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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Main Author: Steven A. E. Miller
Format: Article
Language:English
Published: Wiley 2021-01-01
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.
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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