Analytical shock solutions at large and small Prandtl number
Analytical shock solutions at large and small Prandtl number
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DOI：
10.1017/jfm.2013.262
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发表时间：
2013-05
影响因子：
3.7
通讯作者：
B. M. Johnson
中科院分区：
文献类型：
--
作者：
B. M. Johnson
在$\mathit{Pr}\rightarrow \infty $和$\mathit{Pr}\rightarrow 0$极限（$\mathit{Pr}$为普朗特数）下，导出了流体动力学方程的精确一维解。这些解决方案类似于Becker发现的$\mathit{Pr}= 3/ 4$解决方案，并分析地捕获了理想气体中激波锋面的轮廓。大的$\mathit{Pr}$溶液与贝克尔的溶液非常相似，只是有一个比例因子的区别。小的$\mathit{Pr}$解决方案在质量上是不同的，与一个嵌入等温激波发生在临界马赫数以上。对于恒定粘度和电导率以及由辐射场提供传导的情况，推导出了解决方案。对于完全与密度和温度相关的一般粘度和电导率，在所有三个极限下的方程组可以简化为正交。与有限- $\mathit{Pr}$方程的数值积分相比，解析解的最大误差为$\mathit{O}({\mathit{Pr}}^{- 1} )$为$\mathit{Pr}\rightarrow \infty $, $\mathit{O}(\mathit{Pr})$为$\mathit{Pr}\rightarrow 0$。
Abstract Exact one-dimensional solutions to the equations of fluid dynamics are derived in the $\mathit{Pr}\rightarrow \infty $ and $\mathit{Pr}\rightarrow 0$ limits (where $\mathit{Pr}$ is the Prandtl number). The solutions are analogous to the $\mathit{Pr}= 3/ 4$ solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large- $\mathit{Pr}$ solution is very similar to Becker’s solution, differing only by a scale factor. The small- $\mathit{Pr}$ solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite- $\mathit{Pr}$ equations is $\mathit{O}({\mathit{Pr}}^{- 1} )$ as $\mathit{Pr}\rightarrow \infty $ and $\mathit{O}(\mathit{Pr})$ as $\mathit{Pr}\rightarrow 0$ .