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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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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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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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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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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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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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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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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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Numerical Computation : implementation in internal and

on boundaries

Wall Point

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Numerical Computation : implementation in internal and

on boundaries

Shock Point

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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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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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Example Calculations

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Example Calculations

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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

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Example Calculations

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WEEK 2