Essential Components for High-Fidelity AerodynamicAnalysis and Design

Chongam Kim∗

Abstract

The present talk is composed of two parts. In the first part, essential modules for high-fidelity aerodynamic analysis are discussed. In order to resolve high speed flow physicsaccurately, several high-fidelity shock-stable schemes such as RoeM,AUSMPW+ are pre-sented. So as to achieve high-order spatial accuracy and to incorporate multi-dimensionaleffect, MLP(Multi-dimensional Limiting Process) is also presented. Exploitingthese nu-merical schemes, some applications for two- and three-dimensional internal/external flowanalyses are carried out with various grid systems which can enable the treatment of com-plex geometries.

In the second part of the talk, key components for ASO(Aerodynamic Shape Optimiza-tion) of complex geometries based on GBOM(Gradient Based Optimization Method) arediscussed. Sensitivity analysis is performed by a discrete adjoint approach for turbulentflows to save the computational time cost of design process. In addition, flexible grid de-formation functions using NURBS function are discussed. In order to avoid the problemthat solutions of GBOM are often trapped in local optimum, some remedy by combiningGBOM with global optimum strategy such as surrogate models and GA(Genetic Algo-rithm) is investigated. Lastly, some two- and three-dimensional examples for ASO worksbased on the proposed design methodology are presented.Keywords: RoeM, AUSMPW+, MLP, Adjoint Approach, Overset Mesh, NURBS, ASO

1 Introduction

In general, high-speed flows over complex aircraft geometries show highly non-linear andmulti-dimensional characteristics. Flow analysis and aerodynamic design optimization tech-niques are to be devised or selected by considering these peculiarities carefully. In the presentpaper, ASO(Aerodynamic Shape Optimization) techniques based on high-fidelity flow analy-sis are discussed. Although gradient-based design methodshave difficulties in dealing withhighly non-linear objective functions and potential dangers to be trapped in local optimum,GBOM(Gradient-Based Optimization Method) is still very popular because it is very efficient

∗Associate Professor, School of Mechanical and Aerospace Eng., Seoul National Univ., chongam@snu.ac.kr

1

to find an optimal shape and it can be readily combined within the MDO framework. Typi-cal GBOM consists of the four essential elements: flow solver,sensitivity analysis code, gridgenerator(or grid modifier), and optimization algorithm[1].

Among the four elements, an accurate and efficient flow solveris the first concern becauseit provides pressure distribution and aerodynamic loads such as lift or drag which are intrinsicingredients of the design objective functions. In order to obtain accurate flow solutions effi-ciently, high-fidelity numerical schemes to resolve complex flow phenomena are investigated.In the present study, several high-fidelity shock-stable schemes such as RoeM and AUSMPW+are reviewed[2, 3]. RoeM scheme[2], based on Roe’s flux difference splitting(FDS), is a shock-stable scheme without any tunable parameters while maintaining the accuracy of the originalRoe scheme. AUSMPW+ scheme[3] is an improved version of AUSMPWscheme. By theuse of pressure based weighting functions, AUSMPW+ can reflect both properties of a cellinterface adequately. As a result, oscillations and overshoots behind shocks and near a wall,which are typical symptoms of AUSM-type schemes, are successfully eliminated. Both RoeMand AUSMPW+ schemes are among recently developed advanced schemes for the gas dynam-ics. In addition, to achieve high-order spatial accuracy and to incorporate multi-dimensionaleffects, MLP(Multi-dimensional Limiting Process) are discussed by examining two- and three-dimensional setting[4, 5].

The sensitivity analysis methods commonly used can be summarized by four techniques,i.e., FDM(Finite Difference Method)[6], DD(Direct differentiation)[7], Complex Step Deriva-tive [8], and continuous/discrete adjoint approaches[6, 9]. Among the sensitivity analysis meth-ods, the adjoint approaches are very popular in cases that the design problems require manydesign variables, because the computational time cost is almost independent of the number ofdesign variables. In the present work, sensitivity analysis by discrete adjoint approach is ex-amined on various mesh systems. The importance of the differentiation of turbulent transportequations is investigated by the feasibility study of CTEV(Constant Turbulent Eddy Viscosity)assumption for high-Reynolds number applications. Furthermore, the extension of the adjointapproach to various grid systems such as multi-block for internal flow applications and oversetmesh system for multiple-body external problems is discussed.

For the geometric modification during the design process, various shape functions havebeen used. Most of all, smoothness in shape change and a flexible DOF(degree of freedom) indesign space are top priorities in choosing a shape function. For these reasons, a NURBS(Non-Uniform Rational B-Spline) equation is employed as a new shapefunction and the controlpoints of NURBS surface are used as design variables. NURBS can maintain grid smoothnesssince NURBS equations can preserve a certain level of higher order derivatives at each knot ofsurface. Hence, gradient smoothing which affects the accuracy of the sensitivity is no longernecessary.

Regarding the optimization technique, in order to avoid the problem that solutions of GBOMare often trapped in local optimum in cases that the design space is highly non-linear, someremedy by combining GBOM with global optimum strategy such assurrogate models andGA(Genetic Algorithm) is investigated.

Exploiting all the elements for the high-fidelity flow analyses and ASO, various two- and

2

three-dimensional flow analyses for high-speed flow and two-phase flow are presented. More-over, the ASO results such as high-lift design, wing and wing/body design problems on oversetmesh system are presented.

2 Numerical Schemes for Flow Analysis

For the flow analysis, a numerical representation of inviscid fluxes, namely a numerical fluxfunction at a cell-interface, should guarantee a high levelof accuracy, efficiency, and robust-ness. Today, upwind-based schemes are the main trend of spatial discretization, which may becategorized as either FDS(flux difference splitting) or FVS(flux vector splitting). FDS schemesare generally based on the idea due to Godunov and the Riemann problem is utilized locally.Many researchers have tried to simplify the step of numerical flux calculation, which leadsto the family of Godunov-type schemes or approximate Riemannsolvers, such as Roe’s FDS,HLLEM, Osher’s FDS, and etc. FVS, such as Steger-Warming’s or Van Leer’s, has advantagesin view of robustness and efficiency. However, it is also wellknown that these schemes haveaccuracy problems in resolving shear layer regions due to excessive numerical dissipation. Atthe same time, the AUSM-type schemes(advection upstream splitting method) was proposed byLiou et al., which turn out to be advantageous in several aspects.

In efforts to improve the accuracy and robustness for previous numerical flux schemes, sev-eral high-fidelity flux schemes such as RoeM[2], AUSMPW+[3] arepresented. So as to achievethe high-order spatial accuracy and to incorporate the multi-dimensional effect, MLP(Multi-dimensional Limiting Process)[4, 5] is also presented. Exploiting these numerical schemes,many applications for two- and three-dimensional internal/external flow analyses are carriedout with various grid systems which can enable the treatmentof complex geometries.

2.1 RoeM

RoeM[2] scheme is a newly improved Roe scheme that is free from the shock instability andstill preserves the accuracy and efficiency of the original Roe’s Flux Difference Splitting(FDS).Roe’s FDS is known to possess good accuracy but to suffer from the shock instability, suchas the carbuncle phenomenon. As the first step towards a shock-stable scheme, Roe’s FDS iscompared with the HLLE scheme to identify the source of the shock instability. Through alinear perturbation analysis on the odd-even decoupling problem, damping characteristic is ex-amined and Mach number-based functionsf andg are then introduced to balance damping andfeeding rates in the numerical mass flux, which leads to a shock-stable Roe scheme. In orderto satisfy the conservation of total enthalpy, which is crucial in predicting surface heat transferrate in high-speed steady flows, an analysis of dissipation mechanism in the energy equationis carried out to find out the error source and to make the proposed scheme preserve the totalenthalpy. By modifying the maximum-minimum wave speed, expansion shock and numericalinstability in expansion region is also remedied without sacrificing the exact capturing of con-tact discontinuity. From these analyses, the newly formulated RoeM scheme(Roe scheme withMach number-based function) are proposed as follows:

3

Fj+1/2 =b1 × Fj − b2 × Fj+1

b1 − b2

+b1 × b2

b1 − b2

△Q∗ − gb1 × b2

b1 − b2

×1

1 + |M |B△Q (1)

△Q∗ =

ρρuρvρH

, B△Q =

(

△ρ − f△p

c2

)

1uv

H

+ ρ

0△u − nx△U△v − ny△U

△H

(2)

b1 = max(0, U + c, Uj+1 + c), b2 = min(0, U − c, Uj − c) (3)

where

f =

{

1, u2 + v2 = 0

|M |h, elsewhere(4)

h = 1 − min(Pi,j+(1/2), Pi−(1/2),j , Pi−(1/2),j+1, Pi+(1/2),j+1) (5)

Pi,j+(1/2) = min

(

pi,j

pi+1,j

,pi,j+1

pi,j

)

(6)

and

g =

{

|M |1−min(

pjpj+1

,pj+1

pj), M2 6= 0

1, M2 = 0(7)

2.2 AUSMPW+

Typical symptoms appearing in the application of AUSM-typeschemes in high-speed flows,such as pressure wiggles near a wall and overshoots across a strong shock, can be cured by intro-ducing weighting functions based on pressure (AUSMPW[10]).The main feature of AUSMPWis the removal of the oscillations of AUSM+ near a wall or across a strong shock by introducingpressure-based weight functions. AUSMPW uses the pressure-based weight functionf to treatthe oscillations near a wall andw to remove the oscillation across a strong shock. The startingpoint of AUSMPW is to observe the fact that AUSM+ and AUSMD arecomplementary to eachother. AUSM+ considers the left cell density only while AUSMD takes both cell densities. Thisis thought to be the reason for the numerical oscillations ofAUSM+ and carbuncle phenomenaof AUSMD. Thus, by incorporating the property of the right cell pR, numerical oscillationsnear a wall can be eliminated. A newly improved and refined version of the AUSMPW scheme,called AUSMPW+[3], is developed to increase the accuracy andcomputational efficiency ofAUSMPW in capturing an oblique shock without compromising robustness. With a new defini-tion of the numerical speed of sound at a cell interface, capturing an oblique shock is remarkablyenhanced, and it can be proved that an unphysical expansion shock is completely excluded. TheAUSMPW+ flux at a cell-interface can be summarized as

F 1

2

= M+L c 1

2

ΦL + M−

R c 1

2

ΦR + (P+L |α= 3

16

PL + P−

R |α= 3

16

PR) (8)

4

where(i) for m1/2 ≥ 0

M+L = M+

L + M−

R · [(1 − w) · (1 + fR) − fL]M−

R = M−

R · w · (1 + fR)(ii) for m1/2 ≤ 0

M+L = M+

L · w · (1 + fL)M−

R = M+R + M+

L · [(1 − w) · (1 + fL) − fR]

(9)

with w(pL, pR) = 1 − min(

pL

pR, pR

pL

3)

. fL,R is simplified to

fL,R =

{(

pL,R

ps− 1

)

min(

1,min(p1,L,p1,R,p2,L,p2,R)

min(pL,pR)

)2

, ps 6= 0

0, elsewhere(10)

whereps = P+L pL + P−

R pR. The Mach number and pressure splitting functions of AUSMPW+at a cell interface are as follows.

M± =

(

±14(M ± 1)2, |M | ≤ 1

12(M ± |M |), |M | > 1

(11)

P±|α =

(

±14(M ± 1)2(2 ∓ M) ± αM(M2 − 1)2, |M | ≤ 1

12(1 ± sign(M)), |M | > 1

(12)

2.3 Multi-dimensional Limiting Process (MLP)

Since the late 1970s, numerous ways to control oscillationshave been studied and several limit-ing concepts have been proposed. Most representatives would be TVD, TVB and ENO/WENO.However, most oscillation-free schemes have been based on the mathematical analysis of one-dimensional convection equation and applied to multi-dimensional applications with dimen-sional splitting. Although they may work successfully in many cases, it is insufficient or al-most impossible to control oscillations near shock discontinuity in multiple space dimensions.Thus oscillation control method for multi-dimensional flowphysics is crucial. By suitablyextending the one-dimensional monotonic condition to two-and three-dimensional flows, themulti-dimensional limiting condition can be formulated and, with this limiting condition, amulti-dimensional limiting process (MLP)[4, 5] is proposed.

The starting point is the observation that the dimensional splitting extension does not possessany information on property distribution at cell vertex points, which is essential when propertygradient is not aligned with grid lines. In order to derive the multi-dimensional limiting func-tion, the vertex point value is expressed in terms of variations at a cell-interface. And then, wedetermine the variation to satisfy the multi-dimensional limiting condition using the limitingcoefficientα. The coefficientα possesses the information of multi-dimensionally distributedphysical property. With the coefficientα, we can formulate the multi-dimensional limitingfunction. And, with the multi-dimensional limiting function, a new family of limiting pro-cess to control oscillations in multi-dimensional flows canbe developed. For three-dimensional

5

Figure 1: Comparison of density distributions for shock wave-vortex interaction.

flows,

ΦLi+1/2,j,k = Φi,j,k + 0.5φ(rξ

L,i,j,k, αL, βL)△Φi−1/2,j,k (13)

= Φi,j,k + 0.5max(0,min(αL, αLrξL,i,j,k, βL))△Φi−1/2,j,k

ΦRi+1/2,j,k = Φi,j,k − 0.5φ(rξ

R,i,j,k, αR, βR)△Φi+3/2,j,k

= Φi,j,k − 0.5max(0,min(αR, αRrξR,i,j,k, βR))△Φi+3/2,j,k

whereα is the multi-dimensional restriction coefficient which determines the baseline regionof MLP andβ is the local slope evaluated by a higher order polynomial interpolation. Theinterpolated values ofΦL

i+1/2,j,k andΦRi+1/2,j,k are based on the final form of MLP. Since the

calculations of interpolated values are independent of a numerical flux, MLP can be combinedwith any numerical scheme. Values ofαL,R andβL,R in Eqn. 13 can be summarized as follows.Along theξ-direction, if△Φp

ξ ≥ 0,

αL = g

2max(1, rξL,i,j,k)(Φ

maxp,q,r − Φi,j,k)

(

1 +△Φq

η

△Φpξ

+△Φr

ξ

△Φpξ

)

i,j,k△Φi+1/2,j,k

, αR = g

2max(1, rξR,i,j,k)(Φ

minp,q,r − Φi,j,k)

(

1 +△Φq

η

△Φpξ

+△Φr

ξ

△Φpξ

)

i+1,j,k△Φi+3/2,j,k

(14)whererξ

L,i,j,k =△Φi+1/2,j,k

△Φi−1/2,j,k, rξ

R,i,j,k =△Φi+1/2,j,k

△Φi+3/2,j,kandg(x) = max(1,min(2, x)). Along theη

- andξ -direction, the left and right values at the cell-interfacecan be calculated in the sameway. With β in the form of a third order polynomial and a fifth order polynomial, we finallyobtain MLP3 and MLP5, respectively. For detailed explanation, see Ref. [4, 5]. Figure 1 showsthe computed results of the interaction of normal shock withvortex. A normal shock with theMach number of 1.29 is propagating into a stationary vortex,producing a complex flow pattern.Compared to conventional limiting, MLP can control oscillations across the shock discontinuityand capture local flow structure in detail.

6

3 Aerodynamic Shape Optimization

3.1 Sensitivity Analysis based on Discrete Adjoint Approach

Discrete adjoint variable method is applied to get sensitivity information by fully hand differen-tiating the three-dimensional Euler and N-S equations. Thesymbolic formulation of the discreteresidual R for the steady-state flow equations can be writtenas

{R} = {R(Q,X,D)} = {0} (15)

whereQ is the flow variable vector,X is the position of computational grid andD is the vectorof design variables.

Without evaluating the vectordQ/dD, the sensitivity derivative of the objective function,F = F (Q,X,D), can be calculated as

{

dF

dD

}

=

{

∂F

∂X

}T {

dX

dD

}

+

{

∂F

∂D

}

+ ΛT

([

∂R

∂X

]{

dX

dD

}

+∂R

∂D

)

(16)

if and only if the adjoint vectorΛ satisfies the following adjoint equation.[

∂R

∂Q

]T

Λ +

{

∂F

∂Q

}

= {0}T (17)

The solution vectorΛ is then obtained by solving Eqn. 17 with the Euler implicit method in atime-iterative manner as

(

I

J∆t+

[

∂R

∂Q

])

∆Λ = −

[

∂R

∂Q

]T

Λm −

{

∂F

∂Q

}

, Λm+1 = Λm + ∆Λ

whereI is the identity matrix, andJ represents the Jacobian matrix, and the subscriptVL meansthe van Leer flux Jacobian.

Adjoint formulation on the overset boundary can be similarly derived by slightly modifyingthe conventional adjoint boundary condition, which can be expressed as

[

∂RM

∂QM

]T

ΛM +

[

∂RSF

∂QM

]T

ΛSF +

{

∂FM

∂QM

}T

= {0}T (18)

[

∂RS

∂QS

]T

ΛS +

[

∂RMF

∂QS

]T

ΛSF +

{

∂FM

∂QM

}T

= {0}T (19)

[

∂RM

∂QMF

]T

ΛM +

[

∂RMF

∂QMF

]T

ΛMF +

{

∂FM

∂QMF

}T

= {0}T (20)

[

∂RS

∂QSF

]T

ΛS +

[

∂RSF

∂QSF

]T

ΛSF +

{

∂F S

∂QSF

}T

= {0}T (21)

where the subscriptF indicates fringe cell. The superscriptM andS represents the main-gridand sub-grid domain, respectively. By solving the four equations sequentially, overset boundaryvalue on the main- and sub-grid can be updated. The update procedure of the adjoint variableson the overset boundary(Eqns. 18, 19, 20 and 21) is reverse tothe conventional overset flowanalysis because of the transposed operation in the adjointformulation.

7

3.2 Geometric Modification

In automatic shape design for three-dimensional geometry,a grid generator which guaranteesa sufficient and flexible design space is extremely important. From this point of view, NURBS(Non-Uniform Rational B-Spline) surface equations are attractive as new shape functions. Thebenefits of NURBS have already been mentioned in the previous section. In this section, thedetermination of control points and NURBS blending functions, grid generation process fromthe evaluated NURBS equations are presented. The coordinate vectors of NURBS curve,X(u),are expressed by

X(u) =

∑ni=0 hiPiNi,k(u)

h(22)

where homogeneous coordinateh represents

h =n

∑

i=0

hiNi,k(u) (23)

andPi = (xi, yi, zi) is a position vector of theith control point in three-dimensional space.Homogeneous coordinate acts as a weighting factor for each control point. As the value ofh increases, the corresponding NURBS curve is closer to the control point. In the presentapproximation, all the homogeneous coordinates are set to1 to impose equal weighting for eachcontrol point. The value ofn indicates the number of control point. Also, blending functionsare defined as

Ni,k(u) =(u − ti)Ni,k−1(u)

ti+k − ti+

(ti+k+1 − u)Ni+1,k−1(u)

ti+k+1 − ti+1

, Ni,0(u) =

{

1 (ti ≤ u ≤ ti+1)0 otherwise

(24)whereti(i = 0, 1, 2, . . .) are knot-values, and subscriptk indicates the order of NURBS blendingfunction[11].

The grid points are approximated by the least-square methodwith NURBS. After this, con-trol points and homogeneous coordinates(weighting factors) can be used as design variables.The grid sensitivity is finally evaluated as in Eqns. 25 and 26. For theith control point,

∂X(u)

∂Pi

=hiNi,k(u)

∑ni=0 hiNi,k(u)

(25)

And, for theith weighting factor,

∂X(u)

∂hi

=Ni,k(u)

[

∑nj=0{(Pi − Pj)hjNj,k(u)}

]

[∑n

i=0 hiNi,k(u)]2 (26)

In case of NURBS surface approximation, the overall procedureis similar to NURBS curveapproximation except it is formulated by two-dimensional NURBS equations.

8

-Cp

0

2

4

6

8

10

12

14

Main : Initial

Flap α = 13.1o

Main : Design

Flap α = 14.17o

Main : Design (µT' = 0)

Flap α = 14.03o

Figure 2: Improvement ofClmax for multi-element airfoil

3.3 Feasibility Study of Constant Turbulent Eddy Viscosity Assumptionfor Sensitivity Analysis

The effects of constant eddy viscosity (CEV, hereafter) assumption on the computational timecost of design process and design result are examined. For a turbulent flow over the NLR 7301with flap at a high angle of attack close to stall angle, all thesensitivity gradients of designvariables computed from the CEV assumption are compared withthose from the variable eddyviscosity (VEV, hereafter) assumption. In addition, sensitivity gradients of an objective functioncomputed from thek-ω SST model are compared with those from thek-ω model and the standardk-ε model to study the effects of turbulence model under the CEV assumption.

The objective function is to maximize the lift coefficient, and angle of attack is also includedas a design variable. The initial and designed surface pressure coefficients are compared in Fig.2. For the design case using the adjoint code with the VEV assumption, the angle of attack ischanged from 13.1 to 14.17 deg., while it is changed to 14.03 deg. in the CEV design case.Figure 3 shows that the lift coefficients is increased from 3.2982 to 3.8399 in the VEV designand to 3.8171 in the CEV design, respectively. However, the VEV design calls the flow solver29 times and the sensitivity analysis code 4 times, whereas the CEV design calls the flow solver43 times and the sensitivity analysis code 7 times. For a faircomparison, the same design con-vergence criterion is implicitly imposed on both the VEV andCEV design. Even though theCEV design requires more design iterations than the VEV design, the designed results show al-most the same configuration in engineering sense. Figure 4 shows an improvement ofClmax byavoiding a massive-separated flow over a high-lift airfoil using the present design optimizationtool(the unsteady N-S solver and sensitivity analysis code).

In high-lift optimization with the VEV assumption for theClmax improvement, there are

9

Design iteration

Obj=Cl

Cd

0 1 2 3 4 5 6 72.8

3.2

3.6

4

0.08

0.12

0.16

0.2

0.24

0.28

Cl

Cd

Obj

Cl

Cd ( µT' = 0 )

Obj

Figure 3: Design progress ofClmax improvement for multi-element airfoil.

(a) Massively separated flow during design

(b) Attached flow after design

Figure 4: Comparison of streamlines over high-lift airfoil with flap.

10

29 flow solver calls and 4 adjoint code calls, which corresponds to total 37 flow solver calls.The high-lift design with the CEV assumption requires about 40 percent more computing timethan the VEV design case. The increase of total computing time in the CEV design is mainlyattributed to the inaccuracy of gradient direction.

3.4 Turbulent Internal Flow Design

Adoint approach which has been very successful in external flow problems does not seem tobe applied to highly viscous internal flow problems yet. Thisis mainly due to the difficulty indifferentiating turbulence transport equations [6]. In addition, awfully time consuming work indeveloping a reliable adjoint code may discourage its application to internal flow design prob-lems. For these reasons, most researches on optimal intake design employs parametric studies,GBOM using finite difference method, or other global optimization method based on the mod-eling of design space. We carry out the design optimization of a subsonic intake based on theadjoint approach. The sensitivity analysis for two-equation turbulence model is performed toallow a large number of design variables and to globally modify the intake(or duct) geometry.The computational cost caused by two adjoint equations for turbulence transport equations isside-stepped by the parallelized adjoint method[13]. As a result, the computational cost for sen-sitivity analysis with the parallelized adjoint method is almost equal to that of the flow solver.

The objective function is to maximize the total pressure recovery with minimizing the lossof other performance measures. Thus the maximization of thevolume-averaged total pressure(VATP) is adopted as a useful objective function. The VATP isdefined by

V ATP =

∫

V

P0dv/

∫

V

dv (27)

where∫

Vimplies the volume integration from(x/D = 0) (D: throat diameter) to the outlet

boundary, andP0 is the total pressure in the duct.Figure 5 shows the geometric change between the baseline model and the designed model.

The duct geometry after design is somewhat beyond expectation, and it appears that this kindof shape change can rarely be obtained from conventional design works. Intuitively, three no-ticeable features can be observed from Fig. 5. Most of all, the first curved region at the lowersurface is stretched after design. As a result, the suction effect at the lower surface becomesweak and the size of flow separation is reduced remarkably. Secondly, the exhaust region ofthe designed intake is changed into an elliptical shape froma circular section. This reduces theintensity of swirling flow over the whole duct region. The last one is a smooth bump along thelower surface near the exhaust region, which stabilizes flowinto the engine face.

The performance of the VATP designed model is examined at several off-design conditions.The performance coefficients of the baseline and designed geometry are compared by changingthe mass flow rate into the engine. For each design case, totalpressure recovery, distortion andmass flow rate are compared under the same free stream conditions and back pressure condition.It is observed that the designed model shows a better performance than the baseline model inall the off-design test cases.

11

Figure 5: Streamline comparison (VATP maximization, Up: baseline, Down: designed)

Through design and off-design condition tests, the presentdesign optimization approachusing an adjoint code and NURBS shape modification tool successfully demonstrates that thedesigned model exhibits a good performance in various flightconditions. Even if it may notguarantee the globally optimal shape, several tests at off-design conditions confirm that thedesigned model still yields very desirable performance over a wide range of flow conditions.

3.5 Extension to Complicated Overset Mesh System

The overset grid technique seems to possess several attractive properties which can be beneficialto large scale flow analysis and design optimization. At first, the grid topology to represent thedeforming grid is relatively simple. Secondly, the movement of the sub-domain grid system orthe movement of a local component such as the movement of engine nacelle along the wingsurface can be readily realized without re-generating grid. Thirdly, a high-quality flow solutioncan be obtained by a relatively small number of grid points. Finally, the fully automatic grid-generation is possible because of the simple grid topology.These characteristics of the oversetmesh technique can derive the overall aerodynamic design optimization process to the finalgoal, i.e.,’fully automatic aerodynamic design from the CAD models’

In the present work, the 7 block overset mesh system of DLR-F4 wing/body configuration,which was the test geometry in the1st Drag Prediction Workshop(DPW-I), is considered[15].The total number of mesh point is about 1.22 million. The freestream design conditions are thefree stream Mach number of 0.75 and the angle of attack of 0.0 deg., which corresponds to thecruising condition.

Optimization is performed using the Broydon-Fletcher-Goldfarb-Shanno(BFGS) variablemetric method which is a kind of non-constrained optimization technique. As a standard appli-cation of the overset GBOM tool, a drag minimization with maintaining constant lift coefficient

12

is firstly performed. The objective function is defined by Eq.(30) with the constraint of Eq.(29). To balance the variation of the penalty function in theobjective function, the weightingfactor of the lift constraint is given by the ratio of the liftsensitivity to the drag sensitivity withrespect to angle of attack.

Minimize : CD (28)

Subject to: CL ≥ CL0, CL0 = (Lift Coefficient of the Baseline Model) (29)

F (Objective Function) = CD + Wt × min[0, CL − CL0]. Wt =∂CD

∂α/∂CL

∂α(30)

The total number of design variable is 200, located along 10 different design sections of thewing surface. At each design section, 20 Hicks-Henne functions are used. Three componentblocks - collar block, wing block, and tipcap block - are overlapped on the wing surface. Thedeformation of the overlap surface meshes is carried out by the mapping technique from thewing-surface domain to the planform domain.

After design, the drag reduction of the wing only is about 17%, which is quite reasonableconsidering the drag portion of the fuselage. TheL/D is changed from 32.26 to 36.25 (12.3%increase). It can be observed in Fig. 6 that the shock strength on the wing surface is substantiallydiminished after design. At the section of 33% wing span as shown in Fig. 7, the front shockon the upper surface almost disappears because the shelvingleading-edge relieves strong ex-pansion. Regarding the rear shock, the relatively mild slopearoundx/c = 75% prevents drasticflow expansion after the front shock. The maximum thickness of the wing section is conservedto maintain the lift constraint. As shown at the sections of 84.4% wing span, the position ofthe maximum thickness along the span-wise direction shiftstoward the trailing edge because ofthe twist angle of the baseline wing. Thus the flow region after the maximum thickness is notsufficient enough to transform the rear shock wave into the region of gradual pressure change.Furthermore, the geometric change of the upper wing surfaceafter the maximum thickness re-gion is very limited because the wing section gets closer to the wing-tip. This suggests thatplanform design should be introduced to obtain more refined design solutions such as a shock-free wing. Especially, leading edge sweepback angle can diminish the shock strength on theupper wing surface. Thus planform design variables such as leading edge sweepback, wingspan, taper ratio and twist angle are added. By combining surface design and planform designefficiently, the two-stage design approach is performed. The first stage design is the planformdesign by using global optimization method based on surrogate model combined with geneticalgorithm. The second stage is the wing surface design through the discrete adjoint approachon overset mesh system. This multi-stage, multi-fidelity design strategy incorporating the dis-crete adjoint approach and the overset mesh system is expected to provide optimal aerodynamicshape for complex three-dimensional configurations.

4 Conclusion

The essential modules for high-fidelity aerodynamic analysis and design optimization are dis-cussed. First of all, RoeM and AUSMPW+ schemes are presented asaccurate, efficient and ro-

13

p

0.8727

0.8318

0.7909

0.7500

0.7091

0.6682

0.6273

0.5864

0.5455

0.5045

0.4636

0.4227

0.3818

0.3409

0.3000

p

0.8727

0.8318

0.7909

0.7500

0.7091

0.6682

0.6273

0.5864

0.5455

0.5045

0.4636

0.4227

0.3818

0.3409

0.3000

(a) Baseline Model (DLR-F4) (b) Designed Model

Figure 6: Comparison of Flow Pattern between Baseline Model and Designed Model (Re-designof DLR-F4 Wing/Body Configuration)

X/C

-CP

0 0.25 0.5 0.75 1-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

DLR-F4 (y/Span = 33.1%)

Designed Model (y/Span = 33.1%)

x/c

y

0 0.25 0.5 0.75 1

-0.175

-0.15

-0.125

-0.1

-0.075

Designed 33.1%

Baseline 33.1%

X/C

-CP

0 0.25 0.5 0.75 1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

DLR-F4 (y/Span = 84.4%)

Designed Model (y/Span = 84.4%)

x/c

y

0 0.25 0.5 0.75 1

0.025

0.05

0.075

Baseline 84.4%

Designed 84.4%

Figure 7: Comparison ofCp Curve and Geometric Change (Re-design of DLR-F4 Wing/BodyConfiguration)

14

bust numerical fluxes. As a multi-dimensional limiting strategy which ensures multi-dimensionalmonotonicity and higher-order spatial accuracy, the basicidea of MLP is presented and its per-formance is examined. For the design optimization based on gradient method, the sensitivityanalysis technique using discrete adjoint approach is discussed. Its accuracy with and withoutdifferentiation of the turbulent transport equations is examined through the CTEV assumption.In addition, the sensitivity analysis technique is extended into the overset mesh system to treatcomplex three-dimensional geometries. B-Spline and NURBS functions are introduced as newshape functions to secure the smoothness and high DOF in geometric modification. For ro-bust design optimization without being trapped in local optimum, multi-stage ASO strategy isemployed by combining GBOM for surface design with surrogatemodels/GA for planform de-sign. Through various two- and three-dimensional applications, the capability of the proposedapproach for high-fidelity aerodynamic analysis and designis demonstrated.

Acknowledgements

The author appreciates the financial support by Agency for Defence Development, the BrainKorea-21 Project for the Mechanical and Aerospace Engineering Research at Seoul NationalUniversity, and the “Smart UAV Development Program” of the ‘21th Frontier R&D Program’sponsored by the Ministry of Commerce, Industry and Energy.

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