Abstract
The purpose of this study is to obtain a static pressure contour plot at different Mach
number and to compare them. The Mach number that we reviewed is Mach 0.5, Mach 0.6,
Mach 0.9, Mach 1.2 and Mach 1.5. The results that were obtained are each Mach number
shows different characteristics of the static pressure contour plot. Mach number is the speed
of an object moving through air, or any other fluid substance, divided by the speed of sound
as it is in that substance for its particular physical conditions, including those of temperature
and pressure. It is commonly used to represent the speed of an object when it is travelling
close to or above the speed of sound.
We also want to learn about the pressure coefficient along the upper and lower airfoil
surfaces at the respective Mach number. The pressure coefficient is a dimensionless
number which describes the relative pressures throughout a flow field in fluid dynamics. The
pressure coefficient is used in aerodynamics and hydrodynamics. Every point in a fluid flow
field has its own unique pressure coefficient, Cp.
Lastly we want to obtain the relationship between drag and lift coefficient and Mach
number of the NACA0012. Drag coefficient Cd, is a dimensionless quantity that is used to
quantify the drag or resistance of an object in a fluid environment such as air or water. It is
used in the drag equation, where a lower drag coefficient indicates the object will have less
aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a
particular surface area.
The lift coefficient Cl, is a dimensionless coefficient that relates the lift generated by
lifting body, the dynamic pressure of the fluid flow around the body, and a reference area
associated with the body. A lifting body is a foil or a complete foil-bearing body such as a
fixed-wing aircraft. Lift coefficient may be described as the ratio of lift pressure to dynamic
pressure where lift pressure is the ratio of lift to reference area.
Objectives
1) To compare the contours of static pressure at Mach 0.5, Mach 0.6, Mach 0.9, Mach
1.2 and Mach 1.5.
2) To plot the pressure coefficient (Cp) along the upper and lower airfoil surfaces at
Mach 0.5, Mach 0.6, Mach 0.9, Mach 1.2 and Mach 1.5 respectively.
3) To obtain relationship between drag and lift coefficient and Mach Number of
NACA0012.
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Theory
The pressure coefficient is a very useful parameter for studying the flow of
incompressible fluids such as water, and also the low-speed flow of compressible fluids such
as air. The relationship between the dimensionless coefficient and the dimensional numbers
is
CP=p−p∞12ρ∞V ∞
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Where:
p is the pressure at the point at which pressure coefficient is being evaluated
p∞ is the pressure in the freestream (i.e. remote from any disturbance)
ρ∞is the freestream fluid density (Air at sea level and 15 °C is 1.225 kg / m3)
V ∞is the freestream velocity of the fluid, or the velocity of the body through the fluid
In the flow of compressible fluids such as air, and particularly the high-speed flow of
compressible fluids, ρv2 / 2 (the dynamic pressure) is no longer an accurate measure of the
difference between stagnation pressure and static pressure. Also, the familiar relationship
that stagnation pressure is equal to total pressure does not always hold true. (It is always
true in isentropic flow but the presence of shock waves can cause the flow to depart from
isentropic.) As a result, pressure coefficients can be greater than one in compressible flow.
Cp greater than one indicates the freestream flow is compressible.
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Cl and Cp relationship
The coefficient of lift for an airfoil with strictly horizontal surfaces can be calculated
from the coefficient of pressure distribution by integration, or calculating the area between
the lines on the distribution. This expression is not suitable for direct numeric integration
using the panel method of lift approximation, as it does not take into account the direction of
pressure-induced lift.
Where;
is pressure coefficient on the lower surface
is pressure coefficient on the upper surface
LE is the leading edge
TE is the trailing edge
When the lower surface Cp is higher (more negative) on the distribution it counts as a
negative area as this will be producing down force rather than lift.
Mach number, M is the speed of an object moving through air, or any other fluid
substance, divided by the speed of sound as it is in that substance for its particular physical
conditions, including those of temperature and pressure. It is commonly used to represent
the speed of an object when it is travelling close to or above the speed of sound.
M=Va
Where;
is the Mach number
is the relative velocity of the source to the medium and
is the speed of sound in the medium
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Results
1) Mach number = 0.5
Figure 1: Scale Residuals
Figure 2: Value of Cl and Cd obtained Figure 3 : Static Pressure Contour
Figure 4 : Static Pressure Contour (no fill) Figure 5 : Pressure Coefficient
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Figure 6: Lift Coefficient Figure 7: Drag Coefficient
Figure 8: Mach number along the airfoil Figure 9: Pressure Contour
2) Mach number = 0.6
Figure 10: Scaled Residuals
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Figure 11: Value of Cl and Cd obtained Figure 12: Static Pressure contour
Figure 13: Static Pressure (no fill) Figure 14: Pressure Coefficient
Figure 15: Lift Coefficient Figure 16: Drag Coefficient
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Figure 17: Mach number along the airfoil Figure 18: Pressure Contour
3) Mach number = 0.9
Figure 19: Scaled Residuals
Figure 20: Value of Cl and Cd obtained Figure 21: Static Pressure
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Figure 22: Static Pressure (no fill) Figure 23: Pressure Coefficient
Figure 24: Lift Coefficient Figure 25: Drag Coefficient
Figure 26: Mach number along the airfoil Figure 27: Pressure Contour
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4) Mach number = 1.2
Figure 28: Scaled Residuals
Figure 29: Values of Cl and Cd obtained Figure 30: Static Pressure
Figure 31: Static Pressure (no fill) Figure 32: Pressure Coefficient
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Figure 33: Lift Coefficient Figure 34: Drag Coefficient
Figure 35: Mach number along the airfoil Figure 36: Pressure Contour
5) Mach number = 1.5
Figure 37: Scaled Residuals
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Figure 38: Value of Cl and Cd obtained Figure 39: Static Pressure
Figure 40: Static Pressure (no fill) Figure 41: Pressure Coefficient
Figure 42: Lift Coefficient Figure 43: Drag Coefficient
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Figure 44: Mach number along the airfoil Figure 45: Pressure Contour
Discussions
Mach number Pressure (Pa) Temperature (K)
0.5 85419 296.19
0.6 79439 290.11
0.9 59909 267.64
1.2 41784 241.46
1.5 27601 214.48
Table 1: Relative Pressure and Temperature according to Mach number
For each of the Mach number given, we have to calculate the respective pressure
and temperature according to the Mach number. We calculate it based on two equations
given below;
PoP
=[1+( γ−12 )M 2]γγ−1
ToT
=1+ γ−12M 2
With the given Po of 101325 Pa and T o of 311 K.
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The contour of static pressure;
Figure 46: Mach number 0.5 Figure 47: Mach number 0.6
Figure 48: Mach number 0.9 Figure 49: Mach number 1.2
Figure 50: Mach number 1.5
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The pictures above shows that for Mach number 0.5 and 0.6 the static pressure
around the NACA0012 airfoil is relatively high. It is high at the front of the airfoil, then the
pressure is down a little at the surface of the airfoil, and the pressure goes up again at the
tail of the airfoil.
For Mach number 0.9, the pressure is high at the front of the airfoil and at the tail, but
the pressure is relatively low at the surface body of the airfoil to the tail of the airfoil. For
Mach number 1.2 and 1.5, the characteristic is about the same, except there is a region of
low pressure in front of the airfoil which is like a shock wave form. The pressure is high at
the front tip of the airfoil but the pressure decreases along the surface of the airfoil to the tail
and to the back of the airfoil. The pressure does not build up again.
Relation between lift coefficient and drag coefficient and Mach number
Mach number Lift Coefficient (Cl) Drag Coefficient (Cd)
0.5 4.6467 x 103 1.9181 x 102
0.6 2.0753 x 10-1 7.8653 x 10-3
0.9 1.3531 x 10-1 1.1657 x 10-1
1.2 1.1289 x 10-1 1.0421 x 10-1
1.5 8.3131 x 10-2 1.0271 x 10-1
Table 2: Value of Cl and Cd according to Mach number
From the table given above, it is shown that when the Mach number increases, the value of
lift coefficient (Cl) is becoming smaller. The meaning is the lifting ability of the airfoil is decreasing
when the speed of the air is increase. The value of the drag coefficient (Cd) also becoming smaller.
But it is better for the Cd to be small because, smaller Cd means that the airfoil have less drag when
subjected to a higher Mach number with Cd of 1.0271 x 10-1 (Mach number = 1.5) compared to
the Cd of lower Mach number of 0.5 that yields 1.9181 x 102.
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Conclusion
The objective of the study is achieved because the contours of static pressure are obtained
and compared at Mach 0.5, Mach 0.6, Mach 0.9, Mach 1.2 and Mach 1.5. The graph of
pressure coefficient (Cp) along the upper and lower airfoil surface is also plotted to show the
characteristic of each Mach number. Lastly the relationship between drag and lift coefficient
and Mach number of NACA0012 are obtained.
References
1) http://en.wikipedia.org/wiki/Drag_coefficient
2) http://en.wikipedia.org/wiki/NACA_airfoil
3) http://en.wikipedia.org/wiki/Mach_number
4) http://en.wikipedia.org/wiki/Pressure_coefficient
5) http://www.grc.nasa.gov/WWW/k-12/airplane/mach.html
6) http://www.grc.nasa.gov/WWW/k-12/airplane/dragco.html
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