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THE DESIGN OF A SUPERSONIC NOZZLE BY THE METHOD OF
CHARACTERISTICS
Read Anderson Section 13.3, especially the example discussed in detail on page
- A computer code (Matlab) Is avalable at our web site for the design of the straightening section of a 2-D nozzle, and execute the program for two test section Mach numbers. The flow in the test section must be uniform, parallel to the x- axis, and free from any compression or expansion waves. A reasonable contour for the subsonic section is assumed.
The shape of the expansion section is arbitrary (see p. 591 and the enclosed figure). For simplicity here, take all of the expansion at a sharp corner just downstream of the nozzle throat as illustrated in the sketch below, i.e. set the radius of curvature of the circular arc R , in the figure to zero.
The Mach number at the throat is unity. It may be shown that, for a given Mach number in the test section, the shortest nozzle occurs when the expansion section opens up to an angle equal to one half of the Prandtl-Meyer function ν corresponding to the test section Mach number. The shortest possible nozzle occurs for this condition when all the expansion occurs just downstream of the throat. Thus, the calculation required here corresponds to the design of a shortest possible length nozzle. Because of boundary layer problems, wind tunnel nozzles are normally not designed with the full expansion at the throat.
The nozzles are to be designed for the following test section Mach numbers: 2. and 4.0.
In the figure, the expansion region immediately downstream of the throat is represented by 3 steps. The program must allow for any number of expansion steps, N (N < 10). Take 6 to 10 steps through the expansion turning angle (the more you take, the more accurate the solution).
How does the program work?
- Numbering System: Numbering the nodes using a two-subscript system shown in the enclosed figure simplifies the programming. It is convenient to number the "wall" points on the straightening section using a single subscript system. If you find a more convenient way of numbering the nodes, you may certainly do so.
- Interior Point Calculations: At the interior points (i,j) in the figure below, the flow angle θ , the Prandtl-Meyer function ν , the Mach number and the (x,y) coordinates may all be found, once information at the points (i-1,j) and (i,j-1) are known, using the interior point process. Thus, these points may be computed in a "double DO loop" in a recursive fashion.
(i,j)
(i,j-1)
(i-1,j)
i) Compute first the flow slope θ and ν at (i,j) from the compatibility relations:
θ (^) i,j = [ θ (^) i,j-1 + ν (^) i,j-1 + θ (^) i-1,j - ν (^) i-1,j ] / 2
ν (^) i,j = [ θ (^) i,j-1 + ν (^) i,j-1 - θ (^) i-1,j + ν (^) i-1,j ] / 2
ii) Compute from ν (^) i,j the Mach number and Mach angle μ at (i,j).
iii) Compute the (x,y) locations of the node (i,j) as the intersection of characteristics originating at nodes (i,j-1) and (i-1,j).
dy/dx = ( y i,j - y i,j-1) / (xi,j- xi,j-1) = tan [ (θi,j-1- μi,j-1 + θi,j - μi,j ) / 2 ]
dy/dx = ( y i,j - y i-1,j) / (xi,i- xi-1,j) = tan [ (θi-1,j+ μi-1,j + θi,j + μi,j ) / 2 ]
- Points on the axis: These points have a subscript (i,i) where i varies between 1 and N. require a slight variation of the interior point process, and are best handled in the same "Double DO loop" as the rest of the interior points. At these points, the flow angle θ is zero. consider a characteristic that connects a node (i,i-1) and (i,i). All the information at the node (i,i-1) is known.
(i,i-1)
(i,i) Nozzle Axis
θi,i-1+ νi,i-1 = θi,i + νi,i = νi,i
Once the Prandtl Meyer function values νi,i at these axis points are known, the Mach number at these locations, and the corresponding Mach angles may be found. The (x,y) locations of these axial points are finally found by solving:
yi,i = 0
dy/dx = ( y i,i - y i,i-1) / (xi,i- xi,i-1) = tan [ (θi,i-1- μi,i-1 + θi,i - μi,i ) / 2 ]
- Straightening Section: The straightening section is made of wall points sequentially numbered from 1 to N+1 as shown below.
2 3 N+
(N,2)
(N,N)
(N,1)
(N,0)
a) Assume that the value of M for the given ν lies in an interval between two extremes MLEFT and MRIGHT.
Good starting guesses are: MLEFT = 1 and MRIGHT = 10.
b) Find the mid-point of this interval, MMID = (MLEFT + MRIGHT) / 2
c) Check the width of the interval MRIGHT - MLEFT... If this interval is less than some tolerance, say 0.001, then the Mach number you are seeking is MMID. Exit the iteration loop.
d) Compare given ν with ν(MMID).
If ν is greater than ν(MMID) then you need to search in the right portion of the current interval. Therefore, set MLEFT to MMID.
If ν is less than ν(MMID) then you need to search in the left half of your current interval. Therefore, set MRIGHT to MMID.
e) Go to step (b). Repeat until the criterion in step (c) is satisfied.
