Supersonic flow around the trailing edge of a thin profile

The method of mergeable asymptotic expansions has recently been used effectively in investigations devoted to the study of boundary layer interaction with an external inviscid flow at high subcritical Reynolds numbers Re. The asymptotic analysis permits obtaining a limit pattern of the flow around a solid as Re þ, and determining the similarity and quantitative regularity laws which are in good agreement with experimental results. Thus by using the method of mergeable asymptotic expansions it is shown in [1–4] that near sites with high local curvature of the body contour and flow separation and attachment points, an interaction domain appears that has a small length on the order of Re^-3/8. In this flow domain, which has a three-layer structure, the pressure distribution in a first approximation already depends on the change in boundary-layer displacement thickness, while the induced pressure gradient, in turn, influences the flow in the boundary layer. An analogous situation occurs in the neighborhood of the trailing edge of a flat plate where an interaction domain also appears [5, 6]. The flow in the neighborhood of the trailing edge of a flat plate around which a supersonic viscous gas flows was examined in [7]. Numerical results in this paper show that the friction stress on the plate surface remains positive everywhere in the interaction domain, and grows on approaching the trailing edge. The supersonic flow around the trailing edge of a flat plate at a small angle of attack was investigated in [8, 9], Supersonic flow of a viscous gas in the neighborhood of the trailing edge of a flat plate at zero angle of attack is examined in [10], but with different velocity values in the inviscid part of the flow on the upper and lower sides of the plate. The more general problem of the flow around the trailing edge of a profile with small relative thickness is investigated in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 36–42, May–June, 1981.     

Measurement and analysis of unsteady flow structures in rotor blade wakes

Time-variant data are obtained to investigate the exit flow field from a rotor in a research compressor. In the free-stream region, the instantaneous data are analogous to one another and to the ensemble averaged free-stream results. However, in the wake region, some of the instantaneous signals are similar to one another and to the ensemble averaged wake, but others differ significantly. These variations in the instantaneous data are interpreted and shown to be due to a vortex street structure in the wake. This is accomplished by: (1) developing a mathematical model of the rotor blade exit flow field based on a wake vortex street structure analogous to the unsteady flow field behind bluff bodies due to classical von Karman vortex shedding; and (2) correlating predictions of both the ensemble averaged and instantaneous rotor blade exit flow fields as well as the velocity probability density distributions from this vortex wake flow field model with the corresponding data. The correlation of the ensemble averaged rotor blade exit flow fields is very good and the flow angle distribution correlation excellent. The predicted instantaneous rotor blade exit flow field exhibits many of the flow features found in the data. Also, the probability density distributions for the data and the vortex wake flow field model are analogous to one another.List of symbols N number of rotor revolutions - S _w rotor blade wake width - S _x vortex core horizontal spacing - S _y vortex core vertical spacing - u velocity component parallel to vortex street motion - v velocity component normal to vortex street motion - W instantaneous relative velocity - W _i velocity induced by vortex street - W free-stream relative velocity - W _s velocity of vortex street - x coordinate parallel to vortex street motion - y coordinate normal to vortex street motion - free-stream relative flow angle - _inst instantaneous relative flow angle - vortex strength     

The analysis of unsteady flow structure and low frequency pressure pulsations in the high-head Francis turbines

The paper presents results of numerical simulation of unsteady three-dimensional flow in the two high-head hydraulic turbines. The numerical technique for calculating of low-frequency pressure pulsations in a hydraulic turbine is based on the DES turbulence model and the approach of rotated reference frame. The paper also presents the analysis of the flow structure behind the runner of the turbines, as well as shows the effect of the flow structure on the frequency and intensity of non-stationary processes in the turbines. Besides, the behavior of the pulsations in the hydraulic turbines was defined. Comparison of the calculation results with the experimental data has shown close agreement between theory and experiment.     

Three-dimensional flow structures and associated turbulence in the tip region of a waterjet pump rotor blade

Stereo particle image velocimetry measurements focus on the flow structure and turbulence within the tip leakage vortex (TLV) of an axial waterjet pump rotor. Unobstructed optical access to the sample area is achieved by matching the optical refractive index of the transparent pump with that of the fluid. Data obtained in closely spaced planes enable us to reconstruct the 3D TLV structure, including all components of the mean vorticity and strain-rate tensor along with the Reynolds stresses and associated turbulence production rates. The flow in the tip region is highly three-dimensional, and the characteristics of the TLV and leakage flow vary significantly along the blade tip chordwise direction. The TLV starts to roll up along the suction side tip corner of the blade, and it propagates within the passage toward the pressure side of the neighboring blade. A shear layer with increasing length connects the TLV to the blade tip and initially feeds vorticity into it. During initial rollup, the TLV involves entrainment of a few vortex filaments with predominantly circumferential vorticity from the blade tip. Being shed from the blade, these filaments also have high circumferential velocity and appear as swirling jets. The circumferential velocity in the TLV core is also substantially higher than that in the surrounding passage flow, but the velocity peak does not coincide with the point of maximum vorticity. When entrainment of filaments stops in the aft part of the passage, newly forming filaments wrap around the core in helical trajectories. In ensemble-averaged data, these filaments generate a vortical region that surrounds the TLV with vorticity that is perpendicular to that in the vortex core. Turbulence within the TLV is highly anisotropic and spatially non-uniform. Trends can be traced to high turbulent kinetic energy and turbulent shear stresses, e.g., in the shear layer containing the vortex filaments and the contraction region situated along the line where the leakage backflow meets the throughflow, causing separation of the boundary layer at the pump casing. Upon exposure to adverse pressure gradients in the aft part of the passage, at 0.65–0.7 chord fraction in the present conditions, the TLV bursts into a broad turbulent array of widely distributed vortex filaments.     

Investigation of aerodynamic effects of coolant ejection at the trailing edge of a turbine blade model by PIV and pressure measurements

In order to simulate the thick trailing edges of turbine blades a slotted plate profile together with a newly designed nozzle was installed into the high-speed wind tunnel of the DLR Göttingen. At different supersonic Mach numbers and at four coolant flow rates in the range of 0–2.5% pressure distribution measurements and probe measurements were performed. The flow field was visualized by schlieren photos and the instantaneous velocity field was quantitatively investigated by Particle Image Velocimetry (PIV). The measurements of the velocity field gave an insight into stationary effects, for example the change of shock strength with coolant flow rate, and instationary effects such as the existence of a vortex street in the wake. The PIV technique offers special advantages for the investigation of transonic flow fields, but also yields to special experimental difficulties, which are also described in this article. Measured losses display a maximum at the downstream Mach number 1. This is strongly related to the behaviour of the base pressure. A loss minimum is achieved at moderate coolant flow rates, showing that an optimum coolant flow rate exists. The loss was analysed and separated into the loss contributions from the profile upstream of the trailing edge and the mixing loss due to the coolant flow.     

Numerical evaluation of the aerodynamic influence of the helicopter composite blade trailing edge tabs

In classical composite helicopter rotor blade production, a small flat tab must be formed along the entire trailing edge, in order to enable proper merging of the upper and the lower surface plies during manufacturing. By this, the original airfoil shape is altered. Such fixed tabs have been added in a range of possible angular positions to several existing asymmetrical helicopter airfoils, and their capability to change the moment coefficient about the aerodynamic center of the airfoils was initially analyzed. Although usual tabs are proportionally small, angular domains in which they do not remarkably change the required nearly zero aerodynamic moment, were quantified as very narrow. In the next stage, an algorithm has been defined and implemented: (a) for the determination of optimum angular tab positions for several asymmetrical airfoils, that satisfy the moment requirement (for such airfoils optimum tab direction cannot be known in advance), and (b) for the reduction of the influence of eventual inherent numerical errors of applied software to a minimum. The accuracy of this algorithm has been verified on a symmetrical airfoil, for which the optimum tab position is readily known. In the next step, the tab influence on other aerodynamic airfoil characteristics, and the influence on flight performance of a light helicopter from an on-going project, has been analyzed. Several possible tab design concepts were defined, and some characteristic aspects of their implementation were considered. At the level of preliminary helicopter performance calculations, the influence of the two general outcomes of the tab designs were analyzed, one that preserves initial relative airfoil thickness, and another which leads to its reduction. In the first case, the influence of the slight increase of drag coefficient was taken into account, while in the second one, the decrease of drag coefficient, accompanied with necessary additional strengthening and added blade mass was considered. In both cases applied modifications proved to have moderate direct influence on helicopter flight performance, compared with a hypothetic case that the original airfoil without tab could have been used instead. General conclusions have imposed the need for very careful approach in tab design for asymmetrical airfoils, which must be primarily focused on the tab’s potential remarkable influence on the aerodynamic moment.     

Numerical modelling and flow analysis of a centrifugal pump running as a turbine: Unsteady flow structures and its effects on the global performance

The purpose of this paper is to investigate the flow patterns in a centrifugal pump when it works as a centripetal turbine, with special interest in the unsteady behavior in order to explain the shape of the performance curves. Also, we focus on the determination of the radial thrust and other mechanical loads over a pump‐designed machine. The pump studied is commercial, with single axial suction and a vaneless spiral volute casing. A numerical study has been carried out in order to obtain more information about the flow into the volute and the impeller. A numerical three‐dimensional unsteady simulation has been developed using a commercial code that solves the URANS set of equations with a standard k–ε turbulence model. The results show the non‐axisymmetric flow developed in the volute, responsible for a significant radial thrust; the interaction between the tongue and the impeller, generating force fluctuations; the velocity and pressure distributions inside the impeller; and the exit flow, characterized with post‐rotation and low‐pressure. These flow results allow us to understand the behavior of the machine by comparing it with the pump mode. Complementarily, an experimental study was conducted to validate the numerical model and characterize the pump‐turbine performance curves at constant head. Fast‐response pressure taps and a three‐hole pneumatic pressure probe were employed to obtain a complete data set of non‐stationary and stationary measurements throughout the centrifugal machine. As a result, loss of efficiency or susceptibility to cavitation, detected numerically, was confirmed experimentally. The study demonstrates that the numerical methodology presented here has shown its reliability and possibilities to predict the unsteady flow and time‐mean characteristics of centrifugal pumps working as turbines. In particular, it is shown that the commercial design of the pump allows a reasonable use of the impeller as a turbine runner, due to the suitable adaptation of the inflow distributions to the volute casing. Moreover, the efficiency for the inverse mode is shown to be as high as achieved for the pumping operational mode. In addition, it is concluded that both axial and radial thrusts are controlled, though important unsteady fluctuations—up to 25%—clocked with the blade passing frequency appear beyond the nominal conditions. In that case, a moderate use of the pump as a turbine is recommended in order to minimize risks of fatigue failure of the bearings. Copyright © 2009 John Wiley & Sons, Ltd.     

Stochastic estimation of flow near the trailing edge of a NACA0012 airfoil

A stochastic estimation technique has been applied to simultaneously acquired data of velocity and surface pressure as a tool to identify the sources of wall-pressure fluctuations. The measurements have been done on a NACA0012 airfoil at a Reynolds number of Re _c  = 2 × 10⁵, based on the chord of the airfoil, where a separated laminar boundary layer was present. By performing simultaneous measurements of the surface pressure fluctuations and of the velocity field in the boundary layer and wake of the airfoil, the wall-pressure sources near the trailing edge (TE) have been studied. The mechanisms and flow structures associated with the generation of the surface pressure have been investigated. The “quasi-instantaneous” velocity field resulting from the application of the technique has led to a picture of the evolution in time of the convecting surface pressure generating flow structures and revealed information about the sources of the wall-pressure fluctuations, their nature and variability. These sources are closely related to those of the radiated noise from the TE of an airfoil and to the vibration issues encountered in ship hulls for example. The NACA0012 airfoil had a 30 cm chord and aspect ratio of 1.     

Rollup region of a turbulent trailing vortex issued from a blade with flow separation

This paper presents the results of an experimental investigation on the near field of a tip vortex generated by a blade at moderate incidence. The experiments were conducted at Re=15 000 and the boundary layer over the blade separated around midchord on the upper surface. Laser-Doppler measurements of the turbulent flow (Tu=1.5%) were performed at various stations downstream of the blade. The three components of the mean velocity field and turbulent attributes were quantified at cross-planes, characterizing both the blade wake and the tip vortex structure. This allowed the analysis of the rollup and initial stages of decay of the tip vortex in the light of known theories and models. The axial velocity defect at the center of the vortex core evolved as x⁻¹ log x, without displaying any significant outgrowth imposed by the separated flow upstream. Momentum balances were also carried out at a station downstream to the conclusion of vortex rollup. The approximate axisymmetry of the flow field in the trailing vortex was used to formulate the balances in a cylindrical coordinate system. Among other observations, it was seen that an adverse axial pressure gradient developed in the vortex core, which reinforced the tenacity of the axial velocity defect. In contrast, an area influenced by a favorable pressure gradient was found outside the core.     

Effects of a vectored trailing edge jet on delta wing vortex breakdown

Flow visualization was used to study the effects of a vectored trailing edge jet on the leading edge vortex breakdown of a 65° delta wing. The experimental results indicated that there is little effect of the jet on the leading edge vortex breakdown when the angle of the vectored jet is less than 10°. With the increase of the vectored angle ß, the effect of the jet on the flow becomes stronger, i.e., the jet delays the leading edge vortex breakdown in the direction of the vectored jet, and accelerates breakdown of the other leading edge vortex. Moreover, the effect of the jet control tends to be weaker with the angle of attack.     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

A time marching method for unsteady two-dimensional flow in a blade passage

A time marching method for calculating non-uniform unsteady two-dimensional flow through a blade passage is presented. The various problems encountered in extending time marching techniques into this field are examined, and the method is applied to analyse unsteady wake flows through a compressor and turbine blade row.     

Structure of the flow of a viscous gas near the trailing edge of a flat plate

Solution of the complete system of Navier-Stokes equations forms the basis for a study of the nature of flow of a viscous heat-conducting gas in the neighborhood of a trailing edge of a flat plate. The problem was solved in accordance with a difference scheme of the third order of accuracy [1]. The calculation was carried out under the same conditions as the experiment of [2], in which a plate of finite dimensions (L = 12 cm) had supersonic M = 2, Re, = 1000 gas flow round it. In order to obtain a thickness of the boundary layer which was acceptable for the purpose of making the measurements (of the order of 2 cm), the unperturbed gas was slightly rarefied. In the study of such problems [3–7] it is necessary to use the complete system of Navier-Stokes equations, since in the immediate neighborhood of the trailing edge one of the important assumptions in the theory of the boundary layer, ²u/y^{2 2}u/x², does not hold. As a result the flow upstream near the trailing edge of the plate will depend on the flow immediately behind the edge, since the perturbations propagate both upstream and downstream in this case. The rarefaction of the gas creates additional difficulties in the formulation of the boundary conditions on the plate with flow round it when this problem is studied numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–30, March–April, 1987.     

Flow in the neighborhood of the trailing edge of a plate in a transonic flow of viscous gas

An investigation is made into the influence of the Mach number and the viscosity on the flow in the neighborhood of the trailing edge of a plate. The Mach number is assumed to satisfy m_∞ ² ? 1 = 0(R^?l/5), which corresponds to the regime of transonic interaction. It is shown that if the Mach number is such that ¦M_∞ ² ? 1¦ > O(R^?1/5) the problem in the region of free interaction can be reduced by an appropriate transformation to the already known solutions for an incompressible fluid [5] and supersonic flow [7].     

Experimental investigation of the effects of blowing conditions and Mach number on the unsteady behavior of coolant ejection through a trailing edge cutback

The present paper shows the results of an experimental investigation into the unsteadiness of coolant ejection at the trailing edge of a highly loaded nozzle vane cascade. The trailing edge cooling scheme features a pressure side cutback with film cooling slots, stiffened by evenly spaced ribs in an inline configuration. Cooling air is also ejected through two rows of cylindrical holes placed upstream of the cutback. Tests were performed with a low inlet turbulence intensity level (Tu₁ = 1.6%), changing the cascade operating conditions from low speed (M_2is = 0.2) up to high subsonic regime (M_2is = 0.6), and with coolant to main stream mass flow ratio varied within the 0.5–2.0% range. Particle Image Velocimetry (PIV) and flow visualizations were used to investigate the unsteady mixing process taking place between coolant and main flow downstream of the cutback, up to the trailing edge. For all the tested conditions, the results show the presence of large coherent structures, which presence is still evident up to the trailing edge. Their shape and direction of rotation change with injection conditions, as a function of coolant to mainstream velocity ratio, strongly influencing the thermal protection capability of the injected coolant flow. The Mach number increase is only responsible for a positioning of such vortical structures closer to the wall, while the Strouhal number almost remains unchanged.     

LES investigation on the dependence of the flow through a centrifugal pump on the diffuser geometry

Large-Eddy Simulation is utilized to investigate the rotor–stator interaction within a centrifugal pump. Comparisons are presented across diffuser geometries for two values of the flow-rate. Decreasing the incidence angle on the diffuser blades at off-design is found the main source of higher pressure rise and lower overall values of turbulent kinetic energy within the pump, resulting in efficiency improvement. The impact on the second-order statistics of the flow is especially significant. However, the values of the pressure fluctuations acting on the diffuser blades, defining fatigue loads on them and cavitation phenomena, are found especially affected by the rotor–stator clearance. Results show that at reduced flow-rates the rotation of the diffuser blades around their mid camber is a better option than rotating them around their leading edge. They also suggest that at larger flow-rates the increased incidence on the diffuser blades causes pressure side separation and large shear layers populating the diffuser channels, not affecting substantially the region of interface between impeller and diffuser, but having detrimental effects on the performance of the latter. The rotation of the diffuser blades around their leading edge should be preferred when the pump operates at flow-rates larger than the design one, avoiding decreasing the rotor–stator gap, thus resulting in smoother rotor–stator interaction and lower pressure fluctuations.     

Experimental investigation of a blunt trailing edge flow field with application to sound generation

The unsteady lift generated by turbulence at the trailing edge of an airfoil is a source of radiated sound. The objective of the present research was to measure the velocity field in the near wake region of an asymmetric beveled trailing edge in order to determine the flow mechanisms responsible for the generation of trailing edge noise. Two component velocity measurements were acquired using particle image velocimetry. The chord Reynolds number was 1.9 × 10⁶. The data show velocity field realizations that were typical of a wake flow containing an asymmetric periodic vortex shedding. A phase average decomposition of the velocity field with respect to this shedding process was utilized to separate the large scale turbulent motions that occurred at the vortex shedding frequency (i.e., those responsible for the production of tonal noise) from the smaller scale turbulent motions, which were interpreted to be responsible for the production of broadband sound. The small scale turbulence was found to be dependent on the phase of the vortex shedding process implying a dependence of the broadband sound generated by the trailing edge on the phase of the vortex shedding process.     

Paradox of the blunt edge of an airfoil in an unsteady flow

A paradox of the blunt edge of an airfoil in an unsteady ideal flow is established, which states that the solution of the nonlinear problem of unsteady flow around a bluntedged airfoil subject to strict boundary conditions at this edge is physically meaningless. The paradox is a consequence of the adopted model of the unsteady fluid flow near the blunt edge, which assumes inflection of streamlines. It is established that the solution of the problem by local replacement of the blunt edge by a sharp edge using the hypothesis on the smoothness of streamlines near the trailing edge is physically meaningful.