Concept explainers
Note: The purpose of the following problem is to provide an exercise in carrying out a unit process for the method of characteristics. A more extensive application to a complete flow field is left to your specific desires. Also, an extensive practical problem utilizing the finite-difference method requires a large number of arithmetic operations and is practical only on a digital computer. You are encouraged to set up such a problem at your leisure. The main purpose of the present chapter is to present the essence of several numerical methods, not to burden the reader with a lot of calculations or the requirement to write an extensive computer program.
Consider two points in a supersonic flow. These points are located in a cartesian coordinate system at
The numerical value of
The numerical value of
The numerical value of
The numerical value of
The location of point 3.
Answer to Problem 13.1P
The value of
The value of
The value of
The value of
The location of point 3 is
Explanation of Solution
Given:
The Cartesian coordinate system at
The Cartesian coordinate system at
The numerical value of
The numerical value of
The numerical value of
The numerical value of
The numerical value of
The numerical value of
The numerical value of
The numerical value of
Formula used:
The expression for the Mach number is given as,
Here,
The expression for speed of object is given as,
The expression for speed of sound is given as,
The expression for angle of object is given as,
Calculation:
The speed of sound at point 1 can be calculated,
The speed of object at point 1 can be calculated as,
The Mach number at point 1 can be calculated as,
The angle of object at point 1 can be calculated as,
The constant at point 1 can be calculated as,
The flow constant at point 1 can be calculated as,
The speed of sound at point 2 can be calculated as,
The speed of object at point 2 can be calculated as,
The Mach number at point 2 can be calculated as,
The angle of object at point 2 can be calculated as,
The constant at point 2 can be calculated as,
The flow constant at point 2 can be calculated as,
The angle of object at point 3 can be calculated as given bellow,
The constant at point 3 can be calculated as,
The Mach number at point 3 can be calculated as,
To obtain the other flow variables at point 3, expression is given as,
The pressure at point 2 can be calculated as,
The temperature at point 3 is given as below,
Expression for the temperature,
The speed of sound at point 3 can be calculated as,
The speed of object at point 3 can be calculated as,
The initial velocity of object at point 3 can be calculated as,
The final velocity of object at point 3 can be calculated as,
To locate point 3 expression is,
The average angle at point 2 and 3 along the
The average angle at point 2 and 3 along the
The final angle at point 2 and 3 is given as,
The equation for the point on they-axis is given as,
The average angle at point 1 and 3 along the
The average angle at point 1 and 3 along the
The final angle at point 1 and 3 is given as below,
The equation for the point on the x-axis is given as below,
On solving equation (1) and (2), we will get the point 3 as,
Thus,
Conclusion:
Therefore, the value of
Therefore, the value of
Therefore, the value of
Therefore, the value of
Therefore, the location of point 3 is
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Chapter 13 Solutions
EBK FUNDAMENTALS OF AERODYNAMICS
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