Method of Characteristics for underexpanded flows

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In summary, the conversation discusses the problem of coding a program in MATLAB for a 2D or axisymmetric MOC that does not use an isentropic nozzle. Instead, the user can input two angles to describe the nozzle wall and the program creates a polynomial to describe the wall. However, the problem arises with the use of non-uniform flow nozzles, such as conical nozzles, which result in an 'intercepting shock' after the last expansion wave. The strength of this shock varies and the conversation seeks advice on how to calculate its values using the method of characteristics.
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climberdude
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So I have, what I believe is a pretty specific problem. I am coding up a program in MATLAB for a 2D or axisymmetric MOC (it does both). This code does not use an isentropic nozzle, but rather allows the user to select two angles to describe the nozzle wall, the beginning angle as well as an ending angle at the lip of the nozzle exit (the user also inputs other necessary values to describe the geometry). The program creates a second order polynomial to describe the wall from this input. The reason for this is because it is much simpler to machine these types of nozzles. Past experiments have used conical nozzles which I intend to validate my code with.
My problem lies with the use of the conical nozzle, or any nozzle which does not produce uniform flow. Due to the diverging angle at the exit, an 'intercepting shock' occurs directly after the last expansion wave at the end of the nozzle. This intercepting shock is of variable strength, 0 at the free pressure boundary and very large at the centerline. Does anyone have any experience with this or know how to calculate the values across the shock? Once I can get past the shock, the flow should still be supersonic and the MOC can finish.
 
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I've been advised to use the method of characteristics to calculate the shock, but I am not sure how to go about this. Any help is greatly appreciated!
 

Related to Method of Characteristics for underexpanded flows

1. What is the Method of Characteristics for underexpanded flows?

The Method of Characteristics is a mathematical approach used to analyze underexpanded flows, which occur when a high-pressure gas or fluid expands into a lower pressure environment. It is commonly used in the field of fluid dynamics to study the behavior of gases, liquids, and plasmas.

2. How does the Method of Characteristics work?

The Method of Characteristics works by dividing the flow field into a series of characteristic lines, which represent the paths that fluid particles take as they move through the flow. These lines are calculated using a set of governing equations, and their intersections determine the properties of the flow at any given point.

3. What are the advantages of using the Method of Characteristics for underexpanded flows?

The Method of Characteristics offers several advantages for studying underexpanded flows. It allows for a detailed analysis of complex flows, including shock waves and boundary layer interactions. It also provides important insights into the behavior of the flow, such as pressure and temperature distributions.

4. What are the limitations of the Method of Characteristics?

While the Method of Characteristics is a powerful tool for analyzing underexpanded flows, it does have some limitations. It is most useful for steady-state flows, and may not accurately predict unsteady or turbulent flows. Additionally, it requires significant computational resources and may be time-consuming.

5. How is the Method of Characteristics used in practical applications?

The Method of Characteristics is used in a wide range of practical applications, including aerospace engineering, gas dynamics, and plasma physics. It is commonly used to study the behavior of supersonic and hypersonic flows, and can also be applied to the design of aircraft and rocket engines. In addition, it is often used in the development of numerical models for simulating complex flows.

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