Martin Wosnik

A theory for turbulent pipe and channel flows

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 + R + 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 × 10 2 R + 5 . 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 ] 10 5 .

A similarity theory for the turbulent plane wall jet without external stream

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 ∗ y 1/2 / v , where y 1/2 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 U m , the velocity maximum, and the length scale y 1/2 ; 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 u 2 * , while the normal stresses scale with U 2 m . Also U m ∼ y n 1/2 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, B 1 = ( U m v / M o )/ ( y + 1/2 M o / v 2 ) n = constant where n ≈ −0.528, appears to be required to determine the entire flow for a given source.

Characterising the near-wake of a cross-flow turbine

The performance and detailed near-wake characteristics of a vertical axis, cross-flow turbine (CFT) of aspect ratio 1 were measured in a large cross-section towing tank. The near-wake at one turbine diameter downstream was examined using acoustic Doppler velocimetry, where essential features regarding momentum, energy, and vorticity are highlighted. Dominant scales and their relative importance were investigated and compared at various locations in the measurement plane. Estimates for the terms in the mean streamwise momentum and mean kinetic energy equation were computed, showing that the unique mean vertical velocity field of this wake, characterised by counter-rotating swirling motion, contributes significantly more to recovery than the turbulent transport. This result sheds light on previous CFT studies showing relatively fast downstream wake recovery compared to axial-flow turbines. Finally, predictions from a Reynolds-averaged Navier–Stokes simulation with the commonly used actuator disk model were ...

Hydrofoil Drag Reduction by Partial Cavitation

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

MEASUREMENTS IN HIGH VOID-FRACTION BUBBLY WAKES CREATED BY VENTILATED SUPERCAVITATION

A detailed study of ventilated supercavitation in the reentrant jet regime is being carried out in the high-speed water tunnel at St. Anthony Falls Laboratory, as the hydrodynamics part of an interdisciplinary study on stability and control of high-speed cavity-running bodies. It is aimed at understanding the interaction between a ventilated supercavity and its turbulent bubbly wake, with the goal to provide the information needed for the development of control algorithms. Here Particle Image Velocimetry (PIV) measurements in high void fraction bubbly wakes created by the collapse of ventilated supercavities are reported. Bubble velocity fields are obtained, and shown to submit to the same high Reynolds number similarity scaling as the single-phase turbulent axisymmetric wake. A grayscale technique to measure local average void fraction is outlined. Initial results of a timeresolved PIV experiment (2000 Hz) are also presented.

Velocity Deficit and Swirl in the Turbulent Wake of a Wind Turbine

Energy production data from several of the existing large offshore wind farms indicate that turbine arrays can suffer from a significant overall energy production shortfall, due to wakes generated by turbines upstream interacting with turbines downstream. An experimental investigation of the axial and azimuthal (swirl) velocity field in the wake of a single three-bladed wind turbine with rotor diameter of 0.91 m was conducted. The turbine was positioned in the free stream, near the entrance of the 6 m×2.7 m cross section of the University of New Hampshire (UNH) Flow Physics Facility, a 72-m-long boundary layer wind tunnel. The turbine model was tested at various rotor loading conditions with blade tip-speed ratios up to 2.8. A Pitot-static tube and constant temperature hot-wire anemometry with a multiwire sensor were used to obtain velocity field measurements in the wake of the model turbine up to 20 diameters downstream. The results of an equilibrium similarity theory for the axisymmetric wake with rotation are presented. The measurements obtained were used to examine the validity of the derived scaling functions for streamwise and azimuthal velocity, wake growth, and turbulence.

Numerical Modeling of Probable Maximum Flood Flowing through a System of Spillways

The results of a numerical model study of probable maximum flood PMF flow through a system of spillways consisting of an existing service spillway and a new auxiliary spillway are presented. A commercial computational fluid dynamics CFD code, Fluent, was used to solve the time-dependent Reynolds-averaged Navier-Stokes equations, a standard turbulence lk- model with wall functions, and a water volume of fluid fraction equation. A two-dimensional approach velocity profile was used at the upstream inlet cross section. Water levels, flow splits between the existing and auxiliary spillways, and flow patterns were predicted and compared. A tentative design was chosen, constructed, and tested in a 1:54 scale physical model. Testing results were used to validate the CFD model. Results demonstrate that the CFD model is validated as accurate in the prediction of water levels in the reservoir, the integrated approach used is cost-effective and efficient in optimizing the designs of the auxiliary spillway, the tentative design cannot pass the PMF at the maximum pool level which suggests further modifications being necessary in the physical model. DOI: 10.1061/ASCEHY.1943-7900.0000279 CE Database subject headings: Spillways; Computational fluid dynamics technique; Floods; Numerical models; Simulation; Dam safety; Case studies. Author keywords: Spillways; Computational fluid dynamics technique; Floods; Numerical models; Simulation; Dam safety.

Modeling the near-wake of a vertical-axis cross-flow turbine with 2-D and 3-D RANS

The near-wake of a vertical-axis cross-flow turbine was modeled numerically via blade-resolved k–ω shear stress transport (SST) and Spalart–Allmaras Reynolds-averaged Navier–Stokes (RANS) models in two and three dimensions. The results for each case were compared with the experimental measurements of the turbine shaft power, overall streamwise rotor drag, mean velocity, turbulence kinetic energy, and momentum transport terms in the near-wake at one diameter downstream. It was shown that 2-D simulations overpredict turbine loading and do not resolve mean vertical momentum transport, which plays an important role in the near-wakes momentum balance. The 3-D simulations fared better at predicting performance, with the Spalart–Allmaras model predictions being closest to the experiments. The SST model more accurately predicted the turbulence kinetic energy, while the Spalart–Allmaras model more closely matched the momentum transport terms in the near-wake. These results show the potential of blade-resolved RAN...

Experimental Investigation of Helical Cross-Flow Axis Hydrokinetic Turbines, Including Effects of Waves and Turbulence

The performance characteristics of two cross-flow axis hydrokinetic turbines were evaluated in UNH’s tow and wave tank. A 1m diameter, 1.25m (nominal) height three-bladed Gorlov Helical Turbine (GHT) and a 1m diameter, four-bladed spherical-helical turbine (LST), both manufactured by Lucid Energy Technologies, LLP were tested at tow speeds up to 1.5 m/s. Relationships between tip speed ratio, solidity, power coefficient (Cp ), kinetic exergy efficiency, and overall streamwise drag coefficient (Cd ) are explored. As expected, the spherical-helical turbine is less effective at converting available kinetic energy in a relatively low blockage, free-surface flow. The GHT was then towed in waves to investigate the effects of a periodically unsteady inflow, and an increase in performance was observed along with an increase in minimum tip speed ratio at which power can be extracted. Regarding effects of turbulence, it was previously documented that an increase in free-stream homogenous isotropic turbulence increased static stall angles for airfoils. This phenomenon was first qualitatively investigated on a smaller scale with a NACA0012 hydrofoil in a UNH water tunnel, using an upstream grid turbulence generator and using high frame-rate PIV to measure the flow field. Since the angle of attack for a cross-flow axis turbine blade oscillates with higher amplitude as tip speed ratio decreases, any delay of stall should allow power extraction at lower tip speed ratios. This hypothesis was tested experimentally on a larger scale in the tow tank by creating grid turbulence upstream of the turbine. It is shown that the range of operable tip speed ratios is slightly expanded, with a possible improvement of power coefficient at lower tip speed ratios. Drag coefficients at higher tip speed ratios seem to increase more rapidly than in the non-turbulent case.Copyright

Performance and Near-Wake Measurements for a Vertical Axis Turbine at Moderate Reynolds Number

Experiments were performed with a 1 m diameter, 1 m tall three-bladed vertical axis turbine in a towing tank. Rotor power and drag (or thrust) were measured at a tow speed of 1 m/s, corresponding to a turbine diameter Reynolds number ReD = 106 and an approximate blade chord Reynolds number Rec = λU∞c/ν ∼ 105. Mechanical exergy efficiency estimates were computed from power and drag measurements using an actuator disk approach. Characteristics of the turbine’s near-wake were measured at one turbine diameter downstream. Variation of all three mean and fluctuating velocity components in the vertical and cross-stream directions were measured at peak turbine power output via acoustic Doppler velocimeter. The effect of tip speed ratio on near-wake mean velocity was observed at the turbine center line. Transverse profiles of mean velocity, fluctuating velocity, and Reynolds stresses were also measured at the turbine’s quarter height for two tip speed ratios of interest. Results are compared and contrasted with previous lower Reynolds number studies, and will provide a detailed data set for validation of numerical models.Copyright