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Next: The case of D Up: Upstream Mach number, M1, Previous: The simple procedure Index The Procedure for Calculating The Maximum Deflection PointThe maximum is obtained when . When the right terms defined in (13.20)-(13.21), (13.29), and (13.30) are substituted into this equation and utilizing the trigonometrical and other trigonometrical identities results in Maximum Deflection Mach Number's equation in which is This equation and its twin equation can be obtained by an alternative procedure proposed by someone^{13.16}who suggested another way to approach this issue. It can be noticed that in equation (13.12), the deflection angle is a function of the Mach angle and the upstream Mach number, . Thus, one can conclude that the maximum Mach angle is only a function of the upstream Much number, . This can be shown mathematically by the argument that differentiating equation (13.12) and equating the results to zero creates relationship between the Mach number, and the maximum Mach angle, . Since in that equation there appears only the heat ratio , and Mach number, , is a function of only these parameters. The differentiation of the equation (13.12) yields Because is a monotonous function, the maximum appears when has its maximum. The numerator of equation (13.37) is zero at different values of the denominator. Thus, it is sufficient to equate the numerator to zero to obtain the maximum. The nominator produces a quadratic equation for and only the positive value for is applied here. Thus, the is Equation (13.38) should be referred to as the maximum's equation. It should be noted that both the Maximum Mach Deflection equation and the maximum's equation lead to the same conclusion that the maximum is only a function of upstream the Mach number and the heat ratio . It can be noticed that the Maximum Deflection Mach Number's equation is also a quadratic equation for . Once is found, then the Mach angle can be easily calculated by equation (13.8). To compare these two equations the simple case of Maximum for an infinite Mach number is examined. It must be pointed out that similar procedures can also be proposed (even though it does not appear in the literature). Instead, taking the derivative with respect to , a derivative can be taken with respect to . Thus, and then solving equation (13.39) provides a solution for . A simplified case of the Maximum Deflection Mach Number's equation for large Mach number becomes Hence, for large Mach numbers, the Mach angle is (for k=1.4), which makes or . With the value of utilizing equation (13.12), the maximum deflection angle can be computed. Note that this procedure does not require an approximation of to be made. The general solution of equation (13.36) is Note that Maximum Deflection Mach Number's equation can be extended to deal with more complicated equations of state (aside from the perfect gas model).
This typical example is for those who like mathematics.
Next: The case of D Up: Upstream Mach number, M1, Previous: The simple procedure Index Created by:Genick Bar-Meir, Ph.D. On: 2007-11-21 |