SHORT PAPER

Tze Pei Chong • Shan Zhong

On the transverse growths of turbulent spotsin a heated boundary layer with favorable pressuregradient

Received: 5 February 2012 / Accepted: 19 March 2012 / Published online: 26 April 2012� The Visualization Society of Japan 2012

Keywords Turbulent spot � Liquid crystals � Reynolds analogy � Favorable pressure gradients

1 Introduction

In a non-isothermal flow, the thermal properties of a boundary layer can be related to the momentum transferof fluid. With the absence of pressure gradient and the Prandtl number equal to unity, the skin friction cf canbe related to the Stanton number St by the following relationship: cf/2 = St, which is known as the Reynoldsanalogy. In a transitional boundary layer, the presence of streamwise pressure gradients is known tosignificantly affect the transition zone length. However, much less attention has been paid to the differentbehaviors exhibited by the momentum and thermal boundary layers undergoing transition in non-isothermalflows. It was observed that the extent of transition indicated by the skin friction is smaller than that indicatedby the heat transfer data in the favorable pressure gradient flows. The opposite is true for the data obtained inthe adverse pressure gradient flows (Blair 1982; Sharma 1987).

The above phenomenon has important implications about a reliable prediction of the heat transfer ratefor the design of internally cooled turbine blades and their life span (Mayle 1991). It suggests that whenpressure gradients are present, transition prediction based on aerodynamic parameters will not accuratelypredict the heat transfer process. This indicates the breakdown of Reynolds analogy in transitional flowssuggesting that CFD codes for gas turbine design, which infer the thermal transport directly from themomentum transport, may be inadequate. There are several experimental studies on the overall time-averaged structures of transitional boundary layers to understand this problem (Wang et al. 1996; Bons2005). The authors of this short note had attempted to address this issue at a more fundamental level byexamining the influence of pressure gradients on turbulent spots in terms of their momentum and heattransfer behaviors across the boundary layers using hot/cold-wire anemometry (Chong and Zhong 2006).We found that the spots’ propagation rates deduced from the velocity and temperature data are identical atzero and favorable pressure gradients. However, the heat transfer within a spot is inhibited more than that ofthe momentum transfer in the transverse direction when a favorable pressure gradient is present, whichexplains why the transitional zone length indicated by the heat transfer parameters is longer. To get moreinsight into the flow physics described above, in this note we employed two types of liquid crystals: one

T. P. Chong (&)School of Engineering and Design, Brunel University, Uxbridge, UKE-mail: t.p.chong@brunel.ac.ukTel.: ?44-1895-266370Fax: ?44-1895-256392

S. ZhongSchool of Mechanical, Aerospace and Civil Engineering, University of Manchester, Manchester, UK

J Vis (2012) 15:189–192DOI 10.1007/s12650-012-0130-5

responses primarily to surface temperature changes (TSLC), whereas the other is temperature insensitiveand displays different color upon changes in the shear stresses (SSLC). It is believed that the use of thesetwo types of liquid crystals provides a unique approach to quantify the possible difference in the behavior ofturbulent events that may exhibit in the two distinctly different transfer processes.

2 Experimental set up

The experiment was conducted in an open-circuit wind tunnel with a closed-test section of 200 9 460 mm.The turbulence intensity of the wind tunnel was measured as 0.3 % at a freestream velocity of 28 ms-1.Two flat plates (one of which has a heated surface) with an interchangeable elliptical leading edge were usedfor each type of liquid crystals. Note that the plate surface was only slightly heated to avoid thermalinstability being added to the boundary layer. The flat plate had a total length of 750 mm and was mountedhorizontally across the whole width of the tunnel test section. The favorable pressure gradient was generatedby attaching a wedge-shaped flat sheet to the ceiling of the test section. At the maximum operating velocity,the wedge angle used in the present experiment produces an accelerating flow with a constant pressuregradient parameter of K = 0.24 9 10-6. The velocity and pressure gradient distributions for the zero andfavorable pressure gradient cases are shown in Fig. 1.

3 Results and discussions

The turbulent spots were generated by injecting high frequency air jet through a 0.5 mm hole located at188 mm downstream of the plate’s leading edge at the centerline. The reason for using high spot generatingfrequency is to ensure that the liquid crystal coating reveals the envelope of the wing tips of a train of spots,giving a true indication of the spot average spreading angle at the near wall region. The raw images ofturbulent spots shown by the two types of liquid crystals are shown in Fig. 2. With the SSLC, the laminarregion shows orange color. The region beneath the turbulent spots is associated with higher surface shearstress and appears in green. It can be seen that the spreading angle of the turbulent region decreases as thefavorable pressure gradient increases. With the TSLC, the laminar boundary layer is associated with a vividblue color whereas the region covered by the turbulent spots shows transition from blue to red towards thecenter of the spots. Similar to the results from the SSLC, the spreading angle of the turbulent region alsoappears to decrease with favorable pressure gradient.

Because all of the SSLC images were taken at an oblique angle from a downstream viewpoint, the lateralspreading angles obtained directly from the original images are expected to be considerably larger than theiractual values. Therefore, a perspective transformation of the images is necessary to correct this error beforethe result can be compared with those from the TSLC images which were taken with the camera placingnormal to the test surface. The contours of color intensity (shear stress) and heat transfer coefficient deduced

Fig. 1 Velocity distributions for the zero pressure gradient (open circles) and favorable pressure gradient (open triangles)cases. The figure also shows the K (=m/U2dU/dx) distributions for the zero pressure gradient (closed circles, K = 0) andfavorable pressure gradient (closed triangles, K = 0.24 9 10-6). Note that m is the kinematic viscosity of air and U is thefreestream velocity

190 T. P. Chong, S. Zhong

from the liquid crystal images are shown in Fig. 3. The spot spreading angle was taken as the half includedangle between two boundaries that depict the envelope of the wedge shapes in Fig. 3. It is found that thespreading angle indicated by the SSLC was nearly the same compared to that by the TSLC in the zeropressure gradient case (6.9� and 6.8�, respectively, more in-depth analysis about the validity of these valuescan be found in Chong and Zhong 2005). However, for the favorable pressure gradient case, the spreadingangle of the shear stress footprint (5.9�) was about 19 % larger than that of the thermal footprint of turbulentspots (4.8�).1 Supported by the fact that identical spreading angles were obtained by the SSLC and TSLC in

Fig. 2 Raw images of turbulent spots obtained from (a1, b1) SSLC and (a2, b2) TSLC at pressure gradients of (a1–a2) K = 0;and (b1–b2) K = 0.24 9 10-6. The liquid crystals also exhibited the side wall contaminations from the wind tunnel

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Fig. 3 Contours of (a1, b1) Color intensity obtained from the SSLC raw images (images have been perspectively transformedto correct the error caused by the oblique viewing angle); and (a2, b2) heat transfer coefficients, h (W/m2K) obtained from theTSLC. a1 and a2 represent K = 0; and b1 and b2 represent K = 0.24 9 10-6

1 The main sources of error in determining the relative values of spreading angle obtained from the two types of liquid crystalare the spatial resolution of liquid crystal images and the uncertainty associated with perspective transformation of the imagesfrom SSLC. The spatial resolution is 0.5 mm/pixel and the error due to the perspective transformation is 0.2�. Therefore, thetotal errors in the spreading angles are estimated to be ±0.6� for the TSLC and ±0.7� for SSLC.

On the transverse growths of turbulent spots 191

the zero pressure gradient case, the difference in spreading angles in a favorable pressure gradient flow, asinferred by the same types of liquid crystals, is a valid outcome despite that it is within the overlap of theuncertainty bands.

Because both of the SSLC and TLC experiments were performed in the same wind tunnel with exactlythe same flow conditions, the results suggest that when a favorable pressure gradient is present, the tur-bulence momentum at the near wall is more readily to be radiated to the spanwise extreme of the spots thanheat, in consistent with our previous observations in the main body of the turbulent spots (Chong and Zhong2006). Since a larger spreading angle corresponds to a faster spanwise growth of turbulent spots, a shortertransition zone length will happen. The current results are consistent with the fact that the eddy-transport ofheat is by direct contact of molecules as opposed to the momentum exchange which is through pressureforces acting on the element as a result of contact with the surrounding fluid where no direct contact isrequired (Kays and Crawford 1980). Therefore, the momentum and thermal boundary layers should respondto pressure gradients in different ways.

The results in this note provide a vivid visual approach to support the findings of Blair (1982) and Bons(2005), i.e. that in a transitional flow subject to a favorable pressure gradient the momentum boundary layerestablishes the fully turbulent flow state faster than the thermal boundary layer. This result also suggests thatwhen the turbulent spot-based models are employed to predict a transitional boundary layer, differentempirical relations have to be used to estimate the spot growth rate in order to obtain the correct transitionzone length for the momentum and thermal boundary layers, respectively.

References

Blair MF (1982) Influence of freestream turbulence on boundary layer transition in favourable pressure gradients. ASME TransJ Eng Power 104:743–750

Bons J (2005) A critical assessment of Reynolds analogy for turbine flows. ASME Trans J Heat Transf 127:472–485Chong TP, Zhong S (2005) On the three-dimensional structure of turbulent spots. ASME Trans J Turbomach 127:545–551Chong TP, Zhong S (2006) On the momentum and thermal structures of turbulent spots. ASME Trans J Turbomach

128:689–698Kays WM, Crawford ME (1980) Convective heat and mass transfer, 2nd edn. McGraw-Hill, New YorkMayle RE (1991) The role of laminar-turbulent transition in gas turbine engines. ASME Transaction: Journal of

Turbomachinery 113:509–537Sharma OP (1987) Momentum and thermal boundary layers on turbine airfoil suction surfaces. AIAA paper 87-1918Wang T, Keller FJ, Zhoug D (1996) Flow and thermal structures in a transitional boundary layer. Exp Thermal Fluid Sci

12:352–363

192 T. P. Chong, S. Zhong