Speaker
Description
Background of the Study
Non-equilibrium phenomena can significantly change the heat transfer and aerodynamic characteristics of hypersonic vehicles. Consequently, the design of vehicles entering planetary atmospheres relies heavily on numerical calculations, and in turn these require accurate estimates of physical quantities such as rate constants and intra-molecular interaction parameters. The intra-molecular parameters and rate constants for transport quantities used by numerical solvers for hypersonic flow draw from a range of sources [Par90]. Experimentally generating the flow conditions typical of planetary entry is unattainable in continuous flow facilities, and must be studied using shock tube facilities. Shock tube facilities are comprised of several compression stages using shock heating processes to generate the required flow [MDMG15]. The spatial and temporal variation of the test gas properties are in turn determined by the processes taking place in a shock tube. These processes make the characterization of the test gas state in a shock tube a non-trivial exercise, requiring assumptions about the test gas condition when analysing spectroscopy data. As an example, assumptions about the test gas pressure and temperature were required to determine rate coefficients for high-temperature reactions from spectroscopic data generated by shock tubes [FD61, DL70, Byr59, Par88b]. When radiation measurements in shock tubes are gathered, it is commonly assumed that the test gas has a uniform temperature profile corresponding to the nominal shock speed just upstream of the optical measurement station. However, even during an experiment where the shock is propagating with constant velocity,the growth of the boundary layer on the tunnel wall causes the gas conditions to change along the length of the test slug. Park estimated boundary layer effects by assuming a linear increase of density and both temperatures behind the shock [Par88a]. Recent results have illustrated the importance of shock trajectory on the temperature profile of the test gas,and subsequently the impact on prediction of radiation spectra for equilibrium conditions [SGC + 22]. It has been shown that the flow is in a state of thermal and chemical non-equilibrium [DMG + 12] in many tests performed in high-speed expansion tube experiments. This paper introduces a new method to account for the effect of non-equilibrium phenomena on spectroscopy data in shock tube experiments. The method is demonstrated for experiments in synthetic air using Park’s two temperature model and second-order approximation Chapman-Enskog transport properties.
Methodology
The method applied in this study has two key components, the thermodynamic model, and the numerical model of the shock tunnel.
The thermodynamic model considers thermal and chemical non-equilibrium according to Park’s two temperature model [Par90] with rate-coefficients set by NASA-RP-1232 [GYTL90]. The model assumes the rotational and translational temperatures of the heavy molecules are matched at temperature T . The model also assumes that the vibrational temperature of the molecules, the translational temperature of the electrons, and electronic excitation of the atoms and molecules are equal and are at temperature $T_v$. This assumption is made on the basis that the transfer of energy between the translational motion of free electrons and the vibrational motion of $N_2$ and is very fast, and low-lying electronic states of the molecular species equilibriate quickly with the ground electronic states [Lee84, Lee86]. The dissociation rate coefficients are determined as a function of the geometric mean of the two temperatures $T_a = \sqrt{T T_v}$. Chapman-Enskog theory is used to evaluate the transport properties of the test gas as a multi-component gas mixture [CC70]. Collision integrals requiredfor calculation of these properties come from a variety of sources [HBC64], including intermolecular potential functions such as the Tang-Toennies potential [LW04], differential cross sections [SMI + 95, NMKM88, LKZ04, SGGB95, TN75] and tabulated datasets of ab-initio calculations [LPS90, PSL91, SPL91, SPL00, SPL01]. The tabulated collision integrals are evaluated at the electronic-vibrational temperature $T_v$ for any interactions involving electrons, and at the translational temperature $T$ for all other interactions [GYTL90].
The shock tube flow is modelled as a compressible, one-dimensional unsteady flow. The conditions of the test gas around the centreline are described using the parabolised Navier Stokes equations in cylindrical coordinates. The parabolised equations are simplified by separation of variables, due to the centreline being a streamline. Further simplifications are made exploiting the symmetry conditions around the centreline itself, which require scalar variables and the radial velocity to have zero gradient. Symmetry and separation of variables allow the parabolised Navier-Stokes equations to be written as a set of coupled one-dimensional equations by expanding the solution near the centreline with respect to a small parameter, which represents the distance from the centreline itself. The equations describing the test slug are found to be formally similar to the equations ruling stagnation line problems. Mass continuity requires the centerline derivative of the radial velocity to match the radial velocity at the edge of the boundary layer. The boundary layer at the wall of the shock tube can be determined by using a self-similar solution which matches the local streamwise pressure gradient and centerline scalar quantities to the growth rate of the boundary layer at the wall. The useful test time of the test slug can then be determined by the radial velocity at the edge of the boundary layer. The flow equations, closed by the thermochemistry model equations, are discretised using a second order accurate finite volume method and cast as a large system of coupled algebraic equations. By considering experimental shock trajectory and pressure traces, the system of equations becomes a boundary value problem. The system is then solved using Newton iterations using exact Jacobian matrices.
Conclusion
An numerical method has been presented to calculate the non-equilibrium properties of a shock tube test gas. The method, formally similar to a stagnation line problem with Park’s two temperature model, is based on a version of the parabolised Navier-Stokes equations in cylindrical coordinates. Second-order approximations of the non-equilibrium gas transport properties are evaluated using Chapman-Enskog theory. The centreline solution is coupled to a self-similar boundary layer solution which determines the radial velocity and the test time. History effects on the test slug are simulated using a method based on [SGC + 22]. Therefore, this work allows consideration of shock trajectory effects for a non-equilibrium flow within a shock tunnel.
References
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Summary
A numerical method is developed to calculate non-equilibrium properties of shock tube test gas.
Parabolised Navier-Stokes equations provide the basis for the method, analogous to a stagnation line problem. Gas properties are determined by incorporating Park’s two temperature model with transport properties evaluated using second order Chapman-Enskog theory.
The centreline solution of the shock tube is then coupled to a self-similar boundary layer solution, thus determining the radial velocity and the test time.
