Michael Pagano CFD Analysis of the Supercritical Airfoil RAE-2822 at Transonic Mach Speeds

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CFD Analysis of the Supercritical Airfoil RAE-2822 at Transonic Mach Speeds

Michael Pagano

Department of Mechanical Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada

The study of aerodynamic characteristics for airfoils is an ongoing field of study in mechanical engineering. Airfoils are distinguished by their lift and drag coefficients as well as their stabilities under varying Mach numbers. Airfoils are found to be sensitive to these operating parameters as well as their angle of incidence. Small perturbations of these parameters may induce a different set of physical conditions around the airfoil which in turn determine the stability of the airfoil as well as their ability to provide lift. We must study the flow conditions around airfoils with great detail under many differing set of circumstances in order to make predictable assertions about how each airfoil reacts for a particular set of surroundings. Given these results we may alter the geometry of any airfoil to optimize its stability under a certain set of parameters. Given its interest to the Ph.D project, we study the supercritical RAE-2822 airfoil which provides stability and efficiency at high speeds. We will perform a CFD study on the RAE 2822 airfoil with a chord of 1m at an angle of incidence of under varying flow conditions approaching the Mach regime with Reynolds numbers approaching . It is observed that as the free stream velocity around the airfoil approaches transonic Mach speeds an oblique shock is formed which adversely affects the stability and performance of the airfoil. We choose the shape of the profile using existing literature and perform computations for the Pressure, temperatures as well as velocity profile around the airfoil for varying speeds approaching Mach number. The resulting contours agree well with NASA studies done on the airfoil [1]

I. INTRODUCTION

The most common type of Airfoil is the NACA profile airfoil whose chord profile is recognized by its classic symmetrical shape, curved equally on the top and bottom. Typical NACA airfoils have a camber given by

Where is the maximum camber and is the location in which the airfoil reaches its maximum camber. We observe that is very nearly symmetric for top and bottom profiles with a slight distortion factor for which will provide the airfoil with less curvature underneath the airfoil. The NACA and supercritical airfoils utilize the Bernoulli principle to provide lift. The incoming flow of air separates at the leading edge of the airfoil which then travels quicker on top and slower at the bottom. This is known colloquially as the Magnus Effect which provides lift to an aircraft. The Magnus effect can be directly attributed to the Bernoulli principle

Figure 1: The Camber and Chord for the NACA 2412

where is the pressure of the fluid, the density and its velocity. For the particular case of fluid flow around an airfoil, the Bernoulli equation predicts the fast moving air flowing on the top edge of the airfoil will have an inverse relationship to the pressure at the same location. Therefore we find a low pressure zone where the velocity is moving fast and a high pressure zone at the bottom of the airfoil where the velocity of air is slower. This mechanism produces a pressure differential on the chord which then induces a lift on the aircraft. We shall see that in addition to providing lift this mechanism is also responsible for producing shock waves which will negatively affect its lift. For an airfoil travelling at near Mach speed, the fluid is separated at the leading edge of the airfoil and the section of air which follows the upper path of the airfoil is then accelerated on the top of the airfoil and so must be moving faster than the approaching free stream fluid. Since we are assuming the airfoil to be moving at transonic Mach speeds, the air moving on the top chord is now moving at speeds greater than Mach speed even though the aircraft as a whole has not yet reached Mach speed. Therefore the velocity of the fluid on the top edge of the airfoil can vastly exceed the velocity of the aircraft itself. As a result the supersonic air on the top of the airfoil creates a shockwave which then shears the fluid near the boundary. For the NACA airfoil the shock wave tends to be a normal shock wave which is always observed to be at angles perpendicular to the boundary. This normal shock induces boundary layer separation which produces turbulence. The production of turbulence negatively effects the function of the airfoil as it increases both drag and fuel consumption which in turn also adversely affect the speed and stability of the aircraft. The goal of avionics is to reduce drag and increase stability and fuel efficiency at transonic speeds. The NACA design does not provide reliable results in this velocity regime and so the Supercritical Airfoil has been introduced to mitigate these effects.

Figure 2: Formation of Shock Waves over a NACA airfoil

The Supercritical RAE-2822 Airfoil

Like any other supercritical airfoil the RAE-2822 has a flat top and a more pronounced curve on the bottom. This is in stark contrast to the NACAs more conventional design. The lack of curvature on the top boundary enables the velocity of the fluid to move slowly so as to prevent shock formation from high velocity fluids as the aircraft approaches transonic speeds. As a result the production of a shockwave is delayed and the aircraft gains stability at higher speeds compared to its NACA counterpart. This lowers drag at higher speeds which increases fuel efficiency of the airfoil. This allows airliners and jets to travel to their destinations with less fuel while moving at higher speeds and so quicker. Tests have shown that the RAE 2822 can increase efficiency by up to 15% [2] .The flattened design which is credited with increased efficiency is adversely effected by the lack of lift produced by the airfoil which requires curvature. As a result the geometry of the RAE-2822 is modified with a pronounced curved at the upper side of the trailing edge in order to regain the lift which has been sacrificed by the airfoils flattened design. The increased efficiency and decreased drag have made supercritical airfoils the standard airfoil among modern commercial airliners [2]. The RAE-2822 delays the formation of shock waves however as the aircraft approaches the critical velocity, the air on the top edge of the airfoil becomes supersonic and produces a signature shock wave which is unique to the RAE 2822 and has been well documented by numerous studies. Figure 3 and figure 4 display the shock formation for the RAE-2822 as measured by NASAs JPL [1].

Figure 3: The static pressure contour map for the RAE-2822 [1]

Figure 4: The velocity contours of the RAE-2822 [1]

II. THE AIRFOIL

The RAE-2822 is a well-documented design and so we have the opportunity to build on the standard geometry that are available in public airfoil databases. One such database is provided by the Department of Aerospace Engineering from the University of Illinois Urbana Champaign which provides the coordinate points of over 1000 different types of Airfoils [3]. Here we import the coordinate points of the standard RAE-2822 and tailor them to fit our present CFD analysis. The standard RAE 2822 transonic airfoil has a max thickness of 12.1% at 37.9% of the chord and a max camber at 1.3% at 75.7% of the chord. We choose a chord of 1m and an angle of attack of . Physically, a larger angle of attack increases the fluid velocity over the top of the chord and therefore creates a larger pressure gradient between top and bottom boundaries which increases the lift. An angle of attack of therefore provides a larger lift coefficient than a horizontally aligned airfoil. At we find many reference values to compare our CFD analysis with including a study by the Aerospace Computational Design Laboratory at MIT [4]. This angle also serves as a realistic angle of approach for an airliner on a runway. These specifications are characterized by 129 coordinate data points which are to be used to model the geometry.

Figure 5:The RAE-2822 Coordinate Points

III. GEOMETRY

Before performing our CFD analysis we must create a suitable geometry for the set of 129 coordinate points characterizes the unique shape of the airfoil. We choose to perform our CFD analysis on the RAE-2822 using the ANSYS 17.2 Workbench platform to assemble the geometry. The geometry is initiated using ANSYS DesignModeler. Our primary focus in creating a geometry is to accurately create a sketch of the airfoil and then to create a structured grid in order to lay a mesh over to perform our future CFD analysis. Naturally we would like the mesh to conform to the geometry that we create and therefore our focus in creating the geometry will be split into two parts

i. Creating an environment for the airfoil on a distance scale much larger than the airfoil

ii. Creating the geometry of the RAE-2822 and the region for the mesh around it

The geometry immediately surrounding the airfoil will be more intricate as we intend on laying a very fine mesh around the region of the airfoil. In this region we will take care of the dimensions and the placement of all geometric objects as the future meshing procedure will be very sensitive to the placement and dimensions of all objects near the airfoil. At distance scales much larger than the airfoil we simply need to create a suitable geometric environment to allow fluent to properly compute the solution. At transonic speeds we expect shock formation for the RAE-2822 on the order of where is the length of the chord and so our geometry must include a suitable environment for fluent to resolve the shock.

Region I: Creating an environment on a distance scale much larger than the airfoil

We first import the data points of the modified RAE-2822 from UIUC Airfoil Database into ANSYS DesignModeler (DM) which provide the framework coordinate points of the RAE-2822. In order to properly create the geometry of the airfoil we require DM to join all points together to create a smooth two dimensional curve. The imported coordinate points of the airfoil are not recognized by DM as a smooth continuous curve and so we use the function 3D curve fitting in order to properly join the data points. The ensuing geometry is a trace of the chord of the RAE-2822. Even after this process DesignModeler treats the trace of the top of the airfoil and the trace of the bottom of the airfoil as separate curves and so we must take the necessary steps to join them together creating a smooth trace of the airfoil. We do this using the Draw feature of the software in which we sketch two lines in order to bridge the geometry together. Once this is done we create the surface of the airfoil using the sketch.

Figure 6: The far field geometry

In order to ease computational time of our CFD simulation we would like the environment to be as symmetric around the airfoil as possible. Our assertion are that antisymmetries within the environment may lead to further difficulty in convergence during the computational process. In order to avoid such problems we insist on creating a finely tuned environment that exhibits as many symmetries around the airfoil as possible. The natural shape to choose is a circular geometry around the airfoil . We therefore create a circle of radius around the chord. We make sure that the environment is an order of magnitude larger than the airfoil in order to ensure that the environment does not have any appreciable effect on the experiment. We would like our geometric environment to stretch out far enough from the airfoil so as we may consider it a distance away from the boundary of the airfoil in which case we may impose the free stream boundary conditions at this location. Given the airfoil has a chord of , a distance of around the airfoil will suffice for a suitable free stream location. Experiments have shown [5] that the shock waves produced by a supercritical airfoil do not extent past a distance for a supercritical airfoil of chord which lend credence to our choice of distance scales. We place the circle at a distance of the chord and a vertical distance of which represents of its maximum camber.

Given that we expect airflow behind the geometry of our airfoil to be affected by the motion of the airfoil itself we also need to make we create a suitable environment behind the airfoil in order to properly resolve the patterns of airflow in this region. The circle of radius is insufficient to resolve the airflow behind the airfoil as we expect this regime to exhibit vertical symmetry. We would like our environment to match the symmetries in our experiment so that the computational time of the experiment may be reduced. Since the airflow in this region is expected to be vertically symmetric we will design a rectangular environment. In order to ensure the rectangular environment to be large enough so not to interfere with the computation, we choose the length of the rectangle to be . Next we merge both environments together to create a cohesive and symmetric environment to run our CFD experiment.

Region II: The Geometry of the RAE-2822 and its surroundings

We would like to create a separate geometry in the immediate vicinity of the airfoil so that we can define a refined mesh to resolve the most sensitive region of the airfoil. Using our same assertions as in Region I, we would like the surrounding geometry to be as symmetric as possible in order to ease computational time. Since a circle provides this symmetry we create a circle of radius centered at of the chord. Placing the circle in this location will enable our mesh to be near the shock region which we expect to have a width of of the chord. Positioning the circle at the same point as also enables the entire geometry to represent a scale invariant symmetry so that we have symmetry of our geometry on both large and small scales. We note that there is no need to have the radius of the be an order of magnitude larger than the chord since we now require this circle to be the geometry in which we lay our mesh upon to fully resolve all the details of the flow surrounding the airfoil. As is the case with the environment itself, the flow region at the trailing edge of the airfoil represents a another sensitive region in which the flow is expected to be turbulent. As a result we will inset another rectangular shaped geometry in order to pick up the details of the turbulent region behind the trailing edge of the airfoil. To finalize our geometry we join all surfaces together using the Boolean feature in DesignModeler which allows us to create a cohesive planar geometry while subtracting the portions of the geometry we would like to be removed from our environment.

Figure 7: The geometries of regions I and II

In order to provide future boundary conditions of our experiment to ANSYS Fluent we name the left portion of the geometry to be the INLET which will represent the flow of air to our airfoil. Similarly we define the right portion of the environment to be the OUTLET which will be defined as the region in which air passes after it leaves the trailing edge of the airfoil. Finally the top and bottom portions of the environment are defined as FREE STREAM to allow ANSYS Fluent to define a proper free stream boundary condition. This region therefore represents the far field boundary of our airfoil where we expect the fluid to be stationary and unaffected by the airfoil. In addition to these boundaries we would like to treat the top and bottom of the airfoil as separate boundaries. We expect the flow over the top of the chord to be supersonic and the airflow at the bottom of the chord to be transonic. As a result we would like to track each section as a distinct boundary with the context of our CFD simulation. We employ the Split Edges tool in DesignModeler and split the top and bottom edges. We now define the top edge to be TopChord and the bottom edge BottomChord which will allow the airfoil to be subject to two distinct boundary conditions TopChord and BottomChord.

IV. MESHING

We import the geometry into ANSYS Mesh provided by the Workbench feature of fluent. Our primary objective is to create a mesh that will be suitable to compute the flow features around the airfoil within our CFD analysis. We take care in creating a mesh that is refined in regions which we expect to be very sensitive to our initial conditions. Due to comparisons to previous experimental data from NASA [1] we expect the regions close to the top boundary of the chord to be regions which will carry a significant importance to the flow patterns around the airfoil. It is also expected to find turbulent features due to the significant shear stress produced by the airfoil in regions close to the boundaries. As a result we will create a mesh which is fine in regions close to the airfoil and coarse as we move away from the airfoil to the far field.

Region I: Far Field

We will not create a specific mesh for the far field regions of the airfoil but simply impose constraints on sizing conditions. We will also require the mesh to form a smooth transition to the refined mesh of the region near the airfoil. Therefore we will approach this regime by setting global constraints on the mesh and allowing ANSYS to create a smooth transition between regimes. We set the maximum face size of grid to be which sets the maximum length size of our quadrahedron mesh. It would be possible to reduce the maximum face size between however the increase in computational load is unwarranted as there are few features that require this amount of grid sensitivity at this distance scale away from the airfoil. In order to ensure that the mesh undergoes a smooth transition between the far field (Region 1) and regions close to the RAE-282 (Region II) we must impose a boundary condition at their interface. We impose the boundary condition that the flow conditions on the inner part of the airfoil (Region II) match the flow conditions in the first mesh points of the far field. Mathematically this corresponds to matching the flow conditions at the location of the circle center on the airfoil which represents the separations between both regions.

Region II: Region around the RAE-2822

In this region we lay a fine mesh with inflation around the boundaries of the airfoil. Knowing that the CFD simulation will have sensitivity to this area of the grid we take care in our selection of parameters. Our first step is to split the chord of the airfoil into 525 unique cells and provide sizing to form the foundation of the inflation in which we will lay onto it. The splitting of the chord into unique elements of length allows the mesh to lay onto a foundation of grid points along the geometry. Cutting up the boundary into smaller units of length suggests a finer mesh will be attached to the airfoil. Since we intend on airflow to be moving at transonic speeds which can induce shockwaves we splice the airfoils boundary into 525 elements an unusually large number for a chord of length. Our intentions are to produce a fine mesh in addition to inflation in order to resolve any possible turbulence leading up the shock wave along the chord. We also note that when the 129 coordinate points which represent the geometry of the RAE-2822 were imported into DesignModeler, ANSYS did not create a unified curve as it recognized only sections of the coordinate points as being part of the same geometry. Now that we are splitting the RAE-2822 into 525 unique grid points, the ANSYS mesher will recognize and produce the same defect, it will not regard all 525 elements as being part of the same overall airfoil. Recall that previously we unified the 129 coordinate points of the RAE-2822 by instituting our own lines and curves in order to provide closure to the geometry, we must again follow the same process within the confines of the ANSYS mesh. We split the RAE-2822 into 525 unique elements of the length and then join all fragments together. To ensure that the resulting curve is unified, we employ the sketch to surface feature in the ANSYS mesher. This tool will join and series of lines into a single surface resulting in a unified geometry ready for experiment. Without a unified geometry any CFD simulation as the software will be unable to recognize the airfoil as a single boundary.With a unified geometry we are ready to choose the sizing and inflation parameters of the mesh for Region II. We set the global maximum face size of Region II to be which will allow no grid to have length larger than that maximum value . This is a full order of magnitude less than the maximum face size of Region I which is to be expected since we require the region close to the airfoil to be much more sensitive. Inflation allows us to refine the mesh further in regions which are close to wall boundaries which will increase the accuracy of the simulation. Since the CFD will use the conditions of the airfoils boundary in order to compute the conditions at larger distance scales, the accuracy of the entire simulation will be very sensitive to the computation at the boundary. If the mesh isnt fine enough close to the airfoil, then the CFD simulation may not pick up the detailed features at the wall and so will be unable to build on those conditions as they relate to larger distance scales. Inflation is therefore a necessity in this simulation to provide extra layers of grid points in regions which may be very sensitive and provide better accuracy to all scales in the CFD simulation. Any inflation added to this region will now need to conform to the constraint that its maximum length be . We define 15 layers of inflation beginning at the chord with a growth rate of . The last layer of inflation will therefore need to match the constraining length of , however the first layers of inflation around the airfoil may be significantly smaller than this length scale. Finally we utilise the smooth transition tool in the ANSYS mesher which will force the boundary between the inflation layers and Region II and then the boundary layers between Region II and farfield Region I to have a smooth transition at their respective boundaries. The mesh generated with these parameters create nodes for our CFD simulation.

Figure 8: The region around the RAE 2822

Figure 9: The airfoil

Figure 10: The generated Mesh (407 000 nodes)

Figure 11: Inflation around the airfoil

V. FLUENT SIMULATION

With a suitable geometry and mesh we are ready to import our results into fluent. The default settings are not ideal for a transonic airfoil so we must modify them to allow a proper solution. By default ANSYS fluent does not provide an energy variable for the simulation so we go through the tools setting and make sure it is enabled. The presence of an energy variable will add an extra layer of computational accuracy as at every iteration fluent will compute the energy of the flow at particular location and compare it to the other model variables in order to verify whether the solution is converging or not. This does add additional computational stress to our simulation, however, after previous testing we are assured that the simulation can still run, with a greater accuracy even with an energy variable included in the iterative process.

We would like to define the variables of the environment and the specific boundary conditions applied to them. We choose the environment to be filled with air and we model this fluid using the ideal gas law with specific heat capacity .

Since we expect shock waves to be created on the RAE-2822 we cannot assume the air to be incompressible. The shock wave will produce compressible air which needs to be modeled via an equation of state. As is common practice in gas dynamics, we treat the compressible air with an equation of state equal to the ideal gas law. In theoretical practice treating air as an ideal gas is a necessary approximation as there are often too many unknowns and a closure condition is required to reduce the set of unknowns. The ideal gas law provides that additional closure condition [6]. After various trial runs, our CFD simulation will not provide reliable results unless we model air as an ideal gas in order to provide additional closure to the compressible flow. Modeling air as an ideal gas will add an element of realism to the simulation as the ideal gas law will allow the airs fluctuation of temperature and pressure to have an effect of the airfoil. In the absence of the ideal gas law, we would have to model air as a generic fluid with fixed viscosity and fixed molecular properties which would create a more rigid simulation in which air is unable to be an active ingredient within the physics solutions. With air modeled as an ideal gas, the air is able to be an active participant in the simulation as it is able to expand, contract and provide temperature and pressure fluctuations back to the simulation through the ideal gas law.

In addition to the ideal gas law we also set boundary conditions on the far field flow. In the regime far from the airfoil we do not expect the flow of air to be affected by the motion of the airfoil. As a result we set the initial velocity for length scales far away from the airfoil to be zero, that is . It must be noted that this regime does not represent our free stream velocity. As we have mentioned there are two regions in the geometry: Region I (far field) and Region II (vicinity of the RAE -2822). The free stream velocity is the transonic velocity of the air within the vicinity of Region II whereas the air in Region I is too far away from the airfoil to assume the same boundary condition. We therefore set the velocity of air in the largest distance scales to be 0. There is also the possibility that we may regard the all air in all regimes to be moving at transonic speeds in order to provide uniformity to the environment. We can conclusively state that for velocities this condition will be indistinguishable to setting the far field velocity to be zero. However for simulations with increased velocity within the range the solution will not converge unless we set the far field velocity to be near zero. We surmise that at velocities nearing Mach speed, the shock waves grow and multiply and begin to produce an appreciable effect on the far field conditions which assumes a more important role in the simulation and can no longer be forced to be moving at a set speed.

In order to simulate the airfoil to be moving at transonic speeds we impose a boundary condition on the airflow at the inlet. The physics are invariant to absolute velocity and only depend on the relative velocity between the airfoil and the airflow. We therefore keep the airfoil at a fixed location and have the inlet velocity to be moving at transonic speed toward the airfoil. We therefore set the velocity of the fluid at the location of the inlet to have velocities and which will correspond to our three runs at and speeds respectively. As we shall see there are features unique to each of these speeds as we approach the Mach regime. The velocities of the airflow are only directly along the direction and so there is no component to any initial velocity. In addition to velocity we also set the ambient temperature to be room temperature at . Physically this represents an aircraft which is in the process of taking off and landing while remaining close to the ground level.

To avoid to avoid significant amount of false diffusion we enable Fluents second order upwind option which will limit the amount of nonphysical diffusion which occurs due to numerical roundoff in the streamwise direction [5]. Lastly we enable tracking of the lift coefficient and drag coefficient in the numerical solutions since they quantify the airfoils ability to provide lift and experience frictional drag forces. These quantities will also serve as a measuring stick to compare the lift ability of the RAE-2822 to other airfoils and at different speeds.

We will first run our simulation using the Spalart Allmaras model for turbulence. The Spalart Allmaras model is a simple one equation model for turbulence which provides a transport equation for turbulent eddy viscosity. We choose to run our simulation using the Spalart Allmaras model because it was designed specifically for aeronautical simulations which involve wall bounded flows such as our airfoil. It produces good results for boundary layer flows which are influenced by adverse pressure gradients [7]. Since our airfoil represents a boundary layer flow which will be under significant stress due to the shock wave pressure gradient, we expect the Spalart Allmaras model to conform well in our simulation. Despite its development for aerodynamic simulations, the Spalart Allmaras model was designed to be a low-Reynolds number model which has sensitivities to boundary layer regions which are subject to high Reynolds numbers [7]. Since our simulation will be running for Reynolds numbers up to we need to provide a mesh that can resolve fine details around the boundaries in order to avoid the sensitivities to high Reynolds numbers and properly resolve the boundary. We developed the mesh to be extremely fine with additional layers of inflation since we understood the limitations of the model. We do not expect our simulation to run into difficulties around the boundary. It is important to mention that ANSYS Fluent has modified the Spalart Allmaras model in order to allow the model to be more robust and provide better stability near boundaries for high Reynolds-number flows. This is referred to as the insensitive wall treatment which allows the model to apply to any length scale . In order to provide this additional stability ANSYS has merged the Spalart Allmaras model with a viscous sublayer formulation for and with the log law region for . As the distance away from the wall decreases, the insensitive wall treatment automatically switches from the Standard Spalart Allmaras model for large to the log law region for intermediate and to the linear formulation of the viscous sublayer for small thereby removing the sensitivities to regions close to the wall. Given these modifications it is still recommended to produce a fine mesh around boundary layer flows of airfoils when using the modified Spalart Allmaras model. The transport equation for turbulence kinetic energy is often rewritten in terms of the turbulent kinematic viscosity . The transport equation is given by

Where is the production term given by

in which

and

is a viscous damping term, and are closure constants with the distance from the boundary. is the usual vorticity tensor which is represents the mean rate of deformation

where is related to the curl of the airflow through . Within the transport equation

represents the destruction term which is modeled as

where

and and are part of the closure constants which are given by , , , , and which represent the closure constants for the Spalart Allmaras model in ANSYS Fluent 17.2. As mentioned, the Fluent version of the Spalart Allmaras model introduces the law of the wall in order to lessen the sensitivities that the Spalart Allmaras model has near boundary layers at high Reynolds numbers. The model will activate the law of the wall at distances from the boundary by matching the law of the wall

with the laminar velocity found in the viscous sublayer which is valid for :

where and is the Von Karman constant. These functions provide stability at high Reynolds numbers and so utilizing these two functions for regions close the boundary the modified Spalart Allmaras model will exhibit further stability for boundary layer flows at high Reynolds numbers.

Our second option model to run the simulation is the Shear-Stress Transport (SST) model. This model is a modified model from Wilcox which takes into account the transport of turbulent shear stress whereas the original model does not [8]. This modification has made it significantly more accurate and dependable for airfoils as well as transonic shock waves [7]. The inclusion of the transport of turbulent shear stress in the SST model also allows it to properly predict the onset and quantity of flow separation from smooths surfaces whereas the original model from Wilcox fails in this regime. The original turbulent model is a two equation system which models the transport of turbulent kinetic energy and the specific dissipation rate . The turbulent model is often used in aerodynamics because it incorporates compressibility and shear flow spreading [7]. Since the airflow around the RAE-2822 will be compressible at regions in which we find shock formation, the turbulent model makes it a natural candidate for our simulation. However it has been shown that the model exhibits instabilities for regions outside the shear layer [9]. Therefore solutions for the freestream flow in the far field of the airfoil will have sensitivities to the values of and which do not make it accurate for free shear flows or for regions far away from the boundary[9, 7]. In stark contrast to the model which is reliable for regions close to the boundary of the airfoil, the model exhibits far field. In order to provide stability in each regime, the Shear-Stress Transport (SST) model effectively includes both model for far field flows and the model for regions close to the boundary. The model achieves this by adding both and models together after multiplying each with a blending function which is given by

and

This blending function is one when the computation is near the boundary which activated the model and zero when the computation is in the far field (freestream) which activates the model. This produces a robust model which make it reliable for aerodynamic simulations as well as for airfoils moving at transonic and supersonic speeds. The two equation transport equations are

where represents the production of

which can be computed using the Boussinesq approximation with use of the mean strain rate tensor in the form of . The production of the specific dissipation rate is computed by

Where the constant is given by

The dissipation of turbulent kinetic energy and mean dissipation rate are computed by

and

with

,

for the high Reynolds form of the model [7]. The dissipation for is given by

in addition to employing the standard model for regions close to the airfoil and the model for far field regions the Shear-Stress Transport (SST) model also accounts for the transport of turbulent shear stress. The SST model models the eddy viscosity as

where is the strain rate magnitude and and . The model constants are given as

is and , and , ,

VI. RESULTS

We run the simulation with the Spalart Allmaras model and after 4000 iterations the energy, continuity and velocity equations converge with a residual of . Fluent provides a contour map of the temperature, pressure and velocity which are obtained by creating new contour plots for each parameter which are shown in Figure 12 . The velocity contour map for the transonic velocities exhibits the classic supercritical airfoil shockwave at of the chord. We can confirm these results by obtaining the pressure plot along the chord. We see from the plot that at the is a pressure discontinuity corresponding to the propagation of the shock wave

Figure 12: The pressure contour at 0.72Mach using the Spalart Allmaras model

The results confirm the NASA and MIT experiments [4] [1] for an airfoil moving at transonic speed (Mach

Figure 13: The velocity contour at 0.72Mach using the Spalart Allmaras model

Figure 14:The temperature at 0.72Mach using the Spalart Allmaras model

Figure 15: Pressure Contour at 0.8 Mach using SST

Figure 16:Velocity Contour at 0.8Mach using SST

The results confirm the NASA and MIT experiments [4] [1] for an airfoil moving at transonic speed (Mach conforming identically with their results as they produce the rounded shockwave over the upper and lower cambers that is unique to the supercritical airfoil. The pressure and velocity contours confirm the Bernoulli principle that the velocity of the flow speeds up to supersonic speeds over the top of the airfoil and this coincides with a low pressure shockwave at the same location. On the bottom side of the airfoil the flow remains transonic and so no pressure shock is generated until velocities close to Mach speed. If we compare the pressure values on both sides of the chord we notice that there is overall lower pressure on the top of the airfoil and a high pressure zone on the bottom of the airfoil. This is in agreement with the Bernoulli principle and is the mechanism by which lift is produced on the airfoil. At transonic velocities ( Mach) the lift coefficient generated by the airfoil is . We run the simulation again with 4000 iterations however we increase the speed to 0.8Mach using the SST model. As we increase the Mach number from Mach to Mach, the rounded shock wave transitions into a normal shock and is accompanied by a secondary shock found on the bottom of the airfoil. This is a reasonable since at airfoil velocities which are very close to Mach speed, the air is already transonic and only a small increase in flow velocity will turn it supersonic and generate a shock. As the air flows through the bottom of the airfoil the velocity increases marginally which is enough to turn it supersonic and create a small normal shock. At 0.8Mach the air very nearly becomes supersonic and so generates a small rounded shock while on the top of the airfoil the airflow is now nearing hypersonic speeds and so creates a larger shock than observed at 0.72Mach. We also observe a linear drop in lift coefficient as the lift coefficient decreases to for 0.8Mach speed. The relation of lift coefficient is found to be inversely proportional to speed which is reasonable since shock formation is detrimental to the performance of an airfoil since it increases the drag coefficient . This result is not unexpected and has also been verified by experiment [10]. We also observe the formation of a stagnation point on the leading edge of the airfoil. This provides us with another layer of verification that our simulation is describing the correct physics. Stagnation points are points in the flow field in which the fluid velocity vanishes. Stagnation points are always observed in fluid flow around an airfoil and occur in regions where the static pressure of the fluid is zero. We can confirm the existence of a stagnation point in all runs including 0.72Mach, 0.8Mach and 0.9Mach. We compare the contour plots of the pressure and velocity and observe that the stagnation point is observed just under the leading edge of the airfoil visible with a dark blue spot indication that the velocities are close to . At the same location the pressure plot indicates that the pressure is bright red corresponding to the highest scale pressure on the plot of .

Figure 17: Formation of an Oblique shock at 0.9Mach

Figure 18: Pressure contour at 0.8Mach using the SST model

Figure 19: Stagnation point at 0.8Mach using the SST model

References

[1] J. Slater, "RAE2822 Transonic Airfoil," July 2008. [Online]. Available: https://www.grc.nasa.gov/www/wind/valid/raetaf/raetaf.html.[2] NASA, "NASA Technology Facts," 2000. [Online]. Available: https://www.nasa.gov/pdf/89232main_TF-2004-13-DFRC.pdf.[3] UIUC Applied Aerodynamics Group, "UIUC Airfoil Coordinates Database," Department of Aerospace Engineering, 2017. [Online]. Available: https://m-selig.ae.illinois.edu/ads/coord_database.html. [Accessed 2017].[4] D. L. D. Masayuki Yano, "Turbulent, Transonic Flow over an RAE 2822 Airfoil," American Institute of Aeronautics and Astronautics. [5] M. P. H. Ferziger, "2012," in Computational Methods for Fluid Dynamics, Springer-Verlag, 2002, p. 83.[6] J. D. Anderson, Modern Compressible Flow (3rd ed.), McGraw-Hill, 2003. [7] ANSYS Fluent, "ANSYS Fluent Theory Guide," 04 2017. [Online]. Available: https://uiuc-cse.github.io/me498cm-fa15/lessons/fluent/refs/ANSYS%20Fluent%20Theory%20Guide.pdf. [Accessed 2017].[8] V. Yakhot. a. S. A. Orszag, "Renormalization Group Analysis of Turbulence I Basic Theory," Journal of Scientific Computing, vol. 1, no. 1, pp. 1-51, 1986. [9] D. L. M. a. R. O. Fox, "Solution of Population Balance Equations Using the Direct Quadrature Method of Moments," Aerosol Science and Technology, vol. 36, pp. 43-73, 2005. [10] C. K. T. K. K.Harish Kumar, "CFD ANALYSIS OF RAE 2822 SUPERCRITICAL AIRFOIL AT TRANSONIC MACH SPEED," International Journal of Research in Engineering and Technology, Vols. 2319-1163, no. 2321-7308, 2015.

Lift Coefficient vs Mach Number

0.720.80.90.531959999999999990.493700000000000030.42809999999999998

Mach number

Lift Coefficient