# Presentation: 180030

## Aerodynamic Design Using The Euler and Navier-Stokes Equations

***Juan J. Alonso***
***Department of Aero & Astro***
***Stanford University***
***Sandia-Livermore***
***CSRI Seminar Series***
***May 15, 2001***

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## Outline

***Motivation & background***
***Aerodynamic shape optimization problems***
***Sensitivity / gradient computation for optimization***
*Adjoint-based sensitivity calculation*
*Complex-step formulation*
*Reduced-order models based on Proper Orthogonal Decomposition*
***Sample results and other applications***
***Conclusions***

## Disclaimer

***Although Stanford University is heavily involved in ASCI as an Alliance member, this talk describes some of the work we are doing in our laboratory outside of ASCI***
***Our ASCI project involves the prediction of the complex unsteady flow physics inside jet engines.  Our work within the project involves:***
*Advanced algorithms for unsteady flow integration*
*Efficient nonlinear time-harmonic algorithms for unsteady periodic flow simulations*
*Advanced turbulence modeling*
*Integrated compressor-combustor-turbine simulations*
*Large-scale high-performance parallel implementations*
*Large-scale visualization using parallel rendering in clusters*

## Motivation

***Our efforts at the Aerospace Computing Laboratory are aimed at developing the necessary ******tools****** to facilitate the ******multidisciplinary****** design of aerospace ******systems****** (aircraft, spacecraft, engines, distributed systems)***
***In particular, for aircraft that spend most of their time in the cruise condition, the design of aerodynamic shapes with optimum performance is an absolute necessity to meet requirements of:***
*Range*
*Cruise Mach number*
*Payload / Passengers*
*Takeoff / Landing field length*

## Traditional Aircraft Configurations

## Traditional Design

***For ******traditional aircraft****** configurations where a large experience database has been accumulated over the years, design is evolutionary: a skillful designer can obtain close to optimal solutions through ******trial-and-error***
***Although substantial improvements have been made ******under-the-hood******, the fact remains that aerodynamic configurations for transonic commercial transports have remained virtually unchanged since the B-707***

## B707 vs. A340

***Seemingly identical aircraft***
***Improvements include***
*Active control*
*New Airfoil Concepts*
*Composite structures*
*Propulsion*
*MDO*
***Large economic risks introduce conservatism into new designs***

## Revolutionary Design

***Major advances in aircraft performance can still be obtained with a combination of revolutionary concepts that combine advanced technologies onto a given airframe***
***The field of Multidisciplinary Design Optimization (MDO) where synergism between the participating disciplines in a design is exploited is necessary to obtain dramatic improvements in performance***

## Revolutionary Designs

## Revolutionary Design

***This kind of aircraft has to be designed without the benefit of years of experience.***
***What are the realistic chances that a designer may obtain an optimal design though trial-and-error?  My guess is as good as yours....***
***Automatic ******optimization****** techniques (analysis tools cleverly coupled to constrained nonlinear optimizers) become necessary***
***How can we do this ******efficiently******?***

## Vision

***Effective simulation***

***Simulation-based design***

## Automatic Optimization-Based Design

***The requirements to automate the design process are***
*Efficient, accurate, robust aerodynamic analysis*
*Low cost sensitivity analysis for large design spaces*
*Efficient nonlinear constrained optimization techniques*
***For aerodynamic design problems, the number of design variables is often very large (N ~ 500 - 1,000) and the computation of sensitivities of cost functions and constraints (needed for gradient-based optimizers) typically dominates the CPU time required***
***Must develop ******efficient algorithms for sensitivity calculation****** with large numbers of design variables***

## Automatic Design Based on Control Theory

***One such method is the control theory-based ******adjoint method***
***A ******wing****** is a device to generate lift (with minimum drag) by ******controlling****** the flow***
***Apply theory of ******optimal control****** of systems governed by PDEs (Lions) with boundary control (the wing shape)***
***Merge ******control theory****** and ******CFD***

## Design for Specified Pressure Distribution

***Engineers know desirable characteristics of the surface pressure distribution, say, to prevent boundary layer separation, or delay transition to turbulence***

***Specify target pressure***
***However, not every target pressure is attainable (No body with stagnation pressure everywhere).***
***An optimal control-based method should still work***

## Control Theory Approach to the Design Problem

***Define a cost function, for example***

***or***

***The surface shape is now treated as the control, which is to be varied to minimize *****I*****, subject to the constraint that the flow equations are satisfied in the domain *****D**

## Automatic Shape Design Via Control Theory

***Apply the theory of ******control****** of ******partial differential****** ******equations****** (of the flow) by ******boundary control****** (the shape)***
***Find the ******Frechet derivative****** (infinite dimensional gradient) of the drag (or other performance measure) with respect to the shape by solving the ******adjoint****** equation in addition to the ******flow****** equation***
***Modify the shape in the sense defined by the smoothed gradient***
***Repeat the iterations until the performance value approaches an optimum***

## Formulation of the Control Problem

***In the presence of constraints project g into the admissible trial space.  Accelerate by use of smoothing and Quasi-Newton / Krylov subspace methods***

## Traditional Approach to Optimization

## Traditional Approach to Optimization

## Adjoint-Based Design

## Adjoint-Based Design

## Adjoint-Based Design

## Adjoint-Based Design

***Main advantages:***
*The cost of a **complete gradient** evaluation, **independently** of the number of design variables, can be obtained with just **one flow** solution and **one adjoint** solution*
*Adjoint-based gradients require lower levels of convergence in the flow solver than the finite-difference method*
***Disadvantages***
*An adjoint equation must be developed for each new governing equation set*
*A new boundary condition must be implemented for different cost functions*

## Other Options For Sensitivity Analysis

## Finite-Difference Approximations

## Complex-Step Method

## Complex-Step Method

***Although the cost is still linearly dependent on the number of design variables (with a proportionality constant), the gradient value is step insensitive***
***Great advantage with respect to finite-difference methods in terms of robustness due to step insensitivity***
***We have developed automatic procedures to “complexify” existing FORTRAN, C, and C++ codes***
***Very useful for validation and debugging of more complex analytic methods***

## Proper Orthogonal Decomposition

## Proper Orthogonal Decomposition

## Proper Orthogonal Decomposition

## QSP Program Objectives

***To design a supersonic small aircraft with low enough sonic boom that it can be allowed to fly supersonically over land***
*Cruise Mach number = 2.4*
*TOGW ~ 100,000 lbs*
*Range = 6,000 nmi*
*Maximum overpressure = 0.3 psf*
***Can this be achieved?  Extremely difficult proposition; may lead to relaxed design requirements***

**NLF Concept – Tracy, Kroo**

## Conflicting Program Requirements

***Low boom vs. aerodynamic performance for range and payload***
***Low weight vs. range and payload requirements***
***High Mach number vs. boom and aero requirements***
***Overall, the ******most demanding****** requirement is the ******low boom****** requirement which has been somewhat arbitrarily set based on***
*Early 1990’s HSR acceptability studies (NASA Langley)*
*Gulfstream marketing analysis*

## Fundamentals of Sonic Boom Theory

***Sonic boom on ground has significant lateral influence (typically 25 miles for small aircraft at 30,000 ft)***
***Real atmosphere effects create a region of silence earlier than uniform atmosphere results because of ray refraction due to changes of density with altitude***

## Other Proposed Concepts

***Large number of concepts proposed, but none of them has gone through a detailed optimization process even with simplified linear techniques***
***Unfortunately, for sonic boom, the devil is in the details...***

## Ames HSCT - AIRPLANE

## Ames HSCT - SYN107-MB

## Advanced Algorithms for QSP Design

*** ******MDO: Additional disciplines of critical importance***
*Propulsion*
*Structure/Weights*
*Mission/Trajectory*

- High transonic regime
- Sizes engines, structure

- Boom/wave drag tailoring
- at start cruise

*Multi-point optimization includes boom, cruise performance, off-design constraints*

- Subsonic performance
- key to noise acceptability

## Two Dimensional Sonic Boom Minimization 

***using the Adjoint Method***

- Euler Solver

- Adjoint Solver

- Calculate Gradient

- Modify Grid

***FLO83 flow solver***
*Modified Runge-Kutta time stepping scheme*
*Jameson-Schmidt-Turkel (JST) scheme for artificial dissipation*
*Local time stepping, implicit residual smoothing, multigrid*
***Adjoint method***
*Development of adjoint method for calculation of non-collocated sensitivities*
***Cost function***
*Norm of the pressure difference between current and target pressure distributions*

## Two Dimensional Sonic Boom Minimization 

***using the Adjoint Method***

-   **Flow Condition - Mach Number = 1.8**
-   **Test Cases**
-   _**Case A**_ : Unique Near Field Target Pressure
- A bump is placed on the lower surface of the biconvex airfoil. The near field pressure distribution of the new airfoil is then used as the target pressure distribution for the design of the original airfoil
-    _**Case B**_ : Smoothed Near Field Target Pressure
- Near field target pressure obtained by implicitly smoothing the existing pressure.

## Two Dimensional Sonic Boom Minimization 

***using the Adjoint Method***

**  ****Airfoil Geometry and Mesh Size**
-  Biconvex Airfoil with 5% Thickness Ratio.
-   Fine Mesh Size - 256 by 96 C-mesh

- Near Field Plane

- Location of Target Pressure
- Approximately 6 chord lengths

## Geometry and Near Field Plane Description

- Near Field Plane

-     Target Pressure
-     Initial Pressure

- 6 Chord Lengths

- Biconvex Airfoil

- Fine Mesh 256 by 96 C-mesh
- Mach Number = 1.8

## Initial Near Field Pressure Contours

-     Initial Pressure

-     Target Pressure

- Initial and Target Airfoil Geometry

- Initial and Target Near Field Pressure Distribution

- Pressure Contour Plot

- Near Field Pressure Contour Plot

***Initial Near Field Pressure Contours***
***and Airfoil Geometry - ******(Case A,   Mach Number = 1.8,  Biconvex Airfoil)***

## Pressure Contours and Airfoil Geometry After

-        Target Pressure
-        Pressure after 3
-         Design Iterations

- Initial, Current, and Target Airfoil Geometry

- Initial, Current, and Target Near Field Pressure Distribution

- Pressure Contour Plot

- Near Field Pressure Contour Plot

***Pressure Contours and Airfoil Geometry After***
***3 Design Iterations - ******(Case A,   Mach Number = 1.8,   Biconvex Airfoil)***

## Pressure Contours and  Airfoil Geometry After

- Initial, Current, and Target Airfoil Geometry

- Initial, Current, and Target Near Field Pressure Distribution

- Pressure Contour Plot

- Near Field Pressure Contour Plot

***Pressure Contours and  Airfoil Geometry After***
***20 Design Iterations - ******(Case A,   Mach Number = 1.8,   Biconvex Airfoil)***

## Initial Near Field Pressure Contours and

-         Initial Airfoil

-     Initial Pressure

-     Target Pressure

- Initial and Target Airfoil Geometry

- Initial and Target Near Field Pressure Distribution

- Pressure Contour Plot

- Near Field Pressure Contour Plot

***Initial Near Field Pressure Contours and***
***Airfoil Geometry - ******(Case B,   Mach Number = 1.8,  Biconvex Airfoil)***

## Pressure Contours and Airfoil Geometry After

-       Current Airfoil

-       Initial Airfoil

-       Target Pressure

-        Initial Pressure
-        Pressure after 3
-         Design Iterations

- Initial, Current, and Target Airfoil Geometry

- Initial, Current, and Target Near Field Pressure Distribution

- Pressure Contour Plot

- Near Field Pressure Contour Plot

***Pressure Contours and Airfoil Geometry After***
***3 Design Iterations  - ******(Case B,   Mach Number = 1.8,   Biconvex Airfoil)***

## Pressure Contours and  Airfoil Geometry After

-       Current Airfoil

-       Initial Airfoil

-       Target Pressure

-        Initial Pressure
-        Pressure after 20
-         Design Iterations

- Initial, Current, and Target Airfoil Geometry

- Initial, Current, and Target Near Field Pressure Distribution

- Near Field Pressure Contour Plot

***Pressure Contours and  Airfoil Geometry After***
***20 Design Iterations - ******(Case B,   Mach Number = 1.8,   Biconvex Airfoil)***

- 30% dp/p Reduction

- Pressure Contour illustrates the
- Reduction in Shock Strength on
- Lower Surface

## Application of Viscous Adjoint Design

***Proposed design for a Reno Air Racer***
***Constraint: piston engine + propeller***
***Currently won by WWII aircraft (Bearcat, P51, Hawker SeaFury extensively modified)***
***Typical top speed = 500 mph***

## Application of Viscous Adjoint Design

***Current Design: 550mph***
***“******Cruise” Mach number = 0.78***
***Large airfoil thickness and low sweep***
***C_L = 0.27 (for a 3g turn)***
***Aero design carried out by small team***

## 3D Views

## Design Procedure

***Initial multipoint Euler design (at a variety of C_Ls)***
*Strong normal shock in wing inboard region indicated a need to modify the fuselage-wing intersection*
*Extended fuselage eliminated problems in inboard wing*
***Single point full viscous design using the full NS adjoint***
***Pressure distribution cleanup using NS inverse design***
*Favorable pressure gradient to promote laminar flow run*
***Result: Shock-free pressure distributions with very good off-design behavior***

## Resulting Pressure Distributions

## Resulting Pressure Distributions

## Resulting Pressure Distributions

## Resulting Pressure Distributions

## Complex Configuration Design

***Multiblock version of adjoint-based NS design exists***
***Extensive development required for an industrial strength design method***
*Rewrite of code is already under way*
***Ability to redesign arbitrarily complex configurations including wing, fuselage, pylons, nacelles, diverters (SST)***
***Grid generation is the major bottleneck***
***Parallel implementation is absolutely necessary to reduce wall-clock time ***

## Complex-Step Method ForAero-Structural Design

## Complex-Step Method Accuracy

## Complex-Step MethodFull Gradient

## Complex-Step MethodNatural Laminar Flow Aircraft

## Complex-Step Method

## Proper Orthogonal Decomposition

## Proper Orthogonal Decomposition

## Conclusions

***There are many methods to compute sensitivities to systems governed by the Euler and Navier-Stokes equations, but the adjoint-method is the most computationally efficient by far***
***Other approaches can be used for small dimension design problems or to validate more sophisticated efforts***
***Much work remains to be done in the following areas***
*Coupled system sensitivity analysis (especially if the bandwidth of communication is large such as in aero-structural design)*
*Robustness and accuracy of sensitivity calculation procedures*
*Applications to innovative design concepts*

## Current and Future Work

***Low sonic boom aircraft***
***MDO and aero-structural design***
***Reduced order modeling***
***Extensions to unsteady flows (aeroelastics and turbomachinery)***
***POD for bandwidth reduction in complex multidisciplinary designs***

## Unsteady Flow Simulation of PW6000 Turbine (TFLO)

***Unsteady Flow Simulation of ******PW6000 Turbine (TFLO)***

**Relative Mach Number**

**HPT (1,2,3)**

**LPT (5,6,7)**