External Turbulent Flow through an Airfoil—Mesh Provided Supplement Module to Appendix G Turbulent Compressible Flow: o Spalart-Allmaras model with Energy equation On; o Material—Air as an Ideal gas with Sutherland viscosity model; o Boundary Conditions: Interior Pressure-far-field Pressure-far-field Pressure-far-field Wall-top Fig.1 Fig.2 Wall-bottom 1|Page o o M=0.8 and Angle of Attack is made to vary from 0 to 20 degrees. Full Multi Grid Initialization is employed. Fig.3—Convergence upon solving governing equations o Velocity contours are plotted for ∝= 1°, 2°, 4°, 6°: 2|Page Fig.4--Velocity Contours for ∝= 1° Fig.5--Velocity Contours for ∝= 2° 3|Page Fig.6--Velocity Contours for ∝= 4° Shock Fig.7--Velocity Contours for ∝= 6° Turbulent Incompressible Flow, Case 1 o Modeled Compressible—Pressure Far Field B.C. remains unchanged with Energy Equation On. o M=0.2. Again the angle of attack is made to vary from 0 to 20 degrees. 4|Page Fig.8 o Velocity contours are plotted for ∝= 1°, 2°, 4°, 6°: Fig.9--Velocity Contours for ∝= 1° 5|Page Fig.10--Velocity Contours for ∝= 2° Fig.11--Velocity Contours for ∝= 4° 6|Page Fig.12--Velocity Contours for ∝= 6° Turbulent Incompressible Flow, Case 2: o Material is regarded as incompressible—density constant, viscosity constant, etc. o Spalart-Allmaras model is used again, energy equation is off. o The pressure far-field B.C. is divided into two B.C.s—velocity inlet and pressure outlet. 7|Page Interior Velocity-inlet Pressure-outlet Velocity-inlet Wall-top Fig.13 Fig.14 Wall-bottom In order to manipulate the pressure-far-field boundary and separate it into two distinct zones, first make sure the geometry is visible (Problem Setup—General—Display and select the pressure-far-field-1 surface). Recall the limits of the domain (Problem Setup—General—Scale)— Fig.15. 8|Page Fig.15 Then go to the Adapt—Region. Enter the coordinates as outlined in Fig. 16 and press Mark. Fig.16 Finally proceed to Mesh—Separate and choose Faces. Select Mark under Options and specify the created register and zone as outlined in Fig.17. Then press Separate. Fig.17 The original pressure far field zone is now separated into two zones to be specified as velocity inlet and pressure outlet—Fig.13. 9|Page Fig.18--Convergence Fig.19--Velocity Contours for ∝= 1° 10 | P a g e Fig.20--Velocity Contours for ∝= 2° Fig.21--Velocity Contours for ∝= 4° 11 | P a g e Fig.21--Velocity Contours for ∝= 6° Validation: o Validation is performed for the turbulent compressible and incompressible cases. For the incompressible case both aspects are considered—when the turbulent compressible case is modeled as incompressible by setting the Mach number to 0.2 and keeping the default boundary conditions and when the pressure far-field boundary condition is separated into velocity-inlet and pressure-outlet boundaries. o Lift Theory states that the coefficient of lift is calculated by using Eq.1 where the angle of attack is in radians. 𝐶𝐿 = 2 ∗ 𝜋 ∗ 𝛼 Eq.1 12 | P a g e Coefficient of Lift as a Function of the Angle of Attack 2.500 Coefficient of Lift, C_L 2.000 1.500 Compr. Flow Energy On Lift Theory Valid. 1.000 Incompr. Flow Energy Off Turb.Compr .M=0.2 0.500 0.000 0 5 10 Angle of Attack, Alpha, deg. 15 20 Graph.1 At zero angle of attack for incompressible flow the velocity profile is the same for both walls of the airfoil. The empirical correlation from Abbott and Doenhoff is modeled as incompressible inviscid hence the runoff at the end of the graph—Graph.2. Eq.2 is used to validate where the infinity subscript indicates ref. value and the s subscript—static value. 𝑝 −𝑝∞ 2 ∞ ∗𝑈∞ 𝐶𝑃 = .5∗𝜌𝑠 𝑈 = 1 − (𝑈 𝑠 )2 ∞ Eq.2 13 | P a g e Velocity Profile Vs. Airfoil Length 1.6 1.4 1.2 (u/V)^2 1 Turb. Incompr. Energy Off 0.8 Abbott and Doenhoff Valid. 0.6 0.4 Turb. Incompr. M=0.2 Energy On 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 x/c Graph.2 o By creating a tangent line, its intersection with the coefficient of lift curve obtained at various angles of attack, indicated the most efficient angle of attack. In this case that angle is approximately 8 degrees—Graph.3 14 | P a g e Coefficient of Lift VS. Coefficient of Drag Turbulent Incompressible Airfoil Flow 2.00E+00 1.80E+00 Coefficient of Lift, C_L α=18° α=16° 1.60E+00 1.40E+00 α=14° 1.20E+00 α=10° 1.00E+00 α=8° 8.00E-01 α=6° 6.00E-01 α=4° 4.00E-01 2.00E-01 α=2° 0.00E+00 α=0° 0 0.01 0.02 0.03 0.04 0.05 0.06 Coefficient of Drag, C_D Graph.3 15 | P a g e