Numerical Simulation of the 3D Laminar Viscous Flow on a Horizontal-axis Wind Turbine Blade

The aim of this paper is to describe the methodology followed in order to determine the viscous effects of a uniform wind on the blades of small horizontal-axis wind turbines that rotate at a constant angular speed. The numerical calculation of the development of the three-dimensional boundary layer on the surface of the blades is carried out under laminar conditions and considering flow rotation, airfoil curvature and blade twist effects. The adopted geometry for the twisted blades is given by cambered thin blade sections conformed by circular are airfoils with constant chords. The blade is working under stationary conditions at a given tip speed ratio, so that an extensive laminar boundary layer without flow separation is expected. The boundary layer growth is determined on a non-orthogonal curvilinear coordinate system related to the geometry of the blade surface. Since the thickness of the boundary layer grows from the leading edge of the blade and also from the tip to the blade root, a domain transformation is proposed in order to solve the discretized equations in a regular computational 3D domain. The non-linear system of partial differential coupled equations that governs the boundary layer development is numerically solved applying a finite difference technique using the Krause zig-zag scheme. The resulting coupled equations of motion are linearized, leading to a tridiagonal system of equations that is iteratively solved for the velocity components inside the viscous layer applying the Thomas algorithm, procedure that allows the subsequent numerical determination of the shear stress distribution on the blade surface.     

Boundary layer development in a gas flow with stagnation enthalpy varying across the streamlines

We consider a laminar boundary layer for which the stagnation enthalpy specified in the initial section is variable with height. Such problems arise, for example, for bodies located in the wake behind another body, for hypersonic flow past slender blunted bodies (as a result of the large transverse entropy gradients in the highentropy layer), for stepwise variation of the temperature of a surface on which there is an already developed boundary layer, for sudden expansion of the boundary layer as a result of its flow past a corner of the surface, etc.Strictly, we should in such cases solve the boundary layer equations (if the longitudinal gradients are much smaller than the transverse) with the specified initial distribution of the quantities. However, from the physical point of view, the distributed region may be broken down into two regions, the near-wall boundary layer and an outer region which is a gas flow with constant velocity and the specified initial temperature profile, whose calculation yields the edge conditions for the boundary layer. The boundary between the regions is determined from the condition of adequately smooth matching of the solutions. This approach is much preferable to the first, since it permits avoiding (within the framework of boundary layer theory) the difficulties associated with the presence of a possible singularity at the initial point of the surface due to the discontinuity of the boundary conditions at this point, and also permits using conventional boundary layer theory if the effect of the viscosity in the outer region is not significant. However, this partition requires additional justifications of the possibility of independent determination of the solution in the outer region and the determination of the edge of the boundary layer, considered as the region of influence of the wetted surface. The boundary layer in a nonuniform flow has been considered in several works for a linear initial velocity or temperature profile [1–3].It should be noted that the linear initial enthalpy or velocity profiles for constant gas properties do not undergo changes under the influence of viscosity or thermal conductivity. Thus the fundamental characteristic features noted above which are associated with the presence of the two regions and their interaction in essence cannot be investigated using these examples.In this study we obtain and analyze the exact solutions of the equations of the compressible boundary layer for a power-law variation of the initial stagnation enthalpy profile as a function of the stream function for a constant initial velocity. Here it is shown that the influence of the boundary conditions at the wall are actually localized in the near-wall boundary layer, which is similar in dimensions to the conventional velocity or thermal boundary layers. In the region which is external with relation to this layer, in accordance with the physical picture described above, the solution coincides with the solution of the Cauchy problem for the heat conduction equation, which describes the development of the initial temperature profile in an infinite steady-state flow with constant velocity.It is shown that for the sufficiently smooth initial profiles which are of interest in practice the outer flow undergoes practically no changes until we reach the inner boundary layer, and it may be calculated using the perfect gas laws.     

Effect of incidence angle with wake passing on a film cooled leading edge: A numerical study

This work presents the numerical study of a film‐cooled blade under the influence of wake passing at different incidence angles. The film cooling technology has been proven to be effective to increase the blade life of first turbine stages. However, the leading edge is affected by an high heat transfer rate and cooling this region is difficult. Moreover, separated regions downstream the coolant injection increases the local heat transfer coefficient and can have a detrimental effect in terms of airfoil life. This work analyses how the flow field is affected by the wake passing at different incidence angles (?5, 0, 5) and the impact on heat transfer coefficient. The test case is a linear cascade with two rows of cylindrical holes at the leading edge. Two different holes arrangements are compared in terms of film cooling structures, namely AGTB‐B1 and AGTB‐B2 with 0 and 45^° spanwise inclination. The numerical results show a good agreement with the experiments. A deeper investigation is carried out on AGTB‐B1. The results obtained show that the wake passing and the incidence angle have a strong effect on coolant jets. In particular, there is a significative impact on coolant redistribution near the leading edge. The wake passing has a stronger effect on pressure side, mainly at negative incidence. The predictive approach is based on an U‐RANS in‐house CFD solver using a conventional two‐equations closure. In order to avoid extra turbulence production, critical in the leading edge region, the turbulence model incorporates an extra algebraic equation that enforces a realizability constraint. The unsteady formulation is based on a dual time stepping approach with a sliding plane between the moving bars and the cascade. Copyright © 2009 John Wiley & Sons, Ltd.     

A compressible three-dimensional design method for radial and mixed flow turbomachinery blades

A fully three-dimensional compressible inverse design method for the design of radial and mixed flow turbomachines is described. In this method the distribution of the circumferentially averaged swirl velocity rV _θ on the meridional geometry of the impeller is prescribed and the corresponding blade shape is computed iteratively. Two approaches are presented for solving the compressible flow problem. In the approximate approach the pitchwise variation in density is neglected and as a result the algorithm is simple and efficient. In the exact approach the velocities and density are computed throughout the three-dimensional flow field by employing a fast fourier transform in the tangential direction. The results of the approximate and exact approach are compared for the case of a high-speed (subsonic) radial-inflow turbine and it is shown that the difference between the blade shapes computed by the two methods is well within the manufacturing tolerances. The method was validated by calculating the flow through a designed high-speed radial-inflow turbine by using a three-dimensional inviscid Euler solver. Very good correlation was obtained between the specified and computed rV _θ-distributions.     

A new Navier–Stokes inverse method based on mass‐averaged tangential velocity for blade design

This paper presents a novel viscous inverse method for blade design. In this inverse design method the mass‐averaged tangential velocity and the blade thickness are prescribed, and the corresponding blade profile is sought. The blade profile is then computed iteratively using the discrepancies between the prescribed mass‐averaged tangential velocity distribution and its calculated distribution on an initial blade. The re‐design of an axial rotor blade, starting from an initial arbitrary profile in subsonic flow regimes, demonstrates the merits and robustness of this approach. Copyright © 2008 John Wiley & Sons, Ltd.     

Contribution to the theory of thin-profile trailing edge separation

The conditions of nonsymmetric trailing edge flow with separation are investigated. Solutions of the equations for the interaction zone in the neighborhood of the trailing edge of a thin profile at an angle of attack of the order O(Re^–1/16) in the separated flow regime are constructed numerically. It is shown that for this zone a solution exists up to a certain angle of attack. In all the regimes the value of the friction on the upper surface at the very end of the trailing edge remains a positive quantity. The solution of the equations in the separated flow regimes is found to be nonunique. The flow over the leading edge is assumed to be unseparated, and the separation at the trailing edge, if present, is assumed to be localized in the interior of the boundary layer. The flow over a Kutta profile at zero angle of attack is taken as an example. In this case the satisfaction of the Chaplygin-Joukowsky condition at the trailing edge ensures smooth flow over both the trailing and leading edges.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–59, July–August, 1989.     

Inflow broadband noise from an isolated symmetric airfoil interacting with incident turbulence

Inflow noise from a symmetric airfoil interacting with homogeneous and isotropic turbulence is investigated focusing on the effects of airfoil geometry. The numerical method employed is based on computational aeroacoustic techniques using the high-order dispersion-relation-preserving finite-difference schemes for solving two-dimensional linearized Euler equations. Effects on inflow noise of the airfoil thickness, leading-edge radius, and freestream Mach number are examined by comparing the acoustic power spectrum of the airfoils and their flow field characteristics. Acoustic power levels of airfoils are found to exponentially decrease in the high-frequency range as airfoil thickness increases because incident turbulent velocities are more distorted in the larger stagnation region near the leading edge. This distortion is shown to be related to the slope angle of the streamline of steady mean flow near the leading edge. However, this high-frequency reduction weakens as the Mach number increases due to the decreasing slope angle. In addition, the chordwise velocity component in the incident turbulence contributes more to the radiating acoustic pressure level as the freestream Mach number increases, which also results in less high-frequency reduction at higher freestream Mach number. At fixed airfoil thickness, increasing the leading-edge radius leads to decreases in the acoustic power level, which may also be explained by size variation of the stagnation region around the leading edge. An approximate algebraic formula for acoustic power spectra is derived on the basis of these observations. Acoustic power spectra predicted using this formula are shown to closely follow the numerical results. Finally, the applicability of the algebraic formula and the current numerical methods to more realistic problems are confirmed by comparing their predictions with the measured data.     

Design optimization of axial flow hydraulic turbine runner: Part I—an improved Q3D inverse method

With the aim of constructing a comprehensive design optimization procedure of axial flow hydraulic turbine, an improved quasi‐three‐dimensional inverse method has been proposed from the viewpoint of system and a set of rotational flow governing equations as well as a blade geometry design equation has been derived. The computation domain is firstly taken from the inlet of guide vane to the far outlet of runner blade in the inverse method and flows in different regions are solved simultaneously. So the influence of wicket gate parameters on the runner blade design can be considered and the difficulty to define the flow condition at the runner blade inlet is surmounted. As a pre‐computation of initial blade design on S2m surface is newly adopted, the iteration of S1 and S2m surfaces has been reduced greatly and the convergence of inverse computation has been improved. The present model has been applied to the inverse computation of a Kaplan turbine runner. Experimental results and the direct flow analysis have proved the validation of inverse computation. Numerical investigations show that a proper enlargement of guide vane distribution diameter is advantageous to improve the performance of axial hydraulic turbine runner. Copyright © 2002 John Wiley & Sons, Ltd.     

Study of flow in the vicinity of V-shaped wings formed by the stream surfaces behind a plane shock

In calculating the flow about bodies with plane surfaces and sharp edges it is assumed that in the flow regimes with attached shock the latter may be defined in a section normal to the edge from the corresponding relations for the wedge [1, 2], The solution is taken corresponding to a weak shock on a wedge with supersonic velocity behind it. While in the plane case (wedge) this solution will be the only physically realizable solution, in the case of three-dimensional bodies, when there is a slip velocity along the leading edge, the realization of a second wedge solution with a strong shock is conceivable in the section normal to the leading edge if the total velocity behind the shock (with account for the slip velocity along the edge) is supersonic [3].Relative to the undisturbed stream velocity both of these solutions correspond to a weak shock. We present an example when the solution with a strong shock in the section normal to the edge is possible.     

Multiple laminar-turbulent transition cycles around a swept leading edge

Certain interesting flow features involving multiple transition/relaminarization cycles on the leading edge of a swept wing at low speeds are reported here. The wing geometry tested had a circular nose and a leading edge sweep of 60°. Tests were made at a chord Reynolds number of 1.3 × 10⁶ with model incidence α varied in the range of 3°?18° in discrete steps. Measurements made included wing chord-wise surface pressure distributions and wall shear stress fluctuations (using hot-film gages) within about 10 % of the chord in the leading edge zone. Results at α = 16° and 18° showed that several (often incomplete) transition cycles between laminar-like and turbulent-like flows occurred. These rather surprising results are attributable chiefly to the fact that the Launder acceleration parameter K (appropriately modified for swept wings) can exceed a critical range more than once along the contour of the airfoil in the leading edge region. Each such crossing results in a relaminarization followed by direct retransition to turbulence as K drops to sufficiently low values. It is further shown that the extent of each observed transition zone (of either type) is consistent with earlier data acquired in more detailed studies of direct transition and relaminarization. Swept leading edge boundary layers therefore pose strong challenges to numerical modelling.     

Acoustic receptivity of Mach 4.5 boundary layer with leading-edge bluntness

Boundary layer receptivity to two-dimensional slow and fast acoustic waves is investigated by solving Navier–Stokes equations for Mach 4.5 flow over a flat plate with a finite-thickness leading edge. Higher order spatial and temporal schemes are employed to obtain the solution whereby the flat-plate leading edge region is resolved by providing a sufficiently refined grid. The results show that the instability waves are generated in the leading edge region and that the boundary-layer is much more receptive to slow acoustic waves (by almost a factor of 20) as compared to the fast waves. Hence, this leading-edge receptivity mechanism is expected to be more relevant in the transition process for high Mach number flows where second mode instability is dominant. Computations are performed to investigate the effect of leading-edge thickness and it is found that bluntness tends to stabilize the boundary layer. Furthermore, the relative significance of fast acoustic waves is enhanced in the presence of bluntness. The effect of acoustic wave incidence angle is also studied and it is found that the receptivity of the boundary layer on the ‘windward’ side (with respect to the acoustic forcing) decreases by more than a factor of four when the incidence angle is increased from 0° to 45°. However, the receptivity coefficient for the ‘leeward’ side is found to vary relatively weakly with the incidence angle.      

On the three-dimensional nature of the orthogonal blade–vortex interaction

An orthogonal blade–vortex interaction has been visualised using stereo particle image velocimetry. Significant changes to the vortex axial flow w component velocity are observed during the interaction, with a deceleration on the lower surface of the blade where the vortex axial flow is towards the blade surface. Over this surface the interaction process close to the blade surface spreads the vorticity out to the areas of oppositely signed blade w component, and the results suggest a non-uniform spreading over the leading edge region of the blade, with a tendency for a spanwise transport of vorticity. Over the upper surface of the blade, the vortex axial flow velocity increases and the vortex core shrinks slightly. During the lower surface interaction the vorticity and velocity vectors become significantly realigned with respect to one another, while this is not observed for the upper surface interaction.     

Numerical investigation of the aerodynamics of the near‐slot film cooling

Fluid injection from slot or holes into cross‐flow produces highly complicated flow fields. Physical situations encountering the above problem range from turbine blade cooling to waste discharge into rivers. In this paper, the flow field created by a two‐dimensional slot cooling geometry is examined using the finite volume approach with a second‐order upwind differencing scheme. The time‐averaged Navier–Stokes equations were solved on a collocated Cartesian grid with a two‐equation model of turbulence. Attempting to solve the flow field by assuming a uniform velocity profile at the slot exit leads to inaccurate results, while extending the solution domain improves significantly the results, but proves to be costly, both in memory and in computing time (particularly in the case of multiple holes). A pressure‐type boundary condition, based on uniform total pressure, is developed for the slot exit (easily applied to a three‐dimensional geometry), which yields more accurate results than the widely used uniform velocity assumption. It is also found that the implementation of low Reynolds number turbulence models on this geometry provides no significant differences from the standard k–ε model. Copyright © 2000 John Wiley & Sons, Ltd.     

One method of profiling short plane nozzles

One of the possible methods is considered for profiling short plane nozzles for aerodynamic tubes. The nozzle has a straight sonic line, which allows the subsonic and supersonic sections to be constructed separately. The problem is solved numerically in the plane of a hodograph. In the subsonic region, Dirichlet's problem is formulated for Chaplygin's equation in a rectangle, one side of which is the sonic line. At the present time, two approaches have been defined in papers on calculations of a Laval nozzle, associated with the solution of the so-called “direct” and “inverse” problems (one has in mind a study of the flow in the interconnected region of sub- and supersonic flow). The direct problem determines the flow field in the case of a previously specified contour of the channel wall, the shape of which from technical considerations is obtained with certain geometry conditions. The direct problem can be applied in the construction of the Laval nozzle, if the contour of the inlet section of the channel (generally speaking, quite arbitrary) is chosen so successfully that neither shock compressions nor breakaway zones result in the flow. Although a strictly mathematical theory of the direct problem of the Laval nozzle is only being developed at present, there are still very effective numerical methods for its solution [1, 2]. In the inverse problem (which, by definition, is a problem of profiling), the contour of the nozzle is found with respect to a specified velocity distribution on the axis of symmetry. It is assumed that this quite arbitrary dependence can be selected from the condition of the absence of breakaway zones and shock compressions in the nozzle. By its formulation, the inverse problem is Cauchy's problem which, as is well-known, is incorrect in the classical sense in the ellipticity region — the subsonic section of the nozzle. At present, there are also efficient methods of solving the inverse nozzle problem [3], by interpreting it as an arbitrarily correct problem. Difficulties can arise in the inverse problem, in the provision of short (and, consequently, steep) nozzles because of the sharp increase of the error in the calculation. Together with the stated problems, a procedure can be evolved which is associated with the solution of the correctly posed problem for Chaplygin's equation in the plane of the hodograph. This approach is convenient in that it succeeds a priori in fulfilling the important condition of monotonicity of the velocity at the wall, ensuring (in the absence of shock compressions) nonseparability of the streamline flow at any Reynold's numbers.     

Numerical investigation of the influence of incidence angle on asymmetrical turbine blade model showerhead film cooling effectiveness

The influence of various incidence angles on film cooling effectiveness of an axial turbine blade cascade with leading edge ejection from two rows of cooling holes is numerically investigated. The rows are located in the vicinity of the stagnation line. One row is located on the suction side and the other one is on the pressure side. The predicted pressure field for various blowing ratios (M = 0.7, 1.1 and 1.5) is compared with available experimental results at the design condition. Moreover, the effect of various incidence angles (?10°, ?5°, 0°, 5° and 10°) at three blowing rates is investigated by analyzing the results of both laterally averaged and area averaged values of adiabatic film cooling effectiveness. Numerical results indicate that the incidence angle can strongly affect the thermal protection of the blade at low blowing ratio but becomes less dominant at high blowing ratio. In fact, for the low blowing ratio, a small change in the incidence angle that relates to the design condition can deeply affect the thermal protection of the blade, which is evident from the laterally and area averaged film cooling effectiveness distributions.     

An adjoint method for the calculation of remote sensitivities in supersonic flow

This paper presents an adjoint method for the calculation of remote sensitivities in supersonic flow. The goal is to develop a set of discrete adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the complete formulation and discretization of the discrete adjoint equations. The special treatment of the adjoint boundary condition to obtain remote sensitivities or sensitivities of pressure distributions at points remotely located from the wing surface are discussed. Secondly, we present results that demonstrate the application of the theory to a three-dimensional remote inverse design problem using a low sweep biconvex wing and a highly swept blunt leading edge wing. Lastly, we present results that establish the added benefit of using an objective function that contains the sum of the remote inverse and drag minimization cost functions.     

Euler solutions for self-generated rotor blade-vortex interactions

A finite-difference procedure has been developed for the prediction of three-dimensional rotor blade-vortex interactions. The interaction velocity field was obtained through a non-linear superposition of the rotor flow field, computed using the unsteady three-dimensional Euler equations, and the embedded vortex wake flow field, computed using the law of Biot-Savart. In the Euler model, near wake rotational effects were simulated using the surface velocity ‘transpiration’ approach. As a result, a modified surface boundary condition was prescribed and enforced at each time step of the computations to satisfy the tangency boundary condition. For supercritical interactions using an upstream-generated vortex, accuracy of the numerical results were found to rely on the user-specified vortex core radius and vortex strength. For the more general self-generated subcritical interactions, vortex wake trajectories were computed using the lifting-line helicopter/rotor trim code CAMRAD. For these interactions, accuracy of the results were found to rely heavily on the CAMRAD-predicted vortex strength, vortex orientation with respect to the blade, and to a large extent on the user-specified vortex core radius. Results for the one-seventh scale model OLS rotor and for a non-lifting rectangular blade having a NACA0012 section are presented. Comparisons with the experimental windtunnel data are also made.     

DNS of secondary flows over oscillating low-pressure turbine using spectral/hp element method

This paper investigates the secondary vortex flows over an oscillating low-pressure turbine blade using a direct numerical simulation (DNS) method. The unsteady flow governing equations over the oscillating blade are discretized and solved using a spectral/hp element method. The method employs high-degree piecewise polynomial basis functions which results in a very high-order finite element approach. The results show that the blade oscillation can significantly influence the transitional flow structure and the wake profile. It was observed that the separation point over vibrating T106A blades was delayed 4.71% compared to the stationary one at Re = 51,800. Moreover, in the oscillating case, the separated shear layers roll up, break down and shed from the trailing edge. However, the blade vibration imposes additional flow disturbances on the suction surface of the blade before leaving from the trailing edge. Momentum thickness calculations revealed that after flow separation point, the momentum thickness grows rapidly which is due to the inverse flow gradients which generate vortex flows in this area. It was concluded that the additional vortex generations due to the blade vibrations cause higher momentum thickness increment compared to the conventional stationary LPT blade.     

Receptivity of the boundary layer on a flat plate with a blunted leading edge to steady nonuniformity of the free stream

The flow past a flat plate with a blunted leading edge by a flow of a viscous incompressible fluid with a small spanwise-periodic, steady nonuniformity of the velocity profile is considered. Such a flow simulates the interaction of one type of vortex disturbances of a turbulent external flow with the boundary layer. The solution obtained predicts generation of strong disturbances in the boundary layer, which are similar to the streaky structure observed in the case of high free-stream turbulence. It is shown that the boundary-layer flow on blunted bodies is more sensitive to vortex disturbances than on a plate with a sharp leading edge. Central Aerohydrodynamic Institute, Zhukovskii, 140160. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 93–100, July–August, 2000.     

Effects of modifying the blade trailing edge profile on unsteady pressure pulsations and flow structures in a centrifugal pump

In the present work, the effects of modifying the blade pressure side (EPS profile) on unsteady pressure pulsations and flow structures in a low specific speed centrifugal pump are carried out by experimental and numerical methods. Results are compared to the original trailing edge (OTE profile). Unsteady pressure signals are captured at twenty measuring points at flow rate of 0–1.6Q_d. It is observed that the pump head of the EPS profile is improved for all the concerned working conditions. Pressure amplitudes at the blade passing frequency are compared and discussed in detail. It is found that the EPS profile contributes to pressure pulsation reduction obviously. For all the measured flow rates, pressure amplitudes are attenuated evidently at major measuring positions, especially at high flow rates. As for the mean pressure amplitude of twenty measuring points, pressure amplitude is reduced more than 20% at the nominal flow rate using the EPS profile. From relative velocity distribution, it is found that the uniformity of flow field at the blade outlet region would be improved significantly by the EPS profile. Besides, the corresponding vorticity magnitude at the blade outlet would be reduced compared to the OTE profile. The combined effects contribute to the reduction of pressure amplitude using the EPS profile.