Design of Two-Dimensional Continuous-Curvature Supersonic Nozzles

Abstract

In order to cover continuously a rather wide range of Mach Numbers, several supersonic wind tunnels have flexible-plate nozzles incorporated in their design. The elastic curve of a flexible plate supported at discrete points must have continuous curvature. It is, therefore, desirable that the curvature of the aerodynamic contour also be continuous to enable the flexible plate to conform to the desired contour. A method is presented herein for designing two-dimensional supersonic nozzles having continuous curvature. Approximations are made for determining the length of the nozzle and for correcting the perfect-fluid profile for the effect of the boundary-layer growth. Computing forms are included for making the successive approximations that are necessary to establish the angles of the inflection and characteristic points that will satisfy the desired requirements of nozzle length, throat radius, and continuous third derivative at the inflection point. After these angles are established, the coordinates of the perfect-fluid profile, downstream of the inflection point, can be obtained by the characteristics method. A, B, C, G, O, P, Q, R, S, S', T, U A A* b C D H H k K I M m N n R SYMBOLS = points on the characteristic network of the nozzle as defined in Fig. 2 = flux fraction [see Eq. (19)] = area = throat area = flux fraction [see Eq. (27)] = curvature = distance from throat to the end of the flexible plate = physical half-height of test section = ratio of boundary-layer displacement thickness to momentum thickness [see Eq. (41a) ] = one-half test section height before boundarylayer correction = an integer = an angle defined by Eq. (7) or (8) = length of nozzle measured from the throat to the first point of parallel supersonic flow = overall length of the nozzle measured from the throat to the point of cancellation of the last expansion wave = Mach Number = an integer = boundary-layer velocity-profile parameter = a number, defined by Eq. (34) = Reynolds Number [see Eq. (41a)] R* = radius of curvature at the throat R f * = radius of curvature at the throat corrected for the effects of the throat boundary layer r = an integer s w = coordinate along contour APQ, positive to right (see Fig. 2) h = coordinate along AO, positive downward x = coordinate along centerline of nozzle Yi = coordinate of the initial region corrected for boundary layer y -coordinate normal to the centerline of the nozzle y* = coordinate of throat normal to the centerline of the nozzle 5 = boundary-layer displacement thickness 0 = flow inclination angle 0 W = inclination of flow along contour 0i = inclination of flow crossing AO IJL = Mach angle v = Prandtl-Meyer angle <P = slope angle of characteristic lines defined by Eqs. ( 23) and (29) Subscripts denote point on characteristics network (see Fig. 2).

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“…As the contour of the supersonic nozzle segment between the throat and the inflection point is not specified through the reflection of any characteristic line, arbitrary continuous-curvature profiles are often used in wind tunnels constrained by aspects such as variable-geometry flexible plate designs that may be desired to be used, as previously shown by Sivells [6]. The expansion section of the nozzle (region 3 in Figure 1) induces the greatest magnitude of flow turning in the supersonic portion, resulting in expansion fans or Mach waves that reflect downstream [3].…”

A mechanics model in design of flexible nozzle with multiple movable hinge boundaries

“…After nearly a century of development, the aerodynamicists have founded a series of mature approaches to design nozzle contours for high-Mach number core flows [ 11 , 12 ]. When the sight turned to the engineering reality, a new problem appeared.…”