Small Prandtl Number Research Articles

On the instability of convective flow in cylinder and possible secondary regimes

A new exact solution of equations of free convection is constructed in the framework of the Oberbeck–Boussinesq approximation. The solution contains an independent parameter and describes the flow of a viscous heat-conducting liquid in the vertical cylinder with large radius. Complex rheology and radiative heating are taken into account. The considered problem reduces to the operator equation with strongly nonlinear operator. The solvability of the operator problem is proved. The iterative procedure for finding the free parameter is suggested. Three different classes of solution are obtained with the help of the procedure. The linear stability of all classes of solutions is studied numerically. Critical thermal mode is isolated. Evolution of oscillatory mode depending on Prandtl number is investigated. It is shown that under small Prandtl numbers oscillatory modes decay. If Prandtl numbers are not small a new instability type appears. This instability is connected with growing thermal disturbances. Another instability mechanism is discovered in the short waves domain. In this case the crisis is attributed to growing hydrodynamical disturbances. Secondary regimes arising in the hydrodynamical mechanism of the stability loss are calculated.

A dynamo model of Jupiter’s magnetic field

Jupiter’s dynamo is modelled using the anelastic convection-driven dynamo equations. The reference state model is taken from French et al. [2012]. Astrophys. J. Suppl. 202, 5, (11pp), which used density functional theory to compute the equation of state and the electrical conductivity in Jupiter’s interior. Jupiter’s magnetic field is approximately dipolar, but self-consistent dipolar dynamo models are rather rare when the large variation in density and the effective internal heating are taken into account. Jupiter-like dipolar magnetic fields were found here at small Prandtl number, Pr=0.1. Strong differential rotation in the dynamo region tends to destroy a dominant dipolar component, but when the convection is sufficiently supercritical it generates a strong magnetic field, and the differential rotation in the electrically conducting region is suppressed by the Lorentz force. This allows a magnetic field to develop which is dominated by a steady dipolar component. This suggests that the strong zonal winds seen at Jupiter’s surface cannot penetrate significantly into the dynamo region, which starts approximately 7000km below the surface.

Linear instability analysis of convection in a laterally heated cylinder

AbstractThe three-dimensional instabilities of axisymmetric flow are investigated in a laterally heated vertical cylinder by linear stability analysis. Heating is confined to a central zone on the sidewall of the cylinder, while other parts of the sidewall are insulated and both ends of the cylinder are cooled. The length of the heated zone equals the radius of the cylinder. For three different aspect ratios, $A= 1.92 $, 2, 2.1 ($A=\mathrm{height}$/radius), the dependence of the critical Rayleigh number on the Prandtl number (from 0.02 to 6.7) has been studied in detail. For such a kind of laterally heated convection, some interesting stability results are obtained. A monotonous instability curve is obtained for $A= 1.92 $, while the instability curves for $A= 2 $ and $A= 2.1 $ are non-monotonous and multivalued. In particular, an instability island has been found for $A=2$. Moreover, mechanisms corresponding to different instability results are obtained when the Prandtl number changes. At small Prandtl number, the flow is oscillatory unstable, which is dominated by hydrodynamic instability. At intermediate Prandtl number, the interaction between buoyancy and shear in the base flow plays a more important role than pure hydrodynamic instability. At even higher Prandtl number, Rayleigh–Bénard instability becomes the dominant process and the flow loses stability through steady bifurcation.

α effect in a turbulent liquid-metal plane Couette flow.

We calculate the mean electromotive force in plane Couette flows of a nonrotating conducting fluid under the influence of a large-scale magnetic field for driven turbulence. A vertical stratification of the turbulence intensity results in an α effect owing to the presence of horizontal shear. Here we discuss the possibility of an experimental determination of the components of the α tensor using both quasilinear theory and nonlinear numerical simulations. For magnetic Prandtl numbers of the order of unity, we find that in the high-conductivity limit the α effect in the direction of the flow clearly exceeds the component in spanwise direction. In this limit, α runs linearly with the magnetic Reynolds number Rm, while in the low-conductivity limit it runs with the product Rm·Re, where Re is the kinetic Reynolds number, so that for a given Rm the α effect grows with decreasing magnetic Prandtl number. For the small magnetic Prandtl numbers of liquid metals, a common value for the horizontal elements of the α tensor appears, which makes it unimportant whether the α effect is measured in the spanwise or the streamwise directions. The resulting effect should lead to an observable voltage of about 0.5 mV in both directions for magnetic fields of 1 kG and velocity fluctuations of about 1m/s in a channel of 50-cm height (independent of its width).

Numerical study of the fast rotation effect on the stability of advective flow in a horizontal fluid layer with rigid boundaries at small Prandtl numbers

<< ) using a small perturbation method. In calculations, we apply the grid method for the one-dimensional problem and the differential sweep method. The system of equations and the necessary parameters of the numerical solution are presented for both methods. Neutral curves describing the dependence of the critical Grashof number on the wave number are obtained and their behavior at different Pr values is analyzed. The relationship between the critical Grashof number and corresponding wave number and the Prandtl number is given. It is shown that when the Pr value increases from 0 to 0.3 the flow stability decreases and when it lies within the interval 0.3 – 0.93 the flow stability increases. For the monotonic instability mode Gr 40839.42 c = , the critical Grashof number reaches its lowest value at Pr 0.3 = . The boundary of monotonic instability mode at large Taylor

Secondary flows in a laterally heated horizontal cylinder

In this paper we study the problem of thermal convection in a laterally heated, finite, horizontal cylinder. We consider cylinders of moderate aspect ratio (height/diameter ≈2) containing a small Prandtl number fluid (σ &amp;lt; 0.026) typical of molten metals and molten semiconductors. We use the Navier-Stokes and energy equations in the Boussinesq approximation to calculate numerically the basic steady states, analyze their linear stability, and compute some nonlinear secondary flows originated from the instabilities. All the calculated flows and the stability analysis are characterized by their symmetry properties. Due to the confined cylindrical geometry, -presence of lateral walls and lids-, all the flows are completely three dimensional even for the basic steady states. In the range of Prandtl numbers studied, we have identified four different types of instabilities, either oscillatory or stationary. The physical mechanisms, shear or buoyancy, of the corresponding flow transitions have been analyzed. As the value of the Prandtl number approaches σ = 0.026 the scenario of bifurcations becomes more complicated due to the existence of two different stable basic states originated in a saddle-node bifurcation; a fact that had been overlooked in previous works.

Temporal evolution and scaling of mixing in two-dimensional Rayleigh-Taylor turbulence

We report a high-resolution numerical study of two-dimensional (2D) miscible Rayleigh-Taylor (RT) incompressible turbulence with the Boussinesq approximation. An ensemble of 100 independent realizations were performed at small Atwood number and unit Prandtl number with a spatial resolution of 2048 × 8193 grid points. Our main focus is on the temporal evolution and the scaling behavior of global quantities and of small-scale turbulence properties. Our results show that the buoyancy force balances the inertial force at all scales below the integral length scale and thus validate the basic force-balance assumption of the Bolgiano-Obukhov scenario in 2D RT turbulence. It is further found that the Kolmogorov dissipation scale η(t) ∼ t1/8, the kinetic-energy dissipation rate ɛu(t) ∼ t−1/2, and the thermal dissipation rate ɛθ(t) ∼ t−1. All of these scaling properties are in excellent agreement with the theoretical predictions of the Chertkov model [“Phenomenology of Rayleigh-Taylor turbulence,” Phys. Rev. Lett. 91, 115001 (2003)]10.1103/PhysRevLett.91.115001. We further discuss the emergence of intermittency and anomalous scaling for high order moments of velocity and temperature differences. The scaling exponents \documentclass[12pt]{minimal}\begin{document}$\xi ^r_p$\end{document}ξpr of the pth-order temperature structure functions are shown to saturate to \documentclass[12pt]{minimal}\begin{document}$\xi ^r_{\infty }\simeq 0.78 \pm 0.15$\end{document}ξ∞r≃0.78±0.15 for the highest orders, p ∼ 10. The value of \documentclass[12pt]{minimal}\begin{document}$\xi ^r_{\infty }$\end{document}ξ∞r and the order at which saturation occurs are compatible with those of turbulent Rayleigh-Bénard (RB) convection [A. Celani, T. Matsumoto, A. Mazzino, and M. Vergassola, “Scaling and universality in turbulent convection,” Phys. Rev. Lett. 88, 054503 (2002)]10.1103/PhysRevLett.88.054503, supporting the scenario of universality of buoyancy-driven turbulence with respect to the different boundary conditions characterizing the RT and RB systems.

Analytical shock solutions at large and small Prandtl number

AbstractExact one-dimensional solutions to the equations of fluid dynamics are derived in the $\mathit{Pr}\rightarrow \infty $ and $\mathit{Pr}\rightarrow 0$ limits (where $\mathit{Pr}$ is the Prandtl number). The solutions are analogous to the $\mathit{Pr}= 3/ 4$ solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large-$\mathit{Pr}$ solution is very similar to Becker’s solution, differing only by a scale factor. The small-$\mathit{Pr}$ solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite-$\mathit{Pr}$ equations is $\mathit{O}({\mathit{Pr}}^{- 1} )$ as $\mathit{Pr}\rightarrow \infty $ and $\mathit{O}(\mathit{Pr})$ as $\mathit{Pr}\rightarrow 0$.

Mixed convection above a rotating disk

Mixed convection above a horizontal disk rotating in a semi-infinite fluid is examined when the disk is heated so that its temperature varies quadratically with distance away from its centre. Steady similarity solutions are presented for a range of values of the two dimensionless parameters, a Prandtl number and a Grashof number. The results are corroborated by asymptotic analyses undertaken in the limits of small and large Prandtl number. For finite Prandtl number, the existence of multiple solutions at fixed Grashof number is revealed. The similarity structure for steady solutions fails beyond a critical Grashof number and this is interpreted in terms of a finite-time singularity of the unsteady form of the governing equations.

Turbulent convection in liquid metal with and without rotation

The magnetic fields of Earth and other planets are generated by turbulent, rotating convection in liquid metal. Liquid metals are peculiar in that they diffuse heat more readily than momentum, quantified by their small Prandtl numbers, Pr << 1. Most analog models of planetary dynamos, however, use moderate Pr fluids, and the systematic influence of reducing Pr is not well understood. We perform rotating Rayleigh-Bénard convection experiments in the liquid metal gallium (Pr = 0.025) over a range of nondimensional buoyancy forcing (Ra) and rotation periods (E). Our primary diagnostic is the efficiency of convective heat transfer (Nu). In general, we find that the convective behavior of liquid metal differs substantially from that of moderate Pr fluids, such as water. In particular, a transition between rotationally constrained and weakly rotating turbulent states is identified, and this transition differs substantially from that observed in moderate Pr fluids. This difference, we hypothesize, may explain the different classes of magnetic fields observed on the Gas and Ice Giant planets, whose dynamo regions consist of Pr < 1 and Pr > 1 fluids, respectively.

Magnetic fields generated by hydromagnetic dynamos at the low Prandtl number in dependence on the Ekman and magnetic Prandtl numbers

This article investigates the dependence of hydromagnetic dynamos on the magnetic Prandtl number at low Prandtl number. In all the investigated cases, the generated magnetic fields are dipolar and neither transition to hemispherical dynamos nor weaker magnetic fields (which are less dipole dominated) were observed, although the inertia becomes important. The magnetic field becomes weak in the polar regions (is “convected out of polar regions”) only for low Prandtl numbers, when the inertia becomes important. It is a basic condition. However, whether the magnetic field gets weak in the polar regions (is “convected out of polar regions”) or not depends also on the magnetic Prandtl number. The magnetic Prandtl number has to exceed a minimum value in order to sustain dynamo action. If the magnetic diffusion is small (large magnetic Prandtl numbers) then this phenomenon does not exist but if it is large (small magnetic Prandtl numbers) it exists because the strong magnetic diffusion significantly weakens the magnetic field inside the tangent cylinder. The magnetic diffusion and inertia seem to act in the same direction as to weaken the magnetic field inside the tangent cylinder.

Simulation of mixing different temperature jets using CABARET method

Mathematical modeling of mixing jets with different temperatures discharged from the fuel assemblies of a sodium-cooled fast-neutron reactor BN is presented. Numerical simulations are performed using the CABARET method. The aim is to verify the applicability of the CABARET method to the calculations of liquid flows with a small Prandtl number (~ 0.01) at relatively high Reynolds numbers (~105). For this purpose, the fluid flowing through the geometrically complex channel of the experimental setup has been investigated. Mesh decomposition and finite-difference grid generation techniques are described. The comparison between the numerical and experimental data shows that with the proposed approach the time-averaged local temperatures in the non-isothermal flow of liquid metal can be predicted with a sufficient accuracy, and a good qualitative description of the pulsation characteristics of the flow can be provided.

Numerical Simulation of Mixed Convection in a Rotating Cylindrical Cavity: Influence of Prandtl Number

A numerical study of the flow and heat transfer on a rotating cylindrical cavity solving the mass, momentum, and energy equations is presented in this work. The study describes the influence of the Prandtl number on flow in critical state on a cavity which contains a cooling fluid. Problem studied includes Prandtl numbers 0.001 &lt; Pr &gt; 15, aspect ratios 0.25 &lt; A &gt; 1, and Reynolds numbers 0.001 &lt; Re &gt; 600. Differential equations have been discretised using the finite differences method. The results show a tendency followed by heat transfer as the Reynolds number increases from 300 to 600; in addition, emphasis on the critical values of the Rayleigh number for small Prandtl numbers shows that thermal instability in mixed convection depends on the Prandtl number.

A UNIFYING PICTURE OF HELICAL AND AZIMUTHAL MAGNETOROTATIONAL INSTABILITY, AND THE UNIVERSAL SIGNIFICANCE OF THE LIU LIMIT

The magnetorotational instability (MRI) plays a key role for cosmic structure formation by triggering turbulence in the rotating flows of accretion disks that would be otherwise hydrodynamically stable. In the limit of small magnetic Prandtl number, the helical and the azimuthal version of MRI are known to be governed by a quite different scaling behaviour than the standard MRI with a vertical applied magnetic field. Using the short-wavelength approximation for an incompressible, resistive, and viscous rotating fluid we present a unified description of helical and azimuthal MRI, and we identify the universal character of the Liu limit Ro= -0.8284 for the critical Rossby number. From this universal behaviour we are also lead to the prediction of higher azimuthal wavenumber for rather small ratios of azimuthal to axial applied fields.

CRITICAL FIELDS AND GROWTH RATES OF THE TAYLER INSTABILITY AS PROBED BY A COLUMNAR GALLIUM EXPERIMENT

Many astrophysical phenomena (such as the slow rotation of neutron stars or the rigid rotation of the solar core) can be explained by the action of the Tayler instability of toroidal magnetic fields in the radiative zones of stars. In order to place the theory of this instability on a safe fundament it has been realized in a laboratory experiment measuring the critical field strength, the growth rates as well as the shape of the supercritical modes. A strong electrical current flows through a liquid-metal confined in a resting columnar container with an insulating outer cylinder. As the very small magnetic Prandtl number of the gallium-indium-tin alloy does not influence the critical Hartmann number of the field amplitudes, the electric currents for marginal instability can also be computed with direct numerical simulations. The results of this theoretical concept are confirmed by the experiment. Also the predicted growth rates of the order of minutes for the nonaxisymmetric perturbations are certified by the measurements. That they do not directly depend on the size of the experiment is shown as a consequence of the weakness of the applied fields and the absence of rotation.

Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit

AbstractIn this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\mathit{Pm}= \nu / \eta $. When $\mathit{Pm}\ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour–spectral method, the ‘combined Lagrangian advection method’, formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing $\mathit{Pm}$ for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

Influence of slow rotation on the stability of a thermocapillary incompressible liquid flow in an infinite layer under zero-gravity conditions for small Prandtl number

Instability of a thermocapillary flow arising in a rotating thin infinite liquid layer under zero-gravity conditions is investigated. Both boundaries of the layer are assumed to be plane and free and are subject to the tangential thermocapillary Marangoni force. A convective heat transfer at the boundaries is governed by Newton's law and the temperature of the fluid near the boundaries is a linear function of the coordinates. The axis of rotation is perpendicular to a liquid layer. The rotation is slow, which allows us to neglect the centrifugal force. The examined thermocapillary flow is described analytically, being an exact solution of the Navier–Stokes equations. According to the linear theory of stability the obtained neutral curves depict the dependence of the critical Marangoni number on the wave number at different values of the Taylor number for the small Prandtl number (Pr = 0.1). The behavior of the finite-amplitude perturbations beyond the stability threshold is studied numerically.

The angular momentum transport by standard MRI in quasi-Kepler cylindrical Taylor-Couette flows

The instability of a quasi-Kepler flow in dissipative Taylor-Couette systems under the presence of an homogeneous axial magnetic field is considered with focus to the excitation of nonaxisymmetric modes and the resulting angular momentum transport. The excitation of nonaxisymmetric modes requires higher rotation rates than the excitation of the axisymmetric mode and this the more the higher the azimuthal mode number m. We find that the weak-field branch in the instability map of the nonaxisymmetric modes has always a positive slope (in opposition to the axisymmetric modes) so that for given magnetic field the modes with m>0 always have an upper limit of the supercritical Reynolds number. In order to excite a nonaxisymmetric mode at 1 AU in a Kepler disk a minimum field strength of about 1 Gauss is necessary. For weaker magnetic field the nonaxisymmetric modes decay. The angular momentum transport of the nonaxisymmetric modes is always positive and depends linearly on the Lundquist number of the background field. The molecular viscosity and the basic rotation rate do not influence the related {\alpha}-parameter. We did not find any indication that the MRI decays for small magnetic Prandtl number as found by use of shearing-box codes. At 1 AU in a Kepler disk and a field strength of about 1 Gauss the {\alpha} proves to be (only) of order 0.005.

On the dynamics of 3-D single thermal plumes at various Prandtl numbers and Rayleigh numbers

Three-dimensional (3-D) numerical simulations of single turbulent thermal plumes in the Boussinesq approximation are used to understand more deeply the interaction of a plume with itself and its environment. In order to do so, we varied the Rayleigh and Prandtl numbers from Ra ∼ 105 to Ra ∼ 108 and from Pr ∼ 0.025 to Pr ∼ 70. We found that thermal dissipation takes place mostly on the border of the plume. Moreover, the rate of energy dissipation per unit mass ε T has a critical point around Pr ∼ 0.7. The reason is that at Pr greater than ∼0.7, buoyancy dominates inertia and thermal advection dominates wave formation whereas this trend is reversed at Pr less than ∼0.7. We also found that for large enough Prandtl number (Pr ∼ 70), the velocity field is mostly poloidal although this result was known for Rayleigh–Bénard convection (see Schmalzl et al. [On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 2004, 67, 390--396]). On the other hand, at small Prandtl numbers, the plume has a large helicity at large scale and a non-negligible toroidal part. Finally, as observed recently in details in weakly compressible turbulent thermal plume at Pr = 0.7 (see Plourde et al. [Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction. J. Fluid Mech. 2008, 604, 99--123]), we also noticed a two-time cycle in which there is entrainment of some of the external fluid to the plume, this process being most pronounced at the base of the plume. We explain this as a consequence of calculated Richardson number being unity at Pr = 0.7 when buoyancy balance inertia.

Transient Thermocapillary Convection in a Molten or Weld Pool

This study presents a numerical scenario for the effect of thermocapillary convection on the transient, two-dimensional molten pool shape during welding or melting. Tracing the melting process is necessary to achieve a better and more complete understanding of the physical mechanism of welding. This model is used to simulate a steady state, three-dimensional welding process, by introducing an incident flux with a Gaussian distribution with a time-dependent radius determined by scanning speed and distribution parameter. Aside from presenting the variations of peak surface velocities and temperature, and depth and width of the molten pool with time, the predicted results of this work show that surface velocity and temperature profiles for a high Prandtl number strongly deform in the course of melting. The velocity profile eventually exhibits two peaks, located near the edges of the incident flux and the pool, respectively. Conversely, only one peak velocity occurs near the pool edge for a small Prandtl number. In all cases, surface temperature can ultimately be divided into hot, intermediate, and cold regions. The pool becomes deep due to an induced secondary vortex cell near the bottom of the pool for a small Prandtl number. For a high Prandtl number, the pool edge is thin and shallow, as a result of penetration into the solid near the top surface. The predicted results agree with those obtained using a commercial computer code.