ANALYSIS AND MEASUREMENT OF STOKES LAYER FLOWS IN AN OSCILLATING NARROW CHANNEL *

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770 008,0(6):770-775 ANALYSIS AND MEASUREMENT OF STOKES LAYER FLOWS IN AN OSCILLATING NARROW CANNEL PENG Xiao-xing China Ship Scientific search Center, Wuxi 1408, China, E-mail: me04px@alumni.ust.

Department of Mechanical Engineering, The ong Kong University of Science and Technology, ong Kong, China (ceived November 1, 007, vised January 4, 008) Abstract: The velocities of boundary layer

The focus was on the laminar case where the ynolds number A is much smaller than the transition value.

Then Laser Doppler Velocimeter (LDV) was employed to measure the Stokes boundary layer above an oscillating flat plate and inside the oscillating narrow channel at various numbers.

1 ,0(6): ANALYSIS AND MEASUREMENT OF STOKES LAYER FLOWS IN AN OSCILLATING NARROW CANNEL PENG Xiao-xing China Ship Scientific search Center, Wuxi 1408, China, SU C. T. Department of Mechanical Engineering, The ong Kong University of Science and Technology, ong Kong, China (ceived November 1, 007, vised January 4, 008) Abstract: The velocities of boundary layer flows between two parallel oscillating plates separated by small distance, i.e., in so called narrow channel, were theoretically and experimentally studied. The focus was on the laminar case where the ynolds number A is much smaller than the transition value. The theoretical analysis of the Stokes layer in oscillating flow over a narrow channel was made first. Then Laser Doppler Velocimeter (LDV) was employed to measure the Stokes boundary layer above an oscillating flat plate and inside the oscillating narrow channel at various numbers. At the same time, the phase angle difference along the vertical direction in both analysis and experiment were provided. The good agreements are shown between the measured results and the theoretical solution. Key words: oscillation flow, Stokes layer, Laser Doppler Velocimeter (LDV), narrow channel 1. Introduction Oscillating flow can be found in engineering, such as ocean engineering, environmental engineering, thermal engineering and bio-engineering [1,]. For shallow water waves, the vertical velocity is negligible outside the oscillatory boundary layer near the bottom, where the fluid particle orbits are flat ellipses. Therefore, the flows near the bottoms locally can be modelled as the sinusoidal oscillating flows over a flat bottom. In recent years, bionics is widely studied to reduce the resistance and noise of underwater body. As a basic research Wang et al. [3] numerically studied the turbulence features near an oscillating curved wall. In thermal engineering systems oscillation flows in narrow channel are applied in the cooling of infrared sensors in missile guiding and satellite based surveillance system, as Project supported by the ongkong SAR Government under the RGC (Grant No. 6165/98E), the RIG (Grant No. R195/96.EG15). Biography: PENG Xiao- xing (1963-), Male, Ph.D., Professor well as in the cooling of superconductors, semiconductors and many other civilian applications. Dong et al. [4] used the LES to investigate thermally-stratified turbulent channel flow under temperature oscillation on the bottom wall. In fact, for all the oscillating flow near the well it is necessary to understand the behavior and variation of the Stokes layer. The flow induced in a semi-infinite body of fluid over a harmonically oscillating plate of infinite extent was first analyzed by Stokes [5]. Schilichting [6] gave approximate solutions and showed steady secondary flow with the method of successive approximations. owever, the measurement of velocity inside the Stokes layer has been rare conducted. The thinness of the Stokes layer thickness may be main reason especially in the case of higher oscillation frequency before the Laser Doppler Velocimeter (LDV) is introduced to the flow measurement. In the early work Russell et al. [7] and Carter et al. [8] used dye-tracing techniques to measure the mass transport velocity and only the velocity at the outer edge of the boundary layer was usually presented. Beech [9], Sleath [10],

2 771 wung et al. [11] and Liu et al. [1] presented their measurement results of velocity inside the Stokes layer with the LDV. These studies were limited in the water column above an oscillating solid surface. But the velocity distribution in the oscillating narrow channel is still not found, where the interaction of two Stokes layers need be considered. In addition, phase information was not given in previous measurements. In this work the theoretical analysis of stokes layer in oscillating flow over a narrow channel is made first. Then LDV is employed to measure the Stokes boundary layer above an oscillating flat plate and inside the narrow channel at various numbers. The measurement results are compared with the theoretical solution. At the same time we also provide the experimental results of phase angle difference along the vertical direction.. Analytical solutions Consider the fluid flow in oscillating channel consisting of two parallel infinite plates, as shown in Fig.1, in which A is the displacement amplitude of oscillating narrow channel and = f represents the oscillating frequency. There are three length scales in the oscillating flows in the narrow channel. They are the channel width, the Stokes layer thickness = and the oscillating displacement amplitude A. We can obtain two key non-dimensional parameters defined as A A = = (1) () As ( 1 ), the problem is reduced to that of two separated Stokes layers on the oscillating plates. As is reduced to close to the problem becomes more complicated due to the interaction of two Stokes layers. The solution of the laminar flow at low A can be derived as follow. We define = z, U = u A, = t T (3) The equation and boundary condition for the flow induced by oscillating narrow channel can be expressed as U U = T 1 U(, T) = e +e it it = 1 (4) (5) The solution to Eq.(4) satisfying the boundary condition (5) is 1 cosh( i ) it U(, T)= e + cc.. cosh( i ) (6) To obtain the amplitude and the phase angle the above solution, Eq.(6) is expressed as 1 U T a b cc it (, )= [( +i )e +..]= 1 [ + e i( T + ) +. a b cc.] (7) So the amplitude is ˆ = + U a b and the phase angle is = arctan( b/ a), where a = cosh( ) [1 cos ( )] [sinh( )sinh( )sin( ) sin( ) + cosh( )cosh( ) cos( )cos( )] (8a) Fig.1 Coordinates system of flows induced by an oscillating channel

3 77 b = cosh( ) [1 cos ( )] [cosh( )sinh( ) cos( )sin( ) sinh( )cosh( ) sin( )cos( )] (8b) If we consider the problem under the coordinates moving with the plates, which represents the oscillating flows in the two infinite parallel plates, the solution will be 1 it cosh( i ) U(, T)= [e cosh( i ) it e + cc.. ] (9) The magnitude and phase angle can be written as ˆ = (1 a ) + U b =arctan 1 a b (10) (11) The solution of oscillation flow over a flat plate is just the limit case of oscillation flow over a narrow channel, which can be expressed as / U(, T)=cosT e cos( T )(1) The amplitude and phase angle at different positions can be shown as / AP( )=[(1 e cos ) + / (e sin ) ] 1/ / = arctan e sin( / ) 1 e / cos( / ) where = = z z (13) (14) For the case of flow induced by oscillating flat plate the solution is / U (, T)=e cos( T / ) (15) The amplitude and phase angle of oscillation flow in different position along the z direction can be shown as ˆ ( )=e U z / (16) = (17) The solution clearly shows that the flow induced by oscillating plate decay exponentially with the distance from the plate. In other words, the influence of oscillating plate is just limited to the fluid domain near the plate. And the Stokes layer thickness is the length scale to measure the effect. 3. Experimental setup Experiments based on the LDV measurement were carried out in a water tank facility. The experimental setup consists of the water tank, the oscillation mechanism and a support frame. The experimental model with two flat plates to form a parallel channel was fixed in the support frame and moving with the oscillation mechanism. The LDV used in the experiments was a TSI s IFA 750 LDV system as shown in Fig.. In the model installation process the great attentions were paid on the level for both cases of a flat plate and the two parallel plates. The measurement principle of LDV is based on the Doppler frequency shift. When very fine particles pass though a focus volume of two laser beams, the laser light is scattered by the particles resulting in a slight frequency shift (the Doppler shift) in the scattering light relative to the incident light. In the

4 773 present experiment fine titanium dioxide (TiO ) particles around 10 m in diameter were seeded during the measurements. Fig. A picture of traverse table and fiber-optic probe For the experiments on flows induced by an oscillating flat plate, The LDV measurements were performed over a series of runs of different plate displacement amplitudes (A=10, 50, 81.3mm) and oscillation frequencies ( f = 0.033, 0.087, 0.167, 0.5, 0.5, 0.867z). A is from 0 to , which is in the laminar domain. The velocity profiles were measured over different z locations, and the results were presented by the normalized Stokes layer thickness ( = z / =( z )/ ). In the case of narrow channel, the experiments were carried out at the conditions of two channel gaps ( = 1.5,.5mm), with a series of runs of different displacement amplitudes ( A = 5, 10, 15, 5, 50, 81.3mm) and oscillation frequencies ( f = 0.033, 0.087, 0.133, 0.167, 0.5, 0.333, 0.5, 0.667, 0.867, 1, 1.333, z). So the value of was limited from 0.5 to 40. In the present experiments, the sample size for the velocity measurement was set to be 0000 data points to provide data length of approximately 30 to 60 oscillation periods, which was suitable for phase average process. The measurements of flow fields were conducted in the vertical directions of the oscillation. There were 15 measurement locations for the flat plate and measurement locations for the two parallel plates with the interval of 0. mm. The level of laser beam was adjusted to maintain the parallelity of plate plane and laser plane before the experiments. In our present measurements it was quite difficult to rigorously determine the origin of z coordinate. ence, the origin was determined experimentally by first locating an artificial approximate origin to the LDV measurement system, measuring the velocity profiles and then shifting the origin to best fit the measured velocity profile to the theoretical solution. 4. Measurement results and comparisons with analytical solutions 4. 1 sults for flat plate The experimental results of flow induced by an oscillating flat plate were summarized in Fig.3, and the origin of -coordinate is from the flat plate. The left of Fig.3 is the measurement results of velocity magnitudes inside/near the Stokes boundary layer and the results obtained from the theoretical solution Eq.(16) is plotted as the solid line. The measured phase differences inside/near the Stokes boundary layer were plotted on right of Fig.3, with the solid line representing the results obtained by theoretical solution Eq.(17). The different symbols in the figures show the measurement results at different frequencies of plate oscillation, while the difference oscillation amplitudes with same frequency are shown with the same symbol. But after the normalization of magnitude of velocity and the measurement position, good agreement between the analytical results and experimental data is achieved. When the measurement locations are far from the plate there are some differences between the theoretical solution and experiment data. The reason is that the absolute velocities in these locations are so small (less than 10 mm/s) in the present experiments so that the greater errors are induced in the LDV measurement. Fig.3 Magnitude and phase differences comparisons between the theoretical results and experimental data inside the Stokes boundary layer for plate oscillation in still water

5 774 The experimental results were also converted to those in the moving coordinates that represent the oscillation flow over fixed plate to compare with the theoretical results of oscillating flows over a still flat plate. The experimental magnitudes of velocity inside/near the Stokes boundary layer are plotted in the left of Fig.4. The solid line in the figure represents the theoretical solution expressed as Eq.(13). The measured phase differences inside/near the Stokes boundary layer were plotted in the right of Figure 4. The solid line in this figure represents the theoretical solution shown as Eq.(14). The agreement between the theoretical results and experimental data is excellent for the phase difference. The agreement between the theorrtical solution and experimental data for velocity magnitudes is reasonable though the experimental data are a little bit higher than the theoretical solution at measurement locations far from the plate. The reason is the same as that described above. velocity and phase difference between experimental results and theoretical solution. It should be noted that the origin of -coordinate in the figures is from centerline of the narrow channel. The solid lines in the figures represent the theoretical solutions. The theoretical solution of magnitude in the left of Fig.5 is (1 a) + b and phase angle in the right of Fig.5 is = arctan[ b/(1 a)], where the expressions of a and b can be found in Eq.(8). The different symbols in Fig.5 represent the experimental data at different numbers. Three cases for numbers of.343, and 3.8 were plotted. The agreement between the theoretical soluyion and experimental data is reasonable. Fig.5 Magnitude profiles and phase difference comparisons between the theoretical solution and experimental data For flows between two parallel plates Fig. 4 Magnitude and phase differences comparisons between the theoretical solution and experimental data inside the Stokes boundary layer for oscillation flows over a still plate 4. sults for narrow channel The results presented below were reduced in moving coordinates the same as oscillation flows past narrow channels, though they are for the oscillating narrow channel moving in still water in real experiments Velocity and phase profiles at different Figure 5 shows the comparisons of the profiles of 4.. Velocity and phase in centre plane of narrow channel The velocities amplitude and phase differences at the centre of two parallel plates are given in Fig.6. The results of two different gaps of parallel plates = 1.5mm and =.5mm are presented and all experimental results are normalized by the dimensionless parameter. The solid lines in the figures represent the theoretical solutions Eq.(9) when =0. The agreement between the theoretical solution and experimental data is quite good at the low

6 775 and reasonable for higher numbers. And the velocity amplitude and phase different tend to the incoming values when 35, which imply the effect of two parallel plates on the flow field of centre plane can be ignored as exceeds such value. phase difference in the channel will be identical to the case of flat plate as over 35. It should be pointed out that all flow patterns in the present experimental domain are laminar as A is much smaller than the transition value shown by su et al [13]. Acknowledgement This work was supported by the KUST (Grant No. 654/0E). ferences Fig. 6 Comparisons between the theoretical solution and experimental data for magnitude of velocity and phase difference at centre of two parallel plates 5. Concluding remarks To understand the flow features of oscillating flows in narrow channel the theoretical analysis of velocity distributions in an oscillating channel consisting of two parallel infinite plates has been presented in this paper firstly. The theoretical analysis leads to the results of the amplitude and phase of velocity in the channel and flat plate. Then the LDV has been used to measure the Stokes boundary layers in the channel and flat plate. The measurement results of velocity amplitude and phase of flows induced by either an oscillating flat plates or an oscillating channel of parallel plates have been compared with the theoretical solutions and showed good agreement. Both experimental data and theoretical solutions indicate that the interaction of two Stokes layers is even distinct with the decrease of channel gap. Compared to the case of oscillating flow over flat plate, the phase difference in the Stokes layer decreases with the increase of gap in the case of narrow channel. And velocity magnitudes enlarge with the increase of gap. The velocity magnitudes and [1] DENG Jian, SAO Xue-ming and REN An-lu. Vanishing of three-dimensionality in the wake behind a rotationally oscillating circular cylinder[j]. Journal of ydrodynamics, 007, 19(6): [] LIANG Bing-chen, LI ua-jun and LEE Dong-yong. Bottom shear stress under wave-current interaction[j]. Journal of ydrodynamics, 008, 0(1): [3] WANG Wen-quan, ANG Li-Xiang and YAN Yan et al. Study on turbulence features near an oscillating curved wall[j]. Journal of ydrodynamics, Ser. B, 007, 19(3): [4] DONG Yu-hong, LU Xi-yun and UANG Li-xian. Large eddy simulation of thermally-stratified turbulent channel flow with temperature oscillation on the bottom wall[j]. Journal of ydrodynamics, Ser. B, 004, 16(1): [5] STOKES G. G. On the effect of the internal friction of fluids on the motion of pendulums[j]. Transactions of the Cambridge Philosophical Society, 1851, 9: [6] SCLICTING. Boundary-layer theory[m]. Seventh Edition, New York: McGraw-ill, [7] RUSSELL R. O., OSORIO J. D. C. An experiment investigation of drift profiles in a closed channel[c]. Proc. 6th Conf. Coastal Engng. ASCE, Gainesville, Florid, USA, 1957, [8] CARTER T. G., LIU P. L.-F. and MEI C. C. Mass transport by waves and offshore sand bedforms[j]. J. Waterways, arbor, Coastal Engineering Div., ASCE, 1973, 99: [9] BEEC N. W. Laser Doppler measurements in the oscillatory boundary layer beneath water waves[j]. DSIA Elektronik (Appl. Sci. s.), 1978, A1(3): [10] SLEAT J. F. A. Measurements of mass transport in water waves propagated over a rough bed[c]. 19th Conf. Coastal Engng. ouston Texas, USA, 1984, [11] WUNG.., LIN C. The mass transport of wave propagating on a slopping bottom[c]. th Conf. Costal Engng. Delft, The Netherlands, 1990, [1] LIU P. L.-F., DAVIS M.. and DOWNING S. Wave-induced boundary layers above and in a permeable bed[j]. J. Fluid Mech., 1996, 35: [13] SU C. T., LU X. and KWAN M. K. LES and RANS studies of oscillating flows over a flat plate[j]. Journal of Engineering Mechanics, ASCE, 000, 16():

LES AND RANS STUDIES OF OSCILLATING FLOWS OVER FLAT PLATE

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