Fourier time spectral method for subsonic and transonic...

Acta Mech. Sin.DOI 10.1007/s10409-015-0547-x

RESEARCH PAPER

Fourier time spectral method for subsonic and transonic flows

Lei Zhan1,2 · Feng Liu2 · Dimitri Papamoschou2

Received: 14 September 2015 / Revised: 30 October 2015 / Accepted: 2 November 2015© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag BerlinHeidelberg 2016

Abstract The time accuracy of the exponentially accu-rate Fourier time spectral method (TSM) is examined andcompared with a conventional 2nd-order backward differ-ence formula (BDF) method for periodic unsteady flows. Inparticular, detailed error analysis based on numerical com-putations is performed on the accuracy of resolving the localpressure coefficient and global integrated force coefficientsfor smooth subsonic and non-smooth transonic flows withmoving shock waves on a pitching airfoil. For smooth sub-sonic flows, the Fourier TSM method offers a significantaccuracy advantage over the BDF method for the predic-tion of both the local pressure coefficient and integratedforce coefficients. For transonic flows where the motion ofthe discontinuous shock wave contributes significant higher-order harmonic contents to the local pressure fluctuations,a sufficient number of modes must be included before theFourier TSM provides an advantage over the BDF method.The Fourier TSM, however, still offers better accuracy thanthe BDF method for integrated force coefficients even fortransonic flows. A problem of non-symmetric solutions forsymmetric periodic flows due to the use of odd numbers ofintervals is uncovered and analyzed. A frequency-searchingmethod is proposed for problems where the frequency is notknown a priori. The method is tested on the vortex sheddingproblem of the flow over a circular cylinder.

B Feng [email protected]

1 School of Aeronautics, Northwestern PolytechnicalUniversity, Xi’an 710072, China

2 Department of Mechanical and Aerospace Engineering,University of California, Irvine, CA 92697, USA

Keywords Fourier time spectral method (TSM) ·Pitching airfoil · Transonic flow · Non-symmetric solution ·Computational efficiency ·Vortex shedding flow · Frequencysearch

1 Introduction

Time dependent calculations are needed for various impor-tant applications, such as the study of the internal flow ofturbomachinery, flutter, and jet noise [1–5]. Time accuracyis also needed for large eddy simulation of turbulence-generated noise [6,7] and particle-turbulence interactions[8,9]. In addition to improvements in computer hardware,improvements in numerical methods also play a significantrole in reducing the computational cost of time dependentcalculations. A dual-time stepping method that was origi-nally developed by Jameson [10] has been successful forunsteady flow computations using both the Euler and theNavier–Stokes equations [11]. Although local pseudo-timestepping, residual smoothing, and multigrid method can beused to accelerate convergence, the computational efficiencyof this method is limited. The backward difference formula(BDF) method for the time discretization only provides timeaccuracy of algebraic orders, specifically 2nd-order for theusual fully implicit three-level BDF scheme. Thus, small realtime steps are needed due to the time accuracy requirement.

For periodic flows, an alternative method for real time dis-cretization is to represent flow variables with Fourier series.One of the earliest methods in this category is a linearizedmethod proposed by Hall and Crawley [12]. In this methodeach flow variable is assumed to be a sum of the time-averaged value and a small perturbation. Based on the smallperturbation assumption, the flow governing equation can besimplified and separated into two sets of equations. One is

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L. Zhan et al.

for the time-averaged variables, and the other is for the smallperturbation components. Coefficients in the latter can beobtained by solving the former. Then the small perturbationcomponents are assumed to be harmonic functions with theknown frequency.

To develop an efficient method that can account fornonlinear effects, Ning and He [13] proposed a nonlinearharmonic method. In this method, each flow variable is stillexpressed as a sumof the time-averaged value and a harmonicperturbation. However, when the nonlinear flow equationsare time-averaged, extra “unsteady stress” terms are gener-ated in the time-averaged equations. These extra terms arecomputed from the solution of the unsteady perturbationequations, while the coefficients of the perturbation equa-tions are determined by the solution of the time-averagedequations. The two sets of equations are solved simultane-ously in a strongly coupled manner. Numerical results showthat nonlinear effects, such as oscillation of shock waves, canbe efficiently modeled by the nonlinear harmonic method.

The nonlinear harmonic method was among the earliestmethods that account for a nonlinear effect by reserving extra“unsteady stress” terms in the time-averaged equations.How-ever, the unsteadiness is still evaluated by the first harmonicmode only. Harmonic modes with higher frequencies are notinvolved. Therefore, the effectiveness of this method is lim-ited for large and complex temporal variations. In addition,the formulation of the method requires a lot of effort, espe-cially for three-dimensional viscous unsteady flow problems.To develop amethod that includesmore harmonicmodes andcan be easily applied in engineering practice, Hall et al. [14]proposed the harmonic balance method based on expandingflowvariableswith Fourier series. In thismethod the real timederivative term is approximated by the time spectral opera-tor. Fast Fourier transform (FFT) and its inverse counterpartare used in the time spectral operator. The resulting equa-tions on all instants are solved simultaneously in time domainand they are coupled only through the time spectral operator.Since the approximated real time derivative term reduces to asource term, pseudo-time marching method for steady-statecalculation can be employed. Convergence accelerating tech-niques, such as local time step and the multigrid method, canbe applied during the pseudo-time marching process withoutaffecting time accuracy. The harmonic balance method hasbeen successfully applied in solving two-dimensional vis-cous flow past cascades. Computational results show thatthe method is efficient. Strong nonlinear effects, such asoscillation of shock waves can be accurately predicted ifthe time resolution is sufficiently high. Gopinath et al. [15]applied the harmonic balancemethod to the computation of athree-dimensional unsteady flow insidemulti-stage turboma-chinery. In this computation only one blade passage is usedfor each blade row. The use of the harmonic balance methodgreatly reduced the overall cost of the simulation. The dom-

inant unsteadiness and the main flow features, such as theinteraction between two blade rows, are captured efficiently.In their study, Ekici et al. [16] completed the computationsof three-dimensional periodic inviscid flows around a heli-copter rotor using the harmonic balancemethod. Themethodwas reported to be able to resolve significant unsteady flowfeatures with low computational cost. Since the equationsresulting from the harmonic balance method are well-suitedfor adjoint sensitivity analysis, rotor optimization includingimprovement of aerodynamic performance and noise reduc-tion can be conveniently conducted.

There were also other ways to calculate periodic flowsbased on the Fourier expansion of flow variables. A non-linear frequency domain solver was proposed by McMullenet al. [17], which is equivalent to the harmonic balancemethod. With this solver the governing equations are solvedin the frequency domain rather than the time domain. How-ever, the residual is still calculated in the time domain forsimplicity. Transformations between the time domain andthe frequency domain for both solutions and residuals arerequired. This method has been shown to be efficient insolving transonic flows past a pitching airfoil and the vor-tex shedding flow behind a circular cylinder at rest. A paperby Gopinath and Jameson [18] derived the explicit formulaof the approximated real time derivative using the idea of har-monic balancemethod. Since discrete Fourier transformationand its inverse counterpart are not required in computation,the resulting equations are solved completely in the timedomain. For this reason, this variant of the harmonic balancemethod is called the time spectral method (TSM). We callthis method the Fourier TSM since it is based on the Fourierexpansion of flow variables. The samework byGopinath andJameson [18] also pointed out that stability problem may becaused by the use of even numbers of intervals in a period.Thus, using odd numbers of intervals seems to be favorable.However, we found that non-symmetric solutions are pro-duced for symmetric problems if a period is split into oddnumbers of intervals. This problem has not been reportedbefore. We analyze the reason for this problem, and basedon that, we propose the requirements to ensure symmetricsolutions.

The study by McMullen and Jameson [19] estimated thecomputational efficiency of the nonlinear frequency domainsolver, which is equivalent to the Fourier TSM. The non-linear frequency domain method is demonstrated to be muchmore efficient than the 2nd-order BDFmethod. However, theevaluation is limited to analyzing the error of the computedintegration quantities of the surface pressure coefficient, suchas the lift coefficient and the moment coefficient. To morecomprehensively evaluate the efficiency advantage of theFourier TSM over the 2nd-order BDF method, we here con-duct a systematic error analysis on the computed temporalvariation of the surface pressure coefficient itself.

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For the periodic flows where the frequency is not knowna priori, such as vortex shedding flows, a gradient-basedmethod has been used successfully to search for the fre-quency [17,18]. However, the initial guess of the frequencymust be close to the correct value since the method is locallyoptimal. A new method that updates the frequency basedon Fourier analysis of the lift coefficient before using thegradient-based method is developed. So the initial guessesof the frequency that are far away from the correct value canbe used.

In the following sections, firstly, the Fourier TSM isderived and validated with a transonic flow over the pitchingNACA0012 airfoil. For symmetric flow problems, the non-symmetric solutions produced by the use of odd numbers ofintervals are discussed. Then error analysis on the surfacepressure coefficient and its integration quantities using boththe Fourier TSM and the 2nd-order BDF is conducted fortransonic flows as well as subsonic flows. Finally, the newfrequency search approach is proposed and applied in con-junction with the Fourier TSM to a vortex shedding flowbehind a circular cylinder at rest.

2 The Fourier time spectral method (TSM)

The Navier–Stokes equations for a two-dimensional com-pressible viscous flow can be written as

∂w

∂t+ ∂ f c

∂x+ ∂ gc

∂y− ∂ f v

∂x− ∂ gv

∂y= 0, (1)

where

w=

⎛⎜⎜⎝

ρ

ρuρv

ρE

⎞⎟⎟⎠ , f c=

⎛⎜⎜⎝

ρuρuu+ p

ρvuρEu + pu

⎞⎟⎟⎠ , gc =

⎛⎜⎜⎝

ρv

ρuv

ρvv + pρE v + pv

⎞⎟⎟⎠ ,

(2)

f v =

⎛⎜⎜⎝

0τxxτyx

uτxx + vτyx − qx

⎞⎟⎟⎠ , gv =

⎛⎜⎜⎝

0τxyτyy

uτxy + vτyy − qy

⎞⎟⎟⎠ .

(3)

In the preceding equations, t is time, x and y are positioncoordinates, ρ is density, p is pressure, (u, v) is the localflow velocity. The total energy is E = e + 1

2 (u2 + v2) with

internal energy e = p(γ−1)ρ . γ is the ratio of specific heat.

(u, v) = (u − ub, v − vb) stands for the local convectivevelocity relative to the control surface moving at the velocityof (ub, vb). The components of the viscous stress tensor andthose of the heat flux vector are defined as below

τxx = 2μ

[∂u

∂x− 1

3

(∂u

∂x+ ∂v

∂y

)],

τyx = τxy = μ

(∂u

∂y+ ∂v

∂x

),

τyy = 2μ

[∂v

∂y− 1

3

(∂u

∂x+ ∂v

∂y

)],

qx = − μ

Pr

∂h

∂x,

qy = − μ

Pr

∂h

∂y,

(4)

where μ is the molecular viscosity, which is calculated bySutherland’s law. Pr is the Prandtl number, h = e + p

ρis

enthalpy.If the unsteady flow is periodic, then the flow solution

vector w can be expanded using the Fourier series as below

w(t) =+∞∑−∞

wkeikωt , (5)

whereω is the fundamental angular frequency of the periodicflow. The Fourier coefficient wk can be calculated as

wk = ω

2π

∫ 2πω

0w(t)e−ikωtdt, k = 0,±1,±2, . . . . (6)

If finite harmonic modes are retained in the series of Eq. (5),discrete Fourier transform (DFT) pair can be constructed.Assume a period is equally divided into N intervals, thenthe solution vectors on the left N instants form the followingtime sequence, which is the extended solution vector

w∗ = [w0, . . . , wn, . . . , wN−1]T. (7)

The DFT on the above extended solution vector is written as

wk = 1

N

N−1∑n=0

wne−ikωn�t , (8)

where �t = 2πωN . If N is odd, the inverse DFT (IDFT) is

written as

wn =N−12∑

k=− N−12

wkeikωn�t . (9)

When N is even, the inverse DFT is modified as

wn =N2 −1∑

k=− N2

wkeikωn�t . (10)

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In matrix form, the DFT in Eq. (8) can be expressed as

w∗ = Ew∗, (11)

where E is the DFT operator matrix. w∗ is the Fourier coef-ficient vector of solution. If N is odd,

w∗ =[w0, w−1, . . . , w− N−1

2, w N−1

2, . . . , w1

]T. (12)

When N is even, w∗ becomes

w∗ =[w0, w−1, . . . , w− N

2 +1, w− N2, w N

2 −1, . . . , w1

]T.

(13)

For the IDFT in Eq. (9) or Eq. (10), it can be expressed inmatrix form as

w∗ = E−1w∗, (14)

where E−1 is the IDFT operator matrix.w∗ and w∗ form theDFT pair for flow solution. Since the first-order derivative ofa periodic function is also a periodic function, the DFT pairfor the time derivative of flow solution can be constructed aswell. The corresponding DFT and IDFT are

w(1)∗ = Ew(1)∗ (15)

and

w(1)∗ = E−1w(1)∗, (16)

where the extended vector in time domain w(1)∗= [ ∂w

∂t |0, . . . , ∂w∂t |n, . . . , ∂w

∂t |N−1]T, and w(1)∗ is the Fouriercoefficient vector for the time derivative of flow solution. Thek-th element of w(1)∗ can be calculated from the k-th elementof w∗ through the following relation:

w(1)k = ikωwk . (17)

Written in matrix form, w(1)∗ can be formulated as

w(1)∗ = iωNF w∗, (18)

where NF is a diagonal matrix. If N is odd, NF =diag(0,−1, . . . ,− N−1

2 , N−12 , . . . , 1). When N is even, NF

= diag(0,−1, . . . ,− N2 + 1, 0, N

2 − 1, . . . , 1).In TSM, the extended time derivative vector w(1)∗ can be

calculated from w∗ through DFT and IDFT as below

w(1)∗ = E−1w(1)∗ = iωE−1NF w∗ = iωE−1NF Ew∗ = Fw∗,

(19)

where F denotes the time spectral operator,which is an N×Nmatrix. If N is even, the DFT and IDFT can be replaced byFFT and IFFT so as to evaluate the time spectral operatorefficiently. The element at the l-th row and the n-th columnof the time spectral operator matrix, Fl,n (0 ≤ l, n ≤ N −1),can be expressed as

Fl,n = ω

N

N2 −1∑

k=− N2 +1

ike2πikN (l−n)

=⎧⎨⎩

ω

2(−1)(l−n) cot[π(l − n)

N], l �= n,

0, l = n.

(20)

Note that the fundamental angular frequency of a periodicflow problem ω is an explicitly specified parameter insidethe time spectral operator.

Following the method of lines, the unsteady governingequation (1) can be integrated over grid cells using the cell-centered finite volume method. The governing equations inthe semi-discrete form on N equally spaced instants can becollected together and written as

dw∗

dt+ R(w∗) = 0. (21)

Using the Fourier TSM to solve the above equation, thetime derivative term of the equation can be approximated bythe time spectral operator. The resultant equation becomes

Fw∗ + R(w∗) = 0. (22)

The above equation takes the form of governing equationfor steady problems. There is no explicit time derivative andthe time derivative term reduces to a source term. To solvesuch a system, pseudo-time marching technique is usuallyadopted. Introducing the pseudo-time τ , the final equation tobe solved is

dw∗

dτ+ Fw∗ + R(w∗) = 0. (23)

In Eq. (23), R(w∗) is calculated by the Jameson-Schmidt-Turkel (JST) scheme [20]. The pseudo-time marching pro-cess is conducted by a five-stage explicit Runge–Kuttascheme because of its extended stability range. The fullydiscretized governing equations on all instants are cou-pled only through the time spectral operator and are solvedsimultaneously in time domain as in Ref. [14]. Conver-gence acceleration technique, such as local time stepping andmultigrid method, can be used during pseudo-time marchingwithout losing time accuracy.

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3 Application to the pitching airfoil test case

The periodic flows over a pitching NACA0012 airfoil attransonic flow conditions are frequently used as test casesfor unsteady solvers. For this reason, the Fourier TSM isvalidated by solving an inviscid test case of this kind. Com-putational results using the Fourier TSM is also comparedwith those using the BDFmethod. In the present inviscid testcase, a NACA0012 airfoil is forced to pitch around its quar-ter chord at M∞ = 0.755. The forced pitching movement isgiven by

α = α0 + αm sinωt, (24)

where α0 is the mean angle of attack, αm the pitching ampli-tude. ω is the angular frequency and is related to the reducedfrequency κ as

κ = ωc

2U∞, (25)

where c is the chord length andU∞ is the free-stream veloc-ity. In this case, α0 = 0.016◦, αm = 2.51◦, and κ = 0.0814.The periodic inviscid flow in this test case is calculated on a161 × 33 O-type grid. The convergence criteria is set to be1 × 10−14. To implement the Fourier TSM, three differenttime resolutions are used. One period is equally split into 8,16, or 32 intervals, respectively. For the BDF method, 32 or64 equal intervals are used.

The computed lift and moment coefficient variations withrespect to the angle of attack are shown in Fig. 1. The experi-mental data atReynold’s number Re = 5.5×106 [21] are alsoshown to demonstrate the same general agreement betweenthe computations and the experiment as found in Gao et al.[22,23], despite the noticeable differences due to negligenceof viscous and turbulence effects by the Euler equations. Thepurposeof this paper is to investigate the time spectralmethodvs. the conventional BDF time-marchingmethod in resolving

time evolution for the Euler equations. Therefore, the compu-tational result using the BDF method with 64 time intervalsis used as the benchmark accurate solution in the followingdiscussions. Figure 1a shows that for the lift coefficient, thecomputational result using the Fourier TSM only with eightintervals agrees very well with the accurate solution. For themoment coefficient, Fig. 1b shows that the computationalresults using the Fourier TSM converge fast to the accuratesolution as time resolution increases. However, to predictwell all details of the temporal variation of the moment coef-ficient, at least 16 intervals are needed for the Fourier TSM.Compared to the temporal variation of the lift coefficient,that of the moment coefficient is more complicated. Hence,to reach the same level of accuracy, more intervals (moreFourier modes in the frequency domain) are required for theFourier TSM to calculate the latter.

As for the temporal variation of the surface pressurecoefficient distribution, it is convenient to show it in fre-quency domain. The time-averaged component and the firstthree modes are shown in Figs. 2, 3, 4, and 5. Computa-

Fraction of chord

cp ,

tim

e-av

erag

ed

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

Fourier TSM, 32 intervalsFourier TSM, 16 intervalsFourier TSM, 8 intervalsBDF, 64 intervalsBDF, 32 intervalsExperiment

Fig. 2 Inviscid flow over pitching NACA0012 airfoil: time-averagedsurface pressure coefficient

Angle of Attack ( )

Lif

t co

effi

cien

t c

l

-2 0 2-0.4

-0.2

0

0.2

0.4


Angle of Attack ( )

Mo

men

t co

effi

cien

t c m

-2 0 2

-0.015

-0.01

-0.005

0

0.005

0.01

0.015


a b

Fig. 1 Inviscid flow over pitching NACA0012 airfoil: lift and moment coefficients versus angle of attack. a Lift coefficient. bMoment coefficient

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L. Zhan et al.

Fraction of chord

cp ,

firs

t m

od

e, r

eal c

om

po

nen

t

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1


Fraction of chord

cp ,

firs

t m

od

e, im

agin

ary

com

po

nen

t

0 0.2 0.4 0.6 0.8 1

-0.2

-0.1

0

0.1

0.2Fourier TSM, 32 intervalsFourier TSM, 16 intervalsFourier TSM, 8 intervalsBDF, 64 intervalsBDF, 32 intervalsExperiment

a b

Fig. 3 Inviscid flow over pitching NACA0012 airfoil: first mode of surface pressure coefficient. a Real component. b Imaginary component

Fraction of chord

cp ,

seco

nd

mo

de,

rea

l co

mp

on

ent

0 0.2 0.4 0.6 0.8 1-0.02

0

0.02

0.04

0.06

0.08


Fraction of chord

cp ,

seco

nd

mo

de,

imag

inar

y co

mp

on

ent

0 0.2 0.4 0.6 0.8 1

-0.1

-0.05

0

0.05


a b

Fig. 4 Inviscid flow over pitching NACA0012 airfoil: second mode of surface pressure coefficient. a Real component. b Imaginary component

Fraction of chord

c p ,

thir

d m

od

e, r

eal c

om

po

nen

t

0 0.2 0.4 0.6 0.8 1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08


Fraction of chord

c p ,

thir

d m

od

e, im

agin

ary

com

po

nen

t

0 0.2 0.4 0.6 0.8 1

-0.04

-0.02

0

0.02

0.04


a b

Fig. 5 Inviscid flow over pitching NACA0012 airfoil: third mode of surface pressure coefficient. a Real component. b Imaginary component

tional results as well as experimental data [21] all revealthat the range of the shock wave movement is approxi-mately 0.2 < x/c < 0.6. Within this area, the FourierTSM with eight intervals in a period can only reasonablypredict the time-averaged component of the surface pres-sure coefficient. When time resolution increases to have 16intervals in a period, the Fourier TSM could also generally

resolve the first mode, the second mode and the real com-ponent of the third mode. For the imaginary component ofthe third mode, Fig. 5b shows that the computational resultusing the Fourier TSM with 16 intervals deviates dramati-cally from the accurate solution. To make improvement onthis, 32 intervals are needed. Outside the shock wave move-ment area (x/c < 0.2 or x/c > 0.6), the Fourier TSM with

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only eight intervals could perfectly resolve the time-averagedcomponent and all three modes of the surface pressure coef-ficient. This is because in the shock-free area, the temporalvariation of the surface pressure coefficient is smooth, itsFourier modes with high frequencies decay rapidly. Thus,only retaining the first three modes(corresponding to eightintervals) is sufficient for the Fourier TSM to obtain accuratecomputational results. Since the temporal variation of the sur-face pressure coefficient could be perfectly predicted in mostarea, the Fourier TSMwith only eight intervals is able to cal-culate accurately the integration quantities, in particular thelift coefficient. For the moment coefficient, the accuracy lossof the surface pressure coefficient in the area of shock wavemovement only affects some details of its temporal variation.The general trend and the range of the temporal variation areabout correct.

4 Non-symmetric solutions of symmetric problemsdue to odd numbers of intervals

When the Fourier TSM is applied to solve periodic flowproblems, proper time resolution should be carefully selectedto balance accuracy of the solution and computational cost.Another important issue is whether the number of intervalsis even or odd. Gopinath and Jameson [18] pointed out thatthe use of an even number of intervals may cause stabilityproblem for the Fourier TSM, particularly in the cases wherethe time derivative is significant, such as high RPM turboma-chinery problems. This is because the odd-even decoupledsolution may be allowed when the Fourier TSM is appliedwith an even number of intervals. Thus, using odd numbers ofintervals seems to be favorable. However, we found that theFourier TSMproduces non-symmetric solutions for symmet-ric flowproblems if an odd number of intervals are employed.To illustrate this, a symmetric flow problem is constructedfrom the previous test case and is solved by the Fourier TSM.In this symmetric flowproblem, the airfoil still pitches aroundits quarter chord point. All the flow conditions remain thesame except that the mean angle of attack is set to be zero.Under these flow conditions, the periodic flow past the air-foil is symmetric given that the airfoil is symmetric. For thissymmetric flow, the distribution of the surface pressure coef-ficient on the upper surface should be perfectly symmetricto or overlap that on the lower surface in frequency domain.The computed time-averaged component and the first threeFourier modes are shown in Figs. 6, 7, 8, and 9, respectively.For the Fourier TSM, seven or eight intervals are used. Thesolution obtained by theBDFmethodwith 32 intervals servesas a reference solution. It is observed thatwhen eight intervalsare used, the solution from the Fourier TSM is symmetric,just like the reference solution. This is true for any Fouriermode of the pressure coefficient in any surface area.Whereas,

Fraction of chord

c p ,

tim

e-av

erag

ed

0.2 0.4 0.6 0.8

-0.5

0

0.5

1

Fourier TSM, 7 intervalsFourier TSM, 8 intervalsBDF, 32 intervals

Fig. 6 Symmetric inviscidflowover pitchingNACA0012 airfoil: time-averaged surface pressure coefficient

if seven intervals are employed, non-symmetric solution isproduced by the Fourier TSM, especially in the area whereshock wave moves. Outside the area of shock wave move-ment, the solution is still non-symmetric and the asymmetrybecomes quite weak.

Whether the asymmetry of the solution using seven inter-vals in a certain area is obvious or not actually depends onif seven intervals are sufficient to resolve the local unsteadyflow. The computational results of the previous test case forvalidation have shown that using eight intervals could onlyretain three Fourier modes in the Fourier TSM. This is notsufficient to resolve the shockwavemovement. Thus, the cal-culated surface pressure coefficient exhibits dramatic error inthe area where shock wave moves. Using seven intervals canonly retain the first three Fourier modes in the Fourier TSMas well. Thus, the error of the computed surface pressurecoefficient is also high in the same surface area. This is whythe asymmetry of the solution using seven intervals is obvi-ous. It can be anticipated that if time resolution is increasedso as to retain more Fourier modes in the Fourier TSM, theasymmetry of solutions using odd numbers of intervals willdecrease in the area of shock wave movement.

Actually, the problemof non-symmetric solution of a sym-metric problem due to the use of odd numbers of intervalsis not limited to the Fourier TSM. Numerical experimentshave confirmed that the same problem also happens to theBDF method. The reason for this problem is not associatedwith the specific method of time discretization, but lies in theway independent instants (not including the one at the end ofa period) are distributed in a period. To guarantee symmet-ric solutions for symmetric flow problems, we propose thatthe distribution of independent instants should satisfy the fol-lowing requirements. All independent instants should be ableto be grouped into pairs, and the phase difference between

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Fraction of chord

c p ,

firs

t m

od

e, r

eal c

om

po

nen

t

0 0.2 0.4 0.6 0.8 1

-0.1

-0.05

0

0.05

0.1 Fourier TSM, 7 intervalsFourier TSM, 8 intervalsBDF, 32 intervals

Fraction of chord

c p ,

firs

t m

od

e, im

agin

ary

com

po

nen

t

0 0.2 0.4 0.6 0.8 1

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Fourier TSM, 7 intervalsFourier TSM, 8 intervalsBDF, 32 intervals

a b

Fig. 7 Symmetric inviscid flow over pitching NACA0012 airfoil: first mode of surface pressure coefficient. a Real component. b Imaginarycomponent

Fraction of chord

c p ,

seco

nd

mo

de,

rea

l co

mp

on

ent

0 0.2 0.4 0.6 0.8 1

0

0.02

0.04

0.06

0.08

TSM (7 intervals)TSM (8 intervals)BDF (32 intervals)

Fraction of chord

c p ,

seco

nd

mo

de,

imag

inar

y co

mp

on

ent

0 0.2 0.4 0.6 0.8 1

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

TSM (7 intervals)TSM (8 intervals)BDF (32 intervals)

a b

Fig. 8 Symmetric inviscid flow over pitching NACA0012 airfoil: second mode of surface pressure coefficient. a Real component. b Imaginarycomponent

Fraction of chord

c p ,

thir

d m

od

e, r

eal c

om

po

nen

t

0 0.2 0.4 0.6 0.8 1-0.1

-0.05

0

0.05

0.1 TSM (7 intervals)TSM (8 intervals)BDF (32 intervals)

Fraction of chord

c p ,

thir

d m

od

e, im

agin

ary

com

po

nen

t

0 0.2 0.4 0.6 0.8 1

-0.06

-0.04

-0.02

0

0.02

0.04TSM (7 intervals)TSM (8 intervals)BDF (32 intervals)

a b

Fig. 9 Symmetric inviscid flow over pitching NACA0012 airfoil: third mode of surface pressure coefficient. a Real component. b Imaginarycomponent

each pair of independent instants should be 180◦. Appar-ently, splitting a period into eight equal intervals (or othereven numbers of intervals) automatically satisfies all theserequirements. Hence, the solution for a symmetric problem

is symmetric. If a period is divided into seven equal inter-vals (or other odd numbers of intervals), it is impossible togroup all the independent instants into pairs since the num-ber of independent instants is odd. Thus, the requirements

123


are not satisfied and the solution for a symmetric problem isnon-symmetric.

For the reason discussed above, we choose to use evennumbers of equal intervals in a period for the Fourier TSMas long as the computation can converge. Gopinath also justi-fied that the Fourier TSM using an even number of intervalsis usually stable for problems where the time derivative isrelatively small, such as the flow past pitching airfoils andwings at low forced frequencies [18]. In any of the presentcomputations using the Fourier TSM with even numbers ofintervals, the stability problem is not encountered. A possiblereason is that the frequencies in these flow problems are nothigh enough to trigger the stability problem. In addition tousing even numbers of intervals, we also make sure that thenumber of intervals is a power of 2 since efficient FFT canbe directly used in the Fourier TSM.

5 Error analysis on solutions for transonic andsubsonic flows

For smooth flows, time resolution by the Fourier TSM shouldbe of exponential order of accuracy as compared to the 2nd-order time accuracy of the BDF scheme used in Ref. [11].However, this may not be the case for flows with shockmotion. To evaluate comprehensively the computational effi-ciency of the Fourier TSM and compare it to that of theBDF method, error analysis is conducted to the computedsolutions. The previous test case for validation has shownthat whether or not shock wave movement exists has a greatinfluence on the time accuracy of the Fourier TSM. For thisreason, two flow problems are solved to provide solutions forerror analysis. The first one is still the previous test case forvalidation, in which the NACA0012 airfoil pitches aroundits quarter chord point with M∞ = 0.755, α0 = 0.016◦,αm = 2.51◦, and κ = 0.0814. In this flow problem, shockwaves move back and forth over airfoil surface. For the pur-pose of convenient comparison, the second flow problem isconstructed from the first one. In the constructed flow prob-lem, all the flow conditions remain the same except that thefree stream Mach number is lowered to M∞ = 0.6. Compu-tational results show that the flow of this problem is subsoniceverywhere. Hence, this flow problem is shock-free. To con-duct error analysis, five different time resolutions are used forthe Fourier TSM. Corresponding to these time resolutions, aperiod is split into 4, 8, 16, 32, or 64 equal intervals. Whenthe BDF method is applied, 16, 32, or 64 equal intervals areused in a period. The solution using the Fourier TSM with64 intervals is selected as the accurate solution. To ensurethe reliability of the error analysis, computations using theFourier TSMand theBDFmethod all converge to the residuallevel of 1 × 10−14.

McMullen and Jameson [19] chose to use lift and momentcoefficients as figures-of-merit when evaluating the compu-tational efficiency of the Fourier TSM. As the first step of thepresent error analysis, we follow the same idea to work onthe computed lift and moment coefficients as well. For anycomputed solution, the squared error of the lift coefficient ina period is defined as follows

Error2cl =m∑

n=0

[(Ren(cl) − Ren(cl,a))

2

+ (Imn(cl) − Imn(cl,a))2]

+ma∑

n=m+1

[(Ren(cl,a))

2 + (Imn(cl,a))2],

(26)

where Ren(cl) and Imn(cl) are the real and imaginary com-ponents of the n-th Fourier mode of the lift coefficient forthe given solution. Ren(cl,a) and Imn(cl,a) are the real andimaginary components of the n-th Fourier mode of the liftcoefficient for the accurate solution. m and ma are the num-bers of Fourier modes that the given solution and the accuratesolution include, respectively. The first summation in theEq. (26) represents the aliasing error of the given solution,whereas the second one is a measure of the truncation errorof the same solution. Similarly, for any computed solution,the squared error of the moment coefficient in a period canbe defined as follows

Error2cm =m∑

n=0

[(Ren(cm) − Ren(cm,a))

2

+ (Imn(cm) − Imn(cm,a))2]

+ma∑

n=m+1

[(Ren(cm,a))

2 + (Imn(cm,a))2],

(27)

where Ren(cm) and Imn(cm) are the real and imaginary com-ponents of the n-th Fourier mode of the moment coefficientfor the given solution. Ren(cm,a) and Imn(cm,a) are the realand imaginary components of the n-th Fourier mode of themoment coefficient for the accurate solution. According tothe definitions in Eqs. (26) and (27), the results of erroranalysis for the lift and moment coefficients are shown inFig. 10.

Figure 10a shows that for the flow with shock wave, theerror in the lift coefficient using the BDF method decreasesas the time resolution increases, and the drop rate of the erroris less than 2. In the log-log coordinate system, if the tempo-ral variation of the local pressure coefficient is smooth overthe entire airfoil surface, then the drop rate of its error and theerror of its integration quantities(such as lift coefficient) forthe 2nd-order BDF method should be exactly 2. If a shock

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L. Zhan et al.

Number of intervals

Err

or

in li

ft c

oef

fici

ent

(cl)

20 40 60 80 10010-7

10-6

10-5

10-4

10-3

10-2

10-1

Fourier TSM, with shock wave2nd-order BDF, with shock waveFourier TSM, shock free2nd-order BDF, shock freeLine with slope equal to -2

Number of intervals

Err

or

in m

om

ent

coef

fici

ent

(cm)

20 40 60 80 100

10-7

10-6

10-5

10-4

10-3

10-2

10-1


a b

Fig. 10 Error of the computed solutions. a For lift coefficient. b For moment coefficient

wave occurs in some region of the airfoil surface, the tempo-ral variation of the pressure coefficient is not smooth in theregion. Thus, for the solutions of the 2nd-order BDFmethod,the drop rate of the error in the pressure coefficient and thatof the lift coefficient all decrease below 2. Using the sametime resolution, the error of the lift coefficient for the FourierTSM is lower than that for the BDF method. This indicatesthat to obtain the lift coefficient with the same accuracy level,much fewer intervals are needed for the Fourier TSM. Forinstance, the lift coefficient calculated by the Fourier TSMwith eight intervals is as accurate as the one obtained by theBDF method with 64 intervals. So it can be concluded that ifthe lift coefficient is taken as the figure-of-merit, the FourierTSM is much more efficient than the 2nd-order BDFmethodeven when shock wave occurs.

Figure 10a also shows that in the shock-free case, the droprate of the error in the lift coefficient is exactly 2 for the 2nd-order BDF method. This is an expected result according totheoretical analysis. Under the same time resolution, the errorin the lift coefficient for the Fourier TSM is much lower thanthat for the BDF method. For instance, the lift coefficientcalculated by the Fourier TSM with only four intervals is asaccurate as the one obtained by the BDF method with 64intervals. This means if the lift coefficient is taken as thefigure-of-merit, the Fourier TSM is extremely efficient forthe shock-free case.

Figure 10b shows that the conclusion of the error analysisfor the moment coefficient is similar to that for the lift coef-ficient. However, if the moment coefficient is taken as thefigure-of-merit, the efficiency advantage of the Fourier TSMover the BDF method is weakened especially for the flowwith shock wave. This is because the temporal variation ofthe moment coefficient is much more complicated than thatof the lift coefficient if a shock wave occurs. For this reason,a higher time resolution is required for the Fourier TSM to

resolve the details of the temporal variation of the momentcoefficient.

The above error analysis is made on the integral quantitiesof the surface pressure coefficient. In practice, the computa-tional result of the surface pressure itself is also of greatimportance since it can reflect local flow details and key fea-tures. To evaluate the computational efficiency of the FourierTSM more comprehensively, the error analysis should beconducted on the computed surface pressure coefficient aswell. The following averaged squared error of the surfacepressure coefficient can be defined for a given solution:

Error2cp,averaged = 1

q

q∑s=1

m∑n=0

{[Ren(cp(s)) − Ren(cp,a(s))]2

+ [Imn(cp(s)) − Imn(cp,a(s))]2}

+ 1

q

q∑s=1

ma∑n=m+1

{[Ren(cp,a(s))]2

+ [Imn(cp,a(s))]2},

(28)

where Ren(cp(s)) and Imn(cp(s)) are the real and imaginarycomponents of the n-th Fourier mode of the pressure coef-ficient for the given solution. Ren(cp,a(s)) and Imn(cp,a(s))are the real and imaginary components of the n-th Fouriermode of the pressure coefficient for the accurate solution. qis the number of grid cells over the airfoil surface.

According to the definition in Eq. (28), the results oferror analysis for the surface pressure coefficient is shownin Fig. 11. The average error of the surface pressure coeffi-cient drops as time resolution increases in the similar trendthat can be observed for the lift coefficient or the momentcoefficient. However, for the flow with shock wave, the aver-

123


Number of intervals

Ave

rag

e er

ror

in p

ress

ure

co

effi

cien

t (c

p)

20 40 60 80 100

10-5

10-4

10-3

10-2

10-1


Fig. 11 Averaged error of the surface pressure coefficient

age error of the surface pressure coefficient for the FourierTSM becomes comparable to that for the BDF method if thetime resolution is relatively low (16 or fewer intervals areused in a period). For the flow with shock waves, the spec-trum of the surface pressure coefficient contains significantcomponents of higher modes due to the motion of the dis-continuous shockwave. They only start to decay dramaticallyafter the 7-th Fourier mode (seven modes are included if 16intervals are used in a period). In the shock-free area, thespectrum of the surface pressure coefficient is much simplerand all Fourier modes decay rapidly. When time resolution isrelatively low, the average error of the surface pressure coef-ficient mainly reflects the local error in the region of shockwave movement. In that region if 16 or fewer intervals areused, dominant Fourier modes are not included completelyin the Fourier TSM. Hence, the aliasing error and the trunca-tion error are high. Recall that the lift coefficient has a verysimple spectrum and all of its Fourier modes decay rapidly.Hence, the temporal variation of lift coefficient can be pre-dicted accurately by the Fourier TSM even when the timeresolution is pretty low.

The above error analysis demonstrates that for integralquantities of the surface pressure coefficient (especiallythe lift coefficient) in flows with shock waves, the FourierTSM is much more efficient than the BDF method. If thetemporal variation of the surface pressure coefficient itselfis simulated, the computational efficiency of the FourierTSM decreases, but is still not lower than that of the BDFmethod. For the shock-free flow problems, the Fourier TSMis extremely efficient for both the surface pressure coefficientand its integral quantities.

Given the number of pseudo-time steps needed to reachconvergence at each time instance being about the same, thetotal CPU time required by eachmethod depends on the num-ber of time instances to be computed in a period. The Fourier

TSM, being of exponential order of accuracy, achieves muchhigher time accuracy than the conventional 2nd-order BDFmethod, even for flows with shocks as discussed above. As aresult, to achieve the same level of time accuracy, fewer timeinstances (intervals) in a period are needed for the FourierTSM than for the conventional BDF time-marching method.For example, Fig. 11 shows that the Fourier TSM with eighttime intervals achieves the same accuracy as the BDF with64 time intervals for the shock-free case. In addition, toreach a steady-state periodic solution, the conventional time-marching solver has to go through an initial transient processby marching the real time forward for several (usually five ormore) periods. With the Fourier TSM, however, the steady-state periodic solution is obtained in one shot by solving thecoupled time-space equations without having to go throughan extra transient period. Combining these two factors, theFourier TSM is orders of magnitude faster than the con-ventional time-marching method for periodic flow problemswhere a high-order of accuracy in time is needed. In prac-tice, however, the computational efficiency advantage of theFourier TSM greatly hinges on the efficiency in solving thecoupled system of Eq. (22) and also on the smoothness of thesolutions. For problems where the solution contains discon-tinuities such as shock waves, the advantage of the FourierTSM time may be reduced.

6 Application to the periodic problems withunknown frequency

In the preceding computations of the flows over the pitch-ing NACA0012 airfoil, the frequency of the flow problems isknown since it is equal to the given pitching frequency. Forsuch computations, the Fourier TSM can be directly applied.Recall that the frequency is an explicit factor in the timespectral operator and it must be given for the Fourier TSM.There also exist flow problems in which the frequency is notknown before experiments or computations. A representativeflow problem of this kind is the laminar vortex shedding flowbehind a circular cylinder. Though the flow is known to beperiodic, the frequency is not known a priori. Only a roughestimation can be made by empirical formula. To obtain theaccurate frequency and the unsteady flow field using theFourier TSM or equivalent methods, a frequency searchingprocess must be involved. McMullen et al. [17] proposed agradient based variable time period (GBVTP)method for thefrequency domain method. Gopinath and Jameson [18] pro-posed a similar gradient based method to the Fourier TSM.In the gradient-based method, the frequency is updated usingthe negative gradient of the squared unsteady residual withrespect to frequency as follows

ωl+1 = ωl − α∂ R2

∂ω, (29)

123

L. Zhan et al.

whereα is a coefficient that controls the update and it has to becarefully chosen to guarantee convergence. Given the initialguess of the frequency that is sufficiently close to the correctvalue, the method can find the exact frequency precisely.

To broaden the search range and make the initial guessof frequency less constrained, we propose a new methodthat is based on Fourier analysis of the lift coefficient (oranother local or global flow variable) to estimate the fre-quency before the gradient-based method is used to obtainthe final converged value. For problems like vortex sheddingflow over a circular cylinder, the first Fourier mode of thelift coefficient usually achieves maximum amplitude. Apply-ing Fourier analysis to the lift coefficient, the frequency ofthe Fourier mode with maximum amplitude can be assignedto be the new frequency. When the first mode of the liftcoefficient reaches maximum amplitude, the gradient-basedmethod takes over the duty of searching for the correct fre-quency.

In the present computation, a circular cylinder is fixed inthe laminar flow at M∞ = 0.2 and Re = 180. A 257 × 129O-type mesh is used. The normal distance from the first gridpoint to the surface is 1×10−3, while the cylinder diameter isunity. The far-field boundaryof themesh extends to about 200diameters away from the cylinder. Based on the experiencefrom the preceding computations, eight real-time intervalsare employed when implementing the Fourier TSM. Sincethe present flow problem is shock-free, eight intervals shouldbe sufficient to apply the Fourier TSM. Convergence criteriafor the unsteady flowfield is set to be 1×10−6. The frequencysearch process doesn’t stop until the flowfield converges. Thepresent computational result is compared with experimentaldata [24–27].

The convergence history is shown in Fig. 12a. Using theFourier TSM and the proposed frequency search approach,the residual of the computation can finally drop to the conver-

Table 1 Vortex shedding flow behind circular cylinder

St Time-averaged cd

Williamson [24,25] 0.1919

Roshko [26] 0.185

Henderson [27] 1.336

Current computation 0.1919 1.337

gence criteria. The frequency updating historywith respect tomultigrid cycles is shown in Fig. 12b. The non-dimensionalangular frequency is defined as

ω′ = ωd√p∞ρ∞

, (30)

where d is diameter of cylinder, p∞ the free-stream pres-sure, and ρ∞ the free-stream density. The initial non-dimensional angular frequency is about 0.0237 (the cor-responding reduced frequency is 0.05), while the finalconvergednon-dimensional angular frequency is 0.2852.Theinitial guess of the frequency is far away from the correctvalue.

Finally, the Strouhal number is found and the convergedtime-averaged drag coefficient of the present computationare compared with experimental data in Table 1. For theStrouhal number, the current computational result is veryclose to Williamson’s [24,25] data. For the time-averageddrag coefficient, the current computational result matchesHenderson’s [27] data very well.

In the present computation, Fourier analysis of the liftcoefficient is carried out after every 100 multigrid cycles.The histories of lift coefficient at different time levels withrespect to multigrid cycles are shown in Figs. 13, 14, and 15.Figure 13a shows that during the first 100 multigrid cycles,

Multigrid cycles

Res

idu

al

1000 2000 3000 400010-6

10-5

10-4

10-3

10-2

10-1

100

Multigrid cycles

No

nd

imen

sio

nal

an

gu

lar

freq

uen

cy ω

’

100 101 102 103

0.05 0.05

0.1 0.1

0.15 0.15

0.2 0.2

0.25 0.25

0.3 0.30.35 0.350.4 0.4a b

Fig. 12 Convergence history of the flow over circular cylinder case. a History of residual. b History of computed frequency ω′

123


Multigrid cycles

Lif

t co

effi

cien

t c

l

20 40 60 80 100

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1t=0t=T/8t=2T/8t=3T/8t=4T/8t=5T/8t=6T/8t=7T/8

Fourier mode of cl

Am

plit

ud

e o

f F

ou

rier

mo

de

-1 0 1 2 3 4

0

0.002

0.004

0.006a b

Fig. 13 Flow over circular cylinder: lift coefficient and and its Fourier spectrum in multigrid cycle 1–100. a c1 history in multigrid cycle 1–100.b Spectrum of cl at the end of multigrid cycle 100

Multigrid cycles

Lif

t co

effi

cien

t c l

120 140 160 180 200

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1t=0t=T/8t=2T/8t=3T/8t=4T/8t=5T/8t=6T/8t=7T/8

Fourier mode of cl

Am

plit

ud

e o

f F

ou

rier

mo

de

-1 0 1 2 3 4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14a b

Fig. 14 Flow over circular cylinder: lift coefficient and and its Fourier spectrum in multigrid cycle 101–200. a c1 history in multigrid cycle101–200. b Spectrum of cl at the end of multigrid cycle 200

Multigrid cycles

Lif

t co

effi

cien

t c l

220 240 260 280 300

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t=0t=T/8t=2T/8t=3T/8t=4T/8t=5T/8t=6T/8t=7T/8

t=0t=T/8t=2T/8t=3T/8t=4T/8t=5T/8t=6T/8t=7T/8

Multigrid cycles

Lif

t co

effi

cien

t c l

1000 2000 3000 4000 5000

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

a b

Fig. 15 Flow over circular cylinder: lift coefficient in multigrid cycles after 200. a c1 history inmultigrid cycle 201–300. b c1 history after multigridcycle 300

123

L. Zhan et al.

the lift coefficient at different time levels generally oscillatewith the same phase. However, a phase difference starts toappear near the 100-th multigrid cycle. Figure 13b showsthat at the end of the 100-th multigrid cycle, the third Fouriermode of the lift coefficient achieves maximum amplitude.Since the third Fourier mode is the one with the highest fre-quency that the Fourier TSMwith eight intervals can include,the overall error of this mode is usually the highest amongthe included Fourier modes. For this reason, the frequency ofthe second Fourier mode is assigned as the new frequency.Figure 14a shows that in the next 100 multigrid cycles, thelift coefficient on different time levels continues to oscillatewith phase difference. However, the amplitude of the oscil-lation is growing gradually. Figure 14b shows that at theend of the 200-th multigrid cycle, the first Fourier mode ofthe lift coefficient already achieves maximum amplitude. So

Fraction of period

Dra

g c

oef

fici

ent c d

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

1.2

1.4

Drag due to pressure, originalDrag due to pressure, rebuiltDrag due to skin friction, originalDrag due to skin friction, rebuiltTotal drag, originalTotal drag, rebuilt

Fig. 16 Flow over circular cylinder: drag coefficients in a period

from the 201-th multigrid cycle, the gradient based methodstarts to search the correct frequency. In the third 100 multi-grid cycles, Fig. 15a shows that the lift coefficient on eachtime level stops oscillating gradually and approaches its con-verged value. After that, the lift coefficient on each time levelcompletely stops oscillating and finally converged. This isillustrated in Fig. 15b.

The temporal variations of the total drag coefficient andits components due to pressure and skin friction in a periodare shown in Fig. 16. The original discrete computationalresults, as well as the rebuilt results are plotted. The rebuiltresults are obtained by summing the resolved Fourier modeson 129 equally spaced instants over the obtained period. Itcan be observed that the drag due to pressure is the dominantcomponent of the total drag. Themean value and the variationamplitude of the drag due to pressure are all much larger thanthose of the drag due to skin friction. The temporal variationsof the lift and moment coefficients in a period are shown inFig. 17.

Figure 18 shows the vortex shedding process duringone shedding period. Mach number contours in flood typeand streamlines are plotted using the present computationalresults. Every vortex in this flow is initially generated on thesurface area near the trailing edge, then it grows larger pro-gressively. When the vortex is about the size of the cylinder,it detaches from the surface. After that, the detached vortexsheds away from the cylinder and dissipates in thewake. Thisprocess occurs alternately on upper and lower surface areanear the trailing edge.

7 Conclusion

The Fourier TSM for the unsteady Navier–Stokes equationsis presented and tested for a transonic flow over a pitching

Fraction of period

Lif

t co

effi

cien

t c l

0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8OriginalRebuilt

Fraction of period

Mo

men

t co

effi

cien

t c m

0 0.2 0.4 0.6 0.8 1-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008 OriginalRebuilt

a b

Fig. 17 Flow over circular cylinder: lift and moment coefficients in a period. a Lift coefficient. bMoment coefficient

123


X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

X

Y

0 2 4 6 8

-4

-3

-2

-1

0

1

2

3

4

Ma: 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

a b

c d

e f

g h

Fig. 18 Flow over circular cylinder: mach number contours and stream lines during one vortex shedding period. a t = 0. b t = 18T . c t = 2

8T .

d t = 38T . e t = 4

8T . f t = 58T . g t = 6

8T . h t = 78T

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L. Zhan et al.

NACA0012 airfoil and compared with the results of usinga conventional 2nd-order BDF method. Convergence studyof the time discretization errors with respect to the num-ber of time steps for the two methods show that the FourierTSM offers significantly better accuracy than the second-order BDF method for predicting both the local pressurecoefficient and integrated force coefficients for subsonic peri-odic flows. For transonic periodic flows where the motion ofthe discontinuous shock wave contributes significant higher-order harmonic components to the local pressure, a sufficientnumber of modes must be included before the Fourier TSMprovides an advantage over the BDF method. For example,more than 16 time intervals are needed in a period for theFourier TSM to resolve the surface pressure coefficient inregions of the shock-wave movement for the oscillating air-foil problem. The Fourier TSM, however, still offers betteraccuracy than the BDF method for integrated force coeffi-cients even for flows with shock waves. Computations alsoreveal a problem of non-symmetric solutions occurring forsymmetric periodic flows due to the use of odd numbers ofintervals. This non-symmetry is accentuated for the FourierTSM for transonic flows when an insufficient number of har-monic modes are included.

For problems where the frequency is not known a pri-ori, a search algorithm based on a combination of Fourieranalysis of the computed time-history of a flow quantity anda gradient method based on minimizing the unsteady resid-ual of the Navier–Stokes equations can be used to quicklyand accurately determine the frequency of the flow. Thisfrequency-search method along with the Fourier TSM aresuccessfully demonstrated for the periodic vortex sheddingproblem of the low Reynolds number flow over a circu-lar cylinder. Fourier analysis of the lift coefficient is firstperformed to estimate the unknown frequency before the gra-dient method is used to pin down the final value. The methodworks for initial guesses of the frequency that are far awayfrom the correct value. The Fourier TSM gives excellent pre-diction of the drag value and the Strouhal number comparedto experimental data with as few as only eight time intervalsfor this vortex shedding flow problem.

Acknowledgments The research was supported by the State Schol-arship Fund of the China Scholarship Council (Grant 2009629129).

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