This report provides a concise explanation of the MoC methodology.
A spreadsheet for a computed Mach angle and Prandtl-Meyer function for γ=1.25 and Mach number Mdes 2.4 is shown. The spreadsheet is verified to give the correct values.
A plot of the design of the shape of the nozzle is s...
i. Method of Characteristics Methodology
Method of characteristics is a process that allows for calculation of flow properties within isentropic flow. A streamline is
generated from a point with corresponding positive and negative Riemann invariants. Sections bound by the lines will
have similar flow properties. Generally, these characteristic lines will be straight, however that is under the assumption
of constant Mach number. If the number fluctuates then the characteristic lines can be curved. When two separate
characteristic lines intersect the calculations need to be repeated, as the angle of both the lower and upper characteristic
will shift slightly – altering the area bound by the characteristic line. The following equations are used to find the co-
ordinates where two characteristic lines intersect:
In this case B denotes the negative gradient characteristic line and A is the positive gradient characteristic line.
In flow regimes where there is a line of symmetry the calculation process is simplified, only the flow above the line of
symmetry needs to be accounted for. This centreline acts as a reflecting surface where the wave will turn from a negative
gradient to a positive one. At the point of reflection, the y coordinate will be 0. The following are equations that are used
to find the co-ordinates at the centreline:
For the design of a minimum length nozzle, each point of the wall is designed to cause a wave cancellation. With the
method of characteristics by ensuring the angle of the wall is enough to reduce the reflected angle to 0 will generate a
minimum length nozzle, with the accuracy of the wall contour increasing with the number of characteristic lines used. For
the initial setup of the first points that the characteristic lines stem from, a maximum turning angle needs to be obtained.
This angle is determined by the Prandtl-Meyer angle of the desired exit velocity. The maximum turning angle will be half
the P-M angle.
The following equations are used to find the co-ordinates at the wall for a wave cancellation:
ii.
Mach 2.4 Course Tables Hand Calculation Online Calculator Course Tables -Hand Cal Hand - Online Cal Online Cal - Course Tables
Mach angle 24.6243 24.6243 24.6243 0 0 0
Prandtl-Meyer function 42.7036 42.7036 42.7036 0 0 0
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