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Статті в журналах з теми "High Prandtl"

Автор: Grafiati

Опубліковано: 11 березня 2023

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1

KAMINSKI, EDOUARD, and CLAUDE JAUPART. "Laminar starting plumes in high-Prandtl-number fluids." Journal of Fluid Mechanics 478 (March 10, 2003): 287–98. http://dx.doi.org/10.1017/s0022112002003233.

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Experimental studies of laminar axisymmetric starting plumes are performed to investigate the dependence of the flow on the Prandtl number, focusing on large Prandtl numbers. Thermal plumes are generated by a small electric heater in a glass tank filled with viscous oils. Prandtl numbers in the range of 7–104 were investigated. Experimental conditions are such that viscosity variations due to temperature differences are negligible. Plumes ascend in two different regimes as a function of distance to source. At short distances, the plumes accelerate owing to the development of the viscous boundary layer. At distances larger than about five times the heater size, the ascent velocity is constant and increases as a function of the Prandtl number, as predicted by theory for steady plumes. This velocity is, within experimental error, proportional to the steady plume centreline velocity.

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2

Jin, Y. Y., and C. F. Chen. "Instability of Convection and Heat Transfer of High Prandtl Number Fluids in a Vertical Slot." Journal of Heat Transfer 118, no. 2 (May 1, 1996): 359–65. http://dx.doi.org/10.1115/1.2825852.

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The stability of convective motion of high-Prandtl-number fluids, generated by a lateral temperature difference across a vertical slot with aspect ratio 15, is studied numerically. The Prandtl number range studied is from 50 to 2000. The nonlinear governing equations are solved by a finite difference method. The predicted flow patterns and critical values are in good agreement with the recent experimental results of Wakitani (1994). It is found that the vorticity distribution along the vertical centerline of the slot is a very sensitive indicator of the onset of multicellular flow. The critical Grashof number varies almost inversely with the Prandtl number; consequently, the critical Rayleigh number is essentially independent of the Prandtl number. Heat transfer results show good agreement with the experimentally correlated values, and they are independent of the Prandtl numbers and the flow patterns.

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3

Busse, F. H., M. A. Zaks, and O. Brausch. "Centrifugally driven thermal convection at high Prandtl numbers." Physica D: Nonlinear Phenomena 184, no. 1-4 (October 2003): 3–20. http://dx.doi.org/10.1016/s0167-2789(03)00210-0.

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4

Liang, Ru Quan, Shuo Yang, Fu Sheng Yan, Jun Hong Ji, and Ji Cheng He. "Numerical Study on High Prandtl Number Liquid Bridge." Advanced Materials Research 712-715 (June 2013): 1630–33. http://dx.doi.org/10.4028/www.scientific.net/amr.712-715.1630.

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The overall numerical analysis of liquid bridge for high Pr number fluid and flow field of ambient air under the zero-gravity environment was carried out in the present paper. The paper used level set method of mass conservation to capture two phase interfaces. Not only the free surface deformation was considered, but also the effect of ambient gas was taken into account. Simultaneously, results of stream function in liquid bridge and ambient gas-phase were given.

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5

Kolyshkin, A., and Rémi Vaillancourt. "Stability of internally generated thermal convection in a tall vertical annulus." Canadian Journal of Physics 69, no. 6 (June 1, 1991): 743–48. http://dx.doi.org/10.1139/p91-124.

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The stabilityof a convective fluid motion generated by internal heat sources in a tall vertical annulus is investigated by means of a mathematical model in the cases of both axisymmetric and asymmetric disturbances. The critical Grasshof numbers are computed for several values of the Prandtl number and different sizes of the gap between the cylinders. It is found that, for low Prandtl numbers and large gaps, asymmetric disturbances lead to instability while, in the case of small gaps, instability is associated with axisymmetric disturbances. In both cases, the critical Grasshof number increases as the gap decreases. For high values of the Prandtl number, instability occurs in the form of thermal running waves. The critical Grasshof numbers decrease as the Prandtl number grows. The neutral stability curve has one or two closed loops for sufficiently high Prandtl numbers. It is found that for high Prandtl numbers instability is associated with axisymmetric perturbations at least in the interval 0.05 < R < 1, where R is the ratio of the inner to the outer radii.

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6

Gargano, F., M. Sammartino, V. Sciacca, and K. W. Cassel. "Analysis of complex singularities in high-Reynolds-number Navier–Stokes solutions." Journal of Fluid Mechanics 747 (April 17, 2014): 381–421. http://dx.doi.org/10.1017/jfm.2014.153.

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AbstractNumerical solutions of the laminar Prandtl boundary-layer and Navier–Stokes equations are considered for the case of the two-dimensional uniform flow past an impulsively-started circular cylinder. The various viscous–inviscid interactions that occur during the unsteady separation process are investigated by applying complex singularity analysis to the wall shear and streamwise velocity component of the two solutions. This is carried out using two different methodologies, namely a singularity-tracking method and the Padé approximation. It is shown how the van Dommelen and Shen singularity that occurs in solutions of the Prandtl boundary-layer equations evolves in the complex plane before leading to a separation singularity in finite time. Navier–Stokes solutions, computed at different Reynolds numbers in the range$10^3 \leq Re \leq 10^5$, are characterized by the presence of various complex singularities that can be related to different physical interactions acting over multiple spatial scales. The first interaction developing in the separation process is large-scale interaction that is visible for all the Reynolds numbers considered, and it signals the first relevant differences between the Prandtl and Navier–Stokes solutions. For$Re\geq O(10^4)$, a small-scale interaction follows the large-scale interaction. The onset of these interactions is related to the characteristic changes of the streamwise pressure gradient on the circular cylinder. Even if these interactions physically differ from that prescribed by the Prandtl solution, and they set a possible limit on the comparison of Prandtl solutions with Navier–Stokes solutions, it is shown how the asymptotic validity of boundary-layer theory is strongly supported by the results that have been obtained through the complex singularity analysis.

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7

Chan, C. L., M. M. Chen, and J. Mazumder. "Asymptotic Solution for Thermocapillary Flow at High and Low Prandtl Numbers Due to Concentrated Surface Heating." Journal of Heat Transfer 110, no. 1 (February 1, 1988): 140–46. http://dx.doi.org/10.1115/1.3250444.

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Thermocapillary convection due to nonuniform surface heating is the dominant form of fluid motion in many materials processing operations. The velocity and temperature distributions for the region adjacent to the area of peak surface heating are analyzed for the limiting cases of large and small Prandtl numbers. For a melt pool whose depth and width are large relative to the thermal and viscous boundary layers, it is shown that the most important parameter is the curvature (i.e., ∇2q) of the surface heat flux distribution. The solutions of the temperature and stream functions are presented, some of which are in closed form. Simple, explicit expressions for the velocity and maximum temperature are presented. These results are found to be quite accurate for realistic Prandtl number ranges, in comparison with exact solutions for finite Prandtl numbers. Besides being more concise than exact results, the asymptotic results also display the Prandtl number dependence more clearly in the respective ranges.

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8

Or, A. C. "Chaotic transitions of convection rolls in a rapidly rotating annulus." Journal of Fluid Mechanics 261 (February 25, 1994): 1–19. http://dx.doi.org/10.1017/s0022112094000224.

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Drifting convection rolls in a rapidly rotating cylindrical annulus with conical endwalls exhibit different transitional modes to chaotic flows at different Prandtl numbers. Three transition sequences for Prandtl numbers 0.3, 1.0 and 7.0 are studied for a moderately large Coriolis parameter and a wavenumber near the critical value using an initial-value code. As the Rayleigh number increases, each transition sequence first leads to a vacillating flow, and then to an aperiodic flow, the route of which is Prandtl-number dependent. From the low Prandtl number to the high Prandtl number, the transitions take different routes of torus folding, period doubling, and mode-locking intermittency.

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9

Mkhinini, Nadia, Thomas Dubos, and Philippe Drobinski. "Secondary instability of the stably stratified Ekman layer." Journal of Fluid Mechanics 728 (July 1, 2013): 29–57. http://dx.doi.org/10.1017/jfm.2013.250.

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AbstractThe Ekman flow, an exact solution of the Boussinesq equations with rotation, is a prototype flow for both atmospheric and oceanic boundary layers. The effect of stratification on the finite-amplitude longitudinal rolls developing in the Ekman flow and their three-dimensional stability is studied by means of linearized and nonlinear numerical simulations. Similarities and differences with respect to billows developing in the Kelvin–Helmholtz (KH) unidirectional stratified shear flow are discussed. Prandtl number effects are investigated as well as the role played by the buoyant-convective instability. For low Prandtl number, the amplitude of the saturated rolls vanishes at the critical bulk Richardson number, while at high Prandtl number, finite-amplitude rolls are found. The Prandtl number also affects how the growth rate of the secondary instability evolves as the Richardson number is increased. For low Prandtl number, the growth rate decreases as the Richardson number increases while it remains significant for large Prandtl number over the range of stratification studied. This behaviour is likely a result of the differing amplitudes of the roll vortices. Furthermore, the most unstable wave vector is much lower than for the secondary instability of KH billows. Examination of the energetics of the secondary instability shows that buoyant-convective instability is present locally at high Reynolds and Prandtl numbers but plays an overall minor role despite the presence in the base flow of statically unstable regions characterized by a high Richardson number.

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10

Orvedahl, Ryan J., Michael A. Calkins, Nicholas A. Featherstone, and Bradley W. Hindman. "Prandtl-number Effects in High-Rayleigh-number Spherical Convection." Astrophysical Journal 856, no. 1 (March 20, 2018): 13. http://dx.doi.org/10.3847/1538-4357/aaaeb5.

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11

Yang, Shuo, Ruquan Liang, and Jicheng He. "Velocity field in a high prandtl number liquid bridge." MATEC Web of Conferences 34 (2015): 03004. http://dx.doi.org/10.1051/matecconf/20153403004.

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12

Yang, Shuo, Ruquan Liang, and Jicheng He. "A numerical simulation on high prandtl number liquid bridge." MATEC Web of Conferences 35 (2015): 07002. http://dx.doi.org/10.1051/matecconf/20153507002.

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13

Bergant, Robert, and Iztok Tiselj. "Near-wall passive scalar transport at high Prandtl numbers." Physics of Fluids 19, no. 6 (June 2007): 065105. http://dx.doi.org/10.1063/1.2739402.

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14

Nadim, N., and G. Domairry. "Analytical study of natural convection in high Prandtl number." Energy Conversion and Management 50, no. 4 (April 2009): 1056–61. http://dx.doi.org/10.1016/j.enconman.2008.12.005.

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15

KUHLMANN, H. C., and U. SCHOISSWOHL. "Flow instabilities in thermocapillary-buoyant liquid pools." Journal of Fluid Mechanics 644 (February 10, 2010): 509–35. http://dx.doi.org/10.1017/s0022112009992953.

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The linear stability of the incompressible axisymmetric flow in a buoyant-thermocapillary liquid pool is considered which is heated from above by a heat flux with a parabolic radial profile. Buoyancy forces and radial thermocapillary stresses due to the inhomogeneous surface temperature distribution drive a toroidal vortex. In the absence of buoyancy and for low Prandtl numbers the basic flow becomes unstable typically by a stationary centrifugal instability. At moderate Prandtl numbers the rotational symmetry is broken by hydrothermal waves. In the limit of vanishing Prandtl number two other critical modes are found if the pool is very shallow. One mode is a centrifugally destabilized rotating wave with high azimuthal wavenumber. The other mode is steady and it is driven by the deceleration of the radial inward return flow as it approaches the axis. The deceleration results from an entrainment of fluid into the thin layer of rapid radial outward surface flow. The centrifugal instability of the toroidal vortex flow is assisted by buoyancy in the low-Prandtl-number limit, because the cooling from the sidewall augments the thermocapillary driving. For moderately high Prandtl numbers a stable thermal stratification suppresses the hydrothermal-wave instabilities.

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16

Wang, Wen Lu, Shu Jun Cui, and Jiang Peng Zhao. "The Influence of Lateral Fill on Foundation Bearing Capacity of High-Stacked Culverts." Advanced Materials Research 790 (September 2013): 287–90. http://dx.doi.org/10.4028/www.scientific.net/amr.790.287.

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To find the improved effect on the foundation bearing capacity of lateral fill, combining with the stress characteristics of the foundation soil, the soil force was analyzed. One element of the soil body was taken as the research object, according to the force equilibrium condition of the element of soil,the ultimate bearing capacity could be obtained by calculating torque of one point. The Prandtl and Prandtl-Reissner foundation ultimate bearing capacity formulas were deduced and contrasted. The results showed that the lateral fill could improve the bearing capacity of the culverts foundation when the settlement of the culverts foundation satisfied the design request.

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17

Chen, Y. C., and J. N. Chung. "Stability of Mixed Convection in a Differentially Heated Vertical Channel." Journal of Heat Transfer 120, no. 1 (February 1, 1998): 127–32. http://dx.doi.org/10.1115/1.2830035.

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In this study, the linear stability of mixed convection in a differentially heated vertical channel is investigated for various Prandtl numbers. The results indicate that this fully developed heated flow can become unstable under appropriate conditions. It is found that both the Prandtl number and Reynolds number hold very important effects on the critical Grashof number, wave number, wave speed, and instability mechanism for higher Prandtl numbers. For low Prandtl numbers, the effects from the Prandtl number and Reynolds number are relatively small. The most significant finding is that the local minimum wave numbers can be as high as eight for Pr = 1000, which is substantially higher than those found before for other heated flows. The existence of multiple local minimum wave numbers is responsible for the sudden jumps of the critical wave number and wave speed and the sudden shift of instability type for higher Prandtl numbers. The energy budget analysis shows that the thermal-shear and shear instabilities dominate at both low and high Reynolds numbers for Pr = 0.7 and 7. It is the thermal-buoyant instability for Re < 1365 and shear instability for Re ≥ 1365 for Pr = 100. The thermal-buoyant and mixed instabilities are the possible instability types for Pr = 1000. In general, for mixed convection channel flows, the instability characteristics of differentially heated flows are found to be substantially different from those of uniformly heated flows.

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18

Käpylä, P. J. "Prandtl number dependence of stellar convection: Flow statistics and convective energy transport." Astronomy & Astrophysics 655 (November 2021): A78. http://dx.doi.org/10.1051/0004-6361/202141337.

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Context. The ratio of kinematic viscosity to thermal diffusivity, the Prandtl number, is much smaller than unity in stellar convection zones. Aims. The main goal of this work is to study the statistics of convective flows and energy transport as functions of the Prandtl number. Methods. Three-dimensional numerical simulations of compressible non-rotating hydrodynamic convection in Cartesian geometry are used. The convection zone (CZ) is embedded between two stably stratified layers. The dominant contribution to the diffusion of entropy fluctuations comes in most cases from a subgrid-scale diffusivity whereas the mean radiative energy flux is mediated by a diffusive flux employing Kramers opacity law. Here, we study the statistics and transport properties of up- and downflows separately. Results. The volume-averaged rms velocity increases with decreasing Prandtl number. At the same time, the filling factor of downflows decreases and leads to, on average, stronger downflows at lower Prandtl numbers. This results in a strong dependence of convective overshooting on the Prandtl number. Velocity power spectra do not show marked changes as a function of Prandtl number except near the base of the convective layer where the dominance of vertical flows is more pronounced. At the highest Reynolds numbers, the velocity power spectra are more compatible with the Bolgiano-Obukhov k−11/5 than the Kolmogorov-Obukhov k−5/3 scaling. The horizontally averaged convected energy flux (F̅conv), which is the sum of the enthalpy (F̅enth) and kinetic energy fluxes (F̅kin), is independent of the Prandtl number within the CZ. However, the absolute values of F̅enth and F̅kin increase monotonically with decreasing Prandtl number. Furthermore, F̅enth and F̅kin have opposite signs for downflows and their sum F̅↓conv diminishes with Prandtl number. Thus, the upflows (downflows) are the dominant contribution to the convected flux at low (high) Prandtl numbers. These results are similar to those from Rayleigh-Benárd convection in the low Prandtl number regime where convection is vigorously turbulent but inefficient at transporting energy. Conclusions. The current results indicate a strong dependence of convective overshooting and energy flux on the Prandtl number. Numerical simulations of astrophysical convection often use a Prandtl number of unity because it is numerically convenient. The current results suggest that this can lead to misleading results and that the astrophysically relevant low Prandtl number regime is qualitatively different from the parameter regimes explored in typical contemporary simulations.

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19

LAKKARAJU, R., and MEHEBOOB ALAM. "Effects of Prandtl number and a new instability mode in a plane thermal plume." Journal of Fluid Mechanics 592 (November 14, 2007): 221–31. http://dx.doi.org/10.1017/s0022112007008610.

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The effect of Prandtl number on the linear stability of a plane thermal plume is analysed under quasi-parallel approximation. At large Prandtl numbers (Pr > 100), we found that there is an additional unstable loop whose size increases with increasing Pr. The origin of this new instability mode is shown to be tied to the coupling of the momentum and thermal perturbation equations. Analyses of the perturbation kinetic energy and thermal energy suggest that the buoyancy force is the main source of perturbation energy at high Prandtl numbers that drives this instability.

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20

Guseva, A., R. Hollerbach, A. P. Willis, and M. Avila. "Azimuthal magnetorotational instability at low and high magnetic Prandtl numbers." Magnetohydrodynamics 53, no. 1 (2017): 25–34. http://dx.doi.org/10.22364/mhd.53.1.4.

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21

Liang, Ru Quan, Shuo Yang, Jun Hong Ji, and Ji Cheng He. "Flow Characteristics for High Prandtl Number Fluid under Zero Gravity." Applied Mechanics and Materials 353-356 (August 2013): 3611–14. http://dx.doi.org/10.4028/www.scientific.net/amm.353-356.3611.

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This paper investigated the flow structure in liquid bridge of high Pr Number fluid under zero gravity condition. The free surface deformation and the effects of gas phase around liquid bridge were considered. Navier-Stokes equations coupled with the energy conservation equation were solved on a staggered grid. The two-phase surface was captured by using the mass conserving level set method. The results indicated that location of vortex center move gradually toward the free surface due to thermocapillary convection. The flow velocity nearby the surface of liquid bridge is faster than the internal flow velocity, and the overall velocity level tends to decline with time evolution.

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22

Vigdorovich, I. I., and A. I. Leont’ev. "Energy separation of gases with low and high Prandtl numbers." Fluid Dynamics 48, no. 6 (November 2013): 811–26. http://dx.doi.org/10.1134/s0015462813060124.

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23

Vincent, Alain P., and David A. Yuen. "Thermal attractor in chaotic convection with high-Prandtl-number fluids." Physical Review A 38, no. 1 (July 1, 1988): 328–34. http://dx.doi.org/10.1103/physreva.38.328.

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24

Okino, Shinya, and Hideshi Hanazaki. "Decaying turbulence in a stratified fluid of high Prandtl number." Journal of Fluid Mechanics 874 (July 12, 2019): 821–55. http://dx.doi.org/10.1017/jfm.2019.471.

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Decaying turbulence in a density-stratified fluid with a Prandtl number up to $Pr=70$ is investigated by direct numerical simulation. In turbulent flow with a Prandtl number larger than unity, it is well known that the passive scalar fluctuations cascade to scales smaller than the Kolmogorov scale, and show the $k^{-1}$ spectrum in the viscous–convective range, down to the Batchelor scale. In decaying stratified turbulence, the same phenomenon is initially observed for the buoyant scalar of high $Pr~(=70)$, until the Ozmidov scale becomes small and the buoyancy becomes effective even at the Kolmogorov scale. After that moment, however, the velocity components near the Kolmogorov scale begin to show strong anisotropy dominated by the vertically sheared horizontal flow, which reduces the vertical scale of density fluctuations. An analysis similar to that of Batchelor (J. Fluid Mech., vol. 5, 1959, pp. 113–133) indeed shows that the vertically sheared horizontal flow reduces the vertical scale of density fluctuations, without changing the horizontal scale.

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25

Al-Ali, A. A. "The long wavelength instability of high Prandtl number convective flow." Applied Mathematics Letters 17, no. 4 (April 2004): 393–99. http://dx.doi.org/10.1016/s0893-9659(04)90080-8.

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26

Majumder, Catherine A. Hier, David A. Yuen, Erik O. Sevre, John M. Boggs, and Stephen Y. Bergeron. "Finite Prandtl number 2-D convection at high Rayleigh numbers." Visual Geosciences 7, no. 1 (July 2002): 1–53. http://dx.doi.org/10.1007/s10069-002-0004-4.

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27

Brandenburg, A. "Dissipation in dynamos at low and high magnetic Prandtl numbers." Astronomische Nachrichten 332, no. 1 (January 2011): 51–56. http://dx.doi.org/10.1002/asna.201011478.

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28

NIENHÜSER, CH, and H. C. KUHLMANN. "Stability of thermocapillary flows in non-cylindrical liquid bridges." Journal of Fluid Mechanics 458 (May 10, 2002): 35–73. http://dx.doi.org/10.1017/s0022112001007650.

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The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

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29

Stevens, Richard J. A. M., Detlef Lohse, and Roberto Verzicco. "Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection." Journal of Fluid Mechanics 688 (October 24, 2011): 31–43. http://dx.doi.org/10.1017/jfm.2011.354.

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AbstractResults from direct numerical simulation for three-dimensional Rayleigh–Bénard convection in samples of aspect ratio $\Gamma = 0. 23$ and $\Gamma = 1/ 2$ up to Rayleigh number $\mathit{Ra}= 2\ensuremath{\times} 1{0}^{12} $ are presented. The broad range of Prandtl numbers $0. 5\lt \mathit{Pr}\lt 10$ is considered. In contrast to some experiments, we do not see any increase in $\mathit{Nu}/ {\mathit{Ra}}^{1/ 3} $ with increasing $\mathit{Ra}$, neither due to an increasing $\mathit{Pr}$, nor due to constant heat flux boundary conditions at the bottom plate instead of constant temperature boundary conditions. Even at these very high $\mathit{Ra}$, both the thermal and kinetic boundary layer thicknesses obey Prandtl–Blasius scaling.

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30

BOWERSOX, RODNEY D. W. "Extension of equilibrium turbulent heat flux models to high-speed shear flows." Journal of Fluid Mechanics 633 (August 25, 2009): 61–70. http://dx.doi.org/10.1017/s0022112009007691.

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An algebraic heat flux truncation model was derived for high-speed gaseous shear flows. The model was developed for high-temperature gases with caloric imperfections. Fluctuating dilatation moments were modelled via conservation of mass truncations. The present model provided significant improvements, up to 20%, in the temperature predictions over the gradient diffusion model for a Mach number ranging from 0.02 to 11.8. Analyses also showed that the near-wall dependence of the algebraic model agreed with expected scaling, where the constant Prandtl number model did not. This led to a simple modification of the turbulent Prandtl number model. Compressibility led to an explicit pressure gradient dependency with the present model. Analyses of a governing parameter indicated that these terms are negligibly small for low speeds. However, they may be important for high-speed flow.

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31

Zhou, Biao, Yu Ji, Jun Sun, and Yuliang Sun. "Modified turbulent Prandtl number model for helium–xenon gas mixture with low Prandtl number." Nuclear Engineering and Design 366 (September 2020): 110738. http://dx.doi.org/10.1016/j.nucengdes.2020.110738.

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32

Bascoul, Guillaume P. "Numerical simulations of semiconvection." Proceedings of the International Astronomical Union 2, S239 (August 2006): 317–19. http://dx.doi.org/10.1017/s1743921307000658.

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AbstractUsing a semiconvective model based on thermohaline convection, we investigate the case of an expanding core of a main-sequence massive star. The numerical simulations at high Prandtl number show a flow consistent with the assumption that a dynamically neutral layer sits between the core and the radiative envelope. More simulations at low Prandtl number are needed to infer scaling laws applicable to astrophysical regimes.

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33

Ma, Li, Jing Li, Shui Ji, and Huajian Chang. "High Prandtl number effect on Rayleigh–Bénard convection heat transfer at high Rayleigh number." Heat and Mass Transfer 53, no. 2 (June 17, 2016): 705–9. http://dx.doi.org/10.1007/s00231-016-1849-7.

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34

Kurdyumov, V. N., and E. Ferna´ndez. "Heat Transfer From a Circular Cylinder at Low Reynolds Numbers." Journal of Heat Transfer 120, no. 1 (February 1, 1998): 72–75. http://dx.doi.org/10.1115/1.2830067.

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A correlation formula, Nu = W0(Re)Pr1/3 + W1(Re), that is valid in a wide range of Reynolds and Prandtl numbers has been developed based on the asymptotic expansion for Pr → ∞ for the forced heat convection from a circular cylinder. For large Prandtl numbers, the boundary layer theory for the energy equation is applied and compared with the numerical solutions of the full Navier Stokes equations for the flow field and energy equation. It is shown that the two-terms asymptotic approximation can be used to calculate the Nusselt number even for Prandtl numbers of order unity to a high degree of accuracy. The formulas for coefficients W0 and W1, are provided.

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35

Liang, Ru Quan, Shuo Yang, Jun Hong Ji, and Ji Cheng He. "Flow of High Prandtl Number Fluid under Varying Axial Magnetic Field." Applied Mechanics and Materials 256-259 (December 2012): 2412–15. http://dx.doi.org/10.4028/www.scientific.net/amm.256-259.2412.

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From engineering actual conditions of single crystal grown by floating zone method, Navier-Stokes equations coupled with the energy conservation equation were solved on a staggered grid based on the half floating area physical model. The two-phase surface was captured by using the mass conserving level set method. The internal flow structure of flow field of high Pr number liquid bridge was studied under uniform magnetic field environment in microgravity, which is important to optimize the process of the crystal growth.

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36

Tong, P., and B. J. Ackerson. "Analogies between colloidal sedimentation and turbulent convection at high Prandtl numbers." Physical Review E 58, no. 6 (December 1, 1998): R6931—R6934. http://dx.doi.org/10.1103/physreve.58.r6931.

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37

Masud, J., Y. Kamotani, and S. Ostrach. "Oscillatory Thermocapillary Flow in Cylindrical Columns of High Prandtl Number Fluids." Journal of Thermophysics and Heat Transfer 11, no. 1 (January 1997): 105–11. http://dx.doi.org/10.2514/2.6207.

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38

YANO, Takeshi, and Nobuhide KASAGI. "Direct Numerical Simulation of Turbulent Heat Transport at High Prandtl Numbers." JSME International Journal Series B 42, no. 2 (1999): 284–92. http://dx.doi.org/10.1299/jsmeb.42.284.

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39

Chiba, Shin-ya, Masahiro Omae, Kazuhisa Yuki, Hidetoshi Hashizume, Saburo Toda, and Akio Sagara. "Experimental Research on Heat Transfer Enhancement for High Prandtl-Number Fluid." Fusion Science and Technology 47, no. 3 (April 2005): 569–73. http://dx.doi.org/10.13182/fst05-a746.

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40

Buchlin, E. "Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers." Astronomy & Astrophysics 534 (October 2011): L9. http://dx.doi.org/10.1051/0004-6361/201117890.

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41

Cerisier, P., M. Jaeger, M. Medale, and S. Rahal. "Mechanical Coupling of Convective Rolls in a High Prandtl Number Fluid." Journal of Heat Transfer 120, no. 4 (November 1, 1998): 1008–18. http://dx.doi.org/10.1115/1.2825884.

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Experimental and numerical studies (using the finite element method) were conducted in a rectangular vessel for a high Prandtl number fluid (Pr = 880 at 25°C). The pattern of convective rolls is perturbed by lateral heating of one of the smaller sides of the box. The wave number of the roll pattern is within a narrow range (±10 percent at R = 4.6RC where R and Rc are the Rayleigh number and its critical value of Be´nard convection). The formation of a large roll induced by the lateral heating causes a slight variation in the wave number of the rolls, accompanied by the disappearance of pairs of rolls, or single rolls, depending on the boundary conditions at the other small side of the enclosure. In all instances, this disappearance respects the mechanical coupling between the rolls. Temperature and velocity fields are assessed, as well as the heat transfer. The transient states observed during the experimentation are well reproduced by the two-dimensional numerical model developed for this study.

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42

YIN, Yaochen, Yasutaka NAGANO, and Masato TAGAWA. "Numerical Prediction of Turbulent Heat Transfer in High-Prandtl-Number Fluids." Transactions of the Japan Society of Mechanical Engineers Series B 58, no. 551 (1992): 2254–60. http://dx.doi.org/10.1299/kikaib.58.2254.

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43

YANO, Takeshi, and Nobuhide KASAGI. "Direct Numerical Simulation of Turbulent Heat Transport at High Prandtl Numbers." Transactions of the Japan Society of Mechanical Engineers Series B 63, no. 612 (1997): 2840–47. http://dx.doi.org/10.1299/kikaib.63.2840.

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44

Vasiliev, A., A. Sukhanovskii, and P. Frick. "Turbulent convective flows in a cubic cavity at high Prandtl number." Journal of Physics: Conference Series 754 (October 2016): 022010. http://dx.doi.org/10.1088/1742-6596/754/2/022010.

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45

Poujol, Federico T., Jorge Rojas, and Eduardo Ramos. "Natural convection of a high Prandtl number fluid in a cavity." International Communications in Heat and Mass Transfer 27, no. 1 (January 2000): 109–18. http://dx.doi.org/10.1016/s0735-1933(00)00089-0.

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46

DANIELS, P. G. "On the boundary-layer structure of high-Prandtl-number horizontal convection." Journal of Fluid Mechanics 652 (May 19, 2010): 299–331. http://dx.doi.org/10.1017/s0022112009994125.

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This paper describes the boundary-layer structure of the steady flow of an infinite Prandtl number fluid in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated and the upper surface is assumed to be either shear-free or rigid. In the limit of large Rayleigh number (R → ∞), the solution involves a horizontal boundary layer at the upper surface of depth of order R−1/5 where the main variation in the temperature field occurs. For a monotonic temperature distribution at the upper surface, fluid is driven to the colder end of the cavity where it descends within a narrow convection-dominated vertical layer before returning to the horizontal layer. A numerical solution of the horizontal boundary-layer problem is found for the case of a linear temperature distribution at the upper surface. At greater depths, of order R−2/15 for a shear-free surface and order R−9/65 for a rigid upper surface, a descending plume near the cold sidewall, together with a vertically stratified interior flow, allow the temperature to attain an approximately constant value throughout the remainder of the cavity. For a shear-free upper surface, this constant temperature is predicted to be of order R−1/15 higher than the minimum temperature of the upper surface, whereas for a rigid upper surface it is predicted to be of order R−2/65 higher.

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47

Chiba, Shin-Ya, Kazuhisa Yuki, Hidetoshi Hashizume, Saburo Toda, and Akio Sagara. "Numerical research on heat transfer enhancement for high Prandtl-number fluid." Fusion Engineering and Design 81, no. 1-7 (February 2006): 513–17. http://dx.doi.org/10.1016/j.fusengdes.2005.08.046.

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48

Hier, Catherine A., Stephen Y. Bergeron, David A. Yuen, Erik O. Sevre, and John M. Boggs. "Thermal Convection in Finite Prandtl Number Fluids at High Rayleigh Number." Visual Geosciences 6, no. 3 (November 2001): 1–4. http://dx.doi.org/10.1007/s10069-001-1017-0.

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49

Eckert, E. R. G., and M. Faghri. "Viscous heating of high Prandtl number fluids with temperature-dependent viscosity." International Journal of Heat and Mass Transfer 29, no. 8 (August 1986): 1177–83. http://dx.doi.org/10.1016/0017-9310(86)90149-3.

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50

Davaille, Anne, Angela Limare, Floriane Touitou, Ichiro Kumagai, and Judith Vatteville. "Anatomy of a laminar starting thermal plume at high Prandtl number." Experiments in Fluids 50, no. 2 (July 7, 2010): 285–300. http://dx.doi.org/10.1007/s00348-010-0924-y.

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