Inviscid Flow Theory Research Articles

Investigation of models to estimate flight performance of gliding birds from wakes.

Mathematical models based on inviscid flow theory are effective at predicting the aerodynamic forces on large-scale aircraft. Avian flight, however, is characterized by smaller sizes, slower speeds, and increased influence of viscous effects associated with lower Reynolds numbers. Therefore, inviscid mathematical models of avian flight should be used with caution. The assumptions used in such models, such as thin wings and streamlined bodies, may be violated by birds, potentially introducing additional error. To investigate the applicability of the existing models to calculate the aerodynamic performance of bird flight, we compared predictions using simulated wakes with those calculated directly from forces on the bird surface, both derived from computational fluid dynamics of a high-fidelity barn owl geometry in free gliding flight. Two lift models and two drag models are assessed. We show that the generalized Kutta-Joukowski model, corrected by the streamwise velocity, can predict not only the lift but also span loading well. Drag was predicted best by a drag model based on the conservation of fluid momentum in a control volume. Finally, we estimated force production for three raptor species across nine gliding flights by applying the best lift model to wake flow fields measured with particle tracking velocimetry.

The starting vortices generated by bodies with sharp and straight edges in a viscous fluid

A two-dimensional body that moves suddenly in a viscous fluid can instantly generate vortices at its sharp edges. Recent work using inviscid flow theory, based on the Birkhoff–Rott equation and the Kutta condition, predicts that the ‘starting vortices’ generated by the sharp and straight edges of a body – i.e. the vortices formed immediately after motion begins – can be one of three distinct self-similar types. We explore the existence of these starting vortices for a flat plate and two symmetric Joukowski aerofoils immersed in a viscous fluid, using high-fidelity direct numerical simulations (DNS) of the Navier–Stokes equations. A lattice Green's function method is employed and simulations are performed for chord Reynolds numbers ranging from 5040 to 45 255. Vortices generated at the leading and trailing edges of the flat plate show agreement with the derived inviscid theory, for which a detailed assessment is reported. Agreement is also observed for the two symmetric Joukowski aerofoils, demonstrating the utility of the inviscid theory for arbitrary bodies. While this inviscid theory predicts an abrupt transition between the starting-vortex types, DNS shows a smooth transition. This behaviour occurs for all Reynolds numbers and is related to finite-time effects – there is a maximal time for which the (self-similar) starting vortices exist. We confirm the inviscid prediction that the leading-edge starting vortex of a flat plate can be suppressed dynamically. This has implications for the performance of low-speed aircraft such as model aeroplanes, micro air vehicles and unmanned air vehicles.

Experimental quantification of unsteady leading-edge flow separation

We propose here a method to experimentally quantify unsteady leading-edge flow separation on aerofoils with finite thickness. The methodology relies on the computation of a leading-edge suction parameter based on measured values of the partial circulation around the leading edge and the stagnation point location. We validate the computation of the leading-edge suction parameter for both numerical and experimental data under steady and unsteady flow conditions. The leading-order approximation of the definition of the leading-edge suction parameter is proven to be sufficiently accurate for the application to thin aerofoils such as the NACA0009 without a priori knowledge of the stagnation point location. The higher-order terms including the stagnation point location are required to reliably compute the leading-edge suction parameter on thicker aerofoils such as the NACA0015. The computation of the leading-edge suction parameter from inviscid flow theory does not assume the Kutta condition to be valid at the trailing edge which allows us to compute its value for separated flows. The relation between the leading-edge suction parameter and the evolution of the shear layer height is studied in two different unsteady flow conditions, a fixed aerofoil in a fluctuating free-stream velocity and a pitching aerofoil in a steady free stream. We demonstrate here that the instantaneous value of the leading-edge suction parameter based on the partial circulation around the leading edge is unambiguously defined for a given flow field and can serve as a directly quantitative measure of the degree of unsteady flow separation at the leading edge.

New insights from inviscid modelling of starting flow separation with roll-up

Abstract

Starting Flow by Pitch Acceleration of Flat-Plate Foil

This paper presents the investigation of the starting flow in association with pitch acceleration using transient laminar-flow computation and an unsteady inviscid-flow theory. A thin flat-plate foil pitched at the leading edge is modeled in a two-dimensional incompressible fluid at chord Reynolds numbers of the order of . Several acceleration amplitudes () and acceleration durations () are maneuvered using a modified canonical kinematic equation at reduced pitch rates from 0.196 to 0.763. Comparisons between computational and theoretical data reveal that the formation of the starting vortex is determined by the acceleration amplitude scaled by the square of the convective time. The corresponding starting force coefficient that scales with the dimensionless acceleration amplitude is proportional to the acceleration angle , which decreases with increasing reduced pitch rate. The starting vortex, once formed at the trailing edge, is shed at the end of the acceleration period. The quasi-steady theory that includes only pitch motion as the strength of the dipolar sheet for apparent-mass flow can also yield an acceptable approximation to capture the apparent-mass force coefficient spike in acceleration.

The origin of lift – understanding lift

In this note, an attempt at giving as simple as possible an explanation for the lift of an airfoil is proposed based on an intuitive use of Newton's law on one hand and the application of the well-established inviscid flow theory of thin airfoils on the other. A thin parabolic cambered plate is chosen as profile at zero incidence because the flow it produces minimises viscous effects and the particle paths along the plate, above and below it, have the same length, thus proving the 'equal transit' explanation to be wrong.

Nonlinear dynamics of slender inverted flags in uniform steady flows

A nonlinear fluid-elastic continuum model for the dynamics of a slender cantilevered plate subjected to axial flow directed from the free end to the clamped end, also known as the inverted flag problem, is proposed. The extension of elongated body theory to large-amplitude rotations of the plate mid-plane along with Bollay's nonlinear wing theory is employed in order to express the fluid-related forces acting on the plate, while retaining all time-dependent terms in both modelling and numerical simulations; the unsteady fluid forces due to vortex shedding are not included. Euler-Bernoulli beam theory with exact kinematics and inextensibility is employed to derive the nonlinear partial integro-differential equation governing the dynamics of the plate. Discretization in space is carried out via a conventional Galerkin scheme using the linear mode-shapes of a cantilevered beam in vacuum. The pseudo-arclength continuation technique is adapted to construct bifurcation diagrams in terms of the flow velocity, in order to gain insight into the stability and post-critical behaviour of the system. Integration in time is conducted using Gear's backward differentiation formula. The sensitivity of the nonlinear response of the system to different parameters such as the aspect ratio, mass ratio, initial inclination of the flag, and viscous drag coefficient is investigated through extensive numerical simulations. It is shown that for flags of small aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of sufficiently large aspect ratio, however, the first instability encountered is a supercritical Hopf bifurcation giving rise to flapping motion around the undeflected static equilibrium; increasing the flow velocity further, the flag then displays flapping motions around deflected static equilibria, which later lead to fully-deflected static states at even higher flow velocities. The results exposed in this study help understand the dynamics of the inverted-flag problem in the limit of inviscid flow theory.

Small amplitude oscillations of a shape-morphing plate immersed in a viscous fluid near a solid wall

In this paper, we investigate small amplitude oscillations of a thin shape-morphing plate submerged in a quiescent, Newtonian, incompressible, viscous fluid in the vicinity of a solid wall. The plate undergoes prescribed, time-varying shape-morphing deformation in phase with the transverse oscillation, consisting of periodic curvature to an arc of a circle. As the presence of the solid wall in close proximity of the oscillating plate is known to significantly increase hydrodynamic damping on the structure due to squeeze film effects, shape-morphing is considered as a potential strategy to modulate and control hydrodynamic forces, as well as power dissipation. Via a rigorous formulation based on unsteady Stokes equations and the boundary element method, we show that the shape-morphing strategy is effective in reducing hydrodynamic forces on the structure as well as hydrodynamic power dissipation due to viscous effects. Additionally, such a reduction is enhanced as the gap between the oscillating plate and the solid wall is reduced. We present a comprehensive parametric study which investigates the effect of oscillation frequency, gap size, and imposed curvature on the hydrodynamic performance of the system and support our findings with asymptotics from inviscid flow theory and lubrication theory. These findings are of direct interest and applicability in several cantilever-based sensing and actuation systems, such as atomic force microscopy and microelectromechanical systems in liquids.

Boussinesq global modes and stability sensitivity, with applications to stratified wakes

For the Boussinesq equations, we present a theory of linear stability sensitivity to base flow density and velocity modifications. Given a steady-state flow with small density variations, the sensitivity of the stability eigenvalues is computed from the direct and adjoint global modes of the linearised Boussinesq equations, similarly to Marquetet al.(J. Fluid Mech., vol. 615, 2008, pp. 221–252). Combinations of the density and velocity components of these modes reveal multiple production and transport mechanisms that contribute to the eigenvalue sensitivity. We present an application of the sensitivity theory to a stably linearly density-stratified flow around a thin plate at a$90^{\circ }$angle of attack, a Reynolds number of 30 and Froude numbers of 1, 8 and$\infty$. The global mode analysis reveals lightly damped undulations pervading through the entire domain, which are predicted by both inviscid uniform base flow theory and Orr–Sommerfeld theory. The sensitivity to base flow velocity modifications is primarily concentrated just downstream of the bluff body. On the other hand, the sensitivity to base flow density modifications is concentrated in regions both immediately upstream and immediately downstream of the plate. Both sensitivities have a greater upstream presence for lower Froude numbers.

Cylindrical vortex wake model: skewed cylinder, application to yawed or tilted rotors

AbstractA vortex system consisting of a bound vortex disk, a root vortex and a vortex cylinder is presented and applied for skewed wake situations. Both the longitudinal and tangential components of vorticity of the cylinder are considered. A subset of this system leads to a model, which is commonly used in Blade Element Momentum method codes for yawed conditions. Here, all the components of the full vortex system are analyzed in view of extending Blade Element Momentum models. The main assumptions of the current study are a constant uniform circulation, an infinite number of blades, an un‐expanding wake shape and a finite tip‐speed ratio. The investigation remains within the context of inviscid potential flow theory. The model is derived for horizontal‐axis rotors in general, but results are presented for wind‐turbine applications. For each vortex element, the velocity components in all directions are computed analytically or semi‐analytically for the entire domain. Simplified engineering models are provided to ease the evaluation of velocities in the rotor plane. The predominant velocity components are assessed. Copyright © 2015 John Wiley & Sons, Ltd.

Study on Kelvin–Helmholtz Instability With Heat and Mass Transfer

The effect of heat and mass transfer on the Kelvin–Helmholtz instability between liquid and vapor phases of a fluid has been studied using three different theories: a purely irrotational theory based on the dissipation method, a hybrid irrotational-rotational theory, and an inviscid potential flow theory. These new results are compared with previous results from viscous irrotational theory. The stability criterion is given in terms of the critical value of relative velocity. The system is shown to be unstable when the relative velocity is greater than the critical value of relative velocity; otherwise, it is stable. It is observed that heat and mass transfer has a destabilizing effect on the stability of the system while vapor fraction has a stabilizing effect.

Implementation of a Separated Flow Capability in TRANAIR

It has long been thought by many that separated flows violated the assumptions of integral boundary-layer methods coupled to inviscid flow solvers. In this paper, it is shown how to extend the transpiration theory of the equivalent inviscid flow to essentially arbitrary large separation regions. It is shown through computational examples that integral boundary-layer methods can indeed give useful results up to stall in some situations. The limitations of these methods are also discussed and compared to Reynolds-averaged Navier–Stokes methods.

Wave-power absorption from a finite array of oscillating wave surge converters

Semi-analytical and fully numerical modelling is developed in the framework of the inviscid potential flow theory to investigate the dynamics of a wave farm made by flap-type wave energy converters in the nearshore. The hydrodynamic parameters and the efficiency of the system in typical layouts are calculated with both models. Good agreement is shown between the two approaches. Parametric analysis undertaken with the semi-analytical model allows to identify a near-resonant phenomenon which is responsible for increasing the absorbed power by the single elements of the array. Such result could be used as a preliminary design criterion. The numerical model is then applied to analyse a configuration of practical engineering interest, i.e. an array of two staggered converters. The dynamics arising in this more complex system is explained, showing that non-symmetric layouts can be less effective.

Analysis of Interference Factors of Air Turbine Cascades

This paper reports viscous compressible aerodynamics analysis of two-dimensional cascades (arrays of aerofoils). The study elucidates the effect of cascade parameters such as solidity, σ and angle of attack, α, on the interference between blades. Understanding the interference behaviour is important in the performance analysis of air turbines such as Wells turbine commonly used in oscillating water column (OWC) - wave energy converter (WEC) system. The present analysis shows that isolated two-dimensional lift coefficient modified by interference factors, ko, based on inviscid flow theory is acceptable for low solidity cascades, i.e. σ ≤ 0.5, and that stall occurs at lower angle of attack in cascade compared to isolated aerofoil. When σ > 0.5, ko is shown to depend on both α and σ especially in the post-stall flow. The result suggests that interference factors that is applicable for wide range of flow conditions as in Wells turbines should also include the α effect in the modifier.

Stern End Bulb for Energy Enhancement and Speed Improvement

Unlike the bow bulb, the stern end bulb (SEB) has been used on just a few ships to improve performance. In one of these rare, full-scale applications, a maximum resistance reduction in the 5% to 7% range is claimed. A few applications of SEBs are shown along with some model test data for a Naval Auxiliary ship. The rationale for SEB is discussed along with the hydrodynamic mechanism associated with a SEB. In addition to wave-making reduction, the SEB can reduce eddy-making and possibly improve course-keeping. The results of several fluid flow computations with initial SEB designs are shown for two ship classes: the T-AKE LEWIS and CLARK dry cargo ship and the DDG 51 ARLEIGH BURKE destroyer. The calculations use the Ship Wave Inviscid Flow Theory potential flow computer code and the FreeRans viscous flow free surface computer code. Several SEBs were designed and investigated analytically for the T-AKE class ships, and the best of these is predicted to reduce resistance by 4.5% at 20 knots. In addition, several initial SEB/Stern Flap configurations were designed for the DDG 51 Class Flight IIa destroyers and five configurations, some with just an SEB added to the hull and others with a combined SEB-Stern Flap configuration were model-tested. The examination of these initial efforts led to the design of several new-style combined SEB-Stern flap configurations, the best of which is predicted to save at least 745 Bbls of fuel per ship per year.

A breakup model for transient Diesel fuel sprays

In this paper a breakup model for analysing the evolution of transient fuel sprays characterised by a coherent liquid core emerging from the injection nozzle, throughout the injection process, is proposed. The coherent liquid core is modelled as a liquid jet and a breakup model is formulated. The spray breakup is described using a composite model that separately addresses the disintegration of the liquid core into droplets and their further aerodynamic breakup. The jet breakup model uses the results of hydrodynamic stability theory to define the breakup length of the jet, and downstream of this point, the spray breakup process is modelled for droplets only. The composite breakup model is incorporated into the KIVA II Computational Fluid Dynamics (CFD) code and its results are compared with existing breakup models, including the classic WAVE model and a previously developed composite WAVE model (modified WAVE model) and in-house experimental observations of transient Diesel fuel sprays.The hydrodynamic stability results used in both the jet breakup model and the WAVE droplet breakup model are also investigated. A new velocity profile is considered for these models which consists of a jet with a linear shear layer in the gas phase surrounding the liquid core to model the effect of a viscous gas on the breakup process. This velocity profile changes the driving instability mechanism of the jet from a surface tension driven instability for the currently used plug flow jet with no shear layers, to an instability driven by the thickness of the shear layer. In particular, it is shown that appreciation of the shear layer instability mechanism in the composite model allows larger droplets to be predicted at jet breakup, and gives droplet sizes which are more consistent with the experimental observations. The inclusion of the shear layer into the jet velocity profile is supported by previous experimental studies, and further extends the inviscid flow theory used in the formulation of the classic WAVE breakup model.

Viscous Potential Flow Analysis of Rayleigh-Taylor Instability of Cylindrical Interface

The present paper deals with the study of Rayleigh-Taylor instability at the cylindrical interface using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier-Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. A dispersion relation is derived and stability is discussed in terms of various parameters such as Ohnesorge number, density ratio etc. A condition for neutral stability is obtained and it is given in terms of critical value of the wave number. It is observed that the Ohnesorge number has destabilizing effect while inner fluid fraction has stabilizing effect on the stability of the system.

Viscous potential flow analysis of Kelvin-Helmholtz instability of cylindrical interface

A linear analysis of Kelvin-Helmholtz instability of cylindrical interface is carried out using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier-Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. Both asymmetric and axisymmetric disturbances are considered. A dispersion relation has been obtained and stability criteria are given in the terms of the relative velocity. A comparison between inviscid potential flow and viscous potential flow has been made. It has been observed that Reynolds number and inner fluid fraction both have destabilizing effect on the stability of the system.

Spectral solution methods for free-surface flow: the Rayleigh--Taylor problem

This article describes two spectral methods for solving problems in interfacial fluid mechanics. These are illustrated here for the Rayleigh--Taylor instability, in which a layer of heavy fluid lies above a light fluid. Disturbances to the interface between them are unstable, and grow with time. The first spectral method solves time dependent, free-surface problems in inviscid flow theory. It is capable of following the development of the interface almost to the time at which the curvature at the interface becomes infinite, and the inviscid model then ceases to be valid. A second spectral method is presented, and solves the viscous Boussinesq equations. It shows that the curvature singularity in the inviscid model is associated with regions of large vorticity at precisely these same points on the interface. Consequently, the interface rolls over at these points, forming delicate overhanging structures, leading ultimately to mixing of the two fluid layers. References G. Baker, R. E. Caflisch and M. Siegel, Singularity formation during Rayleigh--Taylor instability. J. Fluid Mech., 252 (1993) 51--78. doi:10.1017/S0022112093003660 G. R. Baker and L. D. Pham, A comparison of blob-methods for vortex sheet roll-up. J. Fluid Mech., 547 (2006) 297--316. doi:10.1017/S0022112005007305 G. K. Batchelor, An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1977). D. E. Farrow and G. C. Hocking, A numerical model for withdrawal from a two-layer fluid. J. Fluid Mech., 549 (2006) 141--157. doi:10.1017/S0022112005007561 L. K. Forbes, M. J. Chen and C. E. Trenham, Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow. J. Comput. Phys., 221 (2007) 269--287. doi:10.1016/j.jcp.2006.06.010 L. K. Forbes and G. C. Hocking, Unsteady draining flows from a rectangular tank. Phys. Fluids, 19, 082104 (2007) 14 pages. doi:10.1063/1.2759891 R. Krasny, Desingularization of periodic vortex sheet roll-up. J. Comput. Phys., 65 (1986) 292--313. doi:10.1016/0021-9991(86)90210-X D. W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. Roy. Soc. London A, 365 (1979) 105--119. doi:10.1098/rspa.1979.0009 P. Ramaprabhu, G. Dimonte, Y.-N. Young, A. C. Calder and B. Fryxell, Limits of the potential flow approach to the single-mode Rayleigh--Taylor problem. Phys. Rev. E, 74, 066308 (2006) 10 pages. doi:10.1103/PhysRevE.74.066308 Lord Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. London Math. Soc., 14 (1883) 170--177. G. I. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, I. Proc. Roy. Soc. London Ser. A, 201 (1950) 192--196. doi:10.1098/rspa.1950.0052 G. Tryggvason and S. O. Unverdi, Computations of three-dimensional Rayleigh--Taylor instability. Phys. Fluids A, 2 (1990) 656--659. doi:10.1063/1.857717

Finite size vortices in the wake of an accelerated bent lamina

A theory of an unsteady incompressible inviscid flow past an accelerated lamina, moving across the fluid, is extended to address a bent (rather than flat) lamina and finite size (rather than point) vortices in its wake. It is shown that when the velocity of the lamina varies with time t as t m , the distance of the vortices from the lamina and the size of their cores grow, initially, as t ( 2 m + 2 ) / 3 and t ( 2 m + 2 ) / 3 exp ( - Ct - ( 8 m + 2 ) / 3 ) , respectively, where C is a strictly positive constant. The growth rate of the core dimensions with time seems to provide a plausible energy argument for the experimentally observed wake asymmetry that culminates the initial symmetric vortex arrangement.