Specific heat effects in two-dimensional shock refractions

Abstract

Compressible mixtures in supersonic flows are subject to significant temperature changes via shock waves and expansions, which affect several properties of the flow. Besides the widely studied variable transport effects such as temperature-dependent viscosity and conductivity, vibrational and rotational molecular energy storage is also modified through the variation of the heat capacity $c_p$ and heat capacity ratio $\gamma$, especially in hypersonic flows. Changes in the composition of the mixture may also modify its value through the species mass fraction $Y_\alpha$, thereby affecting the compression capacity of the flow. Canonical configurations are studied here to explore their sharply conditioned mechanical equilibrium under variations of these thermal models. In particular, effects of $c_p(T,Y_\alpha )$ and $\gamma (T,Y_\alpha )$ on the stability of shock-impinged supersonic shear and mixing layers are addressed, on condition that a shock wave is refracted. It is found that the limits defining regular structures are affected (usually broadened out) by the dependence of heat capacities with temperature. Theoretical and high-fidelity numerical simulations exhibit a good agreement in the prediction of regular shock reflections and their post-shock aerothermal properties.

Appendix: Numerical implementation

Further details on the implementation of the flow solver CREAMS are provided in this appendix. It consists of a high-fidelity fluid dynamics code for fully compressible Navier–Stokes equations. The scheme is a spatial seventh-order accurate WENO and a third-order total variation diminishing Runge–Kutta [51].

The system of equations that describe a three-dimensional, unsteady, compressible, and viscous gas mixture composed of N reactive species, where external forces and radiation are neglected, is expressed as

$$\begin{aligned}&\frac{\partial \rho }{\partial t} + \frac{\partial (\rho u_j)}{\partial x_j} = 0, \end{aligned}$$

(27)

$$\begin{aligned}&\frac{\partial (\rho u_{{i}})}{\partial t} + \frac{\partial (\rho u_{{i}} u_j + p)}{\partial x_j} = \frac{\partial (\rho \tau _{ij})}{\partial x_j}, \quad i = 1, 2, 3, \end{aligned}$$

(28)

$$\begin{aligned}&\frac{\partial (\rho e_{\mathrm t})}{\partial t} + \frac{\partial [(\rho e_{\mathrm t} + p)u_j]}{\partial x_j} = \frac{\partial (\rho \tau _{ij} u_{{i}})}{\partial x_j} - \frac{\partial q}{\partial x_j}, \end{aligned}$$

(29)

$$\begin{aligned}&\frac{\partial (\rho Y_{\alpha })}{\partial t} {+} \frac{\partial (\rho Y_{\alpha } u_j)}{\partial x_j} {=} - \frac{\partial (\rho Y_{\alpha } V_{\alpha j})}{\partial x_j}, \quad \alpha {=} 1, \ldots , N. \nonumber \\ \end{aligned}$$

(30)

The pressure of the gas mixture is given by p, which is related to the density $\rho$ and temperature T of the mixture via the equation of state of a perfect gas, i.e., $p = \rho R T / W$, where $R=8.314~\,{\mathrm {J/(mol \cdot K)}}$ is the universal constant for perfect gases. To close this system of equations in thermodynamics terms, both constant and variable specific heat ratios are considered. What is more, non-reactive inviscid computations without heat conduction and mass diffusion of species are performed with the intention of reproducing ideal flow theoretical conditions and isolating the thermally perfect gas effects. Therefore, the r.h.s. terms in (28)–(30) are set to zero for the simulations carried out.

The numerical setup used to compute a two-dimensional mixing layers follows previous works based on temporally developing [61, 62] and spatially developing [27, 63,64,65] shear layers. The flow is initialized using a hyperbolic tangent for the velocity profile

$$\begin{aligned} u_1 = \frac{U_1+U_2}{2} + \frac{U_1-U_2}{2} \tanh \left( \frac{2 x_2}{\delta _{\omega ,0}} \right) , \end{aligned}$$

(31)

where $U_1$ and $U_2$ are the mean streamwise velocities of the top and bottom streams, respectively, and $\delta _{\omega ,0}$ is the initial vorticity thickness of the shear layer

$$\begin{aligned} \delta _{\omega ,0}= \frac{U_1-U_2}{| \partial { \langle u_1 \rangle _{\mathrm{f}} } / \partial {x_2} |_{\max ,x_1=0} }, \end{aligned}$$

(32)

where brackets indicate Reynolds-averaged quantities and the subscript f is utilized for Favre-averaged quantities. The initial vorticity thickness controls the amount of initial dissipation between the two streams and has an associated Reynolds number

$$\begin{aligned} {\mathrm {Re}}_{\omega ,0}=\frac{(\rho _1 + \rho _2) |U_1 - U_2| {\delta _{\omega ,0}}}{\mu _1 + \mu _2} \end{aligned}$$

(33)

that is set to ${\mathrm {Re}}_{\omega ,0}=640$ in this work following [61, 64, 65]. This condition provides $\delta _{\omega ,0}=2.79 \cdot 10^{-5} \,{\mathrm {m}}$ (air–air shear layer), $1.41 \cdot 10^{-4} \,{\mathrm {m}}$ (ethane–air mixing layer), and $1.74 \cdot 10^{-5} \,{\mathrm {m}}$ (propane–air mixing layer), which are employed to define the grid size ($\varDelta x \approx 0.168 \delta _{\omega ,0}$ for a typical DNS). Although the definition of a vorticity thickness is not compatible with the idealization of the flow, for which the hyperbolic tangent profile would asymptotically degenerate into a step function, $\delta _{\omega ,0}\ne 0$ is still employed for the following two reasons. Firstly, it characterizes the numerical estimated grid size in the absence of diffusion terms in the conservation equations. Secondly, it was previously shown that the choice of $\delta _{\omega ,0}\ne 0$ at the inlet boundary does not fail to reproduce the theoretical inviscid flow field [27].

The computational domain is bounded by an area measuring roughly $800\ \delta _{\omega ,0}\times 800\ \delta _{\omega ,0}$. The region of interest is placed at the top left corner and covers a squared region of dimensions $300\ \delta _{\omega ,0}\times 300\ \delta _{\omega ,0}$, which provides large span for the information to reach the outflow boundaries. This primary area is discretized with a constant grid size $\varDelta x = 0.25\ \delta _{\omega ,0}$ while the rest of the computational domain is progressively meshed with a linear stretch toward the exit with a constant factor of 5%. The boundary conditions that reproduce the mixing or shear layer problem hold a supersonic inlet where Dirichlet conditions are imposed for pressure, temperature, and mixture composition at the left boundary, together with the velocity profile of (31). In addition, the top boundary includes Rankine–Hugoniot jump conditions for an ideal gas mixture. Furthermore, non-reflecting conditions are applied at the far bottom and right boundaries. Finally, a constant CFL number of 0.5 is selected for time integration.

Fig. 9

Numerical simulation values for the transmitted shock angle $\sigma _{\mathrm{t}}$ as a function of the lower-stream Mach number $M_2$. Computations made for air–air mixing layers with $M_1 =2$, $\sigma _{\mathrm{i}}=50^\circ$, and variable specific heat $c_p(T)$. The gas mixture is treated as an inviscid flow as well as a viscous flow using both simplified (ST) and detailed (DT) transport models. Relative differences of transmitted shock angle under the two viscous simulations with respect to the inviscid case

As already mentioned, inviscid numerical simulations have been carried out in this work with the aim of reproducing ideal flow theoretical conditions. We expect that viscous effects will only play a significant role in very specific areas (e.g., regions with high levels of turbulence that may appear ahead of the shock impingement, but whose characteristic scales are smaller than that of the mixing layer thickness [3]). Viscous effects may be also important in determining the vortex roll-up pattern generated downstream, specially in irregular configurations where additional evolutionary scales are present. Vibrational relaxation, on the other hand, may exhibit large-scale effects related to the regular configuration limits [14]. To demonstrate that our inviscid simulations do retain the same behavior observed in real-flows of shear and mixing layers under similar conditions, we have conducted the reciprocal viscous simulations. Figure 9 shows the evolution of the transmitted shock angle with the Mach number of the lower stream for a shear layer of air impinged with a shock wave at $50^\circ$. The flow is assumed to be either inviscid or viscous using both a simplified and a detailed description of the corresponding transport terms which are found at the r.h.s. of (27)–(30). Without getting into too much detail, the simplified description relies upon a mixture-averaged formulation based on the Hirschfelder and Curtiss approximation while the detailed description takes into account differential diffusion, Soret, Dufour, and bulk viscosity effects [51]. It can be readily seen from this figure that the three simulations fall within the same curve as expected. A much closer inspection at the transmitted shock angle relative differences between the two viscous cases with respect to the inviscid one, see Fig. 9, reveals that these differences are well below $\pm 0.5\%$ and suddenly spike at $M_2 \approx 7.5$. After this, the relative errors reduce to roughly 0.5%, but, unfortunately, the lack of additional data beyond $M_2=8$ does not allow us to estimate the trend of the relative error. We believe that the relative error will continue to increase as the value of the transmitted angle diminishes. Indeed, the algorithm used to compute this angle must identify two sufficiently distant points along the transmitted shock: while the first point is always very close to the triple point, the second one might fall outside the squared region of interest (featuring a small grid size) for flattened transmitted shocks, thus deteriorating the accuracy of the measurement.