Development and Assessment of an Isotropic Four-Equation Model for Heat Transfer of Low Prandtl Number Fluids

Abstract

In the simple gradient diffusion hypothesis, the turbulent Prandtl number ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mi>P</mml:mi><mml:msub><mml:mi>r</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math> ) with a constant of 0.85 is difficult to accurately predict for liquid metals having low Prandtl numbers ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mi>P</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> ), while a four-equation model can improve this solution by introducing the turbulence time-scale into the calculation of turbulent thermal diffusivity. However, the four-equation model’s transport form and numerical stability are so complex that suitable commercial code is lacking. Therefore, an isotropic four-equation model with simple Dirichlet wall boundary conditions is built in the present work. Based on the open-source computational fluid dynamics program OpenFOAM, the fully developed velocity, temperature, Reynolds stress, and heat flux of low <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mi>P</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> fluids ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mi>P</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> = 0.01–0.05) in the parallel plane are obtained by numerical simulation. The results show that the time-average statistics predicted using the present four-equation model are in good agreement with the direct numerical simulation data. Then, the isotropic four-equation model is used to analyze the flow and heat of liquid metal ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mrow><mml:mi>P</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> = 0.01) in a quadrilateral infinite rod bundle. The numerical results are compared with the various and available experimental relationships. The Nusselt numbers calculated using the isotropic four-equation model are betweenness the available correlations, while the turbulent Prandtl number model using a constant of 0.85 over predicts heat transfer. More detailed local heat transfer phenomena and distribution of low Pr fluids are obtained using the present isotropic four-equation model.