A Novel Method for Discretizing Gradient Terms in Superfluid Helium Two-fluid Model

In the research of numerical simulation of the two-fluid model for superfluid helium （He II), the issue of numerical oscillations has always been a key concern in the academic community. During the discretization process of the two-fluid model, the traditional cell-centered gradient discretization method has inherent limitations when dealing with temperature gradients. This method is prone to causing numerical oscillations, which not only affect the accuracy of calculation results but also may lead to model instability. When the oscillations caused by the temperature gradient exceed a certain range, the calculation results of the model will deviate from the true values, and in severe cases, it may even lead to model collapse. To address this problem, this paper proposes a face-gradient discretization method for the temperature gradient. By optimizing the discretization approach, it effectively overcomes the numerical oscillations caused by the cell-centered gradient discretization. Specifically, this paper first elaborates on the equations involved in the two discretization methods and the principles of causing numerical oscillations, and then constructs a face-gradient discretization algorithm. Subsequently, on the OpenFOAM^® platform, taking the thermal counterflow phenomenon of He II as the computational object, the face-gradient discretization method is applied. Through the comparative analysis of the numerical simulation results and the analytical solution, the accuracy of the face-gradient discretization method is verified. In addition, to further evaluate the performance of the face-gradient discretization method, the author also compares the numerical simulation results of using the face-gradient discretization method and the cell-centered gradient discretization method. The results show that the cell-centered gradient discretization method will show obvious numerical oscillations during the numerical simulation process, while the face-gradient discretization method can effectively suppress such oscillations. This research provides a more reliable and stable discretization method for the numerical simulation of the superfluid helium two-fluid model, which is conducive to promoting the research progress in related fields.