Strong symmetrical non-Oberbeck-Boussinesq turbulent convection and a possible role of

compressibility

Victor Steinberg and Enrico Segre, Yuri Burnishev

Department of Physics of Complex Systems

Goals of the experiment

• Can strong but symmetric variations of fluid properties with respect to a cell mid-plane alter a scaling of Nu versus Ra?

• Is Pr-dependence of Nu so different from OB-case and strong?

• What is a possible source of this strong Pr-dependence?

Phase diagram of SF6 in the reduced variablesThree isobars are at P=38.2, 37.7, 37.6 bars and at Pr=36,122, 180, respectively (from top to bottom)

Region near CP of the average T in the cellExplored in the experiment

Thick lines on the isobars define the T and ρ distributions at Δused in the experiment

Phase diagram of SF6

Schematic drawing of a convective cell and thermistor suspension for local T measurements in cylindrical cell and without the probe in square cell.

Variations of a simplified non-OB criterion vs Ra for different PrInset: Fractional deviations of δβ and δCp vs Ra

(a) The temperature distributions of the gas propertiesβ and Cp along the cell height, normalized by their values at the cell mid-plane.

(b) Upper plot: the dependence of λ normalized by its value at the cell mid-plane, for ethane and butane. Lower plot: the temperature distribution of λ for SF6 along the cell height, normalized by the values at thecell mid-plane as a function of T-Tm.

Variations of fluid properties

The ratio of the temperature drops across the top and bottom halves of the cell, versus Ra, for four values of Pr.bt /

Temperature profiles in turbulent convection of SF6 near CP: (a) Ra = Pr = 36,¢ = 45 mK. Inset: zoom in of the bottom thermal boundary layer. (b) Ra = Pr = 95,¢ = 25 mK. Inset: zoom in of the bottom thermal boundary layer.

12101.3 13109.1

Measurements of thermal boundary layer and its scaling with Ra and Pr

L/2δ versus Ra for supercritical SF6 far away from CP at three values of Pr = 0.8; 1.5; 3.

The same data in compensated presentation versus Ra for three values of Pr.

01.031.0)2/( RaL

L/2δ versus Ra for SF6 near CP at four values of Pr.

The same data in compensated presentation versus Ra for four values of Pr. Inset: Pr dependence of for four values of Pr.

01.032.0)2/( RaL 01.032.0)2/( RaL

Presentation of the whole dataset on a single plot in the scaled variables: versus Ra.01.023.0Pr)2/( L

Convergence of the iteration procedure to calculate the corrected value of Nu versus the number of iterations N.

Heat transfer measurements

The corrected values of Nu versus Ra for supercritical SF6 far away from CP, for three values of Pr=0.8, 1.5, and 3.

The same data shown in compensated presentation versus Ra, together withthe data of others, taken from literature. Symbols: ■- Niemela1 Γ= 1=2; Pr = 0:7; □ - Niemela2Γ= 1; Pr = 0:7 ; ●-Roche Γ = 1/2; Pr = 1.5 (from Ref.[1]); ○ - Chavanne Γ= 1/2; 0.7 < Pr < 2.0;▲- Ahlers1 0.43 <·Γ<0.98; Pr = 4.4; ∆- Ahlers2 Γ= 0.28; Pr = 4.4 (from Ref. [2]). Our data -- Pr = 0.8, open star - Pr = 1.5, - Pr = 3.0.

309.0NuRa

The corrected values of Nu versus Ra for many values of Pr obtained in the cell with L = 9 cm and the stainless steel top and aluminum bottom plates only.

Corrected values of Nu versus Ra for comparison among cells: stainless steel (SS) (top) and aluminum (bottom), versus copper (Cu) plates at Pr = 4 - upper, Pr = 36 - middle, and Pr = 126 - lower plots.

Corrected values of Nu versus Ra for comparison between the cells with copper plates but different cell heights: L = 9 cm versus L = 4.5 cm at Pr = 4 - upper, Pr = 35 - middle, and Pr = 128 - lower plots.

Compensated plot for the entire data set in the scaled variables versus Ra. Inset: minimization procedure to get the scaling exponent α for Pr dependence.

003.0302.0 NuRa

Pr dependence of the scaled variable for the entire data with Tc =318.733Ktaken from NIST , and the same data with Tc = 318.717 K taken from [23]. We alsoadded on the same plot three data points for Pr = 0.8; 1.5; 3 obtained for SF6 far away from CP.

003.0302.0 NuRa

The entire set of the data collapses onto one line in the scaled variables versus Ra with Tc = 318.733 K taken from NIST007.020.0Pr Nu

Scaling for Nu(Ra,Pr)

007.020.0003.0302.0 Pr)01.0(18.0 RaNuFor Pr in the range 4 - 354

01.023.001.032.0 Pr)05.0(8.02

Ral

L

Scaling for thermal boundary layer L/2l(Ra,Pr)

For Pr in the range11 - 95

Conclusions

• In spite of strong variations of the fluid properties across the cell height, symmetric non-OB turbulent convection exhibits the same scaling of Nu with Ra as the OB turbulent convection but a much stronger Pr dependence.

• The influence of the non-OB effect on the heat transport and found that, for the same Pr, an eight-fold larger non-OB effect does not alter either the value of Nu nor its scaling with respect to Ra.

• Strong symmetric non-OB effect by itself is not responsible for the strong Pr dependence of the heat transport near CP. The possible source of this Pr-dependence is the strongly enhanced isothermal compressibility in the vicinity of CP, which can affect the dynamics of plumes and so the heat transport close to the CP