A theory for fully developed turbulent pipe and channel flows is proposed which extends the classical analysis to include the effects of finite Reynolds number. The proper scaling for these flows at finite Reynolds number is developed from dimensional and physical considerations using the Reynolds-averaged Navier–Stokes equations. In the limit of infinite Reynolds number, these reduce to the familiar law of the wall and velocity deficit law respectively.The fact that both scaled profiles describe the entire flow for finite values of Reynolds number but reduce to inner and outer profiles is used to determine their functional forms in the ‘overlap’ region which both retain in the limit. This overlap region corresponds to the constant, Reynolds shear stress region (30 < y⁺ < 0.1R⁺ approximately, where R⁺ = u_*R/v). The profiles in this overlap region are logarithmic, but in the variable y + a where a is an offset. Unlike the classical theory, the additive parameters, B_i, B_o, and log coefficient, 1/κ, depend on R⁺. They are asymptotically constant, however, and are linked by a constraint equation. The corresponding friction law is also logarithmic and entirely determined by the velocity profile parameters, or vice versa.It is also argued that there exists a mesolayer near the bottom of the overlap region approximately bounded by 30 < y⁺ < 300 where there is not the necessary scale separation between the energy and dissipation ranges for inertially dominated turbulence. As a consequence, the Reynolds stress and mean flow retain a Reynolds number dependence, even though the terms explicitly containing the viscosity are negligible in the single-point Reynolds-averaged equations. A simple turbulence model shows that the offset parameter a accounts for the mesolayer, and because of it a logarithmic behaviour in y applies only beyond y⁺ > 300, well outside where it has commonly been sought.The experimental data from the superpipe experiment and DNS of channel flow are carefully examined and shown to be in excellent agreement with the new theory over the entire range 1.8 × 10² < R⁺ < 5.3 × 10⁵. The Reynolds number dependence of all the parameters and the friction law can be determined from the single empirical function, H = A/(ln R⁺)^α for α > 0, just as for boundary layers. The Reynolds number dependence of the parameters diminishes very slowly with increasing Reynolds number, and the asymptotic behaviour is reached only when R⁺ [Gt ] 10⁵.

Experiments were performed with a large laboratory-scale high solidity cross-flow turbine to investigate Reynolds number effects on performance and wake characteristics and to establish scale thresholds for physical and numerical modeling of individual devices and arrays. It was demonstrated that the performance of the cross-flow turbine becomes essentially R e -independent at a Reynolds number based on the rotor diameter R e_D ≈ 10⁶ or an approximate average Reynolds number based on the blade chord length R e_c ≈ 2 × 10⁵ . A simple model that calculates the peak torque coefficient from static foil data and cross-flow turbine kinematics was shown to be a reasonable predictor for Reynolds number dependence of an actual cross-flow turbine operating under dynamic conditions. Mean velocity and turbulence measurements in the near-wake showed subtle differences over the range of R e investigated. However, when transport terms for the streamwise momentum and mean kinetic energy were calculated, a similar R e threshold was revealed. These results imply that physical model studies of cross-flow turbines should achieve R e_D ∼ 10⁶ to properly approximate both the performance and wake dynamics of full-scale devices and arrays.

A new theory for the turbulent plane wall jet without external stream is proposed based on a similarity analysis of the governing equations. The asymptotic invariance principle (AIP) is used to require that properly scaled profiles reduce to similarity solutions of the inner and outer equations separately in the limit of infinite Reynolds number. Application to the inner equations shows that the appropriate velocity scale is the friction velocity, u∗, and the length scale is v/u∗. For finite Reynolds numbers, the profiles retain a dependence on the length-scale ratio, y+1/2 = u∗y1/2/v, where y1/2 is the distance from the wall at which the mean velocity has dropped to 1/2 its maximum value. In the limit as y+1/2 → ∞, the familiar law of the wall is obtained. Application of the AIP to the outer equations shows the appropriate velocity scale to be Um, the velocity maximum, and the length scale y1/2; but again the profiles retain a dependence on y+1/2 for finite values of it. The Reynolds shear stress in the outer layer scales with u2*, while the normal stresses scale with U2m. Also Um ∼ yn1/2 where n < −1/2 and must be determined from the data. The theory cannot rule out the possibility that the outer flow may retain a dependence on the source conditions, even asymptotically.The fact that both these profiles describe the entire wall jet for finite values of y+1/2, but reduce to inner and outer profiles in the limit, is used to determine their functional forms in the ‘overlap’ region which both retain. The result from near asymptotics is that the velocity profiles in the overlap region must be power laws, but with parameters which depend on Reynolds number y+1/2 and are only asymptotically constant. The theoretical friction law is also a power law depending on the velocity parameters. As a consequence, the asymptotic plane wall jet cannot grow linearly, although the difference from linear growth is small.It is hypothesized that the inner part of the wall jet and the inner part of the zero-pressure-gradient boundary layer are the same. It follows immediately that all of the wall jet and boundary layer parameters should be the same, except for two in the outer flow which can differ only by a constant scale factor. The theory is shown to be in excellent agreement with the experimental data which show that source conditions may determine uniquely the asymptotic state achieved. Surprisingly, only a single parameter, B1 = (Umv/Mo)/ (y+1/2Mo/v2)n = constant where n ≈ −0.528, appears to be required to determine the entire flow for a given source.

Partial cavitation reduces hydrofoil friction, but a drag penalty associated with unsteady cavity dynamics usually occurs. With the aid of inviscid theory a design procedure is developed to suppress cavity oscillations. It is demonstrated that it is possible to suppress these oscillations in some range of lift coefficient and cavitation number. A candidate hydrofoil, denoted as OK-2003, was designed by modification of the suction side of a conventional NACA-0015 hydrofoil to provide stable drag reduction by partial cavitation. Validation of the design concept with water tunnel experiments has shown that the partial cavitation on the suction side of the hydrofoil OK-2003 does lead to drag reduction and a significant increase in the lift to drag ratio within a certain range of cavitation number and within a three-degree range of angle of attack. Within this operating regime, fluctuations of lift and drag decrease down to levels inherent to cavitation-free flow. The favorable characteristics of the OK-2003 are compared with the characteristics of the NACA-0015 under cavitating conditions.