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12.2.1.1: Upstream Mach Number, \(M_1\), and Deflection Angle, \(\delta\)

Again, this set of parameters is, perhaps, the most common and natural to examine. Thompson (1950) has shown that the relationship of the shock

Governing Angle Equation

\[
    \label {2Dgd:eq:Ocubic}
    x^3 + a_1 \, x^2 + a_2 \,x + a_3=0  \tag{18}
\]

where  
\[
    x = \sin^2 \theta
    \label{2Dgd:eq:Ox}  \tag{19}
\]
and
\[
    a_1 & = - \dfrac{{M_1}^2 + 2 }{ {M_1}^2} - k\, \sin ^2 \delta  
    \label{2Dgd:eq:Oa1}  \tag{20}\]
    \[a_2 & = - \dfrac{ 2{M_1}^2 + 1 }{ {M_1}^4 } +
        \left[ \dfrac{(k+1)^2 }{ 4}+ \dfrac{k -1 }{ {M_1}^2} \right]
        \sin ^2 \delta
    \label{2Dgd:eq:Oa2}  \tag{21}\]
   \[ a_3 & = - \dfrac{\cos ^2 \delta }{ {M_1}^4}
    \label{2Dgd:eq:Oa3}  \tag{22}
\]
Equation (18) requires that \(x\) has to be a real and positive number to obtain a real deflection angle. Clearly, \(\sin\theta\) must be positive, and the negative sign refers to the mirror image of the solution. Thus, the negative root of \(\sin\theta\) must be disregarded The solution of a cubic equation such as (18) provides three roots. These roots can be expressed as

First Root

\[
    \label{2Dgd:eq:Ox1}
    x_1 = - \dfrac{1 }{ 3} a_1 + (S +T )  \tag{23}
\]

Second Root

\[
    \label{2Dgd:eq:Ox2}
    x_2 = - \dfrac{a_1 }{ 3}- \dfrac{(S +T )}{2} + \dfrac{i\, \sqrt{3} \, ( S-T) }{2}  \tag{24}
\]

and

Third Root

\[
    \label{2Dgd:eq:Ox3}
    x_3 = - \dfrac{a_1}{ 3} - \dfrac{ (S +T )}{2} - \dfrac{i \,\sqrt{3}\, ( S - T ) }{2}  \tag{25}
\]

Where  
\[
    S = \sqrt[3]{R + \sqrt{D}},
    \label{2Dgd:eq:OS}  \tag{26}
\]
\[
    T = \sqrt[3]{R - \sqrt{D}}
    \label{2Dgd:eq:OT}  \tag{27}
\]
and where the definition of the \(D\) is  
\[
    D = Q^3 + R^2
    \label{2Dgd:eq:OD}  \tag{28}
\]
and where the definitions of \(Q\) and \(R\) are
\[
    Q = \dfrac{ 3 a_2 - {a_1    } ^2 }{ 9}
    \label{2Dgd:eq:OQ}  \tag{29}
\]
and  
\[
    R = \dfrac{ 9 a_1 a_2 - 27  a_3 - 2 {a_1}^3 }{ 54}    
    \label{2Dgd:eq:OR}  \tag{30}
\]
Only three roots can exist for the Mach angle, \(\theta\).  From a mathematical point of view, if \(D>0\), one root is real and two roots are complex. For the case \(D=0\), all the roots are real and at least two are identical. In the last case where \(D<0\), all the roots are real and unequal. The physical meaning of the above analysis demonstrates that in the range where \(D>0\) no solution can exist because no imaginary solution can exist. \(D>0\) occurs when no shock angle can be found, so that the shock normal component is reduced to subsonic and yet parallel to the inclination angle. Furthermore, only in some cases when \(D=0\) does the solution have a physical meaning. Hence, the solution in the case of \(D=0\) has to be examined in the light of other issues to determine the validity of the solution. When \(D<0\), the three unique roots are reduced to two roots at least for the steady state because thermodynamics dictates that. Physically, it can be shown that the first solution (23), referred sometimes as a thermodynamically unstable root, which is also related to a decrease in entropy, is "unrealistic.'' Therefore, the first solution does not occur in reality, at least, in steady–state situations. This root has only a mathematical meaning for steady–state analysis.

Fig. 12.4 Flow around spherically blunted \(30^\circ\) cone-cylinder with Mach number 2.0. It can be noticed that the

These two roots represent two different situations. First, for the second root, the shock wave keeps the flow almost all the time as a supersonic flow and it is referred to as the weak solution (there is a small section that the Second, the third root always turns the flow into subsonic It should be noted that this case is where entropy increases in the largest amount. In summary, if an imaginary hand moves the shock angle starting from the deflection angle and reaching the first angle that satisfies the boundary condition, this situation is unstable and the shock angle will jump to the second angle (root). If an additional "push'' is given, for example, by additional boundary conditions, the shock angle will jump to the third root. These two angles of the strong and weak shock are stable for a two–dimensional wedge (see the appendix of this chapter for a limited discussion on the stability)

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.

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