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ABSTRACT
The present paper, based on the vorticity–velocity formulation of the Navier–Stokes equations, proposes an immersed boundary method for the simulation of heat transfer problems within a geometrically complex domain. The desired boundary conditions are imposed by the direct modification of the initial conditions of vorticity transport and energy equations using smooth interpolations. The time advancement of both transport equations is performed by the explicit fourth-order Runge–Kutta method. One of the main objectives of this paper is to present global smooth interpolations to evaluate the local Nusselt number. The forced convection of moving and fixed circular cylinders, natural convection problem in complex geometries, and the mixed convection between two concentric cylinders—at various Reynolds numbers—are studied.
Nomenclature
| A | = | base circle radius |
| aik | = | coefficients of cubic function |
| B | = | amplitude |
| C | = | number of undulations |
| d | = | cylinder diameter |
| dk | = | Euclidean distance |
| dik | = | Euclidean distance between i, k |
| D | = | solution domain |
| = | D together with boundary | |
| Di | = | inner diameter |
| ez | = | unit vector in z direction |
| F | = | interpolation function |
| g | = | acceleration due to gravity |
| h | = | convective heat transfer |
| i | = | imaginary unit |
| k | = | thermal conductivity |
| keq | = | equivalent thermal conductivity |
| k1 | = | wave number in x direction |
| |k|2 | = | wave number magnitude |
| k⊥ | = | perpendicular wave number |
| L | = | wake length |
| = | linear term in Fourier space | |
| M | = | number of nodes in fluid |
| n | = | normal unit vector |
| N | = | total number of nodes |
| = | nonlinear term in Fourier space | |
| Nc | = | number of points used in the least square fit |
| Pr | = | Prandtl number |
| Q | = | cubic function |
| r | = | radial direction |
| = | radius about point k | |
| Rw | = | radius of influence |
| Ra | = | Rayleigh number |
| Re | = | Reynolds number |
| = | nondimensional time | |
| T | = | total time |
| T | = | temperature |
| u | = | velocity vector |
| U0 | = | cylinder velocity |
| W | = | weight function |
| α | = | thermal diffusivity |
| β | = | volume expansion coefficient |
| φ | = | generic variable |
| Φ | = | weight function for minimization |
| Γs | = | solid boundary |
| ΓD | = | boundary of domain |
| ω | = | vorticity |
| ν | = | kinematic viscosity |
| λ | = | distance ratio |
| Ωf | = | fluid region |
| Subscripts | = | |
| Sol | = | solenoidal parameter |
| Superscripts | = | |
| BC | = | boundary conditioned parameter |
Nomenclature
| A | = | base circle radius |
| aik | = | coefficients of cubic function |
| B | = | amplitude |
| C | = | number of undulations |
| d | = | cylinder diameter |
| dk | = | Euclidean distance |
| dik | = | Euclidean distance between i, k |
| D | = | solution domain |
| = | D together with boundary | |
| Di | = | inner diameter |
| ez | = | unit vector in z direction |
| F | = | interpolation function |
| g | = | acceleration due to gravity |
| h | = | convective heat transfer |
| i | = | imaginary unit |
| k | = | thermal conductivity |
| keq | = | equivalent thermal conductivity |
| k1 | = | wave number in x direction |
| |k|2 | = | wave number magnitude |
| k⊥ | = | perpendicular wave number |
| L | = | wake length |
| = | linear term in Fourier space | |
| M | = | number of nodes in fluid |
| n | = | normal unit vector |
| N | = | total number of nodes |
| = | nonlinear term in Fourier space | |
| Nc | = | number of points used in the least square fit |
| Pr | = | Prandtl number |
| Q | = | cubic function |
| r | = | radial direction |
| = | radius about point k | |
| Rw | = | radius of influence |
| Ra | = | Rayleigh number |
| Re | = | Reynolds number |
| = | nondimensional time | |
| T | = | total time |
| T | = | temperature |
| u | = | velocity vector |
| U0 | = | cylinder velocity |
| W | = | weight function |
| α | = | thermal diffusivity |
| β | = | volume expansion coefficient |
| φ | = | generic variable |
| Φ | = | weight function for minimization |
| Γs | = | solid boundary |
| ΓD | = | boundary of domain |
| ω | = | vorticity |
| ν | = | kinematic viscosity |
| λ | = | distance ratio |
| Ωf | = | fluid region |
| Subscripts | = | |
| Sol | = | solenoidal parameter |
| Superscripts | = | |
| BC | = | boundary conditioned parameter |