Aerodynamic Design Optimization Discussion Group Case 4: Single- and multi-point optimization problems based on the
CRM wing
Lana Osusky, Howard Buckley, and David W. Zingg University of Toronto Institute for Aerospace Studies
1 Information Common to All Problems
1.1 Initial Geometry, Geometric Variation Permitted, and Geometric Con- straints
The baseline geometry is a wing-only geometry with a blunt trailing edge extracted from the Common Research Model (CRM) wing-body configuration, which was the subject of the Fifth Drag Prediction Workshop. An IGES file will be made available to the participants. The extraction of the wing from the wing-body configuration has been accomplished by deleting the fuselage such that the leading edge of the wing root is 120.252 inches from the original symmetry plane, translating the leading edge of the wing root to the origin, and scaling all coordinates by the the mean aerodynamic chord of 275.8 inches. The wing root does not lie exactly on the symmetry plane (y = 0), and a grid generator can also introduce some deviation from a flat symmetry plane. Therefore, once the grid is generated, it should be post-processed to ensure that the entire symmetry plane is at y = 0.
Pitching moments are taken about the point (1.2077, 0, 0.007669) with the origin at the leading edge of the wing root and units in terms of the reference length. The location of the moment centre is taken from the original wing-body geometry, with values scaled by the reference length.
All coefficients are calculated using the projected area as the reference, Sref = Sprojected= 3.407014 squared reference units. Note that the computed value of the projected area will be grid dependent and may not necessarily match the value quoted here.
Section shape changes are permitted in the vertical (z) direction. The trailing edge of the wing is fixed, while the leading edge is free; hence arbitrary wing twist is permitted with the exception of the root section, where both the leading and trailing edges are fixed. The wing planform shape is fixed. The angle of attack is allowed to vary. Geometric constraints include the following.
The internal volume of the wing must be greater than or equal to its initial value. In addition, the thickness must be greater than or equal to 25% of the initial thickness at all locations. The projected area, which is used as the reference area, does not have to be explicitly constrained, as it is calculated based on a zero angle of attack and will not be affected by the permitted shape changes.
1.2 Results to Be Included
• The lift, drag, and pitching-moment coefficients corresponding to the initial and optimized geometries computed on the mesh used for the optimization; the convergence criterion used for the analysis; for multi-point problems, the force and moment coefficients for each operating condition;
• The drag and pitching-moment coefficients corresponding to the optimized geometry on a sequence of refined meshes that shows their mesh convergence with the angle of attack ad- justed to ensure that CL= 0.5; the convergence criterion used for the analysis; for multi-point problems, this should be done for the central point, and all subsequent reporting of force and
• Section shapes comparing initial and optimized geometries at 2.35%, 26.7%, 55.7%, 69.5%, 82.8% and 94.4% span;
• Plots of the coefficient of pressure at the above spanwise stations;
• Spanwise load distributions corresponding to the initial and optimized geometries.
Additional case-specific results are specified for individual optimization problems.
2 Case 4.1: Single-point optimization problem (recommended)
2.1 Objective Function and Constraints
The objective is to minimize the drag coefficient with the lift coefficient constrained to CL = 0.5, and the pitching-moment coefficient constrained to CM ≥ −0.17. Geometric constraints include the following. The internal volume of the wing must be greater than or equal to its initial value. In addition, the thickness must be greater than or equal to 25% of the initial thickness at all locations.
The projected area, which is used as the reference area, does not have to be explicitly constrained, as it is calculated based on a zero angle of attack and will not be affected by the permitted shape changes.
2.2 Flow Conditions
Flow conditions are fully turbulent at a Mach number of 0.85, Reynolds number of 5 million (based on the reference length of the Mean Aerodynamic Chord), and an initial angle of attack of 2.2◦.
3 Case 4.2: Three-point problem with constant Mach number and variable lift coefficient (recommended)
3.1 Objective Function and Constraints
Minimize an approximation to the integral of CD over a range of lift coefficients at a constant Mach number:
minimize
�3 i=1
TiCDi
w.r.t. Wing sectional shape Wing twist
Angle of attack subject to CLi = (CLi)prescribed
CM ≥ −0.17 (at design point 2)
as well as the common geometric constraints given above.
3.2 Operating Conditions
Operating conditions are given in Table 1. The operating conditions at design point 2 are taken from the CRM wing-body case specified in the Fifth Drag Prediction Workshop. Flow conditions are fully turbulent at these operating conditions with an initial angle of attack of 2.2◦.
Table 1: Case 4.2 Design Points
Design Design Point Mach Lift Reynolds Point Weight (Ti) Number Coefficient Number
1 1 0.85 0.450 5.00 × 106
2 2 0.85 0.500 5.00 × 106
3 1 0.85 0.550 5.00 × 106
3.3 Additional Results to Be Included
• CLvs α, CD vs α, CM vs α, and CD vs CLat Mach number 0.85; 1.8◦ ≤ α ≤ 3.8◦; ∆α = 0.1◦
• CD vs Mach number at CL= 0.5; 0.75 ≤ M ≤ 0.90; ∆M = 0.005
4 Case 4.3: Three-point problem with constant lift coefficient and variable Mach number (0.84 ≤ M ≤ 0.86) (recommended)
4.1 Objective Function and Constraints
Minimize an approximation to the integral of CD over a range of Mach numbers at a constant lift coefficient:
minimize
�3 i=1
TiCDi
w.r.t. Wing sectional shape Wing twist
Angle of attack subject to CLi = (CLi)prescribed
CM ≥ −0.17 (at design point 2)
as well as the common geometric constraints given above.
4.2 Operating Conditions
Operating conditions are given in Table 2. The operating conditions at design point 2 are taken from the CRM wing-body case specified in the Fifth Drag Prediction Workshop. Flow conditions are fully turbulent at these operating conditions with an initial angle of attack of 2.2◦. In this case the Reynolds number is constant.
Table 2: Case 4.3 Design Points
Design Design Point Mach Lift Reynolds Point Weight (Ti) Number Coefficient Number
1 1 0.84 0.500 5.00 × 106
2 2 0.85 0.500 5.00 × 106
3 1 0.86 0.500 5.00 × 106
4.3 Additional Results to Be Included
• CLvs α, CD vs α, CM vs α, and CD vs CLat Mach number 0.85; 1.8◦ ≤ α ≤ 3.8◦; ∆α = 0.1◦
• CD vs Mach number at CL= 0.5; 0.75 ≤ M ≤ 0.90; ∆M = 0.005
5 Case 4.4: Three-point problem with constant lift coefficient and variable Mach number (0.82 ≤ M ≤ 0.88) (optional)
5.1 Objective Function and Constraints
Minimize an approximation to the integral of CD over a range of Mach numbers at a constant lift coefficient:
minimize
�3 i=1
TiCDi
w.r.t. Wing sectional shape Wing twist
Angle of attack subject to CLi = (CLi)prescribed
CM ≥ −0.17 (at design point 2)
as well as the common geometric constraints given above.
5.2 Operating Conditions
Operating conditions are given in Table 3. The operating conditions at design point 2 are taken from the CRM wing-body case specified in the Fifth Drag Prediction Workshop. Flow conditions are fully turbulent at these operating conditions with an initial angle of attack of 2.2◦. In this case the Reynolds number has been adjusted to account for the change of altitude needed to maintain constant CL at a given aircraft weight.
Table 3: Case 4.4 Design Points
Design Design Point Mach Lift Reynolds Point Weight (Ti) Number Coefficient Number
1 1 0.82 0.500 5.18 × 106
2 2 0.85 0.500 5.00 × 106
3 1 0.88 0.500 4.83 × 106
5.3 Additional Results to Be Included
• CLvs α, CD vs α, CM vs α, and CD vs CLat Mach number 0.85; 1.8◦ ≤ α ≤ 3.8◦; ∆α = 0.1◦
• CD vs Mach number at CL= 0.5; 0.75 ≤ M ≤ 0.90; ∆M = 0.005
6 Case 4.5: Three-point problem with constant lift and variable Mach number (0.82 ≤ M ≤ 0.88) (optional)
6.1 Objective Function and Constraints
Minimize an approximation to the integral of CD over a range of Mach numbers at a constant lift (as opposed to lift coefficient):
minimize
�3 i=1
TiCDi
w.r.t. Wing sectional shape Wing twist
Angle of attack subject to CLi = (CLi)prescribed
CM ≥ −0.17 (at design point 2)
as well as the common geometric constraints given above.
6.2 Operating Conditions
Operating conditions are given in Table 4. Lift coefficients are adjusted to achieve constant lift equivalent to that at Design Point 2. The operating conditions at design point 2 are taken from the CRM wing-body case specified in the Fifth Drag Prediction Workshop. Flow conditions are fully turbulent at these operating conditions with an initial angle of attack of 2.2◦.
Table 4: Case 4.5 Design Points
Design Design Point Mach Lift Reynolds Point Weight (Ti) Number Coefficient Number
1 1 0.82 0.537 4.82 × 106
2 2 0.85 0.500 5.00 × 106
3 1 0.88 0.466 5.18 × 106
6.3 Additional Results to Be Included
• CLvs α, CD vs α, CM vs α, and CD vs CLat Mach number 0.85; 1.8◦ ≤ α ≤ 3.8◦; ∆α = 0.1◦
• CD vs Mach number at a lift (not a lift coefficient) equal to that at design point 2; 0.75 ≤
7 Case 4.6: Nine-point problem with variable lift and Mach num- ber (optional)
7.1 Objective Function and Constraints
Minimize an approximation to the integral of CD over a range of Mach numbers and lift values:
minimize
�9 i=1
TiCDi
w.r.t. Wing sectional shape Wing twist
Angle of attack subject to CLi = (CLi)prescribed
CM ≥ −0.17 (at design point 5)
as well as the common geometric constraints given above.
7.2 Operating Conditions
Operating conditions are given in Table 5. Design points 1, 4, and 7 produce the same lift, as do design points 2, 5, and 8 as well as 3, 6, and 9. The operating conditions at design point 5 are taken from the CRM wing-body case specified in the Fifth Drag Prediction Workshop. Flow conditions are fully turbulent at these operating conditions with an initial angle of attack of 2.2◦.
Table 5: Case 4.6 Design Points
Design Design Point Mach Lift Reynolds Point Weight (Ti) Number Coefficient Number
1 1 0.82 0.483 4.82 × 106
2 2 0.82 0.537 4.82 × 106
3 1 0.82 0.591 4.82 × 106
4 2 0.85 0.450 5.00 × 106
5 4 0.85 0.500 5.00 × 106
6 2 0.85 0.550 5.00 × 106
7 1 0.88 0.420 5.18 × 106
8 2 0.88 0.466 5.18 × 106
9 1 0.88 0.513 5.18 × 106
7.3 Additional Results to Be Included
• CLvs α, CD vs α, CM vs α, and CD vs CLat Mach number 0.82; 1.8◦ ≤ α ≤ 3.8◦; ∆α = 0.1◦
• CLvs α, CD vs α, CM vs α, and CD vs CLat Mach number 0.85; 1.8◦ ≤ α ≤ 3.8◦; ∆α = 0.1◦
• CLvs α, CD vs α, CM vs α, and CD vs CLat Mach number 0.88; 1.8◦ ≤ α ≤ 3.8◦; ∆α = 0.1◦
• CD vs Mach number at a lift equal to that at design points 1, 4, and 7; 0.75 ≤ M ≤ 0.90;
∆M = 0.005
• CD vs Mach number at a lift equal to that at design points 2, 5, and 8; 0.75 ≤ M ≤ 0.90;
∆M = 0.005
• CD vs Mach number at a lift equal to that at design points 3, 6, and 9; 0.75 ≤ M ≤ 0.90;
∆M = 0.005
