# 一类广义压力下的二维可压缩欧拉方程的特征分解Characteristic Decomposition of the Two-Dimensional Compressible Euler System for a Class of Pressure Laws

DOI: 10.12677/PM.2021.114073, PDF, HTML, XML, 下载: 15  浏览: 63

Abstract: This paper is considering a two-dimensional compressible Euler system for a class of pressure laws, a direct method is used to discuss the sufficient conditions for the existence of the characteristic decomposition and the decompositions of the pressure p and the characteristics Λ± are obtained. It is found that any wave adjacent to a constant state is a simple wave.

1. 引言

$\left\{\begin{array}{l}{\rho }_{t}+{\left(\rho u\right)}_{x}+{\left(\rho v\right)}_{y}=0,\\ {\left(\rho u\right)}_{t}+{\left(\rho {u}^{2}+p\right)}_{x}+{\left(\rho uv\right)}_{y}=0,\\ {\left(\rho v\right)}_{t}+{\left(\rho uv\right)}_{x}+{\left(\rho {v}^{2}+p\right)}_{y}=0,\end{array}$ (1)

${\rho }^{\prime }\left(c\right)=\frac{2c}{{p}^{″}\left(\rho \right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}^{\prime }\left(\rho \right)=\frac{2\rho {p}^{″}{\left(\rho \right)}^{2}-2{\rho }^{\prime }\left(c\right){p}^{\prime }\left(\rho \right){p}^{″}\left(\rho \right)-2\rho {p}^{\prime }\left(\rho \right){p}^{‴}\left(\rho \right)}{{\rho }^{2}{p}^{″}{\left(\rho \right)}^{2}}.$

${\Lambda }_{+}=\mathrm{tan}\alpha ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Lambda }_{-}=\mathrm{tan}\beta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}U=u-\xi ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}V=v-\eta ,$

$\sigma =\frac{\alpha +\beta }{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega =\frac{\alpha -\beta }{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\left(\rho \right)=\frac{2{p}^{\prime }\left(\rho \right)}{\rho {p}^{″}\left(\rho \right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}m=\frac{2{p}^{\prime }\left(\rho \right)-\rho {p}^{″}\left(\rho \right)}{2{p}^{\prime }\left(\rho \right)+\rho {p}^{″}\left(\rho \right)},$

${\partial }_{±}={\partial }_{\xi }+{\partial }_{\eta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{\partial }}_{+}=\mathrm{cos}\alpha {\partial }_{\xi }+\mathrm{sin}\alpha {\partial }_{\eta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{\partial }}_{-}=\mathrm{cos}\beta {\partial }_{\xi }+\mathrm{sin}\beta {\partial }_{\eta }\text{.}$

$\left\{\begin{array}{l}{\left(\rho U\right)}_{\xi }+{\left(\rho V\right)}_{\eta }+2\rho =0,\\ U{U}_{\xi }+V{U}_{\eta }+\frac{1}{\rho }{p}_{\xi }+U=0,\\ U{V}_{\xi }+V{V}_{\eta }+\frac{1}{\rho }{p}_{\eta }+V=0.\end{array}$ (2)

$\left\{\begin{array}{l}\left({c}^{2}-{U}^{2}\right){u}_{\xi }-UV\left({u}_{\eta }+{v}_{\xi }\right)+\left({c}^{2}-{V}^{2}\right){v}_{\eta =0},\\ {u}_{\eta }={v}_{\xi }.\end{array}$ (3)

$\frac{{U}^{2}+{V}^{2}}{2}+{\int }_{{\rho }_{0}}^{\rho }\frac{1}{\rho }{p}^{\prime }\left(\rho \right)\text{d}\rho +\phi =Const,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\phi }_{\xi }=U,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\phi }_{\eta }=V.$

${\Lambda }_{±}=\frac{UV+c\sqrt{{U}^{2}+{V}^{2}-{c}^{2}}}{{U}^{2}-{c}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{C}_{±}:\frac{\text{d}\eta }{\text{d}\xi }={\Lambda }_{±}.$

${\partial }_{±}u+{\Lambda }_{\mp }{\partial }_{±}v=0.$ (4)

$u-\xi =c\frac{\mathrm{cos}\sigma }{\mathrm{sin}\omega },\text{\hspace{0.17em}}\text{\hspace{0.17em}}v-\eta =c\frac{\mathrm{sin}\sigma }{\mathrm{sin}\omega }.$ (5)

2. u，v，c的特征方程

${\stackrel{¯}{\partial }}_{±}u=\mathrm{cos}\left(\sigma +\omega \right)+\frac{\mathrm{cos}\sigma }{\mathrm{sin}\omega }{\stackrel{¯}{\partial }}_{±}c+\frac{c\mathrm{cos}\alpha {\stackrel{¯}{\partial }}_{±}\beta -c\mathrm{cos}\beta {\stackrel{¯}{\partial }}_{±}\alpha }{2{\mathrm{sin}}^{2}\omega },$ (6)

${\stackrel{¯}{\partial }}_{±}v=\mathrm{sin}\left(\sigma +\omega \right)+\frac{\mathrm{sin}\sigma }{\mathrm{sin}\omega }{\stackrel{¯}{\partial }}_{±}c+\frac{c\mathrm{sin}\alpha {\stackrel{¯}{\partial }}_{±}\beta -c\mathrm{sin}\beta {\stackrel{¯}{\partial }}_{±}\alpha }{2{\mathrm{sin}}^{2}\omega }.$ (7)

${\stackrel{¯}{\partial }}_{+}c=-\frac{\mathrm{cos}2\omega }{\mathrm{cot}\omega }+\frac{c}{\mathrm{sin}2\omega }\left({\stackrel{¯}{\partial }}_{+}\alpha -\mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{+}\beta \right),$ (8)

${\stackrel{¯}{\partial }}_{-}c=-\frac{\mathrm{cos}2\omega }{\mathrm{cot}\omega }+\frac{c}{\mathrm{sin}2\omega }\left(\mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{-}\alpha -{\stackrel{¯}{\partial }}_{-}\beta \right).$ (9)

$\left(\frac{1}{{\mathrm{sin}}^{2}\omega }+k\left(\rho \right)\right){\stackrel{¯}{\partial }}_{±}c=\frac{c\mathrm{cos}\omega }{2{\mathrm{sin}}^{3}\omega }\left({\stackrel{¯}{\partial }}_{±}\alpha -{\stackrel{¯}{\partial }}_{±}\beta \right)-\mathrm{cot}\omega .$ (10)

$c{\stackrel{¯}{\partial }}_{-}\beta =\Omega {\mathrm{cos}}^{2}\omega \left(c{\stackrel{¯}{\partial }}_{-}\alpha -2{\mathrm{sin}}^{2}\omega \right),$ (11)

$c{\stackrel{¯}{\partial }}_{+}\alpha =\Omega {\mathrm{cos}}^{2}\omega \left(c{\stackrel{¯}{\partial }}_{+}\beta +2{\mathrm{sin}}^{2}\omega \right).$ (12)

$\Omega =\frac{k\left(\rho \right)\mathrm{cos}2\omega -1}{\left(1+k\left(\rho \right)\right){\mathrm{cos}}^{2}\omega }=m\left(\rho \right)-{\mathrm{tan}}^{2}\omega .$ (13)

${\stackrel{¯}{\partial }}_{-}c=\left(\frac{\mathrm{cot}\omega }{1+k\left(\rho \right)}\right)\left(c{\stackrel{¯}{\partial }}_{-}\alpha -2{\mathrm{sin}}^{2}\omega \right),$ (14)

$c{\stackrel{¯}{\partial }}_{-}\beta =\left(\frac{1+k\left(\rho \right)}{2}\right)\Omega \mathrm{sin}2\omega {\stackrel{¯}{\partial }}_{-}c.$ (15)

${\stackrel{¯}{\partial }}_{+}c=-\left(\frac{\mathrm{cot}\omega }{1+k\left(\rho \right)}\right)\left(c{\stackrel{¯}{\partial }}_{+}\beta +2{\mathrm{sin}}^{2}\omega \right),$ (16)

$c{\stackrel{¯}{\partial }}_{+}\alpha =-\left(\frac{1+k\left(\rho \right)}{2}\right)\Omega \mathrm{sin}2\omega {\stackrel{¯}{\partial }}_{+}c.$ (17)

${\stackrel{¯}{\partial }}_{±}u=±k\left(\rho \right)\mathrm{sin}\left(\sigma \mp \omega \right){\stackrel{¯}{\partial }}_{±}c,$ (18)

${\stackrel{¯}{\partial }}_{±}v=\mp k\left(\rho \right)\mathrm{cos}\left(\sigma \mp \omega \right){\stackrel{¯}{\partial }}_{±}c.$ (19)

3. 特征分解

${\partial }_{-}{\partial }_{+}I-{\partial }_{+}{\partial }_{-}I=\frac{{\partial }_{-}{\Lambda }_{+}-{\partial }_{+}{\Lambda }_{-}}{{\Lambda }_{-}-{\Lambda }_{+}}\left({\partial }_{-}I-{\partial }_{+}I\right).$

${\stackrel{¯}{\partial }}_{-}{\stackrel{¯}{\partial }}_{+}I-{\stackrel{¯}{\partial }}_{+}{\stackrel{¯}{\partial }}_{-}I=\frac{1}{\mathrm{sin}2\omega }\left[\left(\mathrm{cos}\left(2\omega \right){\stackrel{¯}{\partial }}_{+}\beta -{\stackrel{¯}{\partial }}_{-}\alpha \right){\stackrel{¯}{\partial }}_{-}I-\left({\stackrel{¯}{\partial }}_{+}\beta -\mathrm{cos}\left(2\omega \right){\stackrel{¯}{\partial }}_{-}\alpha \right){\stackrel{¯}{\partial }}_{+}I\right]$

$\left\{\begin{array}{l}c{\stackrel{¯}{\partial }}_{-}{\stackrel{¯}{\partial }}_{+}c={\stackrel{¯}{\partial }}_{+}c\left\{\mathrm{sin}2\omega +\frac{1+k\left(\rho \right)}{2{\mathrm{cos}}^{2}\omega }{\stackrel{¯}{\partial }}_{+}c+\left[\left(\frac{1+k\left(\rho \right)}{2}\right)\Omega \mathrm{cos}2\omega +1-{k}^{\prime }\left(\rho \right)\rho \right]{\stackrel{¯}{\partial }}_{-}c\right\},\\ c{\stackrel{¯}{\partial }}_{+}{\stackrel{¯}{\partial }}_{-}c={\stackrel{¯}{\partial }}_{+}c\left\{\mathrm{sin}2\omega +\frac{1+k\left(\rho \right)}{2{\mathrm{cos}}^{2}\omega }{\stackrel{¯}{\partial }}_{-}c+\left[\left(\frac{1+k\left(\rho \right)}{2}\right)\Omega \mathrm{cos}2\omega +1-{k}^{\prime }\left(\rho \right)\rho \right]{\stackrel{¯}{\partial }}_{+}c\right\}.\end{array}$

$\begin{array}{l}\text{​}\text{​}{\stackrel{¯}{\partial }}_{+}\left[k\left(\rho \right)\mathrm{sin}\alpha {\stackrel{¯}{\partial }}_{-}c\right]+{\stackrel{¯}{\partial }}_{-}\left[k\left(\rho \right)\mathrm{sin}\beta {\stackrel{¯}{\partial }}_{+}c\right]\\ =-\frac{k\left(\rho \right)}{\mathrm{sin}2\omega }\left[\mathrm{sin}\beta {\stackrel{¯}{\partial }}_{+}c\left({\stackrel{¯}{\partial }}_{+}\beta -\mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{-}\alpha \right)-\mathrm{sin}\alpha {\stackrel{¯}{\partial }}_{-}c\left({\stackrel{¯}{\partial }}_{-}\alpha -\mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{+}\beta \right)\right].\end{array}$

$\begin{array}{l}\left(\mathrm{sin}\alpha +\mathrm{sin}\beta \right)\frac{c\rho {k}^{\prime }\left(\rho \right)}{{p}^{\prime }\left(\rho \right)}{\stackrel{¯}{\partial }}_{+}c{\stackrel{¯}{\partial }}_{-}c+\mathrm{sin}\alpha {\stackrel{¯}{\partial }}_{+}{\stackrel{¯}{\partial }}_{-}c+\mathrm{sin}\beta {\stackrel{¯}{\partial }}_{-}{\stackrel{¯}{\partial }}_{+}c\\ =-\frac{1}{\mathrm{sin}2\omega }\left[\left(\mathrm{sin}\beta {\stackrel{¯}{\partial }}_{+}\beta -\mathrm{sin}\beta \mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{-}\alpha +\mathrm{cos}\beta \mathrm{sin}2\omega {\stackrel{¯}{\partial }}_{-}\beta \right){\stackrel{¯}{\partial }}_{+}c\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\left(\mathrm{sin}\alpha {\stackrel{¯}{\partial }}_{-}\alpha -\mathrm{sin}\alpha \mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{+}\beta -\mathrm{cos}\alpha \mathrm{sin}2\omega {\stackrel{¯}{\partial }}_{+}\alpha \right){\stackrel{¯}{\partial }}_{-}c\right].\end{array}$ (20)

${\stackrel{¯}{\partial }}_{-}{\stackrel{¯}{\partial }}_{+}c-{\stackrel{¯}{\partial }}_{+}{\stackrel{¯}{\partial }}_{-}c=\frac{1}{\mathrm{sin}2\omega }\left[\left(\mathrm{cos}\left(2\omega \right){\stackrel{¯}{\partial }}_{+}\beta -{\stackrel{¯}{\partial }}_{-}\alpha \right){\stackrel{¯}{\partial }}_{-}c-\left({\stackrel{¯}{\partial }}_{+}\beta -\mathrm{cos}\left(2\omega \right){\stackrel{¯}{\partial }}_{-}\alpha \right){\stackrel{¯}{\partial }}_{+}c\right]$

$\begin{array}{l}\left(\mathrm{sin}\alpha +\mathrm{sin}\beta \right)\frac{c\rho {k}^{\prime }\left(\rho \right)}{{p}^{\prime }\left(\rho \right)}{\stackrel{¯}{\partial }}_{+}c{\stackrel{¯}{\partial }}_{-}c+\left(\mathrm{sin}\alpha +\mathrm{sin}\beta \right){\stackrel{¯}{\partial }}_{-}{\stackrel{¯}{\partial }}_{+}c\\ =\frac{1}{\mathrm{sin}2\omega }\left[-\left(\mathrm{sin}\alpha +\mathrm{sin}\beta \right){\stackrel{¯}{\partial }}_{+}\beta {\stackrel{¯}{\partial }}_{+}c+\left(\mathrm{sin}\alpha +\mathrm{sin}\beta \right)\mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{-}\alpha {\stackrel{¯}{\partial }}_{+}c\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\mathrm{cos}\beta \mathrm{sin}2\omega {\stackrel{¯}{\partial }}_{-}\beta {\stackrel{¯}{\partial }}_{+}c+\mathrm{cos}\alpha \mathrm{sin}2\omega {\stackrel{¯}{\partial }}_{+}\alpha {\stackrel{¯}{\partial }}_{-}c\right].\end{array}$

$\left\{\begin{array}{l}c{\stackrel{¯}{\partial }}_{+}{\stackrel{¯}{\partial }}_{-}\alpha +{M}_{1}{\stackrel{¯}{\partial }}_{-}\alpha =\left[\frac{\mathrm{sin}\left(2\omega \right)}{2}\left(1-3{\mathrm{tan}}^{2}\omega \right)-\frac{2{k}^{\prime }\rho \mathrm{tan}\omega }{{\left(1+k\right)}^{2}\Omega }\right]{\stackrel{¯}{\partial }}_{+}\alpha ,\\ c{\stackrel{¯}{\partial }}_{-}{\stackrel{¯}{\partial }}_{+}\beta +{M}_{2}{\stackrel{¯}{\partial }}_{+}\beta =\left[\frac{\mathrm{sin}\left(2\omega \right)}{2}\left(1-3{\mathrm{tan}}^{2}\omega \right)-\frac{2{k}^{\prime }\rho \mathrm{tan}\omega }{{\left(1+k\right)}^{2}\Omega }\right]{\stackrel{¯}{\partial }}_{-}\beta ,\end{array}$

$\begin{array}{l}{M}_{1}=\frac{1}{\mathrm{sin}\left(2\omega \right)}\left[4{\mathrm{sin}}^{4}\omega \left(1-\Omega {\mathrm{cos}}^{2}\omega \right)-\frac{{k}^{\prime }\rho {\mathrm{sin}}^{2}\left(2\omega \right)}{{\left(1+k\right)}^{2}}-c{\stackrel{¯}{\partial }}_{-}\alpha \\ \text{ }\text{\hspace{0.17em}}\text{ }+\left(1-\frac{1}{2}\Omega {\mathrm{sin}}^{2}\left(2\omega \right)-\frac{2{k}^{\prime }\rho {\mathrm{cos}}^{2}\omega }{{\left(1+k\right)}^{2}}\right)c{\stackrel{¯}{\partial }}_{+}\beta \right],\end{array}$

$\begin{array}{l}{M}_{2}=\frac{1}{\mathrm{sin}\left(2\omega \right)}\left[4{\mathrm{sin}}^{4}\omega \left(1-\Omega {\mathrm{cos}}^{2}\omega \right)-\frac{{k}^{\prime }\rho {\mathrm{sin}}^{2}\left(2\omega \right)}{{\left(1+k\right)}^{2}}+c{\stackrel{¯}{\partial }}_{+}\beta \\ \text{ }\text{\hspace{0.17em}}\text{ }-\left(1-\frac{1}{2}\Omega {\mathrm{sin}}^{2}\left(2\omega \right)-\frac{2{k}^{\prime }\rho {\mathrm{cos}}^{2}\omega }{{\left(1+k\right)}^{2}}\right)c{\stackrel{¯}{\partial }}_{-}\alpha \right].\end{array}$

$\begin{array}{l}c{\stackrel{¯}{\partial }}_{+}\left[\frac{\mathrm{cot}\omega }{1+k\left(\rho \right)}c{\stackrel{¯}{\partial }}_{-}\alpha -\frac{\mathrm{sin}\left(2\omega \right)}{1+k\left(\rho \right)}\right]\\ =\left(\frac{\mathrm{cot}\omega }{1+k\left(\rho \right)}c{\stackrel{¯}{\partial }}_{-}\alpha -\frac{\mathrm{sin}\left(2\omega \right)}{1+k\left(\rho \right)}\right)\left\{\mathrm{sin}2\omega +\frac{1+k\left(\rho \right)}{2{\mathrm{cos}}^{2}\omega }\left[\frac{\mathrm{cot}\omega }{1+k\left(\rho \right)}c{\stackrel{¯}{\partial }}_{-}\alpha -\frac{\mathrm{sin}\left(2\omega \right)}{1+k\left(\rho \right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[-\frac{2}{1+k\left(\rho \right)}\frac{1}{\Omega \mathrm{sin}\left(2\omega \right)}c{\stackrel{¯}{\partial }}_{+}\alpha \right]\left[\frac{1+k\left(\rho \right)}{2}\Omega \mathrm{cos}2\omega +1-{k}^{\prime }\left(\rho \right)\rho \right]\right\},\end{array}$

$\left\{\begin{array}{l}c{\stackrel{¯}{\partial }}_{+}{\stackrel{¯}{\partial }}_{-}\sigma +{N}_{1}{\stackrel{¯}{\partial }}_{-}\sigma =\frac{k\left(\rho \right)+1}{2k\left(\rho \right)}\mathrm{tan}\omega \left(\Omega \mathrm{cos}\left(2\omega \right)-2{\mathrm{tan}}^{2}\omega \right){\stackrel{¯}{\partial }}_{+}\sigma ,\\ c{\stackrel{¯}{\partial }}_{-}{\stackrel{¯}{\partial }}_{+}\sigma +{N}_{2}{\stackrel{¯}{\partial }}_{+}\sigma =\frac{k\left(\rho \right)+1}{2k\left(\rho \right)}\mathrm{tan}\omega \left(\Omega \mathrm{cos}\left(2\omega \right)-2{\mathrm{tan}}^{2}\omega \right){\stackrel{¯}{\partial }}_{-}\sigma ,\end{array}$

${N}_{1}=\frac{k\left(\rho \right)+1}{2k\left(\rho \right){\mathrm{cos}}^{2}\omega }\left[\mathrm{tan}\omega \left(1+2{\mathrm{cos}}^{2}\omega \right)-\frac{2c}{\mathrm{sin}\left(2\omega \right)}\left({\stackrel{¯}{\partial }}_{-}\sigma -\mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{+}\sigma \right)\right]+\mathrm{tan}\omega \left(1-4{\mathrm{cos}}^{2}\omega \right)$

${N}_{2}=\frac{k\left(\rho \right)+1}{2k\left(\rho \right){\mathrm{cos}}^{2}\omega }\left[\mathrm{tan}\omega \left(1+2{\mathrm{cos}}^{2}\omega \right)+\frac{2c}{\mathrm{sin}\left(2\omega \right)}\left({\stackrel{¯}{\partial }}_{+}\sigma -\mathrm{cos}2\omega {\stackrel{¯}{\partial }}_{-}\sigma \right)\right]+\mathrm{tan}\omega \left(1-4{\mathrm{cos}}^{2}\omega \right)$

4. 结论

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