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Next: The Practical Questions and Up: Entrance Mach number, M1, Previous: Entrance Mach number, M1,   Index

#### The maximum location of the shock

The main point in this discussion however, is to find the furthest shock location downstream. Figure (9.16) shows the possible as function of retreat of the location of the shock wave from the maximum location. When the entrance Mach number is infinity, , if the shock location is at the maximum length, then shock at results in .

The proposed procedure is based on Figure (9.16).

1. According to the positive or negative utilizes your root finding procedure.

From numerical point of view, the Mach number equal infinity when left side assumes result in infinity length of possible extra (the whole flow in the tube is subsonic). To overcome this numerical problem it is suggested to start the calculation from distance from the right hand side.

Let denote

Note that is smaller than . The requirement that has to be satisfied is that denote as difference between the maximum possible of length in which the supersonic flow is achieved and the actual length in which the flow is supersonic see Figure (9.15). The retreating length is expressed as subsonic but

Figure (9.17) shows the entrance Mach number, reduces after the maximum length is exceeded.

Solution

The solution is obtained by an iterative process. The maximum for is 0.821508116. Hence, exceed the maximum length for this entrance Mach number. The maximum for is , thus the extra tube is . The left side is when the shock occurs at (flow chocked and no any additional ). Hence, the value of left side is . The right side is when the shock is at the entrance at which the extra is calculated for and is
Normal Shock Input: Mx k = 1.4
Mx My Ty/Tx ρy/ρx Py/Px P0y/P0x
8 0.39289 13.3867 5.56522 74.5 0.00848783

With

Fanno Flow Input: M k = 1.4
M fld P/P* P0/P0* ρ/ρ* U/U* T/T*
0.39289 2.44172 2.74611 1.61362 2.35907 0.423896 1.16406

The extra is Now the solution is somewhere between the negative of left side to the positive of the right side.9.17

In a summary of the actions is done by the following algorithm:

(a)
check if the exceeds the maximum for the supersonic flow. Accordingly continue.
(b)
Guess

(c)
Calculate the Mach number corresponding to current guess of ,

(d)
Calculate the associate Mach number, with the Mach number, calculated previously,

(e)
Calculate for supersonic branch for the

(f)
Calculate the new and improved''

(g)
Compute the new

(h)
Check the new and improved against the old one. If it satisfactory stop or return to stage (b).

shock location are:

 8.0000 1.0000 0.57068 0.32932 1.6706 0.64830
<>

The iteration summary is also shown below

 0 0.67426 0.22574 1.3838 0.74664 0.90000 1 0.62170 0.27830 1.5286 0.69119 0.90000 2 0.59506 0.30494 1.6021 0.66779 0.90000 3 0.58217 0.31783 1.6382 0.65728 0.90000 4 0.57605 0.32395 1.6554 0.65246 0.90000 5 0.57318 0.32682 1.6635 0.65023 0.90000 6 0.57184 0.32816 1.6673 0.64920 0.90000 7 0.57122 0.32878 1.6691 0.64872 0.90000 8 0.57093 0.32907 1.6699 0.64850 0.90000 9 0.57079 0.32921 1.6703 0.64839 0.90000 10 0.57073 0.32927 1.6705 0.64834 0.90000 11 0.57070 0.32930 1.6706 0.64832 0.90000 12 0.57069 0.32931 1.6706 0.64831 0.90000 13 0.57068 0.32932 1.6706 0.64831 0.90000 14 0.57068 0.32932 1.6706 0.64830 0.90000 15 0.57068 0.32932 1.6706 0.64830 0.90000 16 0.57068 0.32932 1.6706 0.64830 0.90000 17 0.57068 0.32932 1.6706 0.64830 0.90000
<>

This procedure is rapidly converted to the solution.

Next: The Practical Questions and Up: Entrance Mach number, M1, Previous: Entrance Mach number, M1,   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21