Flow over a NACA 2412 Airfoil in Converge Studio

Objectives:

• Simulate a flow over NACA 2412 Airfoil.
• Calculate the Drag and Lift coefficient of the Airfoil.
• Simulate the flow for the various angle of attacks and compare the results.
• Simulate the flow using various Turbulence models and compare the results.
• Plot the velocity and pressure plots.

Airfoil:

An airfoil is the cross-sectional shape of a wing, blade (of a propeller, rotor, or turbine), or sail (as seen in cross-section). An airfoil-shaped body when moved through a fluid, produces an aerodynamic force. The component of this force perpendicular to the direction of motion is called the lift force. The component parallel to the direction of motion is called the drag force.

The Aerodynamic force on an airfoil is primarily the result of its angle of attack and shape. The angle of attack is the angle between the oncoming air and the relative line of reference known as the chord line. When oriented at a suitable angle, the airfoil deflects the oncoming air, resulting in a force on the airfoil in the direction opposite to the deflection. This force is known as aerodynamic force and can be resolved into two components: lift and drag. Most foil shapes require a positive angle of attack to generate lift.

This "turning" of the air in the vicinity of the airfoil creates curved streamlines, resulting in lower pressure on one side and higher pressure on the other. This pressure difference is accompanied by a velocity difference, via Bernoulli's principle, so the resulting flowfield about the airfoil has a higher average velocity on the upper surface than on the lower surface.

Airfoils have a huge amount of applications in the real world. A fixed-wing aircraft's wings, horizontal, and vertical stabilizers are built with airfoil-shaped cross sections, as are helicopter rotor blades. Airfoils are also found in propellers, fans, compressors and turbines. Sails are also airfoils, and the underwater surfaces of sailboats, such as the centerboard and keel, are similar in cross-section and operate on the same principles as airfoils.

Drag Coefficient:

The Drag coefficient 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 in which a lower drag coefficient indicates the object will have a less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.

`Cd=(2*Fd)/(rho*V^2*A`

Lift Coefficient:

The Lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a complete foil-bearing body such as a fixed-wing aircraft. Lift coefficient is a function of the angle of the body to the flow, its Reynolds number and its Mach number. The lift coefficient refers to the dynamic lift characteristics of a two-dimensional foil section, with the reference area replaced by the foil chord.

`Cl=(2*Fl)/(rho*V^2*A`

`Cd` and `Cl` are the drag and lift coefficients respectively

Fd and Fl are the drag and lift forces

`rho` is the fluid density

`V` is the freestream velocity of the fluid

`A` is the frontal area of the airfoil

Geometry:

The airfoil used for this simulation is NACA 2412. The coordinates were imported in Solidworks to plot the 3D model of the airfoil.

Coordinates are obtained from - http://airfoiltools.com/airfoil/details?airfoil=naca2412-il

NACA 2412 (to be read as 2 – 4 – 12) describes an airfoil with a camber of 0.02 chord located at 0.40 chord, with 0.12 chord of maximum thickness. The chord length of the airfoil is 1m.

The geometry was imported to converge and a virtual wind tunnel was created around the airfoil to simulate the flow. The should be created in such a way that the recirculation effects should not reach to the boundaries. For this purpose, the inlet is at a distance 25 times the chord length from the leading edge. The outlet is kept at a distance 50 times the chord length from the leading edge.

The dimensions of the Wind tunnel is -

• in X-direction- 7 m
• in Y-direction- 3 m
• in Z-direction- 0.05 m

Boundary Conditions:

At Inlet -

• Dirichlet boundary condition for velocity. The value for velocity was calculated using Reynold's number formula.
• `Re=(rho*V*D)/mu`
• Reynold's number = 200000 as per the problem statement
• Density = 1.1839 `kg*m^-3`
• Viscosity = 1.849e-5 `kg*(m*s)^-1`
• Velocity boundary condition - `31.2357 ms^-1` in x-direction
• Pressure boundary condition - Zero normal gradient

At Outlet -

• Velocity boundary condition - Zero normal gradient
• Pressure boundary condition - 101325 `Pa`

As it is a 2-dimensional simulation, the front and back faces are Two_D

Top and Bottom walls are far-field, hence a symmetry boundary condition is applied.

The Airfoil surface has a Wall type boundary. The velocity and Temperature boundary condition on Wall is kept as Law of Wall.

Initial Conditions:

An initial velocity of `31.2357 ms^-1` is provided in all the regions of the flow domain.

Mesh:

The Base mesh and the Embedding near the airfoil surface were decided based on the `Y^+` value we need for the simulation. `Y^+` is the non-dimensionalized wall distance which tells us where the first cell is in the boundary layer.

`Y^+` calculator: http://www.pointwise.com/yplus/

The desired `Y^+` value was kept 100 and according to the calculator, the `DeltaS` value was 0.00104. This means that the distance from the airfoil body to the first cell should be 0.00104.

Along the Y-axis the base grid was 0.016 m. Along the X and Z direction, as the velocity gradient does not change as rapidly as along Y-direction, the base grid size here was kept as 0.024 m.

• Base Grid along X-direction = 0.024 m
• Base Grid along Y-direction = 0.016 m
• Base Grid along Z-direction = 0.024 m

Mesh embedding was enabled around the airfoil body with a scale of 4 and 3 layers.

With this level of base mesh size embedding, the desired distance between the body and the first cell is achieved.

• First cell distance = Base Grid/(2^Embedding Scale)
• `0.016/2^4=0.001`

Case Setup:

The simulation was performed for four different angles of attack - `1^0, 5^0, 10^0, 15^0`. The Drag and Lift coefficients for these configurations were later compared.

Initially, the `k-omega` `SST` turbulence model is used as it is mostly preferred for external flows. A case is also set up using the `RNG` `k-epsilon` turbulence model to compare the performance of various turbulence models.

A Transient case was set up with simulation end-time of 1.5 sec. The flow through time concept was used to calculate the end time. With a velocity of 31.2357 `ms^-1` and tunnel length of 7 m,

`Time= 7/(31.2357)`= 0.224 sec`

The simulation time should be at least 5 times the flow through time. Hence the end time was taken as 1.5 sec.

Results:

(1) Angle of Attack = `1^0`

Velocity Field

Pressure Field

The above images show the pressure and velocity distribution around the airfoil.

• At the leading edge, as the free stream of air encounters the airfoil, the velocity decreases drastically and due to Bernoulli's principle the pressure increases. This is the Stagnation region where the velocity reduces to almost zero.
• As the flow moves upstream on the top of airfoil, the streamlines follow the curvature of the airfoil. As the streamlines as curved upwards, the pressure near the airfoil is less than that at the far-field (which is atmospheric pressure). Hence, the velocity in this region increases. As the flow progresses further downstream, velocity decreases as there is an increase in pressure.
• At the bottom of the airfoil, it is curved downwards which deflects the streamlines downsides. This implies that pressure near the airfoil is higher than that at far-field (which is atmospheric pressure). Hence, the velocity decreases in this region.
• As we move further, the velocity decreases even more due to boundary layer formation. But the relative increase in pressure is not observed in this region because the Bernoulli's principle cannot be applied in a boundary layer. Towards the trailing edge, as the boundary layer thickens, the velocity decreases.
• The pressure gradient between the upper and lower surface is what produces the lift force.

Animation of Velocity Field

Drag Force

• The average Drag force was calculated to be 1.24 N

Lift Force

• The average Lift force was calculated to be 18.13 N

`Y^+` Value

• The `Y+` value here is 57.76. For the Law of the Wall model, `Y+` should be between 30 and 100. This suggests that the first cell is in the log-law region. A lower value of `Y+` indicates that the first cell is in viscous sublayer where the low Reynold's number flow dominates. As we have a high Reynold's number flow, `Y+` should be between 30-100. This suggests that the grid size selected is right.

(2) Angle of Attack = `5^0`

Velocity Field

Pressure Field

• The flow pattern is the same as expected for the earlier case. The main difference observed is the size of the low-pressure region above the airfoil, which is relatively smaller as compared to `1^0` angle of attack case.

Animation of Velocity Field

Drag Force

• The average Drag force was calculated to be 2.28 N

Lift Force

• The average Lift force was calculated to be 41.59 N

`Y^+` Value

• The `Y+` value here is 56.27

(3) Angle of Attack = `10^0`

Velocity Field

Pressure Field

• The flow pattern is same as expected according to the Bernoulli's principle and concept of bending of streamlines. However, the pressure and velocity field pattern are varying slightly compared to the previous two cases.
• As the flow moves downstream, there is a significant decrease in velocity due to the thickening of the boundary layer.
• Also, the maximum velocity is 51 m/s.

Animation of Velocity Field

Drag Force

• The average Drag force was calculated to be 4.85

Lift Force

• The average Lift force was calculated to be 58.54

`Y+` Value

• The `Y+` value here is 46.27

(4) Angle of Attack = `15^0`

Velocity Field

Pressure Field

The flow characteristics are completely different for an angle of attack of `15^0`.

• At the stagnation point, the velocity reduces to a low value. But as we move further there is an immediate decrease in the velocity of air around the airfoil. This is because the flow separation occurs near the leading edge at such a high angle of attack.
• Due to the separation, a region of recirculation is created where the air fills the vacuum in the space and a condition called Stall occurs. It occurs when the angle of attack exceeds the critical angle of attack of the airfoil. It amounts to a reduced lift and an increase in drag thus reducing the efficiency of the airfoil.

Animation of Velocity Field

Drag Force

• The average Drag force was calculated to be 12.53 N

Lift Force

• The average Lift force was calculated to be 46.11 N

`Y+` Value

• The `Y+` value here is 37.61

Comparison of Drag and Lift coefficients for a various angle of attacks

• The plots above show the drag and lift coefficients at different angle of attacks. Up to `10^0`, drag and lift coefficients increases and reaches the peak at `10^0`, but, beyond that, the lift coefficient significantly reduces whereas the drag coefficient increase.
• In this case, between 10-15 degrees this airfoil stalls, and lift falls off quickly beyond that. The drop in lift can be explained by the action of the upper-surface boundary layer, which separates and greatly thickens over the upper surface at and past the stall angle.
• The thickened boundary layer's displacement thickness changes the airfoil's effective shape, in particular, it reduces its effective camber, which modifies the overall flow field so as to reduce the circulation and the lift. The thicker boundary layer also causes a large increase in pressure drag, so that the overall drag increases sharply near and past the stall point.

Comparison of Various Turbulence Models:

(5) Standard `k-epsilon` Turbulence model - Angle of Attack = `5^0`

Velocity Field

Pressure Field

Drag Force

• The average Drag force was calculated to be 2.39 N

Lift Force

• The average Lift force was calculated to be 42.68 N

`Y+` Value

• The `Y+` value here is 34.01

(6) RNG `k-epsilon` Turbulence model - Angle of Attack = `5^0`

Velocity Field

Pressure Field

Drag Force

• The average Drag force was calculated to be 2.3 N

Lift Force

• The average Lift force was calculated to be 42.39 N

`Y+` Value

• The `Y+` value here is 34.06

• The main difference between the `k-omega` and `k-epsilon` models is the `Y+` values.
• The drag and lift coefficients obtained are nearly the same in all the three scenarios.