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
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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°:
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Fig.4--Velocity Contours for ∝= 1°
Fig.5--Velocity Contours for ∝= 2°
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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.
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Fig.8
o
Velocity contours are plotted for ∝= 1°, 2°, 4°, 6°:
Fig.9--Velocity Contours for ∝= 1°
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Fig.10--Velocity Contours for ∝= 2°
Fig.11--Velocity Contours for ∝= 4°
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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.
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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.
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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.
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Fig.18--Convergence
Fig.19--Velocity Contours for ∝= 1°
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Fig.20--Velocity Contours for ∝= 2°
Fig.21--Velocity Contours for ∝= 4°
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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
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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
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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
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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
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