Flows in horizontal thermohaline convection with differential diffusion

Numerical calculations have been carried out to investigate the physical properties of mantle plumes in highly viscous thermal convection depending on the Rayleigh number (Ra). The Boussinesq approximation was applied in a three-dimensional Cartesian domain filled with isoviscous, purely bottom-heated fluid with infinite Prandtl number. In order to monitor the dynamical behavior of plumes an automatic plume detecting routine was developed based on the temperature between the plume and its surroundings. It was established that as the convection becomes more vigorous with increasing Rayleigh number the average cross-sectional area of an individual plume decreases (∼Ra−2∕3), the vertical velocity in plumes increases (∼Ra2∕3), while the average temperature in plumes is independent of Ra. It means that the volume and the heat transport in an individual plume is independent of the Rayleigh number. The number of plumes forming in the box increases (∼Ra1∕3) which is in accordance with the scale analysis using the energy balance and the conservation of momentum. Furthermore, the Rayleigh number influences the temporal behavior of the average surface heat flow [Nusselt number, Nu0(t)] and the heat advected by plumes [Twp(t)]. The characteristic frequencies of Nu0(t) and Twp(t) increase by ∼Ra2∕3 in agreement with the rate of increase of the vertical velocity in plumes. The characteristic frequencies of Nu0(t) and Twp(t) are between the frequency corresponding to the time necessary for a plume to rise from the bottom to the top of the layer and the frequency of a whole convective cycle. The time series of Twp(t) contain larger amplitudes and higher frequencies than Nu0(t). It was assumed that the heat in the top thermal boundary layer (TBL) propagates by conduction and using Twp(t) as an input at the bottom of the top TBL the amplitude and the frequency of the heat flow series on the surface was calculated. It corresponds very well to the amplitude and the frequency of the observed Nu0(t). The correlation analysis between the time series of the surface Nusselt number and the heat advected by hot plumes showed that the time delay between the time series is equal to the time of the heat propagation by conduction through the TBL. The correlation between time series Twp(t) at different depths demonstrated well that the main heat transfer mechanism in plumes is advection.

Natural convection problem in a Bingham fluid using the operator-splitting method

We report the existence of two new limiting turbulent regimes in horizontal convection (HC) using direct numerical simulations at intermediate to low Prandtl numbers. In our simulations, the flow is driven by a step-wise buoyancy profile imposed at the surface, with free-slip, no-flux conditions along all other boundaries, except along the spanwise direction, where periodicity is assumed. The flow is shown to transition to turbulence in the plume and the core, modifying the rate of heat and momentum transport. These transitions set a sequence of scaling laws that combine theoretical arguments from Shishkina, Grossmann and Lohse (SGL) and Hughes, Griffiths, Mullarney and Peterson (HGMP). The parameter range extends through Rayleigh numbers in the range [ $6.4\times 10^5, 1.92\times 10^{15}$ ] and Prandtl numbers in the range [ $2\times 10^{-3},2$ ]. At low Prandtl numbers and intermediate Rayleigh numbers, a core-driven regime is shown to follow a Nusselt-number scaling with $Ra^{1/6}Pr^{7/24}$ . For Rayleigh numbers larger than $10^{14}$ , the Nusselt number scales with $Ra^{0.225}Pr^{0.417}$ . For these particular regimes, the Reynolds number is found to scale as $Ra^{2/5}Pr^{-3/5}$ for the low-Prandtl-number regime and $Ra^{1/3}Pr^{1}$ for Rayleigh numbers larger than $10^{14}$ . These results embed the HGMP model in the SGL theory and extend the known regime diagram of HC at high Rayleigh numbers. In particular, we show that HC and Rayleigh–Bénard share similar turbulent characteristics at low Prandtl numbers, where HC is shown to be ruled by its core dynamics and turbulent boundary layers. This new scenario confirms that fully turbulent HC enhances the transport of heat and momentum with respect to previously reported regimes at high Rayleigh numbers. This work provides new insights into the applicability of HC for geophysical flows such as overturning circulations found in the atmosphere, the oceans, and flows near the Earth's inner core.

Turbulent Rayleigh–Bénard convection in low Prandtl–number fluids

The origin of instability in enclosed horizontally driven convection

We have proposed an existence of small-scale convection in the mantle wedge in order to explain the alignment of group of volcanoes in the NE Honshu subduction zone, Japan. The roll type thermal convection whose axis is normal to the plate boundaries and “flip-flopping” of rolls (warmer and colder parts exchange their positions, as time passes) are reported. However, the subsequent study shows an existence of non-flip-flopping rolls, also. Thus, in this study, I investigate the possible causes of these differences systematically. I found that along-arc wavelength of small-scale convection is controlled by two length scales which may be associated with the inclined nature of the bottom of the low viscosity mantle wedge where the small-scale convection emerges. When the convection is in early stage of evolvement or the speed of subducting slab is small, the long-wavelength rolls become prominent. As the convection evolves or the speed of subduction increases, short-wavelength rolls tend to take over the long-wavelength rolls. The transition from the long-wavelength to the short-wavelength rolls occurs in a several way. It may occur through the splitting of rolls, adjustment of roll patterns and flip-flopping. If the pattern becomes short-wavelength roll, it becomes stable. The range of existence of flip-flopping is limited and the existence of viscosity jump in the top thermal boundary layer may be required.

Numerical investigations are conducted for free convective laminar flow of a fluid with or without internal heat generation in rectangular enclosures of different aspect ratios and at various angles of inclination. Two principal parameters for this problem are the external Rayleigh number, Ra E , which represents the effect due to the differential heating of the side walls, and the internal Rayleigh number, Ra I , which represents the strength of the internal heat generation. Results are obtained for a fixed external Rayleigh number, Ra E =2 2 10 5 , with internal Rayleigh number, Ra I =0 (without internal heat generation), and also with Ra I =2 2 10 5 (with internal heat generation). Flow patterns and isotherms do not show any significant difference between the cases with and without internal heat generation other than slight shift and changes in stream function and isotherm values as long as the internal Rayleigh number Ra I is less than or equal to the external Rayleigh number Ra E . Local heat flux ratios along the hot and the cold walls decrease monotonically in the flow direction for a major downstream portion. At certain inclinations the local heat flux ratios increase initially and then decrease. The variation of average heat flux ratio is similar for cases with and without internal heat generation, but the corresponding amount of heat transfer is higher through the hot wall and lower through the cold wall with internal heat generation. On the other hand, the convection strength increases as the enclosure shape changes from slender through square to shallow at any particular inclination, but varies mildly with inclination at a particular aspect ratio.

We report high-resolution measurements of the properties of the velocity boundary layer in turbulent thermal convection using the particle image velocimetry (PIV) technique and measurements of the temperature profiles and the thermal boundary layer. Both velocity and temperature measurements were made near the lower conducting plate of a rectangular convection cell using water as the convecting fluid, with the Rayleigh numberRavarying from 109to 1010and the Prandtl numberPrfixed at 4.3. From the measured profiles of the horizontal velocity we obtain the viscous boundary layer thickness δυ. It is found that δυfollows the classical Blasius-like laminar boundary layer in the present range ofRa, and it scales with the Reynolds numberReas δυ/H= 0.64Re−0.50±0.03(whereHis the cell height). While the measured viscous shear stress and Reynolds shear stress show that the boundary layer is laminar forRa< 2.0 × 1010, two independent extrapolations, one based on velocity measurements and the other on velocity and temperature measurements, both indicate that the boundary layer will become turbulent atRa~ 1013. Just above the thermal boundary layer but within the mixing zone, the measured temperature r.m.s. profiles σT(z) are found to follow either a power law or a logarithmic behaviour. The power-law fitting may be slightly favoured and its exponent is found to depend onRaand varies from −0.6 to −0.77, which is much larger than the classical value of −1/3. In the same region, the measured profiles of the r.m.s. vertical velocity σw(z) exhibit a much smaller scaling range and are also consistent with either a power-law or a logarithmic behaviour. The Reynolds number dependence of several wall quantities is also measured directly. These are the wall shear stress τw~Re1.55, the viscous sublayer δw~Re−0.91, the friction velocityuτ~Re0.80, and the skin-friction coefficientcf~Re−0.34. All of these scaling properties are very close to those predicted for a classical Blasius-type laminar boundary layer, except that ofcf. Similar to classical shear flows, a viscous sublayer is also found to exist in the present system despite the presence of a nested thermal boundary layer. However, velocity profiles normalized by wall units exhibit no obvious logarithmic region, which is likely to be a result of the very limited distance between the edge of the viscous sublayer and the position of the maximum velocity. Compared to traditional shear flows, the peak position of the wall-unit-normalized r.m.s. profiles is found to be closer to the plate (atz+=z/δw≃ 5). Our overall conclusion is that a Blasius-type laminar boundary condition is a good approximation for the velocity boundary layer in turbulent thermal convection for the present range of Rayleigh number and Prandtl number.

Stability of natural convection flow adjacent to a vertical plate with uniform heat flux is studied using direct numerical simulation (DNS). Stability of the fully developed boundary layer is examined by adding a perturbation to the simulation in the region of the plate origin, with the resulting traveling waves tracked as they travel up the boundary layer. The approach allows the critical height on the plate at which the traveling waves become unstable to be identified, providing a critical Rayleigh number for the occurrence of instability in this flow. The full details available from the numerical solution also allow the phase speed and amplitude of the most unstable mode to be identified. Flow with Rayleigh number Ra = 1.0 × 1010 and Prandtl number Pr = 6.7 is studied. Finally the DNS results are compared with the results obtained using the conventional linearized stability analysis.

Technical Note Flow characteristics in a heated rotating straight pipe

A theoretical model is developed for steady magnetohydrodynamic viscous flow resulting from a moving semi-infinite flat plate in an electrically conducting nanofluid. Thermal radiation and magnetic induction effects are included in addition to thermal convective boundary conditions. Buongiorno’s two-component nanoscale model is deployed, which features Brownian motion and thermophoresis effects. The governing nonlinear boundary layer equations are converted to nonlinear ordinary differential equations by using suitable similarity transformations. The transformed system of differential equations is solved numerically, employing the spectral relaxation method (SRM) via the MATLAB R2018a software. SRM is a simple iteration scheme that does not require any evaluation of derivatives, perturbation, and linearization for solving a nonlinear system of equations. Effects of embedded parameters such as sheet velocity parameter$\lambda$, magnetic field parameter$\beta$, Prandtl number$Pr$, magnetic Prandtl number$Prm$, thermal radiation parameter$Rd$, Lewis number$Le$, Brownian motion parameter$Nb$, and thermophoresis parameter$Nt$ on velocity, induced magnetic field, temperature, and nanoparticle concentration profiles are investigated. The skin-friction results, local Nusselt number, and Sherwood number are also discussed for various values of governing physical parameters. To show the convergence rate against iteration, residual error analysis has also been performed. The flow is strongly decelerated, and magnetic induction is suppressed with greater magnetic body force parameter, whereas temperature is elevated due to extra work expended as heat in dragging the magnetic nanofluid. Temperatures are also boosted with increment in nanoscale thermophoresis parameter and radiative parameter, whereas they are reduced with higher wall velocity, Brownian motion, and Prandtl numbers. Both hydrodynamic and magnetic boundary layer thicknesses are reduced with greater reciprocal values of the magnetic Prandtl number Prm. Nanoparticle (concentration) boundary layer thickness is boosted with higher values of thermophoresis and Prandtl number, whereas it is diminished with increasing wall velocity, nanoscale Brownian motion parameter, radiative parameter, and Lewis number. The simulations are relevant to electroconductive nanomaterial processing.

In this paper, the steady-state natural convection in a square cavity filled with water–Al2O3 nanofluid in the presence of magnetic fields with variable inclination angles is investigated numerically. The enclosure is subjected to different side-wall temperatures while the top and bottom walls are assumed to be adiabatic. The thermal behavior of enclosure is assessed using a finite volume-based computer program. In order to ensure the accuracy of results, comparisons are also made with a previous published work. In this research, at constant magnetic field strengths, the effect of magnetic field inclination angle on the rate of heat transfer in the square cavity is investigated at the Rayleigh numbers of Ra = 103, 104, 105 and 106. In this work, the Hartmann number ranges from Ha = 0 to 120 and the solid volume fraction varies from φ = 0 to 0.06. Numerical results show that depending on the Rayleigh and Hartmann numbers, the maximum heat transfer rate may occur at magnetic field inclination angles of 45°, 60° or 90° and the effect of magnetic field inclination angle is significant at high values of Rayleigh and Hartmann numbers. It is found that addition of nano-sized solid particles causes higher heat transfer rate when Ra = 103, whereas at Rayleigh number of Ra = 106, a reverse behavior is observed. Results show that at Rayleigh numbers of Ra = 104 and 105, the effect of solid particles addition on the thermal performance of the enclosure depends on the Hartmann number. It is also shown that an increase in the inclination angle causes higher velocity within the enclosure and addition of solid particles leads to suppression of flow field.

Steady, finite-amplitude, rotating magnetoconvection and the mean poloidal circulation of the Earth's core

Broken reaction zone and differential diffusion effects in high Karlovitz n-C7H16 premixed turbulent flames

Differential diffusion effects in counter-flow premixed hydrogen-enriched methane and propane flames