Normal shocks with high upstream pressure

A normal compressive shock wave with supercritical upstream thermodynamic conditions is analyzed using the Soave-Redlich-Kwong equation of state (EoS) relations for real-gas density, enthalpy, and entropy for argon, nitrogen, oxygen, and carbon dioxide. Upstream pressure and temperature vary from 10 to 500 bar and 160 to 800 K. At high pressures, the flow does not follow the calorically perfect-gas behavior. For the perfect gas, the enthalpy and ratio of pressure-to-density are directly proportional to the square of the sound speed, allowing direct substitution of the sound speed in the conservation equations. A thermodynamic function is identified for the real-gas sound speed which is shown to remain as the proper characteristic speed. Although the sound speed does not emerge directly from the conservation equations as it does for a perfect gas, the shock speed goes to this limiting value as shock strength goes to zero. For the real gas, modifications are obtained for Prandtl's relation and the Rankine-Hugoniot relation. The modified real-gas Riemann invariants are constructed and discussed for application to weak shocks. A foundation is presented for use with other cubic EoS, multicomponent flows, and/or for more complex flow configurations. Near-similar solutions are developed by normalization of the variables using critical values for pressure and temperature. These exact solutions are compared with approximate solutions obtained via a linearization of the cubic EoS for deviation from ideal-gas behavior.

Figure 1

Shock upstream and downstream Mach numbers for nitrogen at various upstream pressures $p_1=1$ MPA, 10 MPa, 50 MPa; and $T_1=400$ K. (a) $M_1$; (b) $M_2$.

Figure 2

Shock upstream and downstream Mach numbers for nitrogen at various upstream temperatures and $p_1=10$ MPa. (a) $T_1=200$ K; (b) $T_1=800$ K.

Figure 3

Entropy vs shock Mach number for nitrogen at various upstream conditions. (a) $T_1=400$ K; $p_1=1$, 10, 50 MPa. (b) $p_1=10$ MPa; $T_1=200$, 400, 800 K.

Figure 4

Pressure ratio vs density ratio for nitrogen at various upstream conditions. (a) $T_1=400$ K; $p_1=1$, 10, 50 MPa. (b) $p_1=10$ MPa; $T_1=200$, 400, 800 K.

Figure 5

Four-gas comparison at the same upstream conditions: $T_1=400$ K; $p_1=10$ MPa. (a) Pressure ratio vs density ratio; (b) $M_1$ vs pressure ratio; (c) $M_2$ vs pressure ratio.

Figure 6

Behavior of density ratio, downstream sound speed and velocity for nitrogen at various upstream conditions and pressure ratio: (a) Rankine-Hugoniot curves; (b) downstream sound speed vs pressure ratio; (c) downstream velocity vs pressure ratio; (d) downstream velocity vs upstream velocity.

Figure 7

Various ratios vs pressure ratio: variations from ideal-gas behavior for nitrogen at different upstream pressures. (a) 400 K, 50 MPa; (b) 400 K, 10 MPa; (c) 400 K, 1 MPa.

Figure 8

Various ratios vs pressure ratio: variations from ideal-gas behavior for nitrogen at different upstream temperatures. (a) 200 K, 10 MPa; (b) 800 K, 10 MPa.

Figure 9

Variations from ideal-gas behavior for nitrogen at upstream conditions of 800 K, 6 MPa. (a) Various ratios vs pressure ratio showing departures from unity values; (b) $c^{*2}/\left(u_1{u}_2\right)$ where departure from unity value is a real-gas affect.

Figure 10

Comparison of nitrogen, oxygen, and carbon dioxide for $p_1/p_{\mathrm{c}}=8.00,T_1/T_{\mathrm{c}}=2.00$. (a) Pressure ratio vs density ratio. (b) Downstream Mach number vs upstream Mach number. (c) Upstream Mach number versus pressure ratio. (d) Downstream Mach number versus pressure ratio.

Figure 11

Comparison of exact and approximate methods for nitrogen: $p_1=5\text{MPa},10\text{MPa};T_1=400\text{K}$. (a) Pressure ratio vs density ratio. (b) Downstream Mach number vs upstream Mach number.

Figure 12

Comparisons between exact shock solution and weak-shock approximation for nitrogen at upstream conditions of 400 K, 10 MPa. (a) Velocity jump vs pressure ratio; (b) Rankine-Hugoniot curve.