# Aerodynamic Design Optimization Discussion Group Case 4: Single- and multi-point optimization problems based on the CRM wing

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1 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

2 1 Information Common to All Problems 1.1 Initial Geometry, Geometric Variation Permitted, and Geometric Constraints 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 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 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, ) 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, S ref = S projected = 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 adjusted to ensure that C L =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 moment coefficients should be done based on a refined mesh chosen on the basis of the grid refinement study; Optimization convergence histories: these will depend on the nature of the optimizer, but should show the history of the lift, drag, and moment coefficients with respect to a suitable measure of cost, such as a function evaluation, plus, where possible, measures of optimality and feasibility; the stopping criterion should be given; if the optimization algorithm is stochastic, the optimization convergence history should be an average of a sufficient number of optimizations, which should be given; 1

3 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

4 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 C L =0.5, and the pitching-moment coefficient constrained to C M 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

5 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 C D over a range of lift coefficients at a constant Mach number: minimize w.r.t. subject to 3 T i C Di i=1 Wing sectional shape Wing twist Angle of attack C Li =(C Li ) prescribed C M 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 (T i ) Number Coefficient Number Additional Results to Be Included C L vs α, C D vs α, C M vs α, and C D vs C L at Mach number 0.85; 1.8 α 3.8 ; α =0.1 C D vs Mach number at C L =0.5; 0.75 M 0.90; M =

6 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 C D over a range of Mach numbers at a constant lift coefficient: minimize w.r.t. subject to 3 T i C Di i=1 Wing sectional shape Wing twist Angle of attack C Li =(C Li ) prescribed C M 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 (T i ) Number Coefficient Number Additional Results to Be Included C L vs α, C D vs α, C M vs α, and C D vs C L at Mach number 0.85; 1.8 α 3.8 ; α =0.1 C D vs Mach number at C L =0.5; 0.75 M 0.90; M =

7 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 C D over a range of Mach numbers at a constant lift coefficient: minimize w.r.t. subject to 3 T i C Di i=1 Wing sectional shape Wing twist Angle of attack C Li =(C Li ) prescribed C M 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 C L at a given aircraft weight. Table 3: Case 4.4 Design Points Design Design Point Mach Lift Reynolds Point Weight (T i ) Number Coefficient Number Additional Results to Be Included C L vs α, C D vs α, C M vs α, and C D vs C L at Mach number 0.85; 1.8 α 3.8 ; α =0.1 C D vs Mach number at C L =0.5; 0.75 M 0.90; M =

8 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 C D over a range of Mach numbers at a constant lift (as opposed to lift coefficient): minimize w.r.t. subject to 3 T i C Di i=1 Wing sectional shape Wing twist Angle of attack C Li =(C Li ) prescribed C M 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 (T i ) Number Coefficient Number Additional Results to Be Included C L vs α, C D vs α, C M vs α, and C D vs C L at Mach number 0.85; 1.8 α 3.8 ; α =0.1 C D vs Mach number at a lift (not a lift coefficient) equal to that at design point 2; 0.75 M 0.90; M =

9 7 Case 4.6: Nine-point problem with variable lift and Mach number (optional) 7.1 Objective Function and Constraints Minimize an approximation to the integral of C D over a range of Mach numbers and lift values: minimize w.r.t. subject to 9 T i C Di i=1 Wing sectional shape Wing twist Angle of attack C Li =(C Li ) prescribed C M 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 (T i ) Number Coefficient Number Additional Results to Be Included C L vs α, C D vs α, C M vs α, and C D vs C L at Mach number 0.82; 1.8 α 3.8 ; α =0.1 C L vs α, C D vs α, C M vs α, and C D vs C L at Mach number 0.85; 1.8 α 3.8 ; α =0.1 C L vs α, C D vs α, C M vs α, and C D vs C L at Mach number 0.88; 1.8 α 3.8 ; α =0.1 C D vs Mach number at a lift equal to that at design points 1, 4, and 7; 0.75 M 0.90; M =

10 C D vs Mach number at a lift equal to that at design points 2, 5, and 8; 0.75 M 0.90; M =0.005 C D vs Mach number at a lift equal to that at design points 3, 6, and 9; 0.75 M 0.90; M =

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