (11) dominates over the second one. The neg ative sign of the thirdorder structure function DLLL + (n – 1)DLLN (see (10)) suggests the existence of a direct energy cascade from large scales to small ones, with the energy dissipation becoming crucial. For a linear (in r) longitudinal structure function DLLL, DLLL ∝ r, according to (5), the transverse struc ture function at n = 3 and n = 2 (just as for any n > 1) is (13) D LNN = 1 D LLL . 3 For the secondorder structure functions DLL and DNN for 3D and 2D incompressible turbulence, respectively, the relations (see [27]) D NN = D LL + r d D LL , 2 dr (14) d D NN = D LL + r D LL , dr follow from the zero divergence conditions. For linear (in r) functions, this gives DNN = (3/2)DLL and DNN = 2DLL in the 3D and 2D cases, respectively. Clearly, if
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the zero divergence condition is violated, as for quasi 2D flows, then these relations do not hold. Below, to consider the experimental data, we will need a quantity determined from (5) as a function of r if the functions DLLL and DLNN have been specified (the dimension function): d D D LLL – r LLL dr (15) N ( r ) = 1 + . D LLL – 3D LNN This is because quasi2D flows have the features and peculiarities of both 2D and 3D flows. The latter holds at least in the processes of energy dissipation, which takes place predominantly during the shifts of the horizontal velocity in height. 3. THE LABORATORY EXPERIMENT AND RESULTS A detailed description of the experiments is con tained in [7, 8]. A quasi2D flow was realized in a rect angular 40 × 30 cm2 cuvette filled with an electrically conducting liquid (a weak CuSO4 solution) to a height of 7 mm. There are electrodes on the two opposite lat eral sides of the cuvette that are used to pass a current I through the liquid. The cuvette is mounted on a set of permanent magnets with a staggered pattern of change in magnetic field polarity. Two configurations of the arrangement of magnets are used: 10 × 8 over the cuvette area for rectangular 40 × 36 mm2 magnets and 27 × 23 for circular magnets 14 mm in diameter. To remove the symmetry in the 27 × 23 configuration in the extreme rows of magnets along the electrodes, the successive alternation of poles (…SNSN…) was replaced by (…NNSNNS…) and (…SSNSSN…). When passing a current, the Ampere force acts on the liquid. At a low current, this force produces the same staggered structure of vortices in the liquid that modi fies the largescale motion along the cuvette bound aries in the 27 × 23 configuration, which leads to a nonzero mean flow vorticity (see below). A similar method was used in [31]. The particles on the liquid surface were recorded with a video camera. The experiments were carried out at electric currents I = 75, 100, 150, 200, 400 mA in the 10 × 8 configuration and at I = 500 mA (experi ments I and II) and a constant voltage on the elec trodes but with a decreasing current I (from 1 A) as the electric resistance of the electrolyte increased due to the deposition of part of the conducting component of the solution on the electrodes, such that its concentra tion decreased (experiments III and IV), in the 27 × 23 configuration. For each experiment, we made from N = 11229 to N = 22496 measurements (by the PIV method) of the horizontal velocity field components u and v along the x and y axes with time intervals from T = 4/24 s to T = 8/24 s, which corresponded to records up to two hours. Although the entire cuvette was filmed, the subsequent processing was performed only for the
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(a)
(b) Fig. 1. Averaged (a) velocity field and its perturbations (b) at an arbitrary instant of time for the 10 × 8 configuration. The current I = 75 mA.
central part of the flow in a 19.75 × 15.8 cm2 (27.5 × 22.5 cm2 for I = 400 mA) rectangle with the spatial step Δl = 0.34 cm. The velocity measurement error was about 10%. The timeaveraged velocity fields retain the sym metry specified by the magnetic field in the form of a staggered set of vortices. For each current, these aver aged fields u and v were subtracted from the velocity field for a given instant of time to obtain the nonsta tionary deviations of the velocity from the means (the corresponding 15min means were taken for the 27 × 23 configuration). Figure 1 shows an example of the averaged velocity field and the deviations (I = 75 mA), where the vector with components ( u , v ) (Fig. 1a) and (u', v'), u' = u – u and v' = v – v (Fig. 1b), was drawn at each point in the grid of measurements on identical scales for the two figures. We see from Fig. 1 that the velocity perturbations can exceed the means by almost a factor of 3. The Reynolds numbers Re for the flows can be cal culated from the scale l ≈ 4 cm of the vortices (for the 10 × 8 configuration) generated by an external field at fixed maximum velocity Vm in the averaged field. It was shown in [7] that Vm ~ I2/3 and Vm ≈ 1 cm s ⎯1 for the chosen currents I = 75, 100, 150, 200, and 400 mA. Therefore, Re ≈ 500. However, it should be noted that
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Ω2, s−2 2.4 1.5
0.6 0.5 0.4
0.8
1000
1.0
2.0 1100
1200
0.5 1000
1100
1200
1.6 0.6 1.2 0.4 0
1000
2000
3000
t, s
0.8
0
1000
2000
3000
t, s
Fig. 2. Time dependences of the energy E and entrophy Ω2 of the horizontal velocity field perturbations per unit area for I = 100 mA (the 10 × 8 configuration). The insets show parts of the graphs on an enlarged scale.
the critical Reynolds number is an order of magnitude smaller, because a laminar flow (stationary vortices in staggered order) loses its stability at I = 5 mA (see [7]). Since the changes in Re are small in this case, below we will characterize the flows by the current, as was done in [7–10]. Actually, it is difficult to reach large Reynolds numbers in laboratory experiments with MHD forcing. However, the velocity field perturba tions are distinctly chaotic even at these numbers, so that their statistical characteristics can be studied. We will mention in this connection that when turbulent boundary layers with significant speeds are investi gated, the Reynolds numbers calculated from the scales and sizes of the vortices emerging in the layer turn out to be 100–300 for laboratory flows (see [32]). From the perturbations (u', v') calculated at the grid points, we can find the energy 2 E ( t ) = 1 u dx dy S and entrophy 2 2 Ω ( t ) = 1Ω dx dy S
∫
of the perturbations per unit area (S = 19.75 × 15.8 cm2). An example of the time dependence of these quantities is shown in Fig. 2 for I = 100 mA. The insets in the fig ures show that the characteristic period (integral time scale) is Tp ≈ 15–20 s, so that the total measurement time T of the velocity field is 200–250 integral time scales. Based on the derived deviations u'(x, y, t) and v'(x, y, t) from the means, we calculated the second and thirdorder structure functions by averaging over the area of the rectangular field of measurements and
over time. For this purpose, we took the grid points (xi, yi), (xj, yj), xi = Δli and yi = Δli), that specified the vec tor rij = (xj – xi, yj – yi). We set the condition xj – xi ≥ 0 lest the vectors rij with the opposite sign appear in our calculations when averaging. Based on the specified initial point (xi, yi), we took into account only those j at which the length r ij ≤ Δl(1.0 + (im – 1) × 1.755), where im = 12 or 30. Obviously, the minimum value of r ij is equal to Δl. The projections of the vector u ij' = (u'(xj, yj) – u'(xi, yi), v'(xj, yj) – v'(xi, yi)) onto the direction of rij and onto any direction perpendicular to the vector rij give, respectively, the longitudinal and transverse components of the velocity difference vec L N tor, u ij and u ij . Summing over all i and j in such a way that the vector rij does not emerge from the field of velocity measurements and its length is approximately equal to a given r = Δl(1.0 + (k – 1) × 1.755), 1 ≤ k ≤ im, (summing also over all times of measurements 0 < t ≤ Tk in a given interval Tk) yields the functions aver aged over space and time: L
3
L
D LLL ( r ) = 〈 ( u ( r ) ) 〉 ,
2
D LL = 〈 ( u ( r ) ) 〉 ,
L
2
N
D LNN = 〈 u ( r ) ( u ( r ) ) 〉 , N
(16)
2
D NN = 〈 ( u ( r ) ) 〉 . The main questions that can be considered having experimental data for (16) are: (1) the signs of the third moments specifying the direction of the energy flux in the chosen range of scales; (2) the relations between DLLL and DLNN, DLL and DNN; and (3) the linearity of the third moments in r.
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521
D, (cm/s)3 0.01
DLLL
DLNN
−0.004
0 DLNN −0.01
−0.008
DLLL −0.02
−0.012
(b)
(a) 0
2
6
4
8
0
0.01
0.04
0
0.02
2
6
4
8
DLNN
−0.01
0
DLLL
−0.02
DLNN
−0.02
−0.03
DLLL
−0.04
(d)
(c) 0
2
4 r, cm
6
0
2
4 r, cm
6
Fig. 3. Thirdorder structure functions DLLL(r) and DLNN(r) for currents of 75 (a), 100 (b), 150 (c), and 200 (d) mA. The aver aging over the entire interval of measurements T = 3600 s (the 10 × 8 configuration).
Figure 3 shows DLLL and DLNN for the data averaged over the total measurement time T = 1 h for various currents. It follows from the figure that the structure functions for the chosen intervals r < 7 cm are mainly negative (for I = 200 mA at r > 3 cm) and increase in absolute value with r at r < 6 cm. We also used the aver aging over time intervals τ shorter than T to examine how the means 〈 〉 τ approach the means 〈 〉 T . In LLL
L
3
Fig. 4, the mean D τ (r) = 〈 ( u ( r ) ) 〉 τ are plotted against τ. Each line in Fig. 4 is the τ dependence of L
3
〈 ( u ( r ) ) 〉 τ at fixed r, 3.3 cm < r < 6.8 cm. The figures LLL
show that the mean D τ (r) can have the positive sign for an insufficient averaging time τ, which changes as LLL
τ approaches T. In the time intervals when D τ (r) > 0, the source P(r) (7) (so that P(r) – 2ε > 0) attribut
able to a nonuniform distribution of the vertical veloc ity vz over the layer thickness or to a nonzero horizon tal divergence of the velocity field makes a significant contribution for the quasi2D flow under consider ation. This is especially clearly seen for a relatively weak external force, I = 75 mA, on which the ampli tude of the horizontal velocity in the perturbations u' = u – u depends (Fig. 1): in particular, the functions DLLL (I = 75 mA) determined by them are much smaller (DLLL ~ 0.01 cm3 s ⎯3, Fig. 3a) than those for I = 200 mA (DLLL ~ 0.05–0.1 cm3 s ⎯2, Figs. 3d, 4b). For I = 150, 200, and 400 mA at an averaging time τ > 2000 s, it can be said that there is a transition to a quasiequilibrium regime in which the means change only slightly. For a strong force (I = 200 mA), the structure func tions DLLL and DLNN are negative in the range 3 cm
21 cm. Figure 6b shows the LLL mean D τ (r) as a function of the averaging interval τ for several scales 14.5 cm < r < 18.8 cm. Just as for the currents I = 150 and 200 mA (Fig. 4), the mean LLL D τ (r) take on values equal to those in the equilib rium regime at τ > 2000 s. The secondorder structure functions DLL and DNN for all external forces, I = 75, 100, 150, 200 mA (Fig. 5b), exhibit saturation on the scales of the vorti ces generated by the interaction between the current
and the set of magnets (l ≈ 4–6 cm). In this case, DLL and DNN are essentially equal, (17) D NN = D LL . This experimental result reflects the quasi2D nature of the flow in our experiment, because it differs greatly from relations (14) for the 3D and 2D cases. An anisotropy of the field of (nonsolenoidal) perturba tions manifests itself here, because the generated sys tem of vortices is almost symmetric only during rota tions through π/2. Note also that these structure func tions show a distinct linear dependence on r at low values (less than the pumping scale), DLL ≈ DNN ~ r, for all experiments. Below, we will return to this issue. The relations between the thirdorder structure functions DLLL and DLNN can be estimated using the dimension function (15) N(r). If DLLL in (15) is linear in r, then N = 1. If (13) holds, i.e., DLLL = 3DLNN, then N(r) in (15) becomes indeterminate, irrespective of the linearity of DLLL in r, i.e., the values of N(r) are arbitrary, –∞ ≤ N(r) ≤ ∞. For each r at a fixed time of measurements and for all moments, Fig. 7 (I = 200 mA) plots the quantity D LLL – 3D LNN d = (18) D LLL (where in DLLL and DLNN we did not perform the aver aging over time, as in (16), but only over the area in which the velocity field was measured) along the hori zontal axis and the quantity dD LLL D LLL – r dr N = 1 + (19) D LLL – 3D LNN along the vertical axis.
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STRUCTURE FUNCTIONS OF QUASITWODIMENSIONAL TURBULENCE DLLL, DLNN, (cm/s)3 0.08
DLL, DNN, (cm/s)2 3
(a)
(b)
1
2
0.06
I = 200 mA
0.04
2 1
2 150 mA
0.02
1
100 mA
0
8
−0.02
12
16 r, cm
523
1
75 mA 1
1
2 2 2
−0.04 −0.06
0
2
4
6 r, cm
Fig. 5. (a) Thirdorder structure functions DLLL (1) and DLNN (2) at r < 18 cm. The current is 200 mA. (b) Secondorder structure functions DLL (1) and DNN (2). The 10 × 8 configuration.
DLLL, DLNN, (cm/s)3 6
DτLLL, (cm/s)3 1 (a)
(b) 0
1
4
1000
2000
3000
4000 τ, s
−1 2
0
2
5
10
15
25
−2 30 −3 r, cm
−2 LLL
L
3
Fig. 6. (a) Thirdorder structure functions DLLL and DLNN at r < 27 cm. The current is 400 mA. (b) D τ (r) = 〈 ( u ( r ) ) 〉 τ versus averaging interval τ for various displacements 14.5 cm < r < 18.8 cm for a 400mA current. The 10 × 8 configuration.
The crowdings of points at d = 0 along the N axis show that the data set for DLLL and DLNN satisfies rela tion (13). In this case, there is also a crowding along the N = 1 line, i.e., a significant amount of data indi cates that DLLL is linear in r. The crossshaped crowd ing along the N = 1 and d = 0 lines shows that the over whelming amount of data for DLLL and DLNN corre spond to approximate linearity in r and relation (13). Obviously, the statistics (time averaging) separates out these dependences for the functions in (16), because the density of points in Fig. 7 along the N = 1 and d = 0 directions is at a maximum. Approximate linearity of DLLL in r is also seen from Fig. 5a. Note that these rela
tions are more pronounced at small r, 0 < r < 7 cm (Fig. 7a). In our experiments with the 27 × 23 configuration of circular magnets, the scale of energy input under the Ampere force is significantly smaller (about 2 cm) than that for the experiments considered above. In this case, because of the asymmetry in the arrangement of magnetic poles introduced along the electrodes, the 2D velocity field acquires a nonzero vorticity averaged over the entire cuvette area. Below, we consider the results of our experiments under the action of a 500 mA current, which are distinguished by a slight decrease in energy (per unit area) (experiment I) and a small increase in energy (experiment II), and the
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N
(a)
(b)
N=1
N=1 d
d
Fig. 7. Experimental values of N (19) at various d (18) for I = 200 mA at 0 < r < 7 cm (a) and 7 cm < r < 17 cm (b). The thin segments correspond to N = ±10 and d = ±10. The 10 × 8 configuration.
results of experiments III and IV at a constant voltage on the electrodes but with a decrease in energy due to the reduction in current and the corresponding reduc tion in the Ampere force acting on the liquid. The lower curves in Fig. 8 show the energy per unit area (as in Fig. 2) for experiments I and II at each instant of time. Note that the measurement period T was two hours in experiment I and one hour in experiment II. The energy for these experiments changes because of the asymmetric external force, which results in the growth of jet flows near the cuvette boundaries with longterm amplitude oscillations. Figure 9a shows the areaaveraged vorticity ∂v⎞ dx dy 1 ⎛ ∂u – S ⎝ ∂y ∂x ⎠
∫
E, DτLL, ( cm/s)2 0.26
E, DτLL, ( cm/s)2 0.36
(a)
DτLL(r)
of the velocity field for experiment I. We see from Fig. 9 that its value averaged over time is approxi mately equal to ⎯0.008 s ⎯1, with the vorticity disper sion being about 0.015 s ⎯1. For our experiments in the 10 × 8 configuration with a symmetric arrangement of magnets, the ratio of the timeaveraged vorticity, which is close to zero, to its dispersion was less than LLL 1/10. Figure 9b shows the mean D τ (r) as a function of the averaging interval τ < 7200 s for several scales. Just as for the currents I = 150, 200, and 400 mA LLL (Figs. 4 and 6b), the mean D τ (r) take on values equal to those in the equilibrium regime at τ > 2000 s. The upper lines in Fig. 8 indicate the dependences of the secondorder longitudinal structure functions LL D τ (r) on averaging time τ. According to (8) at n = 2 and the approximate equality (17), which also holds in experiments I and II, they are equal to the energy of the velocity difference at the points separated by dis tance r. On average, the behavior of the curves corre sponds to the change in mean energy per unit area with time (the lower curves in Fig. 8). Figures 10 and 11 for experiments I and II show the thirdorder structure functions DLLL(r) and 3DLNN(r) at r < 12 cm for various averaging intervals. Outside the interval of energy input under the Ampere force r ≈ 2–3 cm, the struc ture functions are negative and, in addition, DLLL(r) ≈ 3DLNN(r), as follows from the comparison of Figs. 10 and 11. This corresponds to Eq. (13). The negative values of the structure functions out side the interval of energy input suggest that the main term on the righthand side of Eq. (6) is the term with the negative sign proportional to the 3D energy dissi pation. It suppresses the possible energy input P(r)
0.22
0.32
0.18
0.28
0.14
0.10
0
DτLL(r)
0.24
E(τ)
2000
4000
6000
τ, s
0.20
(b)
E(τ)
0
1000
2000
3000 τ, s
Fig. 8. The lower curves indicate the time dependences of the energy E(t) of the horizontal velocity field per unit area for I = 500 mA (the 27 × 23 configuration) for experiments I (a) and II (b). The upper curves indicate the dependence of the second LL
L
2
order structure functions D τ (r) = 〈 ( u ( r ) ) 〉 τ on averaging interval τ for various displacements 12.6 cm < r < 16.3 cm for a 500mA current (the 27 × 23 configuration) for experiments I (a) and II (b). JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
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STRUCTURE FUNCTIONS OF QUASITWODIMENSIONAL TURBULENCE Ω, s−1 0.08
DτLLL, (cm/s)3 0.010
(a)
525
(b)
0.06 0.005
0.04 0.02
2000
4000
6000
0
0 −0.02
τ, s
−0.005
−0.04 −0.06 −0.08
−0.010 0
2000
6000 τ, s
4000
Fig. 9. (a) Time dependences of the vorticity (per unit area) of the horizontal velocity field for I = 500 mA (experiment I, the LLL
3
L
27 × 23 configuration). (b) D τ (r) = 〈 ( u ( r ) ) 〉 τ versus averaging interval τ for various displacements 12.6 cm < r < 16.3 cm for experiment I (a 500mA current, the 27 × 23 configuration).
DLLL, (cm/s)3 0.0010
DLLL, (cm/s)3 0.001
(a)
0
4
8
r, cm 12
1 2
r, cm 12
8
−0.001 −0.002
−0.0005 −0.0010
4
0
0.0005
(b)
6
−0.003
4 5
−0.004
1 2
3
3
4
−0.005
Fig. 10. Thirdorder structure functions DLLL(r) at r < 12 cm for various averaging intervals: τ = 2700 (1), 3600 (2), 4500 (3), 5400 (4), 6300 (5), 7200 (6) s for experiment I (a) and τ = 900 (1), 1800 (2), 2700 (3), 3600 (4) s for experiment II (b).
(11) through nonzero 2D divergence of the velocity field and spatial inhomogeneity of the mean velocity field. The quantity εp = ε – P(r)/2 (reduced dissipa tion) can be roughly estimated for experiments I and II using Kolmogorov’s relation D LLL ( r ) ~ εr. From the data in Fig. 10a and 10b, it gives εpI ~ 0.001/10 = 10 ⎯4 cm2 s ⎯ 3 and εpII ~ 0.005/10 = 5 × 10–4 cm2 s ⎯ 3, respectively. The derivative of 〈 E〉 ≈ DLL(r, t) on the righthand side of Eq. (6) can be estimated from the data in Fig. 8: we have –6 – 0.23 δD LL ≈ 0.25 ≈ 3 × 10 7200 δt
for experiment I (Fig. 8a, see the straight line) and –6 – 0.29 δD LL ≈ 0.30 ≈ 3 × 10 3600 δt
for experiment II (Fig. 8b). For these two cases, the values of the reduced dissipation exceed the time derivative of the energy by more than an order of mag nitude. Based on our experimental data, we can also esti mate the contributions of Pv(r) and Pu(r) from (11) to the energy production function P(r) from (7). These quantities are shown in the upper part of Fig. 12a for experiment IV when averaged over a time interval τ < 3600 s. It follows from Fig. 12a that Pv(r) and Pu(r)
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3DLNN, (cm/s)3
0.002
(a)
0.0010
(b)
0.001 4
0
0.0005 0
r, cm −0.001 12 −0.002
8
4
5
3 2 1
−0.004
3 2
−0.0010
4
6 −0.003
4
−0.0005
r, cm 12
8
1
−0.005
Fig. 11. Thirdorder structure functions 3DLNN(r) at r < 12 cm for various averaging intervals: τ = 2700 (1), 3600 (2), 4500 (3), 5400 (4), 6300 (5), 7200 (6) s for experiment I (a) and τ = 900 (1), 1800 (2), 2700 (3), 3600 (4) s for experiment II (b).
Pυ , Pu, cm2/s3 0.004 Pυ 0.002 0 −0.002
DτLL, E, (cm/s)2 Pu 10
20 r, cm
DLLL, 3DLNN, (cm/s)3 0.0025 5 10 15 r, cm 0 −0.0025 III −0.0050 −0.0075 IV −0.0100 −0.0125 −0.0150 (a) −0.0175 −0.0200 −0.0225
0.7 IV
0.6 0.5 0.4
DτLL III
0.3 E
0.2
(b)
0.1 0
1000
2000
3000
4000 τ, s
Fig. 12. (a) Thirdorder structure functions DLLL(r) (thick curves) and 3DLNN(r) (thin curves) for experiments III and IV and functions Pv(r) and Pu(r) (7) for experiment IV when averaged over the entire interval of measurements. (b) Secondorder struc LL
L
2
ture functions D τ (r) = 〈 ( u ( r ) ) 〉 τ versus averaging interval τ for various displacements 12.6 cm < r < 16.3 cm and time depen dences of the energy E(t) of the horizontal velocity field per unit area for experiments III and IV.
correspond in amplitude 0.001–0.002 to the above estimates for εp and the estimate for this experiment that follows from the lower part of Fig. 12a. Note that there is no input of external energy Π(r) for scales r > 2–3 cm. However, the energy input Pd(r) in (11) and (7) is nonzero, since, when averaged over the area of the rectangle in which the horizontal veloc ity was determined, the terms like divx make a nonzero contribution, because the conditions for the velocity becoming zero at the boundaries of this rectangle are violated.
The above relations between the reduced dissipa tion εp and ∂ 〈 E〉 /∂t, εp Ⰷ ∂ 〈 E〉 /∂t, also hold for experiments III and IV (at a constant voltage on the electrodes). Depending on the initial concentration of the solution, we obtain different (in amplitude) struc ture functions DLLL(r) ≈ 3DLNN(r) (Fig. 12a) and dif LLL
ferent (in magnitude) energies E(t) and mean D τ (r) as a function of the averaging interval τ < 2700 s (for III) and τ < 3600 s (for IV) and for several scales 12.6 cm < r < 16.3 cm (Fig. 12b).
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4. CONCLUSIONS Here, we considered the statistical characteristics of turbulent motions in a quasi2D geometry on scales that did not differ greatly from the energycontaining flow scales specified by external forces. Therein lies the difference from a number of experimental studies of similar flows [14–17, 20] where the main scales of motions exceeded greatly the sizes of externally gener ated vortices. In the latter case, the energy flux in the system is realized in the direction of enlargement of the emerging structures with dissipation described by linear (Rayleigh) friction due to bottom effects (see [1, 2]). The viscous 3D energy dissipation in the experi mental works [14–17] is suppressed, because the motion in a thin fluid layer takes place above a layer of heavier nonconducting fluid, so that largescale circu lation motions are produced in the system by the inverse cascade mechanism. A similar effect manifests itself both for 2D flows, which was shown by numeri cal simulations in [33] and explained theoretically in [34], and for 3D flows in thin layers, where direct energy transfer to the vertical smallscale components of motion exists, along with the inverse cascade [35]. In this case, the thirdorder structure function DLLL for the experiments of [14–17] is definitely positive, which reflects the 2D flow structure. This is also dem onstrated by Fig. 6a, where DLLL is also positive for large scales, suggesting the obvious 2D dynamics for such scales. It should also be noted that DLLL and DLNN at I = 200 and 400 mA are also positive at small r: r < 3 cm for I = 200 mA (Fig. 5a), r < 10 cm (Fig. 6a) for I = 400 mA (in the 10 × 8 configuration), and r < 3 cm for experiments I, II, III, IV (in the 27 × 23 configura tion)—Figs. 10, 11, 12a. In fact, this implies that there is energy input on small scales r ~ 1 cm as well, which is then redistributed in the direction of enlargement of the scales by the 2D cascade mechanism. On the one hand, this energy input to the horizontal compo nent of motion is attributable to the appearance of smallscale (of the order of the layer thickness) 3D vortices in the layer due to the adhesion condition at the bottom. It is the horizontal components of these vortices that form smallscale motions on the fluid surface. On the other hand, the energy pumping by the Ampere force Π(r) occurs precisely on these scales and its value can exceed the remaining terms in the balance equation (6). For the case considered above, there are vertical sources and sinks of mass from the surface to the bot tom of the fluid layer in the scales of the vortices gen erated by the interaction between the current and an external magnetic field. This disrupts the 2D flow pat tern, so that P(r) ≠ 0 in Eq. (6). In contrast, the 3D effects leading to energy dissipation on smallscale perturbations return the system to 3D dynamics with the standard negative sign of DLLL for the 3D case. This manifests itself in Figs. 3–6 and 10–12, especially
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under a relatively strong external force (I = 200, 400 mA—Figs. 5, 6; I = 500 mA—Figs. 10, 11; exper iments III, IV—Fig. 12). In fact, this sign is deter mined by the sign of the righthand side of the balance equation (6), in which the terms –2ε, Π(r), and ⎯∂ 〈 E〉 /∂t make a major contribution on the scales r where no external forces act (so that P(r) = 0). As was shown above, the last of the above terms is much smaller than the first. Interestingly, DLLL is negative in the case of soap film turbulence in the experiments [20], where DLLL is negative for scales of the order of the separation between the teeth of the comb introduced into a flow to generate turbulence. In this case, the layer thick ness, just as in our case, is much smaller than the length scale of externally generated motions. In the other case opposite in flow scale, the negativity of the third moments (DLLL + DLNN) manifested itself for geophysical processes, namely for the horizontal velocity components in the range of scales 10–100 km in the upper troposphere and lower stratosphere obtained from ozone concentration measurements (see [36]). Another aspect of the experimentally obtained results is related to approximate linearity of the func tion DLLL in r and the fulfillment of relation (13) for DLLL and DLNN. The crowding of points along the N = 1 and d = 0 lines in Fig. 7 and the data presented in Figs. 5 and 10–12 show approximate linearity in r and satisfactory statistics of the fulfillment of (13). In this case, the DLLL oscillations near the straight line shown in Fig. 5a well corresponds to the behavior of the func tion S3 (but with the opposite sign) from [14] for a flow in which there was no external largescale circulation motion, which corresponds to the experimental con ditions of this work. Finally, let us return to the linear r dependence of the secondorder structure functions DLL and DNN at r smaller than the separation between the magnet cen ters (the external pumping scale). For the same values of r, the thirdorder structure functions DLLL and DLNN are positive for the currents I = 200, 400 mA (in the 10 × 8 configuration; Figs. 3d, 5a, 6a) and all experi ments I–IV (in the 27 × 23 configuration; Figs. 10, 11, 12a) and are close to zero for weak external forces at I = 75, 100, 150 mA (Figs. 3a–3c). This indicates that for the range of scales r under consideration, the term P(r) attributable to the 3D effects and, in part, the external energy input Π(r) make a significant contri bution in the balance equation (6), so that the right hand side of (6) becomes positive. As has been noted above, the contribution of P(r) to Eq. (6) is related to the appearance of smallscale (of the order of or smaller than the layer thickness) 3D vortices (like horseshoeshaped, hairpinshaped ones or streaks in the terminology of a turbulent boundary layer [32]), which contribute to the generation of the horizontal velocity field components. The appearance
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of these vortices is random in time and over the area of the fluid layer. The temporal statistics of the appear ance of such horizontal motion components may be considered to be δcorrelated, so that δcorrelated forces act on the fluid particles at the surface in their smallscale motion, which provide the energy input to them. It is well known (see [27, 37, 38]) that the sec D = ondorder structure functions 2 〈 ( v ( t + τ ) – v ( t ) ) 〉 for the Lagrangian velocity of fluid particles at times t + τ and t have the scaling D = εinτ, where εin is the energy input to the particles and τ is the time interval separating the particles. The Lagrangian particles separated in time by the interval τ, on average, are separated in space by the distance 2 1/2
2 1/2
r = 〈 u 〉 τ , where 〈 u 〉 is the rootmeansquare velocity of the smallscale motions associated with the above 3D structures. Therefore, for the Eulerian sec ondorder structure functions, we obtain 2 – 1/2
D LL ≈ D NN = ε in 〈 u 〉 r, (20) i.e., the dependence linear in r. Note that this transi tion from the Lagrangian characteristics of turbulence to the Eulerian ones is analogous to using Taylor’s freezing hypothesis and the formulas with the root meansquare velocity were used in considering the functions and spectra for a flow between rotating disks (a Karman flow), for example, when passing from the frequency spectra of the velocity field to the spatial ones. The energy input εin through δcorrelated forces is most uncertain in Eq. (20). It should take into account the specificity of the smallscale 3D motions in a vis cous boundary layer above a solid surface. If the mean horizontal velocity in the layer is ~1 cm s ⎯1, then the energy dissipation per unit mass in a viscous boundary layer about 1 mm in thickness is –1 2 2 –3 ∂u 2 ν ⎛ ⎞ ∼ 0.01 ( 1/10 ) ∼ 1 cm s . ⎝ ∂z ⎠
This is not the mean dissipation but precisely the dis sipation that takes place in the above structures in a viscous boundary layer. For such structures to exist, the energy input εin must have the same order of mag 2 1/2
nitude. At 〈 u 〉 ~ 1 cm s ⎯1, the above formula then gives a reasonable estimate for the slope of the curves in Fig. 5b when r 0. Here, it should be noted that the asymptotics (~r) considered for the secondorder structure functions is clearly not of Kolmogorov’s type and includes two dimensional parameters: εin with the 2 – 1/2
dimensions of dissipation and 〈 u 〉 , the root meansquare rate of the velocity pulsations. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project nos. 100500457 and
110501206) and the “Fundamental Problems of Nonlinear Dynamics” Program of the Presidium of the Russian Academy of Sciences. REFERENCES 1. E. B. Gledzer, F. V. Dolzhanskii, and A. M. Obukhov, Systems of the Hydrodynamic Type and Their Applica tions (Nauka, Moscow, 1981) [in Russian]. 2. F. V. Dolzhanskii, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 23, 348 (1987). 3. F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, Sov. Phys.—Usp. 33 (7), 495 (1990). 4. F. V. Dolzhanskii, V. A. Krymov, and D. Y. Manin, J. Fluid Mech. 241, 705 (1992). 5. S. D. Danilov and D. Gurarie, Phys.—Usp. 43 (9), 863 (2000). 6. F. V. Dolzhanskii and D. Yu. Manin, Dokl. Akad. Nauk SSSR 322, 1065 (1992). 7. V. M. Ponomarev, A. A. Khapaev, and I. G. Yakushkin, Izv. Atmos. Oceanic Phys. 44 (1), 45 (2008). 8. V. M. Ponomarev, A. A. Khapaev, and I. G. Yakushkin, Dokl. Earth Sci. 425A (3), 510 (2009). 9. N. F. Bondarenko, M. Z. Gak, and F. V. Dolzhanskii, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 15, 101 (1979). 10. S. D. Danilov, F. V. Dolzhansky, V. A. Dovzhenko, and V. A. Krymov, Chaos 6, 297 (1994). 11. S. D. Danilov, V. A. Dovzhenko, F. V. Dolzhanskii, and V. G. Kochina, JETP 95 (1), 48 (2002). 12. A. R. Cieslik, L. P. J. Kamp, H. J. H. Clercx, and G. J. F. van Heijst, J. HydroEnviron. Res. 4, 89 (2010). 13. R. A. D. Akkermans, A. R. Cieslik, L. P. J. Kamp, R. R. Trieling, H. J. H. Clercx, and G. J. F. van Heijst, Phys. Fluids 20, 116601 (2008). 14. M. G. Shats, H. Xia, H. Punzman, and G. Falkovich, Phys. Rev. Lett. 99, 164502 (2007). 15. H. Xia, H. Punzman, G. Falkovich, and M. G. Shats, Phys. Rev. Lett. 101, 194504 (2008). 16. H. Xia, M. G. Shats, and H. Punzman, in Advances in Turbulence: XII, Ed. by B. Eckhardt (Springer, Berlin, 2009), p. 709. 17. H. Xia, M. Shats, and G. Falkovich, Phys. Fluids 21, 125101 (2009). 18. J. Sommeria, J. Fluid Mech. 170, 139 (1986). 19. H. Kellay and W. I. Goldburg, Rep. Prog. Phys. 65, 845 (2002). 20. A. Belmonte and W. I. Goldburg, Phys. Fluids 11, 1196 (1999). 21. A. E. Gledzer, Izv. Akad. Nauk, Fiz. Atmos. Okeana 39, 466 (2003). 22. S. V. Kostrykin, A. A. Khapaev, and I. G. Yakushkin, JETP 112 (2), 344 (2011). 23. D. K. Lilly, J. Atmos. Sci. 40, 2026 (1989). 24. Yu. I. Portnyagin, Measurements of the Wind at Altitudes of 90–100 km by Terrestrial Methods (Gidrometeoizdat, Leningrad, 1978) [in Russian].
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