1EDUNEX ITB
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01 November 2021
Fakultas Teknik Mesin dan Dirgantara
Characteristics Method
Characteristics Method
AE3110 Aerodynamics 1
Dr. -ing. Mochammad Agoes M. ST. MSc.Ema Amalia, ST., MT.
Pramudita Satria Palar, ST, MT, PhD
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WEEK 9
Outline
➢ Reasons use characteristic method
➢ Philosophy of characteristic method
➢ Determination of Characteristics lines for 2D
➢ Determination of compatibility equations
➢ Numerical Computation
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WEEK 9
Reasons use characteristic method
1. Analytical solution is only for simplified problems
2. For quasi one dimensional flow, the length of the duct for divergent
duct have no calculated, yet
3. Characteristic method is two dimensional flow solution rather than
quasi one dimensional solutiony
x
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WEEK 9
Philosophy of characteristic method
Consider a region of Steady, supersonic flow in X-Y space.
The flowfield can be solved in three steps, as follows:
1. The determination of characteristic lines
2. The determination of compatibility equations which hold along the
characteristics
3. The solution of the compatibility equations point by point along the
characteristics
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WEEK 9
Philosophy of characteristic method
Flow field in x-y
x
y
V
(i,j)
(i,j+1)
(i,j-1)
v
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WEEK 9
Determination of Characteristics lines for 2D
The irrotational compressible flow can be mathematically modelled
as follows:
The change of component of flow velocity in flow field as continuous flow
(continuum) can be written using chain rule:
using Cramer’s Rule we can
obtain the derivative
For any value Δx and Δy, it may
exist a line causing the value
derivative of u is indeterminate
or may even be discontinuous
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WEEK 9
Determination of Characteristics lines for 2D
The result of rate change of velocity can be obtain using Cramer’s Rule
as follows
▪ Characteristic line generated by taking the determinant is equal to
zero showing the solution type of the equation
▪ The characteristic line is drawn in the x-y space.
The Determinant is equal to zero:
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WEEK 9
Determination of Characteristics lines for 2D
8
supersonics
++++
Hyperbolic type
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WEEK 9
2D Characteristics lines formulation
Two slopes of physical characteristics,
CI and CII
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WEEK 9
2D Characteristics lines formulation
Two slopes of physical characteristics,
CI and CII
Substituting intoThe angle θ is considered positive
when turning in a CCW direction. It is
similarly for Mach angle μy
x
right running
characteristic
left running
characteristic
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WEEK 9
Compatibility Equations
y
x
right running characteristic (CI)
left running characteristic (CII)
Prandtl –Meyer Equation
By intergrating
Prandtl –Meyer angle
Along characteristics the above
values are constant
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WEEK 9
Numerical Computation : implementation in internal and on
boundaries
Internal Flow
▪ Points 1 and 2 are known,
namely : ν and θ
▪ Point 3 is unknown variables is
determined by the right running
characteristics from the point 1
and the left running
characteristic from the point 2
How to determine location point 3?
1. Determine the angles θ and ν at the point 3
2. Use straight line approximation to determine the location 3
from points 1 and 2 by computing average angle
From
point 1
From
point 2
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WEEK 9
Numerical Computation : implementation in internal and
on boundaries
Wall Point
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WEEK 9
Numerical Computation : implementation in internal and
on boundaries
Shock Point
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WEEK 9
Numerical Computation : implementation in internal and
on boundaries
Expansion angle for minimum length
▪ The expansion at the point a is Prandtl –
Meyer expansion from initially sonic
conditions :
c
a
d
b
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WEEK 9
Example Calculations
Gas at Mach 1.4349 enters a straight-walled channel that diverges
at an angle of 18o . Determine the two-dimensional flow patter.
Assume the fluid is a perfect gas of constant specific-heat ratio of
1.4. Neglect boundary-layer effect.
14
7
2 5 8
36 9
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WEEK 9
Example Calculations
Compute and Graph the contour of two dimensional minimum length
nozzle for the expansion of air to a design exit Mach number of 2.4