Design of the bell nozzle contour by characteristic method.

1. Characteristic Method
Inviscid & Compressible Flow
Method of obtaining nozzle contours: 1-Characteristic 2-Finite difference 3-Time dependence
Velocity potential equation for irrotational flow in (3-D) and (2-D):
1 −
1
𝑎2
𝜕𝜙
𝜕𝑥
2 𝜕2𝜙
𝜕𝑥2 + 1 −
1
𝑎2
𝜕𝜙
𝜕𝑦
2 𝜕2𝜙
𝜕𝑦2 −
2
𝑎2
𝜕𝜙
𝜕𝑥
𝜕𝜙
𝜕𝑦
𝜕2𝜙
𝜕𝑥𝜕𝑦
= 0
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1 −
∅𝑥
2
𝑎2 ∅𝑥𝑥 + 1 −
∅𝑦
2
𝑎2 ∅𝑦𝑦 + 1 −
∅𝑧
2
𝑎2 ∅𝑧𝑧 −
2∅𝑥∅𝑦
𝑎2 ∅𝑥𝑦 −
2∅𝑥∅𝑧
𝑎2 ∅𝑥𝑧 −
2∅𝑧∅𝑦
𝑎2 ∅𝑧𝑦 = 0
𝑢 =
𝜕𝜙
𝜕𝑥
& v=
𝜕𝜙
𝜕𝑦
& z=
𝜕𝜙
𝜕𝑧

2. 𝜕𝜙
𝜕𝑥
= 𝑢,
𝜕𝜙
𝜕𝑦
=v
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1 −
𝑢2
𝑎2
𝜕𝑢
𝜕𝑥
+ 1 −
𝑣2
𝑎2
𝜕𝑣
𝜕𝑦
−
2𝑢𝑣
𝑎2
𝜕𝑢
𝜕𝑦
= 0
𝜕𝜙
𝜕𝑥
= 𝑓(𝑥, 𝑦)
𝑑𝑓 =
𝜕𝑓
𝜕𝑥
ⅆx +
𝜕𝑓
𝜕𝑦
ⅆ𝑦
𝑑
𝜕𝜙
𝜕𝑥
= 𝑑𝑢 =
𝜕2
𝜙
𝜕𝑥2
ⅆx +
𝜕2
𝜙
𝜕𝑥𝜕𝑦
ⅆ𝑦
𝑑
𝜕𝜙
𝜕𝑦
= 𝑑𝑢 =
𝜕2
𝜙
𝜕𝑥𝜕𝑦
ⅆx +
𝜕2
𝜙
𝜕𝑦2
ⅆy
𝜕𝜙
𝜕𝑦
= 𝑓(𝑥, 𝑦)

3. • Therefore for 2-D:
𝜕𝑢
𝜕𝑥 𝑖,𝑗
=
2𝑢𝑣
𝑎2
𝜕𝑢
𝜕𝑦
− (1 −
𝑣2
𝑎2)
𝜕𝑣
𝜕𝑦
1 −
𝑢2
𝑎2
Velocity of the fluid in the flow field:
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4. • Writing Taylor series for D:
𝑢𝑖+1,𝑗 = 𝑢𝑖,𝑗 +
𝜕𝑢
𝜕𝑥 𝑖,𝑗
Δ𝑥 +
1
2
𝜕2
𝑢
𝜕𝑥2
(Δ𝑥)2+ ⋯
By ignoring higher orders,
𝜕𝑢
𝜕𝑥 𝑖,𝑗
will be obtain.
The determinator should not be equal to zero to the fraction is to be determinable.
In Mach line definition axial velocity is sonic and then:
1 −
𝑢2
𝑎2=0 so u=a=sonic
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5. •
u=a
v
𝑉
𝜇
𝜇
Therefore sin 𝜇 =
𝑎
𝑣
=
1
𝑀
so this is a Mach line meaning
So Mach line is characteristic line
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6. 1 −
𝑢2
𝑎2
𝜙𝑥𝑥 −
2𝑢𝑣
𝑎2
𝜙𝑥𝑦 + 1 −
𝑣2
𝑎2
𝜙𝑦𝑦 = 0
• 𝑑𝑥𝜙𝑥𝑥 + 𝑑𝑦 𝜙𝑥𝑦 = 𝑑𝑢 solve for 𝜙𝑥𝑦
• 𝑑𝑥 𝜙𝑥𝑦 + 𝑑𝑦𝜙𝑦𝑦 = 𝑑𝑣
𝜕2𝜙
𝜕𝑥𝜕𝑦
=
1 −
𝑢2
𝑎2 0 1 −
𝑣2
𝑎2
𝑑𝑥 𝑑𝑢 0
0 𝑑𝑣 𝑑𝑦
1 −
𝑢2
𝑎2 −
2𝑢𝑣
𝑎2 1 −
𝑣2
𝑎2
𝑑𝑥 𝑑𝑦 0
0 𝑑𝑥 𝑑𝑦
=
𝑁
𝐷
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7. • This fraction will become indeterminate when D=0.
• To have an Char-Line, 𝜙𝑥𝑦 had to be indeterminate.
• For given incremental changes, that the denominator goes to zero then 𝜙𝑥𝑦=
discontinuous.
• If D=0,so:
𝑑𝑦
𝑑𝑥
=
−
𝑢𝑣
𝑎2±(
𝑢2+𝑣2
𝑎2 −1)
1−
𝑢2
𝑎2
: Char-Line eq have to be solved
* 𝑢2 + 𝑣2=𝑉2so
𝑢2+𝑣2
𝑎2 =
𝑉2
𝑎2 = 𝑀2 then
𝑑𝑦
𝑑𝑥
=
−
𝑢𝑣
𝑎2±( 𝑀2−1)
1−
𝑢2
𝑎2
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8. 𝐶+
𝐶−
𝑉
𝜇
𝜇
𝜃
Stream line
A
• 𝑢 = 𝑉 cos 𝜃 & 𝑣 = 𝑉 sin 𝜃
By substituting and simplifying,
𝑑𝑦
𝑑𝑥
can be rewritten as:
•
𝑑𝑦
𝑑𝑥
= tan(𝜃 ∓ 𝜇) (char-line Eq)
D tells us how char-lines are located,
what is their orientation and local velocity,
but N does essentially tell us the
relationship of the properties and how they change over the char-line.
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9. • We have 3 conditions:
1. M>1: two real roots, two char-line which means super sonic flow, we got
hyperbolic (PDE).
2. M=1: one real root, one char-line/ Sonic case/ Parabolic PDE.
3. M<1: Imaginary roots/ elliptic PDE.
If N=0: To 𝜙𝑥𝑦 =𝑑u/𝑑y = 𝑑𝑣/𝑑x be finite. Since 0<𝑑u/𝑑x<C
𝑑𝑣
𝑑𝑢
= ∓ 𝑀2−1
𝑑𝑣
𝑣
Which is exactly means P-M expansion angle, i.e.,
𝑑𝑣
𝑑𝑢
= ∓ 𝑀2−1 𝑑𝑣
𝑣
= 𝑑𝜃: this is compatibility EQ
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10. • According to compatibility Eq and based on char-line Eq:
𝑑𝜃 = − 𝑀2−1
𝑑𝑣
𝑣
: 𝐶−
: 𝑅𝑅𝑤
𝑑𝜃 = + 𝑀2−1
𝑑𝑣
𝑣
: 𝐶+: 𝐿𝑅𝑤
By integrating:
𝐶−
≡ 𝜃 + 𝛾 𝑀 = Constant=𝐾−
𝐶+
≡ 𝜃 − 𝛾 𝑀 = Constant=𝐾+
Where K is a constant parameter along each char-line.
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11. • P-M expansion wave angle:
𝜃 = 𝛾 𝑀2 − 𝛾 𝑀1
Where 𝛾 is a P-M function:
𝛾(M)=
𝛾+1
𝛾−1
tan−1 𝛾−1
𝛾+1
(𝑀2 − 1) − tan−1 𝑀2 − 1
𝜃 =
1
2
(𝐾−+ 𝐾+) & 𝛾 =
1
2
(𝐾− − 𝐾+)
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12. • Nozzle definition types according to char-line:
Gradually expansion nozzle
Minimum length nozzle
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14. P
A
B
• Consider the intersection of two characteristic lines A and B at point P, then we
have:
𝑚1 = tan(
𝜃 − 𝑎 𝐴 + 𝜃 − 𝑎 𝑃
2
)
𝑚11 = tan(
𝜃 − 𝑎 𝐵 + 𝜃 − 𝑎 𝑃
2
)
And
𝑦𝑃 = 𝑦𝐴 + 𝑚1(𝑥𝑃 − 𝑥𝐴)
𝑦𝑃 = 𝑦𝐵 + 𝑚11(𝑥𝑃 − 𝑥𝐵)
𝑥𝑃 =
𝑦1 − 𝑦𝐵 + 𝑚11𝑥𝐵 − 𝑚1𝑥𝐴
𝑚11 − 𝑚1
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15. • Inviscid Mach contour
• Diverging length at Mach 3
• RS-25 contour is on MATLAB
(press Ctrl + Clink on it)
Also Solidworks part is here
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18. • 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑟𝑎𝑡𝑖𝑜:
𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑐ℎ𝑎𝑚𝑏𝑒𝑟 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑇𝑒𝑚𝑝 𝑟𝑎𝑡𝑖𝑜: (𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑟𝑎𝑡𝑖𝑜)
𝛾−1
𝛾
• 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑡ℎ𝑟𝑜𝑎𝑡 𝑇𝑒𝑚𝑝 =
2𝛾𝑅.𝑐ℎ𝑎𝑚𝑏𝑒𝑟 𝑇𝑒𝑚𝑝
𝛾−1
𝐶ℎ𝑒𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑡ℎ𝑟𝑜𝑎𝑡 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 =
2
𝛾+1
𝛾
𝛾−1
∗ 2.088
• 𝐶ℎ𝑒𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑡ℎ𝑟𝑜𝑎𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
2𝛾𝑅.𝐶ℎ𝑎𝑚𝑏𝑒𝑟 𝑇𝑒𝑚𝑝
𝛾+1
• 𝐸𝑥𝑖𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑇ℎ𝑟𝑜𝑎𝑡 𝑇𝑒𝑚𝑝. (1 − 𝑇𝑒𝑚𝑝 𝑟𝑎𝑡𝑖𝑜)
• 𝐸𝑥𝑖𝑡 𝑇𝑒𝑚𝑝 = 𝐶ℎ𝑎𝑚𝑏𝑒𝑟 𝑇𝑒𝑚𝑝 . (
𝑃𝑒𝑥𝑖𝑡
𝑃𝑐ℎ𝑎𝑚𝑏𝑒𝑟
)
𝛾−1
𝛾
• 𝐸𝑥𝑖𝑡 𝑠𝑜𝑢𝑛𝑑 𝑠𝑝𝑒𝑒𝑑 = 𝛾𝑅𝑇𝑒𝑥𝑖𝑡
• 𝐸𝑥𝑖𝑡 𝑀𝑎𝑐ℎ =
𝑒𝑥𝑖𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑒𝑥𝑖𝑡 𝑠𝑜𝑢𝑛𝑑 𝑠𝑝𝑒𝑒𝑑
• 𝑃𝑀 =
𝛾+1
𝛾−1
× tan−1 𝛾−1
𝛾+1
𝑀2 − 1 − tan−1 𝑀2 − 1
• 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑤𝑎𝑙𝑙 𝑎𝑛𝑔𝑙𝑒 =
1
2
{𝑃𝑀(𝑒𝑥𝑖𝑡 𝑀𝑎𝑐ℎ)}
• 𝐴𝑛𝑔𝑙𝑒 𝑖𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡 = 2. ( 90 − 𝑀𝑎𝑥 𝑎𝑛𝑔𝑙𝑒 − 𝑓𝑖𝑥 90 − 𝑀𝑎𝑥 𝑎𝑛𝑔𝑙𝑒 )
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19. • MATLAB Coding Method:
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Exit pressure
Pressure ratio
Throat T,P,V
Exit V,T,a,M
Prandtl-Mayer
function to finding
expansion angle-its
recursive
Maximum Wall
angle
Number of division
Calculate position of
the center line
Calculate the wall
positions- its
recursive
Export X,Y into CAD

20. For more information read Rocket Science book
which is written by Mahdi Hossein gholi nejad
and Professor Mofid Gorji.
With special thanks from
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