Benchmark solutions for three-dimensional transient heat transfer in two-dimensional environments via the time Fourier transform.

*(English)*Zbl 1107.80011Summary: The evaluation of heat propagation in the time domain generated by transient heat sources placed in the presence of three-dimensional media requires the use of computationally demanding numerical schemes. The implementation of numerical 3D solutions may benefit from the existence of benchmark solutions to test the accuracy of approximate schemes.

With this purpose in mind, this article presents analytical-numerical solutions to evaluate the heat-field elicited by monopole heat sources in the presence of three different inclusions, namely, a cylindrical circular solid inclusion, a cylindrical circular cavity with null fluxes and a cavity with null temperatures prescribed along its boundary, buried in an unbounded medium. The problem is first subjected to a time and space Fourier transform, which allows the solution to be obtained in the frequency domain as summation of 2D solutions for different spatial wavenumbers. Then, using the inverse Fourier transforms in the wavenumber and frequency domains, the 3D time responses are synthesized. Complex frequencies are used to avoid the aliasing phenomena.

This methodology is first validated calculating the fundamental time solutions for one, two and three dimensions in an unbounded medium. Simulation analyses of these idealized models are then used to study the patterns of heat propagation in the vicinity of the inclusions.

With this purpose in mind, this article presents analytical-numerical solutions to evaluate the heat-field elicited by monopole heat sources in the presence of three different inclusions, namely, a cylindrical circular solid inclusion, a cylindrical circular cavity with null fluxes and a cavity with null temperatures prescribed along its boundary, buried in an unbounded medium. The problem is first subjected to a time and space Fourier transform, which allows the solution to be obtained in the frequency domain as summation of 2D solutions for different spatial wavenumbers. Then, using the inverse Fourier transforms in the wavenumber and frequency domains, the 3D time responses are synthesized. Complex frequencies are used to avoid the aliasing phenomena.

This methodology is first validated calculating the fundamental time solutions for one, two and three dimensions in an unbounded medium. Simulation analyses of these idealized models are then used to study the patterns of heat propagation in the vicinity of the inclusions.

##### MSC:

80M25 | Other numerical methods (thermodynamics) (MSC2010) |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

65T50 | Numerical methods for discrete and fast Fourier transforms |