Turbulent boundary layer over a three-dimensional rough wall

Transcription

1 Turbulent boundary layer over a three-dimensional rough wall Gaetano Maria Di Cicca 1, Andrea Ferrari 2, Michele Onorato 3 1 Dipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino, Italy 2 Dipartimento di Fisica, Università di Torino, Italy* 3 Accademia delle Scienze di Torino, Italy gaetano.dicicca@polito.it, andrea.ferrari@unil.ch, michele.onorato@polito.it Keywords: Turbulence, wall roughness, boundary layer, PIV. SUMMARY. PIV measurements in a turbulent boundary layer over a 3D roughened surface, consisting of pyramidal rows, are presented and compared to those for a smooth wall. The flow behaviour was tested for relatively high roughness and relatively low Reynolds number when the influence of the wall roughness may be expected to be noticeable. Mean flow expressed in velocity defect form, inner scaled streamwise Reynolds normal stress and Reynolds shear stress profiles show excellent agreement in the outer layer between the flow over the rough wall and the smooth surface. Wall-normal Reynolds normal stress profiles, instead, show some small discrepancy, but within the measurement uncertainty. 1 INTRODUCTION Understanding the effect of roughness on wall bounded turbulence is of practical importance in the prediction of a wide range of industrial and geophysical flows. According to the wall similarity hypothesis of Townsend [1] and the subsequent extensions by Perry and Chong [2] and Raupach et al. [3], at sufficiently high Reynolds numbers, outside the roughness sub-layer extending about five roughness heights, k, from the wall, the turbulent motions are independent of the surface conditions. This implies that the mean velocity profiles for smooth and rough walls expressed in velocity-defect form and the turbulent stress profiles, both normalized with respect to inner variables, collapse outside the roughness layer, when displayed in function of the distance from the wall normalized by the boundary layer thickness. A general agreement on the validity of this outer layer similarity hypothesis has not yet been conclusively found. Two main practical problems arise from the validity or not of the Townsend s hypothesis. The first is to clarify if the use of wall functions for rough wall boundary layers in modern flow predictions, based on numerical simulations, is appropriate or not. The second is related to the fact that, as noticed by Jimenez [4], if roughness effects on turbulence extend across the full boundary layer height, the commonly adopted assumption that the small turbulence scales have negligible influence on the large energy containing scales (as for instance it is assumed in LES) may be not a valid assumption. The present work deals with a turbulent boundary layer over a three-dimensional roughened wall and it follows a previously published paper reporting results about a flow over a two-dimensional rough surface [5]. PIV data obtained for a three-dimensional geometry of the wall roughness, having a relatively low value of δ/k (δ/k = 17.2; δ is the boundary layer thickness), are here shown and commented in comparison with smooth wall data. The wall roughness, consisting of pyramidal rows, was tested at relatively low Reynolds number, Re = Re is the Reynolds number based on the boundary layer external velocity and on the momentum thickness,. The *present address: Faculty of Geosciences and Environment, University of Lausanne, Switzerland

2 choice of moderately large values of the roughness height and of low Reynolds number was due to the willingness of generating a flow configuration in which the influence of the wall roughness on the turbulent structures may be expected, according to Jimenez [4], to be noticeable. A large amount of papers has been published on the general subject of a turbulent boundary layer over two- and three-dimensional rough surfaces, since the pioneering work of Nikuradse [6] and later on of Perry et al. [7]. In the following of this section only three significative papers are cited. For a widely exhaustive review see Jimenez [4] and the introduction section of Vesely et al. [5]. Flack et al. [8] observed similarity in mean and turbulence quantities between smooth and rough surfaces beyond the roughness sub-layer, whose height was evaluated 5k or 3k s from the wall (k s is the Nikuradse equivalent sand grain roughness height). They presented results for boundary layer measurements on flat plates covered with sandgrain and woven mesh with the ratio of the boundary layer thickness to roughness height, δ/k, varying from 16 to 110, at Re = Considering the relatively high values of the tested roughness heights, the Authors concluded that a critical roughness height, where the roughness affects all the boundary layer, does not exists. The concept is that extending 5k (or 3k s ) from the wall, the roughness sub-layer begins to occupy an ever increasing fraction of the outer layer, with the outer flow only gradually modified. Vesely et al. [5] tested a two-dimensional roughness consisting of grooves perpendicular to the flow showing excellent agreement in the outer layer between the roughened wall and the smooth wall. Mean flow expressed in velocity defect form and inner scaled streamwise Reynolds stresses were found to overlap within the measurement uncertainty in spite of the relatively high value of the roughness thickness (δ/k =28.6; δ/k s 7) and the relatively low Reynolds numbers, Re θ = 930 and 1360, respectively for the smooth and rough wall. More recently Volino et al. [9] tested turbulent boundary layers over surfaces roughened by periodic small 2D bars and large 2D bars ( /k=161, Re θ =4984 and /k=31, Re θ =4260 respectively) in comparison with a 3D roughened wall (staggered cubes, /k=28, Re θ =4952) and a smooth wall (Re θ =6069). The results show outer-layer similarity between the cases with three-dimensional roughness and smooth walls, and deviations from similarity in cases with small and large twodimensional transverse bars. Differences are most apparent in turbulence quantities. The observed differences between the rough- and smooth-wall results are attributed to large-scale attached eddies which extend from the roughness elements to the edge of the boundary layer. 2 EXPERIMENTAL SET-UP The experiments were carried out in the water tunnel at the Department of Mechanical and Aerospace Engineering of the Politecnico di Torino. This facility is a closed-loop open surface channel with a 350 mm wide, 500 mm high, and 1800 mm long test section. The external freestream turbulence level was 1.2% and the maximum ratio for the boundary layer thickness to the width of the tunnel was 0.15, ensuring that the mean boundary layer on the flat plate was 2D over the central 70% of its width. Measurements were taken on a flat plate with a length of 2050 mm (Fig. 1a), in a region about 1750 mm downstream the leading edge for the canonical boundary layer case and about 500 mm downstream the leading edge for the boundary layer developing over the 3D rough wall. The pressure gradient was null along the test section. The three-dimensional roughness consisted of pyramids inclined 45 degrees with respect to the flow (Fig. 1b). The roughness height and geometry were selected to be significative for the case of moderately small values of /k. At the flat plate leading edge the laminar-turbulent transition was imposed by sandpaper.

3 Measurements were taken, with a PIV System, which consisted of a 1280x1024 pixels high speed NanoSense MKIII CMOS camera and a continuous Spectra-Physics Argon-Ion laser, with a maximum emitted power of 6 W. The maximum acquisition rate of the camera, at full resolution, was 1024 Hz. The laser beam was expanded by a cylindrical lens and focused by a spherical lens, forming a light sheet with a thickness of about 0.5 mm. The water was seeded with spherical silicon carbide particles, 2 µm nominal diameter, which were small enough to follow the flow. The particle density was approximately 0.05 particles per pixel. PIV measurements were taken in the streamwise wall-normal plane (x,y). The physical size of the PIV images in the plane (x,y) was 49.4x39.5 mm 2 (1047x840 viscous units) for the smooth wall and 42x33.6 mm 2 (932x768 viscous units) for the rough wall. In all the cases, the PIV images covered the whole boundary layer thickness. The PIV analysis was done with the LaVision DAVIS 7.2 software to perform the correlations to obtain the velocity fields. The local particle displacements were determined using an adaptive cross-correlation algorithm. The final interrogation window size was 32x32 pixels, with an overlap of 50%. The obtained vectors were validated using a minimum peak height in the correlation, which was 1.2 times the height of the second highest peak. Erroneous vectors were substituted applying a moving average filter with a kernel of 5x5 vectors. Each velocity vector is representative of the mean velocity in an area of 0.62x0.62 mm 2 (13x13 wall units) for smooth wall and 0.52x0.52 mm 2 (12x12 wall units) for rough wall. According to the Ligrani and Bradshaw [10] criterion, a linear dimension of the measurement probe less than 20 wall units is sufficient to resolve the near wall turbulence. (a) (b) FIG.1. a) Experimental setup. b) Roughness geometry. Dimensions in millimeters. The camera acquisition rate was 800 frames per second, but only one image pair each 100 frames were recorded. Therefore the effective acquisition rate of PIV image pairs was 8 Hz statistically independent image pairs were recorded to ensure the convergence of the computed averaged quantities. The flow was assumed to be homogeneous along the x direction in each PIV image. It must be observed that this assumption is acceptable only in the outer layer, while in the immediate vicinity of the roughness elements the flow is certainly inhomogeneous in the x direction. The error in measuring the velocity with PIV depends mainly on the quality of the images, the particle image size, and the displacement of the particles from the first to the second image. Since the particle image size was 2 3 pixels and a good contrast between the particle light intensity and the background light intensity was obtained, the error in locating the correlation peak was

4 estimated to be lower than 0.1 pixels. Taking into account the displacement of particles from the first to the second image of about 16 pixels in the free stream and 11 pixels in the log region for the smooth wall (14 pixels and 7 pixels respectively for the rough wall), the error in measuring the velocity is less than 1%. The error is negligibly small when calculating mean values. 3 RESULTS AND COMMENTS The experimental test conditions are given in Table 1. The values of the Reynolds number, Re θ, are 1471 for the smooth wall and 1234 for the rough surface characterizing the present results in the range of relatively low Reynolds numbers. As it has been mentioned in the introduction, the main goal of the present measurements was to extend, for the case of a 3D rough wall, the results already available in the literature [8] towards the range of moderately low Reynolds numbers, where the wall similarity is not expected to apply, according to Townsend s hypothesis. The external flow velocities are U e =0.485 m/s and U e =0.36 m/s respectively for the smooth and the rough surfaces. It was selected a higher flow velocity for the smooth wall case in order to compare the two flows at similar values of Re θ. The values of the form parameter, H, assure that the two flows are turbulent fully developed. δ is the boundary layer thickness evaluated at U = 0.99U e, where U is the boundary layer mean velocity. The ratio between the boundary layer thickness and the roughness height, /k=17.2, is much lower than the value /k>50 predicted by Jimenez [4] for outer layer wall similarity to be expected to hold, but it is just above the lowest limit of the range (16< /k<110) indicated by Flack et al. [8], in which the outer layer flow is not influenced by the wall roughness. The ratio between the friction velocity and the external flow velocity, U /U e, is about 60% higher for the rough wall case. The coordinate y is the wall-normal distance from the virtual origin. The apex plus indicates, here and in the following, that the quantity has been normalized with respect to the wall units. Table 1 Clauser [11] and Hama [12] found that the effect of roughness on the mean flow was confined to the inner layer (roughness sub-layer), causing only a downward shift in the smooth wall log-law,

5 ΔU +, whose extension is reported in Table 1. They extended the log-law to the case of rough wall boundary layers as U + = 1/k ln((y T + ε) + ) + B - ΔU + (1) where k=0.421 is the Karman constant, B=5.2 is the smooth wall log-law intercept, y T is the distance from the roughness crest, ε is the distance from the roughness crest to the virtual origin and ΔU + is the roughness function. The virtual origin is evaluated according to Perry and Li [13], by essentially performing a least square fit of equation (1) to the measured velocity profile changing the values for U τ, ε and ΔU +. The boundary layer thickness was determined taking into account the virtual origin. The friction velocity,, for the rough wall, has been obtained applying the modified Clauser chart method [13] in the range of the velocity profiles from y + 60 to y/ It is found to increase of about 20% with respect to the smooth wall case. The friction velocity for the smooth wall has been evaluated applying the Clauser chart method in the range from y + 40 to y/ The uncertainty in the evaluation of the friction velocity for the smooth wall and the rough wall has been estimated to be 3% and 5% respectively. Nikuradse [6] defined the equivalent sand grain roughness height, k s, as the uniform sand roughness height that produces the same roughness function in the fully rough regime. The dependence of k s from ΔU may be expressed as (see [3]) ΔU + = 1/k ln(k s + ) 3.5 (2) The high values of k s + in Table 1 insure that the data refer to the case of fully rough regime. Finally δ + is the Karman number. Ideally the Karman number should be matched to investigate outer layer similarity between smooth and rough surfaces; otherwise it would be difficult to discern if differences in velocity and turbulence profiles are due to roughness or to Reynolds number effects [8]. Looking at Table 1, the Karman number for the rough wall is not, unlikely, equal to the one for the smooth wall, but they are at least of the same order of magnitude. Fig.2 Mean velocity profiles.

6 Mean velocity profiles plotted in inner variables are shown in Fig.2 for the two flows. The smooth wall LDV measurements of DeGraaff and Eaton [14] at Re θ = 1430 and Re θ =2900 are also reported for comparison. Data in Fig.2 show satisfactory agreement between the present velocity profiles for the smooth wall and the reference data. The rough surface displays a linear log region shifted below the smooth profile of a quantity corresponding to the roughnesss function, ΔUU +. Despite of the relatively low Reynolds numbers, the log regions are clearly identifiable. The value of ΔU + is reported in Table1. Fig.3 presents the mean velocity profiles in velocity defect form, using classic inner-outer scaling by U τ and δ, respectively for the velocity defect and for the distance from the virtual origin. Also in this figure, as in the following figures 4, 5 and 6, the corresponding smooth wall reference data of DeGraaff and Eaton are reported. The present data for the smooth wall in Fig.3 are in good agreement with the DeGraafff and Eaton low Reynolds number data. The two velocity profiles for the smooth and rough wall satisfactory collapse, starting from about y/ = This suggests that, even in part of the roughnesss sub-layer region, whose height is evaluated for the present roughness geometry as y/ = 5k 0.2, the mean flow is largely insensitive to the surface conditions. Fig.3 Mean velocity profiles in velocity defect form. Fig.4 Streamwise Reynolds normal stress profiles. Overall uncertainty : smooth wall 4.5% ; rough wall 7%.

7 Fig.5 Wall-normal Reynolds normal stress profiles. Overall uncertainty: smooth wall 4.5%; rough wall 7%. The streamwise Reynolds normal stresses <u'u'> + are plotted against y/ in Fig.4. A good agreement is observed between the present data for the smooth wall and the DeGraaff and Eaton data. The comparison between the smooth and the rough wall profiles shows that, for this Reynolds stress component, the flow appears to be quite insensitive to the surface conditions in the outer layer above y/ 0.2. The maximum differences between the two flows are observed around y/ = 0.9 to be of the order of 12%, where the turbulence profiles tend to zero. Moreover, observing the results in Fig.4, the streamwise Reynolds normal stress <u'u'> +, for the rough wall, displays significantly lower values near the wall, suggesting the absence of the strong peak characterizing the smooth wall <u'u'> + profile in the buffer layer, at about y This behaviour has been found in most of the measurements reported in literature and it has been interpreted by Flores and Jimenez [15] as the consequence of the fact that the wall roughness reduces the coherence of the quasi-longitudinal vortices populating the buffer layer and consequently reduces the local production of turbulent kinetic energy. Fig.5 displays the distributions of the wall-normal Reynolds normal stresses, <v'v'> +, as a function of y/. As expected, for the smooth walls, scale effects are to a greater extent evident for the case of this Reynolds stress component, as it is observable looking at the DeGraff and Eaton results for the smooth wall surface. The peak values and the overall distributions increase markedly increasing the Reynolds number. The present smooth wall results are now scarcely comparable with DeGraaff and Eaton data. For y/ > 0.2, the collapse between the smooth wall (Re θ =1471) and the roughened wall (Re θ =1234) appears, for this stress component, not completely satisfactory in the region between y/ = 0.5 and However, the differences are within the evaluated uncertainty. This lack of complete collapse in the outer layer between the present results on smooth and rough walls can not be rationally imputable to the wall geometries, due to the measurement uncertainty, about 7%, and to the even small difference in Re θ. Namely, it has been observed by other Authors (see e.g. Antonia and Krogstad [16], Keirsbulck et al. [17], Vesely et al. [5] for two dimensional roughened surfaces) that the v-component of the velocity is more sensitive to the roughness effects than the u-component. For y/ < 0.15 the wall-normal Reynolds normal stress profile for the rough wall stands above the one for the smooth wall, in agreement with what

8 was reported in the cited references for the case of 2D roughness. Better collapse between smooth and rough wall turbulent stresses are shown in Fig.6, where profiles of Reynolds shear stresses are compared. Fig.6 Reynolds shear stress profiles. Overall uncertainty: smooth wall 4.5%; rough wall 7%. Fig.7 Ratio between the wall-normal and the streamwise Reynolds stress profiles. In order to get free of the high uncertainty in the measurement of the friction velocity U used to normalize the Reynolds stresses in the previous diagrams, Fig.7 displays the ratio between the wall normal and the streamwise Reynolds stresses in function of the distance from the wall. The uncertainty in the measurement of <v'v'>/<u'u'> is less than 1%. In the range from y/ =0.2 to 0.75 the smooth and the rough wall results show a satisfactory collapse. Some difference is observed in proximity of the boundary layer edge, where <v'v'> and <u'u'> approach separately the zero value. The last result and the ones previously shown in Figs.3 to 6, demonstrate that the wall similarity concept between smooth and rough walls may be expected to confidently hold for a typical threedimensional wall roughness, also in the case of a relatively low Reynolds number. As previously commented, the extension of the wall similarity hypothesis to the case of large wall roughness, as large as δ/k = 16, was demonstrated by Flack et al. [8], for Re θ ranging from to Nevertheless, it was not expected by the present authors the quasi complete collapse between mean velocity profiles in velocity defect form and normalized Reynolds stresses over smooth and rough walls, in condition of Reynolds numbers not high enough to imply significant scale

9 separation between the large scale motions and the smaller-scale turbulent eddies. Similar conclusion has been drawn by some of the present authors [5] testing in the same water tunnel at the Politecnico di Torino a two dimensional roughness model ( /k=28.6), at Re = CONCLUSIONS PIV measurements at relatively low Reynolds numbers in a turbulent boundary layer over a 3D roughened surface have been presented and compared to those for a smooth wall. Mean flow expressed in velocity defect form and inner scaled streamwise Reynolds normal stresses show excellent agreement in the outer layer between the roughened wall and the smooth wall, providing support for the wall similarity hypothesis ([1]- [3]). This result occurs in spite of the relatively high value of the roughness thickness ( /k=17.2) and the relatively low Reynolds numbers, Re θ = 1471 and 1234, respectively for the smooth and rough wall. Inner scaled wall-normal Reynolds normal stresses show some discrepancy, but within the measurement uncertainty, between the two flows in the outer layer, in the region y/ = , confirming that the v-component of the velocity is more sensitive to the roughness height, than the u-component as observed also by other Authors. References [1] Townsend, A.A., The Structure of Turbulent Shear Flow, 2nd ed., Cambridge University Press, Cambridge (1976). [2] Perry, A.E. and Chong, M.S., On the mechanism of wall turbulence, J. Fluid Mech., 119, 173 (1982). [3] Raupach, R., Antonia, R.A. and Rajagopalan, S., Rough-wall boundary layers, Appl. Mech. Rev., 44, 1 (1991). [4] Jimenez, J., Turbulent flows over rough walls, Annu. Rev. Fluid Mech., 36, 173 (2004). [5] Vesely, L., Haigermoser, C., Greco, D., and Onorato, M., Turbulent flow and organized motions over a two-dimensional rough wall, Phys. Fluids, 21, (2009). [6] Nikuradse, J., Laws of flow in rough pipes, NASA Technical Memorandum 1292, [7] Perry, A.E., Schofield, W.H., and Joubert, P.N., Rough-wall turbulent boundary layers, J. Fluid Mech., 37, 383 (1969). [8] Flack, K.A., Schultz, M.P., and Connelly, J.S., Examination of a critical roughness height for outer layer similarity, Phys. Fluids, 19, (2007). [9] Volino, R.J., Schultz, M.P. and Flack, K.A., Turbulence structure in boundary layers over periodic two- and three-dimensional roughness, J. Fluid Mech., 676, (2011). [10] Ligrani P.M. and Bradshaw P., Spatial resolution and measurement of turbulence in the viscous sublayer using subminiature hot-wire probes, Exp. Fluids, 5, 407, (1987). [11] Clauser, F.H., Turbulent boundary layers in adverse pressure gradient, J. Aeronaut. Sci., 21, 91 (1954). [12] Hama, F.R., Boundary layer characteristics for rough and smooth surfaces, Trans. Soc. Nav. Archit. Mar. Eng., 62, 333 (1954). [13] Perry A.E. and Li J.D., Experimental support for the attached-eddy hypothesis in zeropressure gradient turbulent boundary layers, J. Fluid Mech., 218, 405 (1990). [14] DeGraaff, D.B. and Eaton, J.K., Reynolds-number scaling of the flat-plate turbulent boundary layer, J. Fluid Mech., 422, 319 (2000). [15] Flores, O. and Jimènez, J., Effect of wall-boundary disturbances on turbulent channel flows, J. Fluid Mech., 566, 357 (2006).

10 [16] Antonia, R.A. and Krogstad, P.A., Turbulence structure in boundary layers over different types of surface roughness, Fluid Dyn. Res., 28, 139 (2001). [17] Keirsbulck, L., Labraga, L., Mazouz, A., and Tournier, C., Surface roughness effects on turbulent boundary layer structures, ASME J. Fluids Eng., 124, 127 (2002).