# Frequency Power Spectra of Global Quantities in Unsteady Magnetoconvection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Direct Numerical Simulations

- (i)
- Real space variables $\mathsf{v}(x,y,z)=({\mathrm{v}}_{1},{\mathrm{v}}_{2},{\mathrm{v}}_{3})$ and $\theta (x,y,z)$ are computed at a given time t using inverse FFT of $\mathsf{v}\left(\mathsf{k}\right)$ and $\theta \left(\mathsf{k}\right)$, where $\mathsf{k}=l{k}_{c}{\mathsf{e}}_{1}+m{k}_{c}{\mathsf{e}}_{2}+n\pi {\mathsf{e}}_{3}.$
- (ii)
- The multiplication of field variables ${\mathrm{v}}_{i}(x,y,z){\mathrm{v}}_{j}(x,y,z)$ and ${\mathrm{v}}_{i}(x,y,z)\theta (x,y,z)$ for ($i,j=1,2,3$) are done at each grid point of the simulation box.
- (iii)
- FFT[${\mathrm{v}}_{i}(x,y,z){\mathrm{v}}_{j}(x,y,z)$] and FFT[${\mathrm{v}}_{i}(x,y,z)\theta (x,y,z)$] are computed using the package FFTW.
- (iv)
- Subsequently, the terms $i{k}_{j}\times FFT\left[{\mathrm{v}}_{i}(x,y,z){\mathrm{v}}_{j}(x,y,z)\right]$ and $i{k}_{j}\times FFT\left[{\mathrm{v}}_{i}(x,y,z)\theta (x,y,z)\right]$ with $j=1,2$ as well as $i{k}_{j}\times FFT\left[{\mathrm{v}}_{i}(x,y,z){\mathrm{v}}_{j}(x,y,z)\right]$ and $i{k}_{j}\times FFT\left[{\mathrm{v}}_{i}(x,y,z)\theta (x,y,z)\right]$ with $j=3$ are computed.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Plot of the total temperature field [$T(x,y,z,t)$] and velocity field [$\mathbf{v}(x,y,z,t)$] at dimensionless time (

**a**) $t={t}_{0}$ and (

**b**) $t={t}_{0}+5$ in the simulation box for a water-based nanofluid [${\mathrm{Ra}}_{nf}=5.0\times {10}^{5}$, ${\mathrm{Pr}}_{nf}=6.4$ and ${\mathrm{Q}}_{nf}=100$]. Colour bars describe the temperature distributions in the convecting fluid. Arrows show the directions of the fluid flow.

**Figure 2.**Plot of the total temperature field [$T(x,y,z,t)$] and velocity field [$\mathbf{v}(x,y,z,t)$] for different values of Chandrasekhar’s number: (

**a**) ${\mathrm{Q}}_{nf}=100$ and (

**b**) ${\mathrm{Q}}_{nf}=250$ in the simulation box for a water-based nanofluid [${\mathrm{Ra}}_{nf}=5.0\times {10}^{5}$ and ${\mathrm{Pr}}_{nf}=6.4$. Colour bars and arrows describe the distribution of total temperature field ($\tilde{T}$) and fluid velocity ($\mathrm{v}$), respectively, at a given time t.

**Figure 3.**Temporal variations of the kinetic energy (E), entropy (${E}_{\mathrm{\Theta}}$) and Nusselt number ($\mathrm{Nu}$) in a water-based nanofluid of Prandtl number ${\mathrm{Pr}}_{nf}=4.0$ for the Rayleigh number ${\mathrm{Ra}}_{nf}=5.0\times {10}^{5}$. The red curves are for the Chandrasekhar number ${\mathrm{Q}}_{nf}=100$, while the blue curves are for ${\mathrm{Q}}_{nf}=400$.

**Figure 4.**Frequency power spectral densities (PSD) of (

**a**) the energy per unit mass [$E\left(f\right)={\left|\mathrm{v}\left(f\right)\right|}^{2}$], (

**b**) the convective entropy per unit mass [${E}_{\mathrm{\Theta}}\left(f\right)={\left|\theta \left(f\right)\right|}^{2}$], and (

**c**) thermal flux [$\mathrm{Nu}\left(f\right)$] in the frequency space for Earth’s liquid outer core ($\mathrm{Pr}\sim 0.1,0.2$ ) and for water-based nanofluids with less than $8\%$ of spherical copper nanoparticles (${\mathrm{Pr}}_{nf}\sim 4.0,6.4$) of for different values of $\mathrm{Ra}$, $\mathrm{Q}$, and $\mathrm{Pr}.$

**Figure 5.**Variation of critical values of the dimensionless frequencies: (

**a**) ${f}_{c}\left(E\right)$, (

**b**) ${f}_{c}\left({E}_{\mathrm{\Theta}}\right)$, and (

**c**) ${f}_{c}\left(\mathrm{Nu}\right)$ for the frequency spectra of kinetic energy [$E\left(f\right)$], entropy spectra [${E}_{\mathrm{\Theta}}\left(f\right)$], and thermal flux [$\mathrm{Nu}\left(f\right)$], respectively, with the Chandrasekhar number [$\mathrm{Q}$] for Prandtl number ($\mathrm{Pr}$) =$0.1$ [red triangles] and $1.0$ [blue circles].

**Table 1.**List of Prandtl number $\mathrm{Pr}$, Chandrasekhar number $\mathrm{Q}$, Rayleigh number $\mathrm{Ra}$, exponents of Kinetic energy$\left(\alpha \right)$, exponents of Entropy $\left(\beta \right)$, and exponents of Nusselt number $\left(\gamma \right)$.

$\mathbf{Pr}$ | $\mathbf{Ra}$ | $\mathbf{Q}$ | Exponent $\mathit{\alpha}$ | Exponent $\mathit{\beta}$ | Exponent $\mathit{\gamma}$ |
---|---|---|---|---|---|

$0.1$ | $7.0\times {10}^{4}$ | 100 | $1.97$ | $1.97$ | $1.96$ |

300 | $1.97$ | $1.97$ | $1.97$ | ||

500 | $1.96$ | $1.96$ | $1.97$ | ||

700 | $1.96$ | $1.97$ | $1.97$ | ||

$0.2$ | $7.0\times {10}^{4}$ | 100 | $1.96$ | $1.97$ | $1.96$ |

300 | $1.97$ | $1.97$ | $1.96$ | ||

500 | $1.96$ | $1.96$ | $1.96$ | ||

$1.0$ | $3.04\times {10}^{6}$ | 300 | $1.96$ | $1.97$ | $1.96$ |

500 | $1.96$ | $1.97$ | $1.96$ | ||

700 | $1.97$ | $1.96$ | $1.96$ | ||

1000 | $1.96$ | $1.97$ | $1.97$ | ||

$2.0$ | $3.04\times {10}^{6}$ | 500 | $1.96$ | $1.96$ | $1.96$ |

700 | $1.96$ | $1.96$ | $1.96$ | ||

1000 | $1.96$ | $1.97$ | $1.96$ | ||

$4.0$ | $5.0\times {10}^{5}$ | 100 | $1.96$ | $1.97$ | $1.96$ |

200 | $1.96$ | $1.97$ | $1.97$ | ||

400 | $1.96$ | $1.97$ | $1.97$ | ||

$6.4$ | $5.0\times {10}^{5}$ | 50 | $1.96$ | $1.97$ | $1.96$ |

100 | $1.97$ | $1.97$ | $1.96$ | ||

250 | $1.97$ | $1.96$ | $1.97$ |

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Das, S.; Kumar, K.
Frequency Power Spectra of Global Quantities in Unsteady Magnetoconvection. *Fluids* **2021**, *6*, 163.
https://doi.org/10.3390/fluids6040163

**AMA Style**

Das S, Kumar K.
Frequency Power Spectra of Global Quantities in Unsteady Magnetoconvection. *Fluids*. 2021; 6(4):163.
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**Chicago/Turabian Style**

Das, Sandip, and Krishna Kumar.
2021. "Frequency Power Spectra of Global Quantities in Unsteady Magnetoconvection" *Fluids* 6, no. 4: 163.
https://doi.org/10.3390/fluids6040163