Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations
Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations
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具有无限普朗特数的可压缩对流的数值解：滞弹性和滞弹性液体模型与精确方程的比较
DOI：
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发表时间：
2019
影响因子：
3.7
通讯作者：
Y. Ricard
中科院分区：
文献类型：
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作者：
J. Curbelo;L. Duarte;T. Alboussière;F. Dubuffet;S. Labrosse;Y. Ricard
我们发展了一种数值方法来求解在无限普朗特数极限下控制完全可压缩对流的方程组。还分析了简化模型，如滞弹性近似和滞弹性液体近似。针对自洽准则对数值格式的测试表明，我们的数值模拟从能量耗散、热传递和熵收支的角度来看是一致的。这项工作考虑了理想气体的状态方程。研究了由于流体的可压缩性而产生的特殊影响，如粘性耗散的比例和粘性力所产生的机械功率对热通量贡献的比例。我们分析了各种模型(完全可压缩模型、滞弹性和滞弹性液体近似)在广泛的无量纲参数范围内所得到的解，并确定了每种近似相对于完全可压缩解的误差。基于热边界层发展的理论基础，我们可以合理地解释完全可压缩模型和滞弹性模型之间的差异，无论是换热还是粘性耗散对可压缩性的依赖。这主要可能是密度变化对热扩散率的影响。基于精确模型和滞弹性模型之间的不同形式的熵平衡，我们发现滞弹性结果收敛到精确解的一个必要条件是乘积$UNICODE[STIX]{x1D716}Q$必须小于1，其中$UNICODE[STIX]{x1D716}$是超绝热温差与绝热差之比，$Q$是超绝热热流与沿绝热流传热的比率。同样的条件似乎也与计算的热通量的收敛有关。关于滞弹性液体近似，我们证实了AnuFriev等人以前的估计。(物理。地球星球。国际文集，第152卷，2005年，第163-190页)发现，当$UNICODE[STIX]{x1D6FC}T{Mathcal{D}}$小于单位时，其结果与完全可压缩模型的结果大体接近，其中$UNICODE[STIX]{x1D6FC}$是等压热膨胀系数，$T$是温度(对于理想气体，$UNICODE[STIX]{x1D6FC}T=1$)和${mathcal{D}}$是耗散数。
We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $unicode[STIX]{x1D716}q$ must be small compared to unity, where $unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $unicode[STIX]{x1D6FC}T{mathcal{D}}$ is small compared to unity, where $unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${mathcal{D}}$ is the dissipation number.
DOI：
10.1088/0004-637x/805/1/62
发表时间：
2015-01
期刊：
The Astrophysical Journal
影响因子：
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作者：
J. Verhoeven;T. Wiesehöfer;S. Stellmach
通讯作者：
J. Verhoeven;T. Wiesehöfer;S. Stellmach