Aerospace Science and Technology 14 (2010) 49–55

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Aerospace Science and Technology

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Upstream condition effects on the evolution of symmetric and asymmetricnear-wakes of a flat plate

Dong-Ha Kim a, Jo-Won Chang b,∗, Hak-Bong Kim a, Myong-Hwan Sohn c

a Department of Aerospace and Mechanical Engineering, Korea Aerospace University, Hanggongdae gil 100 Hwajeon-dong, Deogyang-gu, Goyang city, Gyeonggi-Do, 412-791,Republic of Koreab Department of Aeronautical Science and Flight Operation, Korea Aerospace University, Hanggongdae gil 100 Hwajeon-dong, Deogyang-gu, Goyang city, Gyeonggi-Do, 412-791,Republic of Koreac Department of Aerospace Engineering, Korea Air Force Academy, Ssangsuri, Namil-myun, Chungwon-gun, Chungbuk-Do, 363-849, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 April 2008Received in revised form 6 November 2009Accepted 11 November 2009Available online 14 November 2009

Keywords:Flat plateTurbulent boundary layerUpstream conditionNear-wakeSimilarity profileReynolds shear stress

An experimental study was conducted in order to investigate the influence of upstream conditions withrespect to near-wakes evolution by using a two-dimensional flat plate. A position of xB.L/C = 0.98 for theflat plate with a blunt trailing edge was settled upon as an upstream condition point. Three upstreamconditions (i.e., natural laminar, bypass transitional, and turbulent boundary layers) were applied bytripping wires that were placed on the upper surface of the plate. The free-stream velocity was 6.0 m/sand the local Reynolds numbers based on momentum thickness at the upstream conditions point were259, 303, and 331. The boundary layers and the near-wakes were measured by hot-wire anemometry. Itwas observed that for the symmetric near-wakes, the local similarity profiles of the mean defect velocitymatched well with the traditional similarity profile in the center part, whereas some difference in theouter part occurred due to the different trailing edge shape. For the asymmetric upstream conditions,the center of the local similarity profiles moved to one side, and the amplitude of the local similarityprofiles over the wakes edge varied according to the momentum thickness of the upstream conditions.Also, the collapse of the normalized Reynolds shear stress depended on turbulent quantities included inthe upstream conditions, and these results showed that the evolution of the near-wakes is clearly affectedby upstream conditions.

© 2009 Elsevier Masson SAS. All rights reserved.

1. Introduction

A wake is a complicated flow with various mixing mechanismsand its characteristics can be divided into the inner-wake, near-wake, intermediate-wake, and far-wake [8]. In the far-wake region,the variation of the mean velocity and half-width follows half-power law relations with a given streamwise location from thetrailing edge (x−1/2 for mean velocity, and x1/2 for half-width) [2].This indicates that the far-wake evolution is only dependent on thedownstream distance regardless of upstream condition, and thenthe far-wake or the asymptotic wake allows a similarity analysisof the flow properties. Wygnanski et al. [12] observed that a sim-ilarity of mean flow and turbulent quantities mainly depends onthe wake generators, and suggested an exponential function for themean flow similarity profile. In additional, George [4] proposed inasymptotic flow that self-preservation could be possible dependingon upstream conditions by using proper similarity variables (lengthand velocity scales) even though they have different growth rates.

* Corresponding author. Tel.: +82 2 300 0082; fax: +82 2 3158 1849.E-mail address: jwchang@kau.ac.kr (J.-W. Chang).

1270-9638/$ – see front matter © 2009 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.ast.2009.11.002

This means that a similarity according to the local Reynolds num-ber based on the proper scale of an axisymmetric turbulent wakecould be possible, thus the initial condition, i.e. upstream condi-tion or local Reynolds number, are significant determinants on theevolution of an axisymmetric wake [5].

On the other hand, the inner- and near-wakes are significantlyaffected by upstream flow conditions. Andreopoulos and Bradshaw[1] measured turbulent quantities and mean velocity in a near-wake and suggested an empirical relationship in semi-logarithmicform for the mean velocity. They stated that the mean velocity ofthe inner- and near-wakes is affected by the skin friction scale ofthe boundary layer over the trailing edge. Also, Ramaprian et al.[9] investigated the structure and development of a symmetricand asymmetric near-wake, and showed that the near-wake prop-erties corresponding to the laminar boundary layers change sud-denly after trailing edge separation due to the existence of laminarinstability. Therefore, the boundary layer properties immediatelybefore trailing edge separation ultimately contribute to the near-wake evolution.

As a result, the above mentioned previous study has reportedthe significance of upstream conditions on near-wake evolution.In the present study, it was assumed that the effect of upstream

50 D.-H. Kim et al. / Aerospace Science and Technology 14 (2010) 49–55

Nomenclature

C chord length of a flat plateD trailing edge thickness of a flat plate (15 mm)k kinetic energy in 2-D flow ≈ 3/4(u′ 2 + v ′ 2)

T.I turbulence intensity for boundary layer (%)u mean velocity for boundary layerU local mean velocity for near-wake−u′v ′ Reynolds shear stressx streamwise directiony vertical axis against streamwise directionδ boundary-layer thickness for a flat plate or local half-

width for near-wakeδ∗ displacement thickness for boundary layer

θ boundary-layer momentum thickness or momentumthickness for near-wake (

∫ +∞−∞

uU∞ (1 − u

U∞ )dy)

Rθ Reynolds number based on boundary-layer momen-tum thickness

Subscripts

B.L boundary layer for the flat plateu,B.L boundary layer on upper side for the flat platel,B.L boundary layer on lower side for the flat platei local value for near-wake (u; upper side, l; lower side)W total value for near-waked maximum defect in wakes velocity

Fig. 1. Schematic of test section.

conditions in the near-wake could be simply observed from theexamination of local similarity of the near-wake in order to ob-serve the developing near-wake with respect to upstream condi-tions. However, since a near-wake responds sensitively to upstreamconditions, and because there are many upstream conditions to beconsidered, any studies that consider the influences of upstreamconditions on the near-wake evolution seem to be unique and notuniversal. Accordingly, even if a similarity among measured data isaccomplished, the data may show unique near-wake development,meaning a ‘local similarity profile’, depending on the upstream con-dition.

The objective of the present study is to gain physical insightinto the development of symmetric and asymmetric near-wakesdepending on upstream conditions using a local similarity anal-ysis examination. As mentioned earlier, the results of this studymay be a unique, but they can provide important factors for thenear-wake evolution through consideration of various and typicalupstream conditions. In the present study, the upstream condi-tions were estimated from both the local Reynolds numbers basedon boundary-layer momentum thickness and turbulent quantitiesinvolved in the boundary layers. The position of the upstream con-ditions was selected as the flow properties around the trailing edgeof the plate, and they were controlled by tripping of boundary lay-ers.

2. Experimental apparatus and procedures

The present study was carried out in an open circuit wind tun-nel with a square-cross section of 0.5 m (H) × 0.5 m (W) andlength of 1.4 m (L). Fig. 1 shows a schematic of the test section.The plate model with a blunt trailing edge (without a splitter) wasmade by a duralumin with smooth surface, and chord length andtrailing edge thicknesses were 0.7 m and 15 mm, respectively. Theflat plate model was vertically mounted in the center of test sec-tion, and the gap between the test section side and the plate was

less than 2.5 mm to minimize a three-dimensional effect. The lead-ing edge shape and of the plate was a super-elliptic profile.

An I-type hot-wire probe (Dantec’s 55P14) was employed forthe measurement of the boundary layers, and a cross-type probe(Dantec’s 55R51) was used for the near-wakes measurement. TheI-type hot-wire probe was elaborately calibrated from low to highvelocity, and the measurement for boundary layers were contin-uously executed during 20 s with a sampling rate of 6 kHz andlow pass filter. The boundary layer measurement was limited nearthe wall of the plate owing to the prong thickness of hot-wireprobe, thus the first measurement position from the wall was atabout yB.L = 0.25 mm, and it was compensated for the results.The boundary layers were measured at a position of xB.L/C = 0.98over the trailing edge, and the intervals of the measurements were0.1 mm and 0.5 mm in the inner and outer layers.

A cross-type hot-wire was calibrated from −45◦ to +45◦ , andthe uncertainty during calibration was estimated to be less than0.5%. The probes were adjustable using a two-dimensional traversesystem with a spatial resolution of 0.01 mm. The near-wakes weremeasured at seven downstream stations, x = 15,25,35,45,55,65,and 75 mm from the trailing edge, and the measurement intervalsin the vertical direction were 2.0 mm in the center and 5.0 mmin the outer part. The measurement station of x = 15 mm is veryclose to the trailing edge, thus the measured data may be affectedby recirculation flow. Under the same conditions [6] of presentstudy, the mean velocity which was measured at x = 5 mm wasslightly rectified in the center part of near-wake, however the rec-tification was not observed at x = 15 mm. This result was con-firmed by the near-wake visualization [6], which suggested thatthe recirculation area was generated in the range of x � D .

The free-stream velocity was 6.0 m/s, and the correspondingReynolds number was 2.8 × 105 based on the chord length of theplate. The turbulence intensity under the test free-stream velocitywas about 0.4% in the entrance of the test section. The blockage ofthe flat plate in the test section was less than 2.9% and the free-stream velocity variation was less than 0.2% during the tests.

In the present study, the boundary layer properties at a stationof xB.L/C = 0.98 were controlled by tripping wires with a cylindri-cal shape placed on the upper surface. Fig. 2 shows a schematic ofthe controlled boundary layers. Many attempts were performed toobtain the typical boundary layer properties. In case 1, the trippingwire was not used, and in case 2, the tripping wires of diameter1.0 mm were placed 0.145 m and 0.175 m from the leading edge.In case 3, the tripping wire of diameter 1.0 mm was installed at0.1 m from the leading edge. A tripping wire is known as a use-ful device to control the boundary layer and its size, shape, andthe attached location are very important ingredients to determinethe flow [3]. By using boundary layer tripping, the upstream condi-tions were discernable based on both turbulent quantities included

D.-H. Kim et al. / Aerospace Science and Technology 14 (2010) 49–55 51

Fig. 2. Schematic of upstream conditions for each case.

Fig. 3. Hot-wire signals of the boundary layers (yB.L ≈ 1.0 mm).

in the boundary layer and the momentum thickness. In case 1, anatural boundary layer was allowed on both sides of the plate, thusa natural symmetric near-wake was expected. In that case, we es-timated the error of symmetric properties by the measurement atx = 15 mm downstream (wake region), and it was less than 3.7%.In each case, the flow properties of the upstream conditions willbe discussed in the following section.

3. Result and discussions

3.1. Boundary layers at xB.L/C = 0.98 (upstream conditions)

In the present study, the upstream conditions manifest theboundary layer properties at xB.L/C = 0.98 (over the trailing edgeof the plate), and they were controlled by tripping wires. Fig. 3shows the hot-wire signals of the boundary layers at yB.L ≈1.0 mm and at xB.L/C = 0.98. In case 1, the boundary layer showsa laminar flow without a fluctuation. In case 2, it is a bypasstransitional boundary layer that includes intermittent laminar andturbulent components, and it presents a typical turbulent bound-ary layer with random fluctuations in case 3.

Fig. 4 shows the normalized mean velocity profiles and tur-bulence intensity at xB.L/C = 0.98. Fig. 4a, which shows the nor-malized mean velocity in the boundary layers, presents that theboundary layer in case 3 is the most developed among them. InFig. 4b, the turbulence intensity of the boundary layers in cases 2and 3 is almost similar with maximum value, which was largerthan 0.1U∞ . This is because the boundary-layer development in

Table 1Properties of upstream conditions.

Parameters (unit) Case 1 Case 2 Case 3

Boundary-layer thickness (δu,B.L , mm) 4.5 8.5 12.0Momentum thickness (θu,B.L , mm) 0.64 0.75 0.82Reynolds number based onmomentum thickness (Rθ ) 259 303 331Displacement thickness (δ∗

u,B.L , mm) 1.299 1.278 1.304Shape factor 2.030 1.704 1.590

cases 2 and 3 is intrinsically experiencing a similar developingmechanism, which means a bypass transition elicited by the trip-ping wire. However, the flow structure in cases 2 and 3 are con-siderably different in the temporal domain, as shown in Fig. 3. Theresults of the boundary layers measurement at xB.L/C = 0.98 aresummarized in Table 1.

In the present study, the upstream conditions were estimatedby the momentum thickness and the boundary-layer thicknessbetween the upper and lower side at xB.L/C = 0.98. The corre-sponding values based on the boundary-layer momentum thick-nesses are (θu,B.L/θl,B.L)case 1 ≈ 1.00, (θu,B.L/θl,B.L)case 2 ≈ 1.17, and(θu,B.L/θl,B.L)case 3 ≈ 1.28, and the values with respect to theboundary-layer thicknesses are (δu,B.L/δl,B.L)case 1 ≈ 1.00, (δu,B.L/

δl,B.L)case 2 ≈ 1.89, and (δu,B.L/δl,B.L)case 3 ≈ 2.67. Xiaofeng et al.[13] and Thomas and Xiaofeng [11] employed the normalized pa-rameters based on momentum thickness for near-wake in the flatplate that has a trailing edge with a splitter, and the value (i.e. 2.5)they presented is higher than those in the present study. The dif-ference of both values stemmed from both the trailing edge shapeof the body and upstream condition location (they considered thewakes region as an upstream condition).

3.2. Local similarity profiles of mean defect velocity

Fig. 5 shows the local similarity profiles of the mean velocity forthe three upstream conditions. The presented data was comparedwith Rogers’s similarity profile (RSP) [10]. The ordinate representsthe local mean defect velocity scaled by the maximum defect ve-locity, and the abscissa shows the position of the vertical directionscaled by the total half-width, which is the sum of the local half-width on the upper and lower sides of the near-wakes. The localhalf-width was defined as the distance between the point at whichthe local mean defect velocity is half of the maximum defect ve-locity and y = 0 (center of the plate; see Fig. 2), thus the localhalf-width is different between the upper and lower side in theasymmetric upstream conditions. The legend at the upper rightdenotes the measurement stations scaled by the total momen-tum thickness (θW ), which is the sum of the local momentum

52 D.-H. Kim et al. / Aerospace Science and Technology 14 (2010) 49–55

Fig. 4. Mean velocity distributions and turbulence intensity for the boundary layers.

Fig. 5. Similarity profiles of mean defect velocity.

thicknesses on the upper and lower side of the near-wakes. Thetotal momentum thickness for the near-wakes was increased asthe boundary-layer momentum thickness over the trailing edge in-creased, which indicates that x/θW is decreased if the same down-stream station (x) is considered. From this, the near-wake region,which was classified as the range of x/θW � 25 in Ramaprian andPatel’s study [8], substantially narrows down as the boundary-layermomentum thickness increases.

The mean velocity of near-wakes recovered to the free-streamlevel as it went to downstream. However, in the cases of asym-metric upstream conditions, the mean velocity distribution slowlyshifted to the upper side (positive y-direction), on which the

boundary-layer momentum thickness is larger. This indicates thatthe asymmetric mass flux loss in the upstream conditions yieldsa shift of the mean velocity profiles, and this movement in they-direction balances the mean velocity field when going to down-stream.

In Fig. 5, the mean defect velocity profiles collapse into onecurve in all cases, and the local similarity profile of the mean de-fect velocity matches the RSP in the inner part of the near-wakes,whereas it definitively differs in the outer part. The RSP is, as faras the authors know, the best fit for similarity profile in a sym-metric wake as drawn from the results in Wygnanski et al. [12].In the outer part of the symmetric near-wakes, the difference be-

D.-H. Kim et al. / Aerospace Science and Technology 14 (2010) 49–55 53

tween the present data and RSP seems to be caused by the trailingedge shape between the two studies. That is, the RSP was formu-lated for a flat plate with a splitter, whereas the flat plate witha blunt trailing edge (without a splitter) was used in the presentstudy. Substantially, this means that the mean defect velocity incase of the trailing edge with a splitter is smaller than the casewithout a splitter. On the other hand, the local similarity profilesat x/D = 1.0 (i.e. x/θW = 3.4 for case 1, 3.3 for case 2, and 3.2 forcase 3) almost matches the RSP, as shown in Fig. 5. This is an un-expected result, and it may be induced from the proper ratio of themean defect velocity to the wake width in the recirculation region.

The similarity profiles of the asymmetric near-wakes are shownin Figs. 5b and 5c. The data in the present study fails to collapseto the RSP in the inner part, and the RSP is no longer suitablefor asymmetric cases. It is concluded that the local similarity pro-files of asymmetric near-wakes exhibit an eccentric pattern dueto the asymmetric upstream conditions, and that the eccentricityof the similarity profiles increase as the degree of the asymmetryof the upstream conditions increases.

(U∞ − U )/Ud

= exp[0.252(y/δW − 0.1) − 3.085(y/δW − 0.1)2

+ 1.047(y/δW − 0.1)4 − 0.146(y/δW − 0.1)6] (1)

(U∞ − U )/Ud

= exp[−0.186(y/δW − 0.05) − 3.144(y/δW − 0.05)2

+ 1.294(y/δW − 0.05)4 − 0.182(y/δW − 0.05)6] (2)

The local similarity profiles in case 3 were modified from the RSPformula in Eq. (1), and is shown with the RSP in Fig. 5d. In Eq. (1),the first and sixth orders are added in order to fit the increment ofthe mean defect velocity in the outer part, and the term in paren-thesis is added to correct the movement of the wake center. FromEq. (1), it is possible to observe that the streamwise variations ofthe near-wake center are constant by as much as 10% of the totalhalf-width at the upstream conditions of (θu,B.L/θl,B.L) = 1.28.

In case 2, the difference of the near-wake center cannot beclearly discerned from the RSP (Fig. 5b), but was evaluated byusing the arithmetic mean of the values in the other cases. Fi-nally, it is given in Eq. (2), which has different constants in eachterm compared to Eq. (1). Eq. (2) shows that the near-wake cen-ter constantly shifts by as much as 5% of the total half-width for(θu,B.L/θl,B.L) = 1.17. As a result, the variation of the maximum de-fect velocity position against the boundary-layer momentum thick-ness (y/δW )Ud /(θu,B.L/θl,B.L) can be approximated as 0.352 in thepresent asymmetric near-wakes. In Eqs. (1) and (2), the variationsof the constant for each term are entirely due to the asymmet-ric degree of the boundary-layer momentum thickness. Also, theobservation reveals that the local similarity profiles of the near-wakes corresponding to the laminar boundary layer are differentfrom that of the turbulent or of the transitional boundary layer inthe outer part. This indicates that turbulent quantities containedin the upstream conditions significantly affect a local similarity inthe mean velocity field as well as the difference of the momentumthickness.

3.3. Reynolds shear stress

Reynolds shear stress normalized by the kinetic energy(−u′v ′/k), which was suggested by Rodi [7], was investigatedwhen the kinetic energy is defined as k ≈ 3/4(u′u′ + v ′v ′) in atwo-dimensional flow. Fig. 6 shows the −u′v ′/k and contours of−u′v ′/U 2∞ . In Figs. 6a, 6b, and 6c, the abscissa denotes the verti-cal direction normalized by the local half-width and the ordinate

presents −u′v ′/k, and in Figs. 6d, 6e, and 6f, the abscissa and or-dinate show the normalized streamwise and vertical direction innear-wakes region, respectively.

Fig. 6a shows −u′v ′/k for a symmetric near-wake. A peak of0.3 ∼ 0.4 was observed at x/θW = 3.4, and it rapidly deterioratedas it went downstream. In this case, −u′v ′/U 2∞ (Fig. 6d) almostshows symmetric distributions, and the uncertainty about symmet-ric property seems to come from the setting of the angle-of-attackof the plate. Fig. 6b shows Reynolds shear stress normalized bythe kinetic energy −u′v ′/k in case 2, which shows an asymmetricdistribution. At x/θW = 21.0, −u′v ′/k presents a positive value inentire measurement points, showing the energy transfer direction.Fig. 6e, which shows the contour of −u′v ′/U 2∞ for case 2, gives thephysical insight about the results in Fig. 6b. The peak value dis-tribution on the upper side spreads widely along the streamwisedirection as well as the vertical direction, and the near-wake cor-responding to the laminar boundary layer is clearly shrunken onthe lower side. We can infer, therefore, that the intermittent dis-turbance in the transitional boundary layer is amplified after theseparation at the trailing edge, and the near-wake correspondingto the laminar boundary layer is pliable with respect to the inter-mittent disturbance.

Fig. 6c shows −u′v ′/k for case 3, which the turbulent bound-ary layer was imposed as the upstream condition on the upperside. In this case, −u′v ′/k collapses to a maximum value by 0.2from x/θW = 6.1 on the upper side. This indicates that the de-cay rate of the kinetic energy and of the Reynolds shear stressyields a proportional relationship for the near-wakes correspond-ing to the turbulent boundary layer, and that the local collapse ofthe Reynolds shear stress is clearly affected by the turbulent quan-tities involved in the upstream conditions. The collapse was alsoreported by Thomas and Xiaofeng [11], who noted a collapse by0.3 ∼ 0.4 in −u′v ′/k of the plate with a splitter, and the valueis higher than that of the present data (i.e. 0.2). The differencebetween the two studies is due to the effect of v ′v ′ in the ki-netic energy, which largely increased in the trailing edge withouta splitter. Fig. 6f presents the contour of −u′v ′/U 2∞ in case 3. Asit goes downstream, −u′v ′/U 2∞ of the separated flow from thelaminar boundary layer is almost similar to case 1 (symmetricnear-wake), whereas −u′v ′/U 2∞ of the near-wake come from theturbulent boundary layer is almost constant after about x/D = 2.0.The collapse of −u′v ′/k is accomplished in this region, and the re-sult proves that random and continuous disturbance instigates themore stable near-wake properties.

4. Conclusions

The upstream condition effects on the near-wakes evolution ofa flat plate with a blunt trailing edge (without a splitter) wereinvestigated by measuring the mean velocity and Reynolds shearstress. Three upstream conditions on the evolution of the near-wakes are composed of the laminar (symmetric wake), the bypasstransitional and laminar (asymmetric wake), and the turbulent andlaminar (asymmetric wake) boundary layers.

The local similarity profiles of the mean defect velocity for thesymmetric near-wakes matched well with the traditional similar-ity profile in the center part, whereas some difference in the outerpart occurred due to the different trailing edge shape. The centerof the local similarity profiles for the asymmetric upstream con-ditions moved to one side, and the amplitude of local similarityprofiles over the wakes edge varied according to the momentumthickness of the upstream conditions. For both the symmetric andthe asymmetric near-wakes, the local similarity profiles depend-ing on upstream conditions were observed, and it was concludedthat trailing edge shape and turbulent quantities involved in theupstream condition are very significant factors. The collapse of the

54 D.-H. Kim et al. / Aerospace Science and Technology 14 (2010) 49–55

Fig. 6. Reynolds shear stress distributions (interval 0.00353, - - - negative value, — positive value).

normalized Reynolds shear stress depended on turbulent quantitiesincluded in the upstream conditions, and these results showed thatintermittent disturbance in transitional boundary layer instigatesthe more unstable near-wake properties than the continuous one.

Acknowledgement

This work was supported by grant No. R01-2002-000-00442-0from the Basic Research Program of the Korea Science and Engi-neering Foundation.

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