Research ArticlePrediction of Dynamic Stability Using Mapped ChebyshevPseudospectral Method

Jae-Young Choi ,1 Dong Kyun Im,2 Jangho Park,1 and Seongim Choi 1

1Aerospace and Ocean Engineering Department, Virginia Tech, Blacksburg, VA 24061, USA2Department of Mechanical Design Engineering, Youngsan University, Yangsan, Republic of Korea

Correspondence should be addressed to Seongim Choi; schoi1@vt.edu

Received 22 March 2018; Accepted 20 June 2018; Published 1 August 2018

Academic Editor: Mahmut Reyhanoglu

Copyright © 2018 Jae-Young Choi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A mapped Chebyshev pseudospectral method is extended to solve three-dimensional unsteady flow problems. As the classicalChebyshev spectral approach can lead to numerical instabilities due to ill conditioning of the spectral matrix, the Chebyshevpoints are evenly redistributed over the domain by an inverse sine mapping function. The mapped Chebyshev pseudospectralmethod can be used as an alternative time-spectral approach that uses a Chebyshev collocation operator to approximate thetime derivative terms in the unsteady flow governing equations, and the method can make general applications to bothnonperiodic and periodic problems. In this study, the mapped Chebyshev pseudospectral method is employed to solve three-dimensional periodic problem to verify the spectral accuracy and computational efficiency with those of the Fourierpseudospectral method and the time-accurate method. The results show a good agreement with both of the Fourierpseudospectral method and the time-accurate method. The flow solutions also demonstrate a good agreement with theexperimental data. Similar to the Fourier pseudospectral method, the mapped Chebyshev pseudospectral method approximatesthe unsteady flow solutions with a precise accuracy at a considerably effective computational cost compared to the conventionaltime-accurate method.

1. Introduction

The conventional time-marching approach and time-spectral method have been widely used to solve varioustypes of unsteady flow problems that are governed by par-tial derivative equations. These methods have been devel-oped to approximate the solutions of PDEs by differentapproaches. To estimate the time derivative term in theunsteady flow governing equations, the time-marchingmethod employs a finite difference method, while time-spectral methods used in this study apply a discrete Fourierexpansion and Chebyshev polynomials.

In this study, two different basis functions are used forthe spectral method in order to solve time-dependent prob-lems. The first kind is based on a frequency domain methodby casting a discrete Fourier expansions. Hall et al. [1] firstproposed an application of the frequency-domain-basedapproach to a time-dependent nonlinear flow problem. The

method utilizes the Fourier spectral method to the temporaldiscretization and approximates the time derivative termsin the unsteady transport equations. The transformationto frequency domain from time-domain turns the unsteadyform of the governing equations into the time-independentequations by treating time derivative term as a source term,and this allows ones to solve unsteady state problems assteady-state manner at multiple time instance points [2].The frequency domain-based time-spectral method candeliver very accurate approximation on periodic unsteadyproblems at an efficient computational cost while it satisfiesNyquist theorem. However, the computational efficiencywill become rather slower with an increasing number ofharmonics without knowledge of frequency informationprior [3].

If this is the case and the problem is nonperiodic, the gov-erning equation can be represented with a series of Cheby-shev polynomials. The Chebyshev polynomials are one class

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 2508153, 14 pageshttps://doi.org/10.1155/2018/2508153

of the orthogonal Lagrangian polynomials, and the mainfeatures of the use of the Chebyshev polynomials are itsgeneral application capability to either periodic or nonper-iodic problems compared to the frequency-based time-spectral method [3]. The classical Chebyshev polynomialscan provide remarkable solution resolution near the bound-aries [3], but this feature of the Chebyshev polynomials isnot favorable for current unsteady problems for its highpotential outcome of the numerical instability. To mitigatesuch numerical instability, an application of conformalmappings [4–9] to the traditional Chebyshev pseudospec-tral method is considered. Im et al. [2] have applied confor-mal mapping techniques to unsteady flow problems, and aninverse sine mapping function is employed for even distri-bution of the time instance points over the range for cur-rent problem.

The mapped Chebyshev pseudospectral method can pro-vide the solution accuracy as fine as that of time-accuratemethod at an effective efficiency in computational cost. Imet al. [10] solved two-dimensional airfoil flows under forcedoscillation by the Euler and Navier-Stokes solvers with themapped Chebyshev pseudospectral method. The resultsobtained very similar characteristics compared to those ofthe Fourier pseudospectral method in terms of both solutionaccuracy and computational efficiency.

In this paper, a mapped Chebyshev pseudospectralmethod is expended to three-dimensional case and employedto approximate solutions of the time-dependent flow pastboth two- and three-dimensional bodies under a forced har-monic oscillation. The results are used to calculate dynamicderivatives of the oscillating bodies. The standard NACA0012 and a tailless aircraft, the Lockheed Martin TacticalAircraft Systems-Innovative Control Effector (ICE) configu-ration, [11–13] are subjected to the unsteady flow simula-tions. The solutions are obtained by computation of Eulerand Navier-Stokes equations using time-accurate methodand time-spectral methods.

The details of Chebyshev pseudospectral method with amapping function and the calculation method of dynamicderivatives are explored in Section 2 and Section 3. Thesolutions from the present method are compared withthose from the time-accurate method, the Fourier pseudos-pectral method, and experimental data [12] to validate theaccuracy of the present method in Section 4. Finally, dynamicderivatives are obtained through investigations of theunsteady flow solutions from each of computational methodsin Section 5.

2. Time-Spectral Methods

2.1. Fourier Pseudospectral Method. Frequency domainapproach can provide very accurate solutions for the time-dependent transport equations of conservation of mass,momentum, and energy with a temporal periodicity at an effi-cient computational cost, compared to the traditional time-accurate method. The frequency-based time-spectral methoduses a discrete Fourier expansions to estimate the temporalderivative terms in the unsteady flow governing equations.

I t = ∂U∂t

+ R u = 0, 1

U =

ρ

ρu

ρv

ρE

, 2

where U is the vector of conservative variables and R is theresidual of the spatial discretization of the fluxes.

The time spectral representation of the governing equa-tion can be obtained by the approximation of I, U, and Rwith discrete Fourier expansions truncated at NHth har-monic as follows.

I t ≈ I0 + 〠NH

n=1Ian cos ωnt + Ibn sin ωnt , 3

U t ≈ U0 + 〠NH

n=1Uan

cos ωnt + Ubnsin ωnt , 4

∂U t∂t

≈ 〠NH

n=1ωnUan

sin ωnt − ωnUbncos ωnt , 5

R t ≈ R0 + 〠NH

n=1Ran

cos ωnt + Rbnsin ωnt , 6

where T is the period and ω is the angular frequency. If (3),(4), (5), and (6) are compared to each other, one can findthe relationships between Fourier coefficients, and the result-ing equations can be represented as

I0 = R0 = 0, 7

Ian = ωnUan + Ran = 0, 8

Ibn = −ωnUbn + Rbn = 0 9

The vectors of the Fourier coefficients in (7), (8), and (9)have a size of NT =NH + 1, and a general form of the systemof the equations can be represented as follows:

ωMU + R = 0, 10

where

2 International Journal of Aerospace Engineering

U =

U0

Uc1

Uc2

⋮

UcNH

Us1

Us2

⋮

UsNH

,

R =

R0

Rc1

Rc2

⋮

RcNH

Rs1

Rs2

⋮

RsNH

11

However, a direct calculation of (10) is difficult to solvedue to its highly nonlinearity that requires N3 order of oper-ations [1]. The computational expenses increase rapidly withthe number of harmonics. Hall et al. [1] suggested an alterna-tive way to resolve such problems by casting inverse Fouriertransformation to (10) as shown in

U = FU,R = FR,

12

ωF−1MFU + F−1FR = 0, 13

where F is the Fourier transformation matrix and F−1 isinverse Fourier transformation matrix. The pseudotimederivative term is added to (13) to solve with the pseudotime-stepping, and the final form of the governing equa-tions becomes

∂U∂τ

+ ωDHBMU + R = 0 14

The temporal derivative term is treated as a source term,and the equation becomes a time-independent function.

2.2. Chebyshev Pseudospectral Method. With an absence ofthe frequency term, the Chevyshev pseudospectral methodcan be used for both periodic and nonperiodic problems.The Chebyshev pseudospectral method is based on a seriesof Chebyshev polynomials to approximate the solution asan arbitrary function. The Chebyshev pseudospectral methodcan exhibit general applications to unsteady flow problems,

compared to the frequency pseudospectral method. How-ever, the method itself inherently suffers from clustering ofthe collocation points near the boundaries. This uneven dis-tribution of the collocation points can easily lead to numeri-cal instabilities due to the ill conditioning of the spectralderivative matrix [10]. A need for conformal mapping aroseto mitigate the numerical instabilities, and the conformalmapping functions have been investigated by a number ofresearchers. Im et al. [10] considered several types of map-ping functions in his study of mapped Chebyshev pseudos-pectral method. He considered four different types ofconformal mapping functions: inverse sine mapping func-tion [9, 14], polynomial mapping function [4], tangentfunction [5–7], and hyperbolic sine function [8]. His studyon conformal mapping showed the inverse sine functionwhich represented the most favorable performance in termsof even distribution and convergence characteristics. Thedetail of the other mapping function study can be found in[9, 10, 14].

2.3. Inverse Sine Function. The standard Chebyshev pseudos-pectral method uses a cosine function to provide a distribu-tion of collocation points over the interval −1, 1 .

yi = cos iπ

N15

In this study, inverse sine mapping function is used foreven distribution of the collocation points as it showed themost favorable performance, compared to the other mappingfunctions [10]. The inverse mapping function, proposedby Kosloff and Tal-Ezer [9, 14], can be derived into eithersymmetric or nonsymmetric transformation forms over theinterval −1, 1 , and the nonsymmetric transformation in(16) is considered to deliver more flexibility in the distribu-tion of the collocation points over the domain.

g y, α, β = arcsin αβ + α − β

α + β, 16

 α = cos iπN

,

β = cos iπN

,17

where α and β are the parameters that control the distribu-tion of the collocation points near the boundaries [10, 14].In this study, the values of α and β are used below 0.995.The time derivative term of the Chebyshev collocation oper-ator is used with the inverse sine mapping function anddefined as

dudt

= dydt

dudy

= 1g′

dudy

,

g′ = yi = Tmaxαβ

1 − αy 1 + βy

18

The final form of the mapped Chebyshev collocationoperator becomes

3International Journal of Aerospace Engineering

dudt

= 1g′

dudy

= αβ

Tmax 1 − αy 1 + βyDch 19

2.4. Chebyshev Collocation Operator. Chebyshev pseudospec-tral method uses a series of orthogonal polynomials toapproximate the state and flux variables in the governingequation at each collocation point. However, the collocationpoints are projected onto the interval of −1 ≤ t ≤ 1 byChebyshev-Gauss-Lobatto point, and it results in the undis-tribution of collocation points clustered toward the extremes.

An arbitrary and smooth continuous function u t can berepresented with a series of orthogonal polynomials in thedomain −1 ≤ t ≤ 1 as follows:

uj t = 〠N

n=0anTn t j = 〠

N

n=0an cos

jπnN

, for j = 0, 1, 2,… ,N ,

20

an =2

Ncn〠N

j=0

1cjujTn t j

= 2Ncn

〠N

j=0

uj

cjcos jπn

N, for n = 0, 1, 2,… ,N ,

21

ci =2, if i = 0,N ,1, otherwise,

22

where an in (21) is the Chebyshev coefficient of the functionu t , and it can be defined in a vector form as

a =DT u = 2N

T0 t04

T0 t12 … T0 tN

4T1 t0

2 T1 t1 … T1 tN2

⋮ ⋮ ⋱ ⋮TN t0

4TN t1

2 … TN tN4

u0

u1

⋮

uN

23

The matrix DT in (23) includes the Chebyshev polyno-mials of the function, u . The TN in the matrix representthe Nth degrees of Chebyshev polynomial. The differentialof u can be represented in a vector form as

u ′ =DT b =

T0 t0 T1 t0 … TN t0

T0 t1 T1 t1 … TN t1

⋮ ⋮ ⋱ ⋮

T0 tN T1 tN … TN tN

b0

b1

⋮

bN

,

24

where the elements of b in (24) are the Chebyshev coef-ficients and DT corresponds to the Chebyshev polynomialsof the u′ t . The Chebyshev coefficient u ′ can be repre-sented as

b = G a =

0 1 0 3 0 … G0,N

0 0 4 0 8 … G1,N

0 0 0 6 0 … G2,N

⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮

0 0 0 0 0 … GN ,N

a0

a1

a2

⋮

aN

,

25

ci =2, if i = 0,N ,1, otherwise,

Gij =2jci, if i + j odd,

0, otherwise,

26

where the element Gi,j of the matrix G in (26) is definedby the value of i + j . If the value of i + j is an oddnumber, Gij becomes 2j/ci or turns into 0 otherwise. Ifthe interval is transformed from the interval of −1 ≤ t ≤ 1to the interval 0 ≤ t ≤ Tmax, the differential can be computedby dividing each component of the Gij by Tmax /2. Finally,the time derivatives of an arbitrary function u t can be com-puted as

u′ =DT b =DTG a =DTGDT u =Dch u 27

Here, Dch in 27 represents a Chebyshev collocation oper-ator with the matrix size of N + 1,N + 1 .

2.5. Implementation. The Chebyshev pseudospectral methodwith mapping function is applied to the three dimensionalunsteady Reynolds Average Navier-Stokes equations asfollows:

∂Q∂t

+ ∂ F∂x

+ ∂G∂y

+ ∂H∂z

= 0 , 28

Q = 〠N

n=0QnTn t j , 29

F = 〠N

n=0FnTn t j , 30

G = 〠N

n=0GnTn t j , 31

H = 〠N

n=0HnTn t j , 32

where Q represents the conservative variables, and F, G,and H denote the residual of viscous and inviscid flux vec-tors in the X, Y , and Z direction. These vectors of conserva-tive and flux variables can be approximated with (29), (30),(31), and (32), and the time differential of the Q in theinterval of 0 ≤ t ≤ Tmax can be replaced with the mappedChebyshev collocation operator derived in (28) as

4 International Journal of Aerospace Engineering

1g′

DchQ + ∂ F∂x

+ ∂G∂y

+ ∂H∂z

= 0 , 33

where Q , F , G , and H have the size of N + 1, 1 matrix andeach component is defined at the points which are redistrib-uted by the arcsine mapping function.

Q =

Q0

Q1

Q2

⋮

QN

,

F =

F0F1F2⋮

FN

,

G =

G0

G1

G2

⋮

GN

,

H =

H0

H1

H2

⋮

HN

34

The pseudotime stepping is applied to (33).

∂Q∂τ

+ 1g′

DchQ + ∂ F∂x

+ ∂G∂y

+ ∂H∂z

= 0 35

The diagonalized alternate direction implicit (D-ADI)technique [15] was applied to the Chebyshev pseudospectralmethod for the time integration. In this study, explicit discre-tization of the Chebyshev pseudospectral operator andimplicit discretization of residual term are taken account intothe governing equation with the pseudotime stepping to takeadvantages of the implicit method for the larger time step sizeand the advantage of the explicit method for diagonal domi-nance of the Jacobian matrix as shown in (36). The explicitapplication of the Chebyshev operator removes its influenceon the nondiagonal elements of the matrix, and the sourceterm can be directly added into the flux term at the pseudo-time level [10].

∂Q∂τ

+ 1g′

DchQn+ ∂G

n+1

∂x+ ∂H

n+1

∂y= 0 36

The implicit integration of the spatial flux used the line-arization technique as follows:

Fn+1

= Fn+U ΔQ

n, 37

Gn+1

= Gn+V ΔQ

n, 38

Hn+1

= Hn+W ΔQ

n39

TheU, V, andW in (37), (38), and (39) represent the fluxJacobian matrices with the size of N + 1,N + 1 . Throughoutthe replacements of flux terms in (36) with (37), (38), and(39), (36) becomes

ΔQ n

Δτ + ∂∂x

U ΔQn

+ ∂∂y

V ΔQn

+ ∂∂z

W ΔQn

= −∂ F

n

∂x−∂G

n

∂y−∂H

n

∂z−

1g′

DchQn

40

The flux terms in (37), (38), and (39) are the function ofthe variables at the specific time instance over the domain of0 ≤ t ≤ Tmax, and only diagonal elements of the matrix U, V,and W are considered as below.

U =

∂U0∂Q0

0 … 0

0 ∂U1∂Q1

… 0

⋮ ⋱ ⋮

0 … 0 ∂GN

∂QN

,

V =

∂G0∂Q0

0 … 0

0 ∂G1∂Q1

… 0

⋮ ⋱ ⋮

0 … 0 ∂GN

∂QN

,

W =

∂H0∂Q0

0 … 0

0 ∂H1∂Q1

… 0

⋮ ⋱ ⋮

0 … 0 ∂HN

∂QN

41

5International Journal of Aerospace Engineering

Each of the diagonal element is at different timeinstances and denoted with subscript zero through Nand dependent only on the flow variables in that timeinstance. The key point here is that the diagonal elementsof the simplified diagonal Jacobian matrix are identical tothose of the flux Jacobian matrix in the time-domain solu-tion algorithm. Therefore, computation of additional termsin Jacobian matrices of V and W is not required. Finally,(28) can be represented as (42), employing a diagonallyimplicit method.

I + δxVΔτ + δyWΔτ ΔQn= −

∂Un

∂xΔτ − ∂G

n

∂YΔτ

−∂H

n

∂ZΔτ − Δτ

g′DchQ

n,

42

T0 0 … 00 T1 … 0⋮ ⋮ ⋱ ⋮

0 0 … TN

ΔQ0

ΔQ1

⋮

ΔQN

=

S0S1⋮

SN

, 43

where

Ti = I + δxUiiΔτ + δxViiΔτ + δyWiiΔτ, 44

Si = −∂U

n

i

∂xΔτ − ∂G

n

i

∂yΔτ − ∂H

n

i

∂zΔτ − Δτ

g′DchQ

n

i45

Here, the vector S contains the Chebyshev spectralsource terms and flux terms as shown in (42). The Tnin (45) can be computed by the method proposed byPulliam and Chaussee [15]. Therefore, the final form ofthe governing equations is represented such that themapped Chebyshev collocation operator is treated as asource term in the governing equation in a similar way tothe harmonic source term of the frequency domain method[2, 16].

2.6. Computational Efficiency and Solution Accuracy. Themain advantage of the time-spectral method is the spectral

accuracy and its convergence rate. The time-spectral methodcan achieve an order of accuracy equivalent to that of thetime-accurate method at a considerably effective computa-tional cost. Although the time-spectral solutions can be cor-rupted by aliasing errors if the number of the time samplingpoints is less than doubles of the Nyquist frequency or 3/2 −rule [17], one can find a number of sampling points withwhich the aliasing error is not significant to the time-spectralsolution accuracy through frequency spectrum analysis.

A precise quantification of the aliasing errors is impor-tant to find an optimal number of sampling points. It is idealif the number of sampling points is chosen by the Nyquisttheorem or 3/2 − rule since it can deliver exactly the sameorder of accuracy compared to that of the time-accuratemethod. However, the computational cost highly dependson the number of sampling points, and this will be laggingoff the convergence rate of the time-spectral method with alarger number of sampling points. The total computationalcost will exceed that of the time-accurate method. Thus, thetrade-off between the solution accuracy and the computa-tional cost should be considered.

The number of time instance points is determined by themagnitude of the aliasing error. The aliasing error is quanti-fied by calculating the norm of the discrepancy of the ampli-tude in the frequency contents between the time-accurateand time-spectral methods as follows.

L2,aliasing =〠N

i=1 Ci,TA − Ci,TS2

N, 46

where Ci,TA is the amplitude at each wave number in fre-quency contents of the time-accurate method and Ci,TS isthat of the time-spectral method. In this study, the time-spectral solutions are obtained with the number of timeinstance points to find an appropriate number that main-tains the spectral accuracy at an effective computationalefficiency.

3. Dynamic Derivative Analysis

In a previous study on stability analysis on tailless aircraft[18], stability and control behavior of a tailless aircraft areinvestigated for Innovative Control Effectors (ICE) configu-ration developed by Lockheed Martin. The system matrixof the ICE configuration is developed as follows:

u

w

q

α

=

η∂CA

∂u+ μu0CA0

η∂CA

∂w+ μw0CA0

η∂CA

∂q−w0 η

∂CA

∂α+ g cos θ0

η∂CN

∂u+ μu0CN0

η∂CN

∂w+ μw0CN0

η∂CN

∂q−w0 η

∂CN

∂α+ g cos θ0

η∂CA

∂u+ μu0CA0

η∂Cm

∂w+ μw0Cm0

η∂Cm

∂q−w0 η

∂Cm

∂α+ g cos θ0

0 0 1 0

ΔuΔwΔqΔα

, 47

6 International Journal of Aerospace Engineering

where ∂CA/∂u, ∂CN /∂u, ∂Cm/∂u, ∂CA/∂w, ∂CN/∂w, and ∂Cm/∂w are the static derivatives and ∂CA/∂q, ∂CN /∂q, and∂Cm/∂q are the steady dynamic derivatives. These derivativecan be easily found using steady CFD computations andfinite difference method with rotating flow scheme [19].However, the unsteady dynamic derivative terms, ∂CA/∂α,∂CN/∂α, and ∂Cm/∂α, require time-marching unsteadyCFD computations.

The dynamic derivatives can be approximated basedon the unsteady flow solutions, in which a forced sinusoi-dal oscillation is imposed at the center of gravity of air-craft [20]. From one cycle of unsteadiness during theharmonic oscillation, the mathematical model for theincrement in aerodynamic forces and moments can beexpressed as follows with an assumption of a linear rela-tionship between the aerodynamic properties and flightstates [21].

ΔCk =∂Ck

∂αΔα + c

2V∞

∂Ck

∂αα + c

2V∞

∂Ck

∂qq, 48

where

Δα t = αA sin ωt , 49

Δα t = q = ωαA cos ωt 50

Herein, αA is the amplitude of angle of attack and ω is anoscillation frequency. Equation (48) can be rewritten with(49) and (50) based on an assumption α t = q as shown in(51) [22].

ΔCα t = αAk Ckα + Ckq cos ωt + αACkα sin ωt , 51

ΔCα t = CC cos ωt + CS sin ωt 52

Herein, Ckα and Ckα + Ckq represent stiffness (in-phase)and damping (out-of-phase) components of dynamic deriva-tives, respectively [21]. The determination of the dynamicderivatives can be achieved by comparison of the first har-monic flow solutions from the time-accurate and time-spectral methods. Prior to the comparisons, it was assuredthat the unsteady flow solutions from each method are fullyconverged. The solutions from each method are transformedinto the frequency domain through discrete Fourier trans-form as shown below.

where the vector of X′s and C′s represent the flow solutionsand the Fourier coefficients of the corresponding flow solu-tions in the frequency domain, respectively. Here, the firstFourier coefficients, CC1

and SS1 , are compared with the coef-ficients in (52).

4. Validation of Chebyshev PseudospectralMethod and Dynamic Derivatives

4.1. NACA 0012. A harmonic oscillation is imposed on thequarter chord of NACA 0012 airfoil to validate the accuracy

of Chebyshev pseudospectral method. The computation ofNavier-Stokes equations is performed with reduced fre-quency k = 0 0814, initial angle of attack α0 = 0 016, andpitching amplitude αA = 2 51. The airfoil oscillates in a rangeof −2 494 < α < 2 526. The unsteady flow solutions areapproximated at Mach number M = 0 755 using the time-accurate method and the time-spectral method. A C-typestructured mesh is employed for the NACA 0012 airfoil with4096 points and 4260 elements as shown in Figure 1. TheChebyshev collocation operator with the mapping functionis applied to the time differential term along with D-ADI

CC0

CC1

⋮

CCNH

CS0

CS1

⋮

CSNH

=

cos 2π 0 × 1/NT

2cos 2π 0 × 2/NT

2 … cos 2π 0 ×NT/NT

2

cos 2π 1 × 1NT

cos 2π 1 × 2NT

… cos 2π 1 ×NT

NT

⋮ ⋮ ⋱ ⋮

cos 2πNH × 1NT

cos NHπN × 2NT

… cos 2πNH ×NT

NT

sin 2π 0 × 1NT

sin 2π 0 × 2/NT

2 … sin 2π 0 ×NT

NT

sin 2π 1 × 1NT

sin 2π 1 × 2NT

… sin 2π 1 ×NT

NT

⋮ ⋮ ⋱ ⋮

sin 2πNH × 1NT

sin NHπN × 2NT

… sin 2πNH ×NT

NT

X0

X1

⋮

XNT

, 53

7International Journal of Aerospace Engineering

method for time integration and periodic boundary condi-tions.The spatial numericalflux ineachdirection isdiscretizedusing Roe’s FDS with a third orderMUSCL scheme to achievehigher order accuracy, and turbulent eddy viscosity iscomputed from the two-equation k − ω Wilcox-Durbin +turbulence model [23].

Unsteady flow solutions of oscillating NACA0012 areapproximated by the time-accurate method (TA), frequencypseudospectral method (FPM), and Chebyshev pseudospec-tral method (CPM). In computation with the time-spectralmethods, the spectral accuracy can be contaminated by alias-ing errors due to a lack of number of sampling points. In thisstudy, the aliasing errors are represented by the norm of thediscrepancy in the amplitude at every wave number betweenthe time-accurate method and the time-spectral methodresults. The results are plotted in Figure 2(a). The aliasingerror is quantified for a different number of the time instance

Figure 1: Structured mesh for NACA 0012 airfoil.

10−6

10−5

10−4

10−3

10−2

10−1

100

Am

plitu

de

2 4 6 8 10 120Nth harmonic

3th5th

9th11th

7th

(a) Frequency contents of the mapped Chebyshev

pseudospectral method for NACA0012 under pitchingmotion

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Alia

sing

erro

r

4 6 8 10 122nth

(b) L2 norms of difference in amplitude of frequency contents

between the time-accurate method and the mappedChebyshev pseudospectral method for NACA 0012 under

pitching motion

Figure 2: Frequency analysis for the time-accurate method and thetime-spectral method results.

TAFPMCPM

6 8 10 12 14 16 18 20 224TI

0

1

2

3

4

5

6

7

8

9

10

CPU

tim

e (hr

s)

Figure 3: Computational time for NACA 0012 under pitchingmotion.

8 International Journal of Aerospace Engineering

points, and Figure 2(b) shows that the aliasing errors arereduced with an increasing number of the time instancepoints. This indicates the aliasing errors become less effectiveon the solution reconstruction as the number of the timeinstance points increases, and one can find an optimumnumber, in terms of computational cost, which provides thespectral accuracy fairly equivalent to the order of accuracyof the original wave form. As shown in Figure 2(b), the alias-ing error converges at the time instance points of 15, and thetime-spectral methods give a precise solution accuracy withrespect to the time-accurate solutions. However, the compu-tational time linearly increases with a number of collocationpoints over the interval as shown in Figure 3.

The advantages in the computational efficiency vanishwhen the number of the time instance points is more than50. In this study, the number of the time instance points of15 is chosen as an optimal for the Fourier- and mapped Che-byshev pseudospectral method, and it gives twice faster com-putational efficiency compared to the time-accurate method.

Normal force and pitching moment coefficients are com-pared with those from time-accurate method and the fre-quency pseudospectral method as shown in Figure 4. Thesolid line in black represents the time-accurate method, anddotted and dashed lines in blue correspond to the presentmethod, while the solid line in red represents the frequencypseudospectral method. The experimental data are shownin black circles [11]. With a small number of collocationpoints, the results show a large deviation from those of thetime-accurate method and the frequency pseudospectralmethod results with 15 time instance points. However, theresults of the present method are shown in a good agreementwith the solutions from the other methods as the timeinstance points increases. With 7 or more number of time

instance points, the results from Chebyshev pseudospectralmethod converge to the time-accurate method results. Thetime-spectral method results are also shown in reasonablygood agreement with the experimental data.

Dynamic derivatives are obtained for oscillating NACA0012 from the unsteady flow solutions from the time-accurate method, the frequency pseudospectral method, and

CPM_05CPM_07CPM_09

CPM_11CPM_13CPM_15

ExpTime-accurateFPM_15

−2 −1 0 1 2 3−3Pitching angle

−0.2

−0.2

0

0.2

0.4Cn

(a) Normal force coefficient

CPM_05CPM_07CPM_09

CPM_11CPM_13CPM_15

ExpTime-accurateFPM_15

−2 −1 0 1 2 3−3Pitching angle

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Cm(b) Pitching moment coefficient

Figure 4: Normal force and pitching moment hysteresis of NACA 0012.

Table 1: Dynamic derivatives for NACA 0012 airfoil from normalforce.

Stiffness (in-phase) Damping (out-of-phase)

Time-accurate −7.16 28.1

FPM (15) −7.18 29.1

CPM (4) −8.16 22.3

CPM (6) −7.22 29.0

CPM (8) −7.20 29.1

CPM (14) −7.17 29.7

Table 2: Dynamic derivatives for NACA 0012 airfoil from pitchingmoment.

Stiffness (in-phase) Damping (out-of-phase)

Time-accurate 0.12 −3.96FPM (15) 0.11 −3.79CPM (4) 0.13 −3.09CPM (6) 0.11 −3.60CPM (8) 0.11 −3.75CPM (14) 0.11 −3.82

9International Journal of Aerospace Engineering

the mapped Chebyshev pseudospectral method. The numberof collocation points varied from 5 to 15 for the Chebyshevpseudospectral method, and the frequency pseudospectralmethod is used with 15 time instance points. The resultsare summarized in Tables 1 and 2. The dynamic derivativesfrom normal coefficients are presented in Table 1. With afewer number of collocation points, the values of thedynamic derivatives show some discrepancy compared tothe time-accurate method results. However, they are con-verging to the results from the time-accurate method andthe frequency pseudospectral method as the number of collo-cation points increases.

4.2. ICE Configuration. The mapped Chebyshev pseudospec-tal method is applied to an investigation of the oscillatingthree-dimensional body. The ICE configuration is used toevaluate the accuracy of the present method. The unsteadyflow solutions are obtained by computation of Euler equa-tions with the mapped Chebyshev- and the Fourier colloca-tion operator. A structured mesh is employed with 5.2million elements within 9 blocks as shown in Figure 5. Aforced harmonic oscillation is imposed on the aircraft centerof gravity with reduced frequency of k = 0 1242, with an ini-tial angle of attack of α0 = 0 0, and the amplitude of 4.16 and8.33 degrees. The Mach number is given as 0.0266 which isthe same as the wind tunnel test conditions.

The experiment was performed with the Reynolds num-ber of 0 574 × 106 in Subsonic Air Force Research LaboratoryVertical Wind Tunnel (AFRL) at Wright-Patterson AFB,Ohio [12]. The ICE configuration was tested under variouspitching rate conditions during the longitudinal stability testas presented in Table 3. The magnitude of amplitude variedfrom 4.16 and 8.33 degrees to match pitch rate qc/2V of0.009 and 0.018 at 0 angle of attack [12].

The number of the time instance points varied to inves-tigate the spectral accuracy of the time-spectral methods.The frequency contents of the unsteady solutions from theFourier pseudospectral method and the mapped Chebyshevpseudospectral method are represented in Figure 6(a). Sim-ilar to the pitching NACA 0012 case in the previous section,the discrepancy between the time-accurate method and themapped Chebyshev pseudospectral method decreases as the

number of the time instance points increases. As shown inFigure 6(b), the minimum number of the time instancepoints should be chosen more than five to minimize effectsof the aliasing error on the solution accuracy. The recon-structed solutions from the present method are shown inFigure 7 through dotted and dashed lines in blue. The Fou-rier pseudospectral method solutions are represented in redsolid line, and the time-accurate method is shown in blacksolid line. The reconstructed solutions from the Fourierpseudospectral method and the mapped pseudospectralmethod are shown in a good agreement with that of time-accurate method.

The computational cost is also investigated and com-pared between the time-accurate and the time-spectralmethods as shown in Figure 8. The total computational timefor the time-accurate method is about 333 hours. On theother hand, it linearly increases with the number of the timeinstance points for the mapped Chebyshev pseudospectralmethod and the Fourier pseudospectral method. The compu-tational cost becomes the same order of magnitude when thenumber of time instance points is more than 25. From thefrequency content analysis, the time-spectral methods withthe time instance points more than eleven can provide theorder of solution accuracy equivalent to that of the time-accurate method. This indicates that the mapped Chebyshevpseudospectral method can compute the solution at half costof the time-accurate method for the current application onICE geometry.

To validate the time-spectral methods for the three-dimensional flow, the unsteady flow solutions are comparedwith experimental data as shown in Figure 9. The circle

XX

Y

Figure 5: Structured mesh for ICE configuration.

Table 3: Wind tunnel test conditions for forced oscillation.

Properties Values

Freestream velocity 8.84m/s

Reynolds number 0.574× 106/mAmplitude 4.16 and 8.33 deg.

Reduced frequency 0.1242

Moment reference 38% MAC

10 International Journal of Aerospace Engineering

symbols in black and solid line with diamond symbols in redindicate the experimental data and the Fourier pseudospec-tral method results, respectively. The solid line in blue corre-sponds to the results from the present method. Figure 9shows a good agreement in normal force coefficients withthe experimental data. The pitching moment, however, pre-sents a large deviation at high pitching rate. The issues canbe found in an inaccurate prediction of the flow separationcaused by artificial dissipation term in Euler solver. Thiscan become a significant concern for the calculation of the

complex vortex structures on the delta wing, which areformed from the nose and the leading edge region. As shownin Figure 10(b), the pressure contours are dynamicallychanging as the angle of attack, but it is hard to find an evi-dence of the interactions between the vortex structure andthe surface of the body, which would come out as flow sepa-rations and reattachments as it goes downstream.

The calculation of dynamic derivatives is extended tothree-dimensional case. Unsteady flow solutions of oscillat-ing ICE configuration are obtained using the Fourier

1 2 3 4 5 6 70Nth harmonic

TACPM05CPM07

CPM09CPM11

10−5

10−4

10−3

10−2

10−1

100A

mpl

itude

(a) Frequency contents for ICE configuration

5 6 7 8 9 10 11 124Number of time instances

0

0.01

0.02

0.03

0.04

0.05

0.06

Alia

sing

erro

r

(b) L2 norms of difference in amplitude of frequency contents

between the time-accurate method and the mapped Chebyshevpseudospectral method for ICE configuration under pitching

motion

Figure 6: Frequency analysis for the time-accurate method and the time-spectral method results.

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

FPM_11

CPM_05

CPM_07CPM_11TA

CN

−2 0 2 4−4Pitching angle

(a) Normal force coefficient

FPM_11

CPM_05

CPM_07CPM_11TA

−2 0 2 4−4Angle, V13

−0.1

−0.05

0

0.05

0.1

0.15

CN

(b) Pitching moment coefficient

Figure 7: Normal force and pitching moment hysteresis of ICE configuration.

11International Journal of Aerospace Engineering

pseudospectral method with 11 time instance points and themapped Chebyshev pseudospectral method with 5 to 11 timeinstance points. As shown in Tables 4 and 5, the discrepancyin dynamic derivatives between the current method andtime-accuracy method is decreasing with a larger numberof the time instance points. The out-of-phase and in-phasecomponents of dynamic derivatives show a good agreementwith the Fourier pseudospectral method results with 11 timeinstance points and the time-accurate method.

5. Conclusions

A mapped Chebyshev pseudospectral method is extendedfor three-dimensional unsteady flow problems. While thespectralmethodcanfindunsteadyflowsolutionswithanaccu-racy equivalent to the time-accurate method, the frequency-based pseudospectral approach has been limited to periodicproblems. However, the Chebyshev pseudospectral methodcan be an alternative time-spectral approach with its appli-cation to both periodic and nonperiodic problems. In lightof such potential, the mapped Chebyshev pseudospectralmethod is employed to solve three-dimensional unsteadyperiodic flow in order to validate its spectral solution accu-racy and computational efficiency with those of Fourierpseudospectral method and the traditional time-accuratemethod. The unsteady flow solutions of the flow past NACA0012 and ICE configuration under the forced pitching oscil-lation are obtained by the time-accurate method, the Fourierpseudospectral method, and the mapped Chebyshev pseu-dospectral method. The results from the present methodare compared with the experimental data for validations aswell. The mapped Chebyshev pseudospectral method foundunsteady flow solutions in a good agreement with the other

methods for both two- and three-dimensional cases. Thepresent method also provided a good agreement in normalforce coefficient but showed a discrepancy in some degree inpitching moment coefficient, compared to the experimentaldata. Finally, dynamic derivatives are obtained based on theunsteady flow solutions from the time-accurate, the Fourierpseudospectralmethod, and themappedChebyshev pseudos-pectral method. With 11 time instance points, the values ofdynamic derivatives based on the mapped Chebyshev pseu-dospectral method results are obtained similar to those ofthe time-accurate method and the Fourier pseudospectralmethod at the same number of time instance points. Themapped Chebyshev pseudospectral method showed compu-tational cost equivalent to the Fourier pseudospectralmethod, and about 2.5 and 2 times faster computationalefficiency for NACA 0012 and ICE configuration case,respectively, compared to that of time-accurate method.

TAFPMCPM

6 8 10 12 14 164Number of time instances

0

50

100

150

200

250

300

350

400

CPU

tim

e (hr

s)

Figure 8: Computational time for ICE configuration under pitchingmotion.

TAFPM_11

CPM_11EXP

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

CN

−0.01−0.02 0.01 0.020q_bar

(a) Normal force coefficient

TAFPM_11

CPM_11EXP

−0.04

−0.02

0

0.02

0.04

CM

0.010 0.02 0.03−0.02 −0.01−0.03q_bar

(b) Pitching moment coefficient

Figure 9: Comparison of normal force and pitching momentcoefficient with experimental data.

12 International Journal of Aerospace Engineering

In general, the mapped Chebyshev pseudospectral methodapproximated the unsteady flow solutions for the periodicoscillating body with a reasonable accuracy compared tothe experimental data and provided a good agreement withthose from the traditional time-accurate method and thefrequency pseudospectral as well.

Nomenclature

A: Amplitude of oscillationa: Chebyshev coefficientb: Chebyshev coefficient for the first differentialD: Matrix in harmonic balance equationDch: Chebyshev collocation matrix

DT : Chebyshev coefficient matrixDT : Chebyshev coefficient matrix for the first differentialF: Fourier transform matrixG,H: Fluxesg: A mapping functiong′: Derivative of a mapping functionkc: Reduced frequencyM: Matrix in frequency domain equationMch: Number of reconstruction points by ChebyshevM∞: Freestream Mach numberNhs: Number of harmonicsp, q: Constant of polynomial mapping functionQ: Conservative variablesR: ResidualT : Chebyshev polynomialTmax: Time intervalt: TimeV,W: Flux Jacobian matrices in mapped Chebyshev methody: Variable of a mapping functionα, β: Constant of inverse sine mapping functionαAOA: Angle of attackϕ: Frequencyτ: Pseudotime.

Data Availability

The data for the time-accurate and the time-spectral methodsfor NACA0012 and ICE configuration is not availablebecause it is owned by the university.

Disclosure

This paper is an extended work from previous study [24],and comparisons between the time-accurate method andtime-spectral method are included for solution accuracyand computational efficiency.

1,7

5

4

3

2

6

−0.3−0.25

−0.2−0.15

−0.1−0.05

00.05

0.10.15

0.20.25

0.3

CL

−2 0 2 4−4Angle of attack

(a) Collocation points

1,7

2

3

4

6

5Cp0.80.420.04

−0.34−0.72−1.1−1.48−1.86−2.24−2.62−3

Cp0.80.420.04

−0.34−0.72−1.1−1.48−1.86−2.24−2.62−3

Cp0.80.420.04

−0.34−0.72−1.1−1.48−1.86−2.24−2.62−3

Cp0.80.420.04

−0.34−0.72−1.1−1.48−1.86−2.24−2.62−3

Cp0.80.420.04

−0.34−0.72−1.1−1.48−1.86−2.24−2.62−3

Cp0.80.420.04

−0.34−0.72−1.1−1.48−1.86−2.24−2.62−3

(b) Surface pressure contour

Figure 10: Surface pressure contours of ICE configuration at different collocation points.

Table 4: Dynamic derivatives for ICE configuration from normalforce.

Stiffness (in-phase) Damping (out-of-phase)

Time-accurate −5.65 −2.87FPM −5.79 −2.92CPM (5) −5.39 −2.80CPM (7) −5.79 −2.85CPM (11) −5.78 −2.94

Table 5: Dynamic derivatives for ICE configuration from pitchingmoment.

Stiffness (in-phase) Damping (out-of-phase)

Time-accurate 1.98 1.02

FPM (11) 2.12 1.03

CPM (5) 2.04 0.99

CPM (7) 2.02 1.01

CPM (11) 2.11 1.04

13International Journal of Aerospace Engineering

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

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