Turbulent Rotating Rayleigh–Bénard Convection

The effect of centrifugal force in turbulent rotating Rayleigh–Bénard convection (RRBC) is studied experimentally in an aspect-ratio $\varGamma =1$ cylindrical convection cell and in the ranges of the Froude number $0.004\leq Fr \leq 0.363$ and the Rayleigh number $2.8\times 10^8 \leq Ra \leq 9.5\times 10^9$, and with the Prandtl number fixed at $Pr=4.34$. We use the bulk temperature anomaly to determine the onset Froude number $Fr_c$, beyond which the centrifugal effects cannot be regarded as insignificant. It is found that $Fr_c$ depends on $Ra$ as $Fr_c\sim Ra^{0.53}$, which may be understood qualitatively by the idea of local force balance. For $Fr>Fr_c$, the centrifugal effect is more pronounced for smaller $Ra$, which is also found for larger constant $1/Ro$. This implies that the response of the system to the centrifugal force depends on the flow states, which, in RRBC, is mainly determined by the competition between the buoyancy and Coriolis forces. Detailed analysis of the sidewall temperature signal shows results consistent with those obtained from the bulk temperature. Based on the above results, we propose a different division of the $1/Ro$–$Fr$ phase space than previously suggested. For the heat transport, the results under fixed $1/Ro$ show well-defined $Nu$–$Ra$ scalings, which can provide a better prediction for the heat transport when extrapolating to the unexplored regions in the phase space.

In order to clarify the turbulence behavior in impeller channels of turbomachinery, effects of the Coriolis and centrifugal forces on the turbulent boundary layer on the concave and convex surfaces were studied experimentally, using a rotating channel of rectangular cross section whose aspect ratio was 2 : 1. Velocity profiles and turbulence intensities were also compared with those for a channel having an aspect ratio of 4 : 1. In the present rotating channel it was found that secondary flows generated by the centrifugal and Coriolis forces have a large effect on the time-averaged and turbulence components in the boundary layer along the concave and convex surfaces, and that the suppression of turbulence expected on the lower-pressure side of a two-dimensional flow is not observed because of the accmulation of lower energy fluids and the interference of the fluids with the primary flow.

Technical Note Flow characteristics in a heated rotating straight pipe

The combined effects of system rotation (Coriolis force) and curvature (centrifugal force) on the bifurcation structure of two-dimensional flows in a toroidal geometry of rectangular cross-section are examined. The problem depends on the Rossby number, Ro = U/bΩ, the Ekman number, Ek = ν/b2Ω, the aspect ratio, γ = b/h and the radius ratio, η = ri/ro; here U is the velocity scale, b is the channel width in the spanwise direction, Ω is the rotational speed, (ri, ro) are the inner and outer radii of the duct, h = ro – ri is the channel gap in the radial direction and v is the kinematic viscosity of the fluid. A pseudospectral method is devised to discretize the two-dimensional Navier–Stokes equation in stream-function form. Continuation schemes are used to track the solution paths with Rossby number as the control parameter. Extended systems are used to determine the precise location of the singular points of the discretized system. The loci of such singular points are tracked with respect to curvature of the duct. Unlike the findings of Miyazaki (1973) on the same problem, curvature is found to have profound effects on the solution structure; flow mutations take place through a tilted cusp at (Ro = 7.122, η = 0.678) and a transcritical bifurcation point at (Ro = 1.357, η = 0.349).

In the present paper, we study the combined incompressible and fast rotation limits for the full Navier-Stokes-Fourier system with Coriolis, centrifugal and gravitational forces, in the regime of small Mach, Froude and Rossby numbers and for general ill-prepared initial data. We consider both the isotropic scaling (where all the numbers have the same order of magnitude) and the multi-scale case (where some effect is predominant with respect to the others). In the case when the Mach number is of higher order than the Rossby number, we prove that the limit dynamics is described by an incompressible Oberbeck-Boussinesq system, where the velocity field is horizontal (according to the Taylor-Proudman theorem), but vertical effects on the temperature equation are not negligible. Instead, when the Mach and Rossby numbers have the same order of magnitude, and in absence of the centrifugal force, we show convergence to a quasi-geostrophic equation for a stream function of the limit velocity field, coupled with a transport-diffusion equation for a new unknown, which links the target density and temperature profiles. The proof of the convergence is based on a compensated compactness argument. The key point is to identify some compactness properties hidden in the system of acoustic-Poincar\'e waves. Compared to previous results, our method enables first of all to treat the whole range of parameters in the multi-scale problem, and also to consider a low Froude number regime with the somehow critical choice $Fr=\sqrt{Ma}$, where $Ma$ is the Mach number. This allows us to capture some (low) stratification effects in the limit.

Effects of rotation on heat transfer on leading and trailing sides of gas turbine blades has been extensively studied in the past. It has been established for typical two-pass channel that radially outward flow (first pass) has higher heat transfer on trailing side and lower heat transfer on leading side and vice versa for radially inward flow (second pass). Rotation induces three forces on the coolant flow — Coriolis, Centrifugal and Buoyancy forces. The direction of Coriolis force depends on the relative angle between the coolant flow and the rotation direction, because of which the direction of Coriolis force is different in individual passes — which in turn results in non-uniform distribution of high heat transfer regions on leading and trailing walls. The present study is focused on utilizing the Coriolis force favorably in both the passes by rotating the typical arrangement of two-pass channels by 90°. Firstly, smooth two pass duct (Model A-smooth) having typical arrangement of coolant flow and rotation direction is studied. The second configuration is the corresponding ribbed channel (Model A-ribbed) featuring V-shaped ribs on both leading and trailing walls. The rib-height-to-channel hydraulic diameter ratio was 0.125, rib pitch-to-rib height ratio was 8, and channel aspect ratio was unity. Model B was obtained by rotating the Model A by 90° and changing the coolant inlet port as well. Model B had three configurations — (a) smooth duct, (b) single sided ribbed duct, and (c) double sided ribbed duct. Detailed heat transfer coefficients were measured by transient liquid crystal thermography under rotating conditions. In order to match the direction of Buoyancy force as it exists in actual engines, colder air was passed during the transient experiment. The heat transfer experiments were carried out at a Reynolds number of 20000 and Rotation numbers of 0, 0.05 and 0.1. The Nusselt numbers have been reported in two forms, (a) normalized with respect to Dittus-Boelter correlation for developed turbulent flow in circular duct, (b) normalized with corresponding Nusselt number obtained from smooth channel experiments. The effects of Coriolis force and centrifugal force on heat transfer has been discussed in detail. A new model has been proposed based on the understanding and findings of the present study, which has positive effects of rotation on both leading and trailing walls.

The present paper studies the locations and linear stability of the triangular equilibrium points when both primaries are radiating and considered as heterogeneous spheroid with three layers of different densities. Additionally, we include the effects of small perturbations in the Coriolis and centrifugal forces and potential from a belt (circumbinary disc). It is observed that the positions of the triangular equilibrium points are substantially affected by all parameters (except a perturbation in Coriolis force) involved in the system.The stabilty of motion is found only when 0 < mu < mu_{c}, where mu_{c} is the critical mass value which depends on the combined effect of radiation pressures and heterogeneity of the primaries, small perturbations and the potential from a belt.It is also seen that the Coriolis force and the belt have stabilizing effect,while the centrifugal force, radiation and heterogeineity of the primaries have destabilizing behaviour.The net effect is that the size of the region of stability decreases when the value of these parameters increases where mu is the mass ratio and k_{1} ,k_{2} characterize heterogeneity of both primaries. A practical application of this model could be the study of motion of a dust grain near the heterogeneous and luminous binary stars surrounded by a belt.Finally, we carried out and discuss numerical experiments aiming at computing the positions of triangular points and critical masses of three binary systems: Archid, Xi Booties and Kruger 60.

This paper experimentally investigates the convection in a rapidly rotating tangent cylinder (TC), for Ekman numbers down to $E=3.36\times 10^{-6}$. The apparatus consists of a hemispherical fluid vessel heated in its centre by a protruding heating element of cylindrical shape. The resulting convection that develops above the heater, i.e. within the TC, is shown to set in for critical Rayleigh numbers and wavenumbers respectively scaling as $Ra_{c}\sim E^{-4/3}$ and $a_{c}\sim E^{-1/3}$ with the Ekman number $E$. Although exhibiting the same exponents as for plane rotating convection, these laws reflect much larger convective plumes at onset. The structure and dynamics of supercritical plumes are in fact closer to those found in solid rotating cylinders heated from below, suggesting that the confinement within the TC induced by the Taylor–Proudman constraint influences convection in a similar way as solid walls would do. There is a further similarity in that the critical modes in the TC all exhibit a slow retrograde precession at onset. In supercritical regimes, the precession evolves into a thermal wind with a complex structure featuring retrograde rotation at high latitude and either prograde or retrograde rotation at low latitude (close to the heater), depending on the criticality and the Ekman number. The intensity of the thermal wind measured by the Rossby number $Ro$ scales as $Ro\simeq 5.33(Ra_{q}^{\ast })^{0.51}$ with the Rayleigh number based on the heat flux $Ra_{q}^{\ast }\in [10^{-9},10^{-6}]$. This scaling is in agreement with heuristic predictions and previous experiments where the thermal wind is determined by the azimuthal curl of the balance between the Coriolis force and buoyancy. Within the range $Ra\in [2\times 10^{7},10^{9}]$ which we explored, we also observe a transition in the heat transfer through the TC from a diffusivity-free regime where $Nu\simeq 0.38E^{2}Ra^{1.58}$ to a rotation-independent regime where $Nu\simeq 0.2Ra^{0.33}$.

Simulations of compressibility effects in centrifugal buoyancy-induced flow in a closed rotating cavity

The combined effects of the imposed vertical mean magnetic field ( $B_0$ , scaled as the Alfvèn velocity) and rotation on the heat transfer phenomenon driven by the Rayleigh–Taylor (RT) instability are investigated using direct numerical simulations. In the hydrodynamic (HD) case, as the strength of the Coriolis frequency ( $f$ ) increases, the Coriolis force enhances the mixing of fluids that dampens the growth of the mixing layer height ( $h$ ) and reversible exchanges between the fluids, leading to a reduction in the heat transport, characterised by the Nusselt number ( $Nu$ ). In non-rotating magnetohydrodynamic (MHD) cases, we find a significant delay in the onset of RT instability with increasing $B_0$ , consistent with the linear theory in the literature. The imposed $B_0$ forms vertically elongated thermal plumes that exhibit a larger reversible buoyancy flux due to limited mixing, enabling them to transport heat efficiently between the bottom hot fluid and the upper cold fluid. This leads to enhanced heat transfer in the initial regime of unbroken elongated plumes in non-rotating MHD cases compared to the corresponding HD case. In the turbulent regime of broken small-scale structures, the imposed $B_0$ collimates the flow along the vertical magnetic field lines, reducing vertical velocity fluctuations ( $u_3^{\prime }$ ) and increasing the growth of $h$ . The increased $h$ primarily drives the heat transfer enhancement in the turbulent regime of non-rotating MHD over the corresponding HD case. When rotation is added along with the imposed $B_0$ , the growth and breakdown of vertically elongated plumes are inhibited by the instability-damping effect of the Coriolis force. Consequently, heat transfer is also reduced in the rotating MHD cases compared to the corresponding non-rotating MHD cases. Interestingly, heat transport in rotating MHD cases is enhanced compared to the corresponding rotating HD cases because $B_0$ reduces mixing and mitigates the instability-damping effect of the Coriolis force. The presence of the ultimate state regime $Nu\simeq Ra^{1/2}Pr^{1/2}$ , where $Ra$ is the Rayleigh number and $Pr$ is the Prandtl number, is observed in the non-rotating HD and MHD cases. However, the rotating HD and MHD cases depart from this ultimate state scaling. Furthermore, the dynamic balance between different forces is analysed to understand the behaviour of the thermal plumes. The turbulent kinetic energy budget reveals the conversion of the turbulent kinetic energy, generated by the buoyancy flux, into turbulent magnetic energy.

Inclined turbulent thermal convection in liquid sodium is studied at large Rayleigh numbers based on the results of both experimental measurements and high-resolution numerical simulations. For a direct comparison, the considered system parameters are set to be similar: in the direct numerical simulations (DNS), in the large-eddy simulations and in the experiments, while the Prandtl number of liquid sodium is very small ( ). The cylindrical convection cell has an aspect ratio of one; one circular surface is heated, while the other one is cooled. Additionally, the cylinder is inclined with respect to gravity and the inclination angle varies from , which corresponds to Rayleigh–Benard convection (RBC), to , as in a vertical convection (VC) set-up. Our study demonstrates quantitative agreement of the experimental and numerical results, in particular with respect to the global heat and momentum transport, temperature and velocity profiles, as well as the dynamics of the large-scale circulation (LSC). The DNS reveal that the twisting and sloshing of the LSC at small inclination angles periodically affects the instantaneous heat transport (up to of the mean heat transport). The twisted LSC is associated with a weak heat transport, while the sloshing mode that brings together the hot and cold streams of the LSC is associated with a strong heat transport. The experiments show that the heat transport scales as in both limiting cases (RBC and VC) for Rayleigh numbers around , while any inclination of the cell, , leads to an increase of .

In magnetoconvection, the flow of an electromagnetically conductive fluid is driven by a combination of buoyancy forces, which create the fluid motion due to thermal expansion and contraction, and Lorentz forces, which distort the convective flow structure in the presence of a magnetic field. The differences in the global flow structures in the buoyancy-dominated and Lorentz-force-dominated regimes lead to different heat transport properties in these regimes, reflected in distinct dimensionless scaling relations of the global heat flux (Nusselt number $Nu$ ) versus the strength of buoyancy (Rayleigh number $Ra$ ) and electromagnetic forces (Hartmann number $Ha$ ). Here, we propose a theoretical model for the transition between these two regimes for the case of a static vertical magnetic field applied across a convective fluid layer confined between two isothermal, a lower warmer and an upper colder, horizontal surfaces. The model suggests that the scaling exponents $\gamma$ in the buoyancy-dominated regime, $Nu\sim Ra ^\gamma$ , and $\xi$ in the Lorentz-force-dominated regime, $Nu\sim (Ha^{-2}Ra)^\xi$ , are related as $\xi =\gamma /(1-2\gamma )$ , and the onset of the transition scales with $Ha^{-1/\gamma }Ra$ . These theoretical results are supported by our direct numerical simulations for $10\leq Ha\leq 2000$ , Prandtl number $Pr=0.025$ and $Ra$ up to $10^9$ and data from the literature.

The incompressible viscous steady flow through a helical pipe of circular cross-section rotating at a constant angular velocity about the center of curvature is investigated numerically to examine the combined effects of rotation (Coriolis force), torsion and curvature (centrifugal force) on the flow. The flow depends on the Taylor number T r =[equation], the Dean number D n =[equation], the torsion parameter β 0 =[equation] and the dimensionless curvature of the duct δ , where a is the radius of the helical pipe, Ω T the angular velocity, μ the viscosity, v the kinematic viscosity, G the constant pressure gradient along the pipe axis and β 0 -a parameter related to the torsion τ and curvature δ . When Ω T >0, the rotation is in the direction in which the Coriolis force produces the curvature effect. When Ω T <0, the rotation is in the direction in which the Coriolis force exhibits an opposite effect to that of curvature. The calculations are carried out for -500 ≤ T r ≤ 500, 1500 ≤ D n ≤ 2000 (large Dean number), 0≤ β 0 ≤ 0.4 and 0 < δ ≤ 0.2. The total flux through the duct has a sharp peak at a negative T r .

This paper is the first work on the vibration of a high-speed rotating spherical shell that rotates about its symmetric axis by developing a set of motion governing equations with consideration of both the Coriolis and centrifugal accelerations as well as the hoop tension arising in the rotating shell due to the angular velocity. To the author's understanding, no such work has so far been published on the rotating spherical shell with the Coriolis and centrifugal accelerations as well as the hoop tension, although there have been the works published on the rotating hemispherical shell with consideration of the Coriolis and centrifugal forces. A thin rotating isotropic truncated circular spherical shell with the simply supported boundary conditions at both the ends is taken as an example for the free vibrational analysis. In order to validate the present formulation, comparisons are made with a nonrotating isotropic spherical shell, and a good agreement is achieved since no published data results from open literature are available for comparison on the dynamics of rotating spherical shell. By the Galerkin method, several case studies are conducted for investigation of the influence of the important parameters on the frequency characteristics of the rotating spherical shell. The parameters studied include the circumferential wave number, the rotational angular velocity, Young's modulus of the shell material, and the geometric ratio of the thickness to radius of the spherical shell.

Flow transitions and combined free and forced convective heat transfer in a rotating curved circular tube