Page 144 - 48Fundamentals of Compressible Fluid Mechanics
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106              CHAPTER 6. NORMAL SHOCK IN VARIABLE DUCT AREAS

                                            flow is different from what was discussed before. In this case, no continuous pres-
                                                                                                  continuous pressure
                                                                                            B
                                                                                        a discontinuous point (a shock)
                                            will occur. As conclusion, once the flow becomes supersonic, only exact geometry

                                            sure possibility can exists. Only in one point where
                                            can achieve continuous pressure flow.
                                            exist. If the back pressure,
                                                                      B is smaller than
                                                   In the literature, some refers to a nozzle with area ratio such point b is
                                            above the back pressure and it is referred to an under–expanded nozzle. In the
                                            under–expanded case, the nozzle doesn’t provide the maximum thrust possible.
                                            On the other hand, when the nozzle exit area is too large a shock will occur and
                                            other phenomenon such plume will separate from the wall inside the nozzle. This
                                            nozzle is called an over-expanded nozzle. In comparison of nozzle performance
                                            for rocket and aviation, is that over-expanded nozzle is worse than in the under-
                                            expanded nozzle because the nozzle’s large exit area results in extra drag.
                                                   The location of the shock is determined by geometry to achieve the right
                                                                                        B , is lower than the critical value
                                            (the only value that achieve continuous pressure) a shock occurs outside of the
                                                                                                        than the exact
                                            back pressure. Obviously if the back pressure,
                                            location determined in a such location that after the shock the subsonic branch will

                                            matches the back pressure.
                                            nozzle if needed. If the back pressure is within the range of
                                                                                                   to
                                                   First example is pressed
                                            for academic reasons. It has
                                                                                           troat
                                            to be recognized that the shock                                8)9 :<; =?>@A  BDC$E.F
                                            wave isn’t easily visible (see
                                            for Mach’s photography tech-       	

                 exit
                                            niques).  Therefore, this ex-                       point "e"
                                            ample provides an demonstra-
                                            tion of the calculations for re-                         x  y
                                            quired location even it isn’t real-             "!$#&%  ')(&*,+.-  /1023  4"5$6&7
                                            istic. Nevertheless, this exam-
                                            ple provide the fundamentals to      Fig. 6.2: A nozzle with normal shock
                                            explain the usage of the tools
                                            (equations and tables) that were developed so far.

                                            Example 6.1:
                                            A large tank with compressed air is attached into a converging–diverging nozzle at
                                                                                                            ] and the
                                                            ]. The shock occurs in a location where the cross section area

                                                                                  . Nozzle throat area is 3[
                                                    ]. Calculate the back pressure and the temperature of the flow (It should

                                            exit area is 9[
                                            be noted that the temperature of the surrounding is irrelevant in this case.) Also

                                            is 6[
                                            determine the critical points for the back pressure (point “a” and point “b”.



                                                           and temperature of
                                            pressure
                                              SOLUTION
                                            Since the key word “large tank” was used that means that the stagnation tempera-
                                            ture and pressure are known and equal to the conditions in the tank.
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