Page 128 - 48Fundamentals of Compressible Fluid Mechanics
P. 128

90                                         CHAPTER 5. NORMAL SHOCK

                                                                                         the larger number of iterations
                                            is required to achieve the same accuracy. Yet, for most practical purpose sufficient
                                            (see Figure (5.9(b))). The larger value of the
                                            results can be achieved after 3-4 iterations.


                                            5.3.2.2  Supersonic Issues of Moving Shock
                                            Assuming that gas velocity is supersonic (in stationary coordinates) before the
                                            shock moves, what is the maximum velocity that can be approached before this
                                            model fails. In other words is there point where the moving shock is fast enough
                                            to reduce the “upstream” relative velocity below the speed of sound. This is the
                                            point where no matter what the pressure difference be, the shock velocity cannot
                                            be increased.
                                                   This shock chocking phe-               Shock in A Suddenly Open Valve
                                            nomenon is similar to the chocking                 Maximum M y ’ possible
                                                                                      2.5
                                            phenomenon that will be discussed
                                                                                      2.25       M y (max)
                                            earlier in nozzle flow and will ap-
                                                                                       2
                                            pear in the nozzle flow and in other       1.75
                                            pipe flows models (later chapters).       Maximum M y ’  1.5
                                            It must be noted that in the previ-       1.25
                                            ous case of suddenly and complete
                                                                                       1
                                            closing of valve results in no limit (at
                                                                                      0.75
                                            least from the model point of view).      0.5
                                                                                               The spesific heat ratio, k
                                            To explain this phenomenon look at
                                            the normal shock. Consider when          Thu Aug 24 17:46:07 2006
                                            the shock wave approaches infinity,
                                                             the downstream   Fig. 5.9: The Maximum of Mach number of “down-
                                                                                     stream” as function of the specific heat
                                            Mach number, according to equa-

                                                                                 . One can view this as the source for



                                            the shock chocking phenomenon.

                                            tion (5.38), is approaching to
                                                   To study this limit consider that the maximum Mach number is obtained
                                                                                             . Applying equation (5.23)

                                            to this situation yields
                                            when the pressure ratio is approaching infinity
                                                                                                               (5.64)









                                             From the mass conservation leads into






                                                                                                               (5.65)














                                                                    
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