GT2003-38599me.jhu.edu/lefd/turbo/papers/chow_etal_igti2003.pdfk Turbulent kinetic energy k1 The...

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Proceedings of ASME Turbo Expo 2003 Power for Land, Sea, and Air

June 16–19, 2003, Atlanta, Georgia, USA

GT2003-38599

ON THE FLOW AND TURBULENCE WITHIN THE WAKE AND BOUNDARY LAYER OF A ROTOR BLADE LOCATED DOWNSTREAM OF AN IGV

Yi-Chih Chow, Oguz Uzol, Joseph Katz

Department of Mechanical Engineering

Johns Hopkins University Baltimore, MD 21218

ABSTRACT This paper presents detailed experimental data on the flow

and turbulence within the wake and boundary layer of a rotor blade operating behind a row of Inlet Guide Vanes (IGVs). The experiments are performed in a refractive index matched facility that provides an unobstructed view of the entire flow field. Results of the high-resolution 2D Particle Image Velocimetry (PIV) measurements are used for characterizing the mean flow, Reynolds stresses, turbulent kinetic energy as well as dissipation and production rates. Dissipation and production rates are high and of the same order of magnitude near the trailing edge, and decrease rapidly with increasing distance from the blade. The trend is reversed in the wake kinking region, resulting in elevated turbulence levels, i.e. a turbulent hot spot. One-dimensional spectral analysis shows that, except for the very near wake and hot-spot regions, the turbulence within the rotor wake can be assumed to be isotropic. Also the directions of the maximum shear strain and shear stress are aligned in that region, i.e. consistent with eddy viscosity type Reynolds stress models. The rotor near wake mainly consists of two parallel layers experiencing planar shear with opposite signs as one would expect to find in a 2D wake. However, orientation differences can extend up to 45° near the trailing edge and the hot-spot. Furthermore, there is substantial mismatch in the location of the local maxima of stresses and

strains. The values of 33S are also large there, indicating that

the flow is three-dimensional. Rotor boundary layer measurements focus on a region where the IGV wake intersects with the rotor blade. The impingement of the increased axial velocity region in between the IGV wakes causes the thinning of the boundary layer. This is similar to the effect of a turbulent jet impinging on a flat surface. When viewed in the frame of reference of “non-wake” flow regions, the boundary layer thinning can also be attributed to the suction (or “negative jet”)

effect of the “slip velocity” present in the IGV wake segments. Spectral analysis shows that the turbulence in the rotor boundary layer is highly anisotropic. As a result, the spectra cannot be used for estimating the dissipation rate.

NOMENCLATURE c Rotor chord Cij Transport terms tensor in the turbulent kinetic energy

equation E11 Longitudinal spatial spectra E22 Normal spatial spectra k Turbulent kinetic energy k1 The wavenumber along the direction of the sample line

(longitudinal component of the wavevector). Integral length scale. N Number of instantaneous vector maps ni Orthogonal unit vectors P Production rate of turbulent kinetic energy Sij Strain rate tensor t Time Tij Second-rank tensor u Axial velocity component v Lateral velocity component (almost circumferential,

but the laser sheet is flat)

relV

Velocity magnitude in the rotor frame of reference

x Axial coordinate y Lateral coordinate Greek: α Flow angle ε Dissipation rate φ Rotor phase η Kolmogorov scale λ Taylor microscale


ν Kinematic viscosity ω Vorticity Ω Rotational speed of the turbomachine INTRODUCTION

In multi-stage turbomachines, the wakes of rotor blades are influenced by interaction with wakes generated by upstream stator blades or Inlet Guide Vanes (IGV). It was shown by Uzol et al. [1] that the velocity field non-uniformities due to wake-wake interactions cause 13% variation in the specific work input of the rotor, in the circumferential direction. Furthermore, Chow et al. [2] showed that such interactions cause “kinking” of the rotor wake, which refers to the discontinuities generated in the wake trajectory, and generation of “turbulent hot spots”. The resulting complex unsteady flow field poses challenges to numerical predictions. Detailed experimental data on this complex flow field is essential both for understanding the dominating flow phenomena, as well as for guiding the development of appropriate turbulence models.

Numerous experimental and numerical studies have already examined blade-wake and wake-wake interactions (e.g. Smith [3], Kerrebrock and Mikolajczak [4], Ho and Lakshminarayana [5], Valkov and Tan [6], Van Zante et al. [7], Chesnakas and Dancey [8], Stauter et al. [9], Zaccaria and Lakshminarayana [10], Prato et al. [11], Suryavamshi et al. [12][13], Sentker and Reiss [14]). However, experimental data on the turbulence intensity, Reynolds stresses and relevant length scale distributions in a rotor wake located within a multi-stage environment are limited. Past experiments have been performed using single point measurement techniques, by traversing between blade rows and within blade passages, both in stationary and rotating frames of references. A variety of probes including tri-axial hot-wire, split hot-film, five hole pitot, high response pressure transducer, LDV have been used (e.g. Lakshminarayana and Reynolds [15], Lakshminarayana et al. [16], Reynolds et al. [17], Hah and Lakshminarayana [18], Cherrett and Bryce [19], Witkowski et al. [20], Sentker and Riess [21], Camp and Shin [22]). The same methodologies have been used for probing the boundary layers. For instance, transition in boundary layers of axial compressors and turbines was extensively investigated, both experimentally and numerically, by Halstead et al. [23][24]. Addison and Hodson [25] also measured velocity and turbulence intensity profiles within the boundary layer of a low-speed single-stage, axial flow turbine. We will return to some of these papers as we compare them to the present results.

In recent years 2-D Particle Image Velocimetry (PIV) has been gaining increasing popularity as a means of investigating the flow within turbomachines (Dong et al. [26][27], Day et al [28], Tisserant and Breugelmans [29], Gogineni et al [30], Sanders et al [31], Sinha and Katz [32], Sinha et al. [33], Wernet [34], Balzani et al. [35]). Recently, Uzol et al. [1][36] and Chow et al. [2][37] have introduced a new facility that enables unobstructed PIV measurements within an entire stage

by matching the optical index of refraction of the blades with that of the working fluid. The results demonstrate in great detail the flow complexity and associated turbulence resulting from blade-wake and wake-wake interactions. For example, they show that shearing and “kinking” of a rotor wake occur due to the non-uniform flow field generated by the upstream stator wakes. Shearing of the wake creates a turbulent “hot spot”, a region with elevated turbulent kinetic energy, concentrated vorticity and complex quadruple distributions of mean (ensemble averaged) shear strain and Reynolds shear stress.

The present paper extends our previous work by providing experimental data on the flow and turbulence within the rotor near-wake and boundary layer in more detail and higher resolution. To support effort to address turbulence modeling issues, we include distributions of one-dimensional turbulent energy spectra and estimates of dissipation and production rates. The wake measurements focus on an area starting from the trailing edge up to 90% rotor chord lengths downstream including one hot-spot. We also provide distributions of Reynolds shear stress and strains, and highlight the differences between the alignments of the stress and strain fields. In addition, this paper also presents data on the effect of IGV wakes on the structure of the rotor boundary layer. The discussion that follows demonstrates substantial variations in the near wall turbulence resulting from the non-uniform influx. Spectral analysis, turbulence production and issues related to near-wall turbulence anisotropy will also be discussed. EXPERIMENTAL SETUP AND PROCEDURES Facility

The axial turbomachine test facility enables us to perform complete PIV measurements at any point within an entire stage including the rotor, stator, gap between them, inflow into the rotor and the wake downstream of the stator. The unobstructed optical access is facilitated using a rotor row and an IGV row made of a transparent material (acrylic) that has the same optical index of refraction as that of the working fluid (concentrated solution, 62%- 64% by weight, of NaI in water).

Figure 1. The axial turbomachine facility test section


This fluid has a specific gravity of 1.8, i.e. heavier than water, but has a kinematic viscosity of 1.1x10-6 m2/s, which is very close to that of water. Thus, the blades become almost invisible and as a result, they do not obstruct the field of view and do not alter the direction of the illuminating laser sheet while passing through the blades with minimized reflection from the boundaries. Information related to use and maintenance of the NaI solution can be found in Uzol et al. [36].

The system is driven by a 25 HP rim-driven motor that is connected directly to the 1st stage rotor, preventing the need for a long shaft. The two rotors are connected by a common shaft supported by precision bearings. A shaft encoder and a control system are used for synchronizing our PIV system with the rotor phase. In the present configuration, shown schematically in Figure 1, the 1st stage consists of a rotor row followed by a stator row whereas the second stage consists of a stator (IGV) followed by a rotor. A honeycomb with 6 mm diameter openings occupies the entire gap between the two stator-rows. The purpose of this honeycomb is to reduce the effect of large-scale turbulence generated at the upstream blade rows, and align the flow in the axial direction, consistent with the orientation of the 1st stage stator.

The second stage rotor and IGV blades are made of transparent acrylic. There are 12 rotor and 17 IGV blades, each with a chord length of 50 mm and span of 44.5 mm. The rotor blade thickness varies from 7.37 mm at the hub to 5.62 mm at the tip. The IGV blade thickness is 8.28 mm. The rotor-IGV gap is 0.8 rotor chords and the rotor and IGV pitch-to-chord ratios are 1.34 and 0.97, respectively. The hub-to-tip ratio is 0.708 and the tip clearance is 2.4% of the rotor chord. More details about the facility can be found in Uzol et al. [36], Chow et al. [2][37]. All the present measurements are performed at a rotor blade speed of 500 rpm. The corresponding Reynolds number based on the tip speed and the rotor blade chord is 370,000. In the present experiments, the machine operates close to design point and away from stall condition.

PIV Setup and Experimental Procedure Two-dimensional PIV measurements are performed at 50%

span within the rotor wake and the rotor boundary layer. The light source is an Nd-YAG laser whose beam is expanded to generate a 1 mm thick light sheet. The flow is seeded using 20% silver coated, hollow glass, spherical particles, which have a mean diameter of 13µm and an average specific gravity of 1.6, i.e. slightly below that of the working fluid. The images are recorded by a 2048×2048 pixels2, Kodak ES4.0, 8-bit digital camera. The laser and the camera are synchronized with the orientation of the rotor using a shaft encoder that feeds a signal to a controller containing adjustable delay generators.

High-magnification and high-resolution wake/boundary layer measurements are obtained by recording 1000 images in

order to obtain converged statistics on the turbulence. The realizations are recorded synched to a specific rotor blade and phase. The present sample areas are 15x15 mm2. Data analysis includes image enhancement, removal of blade signature and cross-correlation analysis using in-house developed software and procedures (Roth et al. [38], Roth and Katz [39], Uzol et al. [36]). Additional details about the PIV setup and the data acquisition system can be found in Chow et al. [2,37] and Uzol et al. [36]. A conservative estimate of the uncertainty in mean displacement in each interrogation window is 0.3 pixels, provided the window contains at least 5-10 particle pairs (Roth et al. [38], Roth and Katz [39]). For the typical displacement between exposures of 20 pixels, the resulting uncertainty in instantaneous velocity is about 1.5%. Slip due to the difference between the specific gravity of the particle (1.6) and that of the fluid (1.8) may cause an error of less than 0.2%, i.e. much less than other contributors (Sridhar and Katz [40]). Uncertainty related to inflow variations and timing errors is estimated to be less than 0.6%.

Using the instantaneous measurements, the phase-averaged parameters are calculated using,

u i (x, y, φ) =1

Nui( )

kk=1

N

∑ (1)

( ) ( )i i j1

( , , ) 1

j i jk k

N

k

u u x y u uu uN

φ=

′ ′ = − −∑ (2)

k(x, y,φ) = 3

4′ u i ′ u i( ) (3)

α = tan−1 v u

(4)

ω =∂v

∂x−

∂u

∂y (5)

( , , )12

∂∂+

∂ ∂ = ji

ij

j i

uuS x y

x xφ (6)

where N = 1000 is the number of instantaneous vector maps for each phase, x and y are the axial and lateral coordinates, respectively, and φ is the phase angle. The subscripts i and j take values of 1 and 2, representing the axial (u1=u) and lateral (u2=v, almost circumferential) velocity components, respectively. The 3/4 coefficient of k, the “turbulent kinetic energy”, is used to account for the contribution of the out-of-plane velocity component, assuming that it is an average of the available measured components. Equations 4 and 5 are used for calculating the phase averaged flow angle and (almost radial) vorticity component.


(a) (b) (c) Figure 2. Phase-averaged (a) axial velocity, (b) lateral velocity and (c) vorticity distributions within the rotor wake. The rotor trailing edge is visible at the upper right-hand corner, and the data covers a region up to 90% rotor chord lengths from the trailing edge. Utip=8 m/s and Ω=52.36 rad/s (500 rpm).

RESULTS Rotor Wake

Figure 2 shows the phase-averaged axial and lateral velocity as well as the vorticity distributions within the rotor wake. The vector spacing is 234 µm, which corresponds to 0.4% rotor chord lengths and 30% of the trailing edge thickness. These measurements are performed at mid-span, 500 rpm (Re=370,000-based on the tip speed and the rotor chord), and at the same specific rotor phase. The wake is clearly “discontinuous”, i.e. it is kinked at about (0.28,-0.58). Chow et al. [2] have already shown, using low resolution and less converged data, that the kinked region is characterized by elevated turbulent kinetic energy, concentrated vorticity (as seen in Figure 2c) as well as quadruple distributions of Reynolds shear stress and mean shear strain. These regions have been referred to as “turbulent hot spots”. The mechanism of kinking can be explained as follows: The chopped off IGV wake segments convect at different speeds along the suction and pressure sides of the rotor blade due to the rotor blade circulation. When these phase-lagged segments reach the rotor wake, they are no longer aligned, creating a discontinuity across the rotor wake. Upstream of the rotor wake some of the area is occupied by an IGV wake with a low axial velocity, and the rest contains fluid with relatively high axial momentum. This relative velocity between the wake and non-wake regions,

which is defined as the “slip velocity” in Kerrebrock and Mikolajczak [4] and in Sanders et al [31], causes a non-uniform forcing on the rotor wake and consequently it gets kinked. Variations in the rotor wake thickness and velocity defect along its trajectory due to wake-wake interactions were also observed by previous researchers (e.g. Lockhart and Walker [41]). They even alluded to the existence of a wavy wake. However, previous studies have not presented a detailed picture on the flow structure and turbulence within a kinked wake. The effects of the wake kinking on the flow and turbulence structures in and around the rotor wake, are discussed in detail in the following sections. Near Wake Turbulence intensity:

Figure 3a shows the near wake distribution of turbulent kinetic energy k and streamlines in rotor frame of reference. The turbulence intensity in the wake decays with increasing distance from the blade trailing edge, except for the turbulent hot spot, which is centered at (0.32, -0.65). The turbulence intensity decreases by one order of magnitude from the rotor trailing edge region (within 0%~26% chord lengths downstream) to the wake region beyond the hot spot (close to 90% chord lengths downstream). Also note that the difference in turbulence intensity between the wake and its surrounding is as high as two orders of magnitude.


(a) (b)

(b)

18%~48% chordlengths downstream



Figure 3. Near rotor wake; (a) distributions of turbulent kinetic energy and relative streamlines; and (b) 1-D spatial energy-density spectra along the sample lines indicated in (a). The rotor trailing edge is located at (0,0). Solid line: E11(k1); dotted line: 3/4 E22(k1); dashed line: a -5/3 slope line fits E11(k1). Estimated turbulent kinetic energy dissipation rate, ε (calculated using Equation 6), and Kolmogorov length scale, η, are presented near each line.


One-dimensional spatial energy-density spectra: Several sample lines are chosen in each region, which are

aligned with the most probable directions of turbulence homogeneity (relative streamlines in this case), as shown in Figure 3a. Then, 16 or 32 data points are sampled along these lines to yield the Nyquist wave number of approximately 1.3×104 rad/m. Velocity fluctuation components at each data point are projected to be parallel (denoted as 1) or normal (denoted as 2) to the direction of the corresponding sample line. Then using FFT, we calculate the longitudinal spatial spectra, E11(k1), and normal spatial spectra, 3/4E22(k1), along each line. Results of 1000 instantaneous realizations are then averaged. Further details on filtering and windowing procedures can be found in Doron et al. [42]. By assuming isotropic, homogeneous turbulence in the inertial range, one can estimate the energy dissipation rate, ε, using (Tennekes and Lumley [43])

2/3 5 /311 1 1

18( ) (1.6)

55E k k −= (7)

The Kolmogorov length scale, η, can then be obtained from

1/ 43

=

(8)

where ν = 1.1x10-6 m2/s is the kinematic viscosity. This procedure of obtaining spectra and estimating dissipation was implemented before by us in much simpler flows successfully (e.g. channel flow, isotropic turbulence, where it can be calibrated. Doron et al. [42]), and have shown to give the correct dissipation values.

In isotropic homogeneous turbulence E11(k1)=3/4E22(k1), whereas Figure 3b clearly shows that in some cases, e.g., line 1, located 0%~26% chord lengths downstream of the trailing edge, the two spectra deviate significantly from each other, indicating strong turbulence anisotropy near the rotor trailing edge. A short distance downstream of line 1, the spectra of line 2 shows much more turbulence isotropy, except for the largest scales. Similar behavior of return to isotropy away from the trailing edge region was also observed by Lakshminarayana and Reynolds [15]. The estimated values of ε, shown near each line, decrease by about 4 times from line 1 to line 2, causing also appreciable difference in the Kolmogorov length scales.

In the region located 18%~48% chord lengths downstream, the spectra continue to show isotropy in the inertial (with –5/3 slope) and smaller scales. The scale at which the spectra start to show substantial anisotropy is of particular interest. One can first estimate the integral length scale, i.e. the

characteristic length scale of large eddies using 3/ 2k /≈ . At the site of sample line 1, for example, the values of k are in the 0.23 - 0.31 m2/s2 range, with a mean of 0.255 m2/s2. Thus, can be estimated to be 0.2553/2/39.2x103=3.3 mm, which is about the width of the wake in this region. By inspecting the spectra for line 1, it is found that the wave number, k1, at which the anisotropy becomes apparent ranges between 2000 - 2400

rad/m, which corresponds to wavelengths of 2.6 - 3.1 mm, i.e. very close to at this location. This analysis suggests that in this region the scale range with isotropy extends up to integral length scale.

In the region located 57%~90% chord lengths downstream, where the turbulent hot spot resides, there is substantial anisotropy in the spectra of lines located in the vicinity of the hot spot (lines 1~4), regardless of the fact that this region is located further downstream of the (more isotropic) previous samples. The estimated dissipation rate is higher within the hot spot, consistent with the higher turbulent kinetic energy in this region. Spectra of line 5 and 6 represent the turbulence characteristics at the edge of the wake and outside of the wake, respectively. It is evident that the turbulence level and dissipation rate for these two locations are much lower than those within the wake. In fact, the dissipation rates of lines 5 and 6 are 8 and 30 times lower than the levels at the center of the hot spot, respectively.

To summarize, the reappearance of turbulence anisotropy within the turbulent hot spot, as well as the substantial variability of dissipation rates and length scale along the wake, highly complicate the turbulence structure of the rotor near wake. These complexities pose great challenges to turbulence modeling efforts in turbomachines. On the other hand, the present analysis shows that PIV data can be used for generating critical parameters needed for turbulence modeling (e.g. distributions of both k and ε). Uncertainty of dissipation estimate

Since the dissipation range of the turbulent spectrum cannot be resolved from the present data, due to lack of finer spatial resolution, it is not possible to directly calibrate the procedures of dissipation estimate. However, one can still estimate the uncertainty indirectly from the statistics of the data. Figure 4 shows one of the spectra (Line 1 in 18-48% downstream) along with its standard deviation. To assess the uncertainty of dissipation estimate, the –5/3-slope-line-fit procedure is again performed on both the mean spectrum and the “spectrum+standard-deviation”. The two values for dissipation rate estimate are presented on the figure. By comparing these values, one can conclude that the uncertainty is of the same order of magnitude as the mean value.

Another uncertainty source is associated with the effect of anisotropy. By comparing the dissipation value estimated from the –5/3 slope-line fit to 3/4E22(k1) with the one from E11(k1), we find that in the relatively isotropic regions, e.g. line 2 in 0%~26% chord lengths region, lines 1 and 2 in the 18%~48% chord lengths region, and lines 3, 5, and 6 in 57%~90% chord lengths region, the effect of anisotropy is less than 10%. In the anisotropic regions, the uncertainty can be as high as 30%~60%. However, as will be shown in the boundary layer section, the anisotropy reaches a level in which the estimates can be different by an order of magnitude.


Figure 4. Standard deviation of a sample mean E11(k1) spectrum. Solid line: the mean spectrum; Error bars: standard deviation from the mean (only the top parts are shown for clarity); Upper dashed line: a -5/3 slope line fits the mean spectrum plus standard-deviation; Lower dashed line: a -5/3 slope line fits the mean spectrum; εstd: dissipation value estimated by line-fitting the mean spectrum plus standard deviation; εE11: dissipation value estimated by line-fitting the mean spectrum. Budgets of production and dissipation of turbulent kinetic energy:

The evolution of the turbulent kinetic energy (Pope [44]) is governed by

ijj i j ij ij ijj j

jii i ij

j i

k u k u u S 2 S S Ct x x

uu1 1where k u u , S

2 2 x x

∂ ∂ ∂′ ′ ′ ′+ = − − ν −∂ ∂ ∂

∂∂′ ′= = + ∂ ∂

(9)

where Sij is the strain rate tensor, “ ′ “ is the turbulent fluctuation, overbar indicates ensemble averaging, and Cij includes several transport terms. In Equation (9), the terms

iji ju u S′ ′− , 2 ij ijS Sν ′ ′ , and ijj

Cx

∂∂

are typically referred to as

production rate, dissipation rate, and transport terms of turbulent kinetic energy, respectively. The budget of production and dissipation reveals the interaction between the turbulence and mean flow, and how the turbulence is sustained. Since we have only 2-dimensional data, the out-of-plane components in these terms are unknown (except for S33, which can be inferred by continuity equation). Also, the previous analysis shows strong anisotropy in the critical near wake regions. Therefore, the common approximations to account for these out-of-plane components, which are based on assumed isotropy becomes invalid. As a result, the following analysis is based only on the known in-plane components, and we refer to the data as “two-

dimensional production,” keeping in mind that we are missing some terms. However, as long as we confine ourselves to mid-span, where the out-of plane shear is most likely much lower than the in-plane velocity gradients, the results should at least provide the correct order of magnitude. Figure 5 shows the distribution of 2-D production in the rotor near wake region, along with the results of dissipation results estimate.

Figure 5a shows that the production is enormous along the rotor suction side and around the trailing edge, but is much lower (becoming even slightly negative) along the rotor pressure side. Thus, there are substantial differences between the turbulence in the boundary layers of the suction and pressure sides. In the middle of the rotor wake, close to the trailing edge, there is a low-production area bounded by on both sides by high production layers. This trend is associated with high shear stresses (with opposite signs) and strains on both sides of the center of the wake. At the center of the wake the shear stress is zero but the turbulent kinetic energy is maximum. The dissipation rate estimates appear higher, but of the same order as the mean production averaged along the sample lines (186 m2/s3 and 36 m2/s3, respectively). ε is smaller than the peak production levels at the same distance from the trailing edge. At the18%~48% chord length region (Figure 5b), the production distribution still has the same two-layer pattern. Outside of the rotor wake the production is very low. On the right of the rotor side wake, there is a very low-production area associated with the IGV wakes, as one can see from the distributions of k in Figure 3a. The estimate of ε along the wake centerline, with a value of 39.2 m2/s3 is comparable to the corresponding mean production rate of 19 m2/s3. The dissipation estimated from the lower line, 32.5 m2/s3, is slightly lower than the corresponding mean production, 49 m2/s3, since this time the line coincides with a region of high production. Figures 5c and 5d contain partially overlapping regions and focus on the production in the area of wake kink and turbulent hot spot. Clearly, wake kinking increases the turbulence production, and shifts the point with maximum production to the center of the wake. In this region the production rate is clearly higher than ε. For example, along the lines with ε=13.3 m2/s3, 9.8 m2/s3, 7.1 m2/s3, and 3.7 m2/s3, the mean production rates are of 36 m2/s3, 41 m2/s3, 15 m2/s3, and 16 m2/s3, respectively. This imbalance would cause an increase in the local turbulence level, consistent with the distributions of turbulent kinetic energy and the presence of a hot spot.

To summarize, before the hot spot the rotor near wake is mainly strained by plane shear, and thus, the regions of peak production consist of two layers on both sides of the center of the wake. The dissipation and production rates are of the same order of magnitude. In the hot spot region, the production exceeds the dissipation rate, causing a local increase in turbulence level. Note that outside of the kink there are regions with negative 2-D production rates. Since negative turbulence production is unlikely, the results suggest that in these regions the flow is affected significantly by three-dimensionality.


a b

c d

Figure 5. Two-dimensional production rates (m2/s3) of turbulent kinetic energy at (a) 0%~26%; (b) 18%~48%; (c) 41%~70%; and (d) 57%~90% rotor chordlengths downstream of the rotor trailing edge. The lines used for calculating 1-D spectra are the same as in Figure 3. The numbers show the estimated dissipation rates ε (m2/s3).

Indeed, these regions are the only ones in the near wake and

mid span with significant levels of 33S , which is calculated

using the continuity equation, i.e. 33S = -( 11S + 22S ). As

Figure 6 shows, the rotor wake is affected more by three-dimensionality than the regions located outside of the wake.

The distribution of 33S peaks around the rotor trailing edge and

in the hot spot. Stress-Strain Alignment

To further address issues related to turbulence modeling in turbomachines, the geometric relationship between the Reynolds stresses and strain rates is investigated. Assumption of the alignment of stress-strain fields is fundamental to the eddy-viscosity based turbulence models, which are widely used in turbomachinery computations. The phase-averaged mean strain and Reynolds stresses are second-rank tensors, Tij, and can be

transformed (i.e. projected onto a different Cartesian coordinate system) according to:

k ’l’ ik ’ jl’ ijT C C T= (10)

where Tij are the components in one coordinate system

described by three orthogonal unit vectors in (i = 1-3), Tk’l’ are

the components in a second system with unit vectors ’in , and

’ ’C = ik i kn n is the scalar product of unit vectors aligned with

the Cartesian coordinates. Since we presently have only 2D data, we will perform only 2D rotation, i.e. 3 3 ’=n n . This kind

of transformation enables us to align the local strain rates or the local Reynolds stresses along directions that maximize the shear stress or strain, or along the principal direction, zeroing the shear stress or strain. Figure 7a presents the distribution of normalized transformed shear strain aligned locally with coordinate system that maximizes the local shear strain, i.e.


Figure 6. Distributions of averaged normal strain

33S / , within the rotor wake.

1 2S ′ ′ is maximized. At this orientation 1 1S ′ ′ is equal to

2 2S ′ ′ , the

sum, 1 1 2 2S S′ ′ ′ ′+ , is an invariant, i.e. it does not depend on

orientation, and ( )3 3 33 1 1 2 2 1 1S S S S 2S′ ′ ′ ′ ′ ′ ′ ′= = − + = − (See

Chow et al. [45] for more information). Figure 7b shows the distribution of normalized transformed shear stress aligned locally with coordinate system that maximizes the local shear

stress, i.e. - 1 2′′ ′′′ ′u u is maximized. The arrows in figure 7a and

7b show the local 1′1′ and 1′′1′′ directions, respectively. As a quantitative measure of the relative alignment between the stress and strain fields, Figure 7c shows the distribution of the angle between the maximum shear strain (1′1′) and maximum shear stress (1′′1′′ ) directions, i.e. the angle between the local alignment vectors presented in 7a and 7b.

Some observations can readily be made: 1. Specific areas characterized by the elevated angle

include the near wake close to the rotor trailing edge, the wake segments generated by upstream blades on both sides of the rotor wake, and the region surrounding the center of the hot spot. In a substantial part of the rotor wake (away from the hot spot), the angle is very small. Thus, away from the immediate vicinity of the trailing edge and the hot spot the alignment is consistent with eddy viscosity type Reynolds stress models.

2. Comparing Figure 7a with Figure 6, 1 2S ′ ′ is substantial

almost everywhere in the rotor near wake, whereas

33S ( ( )3 3 33 1 1 2 2 1 1S S S S 2S′ ′ ′ ′ ′ ′ ′ ′= = − + = − ) diminishes in most

of the near wake, except for the region around the rotor trailing edge and in the turbulent hot-spot. These trends imply that the rotor near wake mainly consists of two parallel layers experiencing planar shear with opposite signs (at least in the

available 2-D projection of the flow field), as one would expect to find in a 2-D wake.

3. Figures 7a and 7b show that the Reynolds shear stress peaks at the center of the hot spot, where the shear strain rate diminishes. It means that in this area, trends of the shear strain and stress are contradictory, in contrast to linear eddy-viscosity models. More details on stress-strain alignment in the rotor wake, as well as Sub Grid Scale stress budgets related to Large Eddy Simulations can be found in Chow et al. [45]. Rotor Boundary Layer

Figure 8 shows the results of the high-magnification, high-resolution measurements of the rotor boundary layer, at 500 rpm and at mid-span. The field of view is 15x15 mm2, covering the boundary layer starting from 70% chord up to the trailing edge and the very near wake. Here, the vector spacing is 117 µm (0.2% chord length, 15% of the trailing edge thickness), and the ensemble averages are based on 1000 vector maps. The data presented here is obtained at a different rotor phase than the wake measurements, to ensure that there is an IGV wake impinging on the suction side of the rotor blade.

The rotor suction side and pressure side boundary layers are clearly visible as regions with low relative velocity, high vorticity and shear strain rate, as well as high turbulent kinetic energy and Reynolds shear stress. At this rotor phase an IGV wake segment is intersecting close to the trailing edge of the rotor blade. This incoming wake is evident from the distributions vorticity, turbulent kinetic energy and the Reynolds shear stress. The upper boundary and centerline of the wake extend from (-0.21, 0.03) up to (-0.1, 0.1) and from (0.21, -0.044) up to (-0.03, 0.02), respectively, intersecting with the suction surface. The wake lower boundary extends from (-0.08, -0.09) to (0.03 -0.054) and it intersects with the near wake region. The approximate upper and lower boundaries and center line of the impinging IGV wake are superimposed on the contour plots in Figure 8b to f.

Several flow phenomena occur as a consequence of the wake impingement on the suction side boundary layer. First, immediately below the upper boundary of the IGV wake (upper line), the boundary layer becomes thinner, whereas up to this point, the boundary layer thickness increases monotonically. The thinning is evident from the distributions of relative velocity and axial velocities in Figures 8a and b. Second, within the wake impingement region, starting from x/c=-0.1 and extending up to x/c=-0.05 (just above the wake centerline), the turbulent kinetic energy and Reynolds shear stress levels increase significantly very close to the surface. Third, the distribution of mean strain rate within the impinging wake region becomes very complex (Figure 8d). Thus, wake impingement severely disrupts the uniform and positive mean shear strain distribution that exists along the suction side up to this region, generating a negative shear strain very close to the wall.


a. b. c.

Figure 7. (a) Distribution of shear strain ′ ′1 2S / locally aligned with a coordinate system that maximizes the shear

strain. (b) Distribution of shear stress ′′ ′′′ ′ 21 2 tipu u /U locally aligned with a coordinate system that maximizes the shear

stress and (c) Distribution of magnitude in degrees of the relative angle between the local coordinate system that maximizes the Reynolds shear stress, and a system that maximizes the mean shear strain. The arrows in (a) and (b) indicate the local 1’1’ and 1″1″ directions, respectively.

Shrinking of the boundary layer and the elevated levels of turbulent kinetic energy and Reynolds shear stress in the wake impingement region can also be seen in the profiles shown Figures 9a - c. They show the distributions of wall parallel velocity, turbulent kinetic energy and Reynolds shear stress along four lines, marked A-D in Figure 8a. Figure 9a confirms the increase in boundary layer thickness from A to B, i.e. above the impingement region, and the subsequent decrease in thickness in sections C and D. Figures 9b and c show the increase in the turbulent kinetic energy and Reynolds shear stress levels very close to the surface in sections B and C. Note that the elevated turbulent kinetic energy and shear stress occur within the upper side of the impinging IGV wake, i.e. above the wake centerline, and in the region where the negative vorticity region of the IGV wake impinges on the positive vorticity region within the boundary layer.

Figure 10 shows the distributions of flow and turbulence parameters along the suction side of the rotor blade. The relative velocity magnitude, turbulent kinetic energy and the Reynolds shear stress distributions are presented at 0.6% chord lengths (0.3 mm) away from the surface, i.e. within the boundary layer, and the axial velocity plot is presented at 3.2% chord lengths (1.6 mm) away, i.e. just outside of the boundary layer. The horizontal axis, s/c, is the curvilinear distance from the leading edge along the suction surface of the blade. The upper boundary of the impinging IGV wake is located at s/c=0.86. The axial velocity distribution has a maximum at s/c=0.85, i.e. very close to the upper boundary of the IGV. At the same location, but within the boundary layer, the relative

velocity magnitude is minimal whereas the turbulent kinetic energy and the shear stress distributions have local maxima. These results clearly indicate that, the increased axial velocity between the IGV wakes acts like a jet, impinging on the blade surface. In this jet impingement region the turbulent kinetic energy and the shear stress peak. Below the jet impingement region, i.e. in the IGV wake region, the relative velocity increase, and the boundary layer becomes thinner. These trends are consistent with the characteristics of a turbulent jet impinging on a flat surface (Landreth and Adrian [47], Fairweather and Hargrave [48]). When viewed in the frame of reference of the flow outside the boundary layer and the IGV wake (referred to as “non-wake” regions in the previous section), the boundary layer thinning can also be attributed to the “slip velocity” (Kerrebrock and Mikolajczak [4], Sanders et al [31]) associated with the IGV wake segments. This slip velocity causes a suction (or “negative jet”) effect, as previously reported also by Ho and Lakshminarayana [5], Valkov and Tan [6], and Chow et al. [37]. Clearly, flow non-uniformities associated with the IGV wake segments have a direct impact on the structure of the rotor boundary layers, and on the associated turbulence.

As in the data analysis of the rotor near wake, turbulence energy spectra have been obtained for 6 lines inside and near rotor boundary layer. Figure 11a shows the locations of the sample lines: lines 1~3 are located inside the suction side boundary layer; line 4 is located inside the pressure side boundary layer; lines 5 and 6 are located outside the rotor boundary layer, but inside the upstream IGV wake segments. It is clearly evident that the turbulence inside the rotor boundary


a. b. c.

d. e. f. Figure 8. Phase-averaged (a) relative velocity magnitude (b) axial velocity (c) vorticity (d) mean shear strain (e) turbulent kinetic energy and (f) Reynolds shear stress contours around the rotor trailing edge at 500 rpm. Field of view is 15x15 mm2, the vector spacing is 117 µm (0.2% chord) and averages of 1000 image pairs are shown. Lines in b, c, d, e and f show the approximate boundaries and the centerline of the impinging IGV wake. The lines in (a) indicate the location of velocity profiles shown in Figure 9.

a. b. c. Figure 9. (a) Relative velocity magnitude; (b) turbulent kinetic energy; and (c) Reynolds shear stress profiles in the rotor boundary layer along lines A, B, C and D (Figure 8a). l is the distance normal to the suction surface.


layer is highly anisotropic. As a result, we cannot obtain reliable dissipation rate estimates due to the failure of “local isotropy” assumption, on which the procedure is based. To highlight the

anisotropy effect, the values of 3 / 21

5 / 311 1 1

18( ) (1.6)

55

− E k k and

3 / 21

5 / 322 1 1

183 / 4 ( ) (1.6)

55

−

E k k estimated from E11 and 3/4E22 are

listed in Table 1. One can still clearly observe the orders of magnitude difference between the “dissipation rates” inside and outside of the boundary layer. Saddoughi and Veeravalli [46] and Saddoughi [49], using unique and detailed turbulence measurements within boundary layers conclude that in order to maintain a one decade of locally isotropic inertial subrange, the ratio of the Kolmogorov to mean-shear timescales, *

cS

(≡S(ν/ε)1/2, where S is the mean shear strain), should be no more than approximately 0.01. Clearly, the present case falls well beyond this range. Using line 1 in Figure 11 as an example, the turbulent kinetic energy k in this area is about 0.7 m2/s2 and the mean shear strain S is in the order of 1000 1/s. From the spectra shown in Figure 11b, one is able to estimate the integral length scale for line 1 to be about 1.6 mm. Using

3/ 2 ≈ (Tennekes and Lumley [43]), it gives an estimate for dissipation of ε ≈ 370 m2/s3 (very close to the average of the two values provided in Table 1). Thus, *

cS for line 1 is about

0.06, well beyond the Saddoughi and Veeravalli [46] criterion for local isotropy.

Figure 10. Distributions along the rotor blade suction side. s is the curvilinear distance from the leading edge. Distributions of relative velocity magnitude, turbulent kinetic energy and Reynolds shear stress at 0.6% chord lengths (0.3 mm) away from the surface (within the boundary layer); and distribution of axial velocity at 3.2% chord lengths (1.6 mm) away from the surface (just outside the boundary layer). Upper boundary of the impinging IGV wake is at s/c=0.86.

(a)

(b)

Figure 11. Rotor boundary layer; (a) distribution of 2-dimensional production rates (m2/s3) of turbulent kinetic energy; and (b) 1-D spatial energy-density spectra along the sample lines indicated in (a). The convention used here is the same as Figures 3, 4 and 5. The rotor trailing edge is located at (0,0).


Table 1. Dissipation values (m2/s3) of rotor boundary layer estimated by –5/3 slope-line fit in Figure 11b. C1 is

equal to 18(1.6)

55.

Line 1 Line 2 Line 3 Line 4 Line 5 Line 6 5/3 1 3/ 2

11 1 1 1( ( ) )−E k k C 766 986 758 521 4.51 5.77

5/3 1 3/ 222 1 1 1(3 / 4 ( ) )−E k k C 98.7 84.5 31.3 19.6 3.16 5.09

2-D production 538 453 546 88.1 0.45 -0.35 Nevertheless, it would still be interesting to compare the production rate to the two “dissipation” estimates. Average production values along lines 1, 2, 3, 4, 5 and 6 are presented in Table 1. In lines 5 and 6, the value of Production is very close to zero. In lines 1, 2, 3 and 4 (all within the boundary layer), the values of Production fall between the two values of ε. Near line 2 in Figure 11a, there is a region in which the Production appears to be negative. This suggests that three-dimensional effects are substantial in the region where the wake impinges on the boundary layer.

CONCLUSIONS This paper presents detailed experimental data on the flow

and turbulence within the wake and boundary layer of a rotor blade operating behind a row of Inlet Guide Vanes (IGVs). The experiments are performed in a refractive index matched facility that provides an unobstructed view of the entire flow field. Results of the high-resolution 2D PIV measurements are used for characterizing the mean flow, Reynolds stresses, turbulent kinetic energy as well as dissipation and production rates.

Within the rotor wake, dissipation and production rates are high and of the same order of magnitude near the trailing edge, and decrease rapidly with increasing distance from the blade. The turbulent kinetic energy also decays initially. The trend is reversed in the wake kinking region, where production increases but the dissipation keeps decreasing, resulting in a region with elevated turbulence level, i.e. a turbulent hot spot. One-dimensional spectral analysis shows that, except for the very near wake and hot-spot regions, the turbulence within the rotor wake can be assumed to be isotropic.

The alignment of the Reynolds stresses and mean strain rates within the rotor wake is compared. In a substantial part of the rotor wake, away from the hot spot and immediate vicinity of the trailing edge, the directions of the maximum shear strain and shear stress are aligned, i.e. consistent with eddy viscosity type Reynolds stress models. The rotor near wake mainly consists of two parallel layers experiencing planar shear with opposite signs (at least in the available 2-D projection of the flow field), as one would expect to find in a 2-D wake. However, near the trailing edge, and around the hot spot, orientation differences can extend up to 45°. Furthermore, there is substantial mismatch in the location of the local maxima of

stresses and strains. The values of 33S are also large there,

indicating that the flow is three-dimensional.

Rotor boundary layer measurements focus on a region

where the IGV wake intersects with the rotor blade. It is shown that when the increased axial velocity region between the IGV wakes impinge on the blade surface, the turbulent kinetic energy and the Reynolds shear stress peak within the impingement region, while the relative velocity decreases. Within the IGV wake region the axial velocity is lower, the relative velocity increases, and the boundary layer becomes thinner. The near wall strain rate distribution becomes very complex and the near wall turbulence level increases. This phenomenon is similar to the effect of a turbulent jet impinging on a flat surface. When viewed in the frame of reference of “non-wake” flow regions, the boundary layer thinning can also be attributed to the suction (or “negative jet”) effect of the “slip velocity” within the IGV wake segments. One-dimensional spectral analysis shows that the turbulence in the rotor boundary layer is highly anisotropic. As a result, the spectra cannot be used for estimating the dissipation rate.

ACKNOWLEDGMENTS This project is sponsored in part by ONR, under grant

number N00014-99-1-0965, managed by P. Purtell and Ki-Han Kim, and in part by AFOSR under grant No. F49620-01-1-0010, managed by T. Beutner,. The ONR program is part of a joint effort involving W. Blake (NSWC/Carderock), H. Atassi (U. of Notre Dame) and Y.T. Lee (NSWC/Carderock). We would like to thank Yury Ronzhes and Stephen King for their contribution to the construction and maintenance of the facility, and Y.T. Lee for designing the blades.

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