Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example
Aerodynamic heating due to turbulence significantly affects the operation of high-speed vehicles and the entrainment of fluid by turbulent plumes. In this paper, the heat generated and convected by fluid turbulence is examined by rearranging the Navier-Stokes equations into a single equation for the...
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| Format: | Article |
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Wiley
2021-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/7386215 |
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| author | Steven A. E. Miller |
| author_facet | Steven A. E. Miller |
| author_sort | Steven A. E. Miller |
| collection | DOAJ |
| description | Aerodynamic heating due to turbulence significantly affects the operation of high-speed vehicles and the entrainment of fluid by turbulent plumes. In this paper, the heat generated and convected by fluid turbulence is examined by rearranging the Navier-Stokes equations into a single equation for the fluctuating dependent variables external to unsteady chaotic motions. This equation is similar to the nonhomogeneous heat equation where sources are terms resulting from this rearrangement. Mean and fluctuating quantities are introduced, and under certain circumstances, a heat equation for the fluctuating density results with corresponding mean and fluctuating source terms. The resultant equation is similar to Lighthill’s acoustic analogy and is a “heat analogy.” A solution is obtained with the use of Green’s function as long as the observer is located outside the region of chaotic motion. Predictions for the power spectrum are shown for high Reynolds number isotropic turbulence. The power spectrum decays as the inverse of the wavenumber of the turbulent velocity fluctuations. The developed theory can easily be applied to other turbulent flows if the statistics of unsteady motion can be estimated. |
| format | Article |
| id | doaj-art-9ec96e7b557f40209e996f6ffe05334c |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-9ec96e7b557f40209e996f6ffe05334c2025-02-03T01:25:12ZengWileyJournal of Applied Mathematics1110-757X1687-00422021-01-01202110.1155/2021/73862157386215Analytical Solution of Heat Conduction from Turbulence with an Isotropic ExampleSteven A. E. Miller0University of Florida, USAAerodynamic heating due to turbulence significantly affects the operation of high-speed vehicles and the entrainment of fluid by turbulent plumes. In this paper, the heat generated and convected by fluid turbulence is examined by rearranging the Navier-Stokes equations into a single equation for the fluctuating dependent variables external to unsteady chaotic motions. This equation is similar to the nonhomogeneous heat equation where sources are terms resulting from this rearrangement. Mean and fluctuating quantities are introduced, and under certain circumstances, a heat equation for the fluctuating density results with corresponding mean and fluctuating source terms. The resultant equation is similar to Lighthill’s acoustic analogy and is a “heat analogy.” A solution is obtained with the use of Green’s function as long as the observer is located outside the region of chaotic motion. Predictions for the power spectrum are shown for high Reynolds number isotropic turbulence. The power spectrum decays as the inverse of the wavenumber of the turbulent velocity fluctuations. The developed theory can easily be applied to other turbulent flows if the statistics of unsteady motion can be estimated.http://dx.doi.org/10.1155/2021/7386215 |
| spellingShingle | Steven A. E. Miller Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example Journal of Applied Mathematics |
| title | Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example |
| title_full | Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example |
| title_fullStr | Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example |
| title_full_unstemmed | Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example |
| title_short | Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example |
| title_sort | analytical solution of heat conduction from turbulence with an isotropic example |
| url | http://dx.doi.org/10.1155/2021/7386215 |
| work_keys_str_mv | AT stevenaemiller analyticalsolutionofheatconductionfromturbulencewithanisotropicexample |