Experimental Study Of Fluid Flow Over A NACA 2412 Aerofoil Using Transient Solver By Creating A Virtual Wind Tunnel In Converge CFD & Post-Processing It In ParaView

Abstract:- The experiment is focused on studying the flow characteristics over a symmetric NACA 2412 aerofoil inside a virtually designed low subsonic wind tunnel created using the geometry editing tools available in Converge CFD software & the results obtained will be post-processed using Paraview. The aerofoil designed in Converge Studio will have a span of 1m or 100cm or 1000mm and a chord length of 0.2m or 20cm or 200mm. The virtual wind tunnel which will also be designed in Converge will have a length of 50 times the chord length & breadth of 30 times the chord length of the aerofoil. We will be using a transient solver to analyze the fluid flow over the symmetric aerofoil. The pressure distribution on the airfoil surface was obtained, lift and drag forces were measured, and the mean velocity profiles were obtained over the surface of the aerofoil. Experiments were carried out by varying the angle of attack (AOA) at 1°, 5°, 10° & 15° & stalling of the aerofoil was observed. The experimentation allowed a comparison of flight characteristics between the different AOA of the airfoil. Through this experimental study, we will also be also calculating the drag coefficient `C_D` & lift coefficient `C_L` from the results obtained.

                                                                 

                                                                                                                          Fig 1.1:- Test model & Experimental Setup

 

Introduction To Aerofoils                                                                                                                                                                                                                                                                       

In the early 1800s, Sir George Cayley discovered that a curved surface produced more lift than a similar-sized flat plate. What aviation pioneers discovered, which still holds
true today is that the most efficient way to do this is to use a curved, streamlined shape — the aerofoil. Aerofoils are defined as the cross-sectional shape that is designed with a curved surface giving it the most favorable ratio between lift and drag in flight.  Lift is the component such that the force is perpendicular to the direction of motion and drag is the component parallel to the direction of motion. Airfoils are highly-efficient lifting shapes, able to generate more lift than similarly sized flat plates of the same area, and able to generate lift with significantly less drag. Airfoils have the potential for use in the design of aircraft, propellers, rotor blades, wind turbines, and other applications of aeronautical engineering.

A similar idea is used in the designing of hydrofoils which is used when water is used as the working fluid. The designing of the aerofoil depends on the aerodynamic characteristics which further depends on the weight, speed, and purpose of the aircraft.

The Wright brothers & many of the early designers used “eyeball engineering” in developing or copying the airfoil shape used on their airplanes. From about 1912, airfoil
development and research moved to the wind tunnel laboratories found at the University of Göttingen in Germany, the Royal Aircraft Factory in the United Kingdom, and the National Advisory Committee for Aeronautics (NACA) in the United States. Among other advancements, NACA research & development produced the NACA duct, a type of air intake used in modern automotive applications, the NACA cowling, and several series of NACA aerofoils which are still used in aircraft manufacturing as shown in Fig 1.2.  Since then, the wind tunnel, complex mathematics, and the computer have all played important roles in airfoil development. 

                                                                                      

                                                                                        Fig 1.2:- National Advisory Committee for Aeronautics (NACA) airfoil series.

 

Airfoil Nomenclature                                                                                                                                                                                                                                                                       

The following are the terms that are related to aerofoils. A chord is defined as the distance between the leading edge which is the point at the front of the aerofoil and has maximum curvature and the trailing edge which is the point at the rear of the aerofoil with maximum curvature along the chord line. Leading Edge is the point at the front of the airfoil that has minimum curvature (maximum radius). Trailing-Edge is defined similarly as the point of maximum curvature at the rear of the airfoil. The chord line is defined as the straight line connecting the leading and trailing edges. The upper surface is also known as the suction surface which is associated with high velocity and low static pressure. The Lower surface is also known as pressure surface with higher static pressure. The angle of attack (AOA) is the angle formed between a reference line on a body and the oncoming flow.

                 

                                                                                            Fig 1.2:- NACA aerofoil at a 0-degree AOA & `alpha`- degree AOA.

Based on the shape of the aerofoil, they are classified into cambered aerofoil & symmetric aerofoil. Symmetrical aerofoil has identical upper and lower surfaces such that the chord line and mean camber line are the same producing no life at zero AOA as shown in Fig 1.3. These find applications in most of the light helicopters in their main rotor blades. Cambered aerofoil has different upper and lower surfaces such that the chord line is placed above with large curvature. It is usually designed into an airfoil to maximize its lift coefficient. This minimizes the stalling speed of aircraft using the airfoil. An aircraft with cambered wings will have a lower stalling speed than an aircraft with a similar wing loading and symmetric airfoil wings. It can also generate useful lift at zero AOA. 

                                                                     

 

                                                                                     Fig 1.3:- Pictorial Representation of a Cambered aerofoil & a Symmetric aerofoil.

 

NACA Airfoil Series                                                                                                                                                                                                                                                                         

The early NACA airfoil series, the 4-digit, 5-digit, and modified 4-/5-digit were generated using analytical equations that describe the camber (curvature) of the mean-line (geometric centerline) of the airfoil section as well as the section's thickness distribution along the length of the airfoil. Later families, including the 6-Series, are more complicated shapes derived using theoretical rather than geometrical methods. Before the National Advisory Committee for Aeronautics (NACA) developed
these series, airfoil design was rather arbitrary with nothing to guide the designer except past experience with known shapes and experimentation with modifications to those shapes.

 This methodology began to change in the early 1930s with the publishing of a NACA report entitled The Characteristics of 78 Related Airfoil Sections from Tests in the Variable Density Wind Tunnel. In this landmark report, the authors noted that there were many similarities between the airfoils that were most successful, and the two primary variables that affect those shapes are the slope of the airfoil mean camber line and the thickness distribution above and below this line. They then presented a series of equations incorporating these two variables that could be used to generate an entire family of related airfoil shapes. As airfoil design became more sophisticated, this basic approach was modified to include additional variables, but these two basic geometrical values remained at the heart of all NACA airfoil series.

          A. NACA Four-Digit Series: The first family of airfoils designed using this approach became known as the NACA Four-Digit Series. The first digit specifies the maximum camber (m) in the percentage of the chord (airfoil length), the second indicates the position of the maximum camber (p) in tenths of the chord, and the last two numbers provide the maximum thickness (t) of the airfoil in the percentage of the chord. For example, as in our case for NACA 2412 airfoil has a maximum thickness of 12% with a camber of 2% located 40% back from the airfoil leading-edge (or 0.4c).

         B. NACA Five-Digit Series: The NACA Five-Digit Series uses the same thickness forms as the Four-Digit Series but the mean camber line is defined differently and the naming convention is a bit more complex. The first digit, when multiplied by 3/2, yields the design lift coefficient (cl) in tenths. The next two digits, when divided by 2, give the position of the maximum camber (p) in tenths of the chord. The final two digits again indicate the maximum thickness (t) in the percentage of the chord. For example, the NACA 23012 has a maximum thickness of 12%, a design lift coefficient of 0.3, and a maximum camber located 15% back from the leading edge.

         C. Modified NACA Four- and Five-Digit Series: The airfoil sections you mention for the B-58 bomber are members of the Four-Digit Series, but the names are slightly different as these shapes have been modified. Let us consider the root section, the NACA 0003.46-64.069, as an example. The basic shape is 0003, a 3% thick airfoil with 0% camber. This shape is a symmetrical airfoil that is identified above and below the mean camber line. The first modification we will consider is 0003-64. The first digit following the dash refers to the roundedness of the nose. A value of 6 indicates that the nose radius is the same as the original airfoil while a value of 0 indicates a sharp leading edge. Increasing this value specifies an increasingly more rounded nose. The second digit determines the location of maximum thickness in tenths of the chord. The default location for all four- and five-digit airfoils is 30% back from the leading edge. In this example, the location of maximum thickness has been moved back to a 40% chord. Finally, notice that the 0003.46-64.069 features two sets of digits preceded by decimals. These merely indicate slight adjustments to the maximum thickness and location thereof. Instead of being 3% thick, this airfoil is 3.46% thick. Instead of the maximum thickness being located at 40% chord, the position on this airfoil is at 40.69% chord. 

          D. NACA 6-Series: The naming convention of the 6-Series is by far the most confusing of any of the families discussed thus far, especially since many different
variations exist. One of the more common examples is the NACA 641-212, a=0.6. In this example, 6 denotes the series and indicates that this family is designed for greater laminar flow than the Four- or Five-Digit Series. The second digit, 4, is the location of the minimum pressure in tenths of the chord (0.4c). The subscript 1 indicates that low drag is maintained at lift coefficients 0.1 above and below the design lift coefficient (0.2) specified by the first digit after the dash in tenths. The final two digits specify the thickness in the percentage of the chord, 12%. The fraction specified by a=___ indicates the percentage of the airfoil chord over which the pressure distribution on the airfoil is uniform, 60% chord in this case. If not specified, the quantity is assumed to be 1, or the distribution is constant over the entire airfoil.  

           E. NACA 7-Series: The 7-Series was a further attempt to maximize the regions of laminar flow over an airfoil differentiating the locations of the minimum pressure on the upper and lower surfaces. An example is the NACA 747A315. The 7 denotes the series, the 4 provides the location of the minimum pressure on the upper surface in tenths of the chord (40%), and the 7 provides the location of the minimum pressure on the lower surface in tenths of the chord (70%). The fourth character, a letter, indicates the thickness distribution and mean line forms used. A series of standardized forms derived from earlier families are designated by different letters. Again, the fifth digit indicates the design lift coefficient in tenths (0.3), and the final two integers are the airfoil thickness in the percentage of the chord (15%). 

 

Lift & Drag Force Acting On An Aerofoil                                                                                                                                                                                                                                            

Lift is defined as the component of the aerodynamic force perpendicular to the relative airflow. There are multiple incorrect theories concerning the generation lift. The theory most commonly found in textbooks and pilot training manuals utilizes Bernoulli’s principle which states that for a liquid or gas, areas with high relative velocity create lower pressure systems, and areas with low relative velocity create high-pressure systems. The theory states that airfoils are shaped so that the upper surface is longer than the lower surface; therefore when air molecules are separated by the leading edge of the airfoil, they have a greater distance to travel as they cross the upper surface than along the lower surface. Thus, in order for the air molecules to meet at the trailing edge at the same time, the molecules traveling along the upper surface must be traveling faster than the air molecules along the bottom surface. Since the airflow on the upper surface is faster, Bernoulli’s principle states that a lower pressure system is created. The difference between the low pressure above the airfoil and the higher pressure below causes the lift to occur. 

                                                                            

                                                                                 Fig 1.4:-: Airflow about an airfoil according to the flow turning theory.

However, there are no principles of fluid dynamics stating that two free moving air particles must meet at a single point beyond an obstacle once separated by the obstacle.
The correct theory of lift generation is known as the flow turning theory. It states that the airfoil bends the direction of the airflow around it as the airflow passes over the
the upper surface, and creates a vertical velocity of airflow past the trailing edge. The effect of the airflow bending is due to the viscosity of a fluid and the Coanda effect. As the airfoil bends the airflow near the upper surface, it pulls on the air above it and causes an acceleration of that air down to the airfoil. The pulling of the air causes a low-pressure system to form over the airfoil creating a net force that is lift. Figure 1.5 demonstrates airflow about an airfoil generating lift by the flow turning theory. The other aerodynamic force that affects an airfoil in a wind tunnel is perpendicular to the lifting force, called drag. The airfoil experiences a drag force that opposes the relative motion of the airfoil and has direction parallel to the airflow. Skin friction drag is the friction that occurs between the air molecules and the surface of the airfoil. Form drag is dependent on the overall shape of the airfoil and pertains to the pressure distribution about the airfoil’s surface. As with the lifting force, the airfoil changes the local momentum of the air around it, affecting the velocity and pressure. The resulting pressure distribution produces a force that acts on the airfoil. 

 

Lift & Drag Co-efficient                                                                                                                                                                                                                                                                          

The equations for calculating lift and drag are very similar. The lift that an airfoil generates depends on the density of the air, the velocity of the airflow, the viscosity and compressibility of the air, the surface area of the airfoil, the shape of the airfoil, and the angle of the airfoil’s angle of attack. However, dependence on the airfoil’s shape, the angle of attack, air viscosity, and compressibility are very complex. Thus, they are characterized by a single variable in the lift equation, called the lift coefficient. Due to the complexities of the lift coefficient, it is generally found via experimentation in a wind tunnel where the remaining variables can be controlled. Therefore, the lift equation is given by                                                                                                                                                               `L = 1/ 2 ρV ^2SC_L`

                                       Where,  C_L: lift coefficient, L: lift force, S: relevant surface, q: fluid dynamic pressure, ⍴: fluid density, u: flow speed.

As with lift, the drag of an airfoil depends on the density of the air, the velocity of the airflow, the viscosity and compressibility of the air, the surface area of the airfoil, the shape of the airfoil, and the angle of attack. The complexities associated with drag and the airfoil’s shape, angle of attack, the air’s viscosity, and air’s compressibility are simplified in the drag equation by use of the drag coefficient. The drag coefficient is generally found through testing in a wind tunnel, where the drag can be measured, and the drag coefficient is calculated by rearranging the drag equation 

                                                                                                                                    `D = 1/2 ρV^2AC_D`

                                       Where,  D: drag force, ρ: density of the air, V: velocity of the air, A: body area, and `C_D`is the drag coefficient.

 

Experimental Setup                                                                                                                                                                                                                                                                              

                                                                                     A. Designing The Aerofoil geometry & The Virtual Wind Tunnel In Converge.

We start by creating the profile of the aerofoil geometry by using the X & Y co-ordinate points available in the "Dat files" as shown below. We use these co-ordinate points to plot several vertexes in Converge studio & stitch the geometry by creating triangles using the patch tool method to close all the points as shown in Fig 2.1. Once done it will create a 2D geometry of the NACA 2412 airfoil. We need to convert this 2D geometry into a 3D geometric figure, which is necessary for specifying the inlet, outlet, 2D walls & Top & Bottom walls. We do this by creating an off-set of the already existing 2D aerofoil geometry on the Z-axis, which is then followed by lofting the faces of these 2 2D geometries to create a single wing surface. The bounding box tool available in the geometry editing tools is used to determine the dimensions of any geometry. To understand the dimensions of our airfoil we can enable the bounding box in Converge Studio as shown in the image Fig 2.2. The X-axis of the geometry represents the chord length of the aerofoil, Y-axis represents the height of the aerofoil & the Z-axis represents the wing-span. Another approach for designing the wing geometry will be to make use of a Computer-Aided Designing (CAD) software package & importing the CAD model into Converge Studio as an STL file format. Once the geometry is created we change the normal orientation of the geometry & finally run a diagnostic test to check for errors like intersection errors, open edges, or non-manifold edges.

NACA 2412 Airfoil M=2.0% P=40.0% T=12.0%
  1.000084  0.001257
  0.998557  0.001575
  0.993984  0.002524
  0.986392  0.004086
  0.975825  0.006231
  0.962343  0.008922
  0.946027  0.012110
  0.926971  0.015740
  0.905287  0.019752
  0.881104  0.024079
  0.854565  0.028653
  0.825830  0.033404
  0.795069  0.038260
  0.762469  0.043149
  0.728228  0.048000
  0.692554  0.052741
  0.655665  0.057302
  0.617788  0.061615
  0.579155  0.065609
  0.540008  0.069220
  0.500588  0.072381
  0.461143  0.075034
  0.421921  0.077122
  0.383032  0.078574
  0.344680  0.079198
  0.307289  0.078941
  0.271106  0.077802
  0.236371  0.075794
  0.203313  0.072947
  0.172151  0.069309
  0.143088  0.064941
  0.116313  0.059918
  0.091996  0.054325
  0.070289  0.048257
  0.051324  0.041808
  0.035214  0.035076
  0.022051  0.028152
  0.011907  0.021120
  0.004833  0.014049
  0.000860  0.006997
  0.000000  0.000000
  0.002223 -0.006689
  0.007479 -0.012828
  0.015723 -0.018404
  0.026892 -0.023408
  0.040906 -0.027826
  0.057669 -0.031651
  0.077071 -0.034878
  0.098987 -0.037507
  0.123281 -0.039546
  0.149805 -0.041013
  0.178401 -0.041934
  0.208902 -0.042346
  0.241131 -0.042294
  0.274904 -0.041834
  0.310028 -0.041027
  0.346303 -0.039941
  0.383522 -0.038644
  0.421644 -0.037174
  0.460397 -0.035444
  0.499412 -0.033493
  0.538451 -0.031373
  0.577279 -0.029138
  0.615658 -0.026833
  0.653352 -0.024500
  0.690129 -0.022172
  0.725762 -0.019880
  0.760029 -0.017649
  0.792716 -0.015499
  0.823619 -0.013448
  0.852541 -0.011510
  0.879302 -0.009701
  0.903730 -0.008033
  0.925669 -0.006520
  0.944979 -0.005174
  0.961536 -0.004008
  0.975232 -0.003035
  0.985978 -0.002265
  0.993705 -0.001708
  0.998361 -0.001370
  0.999916 -0.001257

                

                  

                                             Fig 2.1:- Creating the aerofoil face by plotting the X & Y coordinate points & then patching these points to form triangles.

                                                        

                                        Fig 2.2:- Length at X-axis is 0.1m (Chord length), Length at Y-axis is 0.0012m (Height), Length at Z-axis is 0.1m (Wing Span).

                        

                                                                                                         Fig 2.3:- Top & Side view of the NACA 2412 geometry. 

                                                                     

                                                                                                                               Fig 2.4:- Normal Orientation. 

NACA airfoil series is controlled by 4 digits as in our case NACA 2412, which designates the camber, position of the maximum camber, and thickness. If an airfoil number is

then:

• M is the maximum camber divided by 100. In our case M=2 so the camber is 0.02 or 2% of the chord.
• P is the position of the maximum camber divided by 10. In our case P=4 so the maximum camber is at 0.4 or 40% of the chord.
• XX is the thickness divided by 100. In our case XX=12 so the thickness is 0.12 or 12% of the chord.

As we can observe from Fig 2.2, our NACA aerofoil has a length of 0.1m in the X-axis which is the Chord length of the aerofoil, Length of 0.0012m which is the Height of the Aerofoil & Length 0.1m which is the WingSpan. From this information, we need to create a virtual wind tunnel using the tools available in Converge Studio. The dimensions of the virtual wind tunnel will follow a convention i.e. the length will be 50 times the chord length of the aerofoil & the width or height will be 30 times the chord length. For the purpose of our project, we need to set up the Aerofoil at 1°, 5°, 10° & 15° angle of attack (AoA) inside the virtual wind tunnel created using Converge as shown in Fig 2.6. 

 
For our case,

 Chord length (CL) is 0.100008 ~ 0.1 meters, length of our virtual wind tunnel will be  [50*(CL)] i.e. (50*0.1) `=>` 5m & height of our wind tunnel will be [30*(CL)] `=>` 3m.  

 

                                                                                       Fig 2.5:- Airfoil structure inside the Virtual Wind Tunnel created in Converge 

 

       

                                                                       (a)                                                                                                                                                          (b)

     

                                                                        (C)                                                                                                                                                      (d)

                                        Fig 2.6:- Representation of the Aerofoil geometry inside the wind tunnel at (a) 1° (b) 5° (c) 10° & (d) 15° angle of attack (AoA).  

 

Having built the geometry for all four cases, a diagnostic test is conducted to check for any anomaly like intersection errors, nonmanifold problems, open edges, etc within the geometry contour.  If no errors, which is denoted by “green checks”, then we proceed to check the “Normal’s”. Every geometry will have a normal vector which is perpendicular to the geometry, for this problem we use the “Normal Toggle” option to check for the direction of the normal. If normals are pointing outside of the geometry, then it's essential to transform the normal’s to point inside the geometry, where the fluid flow will occur as shown in Fig 2.4. In Converge, all the surface information is stored in the form of 3 entities vertices, edges & triangles. Therefore, any geometry created or exported from other CAD software into converge are converted into triangles & that is the fundamental entity in Converge, where every part of the geometry is assumed to be a triangle. The process of grouping these triangles into boundaries is known as boundary flagging. Each boundary is assigned with a distinct ID as shown in Fig 2.7. For our geometry, we have 6 distinct surfaces Inlet, Outlet, Top&Bottom, Front2D, Back2D & Aerofoil.

 

                            

                                                                                                                                                    Fig 2.7:- Boundary Flagging

 

                                                                                                             B. Setting Up The Flow Physics For The Computation Model                                                                                                                                                                                                                  

The simulations were carried out with the Converge CFD Software. In Converge after editing the geometry, the next stage of the process is to set up the simulation parameters. This step involves specifying certain properties to capture the physics of the problem perfectly, such as materials used, simulation time parameters, solver parameters, initial & boundary conditions to solve the Navier Stokes equation which are PDE’s, defining body forces, type of fluid flow, species involved & grid size of the mesh.

We start setting up the time-based general flow simulation by first defining the Materials involved in the simulation. Since, we are simulation 2 different fluid flow inside a channel we choose ‘gas simulation’ and our choice of species are nitrogen (N2) & oxygen (O2), & we choose our pre-defined mixture as air. For gas simulation parameters we will be using the Redlich-Kwong equation of state with a critical temperature of 133K & critical pressure of 3770000pa. For global transport parameters, we use the default values of ‘turbulent Prandtl number’ and ‘Schmidt number’ which is 0.90 & 0.78 respectively.

The next parameter for our case setup is the Simulation Parameters, which include the ‘run parameters’, ‘simulation time parameters’ & the ‘solver parameters’. For the aerofoil problem, we will be running compressible gas flow using transient solver at full hydrodynamics simulation mode, since our geometry is simple & has no moving part & we also require to solve the NS equations. In addition to this, we will be using ‘density-based’ PISO Navier strokes solver. As for simulation time parameters, we will be running the simulation for an end time of 0.8 seconds with an initial & minimum time step as (1e-05).

 

[ Note:- To find the end time for a transient based simulation,  

                                                                                                          Flow Through Time = Length of Wind Tunnel ÷ inlet velocity

                  We already know the length of the wind tunnel, which is 5 meters & to find the inlet velocity we use the Reynolds number; `Re = (rho VL) / (mu)`

                  where Re = 200000,  ρ = 1.1839 kg/m^3 , V = free stream velocity that we need to find ,  L = 0.1 (chord length),  μ = 1.84e-5 kg/ms.

                  Solving we get the free stream velocity to be V = 31.4 m/s.                                                                                           `therefore` flow through time or end time = `5/31.4` => 0.16 seconds

                  once we get the end time we multiply the value by at least (5) to allow the solution to converge & reach a stable condition]

 

The simulation was run for four different angles of attack (AoA) 1°, 5°, 10° & 15°. Drag & Lift coefficient was calculated for each of the value of angle of attack & the results were then compared by using different turbulent models. The turbulent models used are RNG k-ɛ & k-ω SST. 

As earlier, after designing the geometry, we grouped the triangles and assigned them to a particular boundary ID. Similarly, in Boundary Condition, we group the 6 boundaries & assign each of them to a volumetric region. To do so we add a volumetric region from Initial Conditions & Events & assign the volumetric region (Region 0) to each of the boundaries. These volumetric regions are required to set up the initial conditions for solving PDE's & will have a velocity of 31.4 m/s in the x-direction. Initial conditions are assigned at the volume whereas boundary conditions are given at boundaries. The Inlet & Outlet boundaries are inflow & outflow boundaries respectively, airfoil wall assigned as stationary walls with fixed surface following the ‘Law Of Wall’ & a temperature boundary condition of 300k which is also following the ‘Law Of Wall’. At the inlet boundary of the geometry, we defined a total pressure as Zero Normal Gradient (ZNG), Velocity of 31.4 m/s & temperature of 300K with Air being the species. At the outlet boundary of the geometry, we defined a total pressure of 101325 pa & temperature of 300K with Air being the species if backflow occurs.

The final parameter that we need to setup is defining the Mesh that is the Base grid size, for our case our base grid size along the entire geometry will be 0.017 with permanent embedding near-wall boundary. The scale of embedding was 4 & the number of embedding layers will also be 4 near the Aerofoil boundary this helps in refining particular areas in the domain. For 3D output files, we can calculate by dividing the end time with a number depending on our required output files. [`0.8/0.02 = 40`].

 

                                                                                                         C. Post Processing Results & Outputs For All 4 Cases

The function of Converge studio is to set up the simulation & then create several input files which are then exported to a particular folder. To run these input files we use CYGWIN, a command-line interface that reads these inputs files & solves the complex PDEs of governing equation to generate several output files. These output files are then post-converted into 3D output files which are readable files for ParaView.

 

                                                                                                                           Case 1:- Aerofoil with 1° Angle of Attack (AoA)                                                                                                         

 

 

 

For case1, with a base mesh grid size of 0.017m,  the total number of cells formed for the entire geometry is 53161 cells for the entire time period of the simulation for both the RNG k-ɛ Turbulence Model & k-ω SST Turbulence Model. The smaller the mesh grid size the larger number of cells would be generated. At the boundary of the geometry, we observe a much finer mesh than compared to the entire surface, this is because we used ‘fixed embedding’ to refine the particular areas in the domain.

 

    

    

    

 

The above images represent the pressure & velocity gradient around an aerofoil at a 1° angle of attack inside a virtual wind tunnel. Both pressure & velocity flow field variation across the aerofoil follows Bernoulli's principle. We observe at the leading edge, as the free stream air encounters the aerofoil, the velocity decreases drastically, this is the stagnation region where velocity reduces to almost zero. As the flow moves upstream on the top of airfoil, we can observe the glyph streamlines following the curvature of the airfoil. After the air is split at the wing’s leading edge, we start observing a velocity & pressure difference above and below the wing surface so that the air will reach the trailing edge of the wing at the same time. In general, the aerofoils upper surface is curved so that the air rushing over the top of the wing speeds up and stretches out, which decreases the air pressure above the wing. This is observed in the velocity contour images. In contrast, we observe air flowing below the wing moves in a straighter line, thus its speed and pressure remain about the same (no variation). Since high pressure always moves toward the low-pressure region, the air below the wing pushes the wing upwards, which will generate "Lift force' & an opposing force known as the "Drag Force". 

 

 

 

On simulating transient airflow over the airfoil in CONVERGE, the results appeared to have converged at 0.8 seconds. The above plots represent the pressure force along x-direction which is the "drag force" & pressure force along Y-direction which is the "lift force" for RNG k-ɛ Turbulence Mode. An aerofoil at a low angle of attack generally doesn't produce significant lift & drag. Since it's impossible to take an average of such large amounts of data, for our simplification we perform a time average only for the final 0.1 seconds of the simulation i.e. from 0.8 seconds to 0.9 seconds. This averaged data will provide us the Lift & Drag forces acting on the aerofoil which would help us to calculate the Lift Coefficient & Drag Coefficient for RNG k-ɛ Turbulence Mode. 

                `

     Lift coefficient for 1° Angle of attack using RNG k-ɛ Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
18.5353	1.2754	31.4	0.1	0.29479757485547

 

Drag coefficient for 1° Angle of attack using RNG k-ɛ Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
0.825734	1.2754	31.4	0.1	0.013133

 

 We follow similar steps to find the Lift Coefficient & Drag Coefficient for the aerofoil at 1° angle of attack using the k-ω SST Turbulence Model. 

 

 

 

Lift coefficient for 1° Angle of attack using k-ω SST Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
18.0974	1.2754	31.4	0.1	0.28783292588679

 

Drag coefficient for 1° Angle of attack using k-ω SST Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
0.902494	1.2754	31.4	0.1	0.014354

 

 

                                                                                                                                                                                                                                                  

 

 

                                                                                                                                    Case 2:- Aerofoil with 5° Angle of Attack (AoA)                                                                                                 

 

  

 

For case2, i.e. when aerofoil at 5° angle of attack, with a base mesh grid size of 0.017m,  the total number of cells formed for the entire geometry is 53111 cells for the entire time period of the simulation for both the RNG k-ɛ Turbulence Model & k-ω SST Turbulence Model. The smaller the mesh grid size the larger number of cells would be generated. At the boundary of the geometry, we observe a much finer mesh than compared to the entire surface, this is because we used ‘fixed embedding’ to refine the particular areas in the domain.

 

    

    

    

 

The above images represent the pressure & velocity gradient around an aerofoil at a 5° angle of attack inside a virtual wind tunnel. Both pressure & velocity flow field variation across the aerofoil follows Bernoulli's principle. The flow trends observed are similar to the flow trends for 1°, except the low-pressure region observed at the top of the aerofoil is lower.We observe at the leading edge, as the free stream air encounters the aerofoil, the velocity decreases drastically, this is the stagnation region where velocity reduces to almost zero. As the flow moves upstream on the top of airfoil, we can observe the glyph streamlines following the curvature of the airfoil. After the air is split at the wing’s leading edge, we start observing a velocity & pressure difference above and below the wing surface so that the air will reach the trailing edge of the wing at the same time. In general, the aerofoils upper surface is curved so that the air rushing over the top of the wing speeds up and stretches out, which decreases the air pressure above the wing. This is observed in the velocity contour images. In contrast, we observe air flowing below the wing moves in a straighter line, thus its speed and pressure remain about the same (no variation). Since high pressure always moves toward the low-pressure region, the air below the wing pushes the wing upwards, which will generate "Lift force' & an opposing force known as the "Drag Force". 

 

 

 

On simulating transient airflow over the airfoil in CONVERGE, the results appeared to have converged at 0.8 seconds. The above plots represent the pressure force along x-direction which is the "drag force" & pressure force along Y-direction which is the "lift force" for RNG k-ɛ Turbulence Mode. An aerofoil at a low angle of attack generally doesn't produce significant lift & drag. Since it's impossible to take an average of such large amounts of data, for our simplification we perform a time average only for the final 0.1 seconds of the simulation i.e. from 0.8 seconds to 0.9 seconds. This averaged data will provide us the Lift & Drag forces acting on the aerofoil which would help us to calculate the Lift Coefficient & Drag Coefficient for RNG k-ɛ Turbulence Mode. 

                `

     Lift coefficient for 5° Angle of attack using RNG k-ɛ Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
38.7855	1.2754	31.4	0.1	0.61687004470156

 

Drag coefficient for 5° Angle of attack using RNG k-ɛ Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
1.31126	1.2754	31.4	0.1	0.020855

 

 We follow similar steps to find the Lift Coefficient & Drag Coefficient for the aerofoil at 5° angle of attack using the k-ω SST Turbulence Model. 

 

 

 

Lift coefficient for 5° Angle of attack using k-ω SST Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
37.6569	1.2754	31.4	0.1	0.59892004966604

 

Drag coefficient for 5° Angle of attack using k-ω SST Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
1.45347	1.2754	31.4	0.1	0.023117

 

 

                                                                                              

                                                                                             

 

 

                                                                                                                                    Case 3:- Aerofoil with 10° Angle of Attack (AoA)                                                                                                 

 

 

 

For case3, i.e. when aerofoil at 10° angle of attack, with a base mesh grid size of 0.017m,  the total number of cells formed for the entire geometry is 53121 cells for the entire time period of the simulation for both the RNG k-ɛ Turbulence Model & k-ω SST Turbulence Model. The smaller the mesh grid size the larger number of cells would be generated. At the boundary of the geometry, we observe a much finer mesh than compared to the entire surface, this is because we used ‘fixed embedding’ to refine the particular areas in the domain.

 

    

    

    

 

The above images represent the pressure & velocity gradient around an aerofoil at a 10° angle of attack inside a virtual wind tunnel. Both pressure & velocity flow field variation across the aerofoil follows Bernoulli's principle. We observe at the leading edge, as the free stream air encounters the aerofoil, the velocity decreases drastically, this is the stagnation region where velocity reduces to almost zero. As the flow moves upstream on the top of airfoil, we can observe the glyph streamlines following the curvature of the airfoil. After the air is split at the wing’s leading edge, we start observing a velocity & pressure difference above and below the wing surface so that the air will reach the trailing edge of the wing at the same time. In general, the aerofoils upper surface is curved so that the air rushing over the top of the wing speeds up and stretches out, which decreases the air pressure above the wing. This is observed in the velocity contour images. In contrast, we observe air flowing below the wing moves in a straighter line, thus its speed and pressure remain about the same (no variation). Since high pressure always moves toward the low-pressure region, the air below the wing pushes the wing upwards, which will generate "Lift force' & an opposing force known as the "Drag Force". 

 

 

 

On simulating transient airflow over the airfoil in CONVERGE, the results appeared to have converged at 0.8 seconds. The above plots represent the pressure force along x-direction which is the "drag force" & pressure force along Y-direction which is the "lift force" for RNG k-ɛ Turbulence Mode. An aerofoil at a low angle of attack generally doesn't produce significant lift & drag. Since it's impossible to take an average of such large amounts of data, for our simplification we perform a time average only for the final 0.1 seconds of the simulation i.e. from 0.8 seconds to 0.9 seconds. This averaged data will provide us the Lift & Drag forces acting on the aerofoil which would help us to calculate the Lift Coefficient & Drag Coefficient for RNG k-ɛ Turbulence Mode. 

                `

     Lift coefficient for 10° Angle of attack using RNG k-ɛ Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
52.6222	1.2754	31.4	0.1	0.83693800173505

 

Drag coefficient for 10° Angle of attack using RNG k-ɛ Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
3.76673	1.2754	31.4	0.1	0.05991

 

 We follow similar steps to find the Lift Coefficient & Drag Coefficient for the aerofoil at 10° angle of attack using the k-ω SST Turbulence Model. 

 

 

 

Lift coefficient for 10° Angle of attack using k-ω SST Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
51.5436	1.2754	31.4	0.1	0.81978323951166

 

Drag coefficient for 10° Angle of attack using k-ω SST Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
3.81901	1.2754	31.4	0.1	0.06074

 

                                                                                     

                                                                                   

 

 

                                                                                                                                    Case 4:- Aerofoil with 15° Angle of Attack (AoA)                                                                                                 

 

 

 

For case4, i.e. when aerofoil at 15° angle of attack, with a base mesh grid size of 0.017m,  the total number of cells formed for the entire geometry is 53154 cells for the entire time period of the simulation for both the RNG k-ɛ Turbulence Model & k-ω SST Turbulence Model. The smaller the mesh grid size the larger number of cells would be generated. At the boundary of the geometry, we observe a much finer mesh than compared to the entire surface, this is because we used ‘fixed embedding’ to refine the particular areas in the domain.

 

    

    

    

 

The above images represent the pressure & velocity gradient around an aerofoil at a 15° angle of attack inside a virtual wind tunnel. Both pressure & velocity flow field variation across the aerofoil follows Bernoulli's principle. We observe at the leading edge, as the free stream air encounters the aerofoil, the velocity decreases drastically, this is the stagnation region where velocity reduces to almost zero. As the flow moves upstream on the top of airfoil, we can observe the glyph streamlines following the curvature of the airfoil. After the air is split at the wing’s leading edge, we start observing a velocity & pressure difference above and below the wing surface so that the air will reach the trailing edge of the wing at the same time. In general, the aerofoils upper surface is curved so that the air rushing over the top of the wing speeds up and stretches out, which decreases the air pressure above the wing. This is observed in the velocity contour images. In contrast, we observe air flowing below the wing moves in a straighter line, thus its speed and pressure remain about the same (no variation). Since high pressure always moves toward the low-pressure region, the air below the wing pushes the wing upwards, which will generate "Lift force' & an opposing force known as the "Drag Force". 

 

 

 

On simulating transient airflow over the airfoil in CONVERGE, the results appeared to have converged at 0.8 seconds. The above plots represent the pressure force along x-direction which is the "drag force" & pressure force along Y-direction which is the "lift force" for RNG k-ɛ Turbulence Mode. An aerofoil at a low angle of attack generally doesn't produce significant lift & drag. Since it's impossible to take an average of such large amounts of data, for our simplification we perform a time average only for the final 0.1 seconds of the simulation i.e. from 0.8 seconds to 0.9 seconds. This averaged data will provide us the Lift & Drag forces acting on the aerofoil which would help us to calculate the Lift Coefficient & Drag Coefficient for RNG k-ɛ Turbulence Mode. 

                `

     Lift coefficient for 15° Angle of attack using RNG k-ɛ Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
47.9742	1.2754	31.4	0.1	0.76301315951894

 

Drag coefficient for 10° Angle of attack using RNG k-ɛ Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
14.6165	1.2754	31.4	0.1	0.23247

 

 We follow similar steps to find the Lift Coefficient & Drag Coefficient for the aerofoil at 15° angle of attack using the k-ω SST Turbulence Model. 

 

  

 

Lift coefficient for 15° Angle of attack using k-ω SST Turbulence Model

Lift Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Lift Coefficient (`C_L`)
48.1966	1.2754	31.4	0.1	0.76655035506732

 

Drag coefficient for 15° Angle of attack using k-ω SST Turbulence Model

Drag Force (L)	Density (ρ) in Kg/`m^3`	Velocity (v) in (m/s)	Area in (m)	Drag Coefficient (`C_D`)
14.5912	1.2754	31.4	0.1	0.23207

 

                                                                                      

                                                                                     

 

                                                                                                                      D.   Final Observations, Conclusions & Comparisions     

 

 

.

 

 

Angle Of Attack (AoA) in °	Lift Coefficient (`C_L`) RNG k-ɛ Turbulence Model	Drag Coefficient (`C_D`) RNG k-ɛ Turbulence Model
1	0.2947	0.013133
5	0.6168	0.020855
10	0.8369	0.05991
15	0.7630	0.23247

 

Angle Of Attack (AoA) in °	Lift Coefficient (`C_L`) SST k-ω Turbulence Mode	Drag Coefficient (`C_D`) SST k-ω Turbulence Model
1	0.2878	0.014354
5	0.5989	0.023117
10	0.8197	0.06074
15	0.7665	0.23207

 

   

 

Conclusion:- We successfully ran eight transient state simulation cases for an aerofoil located inside a wind tunnel at 1°, 5°, 10° & 15° AoA using two turbulent models i.e. RNG k-ɛ & k-ω SST, the values obtained for Drag force and Lift force are getting converged well before the assigned end time. Hence, we can decrease the value for the end time for the transient based simulation. From the experiment, we were able to confirm Bernoulli's principle & it was observed that as the AoA increases the Lift force & Drag force acting on the aerofoil also increases simultaneously but after a certain AoA we observe the drag forces start to increase & the lift forces start decreasing, this phenomenon is known as stall.  The drag & lift Coefficients obtained from the pressure force along X-axis (drag) & pressure force along Y-axis (lift) are identical for both the turbulence models. The major difference between the two turbulence models is the Y+ Value.