Three‐dimensional boundary layer on wind turbine blades

This paper describes a three‐dimensional analysis of the laminar boundary layer that develops on the blades of an horizontalaxis wind turbine. The main aim was to investigate a fundamental phenomenon: the effect of rotation on the blade boundary layer of a wind turbine in conjunction with the widely observed phenomenon of stall‐delay. The separation position in retarded flows with pressure gradients is calculated and compared for the rotation and non‐rotation cases. It is concluded that the stall is linearly postponed due to the Coriolis force and the separation point is delayed as a result of increasing rotation speed or decreasing blade spanwise position. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hele Shaw Convection with Imposed Shear Flows: Boundary-Layer Formulation

The nonlinear convection forced by the boundaries of a Hele Shaw cell to align perpendicular to an imposed shear flow was analytically investigated by the boundary-layer method. The imposed shear flow may be a Couette flow that extends throughout the convecting layer or flow confined to a boundary, depending on the geometry of the Hele Shaw cell. This study examined the case in which the imposed shear flow has a boundary-layer structure and its interaction with the convecting interior. Analytical solutions for both the boundary layer and interior were obtained. The study revealed the following.For large aspect ratio A , the interaction of the imposed shear flow and convection is confined to the boundary layer. The boundary layer is a viscous rather than a thermal layer. The results showed that the range of validity of the Hele Shaw equations used in the literature is of order 1/ A ². For an asymptotically large aspect ratio A up to order 1/ A ², the velocity in the y -direction must be zero. The velocity in the x -direction and the z -direction has a parabolic dependence on y , but the temperature perturbation does not depend on y . These results may have implication for convection in porous media.     

The boundary layer in the hub area of wind turbine blades at high wind speeds

At low tip speed ratios, the strongest effect at the inboard half-span, when a vortex like flow occurs, is produced. These 3-D rotational effects contribute to a stall delay as well as higher lift and angles of attack at the inboard sections, as compared to 2-D flow conditions. A greater understanding of the mechanisms and properties of the shear layer close to the upper side surface of blade is sought through the development of an analytic theory and then is completed by CFD computations. The results show that the secondary flow has a slip effect on the blade surface and its circumferential velocity is less than that of blade revolution, particularly important for TSR <3.0. While at the high tip speed ratios the flow around the blade is much as the 2-D one, then at the low tip speed ratios, TSR < 3.0 the flow behaves rather like the impinging flow on a rotating disc. Therefore, the 3-D rotational effects dominate the flow at high wind speeds and a special approach of such flows would be to assume. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cattaneo–Christov heat flux model for three-dimensional flow of a viscoelastic fluid on an exponentially stretching surface

ABSTRACT

In this article, we explore the three-dimensional boundary-layer flow over an exponentially stretching surface in two parallel ways. Constitutive equations of a second-grade fluid are used. Instead of classical Fourier’s law, Cattaneo–Christov heat flux model is employed for the formulation of the energy equation. This model can predict the effects of thermal relaxation time on the boundary layer. The resulting partial differential equations are reduced into ordinary differential equations by similarity transformations. Homotopy Analysis Method (HAM) is employed to solve the non-linear problem. Physical impact of emerging parameters on the momentum and thermal boundary-layer thickness are studied.     

含灰气体激波沿平壁传播时诱导的边界层流动*

本文研究合灰气体激波沿平直壁面传播过程中在壁面附近形成的层流边界层流动。我们依照双连续介质双向耦合模型处理含灰气体激波的波后流动及其诱导的边界层问题,控制方程采用有限差分方法数值求解,给出了激波下游两相流场特性并考虑了含灰气体激波的松弛结构对边界层流动的影响。     

Instabilities of a flexible surface in supersonic flow

The stability of a supersonic boundary layer above a flexiblesurface is considered in the limit of large Reynolds numberand for Mach numbers O(1). Asymptotic theory of viscous–inviscidinteraction has been used for this purpose. We found that fora simple elastic surface, for which deflections are proportionalto local pressure differences, the boundary-layer flow remainsstable as it is for a rigid wall. However, when either dampingor surface inertia is included the flow becomes unstable. Moreover,in a certain range of wave numbers the boundary layer developsmore then one unstable mode. It is interesting that these modesare connected to one another via saddle points in the complex-frequencyplane. A more complex Kramer-type surface is also analysed andin some parameter ranges is found to permit the evolution ofunstable Tollmien–Schlichting waves. The neutral curvesare found for a variety of situations related to the parametersassociated with the flexible surface.     

Free convection flow about a cone under mixed thermal boundary conditions and a magnetic field

A similarity analysis was performed to investigate the laminar free-convection boundary-layer flow in the presence of a transverse magnetic field over a vertical down-pointing cone with mixed thermal boundary conditions. Boundary layer velocity and temperature profiles were determined numerically for various values of the magnetic parameter and the Prandtl number. The results show that the magnetic field suppresses the velocity profiles and increases the skin friction. The temperature profiles were expanded with increasing values of the magnetic parameter resulting in higher surface temperatures. A transformation relating the similarity solutions of the boundary-layer velocity and temperature profiles associated with different values of the mixed thermal boundary condition parameter was obtained.     

Effects of radiation and variable viscosity on unsteady MHD flow of a rotating fluid from stretching surface in porous medium

This work is focused on the study of unsteady magnetohydrodynamics boundary-layer flow and heat transfer for a viscous laminar incompressible electrically conducting and rotating fluid due to a stretching surface embedded in a saturated porous medium with a temperature-dependent viscosity in the presence of a magnetic field and thermal radiation effects. The fluid viscosity is assumed to vary as an inverse linear function of temperature. The Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. With appropriate transformations, the unsteady MHD boundary layer equations are reduced to local nonsimilarity equations. Numerical solutions of these equations are obtained by using the Runge–Kutta integration scheme as well as the local nonsimilarity method with second order truncation. Comparisons with previously published work have been conducted and the results are found to be in excellent agreement. A parametric study of the physical parameters is conducted and a representative set of numerical results for the velocity in primary and secondary flows as well as the local skin-friction coefficients and the local Nusselt number are illustrated graphically to show interesting features of Darcy number, viscosity-variation, magnetic field, rotation of the fluid, and conduction radiation parameters.     

Aerodynamic optimization of Laval nozzle flow with shocks: Numerical investigation of active/passive shock control via expansion fans

Instantaneous ignition in the supersonic part of a 3-D Laval nozzle realized by a well-defined sudden temperature rise across a normal shock is the focus of the present study. Unfortunately, the divergence of the supersonic nozzle part is necessarily smooth. Therefore, the turbulent boundary layer ahead of the shock is thick and causes substantial shock/boundary layer interactions. The non-homogeneous temperature increase across the shock, caused by the boundary layer thickening ahead of the shock and the resulting pre-compression prevents the quasi 1-D evolution of the flow downstream. Additionally, due to multiple boundary layer interactions the single shock disintegrates into a so called pseudo-shock system; i.e., into a sequence of periodic weak compression and expansion regions. To avoid this drawback and to establish homogeneous thermodynamic conditions throughout the entire cross section and flow domain downstream of the shock we apply active and passive control techniques in the area of shock boundary layer interaction. The central idea of the control technique described below is compensation of the thickening of the boundary layer by quantitative appropriate inverse effects, i.e. by superimposing negative and positive pressure gradients in the near wall region close to the shock position. In a first approach the additional expansion fan is created by active suction slots in flow direction and through all sidewalls of the 3-D nozzle. The resulting shock remains straight with exception of the near wall region. Suction creates an effective concave wall curvature. In supersonic flow the resulting local expansion tends to compensate the pre-compression. Because suction in high temperature environment is difficult to realize, we alter the wall curvature to create a negative bump with the same effect on the effective wall curvature. Under these conditions a normal shock in the channel core could be established without active or passive suction. The paper compares these active/passive control techniques with the unmodified setup. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Laminar natural convection boundary layers on horizontal surfaces possessing a circular cut-out

Summary A first-order boundary-layer analysis, including variable properties effects, is given for laminar natural convection on a horizontal surface possessing a circular cut-out. The latter provides the leading edge of the boundary layer and it is shown that, initially, boundary layer development occurs as if the leading edge were straight. Far from the cut-out, boundary layer development is shown to become completely axially symmetric.     

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment-Line Boundary Layer

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier–Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien–Schlichting modes in the classical boundary-layer flow past a compliant surface.     

The theory of micropolar thin elastic shells

A boundary-value problem of the three-dimensional micropolar, asymmetric, moment theory of elasticity with free rotation is investigated in the case of a thin shell. It is assumed that the general stress-strain state (SSS) is comprised of an internal SSS and boundary layers. An asymptotic method of integrating a three-dimensional boundary-value problem of the micropolar theory of elasticity with free rotation is used for their approximate determination. Three different asymptotics are constructed for this problem, depending on the values of the dimensionless physical parameters. The initial approximation for the first asymptotics leads to the theory of micropolar shells with free rotation, the approximation for the second leads to the theory of micropolar shells with constrained rotation and the approximation for the third asymptotics leads to the so-called theory of micropolar shells “with a small shear stiffness”. Micropolar boundary layers are constructed. The problem of the matching of the internal problem and the boundary-layer solutions is investigated. The two-dimensional boundary conditions for the above-mentioned theories of micropolar shells are determined.     

Stall-delay modelling of wind turbine blades

Most aerodynamic design tools for horizontal-axial wind turbines are based on the blade-element momentum theory (BEM). Due to the nature of this theory, the design tools need 2-D steady sectional lift and drag curves as an input. In practice, flow over a wind turbine rotor blade is neither two-dimensional nor steady, and is affected by rotation. Pioneering experiments have identified a consequence: at inboard rotor blade sections stall is delayed. This so-called Himmelskamp effect [1] gives a larger lift than predicted and, as a result, a higher power and loading than expected. Consequently, an aerodynamic model is needed to explain and predict sectional lift and drag under rotating conditions. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Rotation of a Gravitating Sphere in a Monatomic Gas,I

The flow resulting from the steady rotation of a gravitating sphere in a monatomic gas is studied in two special cases. If the rotation is slow, then the motion consists of a swirling flow whose streamlines are concentric circles around the rotation axis, and a slower meridional flow whose structure depends on the strength of the gravitational field. This meridional flow is investigated in detail for small and large field strength. In the latter case, the scale height is small compared to the sphere radius, and most of the fluid is in a thin recirculating layer on the surface. However, the mathematics imposes a severe restriction on the rotation speed if the field is large, which leads to a more general investigation of the flow for very large field strength, i.e. for a thin “atmosphere”. Compressibility effects can be quite sizeable in this final solution. Though a thin recirculating surface layer still exists in only slightly modified form, the fluid outside this layer is much less tenuous than for the case when the rotation is slow. The flow in the region outside the boundary layer involves solution of two quasilinear elliptic equations.     

两种轴对称边界层方程的新解法

本文给出了与Mangler变换相类似的变换,它们将两种过流通道中进口段轴对称层流边界层流动转换成平面层流边界层流动,使问题得到简化.简化后的方程可以用已有的平面层流边界层理论加以解决,从而为人们开辟一条解决轴对称通道进口段流动问题的新途径.     

Effects of slip on steady Bödewadt flow of a non-Newtonian fluid

The steady flow arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip and non-Newtonian fluid characteristics on the momentum boundary layer are discussed in details. It is observed that slip has prominent effect on the velocity field, whereas a predominant influence of the non-Newtonian parameter is observed on the moment coefficient.     

Unsteady flow past a flat plate with suction

A solution is given for the transient response for laminar boundary layer flow past a flat plate to a step-function change in suction velocity. An arbitrary but constant suction velocity normal to the plate is allowed prior to step-change. Using the Laplace transform technique the solutions for the unsteady velocity profile and shear stress are obtained and are graphically sketched when the suction velocity doubles in the stepchange. The results show clear evidence of boundary-layer contraction when suction velocity is increased.     

Spatial eigenmodes of a boundary layer with a triple-deck velocity field

The beforehand unclear relation between the viscous-inviscid interaction and the instability of viscous gas flows is illustrated using three-dimensional boundary-layer perturbations in the case of sub- and supersonic outer flows. The assumptions are considered under which asymptotic boundary layer equations with self-induced pressure are derived and the excitation mechanisms of eigenmodes (i.e., Tollmien-Schlichting waves) are described. The resulting dispersion relations are analyzed. The boundary layer in a supersonic flow is found to be stable with respect to two-dimensional perturbations, whereas, in the three-dimensional case, the modes become unstable. The increment of growth is investigated as a function of the Mach number and the orientation of the front of a three-dimensional Tollmien-Schlichting wave.     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.