An Experimental Investigation of Transitional and Turbulent Channel Flow

Abstract

This thesis is a comprehensive experimental investigation of transitional and turbulent channel flow in the Reynolds number range of Reτ = 55 − 1559. Towards this objective, a new channel flow facility was designed and built, with a very high area contraction ratio of 108. This is achieved by building a channel of cross section 600 mm × 50 mm, length 7320 mm and connecting it to the downstream section of a blower wind tunnel. Two-dimensional (x-y plane along the mid-span of the channel) velocity field is measured using particle image velocimetry, hot-wire anemometry and Pitot tube. The contractions are carefully designed with optimal parameters to have minimal non-uniformity at the exit. This ultra high contraction ratio causes large reduction in the disturbance levels leading to delay in the onset of transition (at Rem = 2050, rather than the usual value of around 1500) and protraction of the extent of the transitional regime by around 4 times (ΔRem = 3150, as opposed to the usual value of around 800). Here Rem is the bulk Reynolds number. A new scaling law for velocity in the transitional region has been obtained based on the present measurements. The mean velocity in the transitional region is scaled by the centerline velocity, and displays a log law given by u/Uc = 0.13ln(y/h) + D1 with a ‘universal’ slope of 0.13. For the relatively quiet channel of the present study, this holds over a range of about 0.3 ≤ y/h ≤ 0.6. The results are compared with experimental and DNS data from literature for the configurations channel, pipe and boundary layer to confirm the universality of the slope, where they are valid over a smaller range. Using this, a relation for skin friction in the laminar-turbulent transitional flow is derived by a methodology similar to that of Prandtl for fully turbulent flow. From this, an expression for pressure drop in a channel (or pipe) as a function of bulk velocity is obtained, consistent with the well known Darcy-Weisbach relation. For the present experiments, pressure drop is given by ΔP ∝ un m, where n = 1.0, 2.5, 1.75 respectively for laminar, transitional and turbulent zones. While the results for laminar and turbulent regimes are already known, the one for the transitional regime is a new result. The scaling of mean velocity in the turbulent region, given by the long known and celebrated log law (Millikan (1938)), is an asymptotic expectation. A stringent test for the validity of the log law for finite Reynolds numbers is the constancy of the so-called diagnostic function in the inertial sublayer. However, for turbulent channel flow, the diagnostic function does not become constant till about Reτ = 5200 (?? and hence the standard log law does not accurately predict the mean velocity for Reτ < 5200. A modified version of the log law is derived from the mean flow momentum equation, which accounts for low Reτ and viscous effects. For closure Townsend (1980) is followed, wherein the turbulent kinetic energy equation is simplified to obtain a mixing-length-like relation between Reynolds shear stress and the mean velocity gradient. The resulting expression for mean velocity is seen to be consistent with the experimental and direct numerical simulation data in the inertial sublayer compared to the standard log law. This is further extended using a composite mixing length estimate for the outer layer along with the inertial sublayer and the agreement of the predicted mean velocity with experimental and direct numerical simulation data is excellent at high Reτ in the outer region. While both the modified and extended log law expressions work reasonably well in predicting the mean velocity from present experiments and DNS of Lee and Moser (2015), the prediction of diagnostic function was not satisfactory. However, when the variation of the structure parameter was accounted for, the prediction of diagnostic function was very good. Overall this vindicates a mixing length model as derived from the turbulent energy equation as proposed by Townsend (1980). Further, instantaneous uniform momentum zones (UMZs) are examined and found to exist in a turbulent channel flow at moderately high Reτ . Next, the scaling of streamwise turbulence intensity (Townsend (1980)) is examined. This scaling which yields a log law for turbulence intensity, seems to occur only at fairly high Reτ in the literature. The notion of active and inactive motions was introduced by Townsend (1961) (see also Bradshaw (1967)). It is proposed here that the non-occurrence of the log-law scaling for turbulence intensity at lower Reynolds numbers such as those of the present experiments are perhaps due to the obfuscatory effect of ‘inactive motions’. By using the so-called episodic description of wall turbulence (Narasimha et al. (2007)), the flow is split into active and inactive parts. The universal or active part of turbulence intensities so separated, display a universal logarithmic slope of -1.26 even at moderate Reynolds numbers while the log-law intercept in non-universal. This is also a vindication of the methodology to separate active and inactive parts followed herein. Conceptually, a connection between episodic descriptions and the active/ inactive description is also established. Next, the distribution of mean and fluctuating spanwise vorticities in the transitional and moderately turbulent regimes are considered. Dimensional mean vorticity profiles plotted for both transitional and turbulent regimes show that mean vorticity increases with Reynolds number close to the wall much more than away from the wall. This is well quantified by an integral quantity, akin to displacement thickness for a boundary layer, called centroid of mean vorticity. This centroid reduces monotonically with Reynolds number showing progressive migration of mean vorticity towards the wall. Likewise for fluctuating vorticity also, a centroid is defined in a similar manner. The centroid of fluctuating vorticity was also found to decrease monotonically with Reynolds number, in both transitional and turbulent regimes, showing increased concentration of fluctuating vorticity towards the wall as the Reynolds number is increased. Probability density function (PDF) of fluctuating spanwise vorticity were seen to be particularly peaky in the core region. Combined with a vanishingly small mean vorticity, a peaky PDF would signify a highly intermittent behaviour. However, we are not able to directly verify the intermittent vortical behaviour from the present study as the vorticity measurement is not time-resolved. Near the wall though, the PDF is less peaky and the mean is also nonzero. Taken together, this indicates, in outer co-ordinates, the vorticity concentration shifts towards the wall with Reynolds number. The (dimensional) vorticity fluctuation increases in the outer region also with Reynolds number, but at a much slower rate compared to the near-wall regions. We anticipate that this tendency is likely to accentuate at very high and ultra high turbulent Reynolds numbers outside the range of present studies. Further, two-point correlation with respect to the wall is measured using hotwire anemometry. Results show that the average inclination angle of the correlated structures decrease with Reynolds number, consistent with the corresponding inward vorticity migration. The next question that is addressed is that of interaction of inner and outer regions of a channel. The footprints of this activity manifest as a very large wavelength activity or very large scale motion (VLSM) in the power spectra of streamwise velocity fluctuation. We propose a scaling to relate the VLSM wavelengths (λ close to the centreline of the channel, with the time scale (T) of the corresponding low frequency activity displayed by the wall-normal velocity at the centerline. Further, the vertical velocity at the centerline has been split using kinematics into three terms. The first term (d/dx(Ucδ∗)) is instantaneous streamwise derivative of mass defect up to the centreline. The second term (hdUc/dx) is acceleration/deceleration of the freestream, is possibly due to the instantaneous response from the other side of the channel. The third term (R h0∂w∂z dy) is due to instantaneous dilation in the spanwise direction and the fourth term (vw) is due to wall transpiration (zero in the present case). The time series of the terms seems to signify a quasi-periodic see-saw like acceleration/deceleration of the streamwise centerline velocity due to the interaction between both the sides of the channel.