Assessment of active flow control techniques to prevent airfoil stall

BACHELOR THESIS TITLE: Assessment of Active Flow Control techniques to prevent airfoil stall DEGREE: Bachelor's Degree in Aerospace Systems Engineering AUTHOR: Martí Armengol Roig DIRECTOR: Fernando Pablo Mellibovsky Elstein SUBMISSION DATE: 07/02/2025 TITLE: Assessment of Active Flow Control techniques to prevent airfoil stall DEGREE: Bachelor's Degree in Aerospace Systems Engineering AUTHOR: Martí Armengol Roig DIRECTOR: Fernando Pablo Mellibovsky Elstein SUBMISSION DATE: 07/02/2025 Abstract Active Flow Control (AFC) techniques are currently being in the spotlight of researchers because of its capabilities to improve significantly the aerodynamic performance of airfoils. This bachelor thesis focuses on the use of Synthetic Jet Actuators (SJA) to improve the performance of the NACA 0012 profile at post-stall angles of attack and high Reynolds numbers. This purpose was carried on by performing Computational Fluid Dynamics (CFD) simulations using pisoFoam, a transient Reynolds-Averaged Navier-Stokes (RANS) solver, and using the Spalart-Allmaras (S-A) turbulence model. The mesh consisted in a hybrid (containing unstructured and structured grids) C-type mesh, widely used for airfoil simulations. The studied airfoil was the NACA 0012 profile in the post-stall angle of attack alpha=19º at a Reynolds number Re=2e6. After performing a mesh dependence study, comparing the results with experimental wind tunnel literature, the SJA actuated simulations were performed. The results concluded that the use of SJA could significantly improve the aerodynamics of the non-actuated case, finding that the non-dimensional frequency F+ and the momentum coefficient C_mu of the jet had a major impact on the behavior of the airflow, specially influenced by the latter. Als meus pares, que m’han donat l’educació i els valors que m’han permès arribar a ser la persona que sóc a dia d’avui. Table of contents 1 INTRODUCTION.................................................................................................................. 8 1.1 Flow Control Techniques ........................................................................................... 8 1.1.1 Passive Flow Control ............................................................................................. 8 1.1.2 Active flow control ................................................................................................. 8 1.2 Objective and Methodology of the study ................................................................... 9 1.2.1 Objective ............................................................................................................... 9 1.2.2 Methodology ......................................................................................................... 9 2 FLUID DYNAMICS BACKGROUND .................................................................................. 12 2.1 Governing equations of Fluid Dynamics ................................................................. 12 2.1.1 Reynolds and Mach numbers .............................................................................. 13 2.2 Turbulence and its modelling .................................................................................. 14 2.2.1 Overview on turbulence ....................................................................................... 14 2.2.2 Types of simulations depending on how turbulence is treated .............................. 15 2.2.3 Turbulence modelling .......................................................................................... 16 2.3 Boundary layer theory .............................................................................................. 17 2.3.1 Laminar and turbulent boundary layers ................................................................ 18 2.3.2 Boundary layer separation and SJAs ................................................................... 19 2.4 CFD important parameters ....................................................................................... 20 2.4.1 Wall 𝒚+ and Law of the wall................................................................................ 20 2.4.2 Courant number .................................................................................................. 21 2.5 Mesh design concepts ............................................................................................. 21 2.5.1 Structured and unstructured grids ........................................................................ 22 2.5.2 Mesh quality metrics ............................................................................................ 22 3 NUMERICAL SETUP ......................................................................................................... 24 3.1 Definition of the baseline and actuated scenarios .................................................. 24 3.1.1 Studied scenarios ................................................................................................ 25 3.2 Solver and boundary conditions .............................................................................. 28 3.2.1 Flow solver .......................................................................................................... 28 3.2.2 Boundary conditions ............................................................................................ 28 3.3 Mesh dependence study .......................................................................................... 30 3.3.1 Time-step configuration ....................................................................................... 30 3.3.2 Meshes without the jet ......................................................................................... 30 3.3.3 Meshes with the jet .............................................................................................. 34 3.3.4 Baseline scenario for the implementation of the jet .............................................. 35 4 RESULTS .......................................................................................................................... 37 4.1 Overview of all the scenarios ................................................................................... 37 4.2 Analysis of each case ............................................................................................... 37 4.2.1 Case 1. 𝑭+=𝟓𝟎,𝑪𝝁=𝟏𝟎−𝟒 ............................................................................ 38 4.2.2 Case 2. 𝑭+=𝟓𝟎,𝑪𝝁=𝟏𝟎−𝟑 ............................................................................ 38 4.2.3 Case 3. 𝑭+=𝟓𝟎,𝑪𝝁=𝟏𝟎−𝟐 ............................................................................ 39 4.2.4 Case 4. 𝑭+=𝟏𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟒 .......................................................................... 39 4.2.5 Case 5. 𝑭+=𝟏𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟑 .......................................................................... 40 4.2.6 Case 6. 𝑭+=𝟏𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟐 .......................................................................... 40 4.2.7 Case 7. 𝑭+=𝟓𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟒 .......................................................................... 41 4.2.8 Case 8. 𝑭+=𝟓𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟑 .......................................................................... 41 4.2.9 Case 9. 𝑭+=𝟓𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟐 .......................................................................... 42 4.2.10 Case 10. 𝑭+=𝟏𝟎𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟒 ...................................................................... 42 4.2.11 Case 11. 𝑭+=𝟏𝟎𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟑 ...................................................................... 43 4.2.12 Case 12. 𝑭+=𝟏𝟎𝟎𝟎,𝑪𝝁=𝟏𝟎−𝟐 ...................................................................... 43 4.3 Results discussion ................................................................................................... 44 4.3.1 Most lift coefficient improvement .......................................................................... 44 4.3.2 Most drag coefficient improvement ...................................................................... 45 4.3.3 Most aerodynamic efficiency improvement........................................................... 45 4.3.4 Final comments ................................................................................................... 46 5 SUSTAINABILITY ANALYSIS ........................................................................................... 47 5.1 Environmental impact .............................................................................................. 47 5.2 Economic impact ...................................................................................................... 47 5.3 Social impact ............................................................................................................ 48 6 CONCLUSIONS AND FURTHER INVESTIGATION .......................................................... 49 6.1 Conclusions .............................................................................................................. 49 6.2 Further investigations .............................................................................................. 49 BIBLIOGRAPHY .................................................................................................................... 51 Index of figures Figure 1.1. Winglet of an Airbus a 320. [1] ................................................................................. 8 Figure 2.1. Representation of the chord of an airfoil. [10] ......................................................... 13 Figure 2.2. Wake created by the wing tip vortices of a commercial airplane. [11] ..................... 14 Figure 2.3. Typical turbulent energy spectrum. [12] ................................................................. 15 Figure 2.4. Boundary layer graphical representation. [19] ........................................................ 18 Figure 2.5. Laminar (a) and turbulent (b) boundary layers. [20] ................................................ 18 Figure 2.6. Boundary layer separation. [19] ............................................................................. 19 Figure 2.7. Pressure of a NACA 0012 airfoil at 𝐴𝑜𝐴=10.15°. .................................................. 19 Figure 2.8. Schematic of a synthetic jet actuator. [21] .............................................................. 20 Figure 2.9. Law of the wall. Dimensionless velocity profile in the proximities of a solid (red line). [22] ......................................................................................................................................... 21 Figure 2.10. Example of a structured grid (left side) and an unstructured grid (right side). [23] . 22 Figure 2.11. Mesh non-orthogonality. [24]................................................................................ 22 Figure 2.12. Visual representation of the skewness vectors. [24] ............................................. 23 Figure 2.13. Visual representation of the smoothness concept. [25] ......................................... 23 Figure 3.1. Numerical domain (a). ........................................................................................... 24 Figure 3.2. Numerical domain (b). ........................................................................................... 25 Figure 3.3. Visual representation of the 𝐶𝑙 and 𝐶𝑑 data gathered by Ladson. ........................... 26 Figure 3.4. Schematic of an SJA. ............................................................................................ 27 Figure 3.5. Mesh near the airfoil. ............................................................................................. 30 Figure 3.6. Boundary layer at the leading edge. ....................................................................... 31 Figure 3.7. Study of different values of 𝜈. ................................................................................. 32 Figure 3.8. Comparison of the non-actuated scenarios without jet implementation. .................. 33 Figure 3.9. Image of the leading edge of the final jet mesh. Jet situated at the high density cell region. .................................................................................................................................... 34 Figure 3.10. Comparison of the non-actuated scenarios with jet implementation. ..................... 35 Figure 3.11. Time evolution of the baseline scenario. .............................................................. 35 Figure 3.12. Fourier transform of the 𝐶𝑙 at 𝛼=19°. ................................................................. 36 Figure 3.13. Streamlines and velocity field of the non-actuated, 𝛼=19° scenario. 𝑡=3.6 𝑠. .... 36 Figure 4.1. Time evolution and Fourier transform of case 1. ..................................................... 38 Figure 4.2. Time evolution and Fourier transform of case 2. ..................................................... 38 Figure 4.3. Time evolution and Fourier transform of case 3. ..................................................... 39 Figure 4.4. Time evolution and Fourier transform of case 4. ..................................................... 39 Figure 4.5. Time evolution and Fourier transform of case 5. ..................................................... 40 Figure 4.6. Time evolution and Fourier transform of case 6. ..................................................... 40 Figure 4.7. Time evolution and Fourier transform of case 7. ..................................................... 41 Figure 4.8. Time evolution and Fourier transform of case 8. ..................................................... 41 Figure 4.9. Time evolution and Fourier transform of case 9. ..................................................... 42 Figure 4.10. Time evolution and Fourier transform of case 10. ................................................. 42 Figure 4.11. Time evolution and Fourier transform of case 11. ................................................. 43 Figure 4.12. Time evolution and Fourier transform of case 12. ................................................. 43 Figure 4.13. Streamlines and velocity magnitude of case 5 at 𝑡=4 𝑠. ...................................... 44 Figure 4.14. Streamlines and velocity magnitude of case 8 at 𝑡=4 𝑠. ...................................... 45 Figure 4.15. Streamlines and velocity magnitude of case 9 at 𝑡=4 𝑠. ...................................... 45 Index of tables Table 3.1. 𝐶𝑙, 𝐶𝑑, and 𝑙/𝑑 data gathered by Ladson. [8] .......................................................... 26 Table 3.2. Baseline scenario average 𝐶𝑙 and 𝐶𝑑. ..................................................................... 26 Table 3.3. Non-actuated cases boundary conditions. ............................................................... 29 Table 3.4. Actuated cases boundary conditions. ...................................................................... 29 Table 3.5. Study of the non-actuated case at α=10.15°.......................................................... 31 Table 3.6. Study of different values of 𝜈. .................................................................................. 32 Table 3.7. 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 and 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚 comparison at 𝛼=14.25°. ........................................... 33 Table 3.8. Non-actuated scenarios without jet implementation. ................................................ 33 Table 3.9. Non-actuated scenarios with jet implementation. ..................................................... 34 Table 4.1. Overview of the configuration of all actuated cases. ................................................ 37 Table 4.2. Important data of case 1. ........................................................................................ 38 Table 4.3. Important data of case 2. ........................................................................................ 38 Table 4.4. Important data of case 3. ........................................................................................ 39 Table 4.5. Important data of case 4. ........................................................................................ 39 Table 4.6. Important data of case 5. ........................................................................................ 40 Table 4.7. Important data of case 6. ........................................................................................ 40 Table 4.8. Important data of case 7. ........................................................................................ 41 Table 4.9. Important data of case 8. ........................................................................................ 41 Table 4.10. Important data of case 9. ...................................................................................... 42 Table 4.11. Important data of case 10. .................................................................................... 42 Table 4.12. Important data of case 11. .................................................................................... 43 Table 4.13. Important data of case 12. .................................................................................... 43 Table 4.14. Summary of all the cases. ..................................................................................... 44 Glossary CFD Computational Fluid Dynamics AFC Active Flow Control AoA Angle of Attack M Mach number Re Reynolds number PFC Passive Flow Control FA Fluidic Actuators ZNMFA Zero Net Mass Flow Actuator GUI Graphical User Interface DNS Direct Numerical Simulations LES Large Eddy Simulations RANS Reynolds-Averaged Navier-Stokes S-A Spalart-Allmaras SST Shear Stress Transport 14 Assessment of Active Flow Control techniques to prevent airfoil stall This is very useful for wind tunnel testing, as the models that are tested are usually much smaller than the prototypes. In addition, this can also be interesting in CFD simulations, as the results obtained for a determinate airfoil will hold for different situations (e.g. different airfoil sizes). There are several dimensionless numbers: e.g. Reynolds (𝑅𝑒), Mach (𝑀), Weber (𝑊𝑒), Froude (𝐹𝑟), etc. Depending on the application, it is very important to keep some of them constant, while the others may not be as critical. In this study, the important numbers to consider are the Reynolds and Mach numbers. A constant Reynolds number implies a similar flow with respect to the relative importance of the inertial and viscous effects. This is the reason why the results presented in this document can be applied in multiple scenarios, but only those in which 𝑅𝑒=2·106 and 𝑀<0.3. The Mach number is also relevant, as for bigger Mach numbers the flow would be compressible. 2.2 Turbulence and its modelling Turbulence is a fascinating physical phenomenon that occurs in all kinds of scenarios, and it can happen in both liquid and solid flows. In aerodynamics it appears everywhere, from the flow inside a jet engine to the wake created when an aircraft is moving. More conventional situations in which turbulence can be found could be water flowing out of a tap, smoke leaving out of a chimney, or wind passing through the buildings of a city. Everywhere there is a fluid, there is turbulence. Figure 2.2. Wake created by the wing tip vortices of a commercial airplane. [11] 2.2.1 Overview on turbulence A turbulent flow consists of eddies (regions of fluid whose direction differs from that of the general flow, creating swirling motions) of different sizes all occurring at the same time in all directions. These can vary in range from being comparable to the characteristic length of the system (the chord in the case of an airfoil) to many orders of magnitude smaller [12]. Turbulence is known to present unsteady and chaotic flow, which is characterized by having a large amount of eddies. Chaotic phenomena are characterized by being extremely sensitive to small perturbations, changing completely the outcome from the initial scenario to the perturbed one. There are two ways of dealing with this:  Solve turbulence by having a very fine mesh so that all eddies, including the smaller ones, are simulated.  Model turbulence completely or partially (depending on the size of the eddies) statistically. Each of these methods has its advantages and inconveniences. This will be elaborated in section 2.2.2. Fluid Dynamics background 15 A very important parameter that characterizes turbulence is turbulent kinetic energy. Figure 2.3 shows that turbulent kinetic energy decays as the wave number (inverse of the eddy size) diminishes. Figure 2.3. Typical turbulent energy spectrum. [12] Eddies can be classified according to their size and the stage of their “life”, fitting into three categories:  Energy containing range: It contains the eddies that are having an injection of energy, which tend to be the bigger ones.  Inertial sub-range: Eddies that are transferring turbulent energy are contained in the inertial sub-range. These are usually mid-sized.  Dissipation range: This range is formed by eddies that are dissipating, moments prior to their “death”. These are the smallest ones, and directly solving them requires very big computational power. It is also known as Kolmogorov scale. 2.2.2 Types of simulations depending on how turbulence is treated There are three different ways of solving flows in CFD depending on how turbulence is treated. 2.2.2.1 Direct Numerical Simulations (DNS) The full kinetic energy spectrum is solved by the CFD simulation in this approach, including the Kolmogorov length scale. It gives the most accurate results that can be achieved by CFD, but it needs very fine meshes and small time-steps. These requirements make DNS the most high computational cost path available, so much that it is used in very concrete scenarios where the other options may not be suitable, or extreme precision is required. Some examples could be turbulence in boundary layers at low to moderate Reynolds numbers. 2.2.2.2 Large Eddy Simulations (LES) LES is a method that was proposed by Joseph Smagorinsky in 1963. The key concept is that only large-sized eddies are solved (typically these belong to the energy containing range or part of the inertial sub-range), while the smaller ones (the rest, mainly in the dissipation range) are modelled. Since solving the smaller eddies requires most of the computational cost, the costs of the simulations (time, size of the files and energy consumption of the computer) is diminished significantly and the accuracy of the results is still quite precise. 2.2.2.3 Reynolds-Averaged Navier Stokes (RANS) Most of the turbulent energy spectrum is solved using time-averaged Navier-Stokes equations (that is, only the larger eddies are solved directly). In order to do so, many different turbulence models have been developed for decades. These can be one equation models (e.g. SpalartAllmaras and Prandtl’s one equation model) or two equation models (e.g. k-epsilon and k-omega). Each model has its advantages and disadvantages. This study has been done with the model 16 Assessment of Active Flow Control techniques to prevent airfoil stall Spalart-Allmaras, as it is specifically developed for aerodynamic applications and it is known to have good performance compared to others. 2.2.3 Turbulence modelling It is usually the case that the computational cost required by the complicated scenarios that must be faced in engineering is extremely big if the simulations are DNS or LES. For this reason, statistical treatment of the flow becomes a very interesting option, as it significantly reduces computational costs associated to those computations. Reynolds was the first person to take this approach, and developed the Reynolds-Averaged Navier-Stokes equations. The idea behind these equations was to apply Reynolds decomposition, whereby an instantaneous quantity is decomposed into a time-averaged and a fluctuating quantity [13]: 𝑈(𝑥,𝑡)=〈𝑈(𝑥,𝑡)〉+𝑢(𝑥,𝑡) (2.7) Where 〈𝑈(𝑥,𝑡)〉 is the time-averaged velocity field, and 𝑢(𝑥,𝑡) is the fluctuating quantity. Now, applying Einstein notation, the RANS equations are: 𝜕〈𝑈𝑖〉 𝜕𝑥𝑖=0 (2.8) 𝜕〈𝑈𝑗〉 𝜕𝑡 +〈𝑈𝑖〉𝜕〈𝑈𝑗〉 𝜕𝑥𝑖=𝜈𝜕2〈𝑈𝑗〉 𝜕𝑥𝑖2−1 𝜌𝜕⟨𝑝⟩ 𝜕𝑥𝑗−𝜕〈𝑢𝑖𝑢𝑗〉 𝜕𝑥𝑖 (2.9) RANS equations introduce the velocity covariance term 〈𝑢𝑖𝑢𝑗〉, often referred to as Reynolds stresses. Notice that if these were to be zero, the RANS equations and the Navier-Stokes equations would be the same. Thus, the difference of the behavior of 𝑈(𝑥,𝑡) with respect to 〈𝑈(𝑥,𝑡)〉 is caused due to the presence of the Reynolds stresses. The RANS equations can be reformulated to introduce the Reynolds stresses as: 𝜌𝜕〈𝑈𝑗〉 𝜕𝑡 +𝜌〈𝑈𝑖〉𝜕〈𝑈𝑗〉 𝜕𝑥𝑖=𝜕 𝜕𝑥𝑖[𝜇(𝜕〈𝑈𝑖〉 𝜕𝑥𝑗+𝜕〈𝑈𝑗〉 𝜕𝑥𝑖)−〈𝑝〉𝛿𝑖𝑗−𝜌〈𝑢𝑖𝑢𝑗〉] (2.10) In this form of the equation, the total stress is composed by the viscous forces stress, the isotropic stress caused by the mean pressure field, and that from the fluctuating velocity field. Moreover, the Reynolds stresses define the turbulent kinetic energy (𝑘), which is the mean kinetic energy per unit mass associated to the fluctuating velocity field, 𝑢(𝑥,𝑡). 𝑘=1 2〈𝑢𝑖𝑢𝑖〉 (2.11) Notice that, for a two-dimensional turbulent flow, there are three equations governing the mean velocity field (the two time-averaged Navier-Stokes and the mean continuity equation), but there are four unknowns since the Reynolds stresses also appear. Thus, the system needs another equation to be closed, which is provided by physical models. 2.2.3.1 Turbulent-viscosity hypothesis The turbulent-viscosity hypotheses is used as a tool to close the RANS equations system in turbulent flows. Introduced by Boussinesq in 1877, it relates the Reynolds stresses to the mean flow in order to close the system of equations. This hypothesis will not be explained in this study, as it is considered a complex subject in fluid mechanics that is out of the scope of this project. If the readers are interested in such topic, it is recommended that they look at references [14], [15]. Fluid Dynamics background 17 Several turbulence models are based on the turbulent-viscosity hypothesis, some examples being Spalart-Allmaras (S-A), 𝑘−𝜀, 𝑘−𝜔, and Menter’s Shear Stress Transport (SST). As mentioned in previous sections, the Spalart-Allmaras model has been chosen to perform the simulations presented in Section 4. 2.2.3.2 Spalart-Allmaras turbulence model The Spalart-Allmaras is a one-equation turbulence (see Equation 2.12) model that solves one modelled transport equation for the kinematic eddy turbulent viscosity, 𝜈, also called the SpalartAllmaras variable. It was specifically developed for aerospace applications, showing good results for boundary layers subjected to adverse pressure gradients. 𝜕𝜈 𝜕𝑡+𝑢𝑗𝜕𝜈 𝜕𝑥𝑗=𝐶𝑏,1(1−𝑓𝑡,2)𝑆󰆹𝜈−[𝑣𝑤,1𝑓𝑤−𝐶𝑏,1 𝑘2𝑓𝑡,2](𝜈 𝑑)2+1 𝜎[𝜕 𝜕𝑥𝑗((𝜈+𝜈)𝜕𝜈 𝜕𝑥𝑗)+𝐶𝑏,2𝜕𝜈 𝜕𝑥𝑖𝜕𝜈 𝜕𝑥𝑖] (2.12) Equation 2.12 [16] and the mathematical background of the S-A model will not be explained in this study, as they have also been considered advanced knowledge out of the scope of this project, but it is encouraged by the author to delve into references [16], [17], [18] to have a deeper understanding of this subject. 2.3 Boundary layer theory When a fluid moves past an object (or an object moves inside a fluid), the molecules next to the surface stick to it. This causes the molecules just above the surface to slow down, as they collide with the ones that are stuck on the surface. At the same time, these molecules slow down the flow near them, and so on. As the distance to the walls of the surface increases, the effect of the wall is diminished, until a height at which the object does not have any effect is reached. Thus, there is a layer of fluid characterized by a velocity gradient, going from zero at the surface of the object (no-slip condition) to the free stream velocity in its outer border. This layer is known as boundary layer, and inside its domain, viscosity forces have a major impact on the behavior of the fluid. This topic is of major relevance in this study, as the boundary layer is critical for the aerodynamics of the airfoil but, since AFC consists in exchanging momentum with the boundary layer, it is even more important in this concrete scenario. Outside of the boundary layer, the viscosity of the fluid plays no role, and inviscid flow can be assumed. The thickness of the boundary layer, 𝛿, is mathematically defined as: 𝛿=𝑦(𝑈=0.99𝑈∞) (2.13) That is, the boundary layer thickness is the distance normal to the wall at which the velocity is 99% of that at the free stream (in which viscous effects are negligible). Figure 2.4. Boundary layer graphical representation. shows a graphical representation of a boundary layer and its thickness. 18 Assessment of Active Flow Control techniques to prevent airfoil stall Figure 2.4. Boundary layer graphical representation. [19] It is important to mention that because of the loss of velocity, a shear stress appears on the surface of the object, according to the mathematical expression: 𝜏𝑠=𝜇𝜕𝑈 𝜕𝑦|𝑦=0 (2.14) It is straightforward to see that steep velocity gradients will cause high values of shear stress, increasing the skin drag. 2.3.1 Laminar and turbulent boundary layers Not all boundary layers present the same behavior, and they are usually classified in two categories: laminar and turbulent boundary layers. On one hand, laminar boundary layers are characterized by having ordered, smooth and layered flow, free of any mixing between successive layers, and they are associated with low Reynolds numbers (see Figure 2.5. Laminar (a) and turbulent (b) boundary layers. On the other hand, turbulent boundary layers (see Figure 2.5. Laminar (a) and turbulent (b) boundary layers. ) have lots of mixing, causing a more uniform flow outside of the immediate region at the wall, and are associated to high Reynolds numbers. Because of the rotational motion of the fluid, vortices and eddies are formed, making them thicker than laminar boundary layers because of the mixing. This fact makes them more resistant to adverse pressure gradients, but it also contributes to having steeper velocity gradients, and thus, causing bigger values of shear stress at the wall, increasing the skin drag. Figure 2.5. Laminar (a) and turbulent (b) boundary layers. [20] When the flow starts to go through the object, the boundary layer is laminar, and there is no turbulence. As the flow advances, some turbulence starts to appear, creating a transition between Fluid Dynamics background 19 a laminar flow and a turbulent flow. Finally, there is a point at which the flow can already be considered fully turbulent. 2.3.2 Boundary layer separation and SJAs The boundary layer separation is defined as the detachment of the boundary layer from the surface of the object. This leads to the creation bigger vortices in the wake. Pressure gradients have a major impact on the detachment of a boundary layer, as they promote or complicate the movement of the fluid:  Negative pressure gradients enable the flow, as they “push” the fluid in the direction of the flow.  Positive pressure gradients (usually referred to as adverse pressure gradients) have the opposite effect, as the flow is moving from a region of low pressure to a region of high pressure. Figure 2.6. Boundary layer separation. [19]shows a visual representation of the detachment of the flow as the fluid suffers the effects of a positive pressure gradient: Figure 2.6. Boundary layer separation. [19] 2.3.2.1 Why Synthetic Jet Actuators are used for delaying boundary layer separation Airfoils suffer an adverse pressure gradient in the extrados once the fluid has passed the low pressure bubble. As already mentioned, this fact promotes the detachment of the boundary layer, which is undesirable, since the lift diminishes and the overall drag is increased. Figure 2.7Figure 2.7. Pressure of a NACA 0012 airfoil at 𝐴𝑜𝐴=10.15°. shows a typical pressure distribution at the extrados of a NACA 0012 airfoil at 𝐴𝑜𝐴=10.15°. Notice how the pressure diminishes abruptly at the leading edge, and soon starts to increase, generating an adverse pressure gradient and contributing to the detachment of the boundary layer. Figure 2.7. Pressure of a NACA 0012 airfoil at 𝐴𝑜𝐴=10.15°. 20 Assessment of Active Flow Control techniques to prevent airfoil stall SJA are membranes that, by vibrating, they cause oscillatory pressure and velocity fields. Thus, they behave as turbulence generators, inducing vortices and eddies in the flow, energizing the boundary layer. This changes the velocity profile at the boundary layer to a more turbulent and “fuller” velocity profile, making it more resistant to adverse pressure gradients. Therefore, the separation of the boundary layer is delayed, reattaching the flow to the object’s surface and substantially improving its aerodynamics (increasing the lift and decreasing the overall drag). Figure 2.8 shows a schematic of a typical SJA at an incidence angle of 90º. Figure 2.8. Schematic of a synthetic jet actuator. [21] 2.4 CFD important parameters There are some important parameters to be considered when performing CFD simulations. Wall y+ is critical for solving correctly the boundary layer, and the Courant number, 𝐶𝑜, is necessary to ensure the selection of an appropriate time-step. 2.4.1 Wall 𝒚+ and Law of the wall When dimensional analysis is applied to the flow at the proximities of an object, it can be observed that its behavior depends only on the characteristics of the surface, no matter the characteristics of the free-stream flow. This discovery, introduced by Nikuradse and Prandtl, is called the Law of the wall, and it is very useful when dealing with turbulent flows. The two variables that are used to formulate this law are the dimensionless length, 𝑦+, and dimensionless velocity, 𝑢+, which are defined by the following expressions: 𝑦+=𝑦 𝜈(𝜏𝑤 𝜌)1 2 (2.15) 𝑢+=𝑢(𝜏𝑤 𝜌)−1 2 (2.16) With this two variables defined, the vertical velocity profile, 𝑢(𝑦), can be normalized, becoming 𝑢+(𝑦+). This normalized velocity profile, despite its chaotic nature, is almost always the same for any turbulent flow at the surface of a solid. The law of the wall is constituted by three different regions, which are the viscous sublayer, the buffer layer, and the log-law region. The viscous sublayer is characterized by 𝑢+ being proportional to 𝑦+, the log-law region by a logarithmic dependence (of 𝑢+ on 𝑦+) , and the buffer layer is a transition layer between the other two. Fluid Dynamics background 21 Figure 2.9. Law of the wall. Dimensionless velocity profile in the proximities of a solid (red line). [22] Figure 2.9 shows in blue the different trends that the dimensionless velocity profiles follows in the viscous sublayer and the log-law region, while the red line is the actual dimensionless velocity profile. The main reason why the law of the wall is so important for RANS CFD simulations is that it can be used as a model for the velocity profile near a wall. This modelling is known as wall functions, and they are different depending on the model of turbulence that is used. Usually, the model requires a specific range of values of 𝑦+ of the first cell to work correctly. In OpenFOAM, SpalartAllmaras can work with 𝑦+∈[1,300], which is known as insensitive wall functions. 2.4.2 Courant number The Courant number (also known as Courant-Friedrichs-Levy number), 𝐶, is a dimensionless value that is used in CFD simulations for ensuring that the time step is in the right range of values for having accurate results. When a time step is too large, it does not capture properly the behavior of the flow, as it exceeds its physical timescales. The Courant condition states that the Courant number should be equal or smaller than one. The Courant number is mathematically defined as: 𝐶= ∆𝑡 ∆𝑥/𝑢≤1 (2.17) Where ∆𝑡 is the time step, ∆𝑥 is the distance between the two closest mesh nodes, and 𝑢 is the velocity at those cells. This condition is imposing is that the flow cannot go across more than one cell in each time step, and thus, the information in each time step is only being transmitted to neighbor nodes. 2.5 Mesh design concepts Having a proper mesh for the application that is being modeled is extremely important in CFD, as it is in other engineering areas, such as structural analysis or telecommunications. This section will give some important concepts to be considered when designing a 2D mesh for an airfoil. 22 Assessment of Active Flow Control techniques to prevent airfoil stall 2.5.1 Structured and unstructured grids These two different approaches can produce three types of meshes: structured grid meshes, unstructured grid meshes, and a hybrid, containing regions with structured grids and others with unstructured grids. Structured grids are characterized by using quadrilateral and hexahedral cells for 2D and 3D geometries, and they follow a clear order. They produce the best accuracy when studying laminar flows, and the convergence CPU time is usually faster, as the solving algorithms are more efficient. It can be complex to use this type of elements when dealing with complex geometries. In the opposite side, unstructured grids are formed by triangular and tetrahedral cells for 2D and 3D geometries, and unlike structured grids, these do not follow a clear order. One of its main advantages is that they can adapt to complex geometries. In fluid dynamics, they can offer better results for turbulent flows, as they are more chaotic and less structured. Boundary layers usually benefit from having structured grids, as even if the flow is turbulent, structured geometries allow the use of geometrical progressions to size the cells, proving to be useful to solve the steep velocity gradients that are present in this region of the flow. Figure 2.10. Example of a structured grid (left side) and an unstructured grid (right side). [23] 2.5.2 Mesh quality metrics There are four important considerations that have to be taken into account when designing a mesh for CFD applications:  Non-orthogonality is the angular deviation between the vector of two adjacent cell centers and the normal of the face shared by the cells. The more orthogonal it is (angles close to zero), the better the quality of the mesh, as high values of this angle cause numerical instability. Figure 2.11. Mesh non-orthogonality. [24]  Aspect ratio is the ratio of a cell’s longest face to the shorter one. The smaller it is (closer to one), the better the quality of the element. This metric usually cannot accomplished in the boundary layer.  Skewness is the normalized distance between a line going from the center of two adjacent cells to the center of the face they share. Fluid Dynamics background 23 𝑆𝑘𝑒𝑤𝑛𝑒𝑠𝑠=|𝑠| |𝑑| (2.14) Figure 2.12. Visual representation of the skewness vectors. [24] The ideal value is zero (center of the faces contained in vector 𝑑).  Smoothness, or expansion rate, defines the transition in size between two contiguous cells. Large values of smoothness add diffusion to the solution, and ideally it should not be bigger than a 20%. Figure 2.13. Visual representation of the smoothness concept. [25] It must be mentioned that these quality metrics cannot be always fulfilled. There may be a few regions of the mesh in which one or two of the metrics are not met because of geometrical reasons, but it is important to consider them and ensure these are controlled exceptions. 30 Assessment of Active Flow Control techniques to prevent airfoil stall 3.3 Mesh dependence study The mesh is responsible of discretizing the flow domain that is to be simulated, making its design a very sensitive task. Mesh dependence studies are critical in any field that involves performing numerical simulations. They are used to study the influence that meshes have on the flow, make changes to optimize the simulations, and get more accurate results. The mesh dependence study was implemented to obtain the mesh that best fitted the experimental data gathered by Ladson, which was pretty accurately accomplished for the lift coefficient, but not as much with the drag. There was also a discrepancy with the stall angle of attack, as the simulated value was higher than the experimental. 3.3.1 Time-step configuration All the simulations performed in subsection 3.3.2 were configured with a time-step ∆𝑡=0.0004, while the simulations in subsection 3.3.3 had a ∆𝑡=0.0001. This was because the jet mesh contains smaller cells, needing a smaller time-step to correctly solve the flow. This was, unfortunately, a mistake. As explained in section 2.4.2, the Courant number (Equation 2.17) determines the maximum value of the time-step for a correctly configured simulation. The error was to use 𝑢=𝑈∞. Since the jet is located at the suction bubble, the amplitude of the velocity field was 𝑢>𝑈∞, leading to Courant numbers 𝐶>1. This mistake was noticed too late to change it. Therefore, it will be mentioned as further work, as it could mean that the boundary layer is not correctly solved, leading to wrong values of the lift and drag coefficients. 3.3.2 Meshes without the jet Several iterations were performed to optimize the mesh at multiple angles of attack. These angles were 𝛼=[10.15,14.15,15.25,16.25,17.25]. After some tests with different mesh configurations, as explained in 3.1, and stablishing the final type of mesh (structured boundary layer, unstructured inner region, and unstructured outer region, see Figure 3.5 and Figure 3.6), the angle of attack 𝛼=10° was chosen to perform the initial adjustments. Figure 3.5. Mesh near the airfoil. Numerical Setup 31 Figure 3.6. Boundary layer at the leading edge. The reason for choosing this angle and not a higher value, which could seem more intuitive at first because of the proximity to the angles that were to be tested with the SJA in the future, is that stall is a complicated phenomenon to study. It happens with some frequency that academic studies show different results on stall, as it may be heavily influenced by many factors that are not easy to control, but at lower angles of attack they usually have very similar results. Therefore, it was assessed that designing a mesh at a moderately high angle of attack would be a better option that at near-stall conditions. 3.3.2.1 Scenarios at 𝛼=10.15° The first thing that was found is that increasing the number of cells of the outer surface (see Figure 3.1 and Figure 3.2) did not have an impact if the chosen values were reasonable (inlet, upper and lower boundaries discretized at ∆𝑥≤2 𝑚), so only the variations that were applied were those of the inner, boundary layer, and wake surfaces. The results in this subsection were obtained using the solver 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚. Number of nodes 𝑪𝒍𝒔𝒊𝒎 𝑪𝒅 𝒔𝒊𝒎 𝑪𝒍 absolute error 𝑪𝒅 absolute error 𝑪𝒍 relative error (%) 𝑪𝒅 relative error (%) 152876 1.1538 0.02315 0.1088 0.00965 10.41 71.48 307541 1.0897 0.02153 0.0447 0.00803 4.28 59.48 394262 1.0541 0.01980 0.0091 0.00630 0.87 46.67 434266 1.0452 0.01945 0.0002 0.00595 0.02 44.07 514389 1.0452 0.01943 0.0002 0.00593 0.02 43.92 Table 3.5. Study of the non-actuated case at α=10.15°. Analyzing the results of this study, presented at Table 3.5, it is straightforward to see that the simulations pretty much converged at around 400000 nodes. For this reason, the 434266 node mesh was chosen to perform the next simulations. With regards to the analysis of the results, it is very easy to see that once the solution converged, the obtained 𝐶𝑙 values were extremely precise, with the chosen mesh presenting only a relative error of 0.02%. Regarding the 𝐶𝑑, the result was extremely different, as it presented a relative error of 40.81%. In order to reduce the error of the drag coefficient, several values of 𝜈 were tried, as modifying this parameter by some orders of magnitude can sometimes improve the calculation of the drag 32 Assessment of Active Flow Control techniques to prevent airfoil stall coefficient. At this point, it still had not been noticed that the time-step configuration was not adequate, so changing to smaller time-steps was not considered. The set of values that were tested were 𝜈=[5·10−3,5·10−4,5·10−5,5·10−6,5·10−7,5· 10−8,5·10−9,5·10−10,5·10−11,5·10−12] (the initial value was 𝜈= 5 · 10−7, which is specified in section 3.2.2. The results of these simulations are presented in Table 3.1 and Figure 3.7. 𝝂 𝑪𝒅 𝒔𝒊𝒎 𝑪𝒅 relative error 𝟓·𝟏𝟎−𝟑 0.04891 262.28 𝟓·𝟏𝟎−𝟒 0.03012 123.09 𝟓·𝟏𝟎−𝟓 0.02125 57.38 𝟓·𝟏𝟎−𝟔 0.01961 45.28 𝟓·𝟏𝟎−𝟕 0.01945 44.07 𝟓·𝟏𝟎−𝟖 0.01944 44.04 𝟓·𝟏𝟎−𝟗 0.01944 43.98 𝟓·𝟏𝟎−𝟏𝟎 0.01946 44.16 𝟓·𝟏𝟎−𝟏𝟏 0.01942 43.87 𝟓·𝟏𝟎−𝟏𝟐 0.01942 43.84 Table 3.6. Study of different values of 𝜈. Figure 3.7. Study of different values of 𝜈. Analyzing the data presented in Table 3.6 and Figure 3.7, it is clear that no significant improvement is observed in the values of 𝐶𝑑, which get much worse for values bigger than 𝜈> 5·10−6. For this reason, it is concluded that this error does not depend on the values of 𝜈. From now on, the error of the simulated 𝐶𝑑 was considered a systematic error, and it was assumed for the rest of the simulations. Two possible explanations, even though there may be more, are proposed in order to explain this phenomenon:  The first hypothesis is the aforementioned possibility of having fluid regions with compressibility effects in the experimental wind tunnel tests, but this is only a theory that should be tested by gathering new experimental data at lower Mach numbers.  The second one is that it was produced because of the too large time-steps problem, explained in subsection 3.3.1. Numerical Setup 33 3.3.2.2 Other values of angle of attack The following results have been computed with the solver 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚, with the only exception of the case at 𝛼=10.15°, which has been exposed in the previous subsection. Since the actuated scenarios were to be simulated using the transient solver, changing to 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 was considered appropriate. 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 produced the same exact average aerodynamic results as 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚, which was verified with the case at 𝛼=14.25° (see Table 3.7). 𝒔𝒐𝒍𝒗𝒆𝒓 𝑪𝒍𝒔𝒊𝒎 𝑪𝒅 𝒔𝒊𝒎 𝒔𝒊𝒎𝒑𝒍𝒆𝑭𝒐𝒂𝒎 1.3870 0.03058 𝒑𝒊𝒔𝒐𝑭𝒐𝒂𝒎 1.3870 0.03058 Table 3.7. 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚 and 𝑠𝑖𝑚𝑝𝑙𝑒𝐹𝑜𝑎𝑚 comparison at 𝛼=14.25°. Once it was seen that the solver was not affecting the results, the following simulations were performed with 𝑝𝑖𝑠𝑜𝐹𝑜𝑎𝑚. The results can be observed in Table 3.8. 𝜶 𝑪𝒍,𝒂𝒗 𝒔𝒊𝒎 𝑪𝒅,𝒂𝒗 𝒔𝒊𝒎 𝑪𝒍 absolute error 𝑪𝒅 absolute error 𝑪𝒍 relative error (%) 𝑪𝒅 relative error (%) 10.15 1.0452 0.01945 0.0002 0.00595 0.02 44.07 14.25 1.3870 0.03058 0.0250 0.00778 1.84 34.12 15.25 1.4793 0.03666 0.0713 0.00756 5.06 25.98 16.25 1.6119 0.04651 0.8589 - 114.06 - 17.25 1.7207 0.06505 - - - - 18.25 1.2248 0.22203 - - - - Table 3.8. Non-actuated scenarios without jet implementation. And its visual representation, with a comparison to the experimental results by Ladson. Figure 3.8. Comparison of the non-actuated scenarios without jet implementation. Notice how these are average 𝐶𝑙 and 𝐶𝑑 values, as vortex shedding started to appear at 𝛼=15°. Table 3.8 and Figure 3.8 clearly show two relevant results: 34 Assessment of Active Flow Control techniques to prevent airfoil stall  The error in the drag coefficient seems indeed to be a systematic error, as its absolute value is pretty similar for all the angles of attack that can be compared, reducing its relative value for the bigger AoA.  Stall occurs later in the simulations. As already mentioned, it is quite usual to not have the same stall AoA, as this phenomenon is very sensitive to many factors. This completes the study of the non-actuated meshes without a specific surface for the jet (until now, the airfoil was divided just into two different surfaces, which were the extrados and the intrados). 3.3.3 Meshes with the jet With an already tested mesh, it was finally possible to implement the jet. To do so, a small part of the extrados near to the leading edge was selected and the jet was implemented following the method explained in section 3.1.1.3 (see Figure 3.9). Figure 3.9. Image of the leading edge of the final jet mesh. Jet situated at the high density cell region. The jet mesh was designed with the same boundary layer thickness characteristics to obtain a similar solution to the meshes without the jet. Regarding the discretization of the surface of the airfoil, it was needed to refine the mesh to correctly capture the behavior of the fluid at the jet and its vicinities. The results gotten for this mesh are presented in Table 3.9 and Figure 3.10. In this case, the errors are expressed with respect to the previous simulations and not the experimental data, as these have been taken as the baseline scenarios for designing this mesh. 𝜶 𝑪𝒍,𝒂𝒗 𝒔𝒊𝒎 𝑪𝒅,𝒂𝒗 𝒔𝒊𝒎 𝑪𝒍 absolute error 𝑪𝒅 absolute error 𝑪𝒍 relative error (%) 𝑪𝒅 relative error (%) 14.25 1.3877 0.03064 0.0007 0.00006 0.05 0.20 15.25 1.4733 0.03648 -0.006 -0.00018 0.41 0.49 16.25 1.6321 0.04742 0.0202 0.00091 1.25 1.96 17.25 1.8480 0.31682 0.1273 0.25177 7.40 387.04 18.25 1.8683 0.34580 0.6435 0.12377 52.54 55.74 19 1.4802 0.42733 - - - - 20 1.3630 0.42841 - - - - Table 3.9. Non-actuated scenarios with jet implementation. Numerical Setup 35 Figure 3.10. Comparison of the non-actuated scenarios with jet implementation. As it can be observed in Table 3.9 and Figure 3.10, this mesh presents quasi-identical behavior compared to the non-jet mesh at 𝛼=[14.25,15.25,16.25]. Even if this is true, it differs again in the stall conditions, which happens for higher values of both 𝐶𝑙 and 𝛼. 3.3.4 Baseline scenario for the implementation of the jet With this, knowing that 𝛼𝑚𝑎𝑥 =18.25°, the baseline for the implementation of the jet has been set at 𝛼=19°. This scenario presented the following 𝐶𝑙 and 𝐶𝑑 time evolution: Figure 3.11. Time evolution of the baseline scenario. Figure 3.11 can be difficult to interpret, as with such a detached boundary layer, vortex shedding presents a complex behavior. In order to better understand what is happening, it can be useful to compute the Fourier Transform of the 𝐶𝑙 to see which are the main frequencies of this mode. In order to do this, the mathematical algorithm 𝑓𝑓𝑡 is used (for more information, see [36]): 36 Assessment of Active Flow Control techniques to prevent airfoil stall Figure 3.12. Fourier transform of the 𝐶𝑙 at 𝛼=19°. Observing Figure 3.12, it can be seen that the main frequencies of this mode are around 𝑓1= 5 𝐻𝑧 and 𝑓2=4 𝐻𝑧, being 𝑓1 slightly more dominant. From now on, 𝑓1 will be considered the vortex shedding frequency, 𝑓𝑣𝑠. To add some statistical data, the main deviation of 𝐶𝑙 and 𝐶𝑑 were computed. The main deviation, 𝜎, is defined as: 𝜎=√∑(𝑥𝑖−𝑥)2 𝑛 𝑖=1𝑛−1 (3.5) Where 𝑥𝑖 is the value of the 𝑖𝑡ℎ point, 𝑥 is the mean value, and 𝑛 is the number of points of the data set. All of these information is summarized in , which defines the baseline for the implementation of the jet. 𝜶(°) 𝑪𝒍,𝒂𝒗 𝒏𝒐 𝒋𝒆𝒕 𝑪𝒅,𝒂𝒗 𝒏𝒐 𝒋𝒆𝒕 𝜼 𝒇𝒗𝒔 (𝑯𝒛) 𝝈𝑪𝒍 𝝈𝑪𝒅 19 1.4802 0.42733 3.4638 5 1.0013 0.30371 In order to illustrate the separation of the boundary layer in this scenario, was generated using ParaView: Figure 3.13. Streamlines and velocity field of the non-actuated, 𝛼=19° scenario. 𝑡=3.6 𝑠. Results 37 4 Results In this section, the reader will find an overview, an analysis case by case, and a discussion about the results of the actuated cases of this study. 4.1 Overview of all the scenarios As explained in 3.1.1.3, with the position of the jet and the incidence angle completely fixed, there are two variables that fully determine the behavior of a SJA. These are the non-dimensional frequency, 𝐹+, and the momentum coefficient, 𝐶𝜇. The range of values of non-dimensional frequencies are 𝐹+=[50,100,500,1000], and the momentum coefficients 𝐶𝜇= [10−4,10−3,10−2]. Notice how the non-dimensional frequencies, which correspond to jet frequencies 𝑓𝑗= [50,100,500,1000], as 𝑐=1 𝑚 and 𝑈∞=1 𝑚/𝑠, have been chosen to fulfill 𝑓𝑗 𝑓𝑣𝑠 = [10,20,100,200]. These frequencies were purposely chosen to see the effects that the nondimensional frequency had at values representing an increase of one and two orders of magnitude (𝐹+=[50,500]), and their first harmonics (𝐹+=[100,1000]). Table 4.1 is a synopsis of all the cases that were analyzed in the next subsection, 4.2. Case 𝑭+ 𝒇𝒋 𝒇𝒗𝒔 𝑪𝝁 1 50 10 10−4 2 10−3 3 10−2 4 100 20 10−4 5 10−3 6 10−2 7 500 100 10−4 8 10−3 9 10−2 10 1000 200 10−4 11 10−3 12 10−2 Table 4.1. Overview of the configuration of all actuated cases. 4.2 Analysis of each case This subsection provides detailed aerodynamic data and graphical resources for all the actuated cases. The parameters that were studied were: lift and drag coefficients, aerodynamic efficiency, vortex shedding frequency and its amplitude. 38 Assessment of Active Flow Control techniques to prevent airfoil stall 4.2.1 Case 1. 𝑭+=𝟓𝟎, 𝑪𝝁=𝟏𝟎−𝟒 Figure 4.1. Time evolution and Fourier transform of case 1. 𝑪𝒂𝒔𝒆 𝑪𝒍,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝑪𝒅,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝜼 ∆𝑪𝒍 ∆𝑪𝒅 ∆𝜼 ∆𝑪𝒍 𝑪𝒍𝒃𝒍(%) ∆𝑪𝒅 𝑪𝒅 𝒃𝒍 (%) ∆𝜼 𝜼𝒃𝒍(%) 1 1.7114 0.4254 4.0230 0.2312 -0.00193 0.5592 15.62 -0.45 16.14 Table 4.2. Important data of case 1. 4.2.2 Case 2. 𝑭+=𝟓𝟎, 𝑪𝝁=𝟏𝟎−𝟑 Figure 4.2. Time evolution and Fourier transform of case 2. 𝑪𝒂𝒔𝒆 𝑪𝒍,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝑪𝒅,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝜼 ∆𝑪𝒍 ∆𝑪𝒅 ∆𝜼 ∆𝑪𝒍 𝑪𝒍𝒃𝒍(%) ∆𝑪𝒅 𝑪𝒅 𝒃𝒍 (%) ∆𝜼 𝜼𝒃𝒍(%) 2 1.706 0.27908 6.1129 0.2258 -0.14825 2.6491 15.25 -34.69 76.48 Table 4.3. Important data of case 2. Results 39 4.2.3 Case 3. 𝑭+=𝟓𝟎, 𝑪𝝁=𝟏𝟎−𝟐 Figure 4.3. Time evolution and Fourier transform of case 3. 𝑪𝒂𝒔𝒆 𝑪𝒍,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝑪𝒅,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝜼 ∆𝑪𝒍 ∆𝑪𝒅 ∆𝜼 ∆𝑪𝒍 𝑪𝒍𝒃𝒍(%) ∆𝑪𝒅 𝑪𝒅 𝒃𝒍 (%) ∆𝜼 𝜼𝒃𝒍(%) 3 1.9755 0.03665 53.9018 0.4953 -0.39068 50.4379 33.46 -91.42 1456.13 Table 4.4. Important data of case 3. 4.2.4 Case 4. 𝑭+=𝟏𝟎𝟎, 𝑪𝝁=𝟏𝟎−𝟒 Figure 4.4. Time evolution and Fourier transform of case 4. 𝑪𝒂𝒔𝒆 𝑪𝒍,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝑪𝒅,𝒂𝒗 𝒂𝒄𝒕𝒖𝒂𝒕𝒆𝒅 𝜼 ∆𝑪𝒍 ∆𝑪𝒅 ∆𝜼 ∆𝑪𝒍 𝑪𝒍𝒃𝒍(%) ∆𝑪𝒅 𝑪𝒅 𝒃𝒍 (%) ∆𝜼 𝜼𝒃𝒍(%) 4 1.5004 0.38373 3.9100 0.0202 -0.0436 0.4462 1.36 -10.20 12.88 Table 4.5. Important data of case 4. 46 Assessment of Active Flow Control techniques to prevent airfoil stall 4.3.4 Final comments It could be argued that the most important aerodynamic performance parameter in this study was the aerodynamic efficiency. The only valid solutions were those that hugely improved this value, and these were, ordering them from the best to the worst:  Case 9: 𝐹+=500, 𝐶𝜇=10−2.  Case 8: 𝐹+=500, 𝐶𝜇=10−3.  Case 6: 𝐹+=100, 𝐶𝜇=10−2.  Case 12: 𝐹+=1000, 𝐶𝜇=10−2.  Case 3: 𝐹+=50, 𝐶𝜇=10−2. It is easy to see that every single case with 𝐶𝜇=10−2 produced a valid solution. In addition, there were solutions with all the values of the non-dimensional frequency. Therefore, it is concluded that, for a NACA 0012 profile at post-stall conditions and 𝑅𝑒=2·106, the momentum coefficient was a more important parameter than the non-dimensional frequency in the domain that was studied (𝐹+=[50,100,500,1000] and 𝐶𝜇=[10−4,10−3,10−2]). It was also noticed that 𝐶𝜇=10−4 may be too low, as not a single scenario with this momentum coefficient provided a good enough result. As for the non-dimensional frequency, it seems safe to state that the best tested option was 𝐹+= 500, and thus, 𝑓𝑗=500 𝐻𝑧 and 𝑓𝑗 𝑓𝑣𝑠 =100. Sustainability analysis 47 5 Sustainability analysis In the current context of innovation in aeronautics, improving aerodynamic efficiency is not only assessed from a technical point of view, but also from a sustainability standpoint. Synthetic Jet Actuators offer a potential solution for controlling airflow and improving aerodynamic performance in post-stall conditions, with possible implications for reducing fuel consumption, lowering operational costs, and improving flight safety. This chapter presents a sustainability study about the elaboration of this project, taking into account three key dimensions: environmental, economic, and social impact. This project required to use several resources, divided in technological and human resources:  Technological resources: o Material: one laptop, one external 2 TB hard drive, and a mouse. o Software: GMSH, OpenFOAM, ParaView, Python, Visual Studio Code, Microsoft Excel 2016, Microsoft Word 2016.  Human resources: one engineering student, dedication time of 300 ℎ. 5.1 Environmental impact It is estimated that an active, high profile laptop consumes approximately 170 W of power. The laptop has been working approximately 300 h with tasks such as mesh design, writing, and other tasks involving human activity. In addition it is estimated that 150 h of simulations have been performed with the author not being actively working. This gives a total of approximately 450 h of use of the laptop for this specific project, giving an energy consumption of 𝐸=170 𝑊· 450 ℎ= 76.5 𝑘𝑊ℎ. According to the most recent data (2023) in reference [37], the carbon dioxide emissions associated to the electric energy consumption were 260 𝑔 𝐶𝑂2/𝑘𝑊ℎ. Thus, the carbon dioxide emissions produced by this project have been: 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 (𝑔 𝐶𝑂2)=260 𝑔 𝑘𝑊ℎ·76.5 𝑘𝑊ℎ= 19.89 𝑘𝑔 𝐶𝑂2. It is considered that there was no environmental impact associated with the human resources. 5.2 Economic impact The laptop that has been used has a current market value of 500 €, the external hard drive bought exclusively for the simulations (which occupied 946 GB of space) was bought for 75 €, and the cost of the mouse was 10 €. As for the software, Microsoft Office 2016 can be currently bought for 10 €. Thus, the economic cost associated to the technology used in this project was 𝐶𝑜𝑠𝑡𝑡𝑒𝑐ℎ= 500+75+10·2=595 €. Regarding the human resources, according to [38], the average salary of a junior engineer in Spain is 13.33€ ℎ. This gives a human resource cost of 𝐶𝑜𝑠𝑡ℎ𝑢𝑚𝑎𝑛 =13.33€ ℎ·300 ℎ=4000 €. Finally, the energy costs have to be considered too. According to [39], electrical energy had an associated cost of 0.2436 €/𝑘𝑊ℎ in June 2024, which will be taken as a reference for the year. Therefore, the energy costs were 𝐶𝑜𝑠𝑡𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑖𝑡𝑦 =0.2436 € 𝑘𝑊ℎ·76.5 𝑘𝑊ℎ=18.64 €. This gave a total cost 𝐶𝑜𝑠𝑡𝑜𝑣𝑒𝑟𝑎𝑙𝑙 =595+4000+18.64=4613.64 €. 48 Assessment of Active Flow Control techniques to prevent airfoil stall 5.3 Social impact The social impact of this project is almost non-existing, and cannot be properly estimated. Nevertheless, there is the possibility that SJA are implemented in commercial and civil aviation in the future. They could contribute to enhance safety of air vehicles and provide new employment positions in universities and the aeronautic industry. Since these would be used to improve the aerodynamic performance of vehicles and this would reduce the fuel consumption, it would be beneficial for the clients that use air transport, making it more accessible. Conclusions and further investigation 49 6 Conclusions and further investigation This chapter provides two sections with the final conclusions and further areas of investigation and improvement. 6.1 Conclusions After the initial bibliographical research and the mesh dependence study, intending to provide the maximum amount of accuracy possible, different discoveries were encountered during the elaboration of this bachelor final project. As for the mesh, it was found that the best design for this concrete application consisted in dividing the numerical domain in four regions:  The outer region, formed by an unstructured grid and coarse elements.  The inner region, also constructed with an unstructured grid, but this time with finer elements.  The wake, made with the same structure as the outer and the inner regions, but with a fine mesh near the airfoil.  The boundary layer, being the only structured-grid surface, characterized by a very fine mesh, and being strongly affected depending on the turbulence model that is used. Regarding the simulations, it was impressive to see the accuracy on the lift coefficient coefficients, as there were angles with a relative error smaller than 1%. Unfortunately, the same thing could not be said about the drag coefficients, which carried a systemic error that could be produced by the inaccuracy in the Courant number, although it is not clear. It was interesting to see too the difficulty of the prediction of the stall conditions, which ended up being significantly higher in the simulations compared to the experimental data by Ladson. The implementation of the jet was definitely the most interesting part of the thesis, as there were noteworthy improvements, reaching increments in the aerodynamic efficiency up to ∆𝜂9 𝜂𝑏𝑙 (%)= 1598.57 % due to achieving the reattachment of the boundary layer, which could be verified using ParaView. In the end, it was found that the values that presented the best improvement were 𝐹+=500 (which corresponded with 𝑓𝑗 𝑓𝑣𝑠 =100) and 𝐶𝜇=10−2. Also, it is worth mentioning that all the simulations with 𝐶𝜇=10−2 accomplished the objective of this project, which was to reattach the boundary layer to the airfoil at post-stall angles of attack. For these reasons, it could be said that the main objective of this project was achieved: the aerodynamic performance of the NACA 0012 profile at post-stall angles of attack and high Reynolds numbers has been tested with satisfying results. Even so, there has been one significant mistake with the Courant numbers, making them bigger than one, that could be the cause of having the systemic error of the drag coefficient. Since this mistake was noticed too late, there was not enough time to make all the simulations again and, for this reason, it will be proposed as an improvement for further investigations. 6.2 Further investigations This project has been a first approach into the use of SJA at high Reynolds numbers. From now on, there are several challenges to take in this interesting subject.  More appropriate Courant numbers could be used to find out if the error of the drag coefficient is related to the inaccuracy of these being to large, and maybe even achieve 50 Assessment of Active Flow Control techniques to prevent airfoil stall better stall conditions in the simulations. In addition, the problem of the wall shear stress could be addressed in order to have better knowledge of the separation of the boundary layer.  An interesting possibility would be to try this technology with 3D flows, making them more realistic, as 2D flows are equivalent to having an infinite 3D wing. Those results could be used as a baseline to implement SJA in real models and test them in wind or water tunnels  Another option could be to make 2D CFD simulations, as in this study, about jets with different incidence angles, or even tangential jets. 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