Approximate analytical calculation of the mach configuration of steady shock waves in a plane constricting channel

Abstract

An approzimate anal~ical model for calculation of the parameters of a steady gas flow inside a plane constricting channel formed by two symmetrically positioned wedges is suggested. A Much configuration of shock waves (triple point) is formed in the channel when the wedge angles are larger than some critical value. The flow calculation in a constricting channel reduces to the solution of the iterative problem for a system of nonlinear algebraic equations. The configurations of shock waves, the slipstream, and the sonic line are described by the proposed model of a gas flow. A comparison of the results obtained using this model allows a fairly accurate calculation of the Much stem and the length of the subsonic-flow region.Flow studies in constricting channels formed by two symmetrically positioned wedges have been performed owing to the interest in the problems of nonunique determination of the transition criteria from regular to irregular (Mach) reflection of an oblique shock wave from the symmetry plane [1][2][3]. Avoiding the problem of the choice of the critical angle of the wedge (the wedge angle is assumed to be larger than the critical value) and assodated problems, we should note that the qualitative pattern of Much reflection in the channel is not yet clear. An engineering approach [4, 5] gives good agreement with experimental results for the height of the Much stem, but underestimates the length of the subsonic region behind the Much stem since the model is approximate [2]. Li and Ben-Dor [6] analytically calculated the interaction of an expansion fan formed on the trailing edge of the wedge with the reflected shock wave and contact discontinuity. After that, the position of the throat of a one-dimensional nozzle is refined for calculation of the height of the Mach stem on the basis of the model [5].We consider a plane channel formed by two symmetrically positioned wedges. A supersonic gas flow comes from the left. Since the problem is symmetric, we consider only the upper half-plane with the wedge ABG and the symmetry axis ON (Fig. 1). In Fig. 1, T is the triple point, AT is the attached shock wave, TO is the normal shock wave, i.e., the Much stem, TF is the oblique shock wave, TE is the slipstream, GF is the first characteristic of the expansion fan, FE is the first characteristic of the expansion fan refracted on the shock wave TF, EN is the sonic line, and 1-4 are the gas-flow regions. The linear dimensions of the problem are the entrance half-section of the channel Y1, the height of the Much stem Ym, the wedge length L, the distance Y. from the axis of symmetry to the slipstream, the length L. of the subsonic-flow region formed by the axis of symmetry and the contact discontinuity, between the normal shock wave TO and the sonic line EN, a~d the distance P between the shock wave TO and the trailing edge of the wedge BG (P > 0 if the Mach stem is located downstream of the trailing edge of the wedge, otherwise P < 0). The angular parameters of the problem are the wedge angle 0, the angle of the attac...

The interaction of a Prandtl–Meyer wave and a quasi-one-dimensional flow region

“…The analogous supposition is accepted in the analysis of flow in the plane narrowing channel between two wedges (Fig. 1b) [4,5].…”

“…The flow downstream of the weakly curved shock j 2 is, as a rule, almost isentropic and can be described by the formulae for Prandtl-Meyer flow with the straight characteristics of the first family [4,5] in both cases shown in Fig. 1 [1].…”

An Approximate Analytical Model of a Jet Flow with Mach Reflection and Pulsed Energy Supply at the Main Shock

“…corresponds to the limiting position of polar IIb of the reflected shock (its contact with polar III, see Figure 3). The flow behind the reflected shock As a rule [7,[10][11][12][13][15][16][17][18] for an approximate analytical description of the flow in region 3 (Figure 1a,b), a model of a quasi-one-dimensional flow with some initial Mach number M 30 directly downstream from the main shock is used. The value M 30 can be determined by Formula ( 9), then M 30 = M 3T , or a similar relation for J 3 = J 3max , which corresponds to the flow at point N (Figure 1) behind the direct shock (then M 30 = M 3N ), or the half-sum of these values.…”

“…Many modern approximate models of flows with Mach reflection [15][16][17] actually replace a fast analytical assessment of the interaction of a rarefaction wave ψ 4 (Figure 1a,b) with a slipstream τ with a calculation by the method of characteristics. However, on the other hand, the reduction in the DCE region of this interaction to a single point D, ignoring the finite length of the interval DC of the slipstream turn in the horizontal direction [7,10], also leads to significant errors in determining the parameters of the shock-wave structure.…”

“…It led to large (up to 50-90%) errors in determining the height of the triple point and other flow parameters. The application of the Grib-Ryabinin model [9] to region two of the supersonic flow, carried out in [10], led to a slight refinement of the results with a simultaneous complication of the mathematical model.…”