Leading-edge Singularity Research Articles

Inviscid Nonlinear Modeling of Vibration-Induced Acoustic Resonance of a Linear Cascade

A nonlinear model is developed to investigate the blade vibration-induced acoustic resonance in a linear cascade. The Euler equations are solved by using high-order computational aeroacoustics techniques. To satisfy no-penetration wall boundary conditions for vibrating blades, a semi-implicit time-accurate body force model is developed. Leading-edge singularity at resonance is remarkably strong, and so the balance of globally high-order quality (less dissipation) and numerical stability is significant. The adaptive discontinuity-capturing filtering is introduced as a local operation to achieve the good balance. To validate our model, the benchmark case of the Parker acoustic resonance induced by the vibrating cascade of flat plates is revisited. Compared to the linearized theory and viscous flow solution, our inviscid model economically achieves a good prediction both for the unsteady loading and the near-field sound pressure for the small blade vibration amplitude of . When it comes to a larger vibration amplitude of , nonlinearity on the unsteady lift and sound pressure appears in our results, and it cannot be considered in the linearized theory. The results, to some extent, are also consistent with the viscous flow solution. The study well verifies the capability of our nonlinear model in predicting this category of linear and nonlinear sound-vortex/structure interaction problems.

On the Edge Singularity of the Actuator Disk Model

AbstractIn this technical brief, the classic actuator disk theory is revisited with a view to shed some light on the singularity of the flow at the edge of the disk where the vortex tube starts and where vorticity is generated. The study is carried out using small perturbation assumption in two-dimensions and simplified boundary conditions in all cases. The problem of the two-dimensional thin cambered plate with constant vorticity distribution is solved and the leading edge singularity is analyzed as it is believed to be relevant to the axisymmetric flow at the actuator disk edge. Next, the velocity components induced by the cylindrical vortex tube of constant vorticity are obtained via the Biot–Savart law and the near edge behavior is investigated. It is shown that the velocity components behavior is consistent with that of the thin cambered plate with constant loading, thus reinforcing the notion that the axisymmetric slip-line behaves as r − R ∝ −xlnx near the disk edge.

On the leading-edge suction and stagnation-point location in unsteady flows past thin aerofoils

Unsteady thin-aerofoil theory is a low-order method for calculating the forces and moment developed on a camber line undergoing arbitrary motion, based on potential-flow theory. The vorticity distribution is approximated by a Fourier series, with a special ‘A0’ term that is infinite at the leading edge representing the ‘suction peak’. Though the integrated loads are finite, the pressure and velocity at the leading edge in this method are singular owing to the term. In this article, the principle of matched asymptotic expansions is used to resolve the singularity and obtain a uniformly valid first-order solution. This is performed by considering the unsteady thin-aerofoil theory as an outer solution, unsteady potential flow past a parabola as an inner solution, and by matching them in an intermediate region where both are asymptotically valid. Resolution of the leading-edge singularity allows for derivation of the velocity at the leading edge and location of the stagnation point, which are of physical and theoretical interest. These quantities are seen to depend on only the A0 term in the unsteady vorticity distribution, which may be interpreted as an ‘effective unsteady angle of attack’. The leading-edge velocity is proportional to A0 and inversely proportional to the square root of leading-edge radius, while the chordwise stagnation-point location is proportional to the square of A0 and independent of the leading-edge radius. Closed-form expressions for these in simplified scenarios such as quasi-steady flow and small-amplitude harmonic oscillations are derived.

Computational methods for investigation of surface curvature effects on airfoil boundary layer behavior

This article presents computational algorithms for the design, analysis, and optimization of airfoil aerodynamic performance. The prescribed surface curvature distribution blade design (CIRCLE) method is applied to a symmetrical airfoil NACA0012 and a non-symmetrical airfoil E387 to remove their surface curvature and slope-of-curvature discontinuities. Computational fluid dynamics analysis is used to investigate the effects of curvature distribution on aerodynamic performance of the original and modified airfoils. An inviscid–viscid interaction scheme is introduced to predict the positions of laminar separation bubbles. The results are compared with experimental data obtained from tests on the original airfoil geometry. The computed aerodynamic advantages of the modified airfoils are analyzed in different operating conditions. The leading edge singularity of NACA0012 is removed and it is shown that the surface curvature discontinuity affects aerodynamic performance near the stalling angle of attack. The discontinuous slope-of-curvature distribution of E387 results in a larger laminar separation bubble at lower angles of attack and lower Reynolds numbers. It also affects the inherent performance of the airfoil at higher Reynolds numbers. It is shown that at relatively high angles of attack, a continuous slope-of-curvature distribution reduces the skin friction by suppressing both laminar and turbulent separation, and by delaying laminar-turbulent transition. It is concluded that the surface curvature distribution has significant effects on the boundary layer behavior and consequently an improved curvature distribution will lead to higher aerodynamic efficiency.

Numerical Investigation of Surface Curvature Effects on Aerofoil Aerodynamic Performance

The prescribed surface curvature distribution blade design (CIRCLE) method optimises aerofoils and blades by controlling curvature continuity and slope of curvature distribution along their surfaces. The symmetrical NACA0012 exhibits a surface curvature discontinuity at the leading edge point, and the non-symmetrical E387 exhibits slope-of-curvature discontinuities in the surface. The CIRCLE method is applied to both aerofoils to remove both surface curvature and slope-of-curvature discontinuities. Computational fluid dynamics analyses are used to investigate the curvature effects on aerodynamic performance of the original and modified aerofoils. These results are compared with experimental data obtained from tests on the original aerofoil geometry. The computed aerodynamic advantages of the modified aerofoil are analysed in different operating conditions. The leading edge singularity of NACA0012 is removed and it is shown that the surface curvature discontinuity affects the aerodynamic performance near the stalling angle of attack. The discontinuous slope-of-curvature distribution of E387 influences the size of the laminar separation bubble at lower Reynolds numbers, and it affects the inherent profile of the aerofoil at higher Reynolds numbers. It is concluded that the surface curvature distribution of aerofoils has a significant effect on aerofoil aerodynamic performance, which can be improved by redesigning the surface curvature distribution of the original aerofoil geometry.

Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanisms

Receptivity studies using direct numerical simulation require computations of equilibrium flow and its response to deterministic excitation. Equivalent flow problem, without heat interaction for zero-pressure gradient boundary layer, has been studied with respect to wall-excitation by a finite difference high accuracy method based on the solution of Navier-Stokes equation in Sengupta and Bhaumik [Phys. Rev. Lett.107, 154501 (2011)] and Sengupta et al. [Phys. Rev. E85, 026308 (2012)]. One of the key features of this study has been that the same methodology is used for computing the equilibrium flow and the disturbance field. Computation of equilibrium flow was performed by solving Navier-Stokes equation to include the leading edge of the plate, so that the effects of leading edge singularity and the growth of the boundary layer is included in the nonlinear framework. When the same methodology is attempted for mixed convection flows past horizontal plate (with Boussinesq approximation to model heat transfer effects) some of the equilibrium flow features could not be explained with linear viscous instability theory results. For horizontal hot flat plate with adiabatic wall conditions, the equilibrium flow could be computed and its receptivity could be correlated with linear spatial theory for lower buoyancy parameter. Here, we focus on receptivity of mixed convection flows to wall excitation for the following cases which do not allow computing the equilibrium flows from the solution of Navier-Stokes equation: (i) Aadiabatic horizontal flat plate cooled significantly at the leading edge only and (ii) strongly heated isothermal wedge flow for a wedge angle of 60°. The cold plate case is particularly interesting as the linear spatial theory indicates enhanced stabilization for higher magnitude of the buoyancy parameter. Results presented for the cold plate case indicates disturbance growth outside the shear layer. This prompted us to re-investigate various mechanisms of instability present for mixed convection flows. There is no definitive study for inviscid mechanism in this respect and the present effort not only fills this void, but also explains the relative roles of viscous and inviscid mechanisms for strong heat transfer effects, within the Boussinesq approximation.

Optimal streaks in a Falkner–Skan boundary layer

This paper deals with the optimal streaky perturbations (which maximize the perturbed energy growth) in a wedge flow boundary layer, which involves a favorable streamwise pressure gradient. These three-dimensional perturbations are governed by a system of linearized boundary layer equations around the Falkner–Skan base flow. Based on an asymptotic analysis of this system near the free-stream and the leading edge singularity, we show that for acute wedge semi-angle, all solutions converge after a streamwise transient to a single streamwise-growing solution of the linearized equations, whose initial condition near the leading edge is given by an eigenvalue problem first formulated in this context by Tumin [Phys. Fluids 13, 1521 (2001)]. Such a solution may be regarded as a streamwise evolving most unstable streaky mode, in analogy with the usual eigenmodes in strictly parallel flows, and shows an approximate self-similarity, which was partially known and is completed in this paper. An important consequence of this result is that the optimization procedure based on the adjoint equations heretofore used to define optimal streaks is not necessary. Instead, a simple low-dimensional optimization process is proposed and used to obtain optimal streaks. Comparison with previous results by Tumin and Ashpis [AIAA J. 41, 2297 (2003)] shows an excellent agreement. The unstable streaky mode exhibits transient growth if the wedge semi-angle is smaller than a critical value that is slightly larger than π/6, and decays otherwise. Thus, the cases of right and obtuse wedge semi-angles exhibit less practical interest, but they show a qualitatively different behavior, which is briefly described in the supplementary material.

Non-Newtonian Natural Convection Along a Vertical Flat Plate With Uniform Surface Temperature

Natural-convection boundary-layer flow of a non-Newtonian fluid along a heated semi-infinite vertical flat plate with uniform surface temperature has been investigated using a four-parameter modified power-law viscosity model. In this model, there are no physically unrealistic limits of zero or infinite viscosity that are encountered in the boundary-layer formulation for two-parameter Ostwald–de Waele power-law fluids. The leading-edge singularity is removed using a coordinate transformation. The boundary-layer equations are solved by an implicit finite-difference marching technique. Numerical results are presented for the case of a shear-thinning fluid. The results indicate that a similarity solution exists locally in a region near the leading edge of the plate, where the shear rate is not large enough to induce non-Newtonian effects; this similarity solution is identical to the similarity solution for a Newtonian fluid. The size of this region depends on the Prandtl number. Downstream of this region, the solution of the boundary-layer equations is nonsimilar. As the shear rate increases beyond a threshold value, the viscosity of the shear-thinning fluid is reduced. This leads to a decrease in the wall shear stress compared with that for a Newtonian fluid. The reduction in the viscosity accelerates the fluid in the region close to the wall, resulting in an increase in the local heat transfer rate compared with the case of a Newtonian fluid.

Unsteady loading on an airfoil of arbitrary thickness

The unsteady loading on an airfoil of arbitrary thickness is evaluated by using the generalized form of Blasius theorem and a conformal mapping that maps the airfoil surface onto a circle. For a blade vortex interaction the results show that the time history of the unsteady loading is determined by the passage of the vortex relative to the leading edge singularity in the circle plane. The singularity lies inside the circle and moves to a smaller radius as the thickness is increased, causing the unsteady loading pulse to be smoothed. The effect of angle of attack is to move the stagnation point relative to the leading edge singularity and this significantly increases the unsteady lift if the vortex passes on the suction side of the airfoil. These characteristics are different for a step upwash gust, which is considered as a simplified model of a large scale turbulent gust. It is shown that the time history of the magnitude of the unsteady loading is almost completely unaltered by angle of attack for the step gust, but it's direction of action rotates forward by an angle equal to the angle of attack, extending an earlier result by Howe for a flat plate in a turbulent flow to airfoils of arbitrary thickness. However spectral analysis of the gust shows that the high frequency blade response is reduced as the thickness of the airfoil is increased.

An experimental investigation of unsteady surface pressure on an airfoil in turbulence—Part 1: Effects of mean loading

An experimental investigation into the response of an airfoil in turbulence is undertaken and the results are presented in a two part series of papers. The effects of mean loading on the airfoil response are investigated in this paper (Part 1) with the likely sources discussed in Part 2. Unsteady surface pressure measurements were made on a NACA 0015 immersed in grid turbulence ( λ / c = 13 % ) for angles of attack, α = 0 – 20 ° , with a dense array of pressure transducers. These measurements reveal a reduction of up to 5 dB in pressure spectral level as the angle of attack is increased for reduced frequencies less than 5. This observed mean-loading effect has never before been measured or shown to occur theoretically. Lift spectra computed from pressure measurements show a similar result. Furthermore, the reduction in lift spectral level appears to have an α 2 dependence. Also, for small angles of attack ( α < 8 ° ) Amiet's zero-mean-loading theory may be useful for predicting the airfoil response since the reduction in spectral level is less than 1 dB here. Based on comparisons at α = 0 ° , Amiet's theory predicts with reasonable accuracy (within 4 dB at low frequency) pressure and lift spectral levels. This theory successfully predicts the shape of both pressure and lift spectra and the decrease in pressure spectral level moving away from the airfoil leading edge. Additionally, Reba and Kerschen's theory, which accounts for non-zero-mean loading using Rapid Distortion Theory, predicts large increases in pressure and lift spectral levels not shown to occur in the measurement. The predicted rise in spectral level appears to result from the flat-plate model with leading-edge singularity which does not fully account for the distortion of the inflow.

Quasiparticle breakdown in a quantum spin liquid

Much of modern condensed matter physics is understood in terms of elementary excitations, or quasiparticles--fundamental quanta of energy and momentum. Various strongly interacting atomic systems are successfully treated as a collection of quasiparticles with weak or no interactions. However, there are interesting limitations to this description: in some systems the very existence of quasiparticles cannot be taken for granted. Like unstable elementary particles, quasiparticles cannot survive beyond a threshold where certain decay channels become allowed by conservation laws; their spectrum terminates at this threshold. Such quasiparticle breakdown was first predicted for an exotic state of matter--super-fluid 4He at temperatures close to absolute zero, a quantum Bose liquid where zero-point atomic motion precludes crystallization. Here we show, using neutron scattering, that quasiparticle breakdown can also occur in a quantum magnet and, by implication, in other systems with Bose quasiparticles. We have measured spin excitations in a two-dimensional quantum magnet, piperazinium hexachlorodicuprate (PHCC), in which spin-1/2 copper ions form a non-magnetic quantum spin liquid, and find remarkable similarities with excitations in superfluid 4He. We observe a threshold momentum beyond which the quasiparticle peak merges with the two-quasiparticle continuum. It then acquires a finite energy width and becomes indistinguishable from a leading-edge singularity, so that excited states are no longer quasiparticles but occupy a wide band of energy. Our findings have important ramifications for understanding excitations with gapped spectra in many condensed matter systems, ranging from band insulators to high-transition-temperature superconductors.

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

AbstractA combined analytical–numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier–Stokes equations. We adopt an asymptotic solution for the local flow around the angular points based on the Stokes flow approximation and a numerical solution for the global flow outside the singular regions using a finite‐volume method. The coefficients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes flow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading‐edge singularity of a semi‐infinite plate. The effect of various influencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the flow around the leading edge can be significantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and efficiency. Based on the numerical results for the leading‐edge singularity, the validity of various classical approximate models for the flow, such as the Stokes approximation, the inviscid flow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading‐edge problem, it is generally applicable to all kinds of angular singularities and all kinds of finite‐discretization methods. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Noise generation by high-frequency gusts interacting with an airfoil in transonic flow

The method of matched asymptotic expansions is used to describe the sound generated by the interaction between a short-wavelength gust (reduced frequency k, with k [Gt ] 1) and an airfoil with small but non-zero thickness, camber and angle of attack (which are all assumed to be of typical size O(δ), with δ [Lt ] 1) in transonic flow. The mean-flow Mach number is taken to differ from unity by O(δ2/3), so that the steady flow past the airfoil is determined using the transonic small-disturbance equation. The unsteady analysis is based on a linearization of the Euler equations about the mean flow. High-frequency incident vortical and entropic disturbances are considered, and analogous to the subsonic counterpart of this problem, asymptotic regions around the airfoil highlight the mechanisms that produce sound. Notably, the inner region round the leading edge is of size O(k−1), and describes the interaction between the mean-flow gradients and the incident gust and the resulting acoustic waves. We consider the preferred limit in which kδ2/3 = O(1), and calculate the first two terms in the phase of the far-field radiation, while for the directivity we determine the first term (δ = 0), together with all higher-order terms which are at most O(δ2/3) smaller – in fact, this involves no fewer than ten terms, due to the slowly-decaying form of the power series expansion of the steady flow about the leading edge. Particular to transonic flow is the locally subsonic or supersonic region that accounts for the transition between the acoustic field downstream of a source and the possible acoustic field upstream of the source. In the outer region the sound propagation has a geometric-acoustics form and the primary influence of the mean-flow distortion appears in our preferred limit as an O(1) phase term, which is particularly significant in view of the complicated interference between leading- and trailing-edge fields. It is argued that weak mean- flow shocks have an influence on the sound generation that is small relative to the effects of the leading-edge singularity.

Transonic Flow Around Optimum Critical Airfoils

Optimum critical nonlifting airfoils that have the highest free stream Mach number $M_\infty $ for a given thickness ratio $\delta $ have recently been presented by Schwendeman, Kropinski, and Cole [Z. Angew. Math. Phys., 44 (1993), pp. 556–571]. The asymptotic shape of these airfoils as $M_\infty \to 1$ and $\delta \to 0$ consists of a sonic arc with a $( x^{2/ 5} )$ nose. In the present paper the transonic potential flow around these asymptotic shapes is analyzed. Asymptotic expansions of the velocity potential function are constructed in terms of $\delta $ in an outer region around the airfoil and in an inner region near the nose. The outer expansion consists of the transonic small disturbance theory, where a leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. The matching between the expansions results in a boundary value problem in the inner region for the solution of a uniform sonic flow about a two-dimensional $( x^{2/ 5} )$ surface. The numerical solution of the inner flow results in the pressure distribution on the nose. Finally, a uniformly valid pressure distribution over the entire airfoil surface is derived.

Subsonic flow around the leading edge of a thin aerofoil with a parabolic nose

Subsonic potential flow around the leading edge of a thin two-dimensional general aerofoil with a parabolic nose at small angles of attack is analysed. Asymptotic expansions of the velocity potential function are constructed at a fixed Mach number in terms of the thickness ratio of the aerofoil in an outer region around the aerofoil and in an inner region near the nose. These expansions are matched asymptotically. The outer expansion consists of linearized aerofoil theory and its second-order problem, where the leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. A boundary-value problem is formulated in the inner region for the solution of a compressible uniform flow with the same Mach number, past a semi-infinite two-dimensional parabola. The analysis shows that the drag of any general aerofoil with a round nose is zero for every subsonic Mach number below the critical Mach number, where the drag component that is calculated by the integration of the outer pressure distribution over the aerofoil is cancelled exactly by the drag of the parabolic nose. The numerical solution of the flow field in the inner region results in the pressure distribution on the parabolic nose. The stagnation point tends to shift toward the leading edge due to the compressibility effects as the Mach number is increased at a fixed angle of attack and camber of the aerofoil. Good agreement is found in the leading edge region between the present asymptotic solution and numerical solutions of the Euler equations. Using the outer linearized solution and the nose solution a uniformly valid pressure distribution on the entire aerofoil surface in a subsonic flow is derived.

Transonic flow around the leading edge of a thin airfoil with a parabolic nose

Transonic potential flow around the leading edge of a thin two-dimensional general airfoil with a parabolic nose is analysed. Asymptotic expansions of the velocity potential function are constructed at a fixed transonic similarity parameter (K) in terms of the thickness ratio of the airfoil in an outer region around the airfoil and in an inner region near the nose. These expansions are matched asymptotically. The outer expansion consists of the transonic small-disturbance theory and it second-order problem, where the leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. Analytical expressions are given for the first terms of the inner and outer asymptotic expansions. A boundary value problem is formulated in the inner region for the solution of a uniform sonic flow about an infinite two-dimensional parabola at zero angle of attack, with a symmetric far-field approximation, and with no circulation around it. The numerical solution of the flow in the inner region results in the symmetric pressure distribution on the parabolic nose. Using the outer small-disturbance solution and the nose solution a uniformly valid pressure distribution on the entire airfoil surface can be derived. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number of the oncoming flow and shape and small angle of attack of the airfoil. The pressure distribution on the upper and lower surfaces of the airfoil is symmetric near the edge point, and asymmetric deviations increase and become significant only when the distance from the leading edge of the airfoil increases beyond the inner region. Good agreement is found in the leading-edge region between the present solution and numerical solutions of the full potential-flow equations and the Euler equations.

Potential flow solution for a yawed surface-piercing plate

This paper presents the results of an analytical investigation of the steady translation of a vertical surface-piercing plate at a small angle of attack. This problem is the antisymmetric equivalent of the symmetric thin-ship problem solved by Michell. The linearized boundary-value problem is transformed into an integral equation of the first kind by the method of Green functions. The Kelvin–Havelock Green function is used to satisfy the linearized free-surface boundary condition and radiation condition. A pressure Kutta condition is imposed at the trailing edge. Effective algorithms are developed to evaluate the hypersingular kernel without recourse to numerical integration. The resulting integral equation is solved by a collocation method with a refined scheme of discretization. After establishing the convergence of the present algorithm, computations are carried out for a surface-piercing rectangular plate of aspect ratio 0.5. The integrated lateral-force and yaw-moment coefficients show good agreement with experimental data. Other parameters of the flow such as pressure distributions, drag, strength of leading-edge singularity and free-surface profiles on the plate are also presented. The incompatibility between the pressure Kutta condition and the linearized free-surface condition does not affect the global solution.

Incompressible flow around porous two-dimensional thin sail wings.

A linearized analysis has been made of a two-dimensional aerodynamic characteristics of infinitely thin, perfectly flexible, possibly porous airfoils. Jacobi polynominals have been used to express pressure distributions and airfoil shapes, considering well over the leading edge singularity. It is found that airfoil shapes become stable with increasing porosity. Furthermore, the effects of porosity on lift and center of pressure are determined as functions of angle of attack and excess length of material over the chord length.

Lifting-Surface Theory for Skewed and Swept Subsonic Wings

A new method is developed for solving the lifting-surface equation for thin subsonic wings which can also be interpreted as a vortex lattice method. The downwash equation is transformed into double integrals involving Cauchy-type singularities in the chordwise and spanwise directions. A technique developed by Lan for airfoil theory is used to reduce both integrals to a double summation. This method properly accounts for the leadingedge singularity, Cauchy singularity, and Kutta condition. The solutions generally compared well with other lifting-surface theories, but with much smaller computational times, and the method was found to be more accurate and converge faster than conventional vortex lattice methods. b c c

Errata: Remarks on Thin Airfoil Theory

Fig. 3 Angles of attack for vanishing leading-edge singularity and for zero lift. References E., High-Speed Aerodynamics, Pergamon Press, London/New York/Paris, 1956, pp. 619-624. 2 Holla, V. S. and Krishnaswamy, T. N., Conically Cambered Triangular Wings, Journal of the Aeronautical Society of India, Vol. 22, May 1970, pp. 94-107. Bera, R. K., Slender Delta Wings with Conical Camber, Journal of Aircraft, Vol. 11, April 1974, pp. 245-247. Evvard, J. C., Effects of Yawing Thin Pointed Wings at Supersonic Speeds, NACA TN 1429, 1947. Hancock, G. J., Note on the Extension of Evvard's Method to Wings with Subsonic Leading Edges Moving at Supersonic Speeds, Aeronautical Quarterly, Vol. VIII, 1957, pp. 87-102. Wagner, B., Untersuchungen uber konische Uberschalltragflachen mit Unterschallvorderkanten, Institut fur Flugtechnik, T. H. Darmstadt, Rept. No. 8/75, 1975.