Nonequilibrium Nozzle Flows Research Articles

Vibrational Relaxation of CO in Nonequilibrium Nozzle Flow, and the Effect of Hydrogen Atoms on CO Relaxation

The vibrational relaxation of carbon monoxide was studied under conditions of rapid nonequilibrium expansion by using a shock tunnel to generate a nozzle flow with stagnation temperatures and pressures of 2000–4500°K and 5–15 atm., respectively. The vibrational temperature of the CO in the supersonic region of the nozzle was obtained from measurements of the first overtone emission at 2.3 μ by using a calibrated infrared detection system. From these data it was determined that the relaxation time of the CO inferred from the expansion experiment is, at most, 5 times smaller than the relaxation time measured behind incident shock waves. This factor of 5 is in sharp disagreement with previously published measurements in CO and N2, which quote factors from 70 to 1000, implying anomalously fast de-excitation of vibration in expanding flows, but is in agreement with other subsequently obtained measurements. Impurities were found to be more important in this type of experiment than in measurements of relaxation time behind incident shock waves mainly because of additional ways of their getting into the flow. Hydrogen atoms were found to have a probability per collision for the de-excitation of CO of P10 = 0.01 to 0.3 for T = 1400–2800°K. It is not certain whether the observed factor of 5 for pure CO is due to residual impurity effects or due to effects of anharmonicity as discussed in recent theories.

Time-dependent solutions of nonequilibrium nozzle flows - A sequel

Time-dependent solutions of nonequilibrium nozzle flows - A sequel

Time-Dependent Analysis of Population Inversions in an Expanding Gas

A time-dependent technique for the numerical solution of convergent-divergent, nonequilibrium nozzle flows has been used to analyze the rapid, vibrational nonequilibrium, supersonic expansion of a mixture of CO2, N2, and H2O, wherein the finite rate molecular energy transfer processes can result in a population inversion between the (001) and (100) vibrational energy levels of CO2. Results for such population inversions are presented. Among these, a comparison has been made between the present results and the recent results of Basov et al.; this comparison indicates that Basov's calculations overestimate the population inversion in an expanding mixture of CO2 and N2. In addition, results are presented from a series of numerical experiments conducted to assess the validity of several simplified methods for computing population inversions.

A time-dependent analysis for vibrational and chemical nonequilibrium nozzle flows

An unsteady technique is presented for the numerical solution of quasi-one-dimensional, vibrational and chemical nonequilibrium nozzle flows including nonequilibrium conditions both upstream and downstream of the throat. This technique is a time-dependent analysis which entails the explicit finite-difference solution of the quasi-one-dimensional unsteady flow equations in steps of time, starting with assumed initial distributions throughout the nozzle. The steady-state solution is approached at large values of time. A virtue of the present time-dependent analysis is its simplicity, which prevails from its initial physical formulation to the successful receipt of numerical results. To exemplify the present analysis, results are given for several cases of vibrational and chemical nonequilibrium expansions through nozzles. A A' A* do a a' cv Nomenclature

On the structure of a class of aerothermodynamic shocks

Some recent work on the existence of vibrational de-excitation shocks (δ-shocks) in expanding non-equilibrium nozzle flows is extended to include situations in which an adiabatic shock (δ-shocks) may be embedded within the de-excitation shock. A discussion of some further properties of the shock solution is given and some examples are worked out. Numerical solutions of the full equations are also presented. These solutions confirm the existence of the δ-shocks but bring to light certain anomalies in the simple approximate solution. The modifications necessary to remove these discrepancies are outlined, and the implications of the numerical results are briefly discussed. Finally, some comments on the nature of the asymptotic solution for an arbitrary rate process are made.

Vibrational Relaxation of CO in Nonequilibrium Nozzle Flow

First Page

Source-flow expansion of a partially ionized gas into a vacuum.

The purpose of this investigation is to provide some theoretical results for use in predicting the flow conditions which could be expected in freejet plasma expansions. The model investigated is a one-dimensional spherically symmetric, supersonic source flow of partially ionized argon, the assumption heing that this model represents a reasonably good approximation for the centerline flow conditions of a freejet. The analysis utilizes kinetic theory equations of change for the three species (ions, electrons, and neutrals) and a relaxation-type collision model. An ellipsoidal Maxwellian distribution function for each of the species is postulated, and using this distribution function equations for the macroscopic flow variables are derived. The reactions considered are two-body ionization and collisional-radiative recombination, with rate constants being those computed by Bates and co-workers. Both the optically thick and optically thin cases are investigated. The equations are integrated numerically for the case of argon. Two types of source flows are investigated: 1) a simple supersonic source flow, with the initial supersonic state assumed to be an equilibrium one; and 2) a nozzle expansion from an equilibrium stagnation state to slightly supersonic conditions, followed by a source flow. In the second case, both equilibrium and nonequilibrium nozzle flows are considered.

Near-frozen quasi-one-dimensional flow II. De-excitation shocks

The possible existence of compression regions, analogous to condensation shocks, in expanding non-equilibrium nozzle flows is noted. For power-law nozzle shapes the structure and position of these ‘de-excitation shocks’ are derived when the relaxation frequency decays algebraically with temperature. The asymptotic limiting solutions downstream of the de-excitation shocks are also discussed. For certain nozzle shapes it appears that this limiting solution is an infinite sequence of such shocks separated in part by regions of near-frozen flow.

Use of Self-Calibrating Catalytic Probes to Measure Free-Stream Atom Concentration

A technique is presented to measure simultaneously the catalytic efficiency (φc) and the atom concentration (α∞) in a dissociated hypersonic free stream. The principle involved is to measure the differential heat transfer by using a combination of axisymmetric and two‐dimensional differential catalytic probes, mounted side by side in an uniform free stream. Using these measured quantities in the appropriate stagnation point heat‐transfer relations for a frozen flow, simple formulas for φc and α∞ are derived, which are independent of such quantities as viscosity, velocity, and density. This self‐calibrating principle is shown to apply even in the viscous shock layer (merged layer) regime. The most suitable geometric proportions for the probes are given. The practical feasibility of this technique has been successfully demonstrated by measuring the gauge‐catalytic efficiency of a silver surface and the free‐stream oxygen atom concentrations in the University of Toronto Institute of Aerospace Studies 11 × 15‐in. hypersonic shock tunnel test section. The free‐stream atom concentration α∞ was measured at five reservoir conditions (14.2 atm ≤ P0 ≤ 40.7 atm, 4530°K ≤ T0 ≤ 4700°K, 0.54 ≥ α0 ≥ 0.32) and varied from 0.30 to 0.16. The measured values of α∞ compared favorably with those computed from coupled vibrational and dissociational nonequilibrium nozzle flow calculations.

Sonic line in nonequilibrium nozzle flows.

Equation for sonic line in nonequilibrium nozzle flows of gas and application in supersonic flow computations

Correlation of the sudden freezing point in nonequilibrium nozzle flows

Correlation of the sudden freezing point in nonequilibrium nozzle flows

Asymptotic solutions in non-equilibrium nozzle flow

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.

Non-equilibrium flow inside a wavy cylinder

The small-perturbation theory for steady, inviscid, non-equilibrium flow is extended to include the transonic speed range. The resulting transonic small-perturbation equation for the velocity potential is solved for flow inside a wavy cylinder. It is shown that this solution gives a representation of transition through sonic speed at the narrowest sections of the cylinder. This extension of the theory is applied to non-equilibrium nozzle flow.

Entropy Changes in Nonequilibrium Flows

The entropy increases arising from chemical nonequilibrium in inviscid flows are investigated by the use of an analytical model. After the validity of the model is examined, the irreversibilities in nonequilibrium expansions are studied. The basic approach to the problem is an analytical study of limiting behavior of the entropy integral followed by numerical evaluation of this integral over finite limits. Small perturbation analyses are employed for investigating near-equilibrium and near-frozen flow in order to justify truncation of the entropy integral for some cases. The conditions necessary for the existence of freezing and for the existence of entropy bounds are established for large expansions. The results of exact numerical solutions for nonequilibrium nozzle flows are compared with those from isothermal approximations for upper bounds on entropy increments. Irreversibilities are also evaluated for relaxation flows in a constant-area duct, a shock-wave reaction zone, and a venturi for a previously frozen flow. It is observed that the entropy rise in the freezing process is practically negligible compared with the total entropy for a wide range of conditions characterizing flows of technical interest. The constant-area relaxation and venturi relaxation flows also show small entropy rises. The shock wave and its accompanying relaxation zone lead to substantial irreversibilities. The entropy rise in the expansion cases is small permitting constant entropy approximations which may be utilized in numerical computation.

INFLUENCE OF CALORIMETER HEAT TRANSFER GAGES ON AERODYNAMIC HEATING

References 1 Bray, K. N. C., Atomic recombination in a hypersonic wind tunnel Fluid Mech. 6,1-32 (1959). 2 Hall, J. G. and Russo, A. L., Studies of chemical nonequilibrium in hypersonic nozzle flows, Cornell Aeronaut. Lab. Rept. AD-1118-A-6 (November 1959). 3 Lordi, J. A., Comparison of exact and approximate solutions for nonequilibrium nozzle flows, ARS J. 32,1285-1286 (1962). 4 Stollery, J. L. and Smith, J. E., A note on the variation of vibrational temperature along a nozzle, J. Fluid Mech. 13, 225236(1962).

Catalysis of recombination in nonequilibrium nozzle flows

The effects of a chemically active additive were investigated for nonequilibrium expansions of a dissociating gas. Numerical solutions were obtained in order to assess the catalytic influence of free radicals formed by bonding between the additive element and atoms of the dissociating gas. High-enthalpy nozzle flows of hydrogen with small quantities of carbon addition were studied over a range of equilibrium reservoir conditions appropriate to nuclear and electrothermal rocket operation. A chemical kinetic model consisting of eight species and twelve reaction steps evolved from studies of composition histories based on various finite-rate reaction schemes. Through two-body abstraction and association reactions between hydrogen atoms and hydrocarbon species, more of the energy in hydrogen dissociation is released than would kinetically be available in the absence of carbon. In some cases, this effect counterbalances the increased molecular weight so that the specific impulse of the system exceeds that of pure hydrogen in finite-rate nonequilibrium flows.