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Frequency Power Spectra of Global Quantities in Magnetoconvection

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Frequency Power Spectra of Global Quantities in Magnetoconvection Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 March 2021 Article Frequency power spectra of global quantities in magnetoconvection Sandip Das and Krishna Kumar* [1] Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India * Correspondence: [email protected] 1 Abstract: We present the results of direct numerical simulations of power spectral densities for 2 kinetic energy, convective entropy and heat flux for unsteady Rayleigh-Bénard magnetoconvection 3 in the frequency space. For larger values of frequency, the power spectral densities for all the −2 4 global quantities vary with frequency f as f . The scaling exponent is independent of Rayleigh 5 number, Chandrasekhar’s number and thermal Prandtl number. 6 Keywords: Magnetoconvection, Turbulent flow, Power spectral densities (PSD) 7 1. Introduction 8 The temporal fluctuations of spatially averaged (or, global) quantities are of interest 9 in several fields of research including turbulent flows [1–4], nanofluids [5], biological 10 fluids [6,7], geophysics [8,9], phase transitions [10,11]. The probability density function 11 (PDF) of the temporal fluctuations of thermal flux in turbulent Rayleigh-Bénard con- 12 vection (RBC) was found to have normal distribution with slight asymmetries at the 13 tails. The direct numerical simulations (DNS) of the Nusselt number Nu, which is a 14 measure of thermal flux, also showed the similar behaviour in presence of the Lorentz 15 force [12]. The power spectral density (PSD) of the thermal flux in the frequency ( f ) −2 16 space [3,12,13] was found to vary as f . In this work, we present the results obtained 17 by DNS of temporal signals of global quantities: spatially averaged kinetic energy per Citation: Das, S.; Kumar, K. 18 unit mass E, convective entropy per unit mass EQ and Nusselt number Nu in unsteady Frequency power spectra of global 19 Rayleigh-Bénard magnetoconvection (RBM) [14–16]. The kinetic energy as well as the −2 quantities in magnetoconvection. 20 entropy vary with frequency as f at relatively higher frequencies. In this scaling Journal Not Specified 2021, 1, 0. 21 regime, the scaling exponent does not depend on the Rayleigh number Ra, Prandtl https://doi.org/ 22 number Pr and Chandrasekhar’s number Q. Received: 23 2. Governing equations Accepted: 24 The physical system consists of a thin layer of a Boussinesq fluid (e.g., liquid metals, Published: 25 melt of some alloys (i.e., NaNO3 melt), nanofluids, etc.) of density r0 and electrical 26 conductivity s confined between two horizontal plates, which are made of electrically Publisher’s Note: MDPI stays neu- 27 non-conducting but thermally conducting materials. The lower plate is heated uniformly tral with regard to jurisdictional 28 and the upper plate is cooled uniformly so that an adverse temperature gradient b is claims in published maps and insti- 29 maintained across the fluid layer. A uniform magnetic field B0 is applied in the vertical tutional affiliations. 30 direction. The positive direction of the z- axis is in the direction opposite to that of the 31 acceleration due to gravity g. The basic state is the conduction state with no fluid motion. Copyright: c 2021 by the authors. 32 The stratification of the steady temperature field Ts(z), fluid density rs(z) and pressure Submitted to Journal Not Specified 33 field Ps(z), in the conduction state [14], are given as: for possible open access publication under the terms and conditions Ts(z) = Tb + bz, (1) of the Creative Commons Attri- rs(z) = r0[1 + a(Tb − Ts(z))], (2) bution (CC BY) license (https:// " 2 # creativecommons.org/licenses/by/ 1 2 B0 Ps(z) = P0 − r0g z + abz + , (3) 4.0/). 2 8m0p Version March© 2021 5, 2021by the submitted author(s). to Journal Distributed Not Specified under a Creative Commons CC BY license. https://www.mdpi.com/journal/notspecified Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 March 2021 Version March 5, 2021 submitted to Journal Not Specified 2 of 8 34 where Tb and r0 are temperature and density of the fluid at the lower plate, respectively. 35 P0 is a constant pressure in the fluid and m0 is the permeability of the free space. 36 As soon as the temperature gradient across the fluid layer is raised above a critical 37 value bc for fixed values of all fluid parameters (kinematic viscosity n, thermal diffusivity 38 k, thermal expansion coefficient a) and the externally imposed magnetic field B0, the 39 convection sets in. All the fields are perturbed due to convection and they may be 40 expressed as: rs(z) ! r˜(x, y, z, t) = rs(z) + dr(x, y, z, t), (4) Ts(z) ! T(x, y, z, t) = Ts(z) + q(x, y, z, t), (5) Ps(z) ! P(x, y, z, t) = Ps(z) + p(x, y, z, t), (6) B0 ! B(x, y, z, t) = B0 + b(x, y, z, t), (7) 41 where v(x, y, z, t), p(x, y, x, t), q(x, y, z, t) and b(x, y, z, t) are the fluid velocity, per- 42 turbation in the fluid pressure and the convective temperature and the induced magnetic 43 field, respectively, due to convective flow. The perturbative fields are made dimension- 44 less by measuring all the length scales in units of the clearance d between two horizontal 45 plates, which is also the thickness of the fluid layer. The time is measured in units of the p 46 free fall time tf = 1/ abg. The convective temperature field q and the induced mag- 47 netic field b are dimensionless by bd and B0Pm, respectively. The magnetoconvective 48 dynamics is then described by the following dimensionless equations: q Pr 2 QPr Dtv = −rp + Ra r v + Ra ¶zb + qe3, (8) q 2 Ra r b = − Pr ¶zv, (9) q 1 2 Dtq = RaPr r q + v3, (10) r · v = r · b = 0, (11) 49 where Dt ≡ ¶t + (v · r) is the material derivative. As the magnetic Prandtl number Pm −5 50 is very small (≤ 10 ) for all terrestrial fluids, we set Pm equal to zero in the above. The 51 induced magnetic field is then slaved to the velocity field. We consider the idealized 52 boundary (stress-free) conditions for the velocity field on the horizontal boundaries. The 53 relevant boundary conditions [14,17] at horizontal plates, which are located at z = 0 and 54 z = 1, are: ¶v ¶v ¶b 1 = 2 = v = b = b = 3 = q = 0. (12) ¶z ¶z 3 1 2 ¶z 55 All fields are considered periodic in the horizontal plane. The dynamics of the flow 56 (as Pm ! 0) is controlled by three dimensionless parameters: (1) Rayleigh number 4 2 2 abgd n sB0 d 57 Ra = , (2) Prandtl number Pr = and (3) Chandrasekhar’s number Q = . nk k r0n 58 The critical values of Rayleigh number Rac and the critical wave number kc are [14]: 2 2 p +kc 2 2 2 2 Rac(Q) = 2 (p + kc ) + p Q , (13) kc q 1 kc(Q) = p a+ + a− − 2 , (14) 59 where 1 ! 3 1 h 1 Q 1 Q 2 1 1 i a± = + ± + − 2 . (15) 4 2 p2 2 p2 4 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 March 2021 Version March 5, 2021 submitted to Journal Not Specified 3 of 8 60 The kinetic energy E and convective entropy EQ per unit is mass are defined as: E = 1 R 2 1 R 2 61 2 v dV and EQ = 2 q dV, respectively. The Nusselt number Nu , which is the 62 ratio of totalp heat flux and the conductive heat flux across the fluid layer, is defined as: RaPr R 63 Nu = 1 + V v3qdV. 64 The system of equations may also be useful for investigating magnetoconvection in 65 nanofluids with low concentration non-magnetic metallic nanoparticles [12]. A homoge- 66 neous suspension of nanoparticles in a viscous fluid works as a nanofluid. As the fluid 67 properties depend on the base fluid and the nano-particles, their effective values may 68 be used for the nanofluid. All fluid parameters are may be replaced by their effective 69 values in the presence of nanoparticles in a simple model. If f is the volume fraction 70 of the spherically shaped nanoparticles, the effective form of the density and electrical 71 conductivity of the nanofluid may be expressed as: r = (1 − f)r f + frp, (16) s = (1 − f)sf + fsp, (17) 72 where r f and sf are the density and electrical conductivity of the base fluid, respectively. 73 Here rp is the density and sp is the electrical conductivity of the nanoparticles. The 74 effective thermal conductivity K [18] is expressed as: " # (Kp + 2K f ) − 2f(K f − Kp) K = K f , (18) (Kp + 2K f ) + f(K f − Kp) 75 where K f and Kp are the thermal conductivity of the base fluid and that of the spherical 76 shaped nanoparticles, respectively. Similarly, the effective specific ccapacity cV may be 77 expressed through the following relation [19]: (rcV ) = (1 − f)(rcV ) f + f(rcV )p. (19) 78 The effective dynamic viscosity m of the nanofluid [20] may also be expressed as: −2.5 m = m f (1 − f) . (20) 79 The relevant values of effective fluid parameters may be used in the set of equations8-11 80 for investigating flow properties in nanofluids. 81 3. Direct Numerical Simulations 82 The direct numerical simulations are carried out using pseudo-spectral method. The 83 perturbative fields are expanded as: ik(lx+my) Y(x, y, z, t) = ∑ Ylmn(t)e cos (npz), (21) l,m,n ik(lx+my) F(x, y, z, t) = ∑ Flmn(t)e sin (npz), (22) l,m,n † † 84 where Y(x, y, z, t) = [v1, v2, p] and F(x, y, z, t) = [v3, q] .
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