Shear layer of inclined jets in crossflow studied with spectral proper orthogonal decomposition and spectral transfer entropy

International Journal of Heat and Mass Transfer xxx (xxxx) xxx

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Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy Pedro M. Milani ⇑, David S. Ching, Andrew J. Banko, John K. Eaton Mechanical Engineering Department, Stanford University, United States

a r t i c l e

i n f o

Article history: Received 22 August 2019 Received in revised form 14 October 2019 Accepted 27 October 2019 Available online xxxx Keywords: Jet in crossflow Film cooling Spectral proper orthogonal decomposition Spectral transfer entropy Flow stability Jet shear layer

a b s t r a c t Several classes of vortical structures are present in a jet in crossflow, and the present paper sets out to investigate one of them: jet shear layer unsteadiness. An inclined jet in crossflow that is relevant to discrete hole film cooling applications is considered, with focus on two velocity ratios, r ¼ 1 and r ¼ 2. Highly resolved large eddy simulations (LES) described and validated in previous work are employed. Power spectra show that shear layer oscillations contain a broader range of frequencies than suggested by previous studies, and two spectral techniques are used to investigate this behavior. Spectral proper orthogonal decomposition (SPOD) shows modes that encompass the hole and the shear layer, suggesting their motions are linked. Then, we develop and employ a technique to establish causality between signals in different locations called spectral transfer entropy (STE), which is adapted from information theory. The results show that unsteadiness in the hole affects the spectral content of the shear layer, confirming the importance of resolving the plenum and the hole in simulations with realistic jet inflow, such as film cooling configurations. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The jet in crossflow consists of an orifice that ejects fluid into a wall-bounded flow [1]. Jets in crossflow are an important configuration in fluid mechanics and they have varied engineering applications, especially in the realm of heat transfer. For instance, in turbomachinery design, turbine blades are cooled by ejecting colder air through holes in their walls which protects the outer surface from the hotter free stream - a technique known as film cooling [2]. The fuel injection in combustors usually resembles a jet in crossflow as well. The schematic of a simple inclined jet in crossflow is shown in Fig. 1. The circular hole has diameter D and is inclined at an angle h with respect to the direction of the main flow (the hole is said to be transverse when h ¼ 90 ). Several distinct parameters control the jet-crossflow interaction. They include Reynolds and Mach numbers, geometry, and upstream conditions such as jet velocity profile and crossflow boundary layer. One crucial parameter is the velocity ratio r defined as

r ¼ U j =U c ;

ð1Þ

with U j and U c being the bulk velocities of the jet and crossflow respectively. In the incompressible regime, in which jet and cross⇑ Corresponding author. E-mail address: [email protected] (P.M. Milani).

flow have the same density, the velocity ratio controls how much jet fluid is being injected and how far the jet penetrates into the crossflow. There is a considerable body of previous work on jets in crossflow. Fric and Roshko [3] presented a landmark study in which they used smoke visualization to identify and describe different vortical structures arising from the interaction of a transverse jet with the crossflow: shear layer vortices, horseshoe vortex, wake vortices, and the counter rotating vortex pair. Subsequent work examined the unsteady behavior of jets in crossflow to elucidate these structures. The works of Kelso et al. [4–6] discussed the origins of the counter rotating vortex pair and argued that the underlying mechanisms change with velocity ratio. Zhong et al. [7] demonstrated the effects of the incoming crossflow boundary layer on flow structures, particularly on the horseshoe vortex. Tyagi and Acharya [8] focused on the dynamics of hairpin vortices in inclined jets, present at low velocity ratios, and their impact on mixing. Other researchers discussed the effects of large scale, organized unsteadiness in the mainstream on a jet in crossflow: for example, Fan et al. [9] and Zhou et al. [10] used different experimental techniques to show that such oscillations of the main flow, even at moderate frequencies, degrade the jet’s performance in film cooling. The current paper focuses on the unsteady behavior associated with a particular vortical structure, the shear layer vortices. Jet shear layer vortices form through Kelvin-Helmholtz instabilities between the jet and the crossflow. This paper deals with

https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Fig. 1. Schematic showing the centerplane of a circular inclined jet in crossflow. The color contour shows the instantaneous concentration of a passive scalar, injected by the jet for visualization.

inclined jets at moderate blowing ratios and the shear layer of interest is the one on the windward side of the jet. An instantaneous snapshot of passive scalar tracer (Fig. 1) shows shear layer oscillations growing into roller vortices that are shed into the wake. Understanding the vortices is important because they have implications in jet control [11] and in the entrainment of mainstream flow into the jet, which reduces effectiveness of film cooling [6]. Megerian et al. [12] studied stability and measured energy spectra of shear layer vortices in a transverse jet with a top-hat mean velocity profile, while Iyer and Mahesh [11] studied the same configuration numerically. However, a typical film cooling hole is not transverse and is fed from a plenum through a short tube, which creates a complex mean velocity profile and significant unsteadiness in the jet. Shear layer vortices were also studied under more practical conditions experimentally and computationally, but such studies have mainly employed flow visualization and thus present only qualitative results [6,13]. An interesting line of work to analyze coherent structures that takes advantage of large computational datasets consists of using model order reduction techniques on high fidelity simulation data to provide a more quantitative view of organized unsteady motions of the flow. These techniques include proper orthogonal decomposition (POD) [14], dynamic mode decomposition (DMD) [15], and spectral proper orthogonal decomposition (SPOD) [16,17]. For example, Kalghatgi and Acharya [18] employed DMD on an inclined jet in crossflow to study the contribution of different coherent motions to the film cooling effectiveness. Rowley et al. [19] applied DMD to a transverse jet in crossflow to identify modes associated with shear layer oscillations and wall oscillations. They showed that they are similar to POD modes, the difference being the time-varying coefficients: they oscillate at a fixed frequency in DMD, but in a range of frequencies in POD. Zhou et al. [20] used DMD to analyze the difference in vortex structures resulting from holes of different shapes in a jet in crossflow. Citriniti and George [21] used SPOD on experimental data of a free round jet to reconstruct the most energetic fluctuations of streamwise velocity, which allowed them to analyze coherent structures in the shear layer quantitatively. SPOD was also employed in many other turbulent flows such as boundary layers [22] and airfoils [23]. Quantitatively analyzing time-dependent signals is useful for obtaining physical insights into vortical structures of jets in crossflow. One popular approach in information theory is called mutual information [24], which determines how much two signals are correlated by quantifying how much information is shared between two distinct signals. This has been used in distinct fields, from image processing [25] to fluid mechanics [26]. When correlation is not enough and one wants to investigate causality between signals, mutual information is not sufficient since it is a symmetric quantity in the two input signals; instead, a related technique called transfer entropy can be employed [27]. Transfer entropy

has been used in a variety of fields, including financial applications [28], physiology [29], and chemical processes [30]. Transfer entropy has also been used in the fluid mechanics community to study interactions of eddies based on their past states [31]. In the present work, we employ transfer entropy to the spectral information of different signals, which we term spectral transfer entropy (STE). The current paper utilizes the large eddy simulations (LES) previously reported and validated in Milani et al. [32] to study oscillations in the jet-crossflow shear layer. The main contribution of the present work lies in analyzing the resulting data of these simulations with distinct techniques, particularly spectral proper orthogonal decomposition and spectral transfer entropy, to show that inhole oscillations are important to determine shear layer spectral content. Three velocity ratios are considered, namely r ¼ 1; r ¼ 1:5, and r ¼ 2. The rest of the paper is structured as follows. Section 2 describes the simulations and Section 3 shows some instantaneous and mean field results. Section 4 describes SPOD and STE and shows results of the spectral analysis in this flow. Finally, Section 5 presents conclusions and ideas for future work. 2. Numerical setup and methods The three large eddy simulations used in the present work were described in detail in Milani et al. [32], who also provide mesh and time convergence studies and show thorough validation against Magnetic Resonance Imaging (MRI) data. This section provides a brief summary of the setup and the reader is referred to Milani et al. [32] for more details. 2.1. Governing equations The filtered continuity and Navier-Stokes equations are solved in the incompressible regime as shown in Eq. (2) and Eq. (3). ui are the Cartesian components of the velocity and p is the pressure. The fluid properties, density q and kinematic viscosity m, are constant. The tilde over the variables indicates filtered quantities. The unresolved turbulent scales are accounted for with the subgrid ~ij ¼ ug ~i u ~ j , which is prescribed using the Vreman scale stress, s i uj  u subgrid scale model [33].

@ u~k ¼0 @xk

ð2Þ

  ~i ~j u ~i @ u ~ ~i @u 1 @p @2u @ ¼ þm  s~ij þ @xj @t q @xi @xj @xj @xj

ð3Þ

The advection-diffusion equation for a passive scalar c also was solved, mostly for the purposes of flow visualization. Therefore, the scalar concentration is set to c ¼ 1 in the jet and c ¼ 0 in the free stream. The Schmidt number is set to Sc ¼ 1 and the subgrid scale ~ j ¼ cu fj  ~cu ~ j consists of the Reynolds analogy model employed for r with a fixed turbulent Schmidt number, Sct ¼ 0:85. Experimentally, the simulation was validated against measurements of a contaminant released by the jet. The filtered scalar equation is shown in Eq. (4).

  ~j ~c @ ~c @ u m @ 2 ~c @ þ ¼  r~ j @xj @t Sc @xj @xj @xj

ð4Þ

The code employed to solve Eqs. (2)–(4) is Vida, from Cascade Technologies. It is second-order accurate in space and utilizes explicit time advancement, with further details in Ham [34]. The time step is chosen such that the Courant-Friedrichs-Lewy (CFL) number based on the smallest mesh dimension and the crossflow bulk

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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velocity

is

1.0,

which

implies

dt=ðD=U c Þ ¼ 5:0  103

for

3

r ¼ 1;dt=ðD=U c Þ ¼ :1  10 for r ¼ 1:5, and dt=ðD=U c Þ ¼ 3:4  103 for r ¼ 2. The simulation is run for 240D=U c to achieve a statistically stationary state, and subsequently run for at least 440D=U c to average statistics. Milani et al. [32] presents more details and shows that the time averaging is sufficient. 2.2. Domain and boundary conditions The geometry considered is the inclined jet in crossflow, whose centerplane is shown in Fig. 1. The hole has a circular cross-section of diameter D and is inclined h ¼ 30 with respect to the streamwise direction. The mainstream consists of a turbulent developing flow in a channel, whose boundary layer thickness just upstream of the hole location is d99 =D ¼ 1:5 and whose freestream turbulence level is about 1%. The total simulation domain is shown in Fig. 2. The channel has a square cross-section of side 8:6D, large enough so that blockage from the side and top wall boundary layers do not affect the jet. Note that the injection hole has length 4:1D and is fed from a rectangular plenum located beneath the channel. This configuration was chosen because it is representative of film cooling applications. The mesh is block-structured and contains exclusively hexahedral elements, numbering 40:1M (r ¼ 1), 43:4M (r ¼ 1:5), or 48:3M (r ¼ 2). The mesh is well resolved near the hole walls and the bottom wall, with yþ values around 1 for the first cell. Throughout the whole jet-crossflow interaction, the subgrid scale viscosity is significantly smaller than the molecular viscosity, so the present LES’s have almost DNS-like resolution in the regions of interest. Thus, from now on, the tilde will be omitted when referring to the filtered quantities output by the LES to simplify the notation. The velocity inlet conditions are prescribed using a method for synthetic turbulence generation based on that from Xie and Castro [35]. The three simulations use the same mean profiles for the main channel inlet condition, which are chosen in order to match the experimental data used for validation as described in Milani et al. [32]. The velocity ratio among the three runs is varied by changing the flow rate into the plenum. The resulting jet Reynolds number is ReD ¼ U j D=m ¼ 2; 900 for the r ¼ 1 case, ReD ¼ 4; 350 for r ¼ 1:5, and ReD ¼ 5; 800 for r ¼ 2. The scalar contaminant is set to c ¼ 0 at the main channel inlet and c ¼ 1 at the plenum inlet; the walls considered are adiabatic, so a no-flux condition is imposed for the scalar at the solid boundaries. 3. Results 3.1. Instantaneous results A 3D instantaneous snapshot is shown for both r ¼ 1 and r ¼ 2 to help visualize the flow structure. Fig. 3 shows those snapshots, with isosurfaces of Q criterion which is used to identify coherent

Fig. 2. 3D view of the full computational domain, with flow inlets/outlet marked and the Cartesian coordinate frame centered on the origin.

3

structures. The Q-criterion is a variable widely used to identify coherent structures, which for incompressible flows reduces to i Q ¼ 0:5 @u @xj

@uj @xi

[36]. Positive values indicate regions where rotation

is larger than shear, thus indicating the presence of a coherent vortex [37]. Fig. 3 confirms that the incoming main flow boundary layer is turbulent with mostly elongated structures, but shows that the turbulence activity intensifies greatly after the jet meets the crossflow, with much finer structures present. The hairpin vortex above the jet (containing only free stream fluid) observed in the r ¼ 1 snapshot and is seemingly just a feature of the turbulent boundary layer. Thin horseshoe vortices around the jet and close to the wall are visible in both cases. It is also noteworthy that, in the present configuration, fine turbulent structures are also present in the hole, upstream of its junction with the bottom wall. In the windward shear layer, fine and chaotic structures dominate, but one can notice thicker and somewhat regularly spaced structures that surround the jet in the bottom plots. These are the shear layer roller vortices as have been previously identified in the literature (e.g. [3]). 3.2. Mean flow results Fig. 4 shows mean velocity streamlines in the r ¼ 1 and r ¼ 2 cases, colored by the local mean contaminant concentration c. The streamlines are all seeded at x=D ¼ 2, spaced uniformly between y=D ¼ 0:05 and y=D ¼ 1:55. Note that closest to the wall at x=D ¼ 2, the jet is separated and the scalar concentration is very low; the streamlines indicate that fluid in that region comes from the upstream boundary layer and flows around the obstacle created by the jet. The streamlines in the jet core originate in the hole and their path suggests that the in-hole velocity is highly threedimensional. The reduction in mean scalar concentration along those streamlines shows that the jet fluid rapidly mixes with the freestream fluid due to turbulence, particularly in the windward shear layer. Fig. 4 also contains streamwise (y  z) planes showing mean scalar concentration at x=D ¼ 2 and x=D ¼ 5. Note that the jet is separated from the wall in both cases. An important feature of the mean scalar contours is their kidney shape, which is caused by the counter rotating vortex pair (CVP) distorting the scalar field. Fig. 5 presents a streamwise plane located at x=D ¼ 2 for the r ¼ 1:5 case, showing mean streamwise velocity color contours  =ðrU c Þ) with vectors for mean secondary flows. Note that the (u CVP is readily visible in the mean flow. It is also important to describe the in-hole velocity for the analysis that follows. Fig. 6 defines a coordinate along the hole (^s is the ^ points radially out and is perpendicular to the axial direction and n z axis) and shows color contour plots of mean axial velocity scaled  s =ðrU c Þ. The symmetry planes show that by the jet bulk velocity, u the flow separates as it moves from the plenum into the hole, and that the separation bubble creates a highly non-uniform mean axial velocity. The in-hole flow is also significantly unsteady as seen before in Fig. 3. The axial planes show that in the wake of separation, along the bottom wall of the hole, the axial velocities are much lower, and that such blockage accelerates the flow along the top wall, with mean velocities on the order of 1:6U j . They also show strong mean secondary flows. Interestingly, the in-hole mean profiles are very similar between all three velocity ratios, which indicates that in the current range of r, these profiles are mainly determined by the plenum-hole junction rather than the main flow characteristics. The only observable difference is that the wake of the separation bubble seems to recover somewhat faster as the velocity ratio goes down. From the jet’s perspective, a change in r modifies the Reynolds number of the flow, and within the fully turbulent regime separated flows around sharp corners tend to be

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Fig. 3. Instantaneous isosurfaces of Q criterion colored by scalar concentration in (a) r ¼ 1 and (b) r ¼ 2. The top plots show isosurfaces of Q criterion at 1.3 U 2c =D2 ; the bottom plots contain a zoomed-in view of isosurfaces at 13 U 2c =D2 .

Fig. 4. 3D view showing mean velocity streamlines colored by mean scalar concentration c for (a) r ¼ 1 and (b) r ¼ 2; 16 streamlines are seeded at x=D ¼ 2 and z=D ¼ 0, uniformly spaced between y=D ¼ 0:05 and y=D ¼ 1:55. Two y  z planes containing contours of c are also shown, at x=D ¼ 2 and x=D ¼ 5.

insensitive to Reynolds number variation [38]. Another effect is the change in the interaction with the crossflow since at higher values of r, the crossflow is moving slower from the jet’s perspective. This could be responsible for changing the recovery of the wake, but clearly the effect is not strong within the factor of 2 variation in the velocity ratio. 4. Analysis 4.1. Power spectra Five distinct points are defined (P1-P5), as shown in Fig. 7. Point P1 is located in the incoming boundary layer, and point P2 is located inside the hole. Points P3-P5 are placed within the windward shear layer of each of the different velocity ratio cases. Their streamwise position x=D is fixed, and the y=D location is moved to better capture the jet boundary. The exact locations are shown in Table 1.

Fig. 8 shows power spectral density of vertical velocity fluctuations, v 0 , probed at each of the 5 points shown in Fig. 7. These curves were computed using Welch’s method [39] from values probed every 0:1D=U j (thus, the maximum resolved frequency is roughly fD=U j ¼ 5). The appropriate dimensionless frequency is the jet Strouhal number, St ¼ fD=U j . In general, the spectra computed in the hole (P2) and in the early shear layer (P3) show mostly broadband oscillations over a range of frequencies, stemming from a highly turbulent flow in both locations. As will be discussed later, spectral information from P2 carries over to P3, which also helps to explain their similar behavior. The spectrum computed in the incoming boundary layer (P1) contains significantly less energy, especially in the higher frequencies. In the two downstream positions in the shear layer, P4 and P5, there is significant difference among different velocity ratios: as r increases, the energy content increases and concentrates in a narrower band of frequencies. Note that these results are not impacted by the synthetic turbulence imposed at the inlets because care was taken to create a long

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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 =ðrU c Þ at y  z planes located at Fig. 5. Color contours of mean streamwise velocity u x=D ¼ 2 for the r ¼ 1:5 LES, with vectors showing in-plane mean velocity. The main flow direction is out of the page. Solid lines indicate mean scalar isocontours at c ¼ 0:2; 0:4; 0:6; 0:8.

development section (see [32]) and because these are highly turbulent regions where most spectral energy is created due to the mean flow separation and shear. Megerian et al. [12] obtained similar plots experimentally, for jets perpendicular to the crossflow at different velocity ratios, shown in their Figs. 7 and 8. Their findings were reproduced numerically by Iyer and Mahesh [11]. Their data for r ¼ 2 show a very narrow frequency peak at around St ¼ 0:7 and its harmonics. Even for r ¼ 2, the data reported in the current work contain a much broader spectrum of frequencies, meaning that the vortex shedding is either not happening at well-defined intervals, or that other structures are contaminating the frequency content of the roller vortices. Part of this could be due to the fact that the present jets are inclined with respect to the main flow, so the effective (vertical) velocity ratio is lower. Plausible sources exciting the broadband instabilities at the shear layer could also be either the incoming boundary layer (which in the present work has thickness of d99 ¼ 1:5D) or oscillations already present in the hole. The latter is a strong candidate due to high-energy and broadband spectrum measured at P2. In the work of Megerian et al. [12], the vertical jet is fed from a nozzle that outputs almost a uniform axial velocity, which is much different than the profile shown in Fig. 6. The next subsections will dive deeper into this question, with further evidence that frequencies in the hole contaminate the shear layer dynamics.

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An interesting phenomenon observed in the spectra of Fig. 8 is that as one moves downstream along the jet, not only do the oscillations become less energetic, but the dominant frequencies seem to decrease. For r ¼ 2, for example, the peak at P4 happens around St ¼ 0:7 while the peak at P5 happens at St ¼ 0:5. A possible explanation for this could be vortex merging, a phenomenon that has been widely observed before in this range of Reynolds number (e.g. [4,40]). To confirm it is also present in the present flow, Fig. 9 shows sequential snapshots which capture two pairs of shear-layer vortices rolling up and merging together. We observe that such merging is more easily identified and happens earlier for the higher velocity ratio case r ¼ 2, when compared to r ¼ 1:5 and r ¼ 1, which is consistent with the summary of Kelso et al. [4]. Another plausible physical explanation for this decrease in the dominant frequency has been suggested in the context of stability of free jet shear layers [41]: as one moves downstream, the shear layer becomes thicker and the velocity differential between the jet core and freestream either stays relatively constant for the first few hole diameters past injection or decreases. This reduces the relevant frequency scaling, so that lower frequencies become more likely to excite the shear layer further downstream. 4.2. Hole effects studied with spectral proper orthogonal decomposition Power spectra have shown that the frequency content in the shear layer considered in this paper is significantly different to what was reported previously. However, the spectra alone are insufficient to explain why. High turbulence levels make it difficult to explain dynamics by examining instantaneous flow snapshots. In an attempt to understand the origin of these differences, we use modal analysis to identify coherent motions. Previous authors that used modal analysis at fixed frequencies, such as dynamic mode decomposition, focused on the dynamics after the jet exit [18,11,19]. We hypothesize that the differences between the present work and earlier findings are due to differences in flow structure inside the jet hole. Therefore, the present work focuses instead on interactions between fluid in the hole and in the shear layer. We use spectral proper orthogonal decomposition because it has several advantages over DMD [17]. SPOD gives a set of modes at a single frequency, whereas DMD can give a collection of modes at many slightly different frequencies over the same frequency range. DMD is also sensitive to noisy data and works best when there are sharp spectral peaks, unlike the current data. Towne et al. [17]

Fig. 6. In-hole velocities for the different velocity ratios. Top figures show symmetry planes that cut through the middle of the channel, and bottom figures show axial planes  s =U j , and vectors show in plane velocities scaled by U j ¼ rU c . in the midpoint along the hole. Contours indicate mean axial velocity, u

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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^ðkjÞ is the Fourier transform coefficient at frequency f k from where q Pw 2 th wi . The cross-spectral density tensor the j window and s ¼ Li¼1 at frequency f k is

bf Q b ; Sf k ¼ Q fk k

ð6Þ

b  is the complex conjugate transpose of Q b f . The eigenvecwhere Q fk k

Fig. 7. Instantaneous scalar concentration at the centerplane of r ¼ 1 case showing the location of 5 different points where spectral content is analyzed. Exact locations given in Table 1.

Table 1 Location of points where spectral information is analyzed. To capture the shear layer, the x location is fixed and y is varied between cases. The comma-separated values in the table show r = 1, r = 1.5, r = 2 respectively. Point

x=D

y=D

z=D

P1 P2 P3 P4 P5

2.0 1.0 0.5 2.0 5.0

0.5 0.5 0.25, 0.30, 0.32 0.9, 1.1, 1.25 1.4, 1.8, 2.0

0 0 0 0 0

showed that SPOD modes are optimally averaged DMD modes, so when a sufficiently long time series of data is available, SPOD is preferred over DMD. SPOD modes wi ðx; f k Þ are a function of space x and operate at a chosen frequency f k . Therefore, the modes show correlated motion at that frequency. The modes are calculated by first arranging the data in each instantaneous flow snapshot into a vector containing all velocity components at each point in space. The time series is divided into N w windows that each contain Lw instantaneous flow snapshots. The windows overlap by 50%. The Fourier transform in time is calculated on each window using a Hanning window function with weights wi . The Fourier transform coefficients at the specified frequency f k from each window are arranged into a matrix

bf Q k

sffiffiffiffiffiffiffiffiffi i Dt h ð1Þ ð2Þ ^k ; . . . ; q ^ðkNw Þ ; ^k ; q ¼ q sN w

ð5Þ

tors of Sf k are the SPOD modes wi . Each SPOD mode contains all three velocity components and is complex, with real and imaginary parts that are phase shifted by a quarter cycle. The eigenvalues of Sf k are proportional to the energy contained in each mode. SPOD modes are ranked so the first mode contains the most energy. Towne et al. [17] gives full details on the method of calculating SPOD modes efficiently. SPOD modes are calculated on instantaneous snapshots of velocity on the centerplane (Z ¼ 0) saved every 0.1 D=U j in the r ¼ 2 case. The window length Lw is set to 256 flow snapshots, giving a frequency resolution of DSt ¼ 0:16, and yielding N w ¼ 64. Fig. 10 shows the first SPOD modes for the r ¼ 2 case at three different Strouhal numbers (St ¼ 0:3; 0:6, and 1:2) which are picked due to the high frequency content shown in Fig. 8. These first modes are converged in the sense that performing similar calculations with only half of the total available data yielded very similar modes and eigenvalues. The streamwise modes, shown in the first row of Fig. 10, are very active inside the hole. For the higher frequencies, which are the most energetic, the streamwise oscillations seem to be advected through the hole, but no longer exist beyond the hole exit. This suggests that such in-hole streamwise motion is not locked to streamwise motions of the shear layer rollers. The second row in Fig. 10 contains the wall-normal velocity modes. At St ¼ 0:6 and St ¼ 1:2, we can observe the signature from the shear layer vortices, which in the main flow region is similar to the SPOD modes presented by Schmidt et al. [41]. Interestingly, these modes also extend down into the hole, which shows that the oscillations in the hole are correlated to the oscillations in the shear layer. In both u and v modes, the strong oscillations in the hole are probably a result of the turbulence in the wake of the separation bubble at the hole entrance seen in Fig. 6. The spanwise modes are generally weak; one can see what might be a hint of the wake vortices at St ¼ 0:3, but as will be discussed next this first mode is not conclusive. To determine whether the first modes are indeed representative of the flow physics, Fig. 11 shows the eigenvalue spectrum. It contains the eigenvalues (a measure of the energy content of the oscillations captured by a particular mode) of the first four

Fig. 8. Vertical velocity power spectra computed at 5 different locations in decibels. The dimensionless frequency used is the Strouhal number based on the jet velocity, St ¼ fD=U j . Locations shown in Fig. 7 and Table 1.

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Fig. 9. Four sequential snapshots showing instantaneous scalar concentration at the centerplane of r ¼ 2 case. Four shear layer vortices, visualized through c, are tracked and seen to merge. Same contours as Fig. 7.

Fig. 10. First SPOD modes for the r ¼ 2 jet in crossflow case, at three different Strouhal numbers. Each row contains one of the Cartesian components of the velocity, and only the real part of the mode is shown since the imaginary part represents the same structures.

modes, calculated at different values of St. As Schmidt et al. [41] pointed out, there is not necessarily any continuity in the structure between the same modes at different frequencies; however, this visualization is useful to determine the variation of modal energy at different values of St, and specifically to determine how much energy the first mode contains relative to other modes. As Fig. 11 shows, there is not much separation between first and second modes at the lower frequencies, particularly St ¼ 0:3, which suggests that the physics associated with the first mode are not dominant there and other dynamics may be contained in lower modes. On the other hand, the separation between modes is much greater around St ¼ 1:2, indicating that dynamics represented by the first mode (displayed in the last column of Fig. 10) contain most of the physical disturbances at that frequency.

4.3. Hole effects studied with spectral transfer entropy The SPOD mode in Fig. 10 shows correlated fluctuations at specific frequencies, but causality is unclear. In this subsection, we adapt a technique from information theory in order to analyze interactions between the hole and shear layer and determine whether the former actually causes the latter. Shannon entropy is a measure of information carried by a signal based on the probability density of the signal [42]. Shannon entropy is defined as

X HðBÞ ¼  Pðbi ÞlogPðbi Þ;

ð7Þ

i

where P ðbi Þ are the discrete probabilities for the event B taking values bi . A probability density function with equally probable events

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Fig. 12. Schematic of overlapping windows to calculate spectral transfer entropy.

v 1 ðtÞ; v 2 ðtÞ, . . ., v i ðtÞ are the velocity series contained in each window. Fig. 11. SPOD eigenvalue spectrum showing energy content of the first four modes as a function of St, in log-log scale.

has a higher Shannon entropy than a probability function with low and high probability events, indicative that more information is needed to convey a signal with more equally probable events. The conditional entropy of events B and A is

  X   P bi ; aj HðBjAÞ ¼  P bi ; aj log   P aj i;j

ð8Þ

and measures the amount of information needed to describe the event B given that A occurred. Information transfer from signal A to signal B is quantified by transfer entropy, given in Eq. (9) [27].

T A!B ðDt Þ ¼ HðBðtÞjBðt  Dt ÞÞ  HðBðtÞjAðt  Dt Þ; Bðt  Dt ÞÞ

ð9Þ

Aðt  DtÞ and Bðt  DtÞÞ are the signals shifted in time by Dt. Transfer entropy tells us how much additional information about signal Bðt Þ is carried by the time-shifted signal Aðt  DtÞ that was not carried by the time-shifted base signal, Bðt  Dt Þ. Transfer entropy can also be considered a measure of the decrease in uncertainty of BðtÞ when the additional signal Aðt Þ is known. We use this idea to examine the relationship between velocity fluctuations in the hole and shear layer. Applying transfer entropy directly did not yield interesting results, probably due to the broadband temporal dynamics arising from small scale turbulent fluctuations. Instead, we want to study coherent structures that oscillate at specific frequencies. So, rather than calculating the transfer entropy based on time series of velocity fluctuations, each time series is converted to the frequency domain to analyze fluctuations occurring at specific oscillation frequencies. We designate this method spectral transfer entropy (STE). To calculate the STE between two points, a time series of the quantity of interest at each point is windowed, such that successive windows are shifted by one data point (shown schematically in Fig. 12). This is the same method used to convert data to the frequency domain in SPOD except that the windows have larger overlap. The maximum overlap is used here so that more data are available when computing the probability density functions. A discrete Fourier transform is performed on each window, and the coefficient at a chosen frequency f k is retrieved, consisting of a complex number with phase and magnitude. This produces a time sequence of Fourier coefficients at a fixed frequency, which we then use to calculate the transfer entropy as shown in Eq. (9); the input to the transfer entropy should be a sequence of real numbers, so we compute the STE with either the magnitude or the phase of the Fourier coefficients. A summary is given in Table 2. There are two parameters for STE. One is the window length Lw . A window length that is too large results in fewer independent windows and highly non-local Fourier coefficients, which increases

statistical noise in the computed probability density functions. Windows that are too small will have insufficient frequency resolution. An analysis of the effect of window length on the resulting spectral transfer entropy is given in the Appendix. We choose to use a window length of 4=f k . The other parameter is the number of bins, N bin , used to compute the probability density functions. Discrete probability density functions are calculated by placing data into equally spaced bins between the minimum and maximum values of each series. The effect of changing the number of bins is also analyzed in the Appendix. For the results shown here, the number of bins is fixed at N bin ¼ 6. Modifying either parameter within a reasonable range does not change the conclusions presented herein. Calculating transfer entropy with Eq. (9) has two issues. The first is that the result is not a normalized quantity. The second is that noise in the computed probability densities results in a spurious positive value of transfer entropy. To deal with these issues, we use a normalized transfer entropy,

T A!B ðDt Þ  T Ashuffled !B ðDtÞ T~ A!B ðDt Þ ¼ ; HðBÞ

ð10Þ

where T~ is the normalized form and Ashuffled is the data A permuted at random. This is similar to the normalization used in [31]. T Ashuffled !B is an estimate of the spurious transfer entropy due to noise which is subtracted from the calculated T A!B . This quantity is then normalized using the unconditional entropy of the signal BðtÞ. In general, one has HðBðtÞÞ P HðBðtÞjBðt  DtÞÞ P HðBðtÞjBðt  Dt Þ; Aðt  DtÞÞ P 0 because conditioning on any new signals (i.e. adding more information) cannot increase the Shannon entropy of a time signal. Therefore, the numerator is always positive and the normalized STE is bounded between zero and unity. Under this normalization, the transfer entropy can be interpreted heuristically as a percent reduction in the variance of Bðt Þ given knowledge of the signal Aðt Þ and after accounting for its self-correlation. If Aðt Þ does not carry any additional information about the signal Bðt Þ, then the variance of Bðt Þ is not reduced, HðBðt ÞjBðt  Dt Þ; Aðt  Dt ÞÞ ¼ HðBðtÞjBðt  Dt ÞÞ, and the transfer entropy is zero. If Aðt Þ provides complete knowledge of Bðt Þ, then Bðt Þ can take only a single value when conditioned on Aðt  Dt Þ; HðBðt ÞjBðt  Dt Þ; Aðt  Dt ÞÞ ¼ 0, and the transfer entropy is as large as possible. Then, the normalized STE reduces to HðBðt ÞjBðt  Dt ÞÞ=HðBðt ÞÞ, which is bounded by 1, but does not necessarily equal unity because of conditioning on Bðt  Dt Þ. In practice, if Dt is much greater than the turbulent autocorrelation time scale, it will indeed approach 1. We first show results for the r ¼ 2 case between points P2 and P3. Fig. 13 shows the STE as a function of the time shift between the velocity series at Strouhal numbers of 0:6; 1:2, and 2:4. Figs. 13 are the STE based on phase information, and Figs. 13 are the STE based on magnitude information.

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Fig. 13. Spectral transfer entropy from P2 to P3 for r ¼ 2.

Fig. 13(a) shows spectral transfer entropy of streamwise velocities, from us at P2 to u at P3 in terms of phase information. Velocities at P2 are in the rotated coordinate system described previously in Fig. 6, with us and un in the axial and radial directions respectively. The STE is close to zero for all time shifts, although it is marginally higher near DtU j =D ¼ 1 at St ¼ 0:6 and St ¼ 1:2. The two points are located a distance of 0:96D apart, so advection at

the jet velocity should make information transfer highest at approximately DtU j =D ¼ 1. Fig. 13(c) shows phase STE from streamwise velocity at P2 to wall-normal velocity at P3. There is a clear peak at DtU j =D ¼ 1 that is highest for St ¼ 1:2. The spectral transfer entropy from wallnormal velocity at P2 to streamwise velocity at P3 shown in Fig. 13 (e) also shows peaks, but the one at the lowest Strouhal number is

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Fig. 14. Spectral transfer entropy from P3 to P2 using phase information of vertical velocities for r ¼ 2.

Fig. 16. Spectral transfer entropy from P3 to P4 using magnitude information of vertical velocities for r ¼ 2.

not significant. The largest peaks are in STE between wall-normal velocities (Fig. 13(g)). The most information is transferred at St ¼ 1:2, followed by St ¼ 0:6. In the power spectra of Fig. 8, the peak in vertical velocity spectra occurs at St ¼ 1:2 for both points, so it is reasonable that the most information is carried at that frequency. The plots of STE on the magnitude (Figs. 13(b), (d), (f), (h)) do not show any significant transfer. In each figure, the spectral transfer entropy at St ¼ 0:6 reaches a plateau due to noise in the probability density functions [43]. The spurious level is higher for lower Strouhal numbers because the time series contains fewer total cycles at lower frequencies. The time series contains 520 cycles at St ¼ 0:6 but 2090 cycles at St ¼ 2:4, so the probability density functions contain more averaging across cycles at higher frequencies, resulting in lower spurious transfer entropy. The spurious levels are also much higher using magnitude than phase because the phase is limited between p and p while magnitude has no such limits; therefore only the probability density functions based on magnitude have low probability tails. When computing

discrete probability functions, the bins near the tails have more noise, resulting in spurious transfer entropy. Results from Fig. 13 show that the spectral content at location P2 causes that of P3 (i.e., is correlated with a time-lag that indicates causality). This transfer of information from P2 to P3 happens only in the phase of the oscillations, but not in their magnitude. Physically, it suggests that incoming disturbances only set off the downstream disturbances, hence dictating their phase without impact on their magnitude. This is reminiscent of a globally unstable behavior, in which forcing triggers an oscillatory response, but does not control the response’s amplitude. The stability of transverse jets in crossflow was studied in Ilak et al. [44] and Peplinski et al. [45], among others. Peplinski et al. [45] showed that their flow is globally unstable for r > 1:6, with modes corresponding to shear layer instabilities. Even though the particular threshold of r for global instability would differ since the jet in Peplinski et al. [45] is transverse and has laminar inflow profiles, the present results suggest that the flow under study is in the same regime.

Fig. 15. Spectral transfer entropy from P3 to P4 for r ¼ 2 based on phase information.

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Fig. 17. Spectral transfer entropy from P4 to P5 vertical velocities for r ¼ 2.

Fig. 18. Spectral transfer entropy based on phase for r ¼ 1.

Fig. 13 analyzes spectral transfer entropy from point P2 to P3. Since STE is a directional quantity, it is different if calculated from P3 to P2. Fig. 14 shows the STE from P3 to P2 based on vertical velocity phase information for the r ¼ 2 case. In stark contrast to Fig. 13(g), there is no significant information transfer in Fig. 14, implying that velocity oscillations in the shear layer lock onto the phase of oscillations in the hole, but not vice versa. STE from P3 to P4 based on phase information is shown in Fig. 15. Peaks are observed at approximately DtU j =D ¼ 3, which corresponds to the time it would take for information to propagate from P3 to P4 (which are about 2:9D apart) at speed around U j . Similar to Fig. 13, the highest information transfer is between vertical velocities in Fig. 15(d). This is consistent with the propagation of shear layer instabilities, which mostly affect the fluctuating ver . Between these points, the tical velocity v 0 relative to the mean v STE based on magnitude has no significant peak for any velocity

components as shown in Fig. 16. Again, these observations suggest vortices are advected downstream throughout the shear layer, which links the phases of P3 and P4, but their magnitude is decoupled due to the perturbation growth under the oscillator-like regime of the flow. STE between P4 and P5 is shown in Fig. 17. For brevity, only results for vertical velocities are shown. The curve based on phase, Fig. 17(a), shows a strong peak which is largest for St ¼ 0:6. This value of St is expected, since it is close to the most energetic frequency in these points as shown in the power spectra of Fig. 8. When the magnitude is used in STE, Fig. 17(b), there is now a noticeable peak which indicates some information transfer in the magnitude of the oscillations. Therefore, knowing the amplitude of oscillations observed at P4 helps determine the amplitude of oscillations at P5 some time later, which implies that vortices maintain some coherency between P4 and P5. In contrast, between

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Table 2 Method of calculating spectral transfer entropy. 1 2 3 4

Each time series is divided into overlapping windows of length Lw The Fourier transform coefficient at the chosen frequency f k is calculated using a Hanning window for each window The phase or magnitude of the Fourier transform coefficient is calculated for each window. The result is a time series of phase or magnitude at a single frequency The transfer entropy between two series is calculated using Eqs. 9 and 10 for various time delays Dt. Each probability density function uses N bin equally spaced bins in each dimension. The result is the spectral transfer entropy as a function of the time delay, defined at a frequency f k for either phase or magnitude

Fig. 19. Spectral transfer entropy from P2 to P3 at St ¼ 1:2 using phase information of vertical velocities (un to v) with various window lengths for r ¼ 2.

Fig. 20. Spectral transfer entropy from P2 to P3 at St ¼ 1:2 using phase information of vertical velocities (un to v) with various number of bins N bin for r ¼ 2.

P3 and P4 vortex merging changes the magnitude of oscillations, so there is no peak observed in the STE based on magnitude (Fig. 16). STE is also examined for the r ¼ 1 case. Most of the conclusions are the same as for the r ¼ 2 case. Fig. 18 shows results for the r ¼ 1 case based on phase of vertical velocities between the same points analyzed in Figs. 13(g), 15(d), and 17(a). The only major difference is that between P3 and P4, the STE based on phase of vertical velocities is slightly larger at St ¼ 1:2 than St ¼ 0:6. This could be explained by the fact that smaller scale vortices maintain coherency for longer in the shear layer at lower blowing ratios, and thus more information is transferred at higher frequencies at r ¼ 1 compared to r ¼ 2. This is consistent with the previous observation that vortex merging is more prevalent in higher velocity ratios [4]. 5. Concluding remarks We have analyzed the unsteady velocity data from the LES’s of Milani et al. [32] to study the windward shear layer in an inclined jet in crossflow. The geometry under consideration contains a short hole which is fed by a plenum, leading to a highly unsteady and non-uniform velocity distribution in the jet. A similar velocity profile might be expected in a realistic film cooling jet, which is

typically fed from a short hole. In most places on a turbine blade however, the jet is also crossflow-fed, which produces its own impact on the jet’s velocity profile [46]. The present configuration does not account for that since our flow is plenum-fed, but it is sufficient to establish impacts from highly non-uniform and unsteady jet velocity into the jet-crossflow shear layer. The power spectra in the shear layer reveal a broad range of frequencies when compared to the results of workers who previously studied canonical transverse jets in crossflow. We apply two distinct spectral techniques to further elucidate this observation. Spectral proper orthogonal decomposition (SPOD), popularized by Towne et al. [17], is a reduced order modeling technique that shows spatial modes that oscillate at fixed frequencies. We show that important modes span both the hole and the shear layer, which suggest that the oscillations in the former are correlated to the unsteadiness in the jet. This, however, does not indicate causality. To study this, we use a technique adapted from information theory, spectral transfer entropy (STE). STE analyzes two signals in spectral space and determines how much information can be gained about one signal by knowing the other signal, at a fixed oscillation frequency. This directional technique indicates that the turbulent motions in the hole change the spectral content of the shear layer. In particular, jet oscillations transfer their phase, but not their magnitude, to the vortices in the shear layer, which might be linked to the globally unstable nature of the flow in which upstream forcing can trigger disturbances without controlling their amplitude. This information transfer does not go the other way, i.e., the shear layer oscillations do not affect the vortices in the hole. The aforementioned phase locking is strongest between vertical velocity components, which in the symmetry plane are the most affected by the shear layer instabilities. The results are similar across all three velocity ratios, and mostly r ¼ 2 is discussed in the present work. One difference is that the phase information transfer is stronger at higher frequencies in r ¼ 1, possibly due to less vortex merging in r ¼ 1, which leads to smaller scale vortices maintaining coherency for longer. These findings are important because they confirm that appropriate simulations of the unsteady behavior of film cooling flows must resolve the flow in the jet hole correctly. This has been suggested by different authors before (e.g. [47]), but we go beyond and show one physical reason why this is true: unsteadiness in the hole contaminates the jet shear layer. This means that previous jet in crossflow simulations that either do not simulate the jet at all [44,45] or simulate the jet with a top-hat or fully developed profile [48,11] might fail to capture dynamics in jets in crossflow that would be relevant to film cooling, especially in the shear layer. Thus, future workers are advised to resolve jet and the plenum, especially in problems where spectral content of the jetcrossflow interaction is important, such as flow control. The technique employed here, spectral transfer entropy, can be applied to other turbulent flow problems whenever the origin of certain spectral content is under investigation. As discussed, baseline levels of noise are present in plots of STE vs time shift, and peaks above that baseline indicate non-negligible information transfer in a particular frequency from a point in the flow to another.

Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972

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Appendix A. Analysis of spectral transfer entropy (STE) parameters The first parameter of STE is the window length Lw , which controls the frequency resolution, number of independent windows, and how non-local each Fourier transform coefficient is. Fig. 19 shows the effect of changing the window length on the STE between points P2 and P3 at St ¼ 1:2 using phase information of vertical velocities. For all window lengths tested, the STE has a peak at approximately DtU j =D ¼ 1. The width of the peak is slightly wider when the window length is longer, which is expected since each Fourier coefficient contains more non-local information. There are also slight differences in the peak value for different window lengths. The window length of 4=f k is chosen for the present work. The second parameter is the number of bins used to compute the discrete probability density functions. Fig. 20 shows the effect of changing the number of bins. When the number of bins is low, the peak near DtU j =D ¼ 1 is smaller because the discrete probability density function is less representative of the true probability density function. For a small number of bins, the STE with a large time shift is almost zero, as is expected since there should be no relation between fluctuations for a large time delay. With a large number of bins, the peak is higher, but there is more noise and the STE for large time shifts converges to a spurious value above zero. The spurious value is due to noise in the probability density functions [43]. The number of bins for the present work is chosen as N bin ¼ 6 since that value results in a clear peak and spurious noise is almost zero. Appendix B. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2019.118972. References [1] K. Mahesh, The interaction of jets with crossflow, Annu. Rev. Fluid Mech. 45 (2013) 379–407. [2] D.G. Bogard, K.A. Thole, Gas turbine film cooling, J. Propul. Power 22 (2006) 249–270. [3] T.F. Fric, A. Roshko, Vortical structure in the wake of a transverse jet, J. Fluid Mech. 279 (1994) 1–47. [4] R.M. Kelso, T.T. Lim, A.E. Perry, An experimental study of round jets in crossflow, J. Fluid Mech. 306 (1996) 111–144. [5] L.L. Yuan, R.L. Street, J.H. Ferziger, Large-eddy simulations of a round jet in crossflow, J. Fluid Mech. 379 (1999) 71–104. [6] E. Sakai, T. Takahashi, H. Watanabe, Large-eddy simulation of an inclined round jet issuing into a crossflow, Int. J. Heat Mass Transf. 69 (2014) 300–311. [7] L. Zhong, C. Zhou, S. Chen, Effects of approaching main flow boundary layer on flow and cooling performance of an inclined jet in cross flow, Int. J. Heat Mass Transf. 103 (2016) 572–581. [8] M. Tyagi, S. Acharya, Large eddy simulation of film cooling flow from an inclined cylindrical jet, J. Turbomach. 125 (2003) 734–742. [9] D. Fan, D.D. Borup, C.J. Elkins, J.K. Eaton, Measurements in discrete hole film cooling behavior with periodic freestream unsteadiness, Exp. Fluids 59 (2018) 37. [10] W. Zhou, M. Qenawy, Y. Liu, X. Wen, D. Peng, Influence of mainstream flow oscillations on spatio-temporal variation of adiabatic film cooling effectiveness, Int. J. Heat Mass Transf. 129 (2019) 569–579. [11] P.S. Iyer, K. Mahesh, A numerical study of shear layer characteristics of lowspeed transverse jets, J. Fluid Mech. 790 (2016) 275–307. [12] S. Megerian, J. Davitian, L. Alves, A. Karagozian, Transverse-jet shear-layer instabilities. Part 1. Experimental studies, J. Fluid Mech. 593 (2007) 93–129. [13] C. Dai, L. Jia, J. Zhang, Z. Shu, J. Mi, On the flow structure of an inclined jet in crossflow at low velocity ratios, Int. J. Heat Fluid Flow 58 (2016) 11–18. [14] G. Berkooz, P. Holmes, J.L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech. 25 (1993) 539–575.

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Please cite this article as: P. M. Milani, D. S. Ching, A. J. Banko et al., Shear layer of inclined jets in crossflow studied with spectral proper erthogonal decomposition and spectral transfer entropy, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.118972