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Bi–global Stability Analysis in Curvilinear

Coordinates

Jinchun Wang, Paul Ziadé, Guoping Huang, and Pierre E.Sullivan

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Citation (published version)

J. Wang, P. Ziadé, G. Huang, and P. E. Sullivan, “Bi-global stability analysis in curvilinear coordinates,” Phys. Fluids, vol. 31, no. 10, p. 105105, Oct. 2019. DOI: 10.1063/1.5118365

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Bi–global Stability Analysis in Curvilinear CoordinatesJinchun Wang(王进春),1, 2 Paul Ziadé,3 Guoping Huang(黄国平),4 and Pierre E.Sullivan5

1)College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu,China 2100162)Turbulence Research Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario,Canada M2L 2N5a)3)Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary, Calgary, Alberta,Canada T2N 1N4 b)4)College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu,China 210016c)5)Turbulence Research Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario,Canada M2l 2N5d)

(Dated: 4 October 2019)

A method is developed to solve bi–global stability functions in curvilinear systems which avoids reshaping of the airfoilor remapping the disturbance flow fields. As well, the bi–global stability functions for calculation in a curvilinear systemare derived. The instability features of the flow over a NACA (National Advisory Committee for Aeronautics) 0025airfoil at two different angles of attack, corresponding to flow with a separation bubble and fully separated flow, areinvestigated at a chord-based Reynolds number of 100,000. The most unstable mode was found to be related to the wakeinstability, with a dimensionless frequency close to one. For flow with a separation bubble, there is an instability plateauin the dimensionless frequency ranging from 2 to 5.5. After the plateau and for increasing dimensionless frequency,the growth rate of the most unstable mode decreases. For fully separated flow, the plateau is narrower than that forflow with a separation bubble. After the plateau, with increased dimensionless frequency, the growth rate of the mostunstable mode decreases and then increases once again. The growth rate of the upstream shear layer instability wasfound to be larger than the downstream shear layer instability.

I. INTRODUCTION

Airfoil performance is often limited or degraded by flowseparation, which is usually associated with loss of lift, in-creased drag, and kinetic energy losses. Thus, many methodsof flow control methods have been developed to suppress oravoid separation entirely1. Exploiting the instability of sepa-rated flow, periodic flow control methods improve on steadycontrol while maintaining the same energy input2. As a pre-cursor to developing control strategies, it is important to un-derstand and quantify the flow stability characteristics of sep-arated flow.

There are three kinds of linear stability analysis approaches:(a) classical stability, (b) bi–global stability and (c) tri–globalstability. Classical stability approaches, often based on theOrr–Sommerfeld equation, assume the basic flow is nonho-mogeneous in only one spatial direction. In the past, manystudies have investigated flow stability by utilizing the Orr–Sommerfeld equation3,4. This is limited by an assumptionof locally parallel flow, which is not altogether applicableto separated flow. Tri–global stability considers the three-dimensionality of the base flow and perturbations5,6. How-ever, there is a non–trivial memory requirement to solve theeigenvalue problem limiting its practical use; e.g., for a casewith 64 mesh points in each direction, 17.6 terabytes of mem-ory is required7.

a)Electronic mail: jinchun.wang@mail.utoronto.cab)Electronic mail: paul.ziade@ucalgary.cac)Electronic mail: hgp@nuaa.edu.cnd)Electronic mail: sullivan@mie.utoronto.ca

The bi–global stability approach considers the non–uniformity of the flow variables in two spatial directions andis a good option for airfoil analysis, as the variation of theflow parameters along the spanwise direction is significantlyweaker than the other two directions. Taking advantage ofthis, the bi–global stability method assumes the perturbationas a wavelike mode in the spanwise direction. Compared withtri–global stability analysis, the solution process of the bi-global stability method is simplified and the required mem-ory is significantly reduced. Despite this simplification, bi–global stability analysis is still computationally expensive, asvery large partial-derivative eigenvalue problems (EVP) mustbe solved. Exploiting the sparsity or developing high-orderfinite-difference schemes are helpful to speed up the solutionprocess. The high-order finite-difference scheme of order-q (FD-q) method was found to significantly outperform allother finite difference schemes in solving classic linear local,bi–global, and tri–global eigenvalue problems based on bothmemory and CPU time requirements8.

Bi–global stability analysis has been used in many appli-cations, e.g., channel flow9–11, flat plate12 and bluff bodyflows13,14. Bi–global stability analysis has also been appliedto flow past a NACA 0012 airfoil at high attack angle atchord-based Reynolds numbers (Rec = u∞c/ν) ranging from400–1000, where u∞ is the freestream velocity, c is the chordlength, and ν the kinematic viscosity15. It was found that thenear wake and far wake instabilities are the two dominant un-stable modes and with increasing wavenumber, the unstablemodes are suppressed. Kitsios et al.16 performed bi–globalstability analysis for flow past a NACA 0015 airfoil at an an-gle of attack of 18 degrees and Rec = 200. Most of the air-foil flows analyzed are at very low Reynolds number, around

J. Wang, P. Ziadé, G. Huang, and P. E. Sullivan, “Bi-global stability analysis in curvilinear coordinates,” Phys. Fluids, vol. 31, no. 10, p. 105105, Oct. 2019.https://doi.org/10.1063/1.5118365

2

200–200016,17. Non-classical stability analyses of Reynoldsnumber flows around 1× 105 are still very limited, and areimportant for unmanned aerial vehicles (UAVs), small-scalewind turbines, and low-speed aircraft, where flow separationis often encountered.

Typically, the input for bi–global stability analysis is thesteady or time-periodic base flow that is non-homogeneous intwo spatial directions at a given Reynolds number18. The baseflow can be stable or unstable. For stable flow, the base flowcan be obtained through solving the governing function of thebase flow. However, it is difficult to compute the base flowfor inherently unstable flows. Normally, selective frequencydamping19 or the mean flow can be used to obtain the baseflow for the analysis of global stability. However, the selectivefrequency damping method is sometimes unable to identify anunstable steady state20. Although the use of the mean flow isnot strictly correct with respect to the formulation of the lin-earized Navier-Stokes equations, solving the eigenmodes ofperturbations about the mean flow helps to identify how per-turbations can grow or delay with respect to the time-averageflow. These growing modes are helpful to aid in the design offlow control strategies20.

For the flow past an airfoil, the bi–global stability equationscannot be directly discretized. To solve the stability equationsin the curvilinear coordinate system, conformal mapping tech-niques are often used16. The mapping between the curvilin-ear and physical coordinates for an airfoil geometry is veryinvolved, requiring 4 mappings. The first maps the rectangu-lar curvilinear grid to cylindrical coordinates. A Joukowskytransformation is then used to convert the cylindrical grid toan airfoil shaped mesh. Then appropriate parameters are se-lected to best represent the airfoil. This airfoil shaped meshis then translated, scaled and finally rotated by the angle ofattack to align with the finite volume mesh to return the co-ordinates in physical space. Sometimes, it is difficult to findsuitable parameters to best represent the shape of a certainairfoil. There is a mapping relationship between the veloc-ity in physical coordinates and curvilinear coordinates. Thus,after solving the bi–global stability equations in curvilinearcoordinates, to show the spatial structure in physical space, aremapping process is needed. To avoid the reshaping of thegeometry of the airfoil and remapping of the flow field, a newmethod for solving the stability equation in the curvilinear co-ordinate system is proposed.

Post processing techniques, such as proper orthogonaldecomposition (POD)21 and dynamic mode decomposition(DMD)22, have been widely applied to analyze the un-steady characteristics of the coherent structure in differentapplications18,23,24. POD can be applied to both instantaneousnumerical and experimental snapshots and the mode of PODis hierarchically ordered with respect to their energy content.The frequency information is contained in the POD tempo-ral coefficient. The spatial distribution of the first two PODmodes was found to be typically coupled in pairs25. And thefirst two modes are also found to be representative of a vor-tex shedding phenomenon which is identified to be inducedby Kelvin–Helmholtz instability26,27.

The present work introduces a new method for solving the

bi–global stability equations in curvilinear coordinates. Thismethod is applied to the flow over a NACA 0025 airfoil todemonstrate its validity. Two states are considered: laminarseparation bubble and fully separated flow, corresponding toangles of attack (AOA) of 5 and 12 degrees, respectively, ata chord-based Reynolds number of 100,000. Stability charac-teristics and flow features are presented in a frequency rangeof F+ = 0− 11(= ωrc/2πu∞), where ωr is the angular fre-quency of the perturbation. The new method is compared withPOD analysis of the fully separated flow.

II. BASE FLOW COMPUTATION METHOD

The numerical computations of the base flow were per-formed using three dimensional large–eddy simulation (LES).The subgrid scale stress tensor,τi j, was modeled with an eddyviscosity approach,τi j = −2υrSi j, where υr is the eddy vis-cosity and Si j is the filtered strain rate tensor. A subgrid scaleturbulence kinetic energy model was employed. The tempo-ral and convective terms were discretized using a second or-der backward implicit time stepping scheme and second or-der total variation diminishing (TVD) scheme, respectively.An adaptive time stepping scheme was employed to main-tain a Courant—Friedrichs—Lewy (CFL) number of C0 < 0.7throughout the domain. The Pressure-Implicit with Splittingof Operators (PISO) algorithm was used for the pressure-momentum coupling. The airfoil surface was defined as ano-slip boundary condition and a periodic boundary conditionwas applied to the lateral boundaries, spaced c/3 apart, wherec is the chord length. The inlet and outlet were assigned lami-nar inflow and zero-gradient outflow conditions, respectively.

The computational domain with respect to the mid-sectionplane of the airfoil and bi–global stability analytic region areshown in Figure 1. The downstream section of the mesh ex-tends 11 chord lengths (11c) to the outlet boundary. Thehalf-height of the mesh is 6c and the lateral boundaries (notshown) are at a distance of c/3. Extending the downstreamsection to 15c resulted in no appreciable change (<1 %). Thecomputations were performed on 64-128 processors using theBlue Gene/Q (BGQ) and General Purpose Cluster (GPC) atScinet28. A block-structured mesh with 32× 106 cells wasemployed with mesh refinement concentrated in the wake andaround the NACA 0025 airfoil which had a chord length c =0.3m. For wall-resolved LES, it is well accepted that the re-quired mesh resolution, which has been achieved in all casespresented, is ∆x+ ≈ 100, ∆y+ ≈ 2 , and ∆z+ ≈ 2029,30. A de-tailed discussion of the validation of the computed flow canbe found in Ziadé and Sullivan31.

III. A NEW METHOD TO SOLVE THE BIGLOBALSTABILITY FUNCTION

The state variables (φ ) can be decomposed into the basestate (φ ) and the perturbation(φ ′)6

φ(x,y,z, t) = φ(x,y)+ εφ′(x,y,z, t) (1)

3

FIG. 1. Computational domain with respect to the mid-section planeof the airfoil

where φ(x,y) indicates the two-dimensional steady base flow,in this study chosen as the time-averaged flow along themidspan obtained by three-dimensional large-eddy simula-tion, and φ ′(x,y,z, t) is the perturbation. The perturbation wasassumed to have a form of

φ′(x,y,z, t) = φ(x,y)ei(β z−ωt)+ φ

∗(x,y)ei(−β z+ωt) (2)

where the ∗ superscript denotes the complex conjugate. Thesecond term is required because φ and ω in general are com-plex, while φ ′ must be real. β is the wavenumber of the struc-ture of the perturbation in the spanwise direction z. The realpart of complex value ω represents the angular frequency andthe imaginary part of ω corresponds to the growth/dampingrate of the associated amplitude function. A positive value ofthe imaginary component, Im(ω), indicates the exponentialgrowth of the perturbation, whereas a negative value of Im(ω)corresponds to the damping of the unstable mode. In the con-text of bi-global stability, the base flow in the spanwise direc-tion z , w, is assumed to be zero. When the above base flowsimplifications and modes of perturbation are substituted intothe Navier-Stokes equations and higher order terms (O(ε2))are neglected, the linearized Navier-Stokes equations are ob-tained:

ux + vy + iβ w = 0 (3)

−uux−vuy− uux− vuy− px +(uxx + uyy−β2u)/Rec =−iω u

(4)

−uvx− vvy− uvx− vvy− py +(vxx + vyy−β2v)/Rec =−iω v

(5)

−uwx− vwy− iβ p+(wxx + wyy−β2w)/Rec =−iωw (6)

where the subscripts denote partial differentiation with respectto the indicated variable. The bi-global stability equations arecast as a partial derivative eigenvalue problem.

Aφ = ωMφ (7)

φ = (u, v, w, p) (8)

A matrix-based approach is the most used method for bi-global stability analysis8,16. A is the spatial discretization op-erator, which is a function of the mesh, base flow, Reynoldsnumber (Rec), and wavenumber (β ). Finite difference meth-ods are often used to determine the expression for A. Thefinite difference method is performed on a set of discrete gridpoints. For the case of flow around the airfoil, the above equa-tions cannot be directly discretized.

To solve the stability equation in the curvilinear coordinatesystem, Kitsios et al. used conformal mapping to transformthe airfoil to a rectangular domain16. In the present work, baseflows are generated in physical space and need to be trans-formed to the curvilinear calculation domain before perform-ing the stability calculation. An O-grid geometry is used togenerate the grid for the stability analysis, as shown in Fig-ure 2. The basic velocity field is interpolated to the bi-globalgrid through a cubic spline interpolation method, on which thebi–global stability analysis will be undertaken.

FIG. 2. Mesh topologies employed for the stability analysis of theNACA airfoil

In order to avoid adding boundary conditions on each ad-jacent grid block, a code was written to merge the grids offour grid blocks into one block, as shown in Figure 3(a). Thecorresponding mesh used for stability analysis in curvilinearcoordinates is shown in Figure 3(b). The airfoil surface (L2)corresponds to j = 1 in the curvilinear coordinate system andthe far field (L1) corresponds to j = N, where N is the num-ber of mesh points in the j direction. First, the relationshipbetween the physical coordinate system (x,y) and the calcula-tion curvilinear coordinate system (i, j) must be established.

i = i(x,y); j = j(x,y) (9)

For corresponding points, they share the same value and sim-plify to

u(xA,yA) = u(iA, jA) (10)

so that when the bi–global stability equations are solved, thevalue of the perturbation in each grid can be obtained withoutremapping.

4

(a)

(b)

FIG. 3. Mesh topologies employed for the instability analyses of theNACA airfoil (a) Merging of O-grid to a single block (b) curvilinearcoordinates (i,j)

The variables in the physical coordinate system can be rep-resented as a function of the curvilinear coordinates. Forexample, considering the continuity equation, ux can be ex-pressed in parametric form in the computed coordinate systemas

ux = ui ∗ ix + u j ∗ jx (11)

Using the approach above, the bi–global stability equations inthe new curvilinear coordinate system are

ui ∗ ix + u j ∗ jx + vi ∗ iy + v j ∗ jy + iβ w = 0 (12)

−(ui ∗ ix +u j ∗ jx)∗ u−(ui ∗ iy +u j ∗ jy)∗ v−(pi ∗ ix + p j ∗ jx)−

u(ui ∗ ix + u j ∗ jx)− v(ui ∗ iy + u j ∗ jy)+{ui ∗ ixx + u j ∗ jxx

+uii ∗ (ix)2 + u j j ∗ ( jx)2 +2ui j ∗ ix ∗ jx +

ui ∗ iyy + u j ∗ jyy + uii ∗ (iy)2 +

u j j ∗ ( jy)2 +2ui j ∗ iy ∗ jy−β2u}/Rec

=−iω u

(13)

−(vi ∗ ix + v j ∗ jx)∗ u−(vi ∗ iy + v j ∗ jy)∗ v−

(pi ∗ iy + p j ∗ jy)−u(vi ∗ ix + v j ∗ jx)−v(vi ∗ iy + v j ∗ jy)+{vi ∗ ixx + v j ∗ jxx

+vii ∗ (ix)2 + v j j ∗ ( jx)2 +

2vi j ∗ ix ∗ jx + vi ∗ iyy + v j ∗ jyy +

vii ∗ (iy)2 + v j j ∗ ( jy)2 +2vi j ∗ iy ∗ jy−β2v}/Rec

=−iω v(14)

−u(wi ∗ ix + w j ∗ jx)−v(wi ∗ iy + w j ∗ jy)−

iβ p+{wi ∗ ixx +

w j ∗ jxx + wii ∗ (ix)2 + w j j ∗ ( jx)2 +

2wi j ∗ ix ∗ jx + wi ∗ iyy + w j ∗ jyy + wii ∗ (iy)2 +

w j j ∗ ( jy)2 +2wi j ∗ iy ∗ jy−β2w}/Rec

=−iωw(15)

It should be noted that the value of ix should be solved throughan inverse transformation.

x = x(i, j); y = y(i, j) (16)

Differentiating, it is possible to obtain

dx = xi ∗di+ x j ∗d j (17)

dy = yi ∗di+ y j ∗d j (18)

di = ix ∗dx+ iy ∗dy (19)

d j = jx ∗dx+ jy ∗dy (20)

5

As a matrix, this becomes

[ix iyjx jy

]=

[y j −x j−yi xi

]∣∣∣∣ xi x j

yi y j

∣∣∣∣ (21)

and the coefficients in the above equations can be solved.0n the airfoil surface (L2), the boundary conditions u = v =

w = 0 are imposed on the perturbation velocities and the com-patibility condition for pressure of a zero-wall-normal gradi-ent is employed32. Since the properties of the perturbationsare not known on the boundary L1 before solving the EVP,the amplitude functions are linearly extrapolated from withinthe domain16. L3 and L4 share the same line, and the interiorboundary condition is imposed.

The bi-global stability equations are cast as a partial-derivative eigenvalue problem in the curvilinear system, eq.7and solved with a purpose–written code. With such large–scale partial–derivative eigenvalue problems, it is compu-tationally more efficient to solve for the eigenvalues near-est a certain point in the complex plane using a shift-and-invert method, such as the implicitly restarted Arnoldi method(IRAM). Thus, the above eigenvalue problem is modified asshown, given a complex shift σ :

(A−σM)−1Mφ = φ/(ω−σ) (22)

The Arnoldi Package (ARPACK) library was used on a clusterwith 500 GB memory. A study of domain size was conductedto ensure the convergence of eigenspectrum.and grid resolu-tion refinement was performed to ensure accurate stability re-sults. The code was validated against the results of Theofilis etal. 32 for Poiseuille flow in a rectangular domain at Re = 100and β = 1 and showed good agreement.

IV. INSTABILITY FEATURES IN FLOW WITHSEPARATION BUBBLE

The base flow for an airfoil angle of attack of 5 degrees andRec = 1×105 is shown in Figure 4. There is a small separationbubble on the airfoil surface near the middle, and the flowreattaches prior to the trailing edge. The time–averaged flowis dominated by a small recirculation zone in this area.

When the Reynolds number is very low (below 2000), theangular frequency of the most unstable mode is close to 0and there is only 1 or even no unstable mode in some cases,which means that setting the ”shift value” σ to zero is theproper choice16. When the Reynolds number is higher, as inthis case, there are many unstable modes. To deploy the fullpower of the Arnoldi algorithm33, several shift values σ arechosen to capture the eigenvalues with the dimensionless fre-quency F+ from 1 to 11. The variation of the growth rateof the most-unstable/least-stable eigenvalue with respect tothe dimensionless frequency is shown in Figure 5. The non-dimensional growth rate ωin is

ωin =ωiδ

∗

2πUr(23)

FIG. 4. The base flow of the NACA 0025 airfoil at AOA = 5◦

where δ ∗ is the thickness of the separation bubble and Uris the characteristic velocity chosen as the maximum reverseflow velocity. The spanwise perturbation wavenumber was setto zero, which examines the stability characteristics of two-dimensional flow. The dimensionless frequency of the mostunstable modes in the whole solved range was found to bearound 1 and the spatial structure of this most unstable mode isshown in Figure 6. Normalizing the complex eigenfunctionsusing the corresponding maximum absolute values, contourplots are generated for the real components of these nondi-mensional eigenfunctions. The alternating velocity perturba-tion originates approximately 2 chord lengths downstream ofthe airfoil. The wake instability mode is dominant and itsfrequency is important for flow control. Many studies haveshown that the best control effect occurs when the excitationfrequency of an active control system is F+ = 1 2. Addition-ally, effective control can be achieved when the dimensionlessexcitation frequency range varies from 0.25 < F+ < 2.0 1,34.In the dimensionless frequency range from 2 to 5.5, there isa plateau where the frequency has little effect on the growthrate of most unstable modes. After the plateau, with increaseddimensionless frequency, the growth rate of the most unstablemodes significantly decreases, as shown in Figure 5.

It can be seen that 39% of all unstable modes are locatedin the dimensionless frequency range of 0.5 to 1.5. With in-creased dimensionless frequency, both the largest growth rateand the number of unstable modes decrease. When the di-mensionless frequency is larger than 6, there are only a fewunsteady modes and the unsteady growth rate is significantlydecreased. This confirms past experimental work that controlis better achieved over this low frequency range.

Monotonically growing modes corresponding to F+ = 0,known as stationary modes, are also found at this Reynoldsnumber. The spatial structure of this mode is shown in Fig-ure 7. While this mode does not fluctuate in time (F+ = 0),

6

ω

FIG. 5. The variation of the growth rate of the most-unstable/least-stable eigenvalue with respect to the dimensionless frequency forβ = 0

it does grow (ωin ≈ 0.032). This kind of stationary mode isself–excited and was also identified in other studies16,35. Ro-driguez et al. 36 demonstrated the requirement of a minimumreverse flow magnitude within the separation bubble to inducethe occurrence of such a global mode was between 7% and 8%of the local free-stream velocity. The maximum amplitude ofthe reverse flow encountered within the separation bubble is∼ 18% of the local free-stream velocity in this case.

V. INSTABILITY FEATURES IN A FULLY SEPARATEDFLOW

The mean velocity magnitude at the midspan of the airfoilat an angle of attack of 12 degrees and Rec = 1×105 is shownin Figure 8. Flow separation occurs shortly downstream of theleading edge. A region of low velocity persists from separa-tion well beyond the trailing edge of the airfoil. In this case,there is no boundary layer reattachment and flow is stalled.Again, the base flow validation is presented in Ziadé et al.31.

The variation of the growth rate of the most–unstable eigen-value with respect to the dimensionless frequency is shown inFigure 9. The spanwise perturbation wavenumber was alsoset to zero. The dimensionless frequency of the most unstablemodes over the range was found in F+ = 1 and F+ = 1.45.The spatial structure of F+ = 1 is shown in Figure 10(a) us-ing the dimensionless real part of the streamwise velocity. Thespatial structure of F+ = 1.45 is similar with the spatial struc-ture of F+ = 1. This structure also exhibits features of thewake type mode. The alternating velocity perturbation origi-nates approximately 1.2 chord lengths downstream of the air-foil, which is closer to the trailing edge than that in 5 degreeflow fields.

(a)

(b)

FIG. 6. Spatial structure of most unstable mode(F+ = 1.0) (a) Eigen-function of Re(u)/Re(u)max (b) Eigenfunction of Re(v)/Re(v)max

FIG. 7. Eigenfunction of Re(u)/Re(u)max) with F+ = 0 (stationarymode)

The instability plateau ranges about from 2 to 3.5, whichis shorter than that in the 5 degree flow field. However, thelargest growth mode did not monotonically decrease with thedimensionless frequency after the plateau. When F+ is near 5,

7

FIG. 8. The base flow of the NACA 0025 airfoil at AOA = 12◦

the largest growth rate is found to reach a minimum. The spa-tial extent of this mode is shown in Figure 10(b). This modeis located along the downstream shear layer and is named thedownstream shear layer mode, which has not received muchattention 37. The largest magnitude of this mode exists in thestreamwise location 0.73c and becomes weaker as it developsdownstream. This modes mainly exists in the streamwise lo-cation 0.2–2.6c. Further increasing the frequency, the mostunstable growth rate increases once again. This feature is dif-ferent from that seen at 5 degrees. The number of unstablemodes in this region is significantly greater than for the 5 de-gree flow fields. The spatial structure of F+ ≈ 10 is shownin Figure 10(c). This mode is associated with the upstreamshear layer instability. The upstream shear layer mode mainlyexists in the streamwise location 0.08–1.32c, and the largestmagnitude, located at the 0.38c, becomes weaker as it devel-ops downstream. Compared with the downstream shear layermode, the position of the largest magnitude of upstream shearmode is farther upstream. Most of the unstable modes arelocated in the dimensionless frequency range from 0.5 to 1.5,similar to the previous case. The number of unstable modes inthe high frequency region (F+ > 6.5) is significantly greaterthan that in the 5 degree flow field.

VI. POD ANALYSIS OF FULLY SEPARATED FLOW

The time-dependent flow field V (x,y, t) can be approxi-mately treated as the sum of average and fluctuating compo-nent:

V (x,y, t) =V (x,y)+V ′(x,y, t) (24)

Using POD, the fluctuating component can be written as thefollowing summation:

V ′(x,y, t) =n

∑i=1

αi(t)Φi(x,y) (25)

ω

FIG. 9. The variation of the growth rate of the most-unstable/least-stable eigenvalue with respect to the dimensionless frequency forAOA = 12◦

The temporal characteristics of mode i are reflected by thePOD temporal coefficients, αi(t).

POD attempts to decouple the spatial and temporal struc-ture of the unsteady flow field as the composition of the vari-ous modes with different amplitudes.

The distribution of energy among the POD modes with re-gards to the transverse velocity is presented in Figure 11. Thefirst and second modes are energetically similar, containingroughly 10 % of the energy each. Modes 1 to 10 are paired;i.e., odd–even modes with energy of similar magnitude. Theenergy ratio of the higher modes are at least one order of mag-nitude less. For example, the energy ratio of the 30th mode isless than 1%. POD provides an orthogonal set of minimalnumber of basis vectors, which is helpful in constructing areduced-order model of the unsteady flow field. Comparedwith the bi–global stability analysis, POD arranges modes ac-cording to energy content. However, here the low energy as-sociated with these modes minimizes the physical associationof the flow structures to the mode. In this case, the flow modesdo not directly correspond to the instabilities previously iden-tified experimentally or computationally.

The spatial structure of the first and second POD mode areshown in Figure 12. It can be seen that these two modes aresimilar but with a phase difference, both positioned down-stream for the trailing edge. With increasing mode num-ber, the spatial structure moves upstream along the airfoil, asshown in Figure 13 (a). The temporal coefficient of the firsttwo modes is shown in Figure 14, where twake is the periodof the wake obtained in the stability analysis. It can be seenthat the coefficients of these two modes also have shift valueabout π/4. Furthermore, the dominant frequency is equal tothe wake frequency previously obtained. Further increasingthe modes, the spatial structure breaks down and becomes

8

(a)

(b)

(c)

FIG. 10. Eigenfunction of Re(u)/Re(u)max (a) F+ = 1 (b) F+ ≈ 5(c) F+ ≈ 10

FIG. 11. Energy ratio of POD eigenvalues for wall normal velocity

FIG. 12. Spatial structure of POD mode 1 and mode 2

very noisy. Compared with POD, bi–global stability providesa clearer separation of the wake instability and shear layer in-stability modes.

9

(a)

(b)

FIG. 13. Spatial structure of POD mode 19 and mode 141 (a) Mode19 (b) Mode 141

VII. CONCLUSIONS

A method is presented to solve the bi–global stability equa-tions in curvilinear coordinates which avoids the reshaping ofthe airfoil or remapping the disturbance flow fields. The bi–global stability equations for a curvilinear system were de-rived. With the validated code, the instability features of air-foil flow at two different angles of attack, corresponding toflow with a separation bubble and fully separated flow, wereinvestigated at a chord-based Reynolds number of 105. Themost unstable mode was found to be related to the wake in-stability with a dimensionless frequency close to one. Forflow with a separation bubble, there is an instability plateau

FIG. 14. POD temporal coefficient of POD mode 1 and mode 2

in the dimensionless frequency range from 2 to 5.5. After theplateau, with increased dimensionless frequency, the growthrate of the most unstable mode decreases. For fully separatedflow, the plateau is narrower than for flow with separationbubble. After the plateau, with increased dimensionless fre-quency, the growth rate of the most unstable mode reaches aminimum and then increases once again. The growth rate ofthe upstream shear layer instability was larger than the down-stream shear layer instability. A stationary global mode wasalso found for these two cases. For stalled flow, the spatialstructure of the wake instability mainly exists downstream,while the upstream shear layer instability zone was upstream1.32 chord lengths. Proper orthogonal decomposition wasperformed on the fully separated flow. The dominant fre-quency obtained using proper orthogonal decomposition is inagreement with the result from the bi-global stability method.Bi–global stability analysis, in comparison to proper orthogo-nal decomposition, better separates the wake and shear layerinstability modes allowing improved insight for control strate-gies.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial supportof the Natural Sciences Engineering Research Council ofCanada, FedDev Southern Ontario and China ScholarshipCouncil. This work was also supported by the National Ba-sic Research Program of China (2014CB239602) and theNational Natural Science Foundation of China (51176072).Computations were performed on the GPC supercomputer atthe SciNet HPC Consortium. SciNet is funded by: the CanadaFoundation for Innovation under the auspices of ComputeCanada; the Government of Ontario; Ontario Research Fund -

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