The flow gradients in the vicinity of a shock wave for a thermodynamically imperfect gas

Abstract

Supersonic rotational planar and axisymmetric flows of a non-viscous, non-heat-conductive gas with arbitrary thermodynamic properties in the vicinity of a steady shock wave are studied. The differential equations describing the gas flow upstream and downstream of the discontinuity surface and the dynamic compatibility conditions at this discontinuity are used. The gas flow non-uniformity in the shock vicinity is described by the spatial derivatives of the gasdynamic parameters at a point on the shock surface. The parameters are the gas pressure, density, and velocity vector. The derivatives with respect to the directions of the streamline and normal to it, and of the shock surface and normal to it, are considered. Spatial derivatives of all gasdynamic parameters are expressed through the flow non-isobaric factor along the streamline, the streamline curvature, and the flow vorticity and non-isoenthalpy factors. An algorithm for determining these factors of the gas flow downstream of a shock wave is developed. Example calculations of these factors for imperfect oxygen and thermodynamically perfect gas are presented. The influence coefficients of the upstream flow factors on the downstream flow factors are calculated. The gas flow in the vicinity of the shock is described by the isolines of gasdynamic parameters. Uniform planar and axisymmetric flows at different distances from the axis of symmetry are examined; the isobars, isopycnics, isotachs and isoclines are used to characterize the downstream flow behind a curved shock in an imperfect gas.

Appendix: Formulae for the influence coefficients \(A_{ij}\) for thermodynamically perfect gas

Below the influence coefficients $A_{ij}$ for a thermodynamically perfect gas are presented. These formulae were previously published in Russian in [1, 2].

$$\begin{aligned} A_{11}= & {} \displaystyle -C_1\frac{1-\varepsilon }{1+\varepsilon } \sqrt{\frac{J+\varepsilon }{(J_m+\varepsilon )^3} [a_p(J_m-J)^2-b_p(J_m-J)+c_p];}\\ A_{12}= & {} \displaystyle -C_1\frac{\chi \, s}{1+\varepsilon } [a_\varTheta (J+\varepsilon )^3+b_\varTheta (J+\varepsilon ) + c_\varTheta ];\\ A_{13}= & {} \chi C_1s \displaystyle {\frac{2(1+\varepsilon J_m)(J+1+2\varepsilon )(J+\varepsilon )}{\gamma (J_m+\varepsilon )};}\\ A_{14}= & {} C_1c(1-\varepsilon )\{[4(J_m-J)(J+\varepsilon ) - (1+\varepsilon J)(J-1)\\&-2(J-1)(J_m-(1-\varepsilon )J)]\sin \varTheta \\&+\chi q(J-1)(1+\varepsilon J)\cos \varTheta \};\\ A_{15}= & {} -\chi C_1(1-\varepsilon )(a_wJ^2-b_wJ-c_w); \\ A_{21}= & {} \displaystyle \chi C_2\frac{s}{J}\left[ f_1+(J_m-1)c^2f_2\frac{1}{\gamma } \right] ;\\ A_{22}= & {} \displaystyle -C_2\frac{c}{J}\left[ \frac{J_m+\varepsilon }{1-\varepsilon }f_1 +(J_m-J)f_2 \right] ;\\ A_{23}= & {} \displaystyle -2C_2\frac{c}{J} \varpi (J+\varepsilon ) [m + (1+\varepsilon )^2J(J_m-J)];\\ A_{24}= & {} \displaystyle \chi C_2 \frac{a}{J}[cf_2\sin \varTheta +\gamma J \sqrt{b}(1+\varepsilon J) \sin (\varTheta +\chi \beta )];\\ A_{25}= & {} \displaystyle C_2\frac{a}{J} \left\{ 2m+J(1+\varepsilon )[(J_m-J) (J+1+2\varepsilon )\right. \\&\left. - (1+\varepsilon J)(J-1)] \right\} , \\ A_{31}= & {} \chi C_3s {\displaystyle \left[ \frac{M^2-1}{\gamma M^2} +\varpi \right] ; \quad A_{32}=-C_3c(\varpi \gamma M^2+q^2);}\\ A_{33}= & {} \displaystyle -C_3c \left[ \varpi +\frac{(1+\varepsilon J)J}{(1-\varepsilon )(J-1)^2 }\right] ;\\ A_{34}= & {} \chi C_3s \sin \varTheta ; \quad A_{35}=C_3q. \end{aligned}$$

Here $\varepsilon =(\gamma -1)/(\gamma +1)$, $J_m=(1+\varepsilon )M^2-\varepsilon$,

$$\begin{aligned}&a = [(J_m-J)(J+\varepsilon )]^{1/2}; \quad b=a^2+(1+\varepsilon J)^2;\\&c = [(J+\varepsilon )/(J_m+\varepsilon )]^{1/2}; \quad q = [(J_m-J)/(J+\varepsilon )]^{1/2}; \\&s = [(J_m-J)/(J_m+\varepsilon )]^{1/2}; \\&a_p = (1-\varepsilon )(3J_m-4-\varepsilon ); \\&b_p = (3-2\varepsilon )J_m^2-2(1-\varepsilon +3\varepsilon ^2)J_m-(4+6\varepsilon -3\varepsilon ^2); \\&c_p = (J_m-1)(1+\varepsilon J_m)(J_m+\varepsilon ); \\&C_1 = \gamma [\sqrt{b}(J-1)]^{-1}; \quad C_2=(1-\varepsilon )J [\sqrt{b^3}(J-1)]^{-1}, \\&C_3 = -\sqrt{b}(1-\varepsilon )(J-1)^2 [J(1+\varepsilon J)^2]^{-1};\\&a_\varTheta = (1-\varepsilon )[J_m+\varepsilon -4(1+\varepsilon )]; \\&b_\varTheta = 2(1+\varepsilon )[(1-3\varepsilon )(J_m+\varepsilon ) - 2(1-\varepsilon ^2)]; \\&c_\varTheta = (1+\varepsilon )^2 (1-\varepsilon ) (J_m+\varepsilon );\\&a_w = 3+\varepsilon ; \quad b_w=3J_m-2-3\varepsilon ; \quad c_w=(1+4\varepsilon )J_m+1; \\&f_1 = mJ+2\varpi (J+\varepsilon )[m+(1+\varepsilon )^2(J_m-J)J];\\&f_2 = 2m-\gamma J[(1+\varepsilon J_m)(J+\varepsilon ) - (1-\varepsilon ^2)(J_m-J)]; \\&m = (1+\varepsilon J)[b-(1+\varepsilon )J(1+\varepsilon J)];\\&\varpi = -(1+\varepsilon J_m)[(1+\varepsilon )(J_m+\varepsilon )]^{-1}. \end{aligned}$$

At the Crocco point, the shock wave intensity $J_c$ is determined from condition $A_{25}=0$, which leads to an equation

$$\begin{aligned} a_3 J_c^3 +a_2 J_c^2 +a_1 J_c +a_0 =0 \end{aligned}$$

with coefficients

$$\begin{aligned} a_0= & {} -2(1+\varepsilon J_m); \quad a_1=1-3\varepsilon -(1+3\varepsilon +4\varepsilon ^2)J_m; \\ a_2= & {} 4+7\varepsilon +\varepsilon ^2-(1+3\varepsilon )J_m; \quad a_3=1+4\varepsilon +3\varepsilon ^2. \end{aligned}$$

At the constant pressure (Thomas) point, the shock wave intensity $J_p$ is determined from condition $A_{15}=0$, which leads to an equation

$$\begin{aligned} (3+\varepsilon ) J_p^2 - (3J_m-2-3\varepsilon ) J_p - (1+(1+4\varepsilon )J_m) = 0. \end{aligned}$$