Luciano Castillo

Zero-Pressure-Gradient Turbulent Boundary Layer

Of the many aspects of the long-studied field of turbulence, the zero-pressure-gradient boundary layer is probably the most investigated, and perhaps also the most reviewed. Turbulence is a fluid-dynamical phenomenon for which the dynamical equations are generally believed to be the Navier-Stokes equations, at least for a single-phase, Newtonian fluid. Despite this fact, these governing equations have been used in only the most cursory manner in the development of theories for the boundary layer, or in the validation of experimental data-bases. This article uses the Reynolds-averaged Navier-Stokes equations as the primary tool for evaluating theories and experiments for the zero-pressure-gradient turbulent boundary layer. Both classical and new theoretical ideas are reviewed, and most are found wanting. The experimental data as well is shown to have been contaminated by too much effort to confirm the classical theory and too little regard for the governing equations. Theoretical concepts and experiments are identified, however, which are consistent-both with each other and with the governing equations. This article has 77 references.

Experimental study of the horizontally averaged flow structure in a model wind-turbine array boundary layer

When wind turbines are deployed in large arrays, their ability to extract kinetic energy from the flow decreases due to complex interactions among them, the terrain topography and the atmospheric boundary layer. In order to improve the understanding of the vertical transport of momentum and kinetic energy across a boundary layer flow with wind turbines, a wind-tunnel experiment is performed. The boundary layer flow includes a 3×3 array of model wind turbines. Particle-image-velocity measurements in a volume surrounding a target wind turbine are used to compute mean velocity and turbulence properties averaged on horizontal planes. Results are compared with simple momentum theory and with expressions for effective roughness length scales used to parametrize wind-turbine arrays in large-scale computer models. The impact of vertical transport of kinetic energy due to turbulence and mean flow correlations is quantified. It is found that the fluxes of kinetic energy associated with the Reynolds shear stresses a...

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 .

Similarity Analysis for Turbulent Boundary Layer with Pressure Gradient: Outer Flow

The equilibrium-type similarity analysis of George and Castillo for the outer part of zero pressure gradient boundary layers has been extended to include boundary layers with pressure gradient. The constancy of a single new pressure gradient parameter is all that is necessary to characterize these new equilibrium turbulent boundary layers. Three major results are obtained: First, most pressure gradient boundary experiments appear to be equilibrium flows (by the new definition), and nonequilibrium flows appear to be the exception. Second, there appear to be only three values of the pressure gradient parameter: one for adverse pressure gradients, one for favorable pressure gradients, and one for zero pressure gradients. Third, correspondingly, there appear to be only three normalized velocity deficit profiles, exactly as suggested by the theory

Influence of external conditions on transitionally rough favorable pressure gradient turbulent boundary layers

Laser Doppler anemometry measurements are carried out in order to investigate the influences of the external conditions on a transitionally rough favorable pressure gradient turbulent boundary layer. The acquired data is normalized using the scalings obtained by the means of equilibrium similarity of the outer flow. The point at hand is to not only understand the interaction between the rough surface and the outer flow but also to include the external pressure gradient as the flow evolves in the streamwise direction. It is found that the velocity profiles show the effects of the upstream conditions imposed on the flow when normalized with the free-stream velocity. However, the profiles do collapse when normalized with U ∞ δ*/δ, demonstrating that this scaling absorbs the roughness effects and upstream conditions. An augmentation in the Reynolds stresses occurs with an increase in the roughness parameter and a decrease due to the external favorable pressure gradient. However, close to the wall, there is an increase due to the favorable pressure gradient while on the outer part of the boundary layer there is a decrease in magnitude due to this imposed effect. The near-wall peak of the ⟨ u 2 ⟩ component is dampened by the surface roughness condition due to the destruction of the viscous sublayer. In addition, the shape of the profile in the inner region tends to flatten due to the surface roughness. The upstream wind-tunnel speed also plays an important role thus creating a Reynolds number dependence on the outer flow of the Reynolds stress components. Furthermore, through 11 consecutive downstream locations, the skin friction coefficient is obtained for smooth and rough favorable pressure gradient data. The skin friction shows dependencies on the Reynolds number, the roughness parameter, and the favorable pressure gradient condition in the transitionally rough regime; while for the fully rough regime, it becomes form drag and the dependencies are on the favorable pressure gradient and the Reynolds shear stress. The external condition effects are isolated with a fixed parameter comparison. Favorable pressure gradient effects slow down the growth of the boundary layer while the surface roughness promotes its growth.

The effects of the upstream conditions on a low Reynolds number turbulent boundary layer with zero pressure gradient

An experiment on turbulent boundary layers with low local Reynolds numbers in a zero pressure gradient has been carried out using the laser-Doppler anemometry technique and a similarity analysis of the Reynolds averaged Navier-Stokes equations. This experiment seeks to investigate the effect of the local Reynolds number and the upstream conditions on the downstream development of the mean flow and the turbulent quantities. It was found that by fixing the upstream conditions (i.e. such as the wind-tunnel speed and trip wire size), the mean velocity profiles and Reynolds stresses tend to collapse in the outer flow even though the Reynolds number, R θ, varied from approximately 700 up to 5321. However, when the upstream conditions were changed and the local Reynolds number was held constant, neither the profiles of the wall-normal Reynolds stress nor those of the Reynolds shear stress collapsed, thus showing the effects of the upstream conditions. Moreover, no effect on the mean velocity profiles or the long...

Effect of Upstream Conditions on the Outer Flow of Turbulent Boundary Layers

It will be shown that the boundary layer develops differently depending on the upstream conditions (e.g., the wind-tunnel speed, size of tripping wire, or recent history). Also, it will be demonstrated that almost all of the Reynolds-number dependence observed in the outer velocity deficit profiles is caused primarily by changes in the upstream conditions and not to the local Reynolds number. The empirical velocity scale of Zagarola and Smits is derived here for boundary layers with and without pressure gradient using similarity principles. This scaling is successful in removing the effect of upstream conditions and the residual dependence on the local Reynolds number from the mean velocity deficit. Even more interesting, it produces only three profiles in turbulent boundary layers, regardless of the strength of the pressure gradient: one for adverse pressure gradient, one for favorable pressure gradient, and one for zero pressure gradient

Separation Criterion for Turbulent Boundary Layers Via Similarity Analysis

By using the RANS boundary layer equations, it will be shown that the outer part of an adverse pressure gradient turbulent boundary layer tends to remain in equilibrium similarity, even near and past separation. Such boundary layers are characterized by a single and constant pressure gradient parameter, and its value appears to be the same for all adverse pressure gradient flows, including those with eventual separation

A dynamic multi-scale approach for turbulent inflow boundary conditions in spatially developing flows

A dynamic method for prescribing realistic inflow boundary conditions is presented for simulations of spatially developing turbulent boundary layers. The approach is based on the rescaling―recycling method proposed by Lund, Wu & Squires (J. Comput. Phys, vol. 140, 1998, pp. 233―258) and the multi-scale method developed by Araya, Jansen & Castillo (J. Turbul., vol. 10, no. 36, 2009, pp. 1―33). The rescaling process requires prior knowledge about how the velocity and length scales are related between the inlet and recycle stations. Here a dynamic approach is proposed in which such information is deduced dynamically by involving an additional plane, the so-called test plane located between the inlet and recycle stations. The approach distinguishes between the inner and outer regions of the boundary layer and enables the use of multiple velocity scales. This flexibility allows applications to boundary layer flows with pressure gradients and avoids the need to prescribe empirically the friction velocity and other flow parameters at the inlet of the domain. The dynamic method is tested in direct numerical simulations of zero, favourable and adverse pressure gradient flows. The dynamically obtained scaling exponents for the downstream evolution of boundary layer parameters are found to fluctuate in time, but on average they agree with the expected values for zero, favourable and adverse pressure gradient flows. Comparisons of the results with data from experiments, and from other direct numerical simulations that use much longer computational domains to capture laminar-to-turbulence transition, demonstrate the suitability of the proposed dynamic method.

Optimizing the Unrestricted Placement of Turbines of Differing Rotor Diameters in a Wind Farm for Maximum Power Generation

This paper presents a new method (the Unrestricted Wind Farm Layout Optimization (UWFLO)) of arranging turbines in a wind farm to achieve maximum farm efficiency. The powers generated by individual turbines in a wind farm are dependent on each other, due to velocity deficits created by the wake effect. A standard analytical wake model has been used to account for the mutual influences of the turbines in a wind farm. A variable induction factor, dependent on the approaching wind velocity, estimates the velocity deficit across each turbine. Optimization is performed using a constrained Particle Swarm Optimization (PSO) algorithm. The model is validated against experimental data from a wind tunnel experiment on a scaled down wind farm. Reasonable agreement between the model and experimental results is obtained. A preliminary wind farm cost analysis is also performed to explore the effect of using turbines with different rotor diameters on the total power generation. The use of differing rotor diameters is observed to play an important role in improving the overall efficiency of a wind farm.Copyright