Inviscid reacting flow near a stagnation point

Abstract

A reacting flow free of molecular transport exhibits noteworthy behaviour in the neighbourhood of a blunt, symmetrical stagnation point. A local analytical study using the Lighthill-Freeman gas model reveals the basic structure of such a flow. Chemical activity is found to affect some, but not all, of the local characteristics of the flow. Unaffected are the pressure and velocity fields near the stagnation point, where the pressure varies quadratically and the velocity varies linearly as in an inert flow. In addition, the stagnation point is found to be in chemical equilibrium for all non-zero reaction rates. On the other hand the density, temperature, and concentration fields are affected by the non-equilibrium reactions. The extent of this effect can be predicted on the basis of a reaction parameter that measures the rate of reaction in terms of the velocity gradient at the stagnation point. A rapidly reacting flow (with reaction parameter greater than unity) approaches the stagnation point with vanishing gradients of density and temperature, whereas a slowly reacting flow approaches with infinite gradients. The flow field is represented mathematically by functions that are regular along the body but non-analytic in the normal direction. Numerical computations support the validity of the local closed-form solution, and provide information on the local effects of the chemical history of the flow.

Reacting flow as an example of a boundary layer under singular external conditions

“…The outer problem is discussed in detail by Conti & Van Dyke (1969). The integration proceeds from the shock wave to the body.…”

“…It is of interest to see how the conventional second-order boundary layer is recovered. The transition occurs at K = 1, which is a singular case as explained by Conti & Van Dyke (1969). Without dwelling on this case, we note that when K > 1, matching to order R-4K requires that the outer expansion (1) be considered at least to the second order, R-t. Now the secondary term is given by the conventional flow due to displacement thickness, and the matching boundary layer will be the conventional secondorder one.…”

“…This problem, governed by equations (A 3), describes the inviscid flow and has been solved by Conti & Van Dyke (1969). I n this section we discuss the main results of that work.…”

“…For this reason we adopt the slowly reacting flows as the basis for our present study. For K < 1, Conti & Van Dyke (1969) Equations (2) describe the inviscid flow near the wall, and we will use them to deduce the behaviour of the boundary layer.…”

On stagnation-point conditions in non-equilibrium inviscid blunt-body flows

“…While this fact may be intuitively obvious, a rigorous proof was lacking, since the analytic solutions were based on certain approximations, and the purely numerical schemes were limited in accuracy. Recently, independent investigations by Stulov & Turchak (1 966) for vibrational relaxation, and Conti & Van Dyke (1966, 1969a for dissociation of a Lighthill gas, established rigorously by means of a local analysis that equilibrium conditions are in fact reached a t the stagnation point. (One should note that the analysis of Stulov & Turchak is not completely correct, as will be indicated later.)…”