Development and Assessment of an Isotropic Four-Equation Model for Heat Transfer of Low Prandtl Number Fluids
Xingkang Su, Xianwen Li, Xiangyang Wang, Yang Liu, Qijian Chen, Qianwan Shi, Xin Sheng, Long Gu
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.