SAROD Aerodynamic Design Optimization Studies at CASDE Amitay Isaacs, D Ghate, A G Marathe, Nikhil Nigam, Vijay Mali, K Sudhakar, P M Mujumdar Centre for Aerospace Systems Design and Engineering Department of Aerospace Engineering, IIT Bombay

SAROD About CASDE 5 years old Master’s program in Systems Design & Engineering MDO MAV Modeling & Simulation Workshops/CEPs/Conferences

SAROD Optimization Studies –Overview Concurrent aerodynamic shape & structural sizing of wing FEM based aeroelastic design MDO architectures WingOpt software Propulsion system Engine sizing & cycle design Intake duct design using CFD

SAROD Intake Design - Background Duct design practice of late 80s – based on empirical rules Problem Revisited – using formal optimization and high fidelity analysis Study evolved with active participation of ADA (Dr. T.G. Pai & R.K.Jolly)

SAROD Problem Formulation Entry Exit Location and shape (Given) Optimum geometry of duct from Entry to Exit ? Objective/Constraints Pressure Recovery Distortion Swirl

SAROD Design Using CFD - Issues Simulation Time CFD takes huge amounts of time for real life problems Design requires repetitive runs of disciplinary analyses Integration & Automation Parametric geometry modeling Grid generation CFD solution Objective/Constraint function evaluation Optimization Gradient Information Finite difference – step size (??), (N DV + 1) analyses required Exact formulations – Automatic differentiation (ADIFOR), Adjoint method, Complex step method – All require source code

SAROD Flow Solver Distortion & Swirl calculation requires NS solution In-house NS Solver Analytical gradients possible Easy to integrate Commercial Solvers (STAR-CD, FLUENT…) Gradients using finite difference only Difficult to integrate FLUENT Inc. S-shaped non-diffusing duct Results validated with a NASA test case ( Devaki Ravi Kumar & Sujata Bandyopadhyay )

SAROD Strategies Reducing Time Parameterization Variable fidelity to shrink the search space Surrogate modeling Meshing Parallel computing Continuation Integration & Automation Wrapping executables and user interfaces Offline analysis (Surrogate models) – semi- automatic

SAROD Our Strategy Variable fidelity Response Surface based design using FLUENT

SAROD Our Methodology Parametrization Low fidelity Analysis DOE in reduced space CFD analysis at DOE points RS for PR & DC 60 Optimization Constraints

SAROD Parametrization Y X Z X Duct Centerline A X Control / Design Variables Y m, Z m A L/3, A 2L/3 Cross Sectional Area

SAROD Y X Z X Duct Centerline A X Control / Design Variables Y m, Z m A L/3, A 2L/3 Cross Sectional Area Parametrization

SAROD Typical 3D-Ducts

SAROD Duct Design - Low Fidelity Low Fidelity Design Rules (Constraints) Wall angle < 6° Diffusion angle < 3° 6 * Equivalent Radius < ROC of Centerline Objective function: pressure recovery No low fidelity analysis for distortion or swirl X 1-MIN X 2-MIN X 2-MAX X 1-MAX

SAROD Optimization Process – Low Fidelity

SAROD Automation for CFD Generation of entry and exit sections using GAMBIT Clustering Parameters Conversion of file format to CGNS using FLUENT Mesh file Generation of structured volume grid using parametrization Duct Parameters (β 1, β 2, α y, α z ) Entry & Exit sections Conversion of structured grid to unstructured format Unstructured CGNS file CFD Solution using FLUENT End-to-end (Parameters to DC 60 ) automated CFD Cycle. Objective/Constraints evaluation Using UDFs (FLUENT) DC 60 CFD Solution Continuation Solution

SAROD Automation for Design Generation of structured volume grid using parametrization Entry & Exit sections Conversion of structured grid to unstructured format CFD Solution using FLUENT Objective/Constraints evaluation Using UDFs (FLUENT) DC 60 Optimization Duct Parameters (β 1, β 2, α y, α z ) Continuation Solution Unstructured CGNS file CFD Solution

SAROD Results: Total Pressure Profile

SAROD Design Space Reduction (0.61, 0.31, 1.0, 1.0) Optimized duct from low fidelity DC P LOSS (-0.4, 1.5, 0.3, 0.6) (0.1, 0.31, 0.2, 0.6) P Poor ductInfeasible duct P – Parameters; P LOSS – Total Pressure Loss

SAROD Optimization Post-processing Distortion Analysis DC 60 = (PA 0 – P60 min ) /q where, PA 0 - average total pressure at the section, P60 min - minimum total pressure in a 60 0 sector, q- dynamic pressure at the cross section. User Defined Functions (UDF) and scheme files were used to generate this information from the FLUENT case and data file. Iterations may be stopped when the distortion values stabilize at the exit section with reasonable convergence levels.

SAROD Huge benefits as compared to the efforts involved!!! Methodology Store the solution in case & data files Open the new case (new grid) with the old data file Setup the problem Solution of ( ) duct slapped on ( ) 3-decade-fall6-decade-fall Without continuation With continuation Percentage time saving 70%30% Continuation Method Generate new case file FLUENT Solution Duct Parameters Old Data file Journal file

SAROD Simulation Time Strategies Continuation Method Parallel execution of FLUENT on a 4-noded Linux cluster Time for simulation has been reduced to around 20%.

SAROD Sequential (Multipoint) Response Surface Approximations

SAROD Sequential (multipoint) Response Surface Methodology Response Surfaces generated in sub-domains around multiple points Surfaces used to march to optimum

SAROD Wing aerodynamic design problem Planform fixed 2 spanwise stations 4 variables for camber 3 variables for geometric pre- twist Maximize cruise L/D Lift constraint

SAROD Design Problem Statement Maximize L/D Sub. to C L =    r +  m  5  -5    r +  m +  t  5  with side constraints,.05  x 1 .33;.001  h 1 .1.05  x 2 .33;.001  h 2 .1 -2    r  5  -2    m  5  -2    t  5 

SAROD Design Tools Lift Calculation: C L from VLM Drag Calculation: C D0 from a/c data C Di from VLM DOE: Design Expert D-optimality Criterion Response Surfaces: Design Expert quadratic/cubic Optimizers : FFSQP

SAROD Overall Design Procedure

SAROD Results - Arbitrary Starting Point 1

SAROD Results - Arbitrary Starting Point 2

SAROD Observations Quadratic model found better than cubic model in subspaces. Global model inadequate. Cost of D-optimality significant SRSA seems to work well!

SAROD GRADIENT INFORMATION BY AUTOMATIC DIFFERENTIATION OF CFD CODES

SAROD User Supplied Analytical Gradients Analysis Code in Fortran Manually extract sequence of mathematical operations Code the complex derivative evaluator in Fortran Manually differentiate mathematical functions - chain rule FORTRAN source code that can evaluate gradients

SAROD Automatic Differentiation for Analytical Gradients Automatically parse and extract the sequence of mathematical operations Use symbolic math packages to automate derivative evaluation Automatically code the complex derivative evaluator in Fortran Analysis Code in FORTARN FORTRAN source code that can evaluate gradients

SAROD Automatic Differentiation for Analytical Gradients Complex Analysis Code in FORTARN FORTRAN source code that can evaluate gradients Automated Differentiation Package eg. ADIFOR & ADIC Euler

SAROD  d(L/D) / d  using ADIFOR d(L/D) / d  using Finite Difference  =0.2 Value % Error  =0.02 Value % Error  =0.002 Value % Error  = Value % Error Comparison of Derivative Calculation Finite Difference vs ADIFOR

SAROD Optimization - ADIFOR vs FD Single design variable unconstrained optimization problem Find  for max. L/D for Onera M6 wing Same starting point; FD step size  init  opt L/D opt CallsTime (min.) ADIFOR FD

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SAROD Problem Statement Ambient conditions: 11Km altitude Inlet Boundary Conditions Total Pressure: Pa Total Temperature: 261.4o K Hydraulic Diameter: 0.394m Turbulence Intensity: 5% Outlet Boundary Conditions Static Pressure: Pa (Calculated for the desired mass flow rate) Hydraulic Diameter: m Turbulence Intensity: 5%

SAROD Duct Parameterization Geometry of the duct is derived from the Mean Flow Line (MFL) MFL is the line joining centroids of cross- sections along the duct Any cross-section along length of the duct is normal to MFL Cross-section area is varied parametrically Cross-section shape in merging area is same as the exit section

SAROD MFL Design Variables - 1 Mean flow line (MFL) is considered as a piecewise cubic curve along the length of the duct between the entry section and merging section x y(x), z(x) 0LmLm L m /2 y(L m /2), z(L m /2) specified C entry C merge r y 1, z 1 y 2, z 2 L m : x-distance between the entry and merger section y 1, y 2, z 1, z 2 : cubic polynomials for y(x) and z(x)

SAROD MFL Design Variables - 2 y 1 (x) = A 0 + A 1 x + A 2 x 2 + A 3 x 3, y 2 (x) = B 0 + B 1 x + B 2 x 2 + B 3 x 3 z 1 (x) = C 0 + C 1 x + C 2 x 2 + C 3 x 3, z 2 (x) = D 0 + D 1 x + D 2 x 2 + D 3 x 3 y 1 (L m ) = y 2 (L m ), y 1 ’ (L m ) = y 2 ’ (L m ), y 1 ” (L m ) = y 2 ” (L m ) z 1 (L m ) = z 2 (L m ), z 1 ’ (L m ) = z 2 ’ (L m ), z 1 ” (L m ) = z 2 ” (L m ) y 1 ’ (C entry ) = y 2 ’ (C merger ) = z 1 ’ (C entry ) = z 2 ’ (C merger ) = 0 The shape of the MFL is controlled by 2 parameters which control the y and z coordinate of centroid at L m /2 y(L m /2) = y(0) + (y(L) – y(0)) α y 0 < α y < 1 z(L m /2) = z(0) + (z(L) – z(0)) α z 0 < α z < 1

SAROD Area Design Variables – 1 Cross-section area at any station is interpolated from the entry and exit cross- sections A(x) = A(0) + (A(L m ) – A(0)) * β(x) corresponding points on entry and exit sections are linearly interpolated to obtain the shape of the intermediate sections and scaled appropriately P section = P entry + (P exit - P entry ) * β

SAROD Area Design Variables - 2 A 0 + A 1 x + A 2 x 2 + A 3 x 3 0  β < β 1 B 0 + B 1 x + B 2 x 2 + B 3 x 3 β 1  β  β 2 C 0 + C 1 x + C 2 x 2 + C 3 x 3 β 2 < β  1 β = x β(x) 0 LmLm L m / L m /3 β1β1 β2β2 β(L m /3) and β(2L m /3) is specified β variation is given by piecewise cubic curve as function of x

SAROD Turbulence Modeling Relevance: Time per Solution Following aspects of the flow were of interest: Boundary layer development Flow Separation (if any) Turbulence Development Literature Survey S-shaped duct Circular cross-section Doyle Knight, Smith, Harloff, Loeffer  Baldwin-Lomax model (Algebraic model) Computationally inexpensive than more sophisticated models Known to give non-accurate results for boundary layer separation etc. Devaki Ravi Kumar & Sujata Bandyopadhyay (FLUENT Inc.)  k-  realizable turbulence model Two equation model

SAROD Turbulence Modeling (contd.) Standard k-  model Turbulence Viscosity Ratio exceeding 1,00,000 in 2/3 cells Realizable k-  model Shih et. al. (1994) Cμ is not assumed to be constant A formulation suggested for calculating values of C1 & Cμ Computationally little more expensive than the standard k-  model Total Pressure profile at the exit section (Standard k-  model)

SAROD Results Mass imbalance: 0.17% Energy imbalance:0.06% Total pressure drop:1.42% Various turbulence related quantities of interest at entry and exit sections: EntryExit Turbulent Kinetic Energy (m 2 /s 2 ) Turbulent Viscosity Ratio y + at the cell center of the cells adjacent to boundary throughout the domain is around 18.

SAROD Flow Separation

SAROD Flow Separation