Application of main flow data in the determination of boundary shear stress in smooth closed ducts

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1 University of Wollongong Research Online Faculty of Engineering - Papers (Archive) Faculty of Engineering and Information Sciences 01 Application of main flo data in the determination of oundary shear stress in smooth closed ducts Yu Han University of Wollongong, yh916@uo.edu.au Shu-Qing Yang University of Wollongong, shuqing@uo.edu.au Nadeesha Dharmasiri University Of Wollongong Pulication Details Han, Y., Yang, S. & Dharmasiri, N. (01). Application of main flo data in the determination of oundary shear stress in smooth closed ducts. In E. D. Loucks (Eds.), World Environmental and Water Resources Congress 01: Crossing Boundaries, Proceedings of the 01 Congress (pp ). United States: American Society of Civil Engineers. Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Lirary: research-pus@uo.edu.au

2 1175 Application of Main Flo Data in the Determination of Boundary Shear Stress in Smooth Closed Ducts Yu Han 1, Shu-Qing Yang, Nadeesha Dharmasiri 3 1 School of Civil, Mining & Environmental Engineering, University of Wollongong, NSW 5, Australia, PH (+61) ; yh916@uo.edu.au. School of Civil, Mining & Environmental Engineering, University of Wollongong, NSW 5, Australia; shuqing@uo.edu.au. 3 School of Civil, Mining & Environmental Engineering, University of Wollongong, NSW 5, Australia; ncdkl861@uo.edu.au. Astract Boundary shear stress plays an important role in momentum and mass transfers, ut it is very difficult to determine experimentally. This paper develops a ne method hich can evaluate the oundary shear stress using the main flo data ased on Momentum Balance Method (MBM). A theoretical relationship eteen the oundary shear stress and parameters of main flo region is estalished. The otained results from MBM agree ell ith other methods, indicating that the ne method is reasonale and reliale. The significant contriution of MBM is that the local oundary shear stress accurately and quickly can e determined y application of this ne method. Hence, MBM may provide a useful tool for hydraulic engineers ho need a simple and easy ay to measure the oundary shear stress. Suject Headings: Boundary shear; Reynolds stress, Secondary flo; Turulent flo; Velocity distriution. 1 Introduction Flo in a straight, rectangular closed duct has een attracted considerale attention from many researchers due to its simple oundary geometry and nonexistence of free surface effects. It is generally accepted that there are many similarities eteen flo in a rectangular open channel and closed duct [1-]. Knoledge of the distriution of oundary shear stress around the etted perimeter of an open channel or closed duct is a matter of fundamental importance in many hydraulic equations concerning resistance, sediment, dispersion, or cavitations prolems [3-4]. Most idely used method of skin friction measurements in turulent oundary layer is the application of Preston tue[5]. It is understandale that Preston tue provides an easier ay to operate in skin friction measurements, hoever, it also presents an ostruction to the flo. Some measurements of fluctuating oundary shear stress in all-ounded flos have utilized heat transfer or mass transfer shear proes[6]. The main idea of operation of a heated-element proe is to kno the rate of removal of heat y using a small metal film or ire. Hence, the instantaneous fluctuations of the all shear stress can e measured hen the proe is placed very near to or in contact ith the all. Using these intrusive proes, the difficulties are dominated y using these intrusive methods in reversing flos, vortices and highly turulent flos. The hot-ire technique has also een used for the detailed investigation of all shear stress [7]. Hoever, the determination of the instantaneous skin-friction from hotire measurements of the velocity gradient in the viscous sulayer is strongly

3 World Environmental and Water Resources Congress 01: Crossing Boundaries ASCE affected y the proximity of the all [8]. More recently, all-shear-stress measurements ere carried out using liquid-crystal coating, hich is thin enough to detect the local all-shear-stress distriution [9]. The near oundary region is the source of turulent eddies, and it is also the place here turulent energy is dissipated into thermal energy due to resistance [10]. Boundary region is the most difficult layer from hich to otain reliale experimental data. Thus, the accurate velocity data is very difficult to acquire relative to the data in the main flo region. The purpose of this paper is to present a theoretical relationship asis for oundary shear stress and main flo ased on the momentum alance or the Momentum Balance Method (MBM), and this ne method as verified using the experimental data, also the otained results are compared ith other methods, e.g. Log-la method (LLM), Reynolds shear stress method (RSM) and Viscous su-layer Method (VLM). It is found that the agreement is reasonaly acceptale. Thus this novel method can e developed to determine the oundary shear stress using the main flo data in order to avoid the viscous su-layer measurements. Theoretical considerations Yang [11] estalished the momentum alance equation hich is validated for a steady, uniform, and fully developed turulent flo (see Figure 1). For any aritrary shapes of channel or closed duct, taking the element acd as an example, the gravitational force component for a unit length in the x-direction has to e alanced y the friction force on the interface of element acd : y z x a 3 d u 4 1 c Figure 1. The calculation grids for aritrary cross-sectional channel and coordinate system c d a ρ gaacd S f = ds = ds ds ds ds s c 3 + d 4 (1) a here = density of fluid; g = gravitational acceleration; A acd = area of flo region acd shon in Fig. 1; S f = the energy slope; = the shear stress on the interface of acd ; s = the length of the interface, e.g. c, cd and da in Fig.1; 1,, 3= shear stresses on the interface c, cd and da respectively; and 4 = the oundary shear stress to e determined. In streamise direction, ρgaacd S f denote

4 1177 the gravitational force component for a unit length. The shear stress on the interface can e expressed as follos [1]: u ' xn = μ ρuv ρu' v () here ρ uv = the momentum flux caused y mean flo ( n represents the normal direction of the interface); and v n = the velocity component of secondary flo normal to the interface. In the Cartesian coordinate system, Eq. can e reritten in the folloing form [13]: u xy = μ ρuv ρu'v' (3a) y u (3) xz = μ ρu ρu'' z here xy, xz = the shear stress on the horizontal and vertical interface, respectively; u and u, v and v, and are the mean and fluctuating velocities in x, y and z directions, respectively. In the main flo region, the viscosity effect can e ignored. Therefore, for an aritrary control volume (see Figure and Figure 3), the averaged sideall shear stress ( y, y+ Δy) (acting on oundary from y to y+ y on the y-axis) can e computed and Eq. 1 can e ritten in the folloing form: z z y+δy y+δy ρ = ds = dz + dz + dy + dy (4) s gas ) xy1 xy xz ( y, y+δy 0 0 y y here A = the shadoed area (see Figure and Figure 3); xy1, xy = the shear stress on the upper and loer horizontal interfaces, respectively; xz = the shear stress on the vertical interface from the point (z, y) to the point (z, y+ y). From experiments, the first three terms can e calculated from the measured 3D velocities, thus the sideall averaged shear stress from y to y + y could e determined y using this momentum alance equation. Likeise, the ed shear stress can e evaluated from a vertical strip. Thus the all shear stress can e calculated. y y + Δ xz ( y, y + Δy) dy y xz ( y + Δy / ) Δy (5) When e started to calculation, one can set at the same strip ut choosing different Momentum alance elements to uild the equation, thus the averaged calculation value ould e used. To check the MBM in application of determination of oundary shear stress, one must kno the local friction velocity u * along the etted perimeter ith different methods, such as: Log-la method (LLM): For a smooth oundary, the shear velocity u * can e determined from the measured mean velocity distriution u (y) in conjunction ith the logarithmic la. u u* y =.5ln u* ν (6)

5 1178 y y/h / 0.6 (0,y+Δy) (0,y) h/ 0. (0,0,0) (z,0) (z+δz,0) x z z/h Figure. Orientation of test channel and calculation grids in measured crosssection y Tracy (1965) 5mm 5mm y/h (0, y+δy) (0, y) (0, 0) / h (z, 0) (z+δz, 0) 15.4 mm 15.4 mm 15.4 mm 15.4 mm z/h Figure 3. Orientation of test channel and calculation grids in measured crosssection y Tracy (1976) here ν = kinematic viscosity. Eq. 6 is valid only in the inner region [14]. Reynolds shear stress method (RSM): u can e determined from the measured Reynolds stress distriution[15]: xy u y = u' v' + ν = u* (1 ) (7) ρ y h here ρu'v' = the Reynolds shear stress; h = ater depth. In near oundary region y 0, the measured Reynolds shear stress is constant, thus the oundary shear stress is the Reynolds shear in the constant shear stress layer. Viscous su-layer Method (VLM): It is ell knon that the velocity in the viscous sulayer follos the linear distriution, i.e., u yu = * (8) u* ν 3 MBM application to Smooth Closed Duct Flos In this section, to computational modules are comprised to prove the MBM conception, hich are ased on the experimental components carried out on a rectangular test duct and a complex oundary duct y Tracy [16-17] respectively. In these to test ducts, the air as supplied to a ind tunnel. In Tracy s (1965) experiment, the measurements ere otained y the rectangular closed duct ( =

6 mm, h = 81mm). In the original paper the value of secondary currents are discernile only for the region of 0 < y / < 1and 0 < z / < 1, so e ill mainly focus on this domain. The ed shear stress distriution on the side all and the ottom are in unity in this domain. Within Tracy s (1976) experiments regarding complex oundary, the flo characteristics could e analyzed y dividing the duct into to simple rectangular ducts as upper and loer parts. Experimental data [17] ere only reported velocity measurements in loer part, hence, measurements in the compound channel only the loer part ere used. If this loer duct as cut horizontal and vertically don its centre line, and then the flo in marked area ould e the same as a rectangular duct. Thus for the rectangular duct illustrated (see Figure 3), the horizontal plane of symmetry, having zero shear stress, is analogous to the free surface, the calculation domain of these to ducts provide the physical model of a rectangular open channel. It is ovious the aspect ratios of these to models are equal to 1. The parameters measured y Tracy [16-17] in calculation domains include u, v,, u 'v' and u '', thus this dataset can e used for our purpose. In performing the computations, Tracy [16-17] datasets reported in literatures are used in the (see Figure ) and 10 8 (see Figure 3) mesh grid respectively. Studies on the distriution of the oundary shear stress ere started to evaluate the friction velocity u * along the etted perimeter in closed duct. Hence, LLM, VLM and RSM have een performed to investigate the value of u * along the etted perimeter. As MBM outlined in theoretical considerations, the averaged local shear for each strips (see Figure and Figure 3) ould e further explored from Eq. 1 to 5. It is noted each velocity profiles represented the velocity u varied ith the horizontal distance z at a constant value of y/h, hich can determine the u * results y Eq. 6 for each value of y/h, and the range are from 0. to 0.6 [16] and from to [17] respectively. Folloing this, the value of can e computed using ρu * along the sideall, thus, the otained oundary shear stress using LLM is normalized y the mean all shear stress, i.e., / and the data are presented in Figures 4 and 5 respectively LLM 0. VLM RSM MBM (Tracy 1965) y/h Figure 4. Comparison of averaged sideall shear stress distriutions eteen MBM and other three methods ased on Tracy s (1965) data.

7 MBM Tracy (1976) LLM RSM VLM y/h Figure 5. Comparison of averaged sideall shear stress distriutions eteen MBM and other three methods ased on Tracy s (1976) data. The velocity u value closed to the all (z/h < 0.0 and z/h < ) ould e used u zu* in calculation y VLM ( = ) respectively. Here, it appears from VLM that the u* ν friction velocity u* ould e computed, and define the oundary shear stress = ρu * along the duct perimeter (on the sideall), then they otained results are presented (see Figure 4 and Figure 5) as /. The measured the Reynolds shear stress near the oundary can e used to determine the oundary shear stress y RSM, i.e.: z u' ' = u* (1 ) u* (9) h This simplified methodology is also proposed y Nikora [18]. The distriution of u '' relative to u e shon from Tracy s paper [16-17], therefore, ρu'' can e * used to define all shear stress, here all the measurements of u '' closed to the all (z/h<0.0 and z/h<0.0167) ere used in computation respectively. The data points (see Figure 4 and Figure 5) also sho the variation of / along the duct perimeter made dimensionless hich is computed y RSM LLM 0. VLM RSM MBM (Tracy 1965) z/h Figure 6. Comparison of averaged ed shear stress distriutions eteen MBM and other three methods ased on Tracy s (1965) data

8 MBM (Tracy 1976) LLM RSM VLM z/h Figure 7. Comparison of averaged ed shear stress distriutions eteen MBM and other three methods ased on Tracy s (1976) data Similarly, LLM, VLM and RSM are also applicale to assess the oundary shear stress distriution along the ed. Based on Tracy s measurement data in the oundary region, like velocity and Reynolds shear stress, one can determine the u * values for each z / h using LLM and VLM. Folloing the calculation of = ρu *, thus, the variation of ed shear stress / along duct perimeter are presented (see Figure 6 and Figure 7). As the distriution of u 'v' relative to u* e shoed in literature, it is used to investigate the oundary shear stress along the ed y RSM. The comparison shos that MBM orks ell and agrees ith other methods. The ed shear stress rises from the corner and approaches stale value hen it is nearly the symmetry plane. Fig. 8 and Fig. 9 shos the comparison eteen the oundary shear stress otained y MBM and other researchers measured results, including Larence [19], Leutheusser [0] and Knight and Patel [1]. It is seen that the relative ed shear stress / of MBM agrees reasonaly ell ith these mentioned data. These experiments ere also conducted in the long rectangular ducts controlled y airflo through the system. In their experiments, values of local all shear stress ere deduced from the readings of Preston tues, hich ere placed in a direct contact ith the oundary. Experimental measurements for all shear stress are made in a dimensionless form, i.e. / as ell. It is seen that the relative ed shear stress / of MBM agrees reasonaly ell ith these mentioned data.

9 MBM Tracy (1965) Knight 1985 Leutheusser 1963 Larence z/h Figure 8. Comparison of averaged allside shear stress distriution eteen MBM and other researcher s data for same aspect ratio MBM Tracy (1976) Knight 1985 Leutheusser 1963 Larence z/h Figure 9. Comparison of averaged ed shear stress distriution eteen MBM and other researcher s data for same aspect ratio To validate the accuracy of MBM, the relative error eteen MBM calculations (suscript c) and other methods (suscript m) has een defined as E = c - m / m. The sideall shear stress has the averaged E of.5%, 1.9% and 3.3% eteen MBM and LLM, VLM, RSM respectively (see Figure 4); The sideall shear stress has the averaged E of -.6%, 0.1% and 0.3% eteen MBM and LLM, VLM, RSM respectively (see Figure 5); In Figure 4, the maximum value of E (=15.5%) occurs in the strip of Tracy s to groups of experimental data prove MBM can provide a reasonaly good estimation for the distriution of all shear stress. Figure 6 shos the comparison along the ed y Tracy s 1965 dataset, it presents MBM orks ell and agrees ith other methods (the averaged E=-5.17%, -3.1% and 0.98% eteen MBM and LLM, VLM, RSM respectively), the maximum E= 13%. Similarly, Figure 7 shos the calculation results from Tracy s 1976 data, the comparison shos that MBM orks ell and agrees ith other methods (the averaged E = -0.9%, 0.9 % and -0.3% eteen MBM and LLM, VLM, RSM respectively), the maximum E = 13.8% occurred in the corner region. The

10 1183 distriution of / is similar to the distriution of /,so e only compared / ith others experimental data as presented (see Figure 8 and Figure 9), the maximum averaged E occurred near its centre line. It should also e noted that the main flo values near the plane of symmetry used in the calculation are greater, hence a small error in measurements ill result in significantly inaccurate. It should also e noticed that our model relied on Tracy s data, hich calculates only certain layer due to the data limited, therefore, the grids cannot e detailing. The accuracy also depends on the main stream velocity on each grid points. Therefore, dividing the grids can lead to calculation accuracy. The present model simulates this ehavior fairly ell, and is actually in etter agreement ith present experiment data. 4. Conclusions A novel method (MBM) has een developed for determining oundary shear stress using main flo data ased on the concept of momentum alance. The existing experimental data in smooth closed duct [16-17] have een ell applied in MBM. In terms of the average all or ed shear stress, comparison eteen three existing methods and other researcher s experimental data are shon MBM is possile to measure local oundary shear stress accurately and quickly under field conditions. The folloing conclusions can e dran from this study: 1) MBM outlined the local oundary shear stress for turulent flo over the etted area can e conveniently analyzed in terms of the volume of flo toards a particular etted oundary area. Thus, the main flo data that are important for calculation accuracy, on the asis of this concept, the flo cross-sectional area in different flo geometry can e divided and the grids used uild momentum alance equation. ) This study used smooth rectangular duct flos to test the MBM, and other existing methods ere used for comparison including LLM, RSM and VLM. These comparisons shoed that the MBM is comparale ith these existing methods. When the estimated shear stress y MBM as compared ith Larence (1960), Knight and Patel (1985) and Leutheusser (1963), it presents this method orks ell and agrees ith these existing data. No empirical coefficient or assumption is involved in the method. This method s reliaility mainly depends on the accuracy of velocity measurement in the main flo region. 3) For smooth rectangular duct flos, MBM can assess the distriution of oundary shear stress.its performance is as good as other methods. The applicaility of MBM could e extended to complex oundary conditions. The next step is to prove its applicaility in complex flos.

11 1184 References: [1] D. W. Knight and H. S. Patel, "Boundary Shear in Smooth Rectangular Ducts," Journal of Hydraulic Engineering, vol. 111, pp. 9-47, [] S. Y. Lim and S. Q. Yang, "Simplified model of tractive-force distriution in closed conduits," Journal of Hydraulic Engineering-Asce, vol. 131, pp. 3-39, Apr 005. [3] S. Dey and M. F. Lamert, "Reynolds stress and ed shear in nonuniform unsteady open-channel flo," Journal of Hydraulic Engineering-Asce, vol. 131, pp , Jul 005. [4] A. Divedi, et al., "Hydrodynamic Forces Generated on a Spherical Sediment Particle during Entrainment," Journal of Hydraulic Engineering, vol. 136, pp , 010. [5] J. H. Preston, "The Determination of Turulent Skin Friction y Means of Pitot Tues," Journal of the Royal Aeronautical Society, vol. 58, pp , [6] A. Wietrzak and R. M. Luepto, "Wall shear stress and velocity in a turulent axisymmetrical oundary-layer," Journal of Fluid Mechanics, vol. 59, pp , Jan [7] H. H. Bruun, Hot-ire anemometry: principles and signal analysis: Oxford University Press, [8] H. H. Fernholz, et al., "Ne developments and applications of skin-friction measuring techniques," Measurement Science and Technology, vol. 7, pp , [9] D. C. Reda, et al., "Ne methodology for the measurement of surface shear stress vector distriutions," Aiaa Journal, vol. 35, pp , Apr [10] N. Chien, and Wan, Z., Mechanics of sediment transport: Reston, Virginia., [11] S.-Q. Yang, "Depth-Averaged Shear Stress and Velocity in Open-Channel Flos," Journal of Hydraulic Engineering-Asce, vol. 136, pp , Nov 010. [1] S. Q. Yang, "Conditionally averaged turulent structures in D channel flo," Proceedings of the Institution of Civil Engineers-Water Management, vol. 163, pp , Fe 010. [13] J. Guo, et al., "Modified log-ake la for zero-pressure-gradient turulent oundary layers," Journal of Hydraulic Research, vol. 43, pp , [14] I. Nezu, and Nakagaa, H., Ed., Turulence in open-channel flos. Rotterdam, Netherlands: Balkema Pulishers, 1993, p.^pp. Pages. [15] I. Nezu and W. Rodi, "Open channel flo measurements ith a laser doppler anemometer," Journal of Hydraulic Engineering-Asce, vol. 11, pp , May [16] H. J. Tracy, "Turulent flo in a three-dimensional channel," J. Hydraul. Div.Am. Soc. Civ. Eng, vol. 91, pp. 9-35, [17] H. J. Tracy, "The structure of a turulent flo in a channel of complex shape," U.S. Geological Survey, Prof. Paper 983, 1976.

12 1185 [18] V. I. Nikora and D. G. Goring, "Fluctuations of Suspended Sediment Concentration and Turulent Sediment Fluxes in an Open-Channel Flo," Journal of Hydraulic Engineering, vol. 18, pp. 14-4, 00. [19] L. C. Hoagland, "Fully developed turulent flo in straight rectangular ductssecondary flo, its cause and effect on the primary flo (thesis). Technical Report No.," OOR United StatesTue Fe 05 15:11:00 EST 008DTIE; NSA English, [0] H. J. Leutheusser, "Turulent Flo in Rectangular Ducts," Journal of the Hydraulics Division, Proc. of ASCE, vol. 89, pp. 1-19, 1963.