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Natural convection of a heated paddle wheel within a cross-shaped cavity filled with a nanofluid: ISPH simulations

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Abstract

The natural convection resultant from the uniform circular rotation of the paddle wheel in a cross-shaped porous cavity filled by \(\hbox {Al}_{2}\hbox {O}_{3}{-}\hbox {H}_{2}\hbox {O}\) was simulated by the ISPH method. A cross-shaped cavity’s two vertical area is saturated with a homogeneous porous medium, whereas the entire horizontal area is saturated with a heterogeneous porous medium. The paddle wheel rotates with a uniform circular velocity around the cavity’s center. The paddle wheel’s entire integrated body has temperature \(T_\mathrm{h}\). The temperature is set on the inside walls of a cross-shaped cavity \(T_\mathrm{c}\). The present geometry can be used to analyze and comprehend the thermo-physical behaviors of electronic motors. Angular velocity is set to \(\omega =7.15\), and thus, the natural convection case is only evaluated due to the low speed of inner rotating shape. The simulation results are graphically represented for temperature distributions, velocity fields, and tabular representations for the average Nusselt number. The important parameter ranges are the Rayleigh number \((10^{3} \le \hbox {Ra} \le 10^{6})\), paddle wheel length \((2.5 \le L_\mathrm{P} \le 14)\), nanoparticles parameter \((0 \le \phi \le 0.05)\), and Darcy parameter \((10^{-3} \le \hbox {Da} \le 10^{-5})\). The results show that increasing the length of the paddle wheel increases heat transfer and nanofluid movements within a cross-shaped cavity. In addition, increasing the Rayleigh number improves heat transfer and the nanofluid speed inside a cross-shaped cavity. When the Darcy parameter is reduced, the fluid flow is restricted to the rotating inner shape. The value of \({\overline{\hbox {Nu}}}\) powers as the length of the paddle wheel and \(\phi \) are increasing.

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Abbreviations

\(C_\mathrm{p}\) :

Heat capacity

Da:

Darcy parameter

\(D_\mathrm{eff}\) :

Effective diffusivity

\(\mathrm {g}\) :

Gravitational acceleration

\(H_\mathrm{C}\) :

Height of a cross-shaped cavity

\(L_\mathrm{C}\) :

Length of a cross-shaped cavity

\(K_{0}\) :

Permeability

\(K_\mathrm{B}\) :

Boltzmann’s coefficient

k :

Thermal conductivity

\(L_\mathrm{P}\) :

Paddle wheel length

P :

Dimensionless pressure

\(T_\mathrm{rf}\) :

Water freezing point

Pr:

Prandtl number

m :

Mass

t :

Dimensional time

T :

Dimensional temperature

\(r_{ij}\) :

Distance between particles

U, V :

Dimensionless velocities

Ra:

Rayleigh number

u, v :

Dimensional velocities

\(u_\mathrm{B}\) :

Brownian velocity

h :

Smooth length

p :

Dimensional pressure

\(R_\mathrm{c}\) :

Radius of a circular cylinder

\({\overline{\hbox {Nu}}}\) :

Average Nusselt number

x, y :

Dimensional coordinates

W :

Smooth function

X, Y :

Dimensionless coordinates

\(\alpha \) :

Effective thermal diffusivity

\(\alpha _\mathrm{d}\) :

Constant of a smooth function

\(\beta \) :

Thermal expansion coefficient

\(\eta \) :

Constant (0.0001 h)

\(\eta _{1}\) :

Alter rate of \(\ln (K)\) in X-axis

\(\eta _{2}\) :

Alter rate of \(\ln (K)\) in Y-axis

\(\phi \) :

Nanoparticles parameter

\(\varepsilon \) :

Porosity

\(\gamma \) :

Relaxation coefficient

\(\Phi _{i}\) :

Any function

\(\rho ^\mathrm{num}\) :

Density in SPH form

\(\mu \) :

Dynamic viscosity

\(\tau \) :

Dimensionless time

\(\theta \) :

Dimensionless temperature

\(\rho \) :

Density

c:

Cold

h:

Hot

f:

Fluid

p:

Porous medium

nf:

Nanofluid

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant Number (RGP.1/254/42). This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha, 61413, Saudi Arabia

    Abdelraheem Mahmoud Aly

  2. Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

    Abdelraheem Mahmoud Aly

  3. Electrical Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, P.O. Box 11991, Wadi Addwasir, Saudi Arabia

    Ehab Mahmoud Mohamed

  4. Electrical Engineering Department, Faculty of Engineering, Aswan University, P.O. Box 81542, Aswân, Egypt

    Ehab Mahmoud Mohamed

  5. Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia

    Noura Alsedais

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Aly, A.M., Mohamed, E.M. & Alsedais, N. Natural convection of a heated paddle wheel within a cross-shaped cavity filled with a nanofluid: ISPH simulations. Arch Appl Mech 91, 4441–4458 (2021). https://doi.org/10.1007/s00419-021-02019-8

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