A new energy-consistent discretization of the viscous dissipation function in incompressible flows is proposed. It is implied by choosing a discretization of the diffusive terms and a discretization of the local kinetic energy equation and by requiring that continuous identities like the product rule are mimicked discretely. The proposed viscous dissipation function has a quadratic, strictly dissipative form, for both simplified (constant viscosity) stress tensors and general stress tensors. The proposed expression is not only useful in evaluating energy budgets in turbulent flows, but also in natural convection flows, where it appears in the internal energy equation and is responsible for viscous heating. The viscous dissipation function is such that a consistent total energy balance is obtained: the 'implied' presence as sink in the kinetic energy equation is exactly balanced by explicitly adding it as source term in the internal energy equation. Numerical experiments of Rayleigh-B\'enard convection (RBC) and Rayleigh-Taylor instabilities confirm that with the proposed dissipation function, the energy exchange between kinetic and internal energy is exactly preserved. The experiments show furthermore that viscous dissipation does not affect the critical Rayleigh number at which instabilities form, but it does significantly impact the development of instabilities once they occur. Consequently, the value of the Nusselt number on the cold plate becomes larger than on the hot plate, with the difference increasing with increasing Gebhart number. Finally, 3D simulations of turbulent RBC show that energy balances are exactly satisfied even for very coarse grids; therefore, we consider that the proposed discretization forms an excellent starting point for testing sub-grid scale models.

翻译：本文提出了一种不可压缩流动中粘性耗散函数的新型能量一致性离散方法。该方法通过选择扩散项的离散格式与局部动能方程的离散格式，并要求连续恒等式（如乘积法则）在离散层面得到复现而实现。所提出的粘性耗散函数对于简化（恒定粘度）应力张量及一般应力张量均具有二次严格耗散形式。该表达式不仅可用于评估湍流中的能量收支，更适用于自然对流——在自然对流中，粘性耗散出现在内能方程中，是粘性加热的成因。该粘性耗散函数使得总能量守恒得以精确实现：其作为动能方程中"隐含"的汇项，与内能方程中显式添加的源项达到精确平衡。瑞利-贝纳德对流（RBC）与瑞利-泰勒不稳定性的数值实验证实，采用该耗散函数时，动能与内能之间的能量交换被精确保持。实验还表明，粘性耗散不影响失稳形成的临界瑞利数，但会显著影响失稳后的发展过程。因此，冷板上的努塞尔数大于热板的值，且差值随格布哈特数增加而增大。最后，湍流RBC的三维模拟显示，即使在极粗网格上也能精确满足能量守恒；据此，我们认为所提出的离散格式为亚格子模型的测试提供了极佳起点。