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Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles ------Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles

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Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles ------Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles UOT Mechanical Department / Aeronautical Branch Gas Dynamics Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles -------------------------------------------------------------------------------------------------------------------------------------------- Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles 16.1 Exit Flow for Underexpanded and Overexpanded Supersonic Nozzles The variation in flow patterns inside the nozzle obtained by changing the back pressure, with a constant reservoir pressure, was discussed early. It was shown that, over a certain range of back pressures, the flow was unable to adjust to the prescribed back pressure inside the nozzle, but rather adjusted externally in the form of compression waves or expansion waves. We can now discuss in detail the wave pattern occurring at the exit of an underexpanded or overexpanded nozzle. Consider first, flow at the exit plane of an underexpanded, two-dimensional nozzle (see Figure 16.1). Since the expansion inside the nozzle was insufficient to reach the back pressure, expansion fans form at the nozzle exit plane. As is shown in Figure (16.1), flow at the exit plane is assumed to be uniform and parallel, with . For this case, from symmetry, there can be no flow across the centerline of the jet. Thus the boundary conditions along the centerline are the same as those at a plane wall in nonviscous flow, and the normal velocity component must be equal to zero. The pressure is reduced to the prescribed value of back pressure in region 2 by the expansion fans. However, the flow in region 2 is turned away from the exhaust-jet centerline. To maintain the zero normal-velocity components along the centerline, the flow must be turned back toward the horizontal. Thus the intersection of the expansion fans centered at the nozzle exit yields another set of expansion waxes, just as did the reflection of the expansion fan from a plane wall (reflected Pradtl-Myer waves. The second expansion, however, produces a pressure in region 3 less than the back pressure, so the expansion waves reflect from the external air as oblique shocks. These compression waves produce a static pressure in region 4 equal to the back pressure, but again turn the flow away from the centerline. The intersection of the oblique shocks from either side of the jet then requires another set of oblique shocks to turn the flow back toward the horizontal, with the shocks reflecting from the external air as expansion waves. 1-7 ch.16 Prepared by A.A. Hussaini 2013-2014 UOT Mechanical Department / Aeronautical Branch Gas Dynamics Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles -------------------------------------------------------------------------------------------------------------------------------------------- The process thus goes through a complete cycle and continues to repeat itself. The flow pattern discussed appears as a series of diamonds, often visible at the exit of high - speed rocket nozzles. Theoretically, the wave pattern should extend to infinity. Actually, however, mixing of the jet with ambient air along the jet boundaries eventually causes the wave pattern to die out. Flow at the exit of an overexpanded nozzle is shown in Figure (16.2). Since the exit- plane pressure is less than the back pressure, oblique shock waves form at the nozzle exit. The intersection of these shocks at the centerline yields a second set of oblique shocks, which in turn reflect from the ambient air as expansion waxes. Thus, except for being out of phase with the wave pattern from the underexpanded nozzle, the jet flow of the overexpanded nozzle exhibits the same characteristics as the underexpanded nozzle. Example 16.1 A supersonic nozzle is designed to operate at Mach 2.0. Under a certain operating condition, however, an oblique shock making a 45° angle with the flow direction is observed at the nozzle exit plane, as in figure (16.3). What percent of increase in stagnation pressure would be necessary to eliminate this shock and maintain supersonic flow at the nozzle exit? Solution From isentropic table, for gives ⁄ . The component of normal to the oblique wave is . From normal shock table, ⁄ . Therefore, with the oblique shock, the ratio With the shock, is equal to ( ⁄ ) ⁄ For supersonic exit flow with no shocks (perfectly expanded case), ( ⁄ ) ( )⁄ Thus, an increase of in stagnation pressure is required. 2-7 ch.16 Prepared by A.A. Hussaini 2013-2014 UOT Mechanical Department / Aeronautical Branch Gas Dynamics Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles -------------------------------------------------------------------------------------------------------------------------------------------- 16.2 Plug Nozzle The thrust developed by a nozzle is dependent on the nozzle exhaust velocity and the pressure at the nozzle exit plane. In a jet propulsion device, when an exit-plane pressure greater than ambient gives a positive contribution to the thrust of the device, whereas when an exit-plane pressure less than ambient gives a negative thrust component. ̇ ( ) ( ) When a supersonic nozzle is operating in the under- or overexpanded regimes, with flow in the nozzle independent of back pressure, the exit velocity is unaffected by back pressure ( ). Thus, over this range of back pressures, Eq. (16.01) shows that the greater thrusts are developed in the underexpanded case ( ), and the lesser in the overexpanded case ( ). A plot of thrust versus back pressure for a converging-diverging nozzle is shown in Figure 16.4. For back pressures greater than the upper limit indicated, a normal shock starts to appear in the diverging portion of the nozzle, the exit velocity becoming subsonic, and this analysis no longer applies. The plug nozzle ( figure 16.5) is a device that is intended to allow the flow to be directed or controlled by the ambient pressure rather than by the nozzle walls. In this nozzle, the supersonic flow is not confined within solid walls, but is exposed to the ambient pressure. Plug nozzle operation at the design pressure ratio is depicted in Figure 16.6. Figure 16.6a shows the expansion wave pattern and part b shows the streamlines at the nozzle exit. The annular flow first expands internally up to at the throat. The remainder of the expansion to the back pressure occurs with the flow exposed to ambient pressure. Since the throat pressure is considerably higher than the 3-7 ch.16 Prepared by A.A. Hussaini 2013-2014 UOT Mechanical Department / Aeronautical Branch Gas Dynamics Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles -------------------------------------------------------------------------------------------------------------------------------------------- back pressure, a Prandtl Meyer expansion fan is attached to the throat cowling as shown. The plug is designed so that, at the design pressure ratio, the final expansion wave intersects the plug apex. Thus, under this operating condition, the pressure at the plug wall decreases continuously from throat pressure to ambient pressure, just as with the converging-diverging perfectly expanded nozzle. To produce a maximum axial thrust, it is necessary for the exit flow to have an axial direction. Therefore, the flow at the throat cowling must be directed toward the axis so that the turning produced by the expansion fan will yield axial flow at the plug apex. For the underexpanded case, the operation of the plug nozzle (Figure 16.7) is similar to that of the converging- diverging nozzle. The pressure along the plug is the same as for the design case, just as the static pressure along the converging-diverging nozzle wall is the same as for the perfectly expanded case. With a lower back pressure than that for the design case depicted in Figure 16.6, the flow continues to expand after the apex pressure, yielding a non-axial jet velocity component, just as with the underexpanded supersonic converging-diverging nozzle. The major improvement to be derived from the plug nozzle occurs with the overexpanded mode of operation. This is significant, in that a rocket nozzle, for example, accelerating from sea level up to design speed and altitude, must pass through the overexpanded regime. With the ambient pressure greater than the design back pressure, the flow expands along the plug only up to the design back pressure. The final wave of the expansion fan centered at the cowling intersects the plug at a point upstream of the apex. As shown in Figure 16.7, the outer boundaries of the exhaust jet are directed inward. Further weak compression and expansion waves occur downstream of the point of impingement of the final wave from the fan; the strength and location of these waves are dependent on the plug contour. Thus the expansion along the plug is controlled by the back pressure, whereas the converging-diverging nozzle expansion is controlled by nozzle geometry. A plot of pressure along the plug surface versus x is given in Figure 16.8. The pressure along the plug surface does not decrease below ambient, so there is not a negative thrust term 4-7 ch.16 Prepared by A.A. Hussaini 2013-2014 UOT Mechanical Department / Aeronautical Branch Gas Dynamics Chapter Sixteen / Plug, Underexpanded and Overexpanded Supersonic Nozzles -------------------------------------------------------------------------------------------------------------------------------------------- due to pressure difference. As a result, the plug nozzle provides improved thrust over the converging-diverging
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