Turbulent boundary layer measurements above a smooth wall and sandpaper roughness are presented across a wide range of friction Reynolds numbers, δ + 99 , and equivalent sand grain roughness Reynolds numbers, k + s (smooth wall: 2020 δ + 99 21 430, rough wall: 2890 δ + 99 29 900; 22 k + s 155; and 28 δ + 99 /k + s 199). For the rough-wall measurements, the mean wall shear stress is determined using a floating element drag balance. All smooth-and rough-wall data exhibit, over an inertial sublayer, regions of logarithmic dependence in the mean velocity and streamwise velocity variance. These logarithmic slopes are apparently the same between smooth and rough walls, indicating similar dynamics are present in this region. The streamwise mean velocity defect and skewness profiles each show convincing collapse in the outer region of the flow, suggesting that Townsend's (The Structure of Turbulent Shear Flow, vol. 1, 1956, Cambridge University Press.) wall-similarity hypothesis is a good approximation for these statistics even at these finite friction Reynolds numbers. Outer-layer collapse is also observed in the rough-wall streamwise velocity variance, but only for flows with δ + 99 14 000. At Reynolds numbers lower than this, profile invariance is only apparent when the flow is fully rough. In transitionally rough flows at low δ + 99 , the outer region of the inner-normalised streamwise velocity variance indicates a dependence on k + s for the present rough surface.

Well-resolved measurements of the streamwise velocity in zero pressure gradient turbulent boundary layers are presented for friction Reynolds numbers up to 19, 670. Distinct from most studies, the present boundary layers undergo nearly a decade increase in Reynolds number solely owing to streamwise development. The profiles of the mean and variance of the streamwise velocity exhibit logarithmic behavior in accord with other recently reported findings at high Reynolds number. The inner and mid-layer peaks of the variance profile are evidenced to increase at different rates

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A refined scaling analysis of the two-dimensional mean momentum balance (MMB) for the zero-pressure-gradient turbulent boundary layer (TBL) is presented and experimentally investigated up to high friction Reynolds numbers, $\unicode[STIX]{x1D6FF}^{+}$. For canonical boundary layers, the mean inertia, which is a function of the wall-normal distance, appears instead of the constant mean pressure gradient force in the MMB for pipes and channels. The constancy of the pressure gradient has led to theoretical treatments for pipes/channels, that are more precise than for the TBL. Elements of these analyses include the logarithmic behaviour of the mean velocity, specification of the Reynolds shear stress peak location, the square-root Reynolds number scaling for the log layer onset and a well-defined layer structure based on the balance of terms in the MMB. The present analyses evidence that similarly well-founded results also hold for turbulent boundary layers. This follows from transforming the mean inertia term in the MMB into a form that resembles that in pipes/channels, and is constant across the outer inertial region of the TBL. The physical reasoning is that the mean inertia is primarily a large-scale outer layer contribution, the ‘shape’ of which becomes invariant of $\unicode[STIX]{x1D6FF}^{+}$ with increasing $\unicode[STIX]{x1D6FF}^{+}$, and with a ‘magnitude’ that is inversely proportional to $\unicode[STIX]{x1D6FF}^{+}$. The present analyses are enabled and corroborated using recent high resolution, large Reynolds number hot-wire measurements of all the terms in the TBL MMB.