Numerical Investigation of Active Flow Control of

Numerical Investigation of Active Flow Control of Low-Pressure Turbine Endwall Flow S. Romero∗ and A. Gross† Mechanical and Aerospace Engineering Department, New Mexico State University, Las Cruces, NM 88003

Simulations of the flow through a low-pressure turbine passage with highly-loaded L2F profiles were performed for an inlet Mach number of 0.1 and a chord-based Reynolds number of 100,000. An analysis of the mean flow reveals a horseshoe vortex at the junction. One leg of the horseshoe vortex develops into a passage vortex. Limiting skin-friction lines indicate that a strong secondary flow displaces the flow on the blade suction surface away from the endwall. Time-dependent data recorded in the junction flow region provide evidence of a low-frequency unsteadiness of the horseshoe vortex. A dynamic mode decomposition indicates that both the passage vortex and the suction side corner flow maybe susceptible to unsteady flow control. Various steady and unsteady flow control strategies were investigated for countering the secondary flow and weakening the passage vortex. Steady blowing against the secondary flow with large momentum coefficient lowers the total pressure losses. Unsteady blowing aimed at a reduction of the coherence of the passage vortex is less effective but much more efficient than steady blowing. A technique for channeling air from the pressure side to the suction side to directly counter the secondary flow was explored but requires further optimization to be successful.

I.

Introduction

Advances in modern turbofan engines are driven by both practical considerations (e.g. increased efficiency and performance, reduced operating and maintenance costs, etc.) and scientific advancement (e.g. improved understanding of flow physics, new materials, etc.). Turbofan engines are very common in commercial and military aviation which explains the continued strong demand for lighter and more efficient engines. For high bypass-ratio turbofan engines, the fan produces most of the thrust. As a result, the low-pressure turbine (LPT), which drives not only the fan but also the low-pressure compressor, has to extract an enormous amount of energy from the flow. Axial flow turbines generally consist of several stages. A turbine stage is illustrated in Fig. 1. The stage efficiency, η=

1 − (T03 /T01 ) , 1 − (p03 /p01 )(γ−1)/γ

(1)

can exceed 90% in modern engines. Because the LPT rotor speed and Mach number are relatively low, the work per stage is limited. Hence, multiple stages are required which imposes a weight penalty.1 In fact the LPT can account for as much as one third of the total engine weight.2 Because of its large weight, the LPT is a prime target for efforts aimed at a reduction of the total engine weight. One way to reduce the LPT weight is to increase the stage loading which would allow for a wider blade spacing and thus a reduced solidity. The Zweifel number, S Z=2 cos2 α2 (tan α1 − tan α2 ) , (2) Cax is commonly used for quantifying the blade loading. This parameter measures the amount of available work that can be extracted from a turbine blade. Starting from the Pratt and Whitney Pak-B profile, new LPT profiles with much higher Zweifel number, such as the front-loaded L2F,3 have been developed at the Air Force Research Laboratory (AFRL). Front-loading weakens the adverse pressure gradient, thus delaying ∗ Graduate † Assistant

Research Assistant. Member AIAA. Professor. Senior Member AIAA.

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Rotor

Stator

𝛼1

𝑢2 𝑢1

𝛼2 U

𝑝01 𝑇01

𝑝02 𝑇02

𝑝03 𝑇03

1

2

3

Figure 1: Turbine stage. or preventing laminar flow separation from the suction surface. The high loading of the L2F allows for a 38% reduction in blade count over the Pak-B profile. Unfortunately, the L2F was found to induce a strong secondary flow that redirects endwall momentum in the pitchwise direction. Secondary flows typically drain energy from the primary flow and thus incur losses. Detailed studies and review papers on secondary flow losses in turbines can be found in the literature (e.g. Refs. 4–7). Lyall et al.8 demonstrated that the high stagger angle of the L2F produces a strong pressure blockage near the leading edge. As a result of the blockage, fluid from inside the passage is entrained into the horseshoe vortex system at the junction which intensifies the secondary flow. Lyall et al.8 contoured the L2F (endwall fillet) to reduce the stagger angle near the endwall and named the new blended profile L2F-EF. For Reynolds numbers below 50,000 the contouring reduced the total pressure losses by as much as 13%. For typical cruise Reynolds numbers above 50,000, the contouring was less effective. Marks et al.9 investigated the effect of Reynolds number and blade geometry on the total pressure losses. The endwall total pressure losses were found to increase with Reynolds number. Out of a number of profile modifications for the L2F, the L2F-EF provided the most substantial total pressure loss reductions at Re = 100, 000. Based on detailed particle image velocimetry (PIV) measurements for a L2F passage at Re = 100, 000, Bear et al.10, 11 determined that the bulk of the total pressure losses were generated near the endwall and in the wake. In agreement with earlier measurements by Sangston et al.,7 Bear et al. noted that the normal components of the deformation work were small compared to the shear components. Insight into the endwall flow physics has also been obtained from high-fidelity numerical simulations. Implicit large eddy simulations (ILES) for a chord-based Reynolds number of 100,00012–15 were in good agreement with PIV measurements16 and oil-flow visualizations.9, 14 Romero and Gross15 showed that lowReynolds number laminar simulations can serve as a viable alternative to ILES. The reduced computational cost of the low-Reynolds number simulations allowed for a computationally inexpensive evaluation of different active flow control (AFC) strategies. It was demonstrated that unsteady endwall-normal blowing can effectively alter the position of the passage vortex resulting in a maximum total pressure loss reduction of 8%. Low-pressure turbine endwall losses and methods for their active control have been investigated by a large number of researchers. The general goal of the control is to increase the stage efficiency (Eq. 1) by raising the total pressure ratio. Due to the complexity of the LPT endwall flow, a successful application of AFC can be challenging. Bloxham and Bons17 employed vortex generator jets (VGJs) to target both profile and endwall losses for a linear cascade of aft-loaded blades at Re = 50, 000. A row of VGJs that extended up to midspan was placed on the suction surface at x = 0.59Cax . The jets were pitched and skewed by 30 and 90deg, respectively. In addition, steady suction was applied through an array of endwall holes. The suction pulled high momentum free-stream fluid into the near wall region, thus reducing flow separation. The combination of VGJs and suction holes constituted an overall zero-net mass flux control system. The area-averaged total 2 of 27 American Institute of Aeronautics and Astronautics

pressure losses were reduced by up to 28%. The majority of the loss reduction was attributed to the VGJs. Benton et al.18 carried out wind tunnel experiments for a linear L2F cascade at Re = 80, 000. Steady VGJs were placed at 60% axial chord on the suction surface. A pitch and skew angle of 30deg were chosen such that the VGJs opposed the rotation of the passage vortex while still issuing the majority of the injected momentum in the streamwise direction. Maximum total pressure loss reductions of 42% were reported for a jet momentum coefficient of cµ = 0.04. In a follow up paper, Benton et al.19 reported on experiments with pulsed VGJs in the junction flow region. Particle image velocimetry measurements in an outflow plane showed that the lateral motion of the passage vortex was reduced for a forcing frequency of f = 1.0 (made dimensionless with the axial chord length and the inlet velocity). For f = 0.4 the passage vortex motion was amplified, resulting in enhanced mixing and reduced total pressure losses. This paper builds on and continues earlier LPT research12–15 by making contributions in three areas, the understanding of the steady and unsteady flow field, the characterization of the total pressure losses, and the development and evaluation of AFC strategies. First, the setup of the simulations and the postprocessing techniques are explained. In the results section, the time-averaged and unsteady uncontrolled flow are analyzed and, based on the analysis, six different AFC strategies are proposed and evaluated. Finally, a brief summary and conclusions are provided.

II. A.

Methodology

Numerical Method

The compressible Navier stokes equations in the finite volume formulation were solved with a research computational fluid dynamics code by Gross and Fasel.20, 21 The convective terms were discretized with a ninth-order-accurate WENO scheme. A forth-order accurate discretization was employed for the viscous terms. The simulations were advanced in time with a second-order-accurate implicit Adams-Moulton scheme. To allow for a comparison with the AFRL experiments (e.g. Refs. 9–11) upstream wakes and surface roughness were not considered. The reference Mach number for the simulations was Mref = 0.1. This Mach number is low enough to satisfy the incompressible flow assumption. The Prandtl number was P r = 0.72 and the molecular viscosity, µ, was calculated using Sutherland’s law. The inlet temperature was 300K. All flow quantities were made dimensionless with the reference values listed in Tab. 1. Parameter Length scales (x, y, z) Velocities (u, v, w) Density (ρ) Pressure (p) Temperature (T ) Specific energy (e) Time (t)

Reference value Axial chord length, Cax Inlet velocity, uref Inlet density, ρref ρref u2ref Inlet temperature, Tref u2ref Cax /uref

Table 1: Reference quantities.

B.

Simulated Cases

All simulations are for a linear cascade with L2F airfoils that has been investigated experimentally at AFRL.9–11 The axial chord, pitchwise spacing, and blade span are Cax = 6in, S = 1.221Cax , and H = 4.167Cax , respectively. The L2F blade has a design inflow angle of 35deg, a stagger angle of 34.5deg, and an exit angle of approximately −60deg. The L2F was developed at AFRL for investigating the aerodynamics of low-Reynolds number high-lift airfoils.3 The reference Reynolds number for the simulations (based on axial chord and inlet velocity) was 100,000. Simulations were carried out without flow control (uncontrolled case, baseline case) and using six AFC strategies. First, the baseline case was advanced in time from 0 to 18. The t = 18 solution for the baseline case then served as “starting point” (initial condition) for the controlled cases. The cases with flow control were advanced in time over a time interval of six before the time-averaging was initiated. Time averages

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were then computed over time intervals of six for the controlled cases and 10 for the baseline case. The total simulation times were 6 + 6 = 12 for the controlled cases and 18 + 6 + 10 = 34 for the baseline case. The computational timestep was 0.001. C.

Computational Grid

A two-dimensional (2-D) multi-block structured grid (shown in Fig. 2) was generated with a Poisson grid generator.22 The grid consists of nine blocks and is periodic in the pitchwise direction. The axial extend of blocks 1-7 is −1.554 ≤ x/Cax ≤ 2 whereas blocks 8 and 9 reach upstream to x = −3.262 and −3.747, respectively.

Figure 2: Computational grid with block numbers (top, every other grid point is shown in each direction). Detail of leading edge (bottom-left) and trailing edge (bottom-right). Because for z ≥ H/4 the mean-flow is approximately 2-D, only a quarter-span of the cascade was meshed to reduce the computational expense of the simulations. A comparison of the mean flow fields obtained from a half- (not shown) and quarter-span simulation showed that the restriction of the spanwise domain extent had no impact on the flow near the endwall. The 2-D grid for blocks 1-8 was extruded in the spanwise direction over the full quarter-span using a grid stretching function that clusters cells near the endwall. The spanwise grid extent of block 9 was 0.07349 and cells were clustered near both spanwise boundaries. Block 9 was employed for an independent turbulent temporal channel flow simulation that provided unsteady inflow data for the spatial LPT simulations.13, 15 The total number of cells was 22.4 million. The wall-normal grid line spacing in wall units, ∆y + , at the wall was less than unity as recommended for ILES simulations23, 24 (Fig. 3). D.

Boundary Conditions and Volume Forcing

For block 9, (turbulent channel flow simulation) the walls were considered to be iso-thermal (to remove the heat added by viscous dissipation) and streamwise flow periodicity was enforced. A volume forcing term was added to compensate for the streamwise pressure drop,13, 15 ∂p 1 1 = ρvb2 f . ∂s 2 Dh

4 of 27 American Institute of Aeronautics and Astronautics

(3)

Figure 3: Spanwise-averaged wall-normal grid resolution in wall units. Here, vb is the bulk velocity, f is the friction factor, and Dh = 2h is the hydraulic diameter. Marks et al.9 provided boundary layer measurements at a streamwise distance of s = −1.5Cax (9in) upstream of the cascade (x = −1.5Cax cos 35deg = −1.23Cax ). The volume force was adjusted until the endwall boundary layer thickness measurements at x = −1.23 were approximately matched.13 All other walls were considered as adiabatic. Flow periodicity was imposed in the pitchwise direction. Non-reflecting boundary conditions25 were applied at the inflow and outflow boundary. Symmetry conditions were enforced at the quarter-span boundary. E.

Total Pressure Loss Coefficient

The dynamic and total pressure are, 1 ρ(u2 + v 2 + w2 ) 2 ρT p0 = p + q = +q. γM 2 q=

(4) (5)

For the chosen non-dimensionalization, the inflow values become 1 2

(6)

1 1 + . γM 2 2

(7)

p0 − p0,ref . qref

(8)

qref = p0,ref = The total pressure coefficient is defined as cp,0 =

The difference between the inlet and outlet total pressure coefficient is referred to as total pressure loss coefficient, cp,0,in − cp,0,out . (9) Area and mass averages of a mean-flow quantity, ϕ(y, z), for a fixed axial location (x = const.) can be computed as, Z Z 1 ϕA = ϕ dy dz (10) A H S Z Z 1 ϕ00 = ρ|~v | ϕ dy dz , (11) m ˙ H S respectively. Here, |~v | is the magnitude of the streamwise velocity. Setting ϕ = cp,0 in Eq. 11 gives P Z Z v |cp,0 Vi 1 1 X 00 i ρ|~ ρ|~v |cp,0 dy dz ≈ ρ|~v |cp,0 Vi = P , cp,0 = m ˙ H S m ˙ i ρ|~ i v |Vi 5 of 27 American Institute of Aeronautics and Astronautics

(12)

where the sums are taken over the volumes, Vi , of the cells that are directly adjacent to the inflow and outflow boundary. The mass-averaged total pressure loss coefficient can then be computed as Ytot = c00p,0,in − c00p,0,out .

(13)

A similar expression for the area-averaged total pressure loss coefficient can be obtained by considering Eq. 10 instead of Eq. 11. F.

Proper Orthogonal Decomposition

For a given number of modes and a kernel based on the kinetic energy, the proper orthogonal decomposition (POD)26 of a time-dependent flow field captures more of the flows kinetic energy than any other decomposition. For the present results the “snapshot” method by Sirovich27 was employed. For discrete temporal data this method is computationally more efficient than the original method. The instantaneous velocity fields, vn , are stored in an array, V0N = [v0 v1 . . . vN ] , (14) where 0 ≤ n ≤ N . A singular value decomposition is employed to obtain a low-rank approximation of the data,28 V0N = U ΣW T . (15) Here, U = V0N W Σ−1 are the POD modes and ΣW T are the POD time-coefficients. An eigenvalue decomposition of the temporal correlation matrix is performed, T

V0N V0N = W ΣT U T U ΣW T = W Σ2 W T .

(16)

T

The matrix V0N V0N is symmetric and therefore has only real eigenvalues. The entries on the main diagonal Σ2 are referred to as POD eigenvalues. The eigenvalue magnitudes are identical to two times the kinetic energy content of the respective modes. The POD decomposition allows for an expression of the velocity data in the form, N X ai (t)σi , (17) vn = i=0

with POD time-coefficients, ai , and POD modes, σi . G.

Forward Dynamic Mode Decomposition

The DMD is based on the premise that modes are exponentially growing or decaying in time. In AFC applications, unstable modes are of particular interest, as they can be exploited for making an AFC strategy more effective and efficient. For the forward DMD, a linear mapping from timestep n to timestep n + 1 is assumed,28, 29 vn+1 = Avn , (18) with constant matrix A. For data from non-linear simulations, this assumption amounts to a linear tangent approximation. The assumption is supposed to hold for all timesteps such that one can also write V1N = AV0N −1 ,

(19)

Y = AX .

(20)

or to simplify notation, The dimensions of A are N × N . Because of the large size of the matrices, a direct inversion A = Y X −1 is not feasible. Rank deficiencies in V0N −1 can be accounted for via a restriction to a limited projection basis of POD modes (with eigenvalues above a threshold, i.e., keeping only the R largest POD modes), X = V0N −1 ' UR ΣR WRT .

(21)

Different from section F, the POD is performed for 0 ≤ n ≤ N − 1 and not 0 ≤ n ≤ N . From this, V1N T N UR V1 WR Σ−1 R

= AV0N −1 = AUR ΣR WRT =

URT AUR

.

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(22) (23)

Equation 20 is projected on the POD basis, Y URT Y Y˜

= AX = URT AUR URT X ˜, = A˜X

(24)

˜ T ˜ where A˜ = URT AUR = URT V1N WR Σ−1 R (and A = UR AUR ). Because A (dimensions: R × R) is not symmetric the eigenvectors and eigenvalues, A˜ = DΛD−1 , (25) can be complex. With this vn+1 = Avn = UR DΛD−1 URT vn ,

(26)

vn = UR DΛn D−1 URT v0 ,

(27)

is obtained or, by induction, n

where Λ = Λ × Λ · · · × Λ. A simple manipulation    q   λi t t ln λ = exp ln λ2r + λ2i + iatan = e−iωt , λn = exp ∆t ∆t λr

(28)

provides the frequency and temporal growth rate, λi 1 atan ∆t λr q 1 ln λ2r + λ2i . ∆t

ωr

= −

(29)

ωi

=

(30)

Here, ∆t is the time interval between the “snapshots”. The matrix D is scaled by a diagonal matrix B = diag[b0 , b1 , . . . ] such that, v0 = UR DB [1, 1, . . . ]T = UR D0 [1, 1, . . . ]T .

(31)

Matrix B is obtained by solving the linear system, URT v0 = DB [1, 1, . . . ]T . With this scaling Eq. 27 simplifies to vn = UR DΛn D−1 URT v0 = UR D0 [λn0 , λn1 , . . . ]T . (32) The DMD allows for an expression of the flow data in the form, vn =

R X j=0

λnj dj =

R X

e−iωj t dj .

(33)

j=0

The DMD modes, dj , are the columns of UR D0 and the DMD eigenvalues, λj , are the entries on the main diagonal of Λ. H.

Total Dynamic Mode Decomposition

According to Hemati et al.,30 Eq. 20 introduces a bias which results in a systematic (and erroneous) reduction of the predicted temporal growth rates. The bias was traced back to the observation that the relationship Y = AX is in most instances not satisfied accurately. For the forward DMD the error EY in Yexact +EY = AX is minimized. For the backward DMD (not discussed here) the error EX in Y = A(Xexact +EX ) is minimized. For the total dynamic mode decomposition (TDMD)30, 31 the forward and backward DMD are combined. The assumption is made that error occurs on both sides, (Yexact + EY ) = A(Xexact + EX ). To find a least squares solution that minimizes both errors the equation is rearranged, " # X [A − I ] = 0. (34) Y The matrix [X Y ]T has dimensions 2N × N . A projection onto a POD subspace with dimension R < N/2, V0N = UR ΣR WRT , 7 of 27 American Institute of Aeronautics and Astronautics

(35)

is performed. Different from Eq. 21 the POD modes are computed for the entire time interval (all “snapshots”). The projection results in " # ˜ X [A˜ − I ] =0 (36) Y˜ ˜ = U T X (dimensions: R × N ), Y˜ = U T Y (dimensions: R × N ), and A˜ = U T AUR (dimensions: with X R R R R × R). A singular value decomposition is performed, " # ˜ X = U ΣV T , (37) Y˜ to obtain the least squares approximation that best minimizes both EX and EY . The matrices U and Σ (with dimensions 2R × 2R) are obtained from " #" #T ˜ ˜ X X = U Σ2 U T . (38) Y˜ Y˜ Only the leading R singular values are retained, " # " #" ˜R X U U Σ1 11 12 = U ΣR V T = ˜ YR U21 U22 0

0 0

#"

V1T V2T

#

" =

U11 Σ1 V1T U21 Σ1 V1T

# .

(39)

˜ 11 Σ1 V T is obtained which can be solved for A, ˜ From Eq. 36, U21 Σ1 V1T = AU 1 −1 A˜ = U21 U11 .

(40)

The remaining steps are identical to the forward DMD (starting with Eq. 25). I.

Flow Control Parameters

The AFC strategies discussed in this paper rely on the addition (or removal) of momentum through blowing (or suction) slots and holes. The ratio of the maximum blowing or suction velocity, uj,max , and the inlet velocity, uref , is referred to as blowing ratio, B=

uj,max . uref

(41)

Negative values of B indicate suction whereas B > 0 indicates ejection of fluid. The momentum coefficient, R ρj u2j dA , (42) cµ = 1 2 2 HSρref uref cos(αin ) is defined as the ratio of the momentum flux introduced by the AFC and one-half of the cascade inlet momentum flux (assuming that the control is applied at both endwalls). Similarly, a mass flow coefficient can be defined as R ρj uj dA cm = 1 , (43) 2 HSρref uref cos(αin ) where the integral is taken over the slot or hole area, Aj , which is defined normal to uj . The density was assumed to be constant, ρj = ρref . For the results shown in this paper, circular holes with diameter d or slots with width b and length ` were considered. The respective areas are Aj = πd2 /4 and Aj = b `. A Gaussian velocity distribution was assumed over the holes and slots. For example, for steady blowing through a circular orifice, the velocity varied according to 2

uj = ecr uj,max .

(44)

For harmonic blowing the maximum jet velocity was scaled with (1 − cos ωt)/2. Following common practice, the pitch and skew angle were used to define the jet orientation. The pitch angle, φ, is the angle between the jet and the blade surface. The skew angle, θ, is the angle between the jet and the endwall. 8 of 27 American Institute of Aeronautics and Astronautics

III. A. 1.

Results

Uncontrolled Flow Instantaneous Flow Field

Instantaneous flow visualizations of the uncontrolled flow using the Q-criterion by Hunt,32 Q=


1 Wij Wij − Sij Sij , 2

(45)

are provided in Fig. 4. The Q-criterion indicates areas where rotation dominates strain. The endwall boundary layer upstream of the cascade appears to be fully turbulent. The laminar flow on the blade suction surface separates approximately at the beginning of the uncovered turning and reattaches shortly thereafter. Spanwise coherent structures which quickly lose their coherence in the streamwise direction are observed in the reattachment region. The suction side corner flow is highly turbulent. The passage vortex is hidden in the turbulent endwall boundary layer. A “quiet” region can be observed above the passage vortex. This region can be explained by the secondary flow which transports laminar flow from inside the passage towards the endwall. The quiet region extends downstream into the wake. Figure 4d reveals significant flow three-dimensionality near the endwall.

a)

b)

c)

d)

Figure 4: Iso-contours of a), b), c) Q = 10 and d) Q = 1000 colored by velocity magnitude. For d), view is in the negative y-direction. 2.

Mean-Flow

Mean-flow visualizations are provided in Fig. 5. Shown are iso-surfaces of the Q-criterion colored by the vorticity in the direction of the cascade outflow. The horseshoe vortex (structure no. 1) is wrapped around the leading edge of the blade (i.e. junction) and has two distinct “legs”. The pressure side leg of the horseshoe vortex is strengthened by the secondary flow and develops into the passage vortex (structure no. 2). The passage vortex impinges onto the blade suction surface where it interacts with structure no. 4. Structure no. 4 originates from a longitudinal structure at the endwall. Another vortical structure (no. 5) is seen to

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originate from the corner. Structures 4 and 5 have the same sense of rotation as the passage vortex. The suction side leg of the corner vortex develops into a weak corner flow vortex (structure no. 6) with opposite sense of rotation from the passage vortex. A “shed vortex” (structure no. 7) with positive vorticity can be seen downstream of the trailing edge.

a)

b)

c)

d)

Figure 5: a), b), c), d) Iso-contours of Q = 30 colored by vorticity in outflow direction. For d), view is in the negative y-direction. The wall pressure coefficient, cp , for the L2F (at quarter-span) and the Pak-B8 are compared in Fig. 6. Because the L2F is more loaded than the Pak-B, the area enclosed by the cp -curve is larger. The suction peak is at x ≈ 0.2 for the front-loaded L2F and x ≈ 0.7 for the Pak-B. Although the front-loading weakens the adverse pressure gradient, a pressure plateau is observed for 0.4 ≤ x ≤ 0.65. The plateau can be associated with a laminar separation bubble. This bubble, which is also clearly seen in Fig. 4d, is short (compared to the axial chord) and shallow.

a)

b)

Figure 6: a) Wall pressure coefficient at quarter-span. b) Iso-contours of static pressure and streamlines (black) in z = 0.70133 plane as well as skin-friction lines (white). Endwall skin-friction lines superimposed on iso-contours of the static pressure field and streamlines at

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z = 0.70133 are shown in Fig. 6b. At this distance from the endwall, the flow is almost 2-D (small spanwise velocity component) and inviscid outside of the boundary layer. The curvature of the inviscid flow inside the cascade leads to a pitchwise pressure gradient, ρv 2 ∂p = , ∂r r

(46)

which is imprinted on the endwall. The velocity in the endwall boundary layer is too low for the centrifugal acceleration to balance the pressure gradient. As a result, a secondary flow from the pressure side to the suction side develops along the endwall. This flow is illustrated by the skin-friction lines in Fig. 6b. The secondary flow leads to inflectional pitchwise velocity profiles that strengthen the pressure side leg of the horseshoe vortex, resulting in a highly coherent passage vortex. According to Rayleigh’s inflection point theorem, inviscid velocity profiles are a necessary (but not sufficient) condition for instability. Cross-flow instability may explain the longitudinal structures shortly downstream of the cascade inlet (structures no. 3 in Fig. 5). Iso-contours of the skin-friction coefficient magnitude, skin-friction lines, and iso-surfaces of Q = 60 are presented in Fig. 7. The skin-friction coefficients in the x-, y- and z-directions are, cf,x =

µref ∂u ∂η 1 2 2 ρref uref

,

cf,y =

∂v µref ∂η 1 2 2 ρref uref

,

cf,z =

µref ∂w ∂η 1 2 2 ρref uref

,

(47)

where η denotes the surface-normal coordinate. In dimensionless form cf,x =

2 ∂u , Re ∂η

cf,y =

2 ∂v , Re ∂η

cf,z =

2 ∂w , Re ∂η

are obtained. The skin-friction coefficient magnitude is, q cf = c2f,x + c2f,y + c2f,z .

(48)

(49)

Limiting skin-friction lines (separation, S, and reattachment, R) as well as saddle points (A) are marked

b)

c)

a)

Figure 7: Iso-contours of skin-friction coefficient magnitude, skin-friction lines, and iso-surfaces of Q = 60. Detail of b) junction flow region and c) blade suction surface (view normal to outflow). by capital letters in Fig. 7. The primary separation line, S1 , marks the endwall boundary layer separation 11 of 27 American Institute of Aeronautics and Astronautics

ahead of the leading edge. Another separation line, S2 , results from the horseshoe vortex and in extension the passage vortex. Saddle points, A1 and A2 , mark the intersections of the two separation lines with the stagnation line. Details of the junction flow region are provided in Figure 7b. Separation line S2 is a direct consequence of the horseshoe vortex. Temporal animations of the time-dependent data reveal that this line is shifting upstream and downstream due to the unsteady nature of the horseshoe vortex. A line of reattachment, R1 , is seen closer to the leading edge. The skin-friction coefficient magnitude is lower between S2 and R1 than between R1 and the blade. Similar lines of separation and reattachment as well as wallshear stress distributions in the junction region have been observed in experiments for other LPT blades33, 34 and earlier L2F simulations.14 The separation line S5 and reattachment line R3 outline the 2-D laminar separation bubble on the suction surface. A region of separated flow adjacent to the endwall with separation line S4 and reattachment line R2 (Fig. 7c) can be explained by the secondary flow which spreads in the spanwise direction away from the endwall after “impacting” on the suction surface. A detailed comparison of numerical skin-friction lines with experimental oil-flow images revealed that the mean-flows obtained from simulation and experiment are in good agreement.14 Further insight into the behavior of the mean passage vortex was obtained by computing its circulation, I X Γ= ~v · d~r ≈ (ui dxi + vi dyi + wi dzi ), (50) ∂S

i

for several equidistant planes normal to the chord-line (Fig. 8a). For each plane, the line of integration was defined manually based on visualizations of the Q-criterion (e.g. Fig. 8c). The circulation increases from plane 2 to plane 11 and reaches a maximum for plane 14 (Fig. 8b). As explained earlier, this stengthening of the passage vortex is due to the secondary flow. Downstream of plane 14 the passage vortex is being dissipated and the circulation decreases rapidly.

b)

a)

c) Figure 8: a) Chord-normal planes (only even planes from 2 to 22 are shown). b) Passage vortex circulation. c) Contours of Q-criterion and lines of integration (for planes 6, 8, 10, 12, and 14).

3.

Junction Flow

Mean-flow visualizations in a plane normal to the endwall that bisects the horseshoe vortex are shown in Fig. 9. This plane, here denoted as “horseshoe vortex plane” (HSVP), has an inclination of θ = 37deg with 12 of 27 American Institute of Aeronautics and Astronautics

respect to the x-direction and intersects the blade leading edge at x = 0. The vorticity normal to the plane, ω, was computed as ω = −ωx sin θ +ωy cos θ. Iso-contours of ω in Fig. 9a reveal three vortical flow structures. A weak vortex, B, and a saddle point, A, can be observed far upstream of the junction. These structures are related to separation line S1 in Fig. 7a. Structure D can be associated with the horseshoe vortex. The horseshoe vortex causes the incoming flow to separate, resulting in another saddle point, C (separation line S2 in Fig. 7b). The vorticity associated with structure D is “smeared out” in the streamwise direction likely as a consequence of the bi-modal behavior of turbulent junction flows.33, 35 The horseshoe vortex lowers the static pressure at the leading edge and “pulls” air into the junction flow region (Fig. 9b). Finally, a small vortex, E, can be observed between the horseshoe vortex and the blade leading edge (Fig. 9a). This vortex can be associated with line R1 in Fig. 7b.

a)

b)

Figure 9: Iso-contours of a) vorticity and b) static pressure and streamlines in HSVP computed from the time-averaged flow. The bi-modal behavior is characterized by a more or less random (in time) motion of the horseshoe vortex away from and back towards the junction.33 A sequence of snapshots of the flow in the HSVP illustrates this phenomenon (Fig. 10). In contrast to the time-averaged flow, the junction flow region is populated by multiple vortical structures that are traveling both upstream and downstream. The dominant structure with positive vorticity corresponds to the horseshoe vortex. At certain time instances (e.g. 3, 4.8, 6.2) the horseshoe vortex is very coherent and induces a strong reverse flow at the endwall (backflow mode). At other time instances, the horseshoe vortex is weaker and the reverse flow is reduced (zero flow mode).

Figure 10: Iso-contours of in-plane vorticity (−200 < ω < 200) for HSVP.

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Unsteady velocity data were recorded at four different locations (i.e. probes in Tab. 2 and Fig. 11). The distance from the endwall (z-coordinate) was chosen such that the probes were located inside the turbulent endwall boundary layer. Probe 1 was placed upstream of the forward saddle point and represents the undisturbed approach endwall boundary layer. To capture the unsteadiness of the junction flow, probe 2 was located near the leading edge. Probes 3 and 4 were situated underneath the passage vortex. Probe 1 2 3 4

x, y, z coordinates -0.3, 1.521, 0.025 -0.006493, 0.6604, 0.025 0.444, 1.519, 0.05 0.878, 0.747, 0.05

Table 2: Probe locations.

Figure 11: Iso-surfaces of Q=60 and probe locations. The instantaneous kinetic energy of the velocity fluctuations, u0i u0i /2, for the different probes is plotted in Fig. 12a. Fourier spectra and spectral density estimations are shown in Figs. 12b and 12c. The spectral density estimates were computed with the maximum entropy method by Ghil et al.36 with M=150. Probe 1 upstream of the leading edge is relatively quiet with no large-scale excursions from the mean. As expected for a turbulent boundary layer, the spectrum is relatively broad. Probe 2, which is located near the junction, captures large-scale unsteadiness at the low-frequency end of the spectrum. The spectrum for probe 2 shows significant frequency content for f ≈ 0.5. For the conditions of the AFRL experiments,9 Re = 100, 000, ∗ Cax = 6in = 152.4mm, and ν ∗ = 1.5 × 10−5 m2 /s, a frequency of f = 0.5 corresponds to a dimensional frequency of ν∗ f ∗ = f × Re × ∗ 2 ≈ 32Hz . (51) Cax The quiet regions or valleys in the signal of probe 2 seem to be correlated with time instances when the zero flow mode is prevalent. Since one leg of the horseshoe vortex develops into the passage vortex, the horseshoe vortex unsteadiness is propagated along the passage vortex. As a result, reminiscences of the probe 2 unsteadiness are also observed for probe 3 (circled events in Fig. 12a). 4.

Dynamic Mode Decomposition

The TDMD was employed to identify temporally growing modes that may be exploited for an effective and efficient AFC. Towards that end pulse disturbances were introduced at two different locations in the flow field as indicated in Fig. 13. Pulse disturbances excite a broad spectrum that may include unstable frequencies and wavelengths. The pulses were localized over spherical volumes of radius r = 0.05 centered 14 of 27 American Institute of Aeronautics and Astronautics

a)

b)

c)

Figure 12: a) Instantaneous kinetic energy of velocity fluctuations. b) and c) Spectral density estimations (black lines) and Fourier transforms (grey lines). at x, y, z = 0.561, 0.781, 0.05 for pulse 1 and x, y, z = 0.09, 0.6, 0.05 for pulse 2. A velocity increment ud = u ˆ exp (cr−2 ) was added to the initial condition (“start-up” flow field for simulations) where r is the distance from the center of the pulse. The amplitude for both pulses was u ˆ = 1. The undisturbed flow, the flow with pulse disturbance 1, and the flow with pulse disturbance 2 were then advanced in time over a time interval of 6 and 300 snapshots, equally distributed over the time interval, were saved. The differences between the time series for the disturbed and undisturbed flow were analyzed with the TDMD. For both cases, the first ten TDMD modes were analyzed. As for the POD, traveling waves are resolved by mode pairs. For pulse 1, mode pairs 0&1, 3&4, 5&6, and 7&8 appear to capture traveling waves in the corner region that may be exploited for increasing the mixing between the core flow and the secondary flow. The associated frequencies are f = ∆tωr /(2π∆t) =4.3, 0.7, 2.2 and 1.7, respectively where ∆t=0.02 is the spacing between the snapshots. Mode pair 7&8 is most strongly amplified and would be the first choice for an unsteady actuation. For the second pulse, mode pairs 0&1 and 4&5 capture traveling waves in the vicinity of the passage vortex that may be exploited for a control aimed at an earlier diffusion of the passage vortex. The associated frequencies are f = ωr ∆t/(2π∆t) =1.3 and 0.6, respectively. B.

Active Flow Control

A summary of the flow control cases is presented in Tab. 3. The different cases were classified with a four digit notation, X − |{z} X − |{z} X − |{z} N . |{z} S or U

E or B

S or H

Case #

Here, S or U refers to a steady or unsteady actuation, E or B means that the actuation is either at the endwall or on the blade suction surface, and S or H refers to a slot or hole, respectively. Outlines of the surface areas over which the actuation was applied are shown in Fig. 15. Cases SBS1 and SBS2 were motivated 15 of 27 American Institute of Aeronautics and Astronautics

mode 0 u = −5.1 × 10−4

mode 0 |v| = 3.6 × 10−5

mode 2 mode 4 u = 3.8 × 10−4 v = −3.8×10−4

mode 2 |v| = 2.8 × 10−5

mode 6 u = 3.8 × 10−4

mode 4 |w| = 6.2 × 10−5

mode 8 u = 6.8 × 10−4

mode 8 |w| = 1.6 × 10−6

Figure 13: TDMD analysis of pulse disturbances. Velocity iso-surfaces as indicated.

Pulse 1

Pulse 2

Figure 14: TDMD eigenvalues. Case Uncontrolled Steady, blade, slot (SBS1) Steady, blade, slot (SBS2) Airflow channel (SBS3) Steady, endwall, slot (SES1) Unsteady, endwall, hole (UEH1) Unsteady, endwall, hole (UEH2)

f 1 1

B 2.5 3.52 1 0.76 0.5

cµ × 10−2 4 7.93 0.65 0.02 0.02

cm × 10−2 0.85 1.19 0.44 0.04 0.04

Table 3: Flow control parameters.

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φ 30◦ 30◦ 30◦ 0◦ 0◦ 30◦

θ -30◦ -30◦ 0◦ 90◦ 90◦ 35◦

d or b 0.028 0.028 0.028 0.1 0.04 0.023

` 0.269 0.269 0.269 0.256 0.055

by experiments by Benton et al.18 who employed steady jets with cµ = 4% for countering the secondary flow. Since the jets cannot easily be resolved in the simulations, they were modeled with a slot. Steady flow control requires compressor bled air which lowers the compressor efficiency. High-lift devices on airplanes (such as slotted flaps) channel air from the pressure side to the suction side and inject it in the wall-normal direction. This motivated case SBS3 where air is channeled through a conduit from the pressure side to the suction side. The channel was resolved with 103 × 7 × 60 cells (the boundary layers inside the channel were not resolved). The conduit had the same spanwise extent as the blowing slot for cases SBS1&SBS2. Since case SBS3 requires no external energy input, it may be referred to as a “passive” flow control strategy. Case SES1 was motivated by the endwall contouring of the L2F-EF profile, which reduces the total pressure losses through a modification of the secondary flow.8 The intent of case SES1 was to simulate the endwall fillet through “fluidic shaping”. For this configuration, steady wall-normal blowing through a slot close to the pressure surface (Fig. 15d) was employed. It was speculated that since the resulting wall jet opposes the secondary flow, it should result in flow separation and the formation of a separation bubble that mimics the shape of the endwall fillet. Cases UEH1 and UEH2 were designed based on the results of the TDMD analysis for the pulse disturbance 2. Mode pair 0&1 has a traveling wave structure that follows the passage vortex (Fig.13) and is relatively strongly amplified (Fig. 14). The frequency associated with mode pair 0&1 is 1.3 and thus roughly three times larger than the peak at f ≈0.5 in the probe 2 spectrum (Fig. 12b). It was decided to use a forcing frequency of one for cases UEH1 and UEH2.

a) SBS1, SBS2

b) SBS3

e) UEH1

f) UEH2

d) SES1

Figure 15: Skin-friction lines for time-averaged uncontrolled flow. Red outlines illustrate areas where flow control was applied. For case b), red line outlines channel from pressure to suction surface.

1.

Instantaneous and Time-Averaged Flow Fields

Instantaneous iso-surfaces of the Q-criterion are presented in Figs. 16 & 17. Mean-flow visualizations are provided in Figs. 18 & 19. The top-down view on the blade suction surface for cases SBS1 and SBS2 reveals that the steady blowing is directly countering the spreading of the corner flow region. A similar but less pronounced effect is observed for case SBS3. For case SBS3 the skew angle of the injected fluid is not fixed (as for cases SBS1 and SBS2) and the injected momentum is not directly opposing the secondary flow. Iso-contours of the static pressure reveal the streamwise pressure drop in the channel (Fig. 20a). The slot forcing for cases SBS1 and SBS2 also leads to new asymmetric (because of the angled injection) necklace-like vortices, with a predominant leg near the outboard part of the slot. For both cases, the passage vortex trajectory is altered. For case SBS2, the momentum coefficient is large enough to displace the passage vortex considerably away from the blade suction surface. For case SES1 the “quiet” region (for the uncontrolled flow) underneath the blade is no longer quiet. The strong blowing normal to the endwall introduces significant unsteadiness. This can be explained by the fact that large velocity gradients increase the turbulence production. Interestingly, the coherence of the passage vortex is reduced for case SES1. According to Fig. 20b, the actuation introduces longitudinal structures with positive and negative vorticity

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SES1

SBS3

SBS2

SBS1

Uncontrolled

that appear to cancel out the passage vortex. The instantaneous flow fields for cases UEH1 and UEH2 and the uncontrolled flow seem very similar. For case UEH2 the coherence of the passage vortex is reduced compared to the uncontrolled flow.

Figure 16: Instantaneous iso-surfaces of Q = 25 (left, and center) and Q = 1000 (right; block 1; view in negative y-direction.

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UEH1 UEH2 Figure 17: Instantaneous iso-surfaces of Q = 25 (left, and center) and Q = 1000 (right; block 1; view in negative y-direction.

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Uncontrolled SBS1 SBS2 SBS3 SES1 Figure 18: Mean-flow iso-surfaces of Q = 50 (left, and center) and Q = 10 (right; block 1; view in negative y-direction.

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UEH1 UEH2 Figure 19: Mean-flow iso-surfaces of Q = 50 (left, and center) and Q = 10 (right; block 1; view in negative y-direction.

a)

b)

Figure 20: a) Iso-contours of static pressure for case SBS3 at z = 0.1885. b) Iso-surfaces of Q = 50 colored by vorticity in x-direction (case SES1).

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2.

Total Pressure Losses

The mass-averaged total pressure loss coefficients for the different cases are provided in Tab. 4. The losses were computed with respect to the quarter-span passage considered here. Interestingly, for case SBS1 the total pressure losses are increased while for for case SBS2 they are reduced by more than 20%. For case SBS3 the injected fluid does not counter the secondary flow and the total pressure losses are increased by 14%. A mild reduction of 2% is obtained for cases UEH1 and UEH2. When considering that the momentum coefficient for case UEH1 is 400 smaller than for case SBS2, it might be argued that unsteady forcing should be favored over steady blowing because of its greater efficiency. No significant total pressure loss reduction is obtained for case SES1. Case Uncontrolled SBS1 SBS2 SBS3 SES1 UEH1 UEH2

c00p,0,in 0.67031 0.71527 0.72496 0.66824 0.69611 0.67227 0.67158

00 qin 0.47375 0.46982 0.46843 0.47408 0.47140 0.47376 0.47353

c00p,0,out 0.48312 0.40895 0.57726 0.45413 0.50977 0.48943 0.48750

00 qout 1.1029 1.0368 1.1081 1.0869 1.1084 1.1091 1.1060

Ytot 0.18719 0.30632 0.14770 0.21411 0.18634 0.18284 0.25894

Ytot /Ytot,U (+63.6%) (-21.1%) (+14.4%) (- 0.1%) (- 2.3%) (- 1.7%)

Table 4: Mass-averaged total pressure loss coefficient.

a) Baseline

b) SBS1

c) SBS2

Figure 21: Iso-surfaces of Q = 30 colored by vorticity in outflow direction. Outlet plane (OP) is flooded by total pressure loss coefficient, cp,0,in − cp,0,out . Iso-contours of the vorticity in the direction of the outflow, ωout , and the total pressure loss coefficient, cp,0,in − cp,0,out , for an outlet plane at, x = 1.5Cax = const., are shown in Figs. 21-23. The passage vortex and the “shed” vortex have the opposite sense of rotation and appear in Fig. 22 as regions of negative and positive vorticity respectively. For the uncontrolled flow, the total pressure loss coefficient reaches large values in the wake of the blade (z/H > 0.2 and y/S ≈ 0.32) and near z/H ≈ 0.08 and y/S ≈ 0.5. As illustrated in Fig. 21a, the second loss peak can be associated with the “shed vortex”. For case SBS1 the passage vortex is strengthened by the actuation and the losses increase. In contrast, for case SBS2 (twice the momentum coefficient compared to case SBS1) the shed vortex is missing and the passage vortex is more compact. The skewed injection also introduces vorticity that leads to the emergence of a new vortical structure underneath the passage vortex with opposite sense of rotation (Fig. 21c). Overall, for case SBS2 the total pressure losses are more localized and generally reduced compared to the uncontrolled flow. For case SBS3 the loss core near the endwall (z/H = 0.05 and y/S=0.5) is stronger while for case SES1 the strength of the passage vortex is reduced compared to the baseline case. Finally, relative to the uncontrolled flow, the loss peak near the endwall is increased for case UEH1 and reduced for case UEH2.

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Baseline

SBS1

SBS2

SBS3

SES1

UEH1

UEH2

Figure 22: Iso-contours of vorticity in direction of outflow (outlet plane).

Baseline

SBS1

SBS2

SBS3

SES1

UEH1

UEH2

Figure 23: Iso-contours of total pressure loss coefficient and velocity vectors (outlet plane). 3.

Proper Orthogonal Decomposition

The POD was employed to extract the dominant coherent flow structures from the unsteady flow fields. A total of 150 snapshots for blocks 1 and 2 evenly distributed over a time interval of 3 were analyzed. The POD was performed for the uncontrolled flow and for case UEH1. The POD spectra (Fig. 24) illustrate that the majority of the energy of the flow fields is captured by the first six modes. The time-coefficients for the first six modes are presented in Fig. 25 and the POD modes are shown in Figs. 26 & 27. For the uncontrolled flow, modes 1-4 capture long wavelength structures that are aligned with the passage

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vortex. According to Gross et al.,37 as a result of the bimodal behavior of the junction flow, the passage vortex intermittently loses its coherence. This behavior is captured by the leading POD modes. For case UEH1, the magnitude of eigenvalues 1-15 is lowered and the frequency content of the time-coefficients is higher compared to the uncontrolled flow (Fig. 25). This suggests that the passage vortex is overall moving less than for the uncontrolled flow. Modes 1-4 capture shorter wavelength structures. The dominant period of modes 3&4 is about one and thus identical to the period of the actuation. The mode 3&4 visualizations reveal an increasing mode amplitude along the trajectory of the passage vortex (Fig. 27) which is an indication of disturbance amplification.

Figure 24: POD eigenvalues.

Figure 25: POD time coefficients for uncontrolled flow (top) and case UEH1 (bottom).

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Figure 26: POD modes for uncontrolled flow. Iso-contours of Q-criterion and skin-friction lines for timeaveraged flow.

Figure 27: POD modes for case UEH1. Iso-contours of Q-criterion and skin-friction lines for time-averaged flow.

IV.

Conclusions

Implicit large-eddy simulations of the flow through a LPT cascade with L2F airfoil were carried out for a Reynolds number based on inlet velocity and axial chord of 100,000. An analysis of the time-averaged flow field revealed a spanwise flow over the suction surface away from the endwall that provided the motivation for a steady flow control that directly opposed the secondary flow and reduced the total pressure losses by 21%. In practice, this control strategy is less attractive because the required momentum coefficient is very large (cµ = 8%). The loss reduction normalized with the momentum coefficient is 21/8=2.6/cµ . To reduce the control effort, it was decided to channel air from the pressure to the suction surface of the blade and to inject it in the downstream direction (similar to slotted flaps on aircraft wings). This approach was not successful, likely because the injected fluid was not directed towards the endwall (and did thus not directly counter the secondary flow). A successful passive flow control with endwall fillet8 motivated a simulation with steady blowing at the junction (“fluidic shaping” of the endwall). This strategy had no noticeable effect on the total pressure losses. Instantaneous visualizations of the junction flow region revealed a low-

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frequency unsteadiness (f ≈ 0.5) of the horseshoe vortex that is typical for turbulent junction flows (bimodal behavior). Amplified temporal modes with a frequency of f ≈ 1 were detected with the total dynamic mode decomposition. By forcing the unsteady mode with a low momentum coefficient (cµ = 0.02%), the coherence of the passage vortex was reduced and the total pressure losses were lowered slightly (-2.3%). The loss reduction normalized with the momentum coefficient is 2.3/0.02=115/cµ and thus much higher than for the steady blowing. A proper orthogonal decomposition (POD) indicates that the flow control suppresses the low-frequency unsteadiness of the passage vortex (related to the bimodal behavior of the junction flow region) and introduces disturbances that are amplified along the trajectory of the passage vortex which may explain the earlier diffusion of the passage vortex for the controlled case.

V.

Acknowledgments

This research was funded by the Air Force Office of Scientific Research (AFOSR) under grant number FA9550-15-1-0139 with Dr. Douglas Smith serving as program manager. High Performance Computing (HPC) resources were provided by the Department of Defense HPC Modernization Program.

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23 Rizzetta, D.P., Georgiadis, N.J., and Fureby, C., “Large-Eddy Simulation: Current Capabilities, Recommended Practices, and Future Research,” AIAA-Paper AIAA-2009-948. 24 Piomelli, U., and Balaras, E., ”Wall-Layer Models for Large-Eddy Simulations,” Ann. Rev. Fluid Mech., Vol. 34, 2002, pp. 349-374. 25 Gross, A., and Fasel, H.F., “Characteristic Ghost-Cell Boundary Conditions,” AIAA J., Vol. 45, No. 1, 2007, pp. 302-306. 26 Lumley, J.L., “The Structure of Inhomogeneous Turbulence,” Atmospheric Turbulence and Wave Propagation,, Ed. Yaglom, A.M., Tatarski, V.I., 1967, pp. 166-178. 27 Sirovich, L., “Turbulence and the Dynamics of Coherent Structures, Part I-III,” Quarterly of Applied Mathematics, Vol. 45, 1987, pp. 561-590. 28 Rowley, C.W., Mezi´ c, I., Bagheri, S., Schlatter, P., and Henningson, D., “Spectral Analysis of Nonlinear Flows,” J. of Fluid Mech., Vol. 641, 2009, pp. 115-127. 29 Schmid, D., “Dynamic Mode Decomposition of Numerical and Experimental Data,” J. Fluid Mech., Vol. 656, 2010, pp. 5-28. 30 Hemati, M.S., Rowley, C.W., Deem, E.A., and Cattafesta, L.N., “De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis of noisy datasets,” Theor. Comput. Fluid Dyn., Springer, 2017. doi:10.1007/s00162-0170432-2. 31 Dawson, S., Hemati, M.S., Williams, M.O, and Rowley, C.W., “Characterizing and Correcting For the Effect of Sensor Noise in the Dynamic Mode Decomposition,” Exp. Fluids, Vol. 57, Springer, 2016. 32 Hunt, J.C.R., Wray, A.A., and Moin, P., “Eddies, Streams, and Convergence Zones in Turbulent Flows,” Center for Turbulence Research, Stanford, CA, 1988. 33 Devenport, W.J., and Simpson, R.L., “Time-Dependent and Time-Averaged Turbulence Structure Near the Nose of a Wing-Body Junction,” J. of Fluid Mech., Vol. 210, 1990, pp. 22-55. 34 Sieverding, C.H., “Recent Progress in the Understanding of Basic Aspects of Secondary Flows in Turbine Blade Passages,” J. Eng. Gas Turbines Power, Vol. 107, No. 2, 1985, pp. 248-257. 35 Ghosh, K., and Goldstein, R.J., “Effect of Inlet Skew on Heat/Mass Transfer From a Simulated Turbine Blade,” J. of Turbomach., Vol. 134, No. 5, 2012, pp. 05104201-05104211. 36 Ghil, M., Allen, R.M., Dettinger, M.D., Ide, K., Kondrashov, D., Mann, M.E., Robertson, A., Saunders, A., Tian, Y., Varadi, F., and Yiou, P., “Advanced Spectral Methods for Climatic Time Series,” Rev. Geophys., Vol. 40, No. 1, 2002, pp. 3.1-3.41. 37 Gross, A., Marks, C.R., and Sondergaard, R., “Numerical Investigation of Low-Pressure Turbine Junction Flow,” AIAA J., Vol. 55, No. 10, 2017, pp. 3617-3621.

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