Lab 2

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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