Alexander J. Smits

Mean-flow scaling of turbulent pipe flow

Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×10 3 to 35×10 6 . Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60 y + <500 or y + R + , the outer limit depending on whether the Karman number R + is greater or less than 9×10 3 ; and a log law for 600 y + R + . The log law is only evident if the Reynolds number is greater than approximately 400×10 3 ( R + >9×10 3 ). Von Karmans constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600 y + R + , and the additive constant was shown to be 6.15 when the log law is expressed in inner scaling variables. A new theory is developed to explain the scaling in both overlap regions. This theory requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. A reasonable candidate for the outer velocity scale is the velocity deficit in the pipe, U CL − Ū , which is a true outer velocity scale, in contrast to the friction velocity which is a velocity scale associated with the near-wall region which is ‘impressed’ on the outer region. The proposed velocity scale was used to normalize the velocity profiles in the outer region and was found to give significantly better agreement between different Reynolds numbers than the friction velocity. The friction factor data at high Reynolds numbers were found to be significantly larger (>5%) than those predicted by Prandtls relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×10 3 and 35×10 6 , and includes a term to account for the near-wall velocity profile.

Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues

Wall-bounded turbulent flows at high Reynolds numbers have become an increasingly active area of research in recent years. Many challenges remain in theory, scaling, physical understanding, experimental techniques, and numerical simulations. In this paper we distill the salient advances of recent origin, particularly those that challenge textbook orthodoxy. Some of the outstanding questions, such as the extent of the logarithmic overlap layer, the universality or otherwise of the principal model parameters such as the von Karman “constant,” the parametrization of roughness effects, and the scaling of mean flow and Reynolds stresses, are highlighted. Research avenues that may provide answers to these questions, notably the improvement of measuring techniques and the construction of new facilities, are identified. We also highlight aspects where differences of opinion persist, with the expectation that this discussion might mark the beginning of their resolution.

Further observations on the mean velocity distribution in fully developed pipe flow

The measurements by Zagarola & Smits (1998) of mean velocity profiles in fully developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the uncertainties due to velocity gradient corrections. A new static pressure correction (McKeon & Smits 2002) is used in analysing all data and leads to significant differences from the Zagarola & Smits conclusions. The results confirm the presence of a power-law region near the wall and, for Reynolds numbers greater than 230×10^3 (R+ >5×10^3), a logarithmic region further out, but the limits of these regions and some of the constants differ from those reported by Zagarola & Smits. In particular, the log law is found for 600<y+ <0.12R+ (instead of 600<y+ <0.07R+), and the von Karman constant κ, the additive constant B for the log law using inner flow scaling, and the additive constant B∗ for the log law using outer scaling are found to be 0.421 ± 0.002, 5.60 ± 0.08 and 1.20 ± 0.10, respectively, with 95% confidence level (compared with 0.436±0.002, 6.15±0.08, and 1.51±0.03 found by Zagarola & Smits). The data also confirm that the pipe flow data for ReD ≤ 13.6×10^6 (as a minimum) are not affected by surface roughness.

Scaling of the streamwise velocity component in turbulent pipe flow

Statistics of the streamwise velocity component in fully developed pipe flow are examined for Reynolds numbers in the range 5.5 x 10^4 ≤ ReD ≤ 5.7 x 10^6. Probability density functions and their moments (up to sixth order) are presented and their scaling with Reynolds number is assessed. The second moment exhibits two maxima: the one in the viscous sublayer is Reynolds-number dependent while the other, near the lower edge of the log region, follows approximately the peak in Reynolds shear stress. Its locus has an approximate (R^+)^{0.5} dependence. This peak shows no sign of ‘saturation’, increasing indefinitely with Reynolds number. Scalings of the moments with wall friction velocity and

Experimental study of three shock wave/turbulent boundary layer interactions

(U_{cl}-\overline{U})

The effect of short regions of high surface curvature on turbulent boundary layers

are examined and the latter is shown to be a better velocity scale for the outer region, y/R > 0.35, but in two distinct Reynolds-number ranges, one when ReD 7 x 10^4. Probability density functions do not show any universal behaviour, their higher moments showing small variations with distance from the wall outside the viscous sublayer. They are most nearly Gaussian in the overlap region. Their departures from Gaussian are assessed by examining the behaviour of the higher moments as functions of the lower ones. Spectra and the second moment are compared with empirical and theoretical scaling laws and some anomalies are apparent. In particular, even at the highest Reynolds number, the spectrum does not show a self-similar range of wavenumbers in which the spectral density is proportional to the inverse streamwise wavenumber. Thus such a range does not attract any special significance and does not involve a universal constant.

Roughness effects in turbulent pipe flow

The paper presents a systematic study of the supersonic flow of a turbulent boundary layer over several compression-corner models. The wind tunnel and the compression-corner models (ramps fitted with aerodynamic fences to minimize three-dimensional effects) were identical with those used by Settles et al. (1979); constant-temperature hot-wire anemometry was used for the mass-flow measurements. The turning angles used for the compression corners were 8, 16, and 20 deg. In all three flow cases, the shock wave/turbulent flow interaction did amplify the turbulent stresses dramatically, with amplification increasing with increasing turning angle. However, different stress components were amplified by different amounts.

Horseshoe vortex systems resulting from the interaction between a laminar boundary layer and a transverse jet

Measurements, including one-point double, triple or quadruple mean products of velocity fluctuations, have been made in low-speed turbulent, boundary layers on flat surfaces downstream of concave or convex bends with turning angles of 20 or 30 degrees, the length of the curved region being at most 6 times the boundary-layer thickness at entry. These short bends approximate to ‘impulses’ of curvature, and the object of the work was to investigate the impulse response of the boundary layer, essentially the decay of structural changes downstream of the bends. The work can be regarded as a sequel, with much more detailed measurements, to the study by So & Mellor (1972, 1973, 1975) who investigated the response to step increases of curvature: turbulent boundary layers being nonlinear systems, responses to several kinds of curvature history are needed to assemble an adequate description of the flow. The most striking feature of the ‘impulse’ response is that the decay of the high turbulent intensity found at exit from the concave bends is not monotonic; the Reynolds stresses in the outer layer collapse to well below the level at entry, and are still falling slowly at the end of the test rig although in principle they must recover eventually. On the convex (stabilized) side the flow recovers, monotonically in the main, from a low level of turbulent intensity at the exit. The pronounced second-order response on the concave side can be explained qualitatively by interaction between the shear stress and the mean shear and is not peculiar to curved flows, but in the present cases the response is complicated by large changes in the dimensionless structure parameters related to double or triple mean products of velocity fluctuations. Strong spanwise variations, due presumably to longitudinal vortices, further complicate the flow in the concave bends, and decay only very slowly downstream.

The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel

Mean flow measurements are presented for fully developed turbulent pipe flow over a Reynolds number range of

Constant temperature hot-wire anemometer practice in supersonic flows

57\,{\times}\,10^3