Uniform Incompressible Flow Research Articles

Aero-/hydro-elastic stability of flexible panels: Prediction and control using localised spring support

We study the effect of adding localised stiffness, via a spring support, on the stability of flexible panels subjected to axial uniform incompressible flow. Applications are considered that range from the hydro-elasticity of hull panels of high-speed ships to the aero-elasticity of glass panels in the curtain walls of high-rise buildings in very strong winds. A two-dimensional linear analysis is conducted using a hybrid of theoretical and computational methods that calculates the system eigen-states but can also be used to capture the transient behaviour that precedes these. We show that localised stiffening is a very effective means to increase the divergence-onset flow speed in both hydro- and aero-elastic applications. It is most effective when located at the mid-chord of the panel and there exists an optimum value of added stiffness beyond which further increases to the divergence-onset flow speed do not occur. For aero-elastic applications, localised stiffening can be used to replace the more destructive flutter instability that follows divergence at higher flow speeds by an extended range of divergence. The difference in eigen-solution morphology between aero- and hydro-elastic applications is highlighted, showing that for the former coalescence of two non-oscillatory divergence modes is the mechanism for flutter onset. This variation in solution morphology is mapped out in terms of a non-dimensional mass ratio. Finally, we present a short discussion of the applicability of the stabilisation strategy in a full three-dimensional system.

On the aeroelastic stability and bifurcation structure of subsonic nonlinear thin panels subjected to external excitation

Dynamic behavior of panels exposed to subsonic flow subjected to external excitation is investigated in this paper. The von Karman’s large deflection equations of motion for a flexible panel and Kelvin’s model of structural damping is considered to derive the governing equation. The panel under study is two-dimensional and simply supported. A Galerkin-type solution is introduced to derive the unsteady aerodynamic pressure from the linearized potential equation of uniform incompressible flow. The governing partial differential equation is transformed to a series of ordinary differential equations by using Galerkin method. The aeroelastic stability of the linear panel system is presented in a qualitative analysis and numerical study. The fourth-order Runge-Kutta numerical algorithm is used to conduct the numerical simulations to investigate the bifurcation structure of the nonlinear panel system and the distributions of chaotic regions are shown in the different parameter spaces. The results shows that the panel loses its stability by divergence not flutter in subsonic flow; the number of the fixed points and their stabilities change after the dynamic pressure exceeds the critical value; the chaotic regions and periodic regions appear alternately in parameter spaces; the single period motion trajectories change rhythmically in different periodic regions; the route from periodic motion to chaos is via doubling-period bifurcation.

On the direct determination of the eigenmodes of finite flow–structure systems

A new method for directly determining the eigenmodes of finite flow–structure systems is presented using the classical problem of the interaction of a uniform incompressible flow with a flexible panel, held at both ends, as an exemplar. The method is a hybrid of theoretical analysis and computational modelling. This method is contrasted with Galerkin and travelling-wave methods, which are most often used to study the hydroelasticity of such systems. The new method does not require ana prioriapproximation of perturbations via a finite sum of modes. Instead, the coupled equations for the wall–flow system are used to derive a single matrix equation for the system that is a second-order differential equation for the panel-displacement variable. This is achieved in this exemplar by applying a combination of boundary-element and finite-element methods to the discretized system. Standard state-space methods are then used to extract the eigenmodes of the system directly. We present the results for the stability of the case of an unsupported flexible plate, elucidating its divergence and flutter characteristics, and the effect of energy dissipation in the structure. We then present the results for the case of a spring-backed flexible plate, showing that its motion is dominated by travelling waves. Finally, we illustrate the versatility of the approach by extracting the stability diagrams and modes for a panel with spatially varying properties and a panel with non-standard boundary conditions. In doing so, we show how spatial inhomogeneity can modify the energy exchanges between flow and structure, thereby introducing a single-mode flutter instability at pre-divergence flow speeds.

Vortex Induced Vibrations of a Pair of Cylinders at Reynolds Number 1000

Vortex induced vibrations of two equal-sized cylinders in tandem and staggered arrangement placed in uniform incompressible flow is studied. A stabilized finite element formulation is utilized to solve the governing equations. The Reynolds number for these 2D simulations is 1000. The cylinders are separated by 5.5 times the cylinder diameter in the streamwise direction. For the staggered arrangement, the cross-flow spacing between the two cylinders is 0.7 times the cylinder diameter. In this arrangement, the downstream cylinder lies in the wake of the upstream one and therefore experiences an unsteady inflow. The wake looses its temporal periodicity, beyond a few diameters downstream of the front cylinder. The upstream cylinder responds as an isolated single cylinder while the downstream one undergoes disorganized motion. Soft-lock-in is observed in almost all the cases.

FLOW-INDUCED OSCILLATIONS OF TWO CYLINDERS IN TANDEM AND STAGGERED ARRANGEMENTS

A stabilized finite element formulation is employed to study flow-induced oscillations of a pair of equal-sized cylinders in tandem and staggered arrangement placed in uniform incompressible flow. Computations, restricted to 2-D, are carried out for Reynolds numbers 100 for various values of the structural frequency of the oscillator. The cylinders are separated by 5·5 times the cylinder diameter in the streamwise direction. For the staggered arrangement, the cross-flow spacing between the two cylinders is 0·7 times the cylinder diameter. In this arrangement, the downstream cylinder lies in the wake of the upstream one and therefore experiences an unsteady in-flow. Since the spacing between the two cylinders is beyond the critical value for proximity interference, it is expected that the upstream cylinder behaves like an isolated single cylinder, while the downstream one experiences wake-induced flutter. The Re=100 flow leads to a very organized wake and large amplitude motion is observed for the downstream cylinder. The trajectory of the upstream cylinder resembles a figure-of-eight. The downstream cylinder shows a similar behaviour for the tandem arrangement. However, for the staggered arrangement, the trajectory of the rear cylinder resembles a tilted oval. Soft lock-in is observed in almost all the cases.

A Study on Unsteady Aerodynamic Characteristics of Oscillating Cascades with Arbitrary Blade Profiles. 3rd Report. Unstalled Flutter Characteristics of Three-Dimensional Cascades. Theory and Numerical Result.

Aerodynamic characteristics of three-dimensional unstalled cascade oscillating in a uniform incompressible flow are predicted numerically. Unsteady formulation is based on a small perturbation approach with use of potential equations expressed on a moving computational grid. The governing equations are discretized by the finite element method. A parametric study on unsteady aerodynamic characteristics is conducted for the three-dimensional rotor cascade of a low-speed axial-flow compressor. Flutter characteristics of blades oscillating with first three eigenmodes, i.e., first bending mode, second bending mode, and first torsion mode, are shown, and the distribution of flutter excitation along the blade span is described.

A Study on Unsteady Aerodynamic Characteristics of Oscillating Cascades with Arbitrary Blade Profiles. (Two-Dimensional Cascades Oscillating with Blade Section Deformation).

Aerodynamic characteristics of oscillating cascades of blades accompanied by elastic deformation of blade sections in a uniform incompressible flow are predicted numerically. Unsteady formulation is based on a small perturbation approach with the use of potential equations expressed on a moving computational grid. The governing equations are discretized by the finite element method. Modal shapes of blade oscillation are obtained first and a parametric study on unsteady aerodynamic characteristics is conducted for the two-dimensional cascade of the tip section of a test compressor. Furthermore, the oscillation displacement of the blades is divided into four components, i.e., two translational components (normal and tangential to the blade chord), a rotational component, and a section-deformation component, in order to analyze the contribution of each component to unsteady aerodynamic work which causes flutter excitation. It is shown that the effect of the blade section deformation is very significant in determining the unsteady characteristics of oscillating cascades.

A Study on Unsteady Aerodynamic Characteristics of Oscillating Cascades with Arbitrary Blade Profiles. 2nd Report, Two-Dimensional Cascades Oscillatingwith Blade Section Deformation.

Aerodynamic characteristics of oscillating cascades of blades accompanied with elastic deformation of blade sections in a uniform incompressible flow are predicted numerically. Unsteady formulation is based on a small perturbation approach with use of potential equations expressed on a moving computational grid. The governing equations are discretized by the finite element method. Modal shapes of blade oscillation are obtained first and a parametric study on unsteady aerodynamic characteristics is conducted for the two-dimensional cascade of the tip section of a test compressor. Furthermore, the oscillation displacement of the blades is divided into four components, i.e., two translational components (normal and tangential to the blade chord), a rotational component, and a section-deformation component, in order to analyze the contribution of each component to unsteady aerodynamic work which causes flutter excitation. It is shown that the effect of the blade section deformation is very significant in determining the unsteady characteristics of oscillating cascades.

Fluid loading with mean flow. I. Response of an elastic plate localized excitation

The response to localized forcing of a fluid-loaded elastic plate is studied in the case when there is uniform incompressible flow over the plate. Absolute instability of the fluid-plate system is found when the dimensionless mean velocity U exceeds a threshold U c which is found exactly. For U < U c the system is convectively unstable for 0 < ω <ω s ( U ), neutrally stable, with anomalous features, for ω s ( U ) < ω < ω p ( U ) and stable, with conventional features, for ω > ω p (U), ω being the excitation frequency: here asymptotic expressions are found for the frequencies and for the wavenumbers and amplitudes of the waves found upstream and downstream of the excitation. A significant feature is that Re A 0 < 0 throughout 0 < ω < ω p , A 0 being the drive admittance (velocity at the point of application of the force); this means that throughout the convectively unstable and the anomalous neutral frequency ranges, the exciting force must absorb energy. An exact energy equation is derived, and shown to require the introduction of a new fluid-plate interaction flux UnO t , where O is the fluid potential and n the plate deflexion. The energy equation is used to illuminate properties of the convectively unstable and neutral waves, to verify the property Re A 0 < 0 and to trace the waves responsible for this. Anomalous features in the frequency range ω s (U) < ω < ω p ( U ) are investigated further from the viewpoint of the theory of negative energy waves, and it is found that not only can some wave modes in this frequency range have negative energy, but also group velocity in an inward direction (towards the excitation). It is argued that this does not contradict the outward group velocity ‘radiation condition’ of M. J. Lighthill, because that condition refers expressly to circumstances in which the excitation is the sole source of all the wave energy, whereas here the excitation acts also as a scatterer, transferring energy from the mean flow to the wave field.

The stability of cantilever panels in uniform incompressible flow

The stability of large aspect ratio cantilever panels in uniform incompressible flow is studied by employing linearized plate and potential flow theories together with asymptotic expressions for the generalized pressures. It is assumed that if the downstream wake effects on the panel stability are insignificant then the results will be almost independent of where these effects are neglected in the analysis. Accordingly two approximations for the pressure, corresponding to different ways of neglecting the wake effects, are compared, and it is concluded that such effects cannot be ignored when the mass ratio is large. The results also demonstrate that a two-dimensional analysis (infinite aspect ratio) may yield an overestimate of the critical flow velocity for finite width panels.

The stability of partially rigid two-dimensional surfaces in uniform incompressible flow

A theoretical analysis is made of the stability of a partially rigid two-dimensional surface embedded in the uniform flow of an incompressible inviscid fluid. Membranes, simply supported panels and clamped panels, attached at their leading and trailing edges to rigid flat extensions aligned with the undisturbed flow direction, are considered and numerical results are obtained, by using the Galerkin method, showing how the stability varies with the change in length of the upstream and downstream rigid elements. Similar results are obtained for a cantilever panel attached to a leading edge rigid surface modelling the aerofoil or splitter plate used in experiments. The effects of structural damping are included where appropriate and comparisons made with other relevant theoretical and experimental results.

Second Approximation to the Laminar Boundary-Layer Flow over a Parabola

The method of matched asymptotic expansions is applied to the problem of a parabolic cylinder in a uniform incompressible flow at high Reynolds numbers. The equations and the boundary conditions for the first-order boundary layer, the second-order outer flow and the second-order boundary layer are formulated in terms of the parabolic co-ordinate system. The equations are solved numerically by means of finite-difference approximation. The results for skin friction and surface pressure are compared with the available numerical solutions of the full Navier-Stokes equations. The application of the technique of rational fraction to the two-term asymptotic expansion for skin friction shows that a further approximation will give a better result.

On the aeroelastic instability of two-dimensional panels in uniform incompressible flow

A theoretical and experimental study is presented of the aeroelastic instability of a panel with various boundary conditions on its leading and trailing edges, exposed to air flow over its upper surface or on both sides. The flow is incompressible and two-dimensional (no span-wise deformation of the panel). The case of a panel clamped at its leading edge and free at its trailing edge is investigated both theoretically and experimentally. Agreement between theory and experiment is fair. The aerodynamic theory of steady non-circulatory flow is applied for panels fixed at both ends, and the quasi-steady and full unsteady aerodynamic lifting theories to a cantilevered panel (free at the trailing edge). The analogy with incompressible flow through a long slender tube is pointed out. Theory shows that a panel with both ends fixed loses its stability by divergence, while the instability of a cantilevered panel is of a flutter type. The latter is also confirmed experimentally.

THE DAMPING EFFECT OF DISTRIBUTED AND CONCENTRATED RESISTANCES ON SMALL PERTURBATIONS IN A UNIFORM FLOW

The damping effect of a distributed resistance on small stationary harmonic perturbations in a uniform incompressible flow is theoretically analysed for the two-dimensional case. A quadratic friction law is assumed. A solution is obtained for the damping factor and simplified versions are found for the cases of widely distributed resistances and concentrated resistances. These solutions have applications to the damping effect of flat plate heaters used in blow-down tunnels and gauze wire screens respectively. It is shown that there is good agreement between the theoretical result for a concentrated resistance and experimental results achieved on gauze screens by previous investigators.

An Approximate Solution of the Laminar Boundary Layer on A Rotating Body of Revolution in Uniform Compressible Flow

We attempt to investigate a problem of laminar boundary layer on rotating bodies of revolution in compressible flow such as spinning projectiles in the air. These three-dimensional boundary layers which appear in this paper are consisted of the thermal boundary layer and velocity boundary layers in meridian and in rotational directions about the rotating body of revolution. Recently, velocity boundary layer calculations on a rotating body of revolution in uniform incompressible flow were treated by H. Schlichting in Ing. Archiv Vol. 21, No. 4 (1953). Present paper is an extension of the H. Schlichting's investigation to the compressible case and an approximate integration of the momentum equation of transformed compressible co-ordinates using Karman-Pohlhausen approximate method. As a result, we obtain the distributions of the boundary layer thickness and determine the effects of the rotation and compressibility on the separation point. As an example of this method of calculation, we treat a rotating sphere in compressible flow and determine the position of separation point as a function of rotating velocity and Mach number of uniform flow.