1 Introduction. Measurement of Velocity Field in Thermal Turbulence in Mercury by Ultrasonic Doppler Method

Transcription

1 Measurement of Velocity Field in Thermal Turbulence in Mercury by Ultrasonic Doppler Method Takashi Mashiko, Yoshiyuki Tsuji, Takatoshi Mizuno, and Masaki Sano Graduate School of Science, University of Tokyo, Tokyo Graduate School of Engineering, Nagoya University, Nagoya With the use of the ultrasonic velocimetry, we measured the velocity profiles in thermal turbulence in mercury. The measurement, which is different from the traditional measurement of time series of the local velocity, has brought about some intriguing results. We observed the inversion of the flow direction near the boundary plate, which seems the appearance of the velocity boundary layer. We also elucidated the nature of the fluctuation in thermal turbulence through the principal component analysis. The macroscopic flow structure and its vertical movement were observed. In addition, the energy spectrum and the velocity structure functions were calculated without employing Taylor s frozen-flow hypothesis for the first time in thermal turbulence, which appeared to be consistent with the Bolgiano-Obukhov theory. 1 Introduction Thermal turbulence has long been studied as a typical example of complex physical systems. Nevertheless, many problems have as yet remained open and have challenged researchers to further works both theoretically and experimentally. Among them, scaling laws for the heat transport, 1, 2) the behavior of the large-scale circulation, 3 8) and the nature of the boundary layers (BLs) 8 14) can be given as important examples. These are related to the problem of the existence of an ultimate state that may appear when the Rayleigh number (Ra) becomes very high ) For another example, the scaling behavior of the fluctuations in thermal turbulence has been studied actively too. Theoretically, the energy spectrum of the form E(k) k 11/5 was predicted in the Bolgiano-Obukhov theory 22, 23) for thermal turbulence, unlike the Kolmogorov 1941 (K41) spectrum E(k) k 5/3 for isotropic turbulence. For the above problems, it is expected to be a very effective approach to grasp the velocity field precisely. However, there have been few works which measured the velocity field with sufficient accuracy both in space and time, as it has been difficult to measure the velocity at a number of points simultaneously. Thus alternative methods have been utilized. For example, measurements of the velocity at one point after another have been used instead of simultaneous multipoint measurements. But in those cases, obtained velocity profiles are, of course, not instantaneous ones, and cannot be used for further analyses such as the derivation of the spatial characteristics of the fluctuation. And the frequency power spectrum P(f ) instead of E(k) has been calculated, then regarded as equivalent with E(k), employing Taylor s frozen-flow hypothesis. However, Taylor s hypothesis is not always adequate in thermal turbulence. Therefore, instantaneous measurement of the velocity field is highly desired to investigate the above-mentioned fundamental questions in thermal turbulence. In this paper, which is supplementary to our previous report, 24) we report the instantaneous measurement of the velocity profiles in thermal turbulence by using the ultrasonic velocimetry (USV). As there have been no similar studies so far, the present measurement can be referred to as a new trial in thermal turbulence. Another characteristic of the present work is the use of liquid mercury as the working fluid. As mercury has low Prandtl number (Pr at 20 C), it is expected that highly turbulent states with high Reynolds number 1

2 (Re) are easily realized, in which the velocity BL is thinner than the temperature BL. 10, 12, 13, 17) Therefore liquid mercury is a suitable fluid to investigate highly turbulent states. In the present work, we attained the highest Ra (> ) so far realized for low Pr fluids, and high Re (> 10 5 ), where Re is defined by Re = v L/ν, with v being the mean velocity and L the cell size. 2 Experiment The principle of the USV (or often referred to as the ultrasonic Doppler method) is as follows. 25) A transducer emits ultrasonic pulses (bursts) of several wavelengths with a pulse-repetition period of T pr, and receives echoes scattered by tracers in the fluid. The echo signal for the nth burst is well approximated by s n (t) A n (t) cos[2πf 0 t + φ n (t)], where f 0 is the pulse frequency, t is the elapsed time measured after the burst emission, and φ n (t) is the phase shift which can be calculated by complex demodulation. Since the Dopplershift frequency of the signal s n (t)isgivenby f = φ n (t)/2πt pr = (φ n+1 (t) φ n (t))/2πt pr, the velocity component is calculated by v n (t) = c s φ n (t)/4πf 0 T pr, where c s is the sonic speed in the fluid. Then v n (t) is converted to the spatial profile v n (x), using the relation t = 2x/c s. Here we consider the effect of the temperature field T(x, t) on the USV measurement. The most noticeable effect is that c s varies as a function of T. In our present work, we fixed c s at a constant c s,mean which is the value in mercury at the mean temperature T mean of the top and bottom plates (T top and T bot ). It is reported that in mercury thermal turbulence with Ra = , the velocity BL is thinner than the temperature BL, and both get thinner as Ra increases and become thinner than 4 mm at Ra = , 13) In the present experiment with much higher Ra ( ), the BLs must be much thinner, and almost all of the region in the cell is turbulent with a resultant time-averaged temperature, T = T mean, which seems to justify setting c s at c s,mean. But considering that T may deviate from T mean in each sampling (which is a result of the short sampling time of the present USV measurement, s) and that even T is different from T mean in the nearest regions to the plates, we now estimate the temporal or local error of the calculated velocity v. Considering the case will be the most critical test, in which T becomes T top or T bot. T top and T bot are respectively the lowest and highest possible temperature of mercury in the cell, and correspondingly c s takes the highest and lowest value. Even in such a case, however, the deviation of c s from c s,mean is the order of 1%, and the error of v is of the same magnitude, because v is proportional to c s as mentioned above. We must notice also the possibility that the deviation of c s may cause an error in the conversion of v(t) to v(x) through t = 2x/c s. Such an error is to appear as a variance of the scale x of the velocity profile v(x). But in the present measurement, we found that the measured velocity profile v(z, t) nicely takes the value of zero at z = 306 mm which corresponds to the surface of the bottom plate and is the farthest point from the USV transducer (see Fig. 1 and 3). Hence, we conclude that the effect of the fluctuation of the temperature field on the present USV measurement is negligible. In Fig. 1, we show the apparatus in which we filled the mercury. It is a cylindrical cell with chromium plated copper top and bottom plates of d = 306 mm in diameter and stainless steel sidewall of L = 612 mm in height. The aspect ratio Γ d/l, is thus 0.5. The top plate is cooled by water via 210 vertical pipes. The temperature of the cooling water was regulated by a proportional-integral-differential (PID) controller. The bottom plate was heated by four resistance coils driven by four direct current power supplies. The cell is surrounded by a jacket and the heat loss is suppressed to less than 0.1%. In the top plate, the USV transducer was embedded along the center axis (which hereafter we call z axis, with its origin set at the cell center). Under the control of the ultrasonic velocimeter (UVP-X-1, MET-FLOW), the transducer emits ultrasonic pulses of f 0 = 4 MHz downward, and then receives the echoes. This enables us to obtain the velocity at 128 points, which are located every z mm along z axis simultaneously. The resolution of the measurable velocity is mm/s, and the spatial resolution z can be varied from 0.72 ( 2 wavelength) to 5.07 mm, while the transverse resolution (the width of the ultrasonic beam) is 5 mm. As previously explained, the USV method requires the existence of a tracer. In our measurement, we mixed a powder of gold-palladium alloy with the mercury as the tracer. The typical size of the powder was µm, and its density was adjusted to coincide with that of mercury, g/cm 3,at45 C which is the mean temperature of the two plates at the highest Ra attainable in our measurement. 1 g of the powder per 1 liter of the fluid was mixed. This concentration was enough to enable us to measure the velocity profile all over the cell along the z axis. At the same time, it was less than 10 4 in weight ratio, and hence the original 2

3 cooling water inlet outlet heat insulating jacket z 306 ultrasound transducer copper stainless steel L = 612 mm 0 heater d = 306 mm heat insulator -306 Figure 1: The cell in which liquid mercury is filled. The USV transducer is embedded in the top copper plate. purity of the mercury (99.99%) was not impaired, which allowed us to ignore the changes of the physical properties of the fluid. 3 Velocity Profiles At first, we show in Fig. 2 a typical spatiotemporal plot of the velocity field. This plot was obtained at Ra = , and Re = The horizontal axis denotes the time which spans a period of 134 s consisting of 1024 successive records, while the vertical one is the z axis which spans over 612 mm. The light and dark regions correspond to the upward and downward flows, respectively. We notice that the upward flow is dominant in the upper half while the downward flow is dominant in the lower half of the cell. z t mm/s Figure 2: Spatiotemporal plot of the velocity field at Ra = The light and dark regions correspond to the upward and downward flows respectively. 3

4 In Fig. 3, we show the time-averaged profile of the velocity, v(z), which was calculated from samples. This profile again suggests the above-mentioned tendency; upward flow in the upper half and vice versa. Looking at the shape of v(z), we can imagine the form of the macroscopic flow which is often referred to as the mean flow. In the top-left inset of Fig. 3, we show two simplest examples of the imaginable mean flow. A similar form to (a) was proposed by Qiu and Tong 6) in an experiment in a Γ=0.5 cell filled with water using laser Doppler velocimetry (LDV) at Ra = However, in our measurement we observed the inversion of the flow direction near the top plate in addition, which was not reported by them. This phenomenon is shown in the bottom-right inset of Fig. 3, in which the inversion can be seen at the position 13 mm below the top plate (indicated by an arrow). This local profile in the inset was obtained by the measurement with the finest spatial resolution of z = 0.72 mm, and is identical with the solid line superposed on the whole profile in the main figure which was obtained with z = 5.07 mm. This inversion indicates the existence of two kinds of flows; the macroscopic upward flow and the tiny downward flow emitted from the cold top plate. We presume that the position of the inversion is directly related to the thickness λ v of the velocity BL. The behavior of λ v, together with that of λ T (the thickness of the temperature BL) is one of the most important information for determining how is the ultimate state of thermal turbulence is. 9, 12, 13, 20) It was found that the thickness of the layer in which the tiny downward flow is observed decreases as Ra increases. We are going to elucidate the behavior of this layer precisely in further measurements (a) z (b) z <v> [mm/s] z [mm] Figure 3: Time-averaged velocity profile v(z) obtained with z = 5.07 mm. Topleft inset: Simple examples of the macroscopic flow structure imaginable from v(z). Bottom-right inset: v(z) near the top plate obtained with z = 0.72 mm, which is also represented by a solid line in the main figure. 4 Principal Component Analysis We showed the mean profile v(z) and examples of imaginable mean flow structures in Fig. 3. But looking at Fig. 2 carefully, we notice that the velocity profile is never stationary but highly fluctuating. To investigate the behavior of the velocity field in detail, we performed the principal component analysis (PCA) which is often referred to as the Karhunen-Loève transformation or still other names. In the process of PCA, at first the covariance matrix C V t V/N is constructed, where each column of N M matrix V is a series of the fluctuating part of the velocity profiles, v(z m, t n ) v(z m ) (m = 1, 2,...,M; n = 1, 2,...,N). Next the eigenvalues of C, µ i (i = 1, 2,...,M), are calculated so that they satisfy µ 1 µ 2... µ M. Then, the eigenvector ξ 1 is the most contributing mode to the fluctuation, and ξ 2 comes the next, and so on, where ξ i is the corresponding eigenvector of µ i. And under normalization, the eigenvalue µ i also represents the contribution of the mode ξ i to the total fluctuation. In summary, series of the velocity profiles are decomposed as v(z m, t n ) v(z m ) = M i=1 a i(t n )ξ i (z m ). 4

5 In Fig. 4 we show the first three principal components, ξ 1 (z), ξ 2 (z), and ξ 3 (z), obtained through PCA with M = 128 and N = for the velocity field at Ra = ξ 1 (z) has a single-peaked shape, which represents the sloshing motion of the whole flow. ξ 2 (z) has a shape similar to the mean profile v(z), which represents the motion consisting of two opposite flows; upward flow in the upper half and downward flow in the down half of the cell, or vice versa. Roughly speaking, we found that the ith eigenmode ξ i has i peaks, and the larger i becomes, the smaller its spatial scale is. The contribution of ξ 1 (z), ξ 2 (z), and ξ 3 (z) are 0.36, 0.18, and 0.10, respectively. Thus about 2/3 of the total fluctuation is explained by these three principal modes. Especially, the broadest dynamics is to be described by ξ 1 (z) together with v(z). The eddies, which we imagined from v(z) in Fig. 3, then should show a sloshing motion. Such a picture appeared to be consistent with the double-peaked velocity probability-distribution-functions PDF(v) and the conditional average of the velocity profile v(z) ± v(z) v(0) = v ± peak.24) 0.15 x 1 (z), x 2 (z), x 3 (z) [a.u.] z [mm] Figure 4: First three principal components. The solid, dashed, and dotted lines correspond to ξ 1, ξ 2, and ξ 3, respectively. We can reconstruct the velocity field by selecting appropriate modes. Of course, the original profile recovers when all the 128 eigenmodes are used for reconstruction. In Fig. 5, we show the reconstructed fields corresponding to Fig. 2. Figure 5 (top) shows v + 2 i=1 a i(t)ξ i (z), which represents the large-scale dynamics. We can see some slowly-fluctuating nature without clear periodicity. Figure 5 (bottom), on the other hand, shows 30 i=3 a i (t)ξ i (z). ξ i s with i = 3,...,30 were found to be in the scaling range shown in Fig. 6, hence this plot visualizes the cascade dynamics in the inertial range. Although we notice that the mean profile v(z) shown in Fig. 3 is somewhat asymmetric, after the subtraction of the mean profile we find here that the dynamics in the inertial range is symmetric as well as the main eigenmodes shown in Fig. 4. Therefore, we presume that in thermal turbulence there exists a universal cascading range with symmetry in spite of the large-scale dynamics which may be asymmetric. 5 Energy Spectrum From velocity profiles v(z), we can calculate the energy spectrum E(k). In Fig. 6 we show E(k) calculated from v(z) s obtained in measurements with three different three z s; z = 3.62, 2.17, and 0.72 mm. In each measurement, the center of the range is set at z = 0 mm, and profiles were sampled. In Fig. 6, data from three z s overlap each other, and the transverse resolution is indicated by an arrow. To be sure that the mean profile v(z) should not affect E(k), we calculated E(k) after the subtraction of v(z) from v(z) (circles). To be more careful, we also calculated E(k) after the slow dynamics due to the first two eigenmodes, ξ 1 and ξ 2, were subtracted in addition to v(z) (crosses). For both plots, there exists a scaling range in which the power law E(k) k β holds. By fitting, we found β = 2.15 ± 0.02 and 2.22 ± 0.02 for the circles and the crosses, 22, 23) respectively. In Fig. 6, the scaling law with the exponent of 11/5, derived in Bolgiano-Obukhov theory 5

6 Figure 5: Reconstructed velocity fields. Large-scale slow dynamics reconstructed from v(z), ξ 1, and ξ 2 (top), and energy-cascading dynamics reconstructed from ξ 3,...,ξ 30 (bottom). for thermal turbulence, is also represented by a solid line. Both values mentioned above are close to 11/5, and distinct from 5/3 for the isotropic turbulence. From instantaneous velocity profiles, we can also calculate the structure functions S n (r) defined by S n (r) δv(r) n v(x + r) v(x) n. We found a range in which scaling relations S n (r) r ζ n hold, and ζ 2 = 1.18, ζ 3 = 1.58, ζ 4 = 2.03, ζ 5 = 2.29, and ζ 6 = ) Being equivalent with E(k), S 2 (r) r 6/5 is expected theoretically. Now we again obtained a result which is consistent with the theory E(k) [a.u.] µ k -11/5 P(f) [a.u.] µ f -5/3 µ f -11/ f [Hz] k [1/mm] Figure 6: Energy spectrum E(k) calculated from v(z) after subtracting the mean profile (circles) and the first two eigenmodes in addition (crosses). By fitting, β = 2.15 ± 0.02 and 2.22 ± 0.02 were obtained for E(k) k β, respectively. The arrow indicates the lateral resolution (5 mm). Inset: Frequency power spectra P(f ) s at z = 0, 50,...,250 mm (from bottom to top). As mentioned in the introduction, the difficulty of the multipoint measurement of the velocity obliged researchers to calculate the frequency power spectrum P(f ) instead of the energy spectrum E(k). To regard P(f ) as equivalent with E(k), Taylor s frozen-flow hypothesis needs to be employed, which requires that the magnitude of the velocity fluctuation v rms be much smaller than that of the mean velocity v. However, this condition is not generally satisfied in thermal turbulence, then Taylor s hypothesis is not valid. In fact, we calculated P(f ) in our measurement as well as E(k), and found that the Taylor s hypothesis is questionable even when Ra is high enough for the mean flow to exist stably. In the inset of Fig. 6, P(f ) s at positions located 6

7 every 50 mm along z axis, z = 0, 50,...,250 mm are shown. Power laws with exponents of 11/5 (the dashed line) and 5/3 (the dotted line) are also shown for comparison. The slope of P(f ) was found to depend on the position of the measurement. At z = 0 mm, the slope is 2.14 ± 0.03 which is close to 11/5. But at the cell center P(f ) has no relation to E(k) since the mean velocity v(0) is nearly zero. At z = 250 mm, around which v takes the maximum but v rms / v remains near 1, the slope is 1.72 ± Therefore, we conclude that P(f ) is different from E(k) in thermal turbulence. 6 Conclusion We adopted the USV to measure the velocity field in thermal turbulence for the first time and obtained some new results. The mean velocity profile and the results from PCA allowed us to imagine the dynamics of the large-scale circulation. The inversion of the flow direction near the boundary plate was found, which encourages us to investigate its behavior further toward the problem of the ultimate state of thermal turbulence. We observed the dynamics of the scale-separated velocity field in spatiotemporal plots, where we expected the existence of the universal energy-cascading dynamics in spite of the behavior of large-scale motions. The energy spectrum, and equivalently the structure functions, were directly calculated from the measured instantaneous velocity profiles, without the use of Taylor s hypothesis, which appeared to be consistent with the Bolgiano-Obukhov theory. We thank Y. Takeda for helpful discussions and valuable comments. This work was supported by a Japanese Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (No ). References 1) Castaing, B. et al., Scaling of hard thermal turbulence in Rayleigh-Bénard convection, J. Fluid Mech. 204 (1989), pp ) Siggia, E. D., High Rayleigh number convection, Annu. Rev. Fluid Mech. 26 (1994), pp ) Sano, M., Wu, X. Z., and Libchaber, A., Turbulence in helium-gas free convection, Phys. Rev. A 40 (1989), pp ) Villermaux, E., Memory-induced low frequency oscillations in closed convection boxes, Phys. Rev. Lett. 75 (1995), pp ) Cioni, S., Ciliberto, S., and Sommeria, J., Strongly turbulent Rayleigh-Bénard convection in mercury: Comparison with results at moderate Prandtl number, J. Fluid Mech. 335 (1997), pp ) Qiu, X.-L. and Tong, P., Large-scale velocity structures in turbulent thermal convection, Phys. Rev. E 64 (2001), pp ) Sreenivasan, K. R., Bershadskii, A., and Niemela, J. J., Mean wind and its reversal in thermal convection, Phys. Rev. E 65 (2002), pp ) Verzicco, R. and Camussi, R., Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell, J. Fluid Mech. 477 (2003), pp ) Belmonte, A., Tilgner, A., and Libchaber, A., Boundary layer length scales in thermal turbulence, Phys. Rev. Lett. 70 (1993), pp ) Takeshita, T., Segawa, T., Glazier, J. A., and Sano, M., Thermal turbulence in mercury, Phys. Rev. Lett. 76 (1996), pp ) Xin, Y.-B., Xia, K.-Q., and Tong, P., Measured velocity boundary layers in turbulent convection, Phys. Rev. Lett. 77 (1996), pp

8 12) Naert, A. and Segawa, T., and Sano, M., High-Reynolds-number thermal turbulence in mercury, Phys. Rev. E 56 (1997), pp ) Segawa, T., Naert, A., and Sano, M., Matched boundary layers in turbulent Rayleigh-Bénard convection of mercury, Phys. Rev. E 57 (1998), pp ) Xia, K.-Q. and Zhou, S.-Q., Temperature power spectra and the viscous boundary layer in thermal turbulence: The role of Prandtl number, Physica A 288 (2000), pp ) Kraichnan, R. H., Turbulent thermal convection at arbitrary Prandtl number, Phys. Fluids 5 (1962), pp ) Howard, L. N., Heat transport by turbulent convection, J. Fluid Mech. 17 (1963), pp ) Shraiman, B. I. and Siggia, E. D., Heat transport in high-rayleigh-number convection, Phys. Rev. A 42 (1990), pp ) Glazier, J. A., Segawa, T., Naert, A., and Sano, M., Evidence against ultrahard thermal turbulence at very high Rayleigh numbers, Nature 398 (1999), pp ) Niemela, J. J., Skrbek, L., Sreenivasan, K. R., and Donnelly, R. J., Turbulent convection at very high Rayleigh numbers, Nature 404 (2000), pp ) Grossmann, S. and Lohse, D., Scaling in thermal convection: A unifying theory, J. Fluid Mech. 407 (2000), pp ) Chavanne, X. et al., Turbulent Rayleigh-Bénard convection in gaseous and liquid He, Phys. Fluids 13 (2001), pp ) Bolgiano, Jr., R., Turbulent spectra in a stably stratified atmosphere, J. Geophys. Res. 64 (1959), pp ) Obukhov, A. M., (in Russian), Dokl. Akad. Nauk SSSR 125 (1959), pp ) Mashiko, T., Tsuji, Y., Mizuno, T., and Sano, M., Instantaneous measurement of velocity fields in developed thermal turbulence in mercury, Phys. Rev. E 69 (2004), pp ) Takeda, Y., Velocity profile measurement by ultrasound Doppler shift method, Int. J. Heat Fluid Flow 7 (1986), pp