ISSN 10637850, Technical Physics Letters, 2011, Vol. 37, No. 12, pp. 1154–1157. © Pleiades Publishing, Ltd., 2011. Original Russian Text © N.V. Semin, V.V. Golub, G.E. Elsinga, J. Westerweel, 2011, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2011, Vol. 37, No. 24, pp. 26–34.
Laminar Superlayer in a Turbulent Boundary Layer¶ N. V. Semina, b, c *, V. V. Goluba, G. E. Elsingac, and J. Westerweelc a
Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow, 125412 Russia Moscow Institute for Physics and Technology, Dolgoprudny, Moscow oblast, 141700 Russia c Delft University of Technology, 2628CA Delft, The Netherlands *email:
[email protected]
b
Received August 16, 2011
Abstract—We confirm the existence of a sharp boundary between the turbulent and nonturbulent fluid at the outer region of a zeropressuregradient turbulent boundary layer at a low Reynolds number. For the first time, using the Tomographic Particle Image Velocimetry technique, we determine mean statistical parame ters of the boundary: its thickness and relative velocity jump. DOI: 10.1134/S1063785011120285
Widespread classical turbulent flows are bounded by an external fluctuating potential flow. Since the beginning of the last century a long discussion is ongo ing about how nonturbulent fluid is entrained by tur bulent flows [1–10]. Flow visualization [2] revealed the existence of large organized coherent structures in the instantaneous flow structure of a plane turbulent mixing layer (TML). Large eddies entrain the irrota tional fluid from the external flow and transfer it inside of a growing turbulent flow. Until recently, this entrainment mechanism was considered common for all turbulent shear flows [1–5]. Practically, it implies that large eddies produce macroscopic Reynolds shear stress. Similarly to the viscous shear stress in laminar flows, it entrains via friction increasing amounts of fluid inside the turbulent flow. In another point of view, originating in [6], it was suggested that the turbulent part of the flow could grow and transfer vorticity only in the direct contact with a nonturbulent part through a viscous shearing mechanism. The interface between the turbulent flow and a fluctuating external potential flow was hypothe sized to comprise socalled laminar superlayer (LS). Laminar here is understood in a sense that present in it fluctuations are irrotational. As a result, shear Rey nolds stresses are zero. LS is continuously propagating in the irrotational part of the flow and transferring it vorticity. This hypothesis got support from measure ments of intermittency and the analysis of schlieren photographs, where a thin shear layer could be seen at the interface with the external flow. Control volume analysis predicted a nonzero jump in the mean longi tudinal velocity component across the LS. Attempts to measure this quantity were unsuccessful due to the inability at that time to conduct satisfactory measure ments of the vorticity field in turbulent flows. ¶
The article was translated by the authors.
Recently, several theoretical, experimental, and numerical studies [7–10] have shown quantitatively that, in contrast to the TML, contribution of large eddies to the entrainment in a developed turbulent wake (DTW) and free turbulent jet (FTJ) is only about 10%. The main effect comes from small eddies. They form a sharp wavy boundary between the turbulent and nonturbulent fluid. Thus, the LS hypothesis received some evidence in a number of turbulent flows. In this regard, a question about the external struc ture of a turbulent boundary layer (TBL) and to what extent it is similar to other boundaryfree turbulent shear flows is of interest. In this paper, for the first time, a detailed study of the LS in a TBL was con ducted by measuring the 3dimensional flow structure of the velocity field. The main goal is to determine the statistical properties of the spreading LS. This may lead to qualitative or gross quantitative description of the phenomenon. Measurements of the zeropressure gradient TBL (ZPG TBL) have been done in a water tunnel with a 5 m long optically transparent test section with a cross section of 0.6 × 0.6 m2. The schematic of the experi mental setup is shown in Fig. 1a. The water is lead into the test section via a settling chamber with honey combs and wire meshes followed by a 6 : 1 area ratio contraction. As a consequence, a homogeneous flow is created with the axial free stream intensity measured σU < 0.5%. To compensate for a gradual acceleration of the flow downstream due to growth of the boundary layer and reduction of the effective crosssection, an adjustable bottom has been installed. With this bottom the bulk velocity over the whole measuring section is maintained constant within 0.5% of the mean velocity of 0.20 m/s. TBL is tripped 2.5 m upstream from the measurement location with a cylinder of 5 mm diam eter. This fixed the laminarturbulent transition.
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Fig. 1. (a) Schematic of the experimental setup: (1) test section of the water tunnel; (2) CCD cameras; (3) laser; (4) lasersheet optics; (5) laser sheet. (b) Several instantaneous profiles of the x component of the velocity in different equidistant spanwise cross sections. (c) Mean velocity profilea: (1) this work; (2) LDA measurements [12]; (3) logregion; (4) DNS data for Reynolds num ber Reθ = 1410 [13]. (d) Reynolds stresses plotted in inner scaling (U∞, ν): (1–6) 〈uu〉, 〈vv〉, 〈ww〉, 〈uv〉, 〈uw〉, and 〈vw〉, respec tively, obtained in this work with the help of TPIV; (7–9) 〈uu〉, 〈vv〉, and 〈uv〉, respectively, obtained with the help of LDA [12]; (10) DNS data for Reynolds number Reθ = 1410 [13].
At the measurement location the boundary layer thickness δ99 is 68 mm. We estimate it as a coordinate where the value of the mean velocity profiles reaches 99% of the free stream velocity, U∞ = 0.203 m/s: 〈U(δ99)〉 = 0.99 U∞. The Reynolds number based on the momentum thickness, θ = 7 mm, and the free stream velocity, is 1370. The friction Reynolds num ber, Reτ ≡ Uτδ99/v, is 600. Here, Uτ = 9.2 mm/s is the friction velocity, v = 1.05 mm2/s is the kinematic vis cosity. The measurements are done with the help of the TPIV technique [11] in a volume spanning 0.05δ99 × 1.1δ99 × 1.1δ99. Coordinate axes are aligned with the streamwise (x), wallnormal (y), and wallparallel (z) directions. U, V, and W stand for the instantaneous velocity components along the streamwise, normal, and spanwise directions, respectively. One thousand instantaneous realizations (U, V, W) of the ZPG TBL has been obtained of size 2 × 82 × 97 each. Figures 1c and 1d present statistical properties of the ZPG TBL flow obtained with the help of TPIV. For comparison, reference measurements [12] of the same flow in the same facility done with the help of Laser Doppler Anemometry (LDA) and the data from the Direct Numerical Simulation (DNS) [13] for the TECHNICAL PHYSICS LETTERS
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similar Reynolds number Reθ = 1410 are plotted. When approaching the wall y+ < 20, a significant dif ference between TPIV data and data from other sources (LDA, DNS) is observed. Here, y+ ≡ yUτ/ν is an universal normal coordinate. In the region y+ > 20 there is a collapse between the data sets within the measurement error. Small deviations between TPIV/LDA data points and the DNS data can be explained because of the differences in initial and boundary conditions, which are inevitably present between experiments and simulations. The spatial res olution of the TPIV measurements is estimated according to the vonNeumann criterion as 2Δy+ = 16. This is comparable to the Kolmogorov scale η+ ~ 10 [14]. Thus, we can state, that main scales of motion from η to δ99 are correctly resolved. Figure 1b shows an example of several instanta neous profiles of the xvelocity component (U) at dif ferent equidistant spanwise crosssections. Solid verti cal lines designate the freestream velocity. Local mean velocity becomes equal to the freestream veloc ity not gradually, but through a characteristic shear layer. We will focus at this layer and determine its sta tistical properties. 2011
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Fig. 2. (a) Map of the modulus of |ω| · (U∞ – U). (b) Example of an instantaneous velocity field in TBL: vectors show y and z components of velocity (V, W); only each second vector is plotted; gray color map designates the x component of velocity (U); solid black line shows the LS location as obtained from the analysis of Fig. 2a. (c) Magnitude of the conditionally averaged vor ticity |ω|1. (d) Conditionally averaged x component of velocity 〈U〉1.
To determine the location of the LS, threshold based criteria for vorticity and local velocity fluctua tions are used [6–9]. Also, a combined criterion has been proposed [15]. We use a threshold criterion for the product of the local vorticity magnitude and the local velocity fluctuation: |ω| · (U∞ – U), both available from the TPIV data. The utmost points from the wall with the value of |ω| · (U∞ – U) above the threshold are taken as the location of the LS (Fig. 2a). For the 2 threshold we have taken (1–10) U τ /δ99. Figure 2b shows an instantaneous velocity field. The solid line designates to the location of the LS. As a next step, we compute conditional statistics of the xvelocity com ponent and vorticity components relative to the LS. For that we introduce a new coordinate system (z1, y1), where (z1) is normal to the LS and (y1) is tangent to it. In the new coordinate system the location of the LS is exactly at (0,0). The conditional averaging, which we designate as 〈·〉1, is done first along the local direction (z1), and then in time. In Fig. 2c the modulus of the conditionally aver aged vorticity |ω|1 relative to the location of the LS is presented. It is essential that the vorticity becomes
zero when crossing the LS (y > y1). The result does not depend on the threshold value in the used range. This indicates that the procedure to determine the position of LS is correct. The average location of the LS y1 = (0.6 ± 0.05)δ99. In Fig. 2d the conditionallyaveraged relative to the LS xvelocity component 〈U〉1 is presented. We define the thickness of the layer Δ1 as the distance between the LS boundary (y – y1 = 0) and a jump in the slope of the curve (y – y1 ≈ 0.1δ99). It is found that Δ1 = (0.09 ± 0.02)δ99. The relative velocity jump is equal to ΔU = (1.6 ± 0.1)Uτ. If we take into account that the integral length scale of turbulence for the TJ and DTW are of the order of their halfwidth [3–5], and for the TBL it is about δ99, then there is a similarity of the graph in Fig. 2d and the thickness Δ1 for the TBL and similar studies in the TJ and DTW [6, 7]. The results confirm that, at first, viscous effects in the TBL are not confined to the wallregion only. Sec ondly, the outer structure of the TBL at low Reynolds numbers is indeed similar to that of a turbulent jet and wake with comparable integral scale values. Thus, the result is a quantitative argument in favor of a well
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known hypothesis about the similarity between the outer structure of the TBL and DTW, which lies in the basis of the semiempirical law for the mean velocity profile in the TBL [16]. It should be noted that the Reynolds numbers in this paper and in [7–10] are low. Therefore, strictly speaking, all the conclusions are applicable only to these values of Reynolds numbers. The results confirm the conclusions of theoretical studies [6, 9], and indi cate that, in particular, viscous effects in the TBL are not limited to areas close to the wall, as assumed in the classical theory [3–5]. REFERENCES 1. J. S. Turner, J. Fluid Mech. 173, 431 (1986). 2. G. L. Brown and A. Roshko, J. Fluid Mech. 64, 775 (1974). 3. A. S. Monin and A. M. Yaglom, Statistical Hydrome chanics (Gidrometeoizdat, St. Petersburg, 1992) [in Russian]. 4. S. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000).
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5. E. U. Repik and Yu. P. Sosedko, Turbulent Boundary Layer (Fizmatlit, Moscow, 2007) [in Russian]. 6. S. Corrsin and A. L. Kistler, FreeStream Boundaries of Turbulent Flows (NACA, 1954). 7. D. J. Bisset, J. C. R. Hunt, and M. M. Rogers, J. Fluid Mech. 451, 383 (2002). 8. J. Westerweel, C. Fukushima, J. M. Pedersen, and J. C. R. Hunt, J. Fluid Mech. 631, 199 (2009). 9. J. C. R. Hunt, L. Eames, and J. Westerweel, J. Fluid Mech. 554, 499 (2006). 10. M. Holzner, A. Liberzon, and N. Nikitin, J. Fluid Mech. 598, 465 (2008). 11. G. Elsinga, F. Scarano, B. Wieneke, and B. van Oud heusden, Exp. Fluids 41, 933 (2006). 12. M. Harleman, R. Delfos, J. Westerweel, and T. van Ter wisga, Int. Workshop “Progress in Wall Turbulence” (April, 21–23, 2009, Lille, France). 13. P. R. Spalart, J. Fluid Mech. 187, 61 (1988). 14. M. Stanislas, L. Perret, and J. Foucaut, J. Fluid Mech. 602, 327 (2008). 15. R. K. Anand, B. J. Boersma, and A. Agrawal, Exp. Flu ids 47, 995 (2009). 16. D. Coles, J. Fluid. Mech. 2, 191 (1956).
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