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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:21 No:01 1
The computational Fluid Dynamics is governed by the following equations [7]: the continuity equation :
𝜕𝜌
𝜕𝑡+
𝜕
𝜕𝑥𝑗
(𝜌𝑈𝑗) = 0 … … … … … … . (1)
Momentum equation:
𝜕
𝜕𝑡(𝜌𝑈𝑖) +
𝜕
𝜕𝑥𝑗
(𝜌𝑈𝑖𝑈𝑗) = −𝜕𝑝
𝜕𝑥𝑖
+𝜕𝜏𝑖𝑗
𝜕𝑥𝑗
… … … … … … … . . (2)
And the energy equation:
𝜕
𝜕𝑡(𝜌ℎ) +
𝜕
𝜕𝑥𝑗
(𝜌𝑈𝑗ℎ) =𝜕𝑝
𝜕𝑡+ 𝑈𝑗
𝜕𝑝
𝜕𝑥𝑗
+ 𝜏𝑖𝑗
𝜕𝑈𝑗
𝜕𝑥𝑗
−𝜕𝑞𝑖
𝜕𝑥𝑖
… … … … . (3)
Geometry & Grid Generation
The Ansys-Fluent 14.5 finite element program is used to analyze the NACA0015 airfoil with a chord of 1m. To create the airfoil
geometry, the coordinates were taken from [8] . For airfoil flow analysis, the C mesh domain was selected and a structured mesh
called "mapped face mesh" was generated. This method is very time-consuming to generate high-quality meshes and is not suitable for
complex meshes. As shown in figure (1), the dimension of the arc radius (R1) is set to 12.5m, while the sides of the other two squares
(H2) are set to 20m. The airfoil is discretized into 149,252 elements with 150268 nodes. The mesh model shown in Figure (2) and
Figure (3) and the mesh details shown in Table (1). Figure 3 shows a mesh of airfoil with C domains. The mapped mesh is created on the entire domain. The cross section near the airfoil is developed to be fine and coarser at the farther away from the airfoil. For this
kind of airfoil, a quadratic element is used. In some areas away from the airfoil, the mesh must also be fine.
Fig. 1. Computational Domain
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:21 No:01 3
Figure 7. demonstrates the distribution of pressure coefficient (cp) on both surfaces of Airfoil at =2.5°. At 𝑀∞ = 0.6, the flow
extends along the leading edge and then begins to slow down, which is the typical subsonic behavior. At 𝑀∞ = 0.7 the flow continues
to expand after going around the leading edge, and it returns to subsonic speed through a shock wave on the upper side at (x/c=0.348)
a cross which there is an extremely rapid rise in pressure coefficient., the shock moves aft, becoming much stronger as 𝑀∞ increases
further. This leads to formation of bow shock, where the pressure coefficient is high compared to other region.
Fig. 7. distribution of Pressure coefficient (Cp) on the upper and lower sides of the Airfoil NACA0015 at =2.5°
3.WALL SHEAR STRESS
Figure 8. shows Effect of increasing 𝑀∞on the wall shear stress at =2.5°.It can be seen that the shear stress of the wall is proportional to the gradient of speed at the wall. This means that higher speeds cause greater shear stress on the wall. Consequently, the suction
area (upper surface) generates more shear stress on the wall than the pressure area (lower surface) of the airfoil. As the 𝑀∞increases,
the effect of wall shear stress on the leading edge and on the middle of the airfoil is more significant.
Fig. 8. Effect of increasing 𝑀∞on the wall shear stress at =2.5°
4- LIFT AND DRAG COEFFICIENTS
Figure 9. shows the effect of increasing 𝑀∞ on lift coefficient at =2.5°. It can be noticed that the lift coefficient decreases due to the formation of shock wave, while lift increases again on intrados. The loss of lift is due to the separation of the boundary layer on the
upper surface of airfoil. As the 𝑀∞ increases, the shock wave moves the back and attached to the trailing edge, the amount of
separation decreases, and the airfoil recovers part of its lift, until the free stream becomes supersonic, after this point, the lift gradually
decreases again.
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:21 No:01 7
Fig. 9. Effect of increasing 𝑀∞ on lift coefficient at =2.5°
Figure 10. shows the Effect of increasing 𝑀∞ on CD
coefficient at =2.5°.The increasing in drag coefficient is due to the normal shock wave behind the supersonic airfoil
flow areas, which typically begin to appear at (𝑀∞= 0.7). It
is clear that the drag coefficient is relatively high near
(𝑀∞=0.9). There are two reasons that can cause to increase
drag; firstly, a shockwave causes increase in static pressure,
higher Mach number, higher increase in static pressure.
Therefore, by definition of drag, increase in static pressure will cause to increase drag . Secondly, Shockwaves can
cause separation in the flow, it means that the smooth flow
over the body is disturbed and flow is no more attached to
body neatly. This results in decrease in lift. The flow is
disturbed due to sudden drop in velocity and decrease in
velocity means decrease in energy of the flow. Just a
background on separation: when flow is going around a
body, it loses energy because it has to overcome skin
friction. If the energy decreases by decreasing velocity then
the drag will be increased because velocity and pressure are
inversely related, decrease in velocity causes increase in
pressure which causes increase in drag. As 𝑀∞increases, the
flow is supersonic all around the body (with the exception
of a small area near the stagnation point on the leading edge). There is a bow shock wave around the airfoil nose,
most of the airfoil is in supersonic flow. The flow begins to
be realigned parallel to the body surface and stabilizes, and
the shock-induced separation decreases. This condition
results in a lower drag-coefficient.
Fig. 10. Effect of increasing 𝑀∞ on CD coefficient at =2.5°
Validation of the Simulation process
In order to validate the computational results obtained in this study, Pressure coefficients at (=0°,-4°) and (𝑀∞= 0.675, 0.777, 0.702) are compared with the results of experimental work [10] as shown in figure 11. It can be seen that there is a good agreement between
the computational and experimental results. The small variation in results is due to variation in grid sizing, operating condition,
geometrical parameters, etc. but the obtained result shows the same trend so that the results are suitably verified.
Figure 11. Comparison of Cp values between computational and experimental results for Cp for NACA 0015.
CONCLUSIONS
It is evident form the data obtained from the simulated flow
over airfoil NACA 0015 is that:
1- As the 𝑀∞increases, shock waves appear in the
flow region. When 𝑀∞ increases further, the shock
becomes much stronger and moves aft rapidly
leading to the creation of bow shock, where the
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:21 No:01 8