可压缩Navier-Stokes方程中的等温壁：仿射熵、弹道能量与熵稳定离散格式
Isothermal walls in the compressible Navier-Stokes equations: affine entropy, ballistic energy, and entropy-stable discrete formulations
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中文总结 AI 辅助
针对可压缩Navier-Stokes方程，提出熵稳定的等温壁条件，通过仿射熵修正和弹道能量实现壁面贡献消除，并验证了高阶收敛与复杂流动适用性。
中文摘要 AI 辅助
我们在连续和半离散层面为可压缩Navier-Stokes方程发展了熵稳定的等温壁条件。对于恒定正温度下的静止、不可渗透、无滑移壁面，将总能密度除以壁温加到经典凸熵上，可消除壁面贡献而不抑制热传递。这种仿射修正保持了熵Hessian、相对熵和体积熵产生。它正比于总弹道能量密度，并在所述假设下基于初始场产生一个条件性的$L_2$型界。两种壁面公式使用具有求和-分部和同时逼近项的公共间断Galerkin离散。两者在无惩罚时给出零自适应壁面贡献，在有惩罚时给出非正贡献。完整残差公式惩罚完整的壁面数据残差，并通过其完整的数值壁面通量平衡流体能量。傅里叶能量精确公式将熵中性斜校正与动量投影惩罚相结合，使得在每个壁面节点和空间分辨率下，完整的数值外能通量等于单侧数值傅里叶热通量。逐点测试验证了局部恒等式。具有熵稳定内部耦合的制造解研究证明了在离散$L_2$范数下与阶$p_s+1$一致的收敛速率，其中$p_s$是解多项式次数。热弛豫计算验证了半离散能量和自适应熵平衡。三维计算证明了封闭域中的平衡闭合、曲壁迹线的收敛性，以及对复杂超声速分离流的适用性。
英文摘要
We develop entropy-stable isothermal-wall conditions for the compressible Navier--Stokes equations at continuous and semi-discrete levels. For a stationary, impermeable, no-slip wall at constant positive temperature, adding total-energy density divided by wall temperature to the canonical convex entropy cancels the wall contribution without suppressing heat transfer. This affine modification preserves the entropy Hessian, relative entropy, and volume entropy production. It is proportional to total ballistic-energy density and yields a conditional $L_2$-type bound based on the initial field under the stated assumptions. Two wall formulations use a common discontinuous Galerkin discretization with the summation-by-parts property and simultaneous approximation terms. Both give zero adapted wall contribution without a penalty and a nonpositive contribution with it. The complete-residual formulation penalizes the full wall-data residual and balances fluid energy through its complete numerical wall flux. The Fourier-energy-exact formulation combines an entropy-neutral skew correction with a momentum-projected penalty so that the complete numerical outward energy flux equals the one-sided numerical Fourier heat flux at every wall node and spatial resolution. Pointwise tests verify the local identities. Manufactured-solution studies with entropy-stable interior coupling demonstrate convergence rates consistent with order $p_s+1$ in the discrete $L_2$ norm, where $p_s$ is the solution polynomial degree. Thermal-relaxation calculations verify the semi-discrete energy and adapted-entropy balances. Three-dimensional calculations demonstrate balance closure in a closed domain, convergence of curved-wall traces, and applicability to a complex supersonic separated flow.
发表机构
- King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
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