Trailing Edge Of Plate Research Articles

The effect of inertial inhomogeneity on the flutter of a cantilevered flexible plate

We study the effect of adding discrete structural mass on the linear stability of an otherwise homogeneous cantilevered-free flexible plate immersed in uniform axial flow. The methods of Howell et al. that mixed numerical simulation with eigenvalue analysis are simply extended for the present study. An ideal two-dimensional flow is assumed wherein the rotationality of the boundary-layers is modelled by vortex elements on the solid–fluid interface and the imposition of the Kutta condition at the plate's trailing edge. The Euler–Bernoulli beam model is used for the structural dynamics. It is shown that addition of mass to the plate can be either stabilising or destabilising, depending upon the location of the added mass, and how its inclusion modifies the energy exchanges of the corresponding homogeneous structure. Our results therefore suggest a straightforward means by which the critical flow speed at which low-amplitude flutter sets in can be passively controlled in engineering applications.

Mixing Layers in Sink Flow: Effect of Length of Flight on Mixing in a Channel Downstream

We have experimentally and analytically studied transport of a passive scalar in the wake of a thin flat plate located at the centerline of a planar contraction with flat walls. The constant Launder parameter in the contraction, K = 6.25 ×10 − 6, was twice the value required for a turbulent boundary layer to relaminarize. In addition to the mixing analysis inside the contraction, layer mixing is also investigated downstream, where the flow continues inside a constant cross-section channel. In order to generate the passive scalar, the airflow above the plate was heated and the temperature stratification in the wake was traced by measuring the temperature field using constant current anemometry. Using different plate lengths, we found that the degree of mixing, obtained at a given position in the straight channel, is a function of the distance from the plate trailing edge to the contraction outlet. For a plate which does not protrude into the straight channel, we demonstrate the existence of an optimal trailing edge-contraction outlet distance that results in the lowest possible degree of mixing at a given downstream position in the straight channel. This finding is also supported by a semi-empirical relationship based on our developed self-similar solution for mixing layers in planar contractions.

A fluidelastic model for the dynamics of cantilevered plates with an additional spring support in axial flow

A fluidelastic model has been developed for studying the dynamics of cantilevered plates in axial flow, with an additional spring support at the plate trailing edge. The plate is supposed to be two-dimensional; a nonlinear equation of motion based on the inextensibility condition is used to account for the possible large vibration amplitude. For the fluid-dynamic part, an unsteady lumped vortex model is used for calculating the fluid loads acting on the plate undergoing deformation; the longitudinal displacement of the plate associated with lateral motions is also taken into account. The fluidelastic model is first validated through comparisons with experimental data published for systems without the spring support; it is then used to investigate the dynamics of cantilevered plates in axial flow with an additional spring support at the plate trailing edge. Some interesting results of the dynamics of the fluid–structure system are presented.

On the fluid–structure interaction of a splitter plate: vibration modes and Reynolds number effects

Previous work (Eloranta et al. in Exp Fluids 39:841–855, 2005) has shown that flow separation from the trailing edge of a splitter plate in a convergent channel involves a fluid–structure interaction (FSI), which modifies the fundamental instability related to vortex shedding. Under certain conditions, the FSI induces cellular vortex shedding from the trailing edge. This paper reports detailed measurements of the plate vibration mode and studies the effect of the Reynolds number on the FSI. Experimental techniques including laser vibrometer and digital imaging are used to measure the response of the plate and particle image velocimetry is used to measure the flow field in the near wake. Combining data from these techniques, the development of the vibration frequency and mode can be addressed together with the imprint of the vibration mode in the flow. The results show that over most of the Reynolds numbers measured, the plate vibrates in a distinct mode characterized by a spanwise standing wave along the plate trailing edge. The vibration frequency and the spacing between the nodes of the standing wave depend on the Reynolds number. As the Reynolds number is increased, the frequency of the dominant vibration mode does not increase linearly. The plot of the vibration frequency as a function of the Reynolds number shows that the vibration tends to lock to a rather constant frequency over of range of Reynolds numbers. After certain Reynolds number if threshold is exceeded, the frequency jumps to a new level, which also involves a new vibration mode.

Fluid–structure interaction of a splitter plate in a convergent channel

The fluid–structure interaction (FSI) of a splitter plate in a convergent channel flow is studied by measuring both the flow field and the plate vibration. Particle Image Velocimetry (PIV) measurements show that the wake generated by the plate is characterized by cellular vortex shedding. Mean and RMS velocities presented in the plane normal to the main flow direction visualize clearly the cellular structure and related secondary flows. To evaluate the energy and spatial organization of the vortex shedding, spectral and correlation estimation methods are adapted to the PIV data. By presenting the spanwise variation of the streamwise spectra along the trailing edge, the nature of the cellular vortex shedding becomes evident. 2D space-correlation function reveals that the shedding in two neighboring cells occurs in a 180-degree phase shift. The vibration of the plate is studied with Digital Imaging (DI) and Laser Vibrometer (LV). The DI is based on images measured by the PIV system. An image-processing algorithm is used to detect the plate tip location and velocity simultaneously with the estimation of the fluid velocity field. The LV is used for the time-resolved measurement of the plate vibration. The results show that the plate vibrates in a very distinct mode characterized by a spanwise standing wave along the plate-trailing edge. This mode, in turn, causes the cellular vortex shedding.

Shock wave interaction with a viscous wake in supersonic flow

A theoretical investigation of the breakdown of the viscous wake downstream of a flat plate in supersonic flow is performed in this paper based on the large Reynolds number ($Re \to \infty $) asymptotic analysis of the Navier–Stokes equations. The breakdown is provoked by an oblique shock wave impinging on the wake a small distance $l_s$ downstream of the plate trailing edge. Two flow regimes are considered. In the first $l_s$ is assumed to be an $O(Re^{-3/8})$ quantity in which case the shock impinges on the wake within the region of viscous–inviscid interaction that is known to occupy a vicinity of the trailing edge with longitudinal extent of $O(Re^{-3/8})$. Under these conditions the interaction process may be described by the equations of the triple-deck theory. To obtain a numerical solution of these equations we used a rapid matrix Thomas technique in conjunction with Newton iterations. The results of the calculations not only predict the wake breakdown near the shock location but also reveal a hysteresis behaviour of the flow as the shock is moved downstream, giving rise to three solution branches. The second part of the paper is concerned with the flow regime when the shock interacts with the wake further downstream of the trailing edge triple-deck region: $l_s \gg Re^{-3/8}$. In this case the fluid motion proves to be inviscid to leading order not only in the upper deck of the interaction region but also everywhere inside the wake. Due to this simplification the interaction problem can be reduced to a single integro-differential equation governing the pressure distribution along the interaction region. With known pressure the Bernoulli equation may be used to find the velocity field. The Bernoulli equation also allows us to formulate a simple criterion which may be used to predict the onset of wake breakdown. We found that viscosity becomes important again in a smaller vicinity of the breakdown point where the flow reversal takes place. It is remarkable that the viscous–inviscid interaction problem governing the flow in this vicinity admits an analytical solution.

Bluff body drag control by boundary layer disturbances

The experimental results are presented for the influence of thin plates and dimples arranged over the upper surface of a coach model on base pressure variations, turbulent wake characteristics, and body drag. The influence of the plate height, the plate spacing, the attack angle, and the distance from the plate trailing edge to the model base on pressure distributions over the model and its back panel are studied. It is shown that rather small disturbances of the boundary layer by plates cause the base pressure to increase up to 6% and model drag to decrease by as much as 3%. However, the effect of the dimples on the base pressure appears to be much less.

Asymptotic solutions of the wake induced by a streamwise moving flat plate in a channel

The laminar flow at large Reynolds number in the wake of a moving flat plate in a channel is investigated. Numerical and analytical analysis are carried out and their results are compared. The transition of the initial Couette flow at the trailing edge to the Poiseuille–Couette in the far wake is established. It is demonstrated that the flow is accelerated in a thin layer at the downstream of the plate. This acceleration is induced by the slipping of the fluid on the plate trailing edge. Outside this layer area the flow is decelerated in order to maintain the flow rate.

Trailing-Edge Noise Prediction Using Large-Eddy Simulation and Acoustic Analogy

The filtered Navier-Stokes equations are used to perform the large-eddy simulation of the unsteady incompressible flow around the blunt trailing edge of a thick flat plate. The computed flow exhibits a three-dimensional vortex shedding mechanism. The frequency domain of this mechanism is in agreement with experiments and theory. In that frequency domain normalized wall-pressure levels are favorably compared to spectra measured at the blunted trailing edge of an airfoil. The far-field radiated noise is first computed via Curle's formulation, the solution of Lighthill's equation for flows embedding solid bodies. Then the theory developed by Ffowcs Williams and Hall is considered. This formulation expresses the noise generated by turbulence passing over the edge of an infinite half-plane. The suitability of this theory to the case for a thick plate is discussed. The normalized spectra of the radiated noise predicted by both methods are compared, in the frequency domain of the vortex-shedding mechanism, to airfoil noise measurements in an anechoic facility.

The turbulent near wake at a sharp trailing edge

The problem of a turbulent boundary layer that evolves into a wake flow at the sharp trailing edge of a thin flat plate is considered; the formal structure of the near-wake flow is investigated using matched expansions in the limit of infinite Reynolds number. The symmetric turbulent near wake is shown to develop a two-layer structure which is independent of turbulence model. The general asymptotic analysis shows that a thin layer at the wake centreline grows linearly with distance from the trailing edge while the centreline velocity varies logarithmically in a manner that is supported strongly by experimental measurements. The relatively thick outer layer of the near-wake flow is undisturbed by the evolution of the inner layer to leading order. An additional region near the trailing edge is required to resolve a non-uniformity in transverse velocity. The general asymptotic results are used to guide the development of a zonal turbulence model for the near wake in the form of a simple eddy viscosity formula. Analytic profiles for velocity and Reynolds stress are obtained for the near-wake region; these profiles are shown to provide accurate representations of available near-wake experimental data.

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/∂y2 ≫ ∂2u/∂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.

Numerical analysis of boundary-layer-type flow including singular points. Employment of partially parabolized Navier-Stokes equations.

The efficient method to solve two-dimensional, incompressible, steady, partially parabolized Navier-Stokes equations has been presented. The method has been extended from Keller's BOX method for boundary-layer equations. The Israeli's source term, a streamwise direction staggered grid and a multi-grid procedure have been applied in order to obtain numerical stability and fast convergence. Quantitative comparisons with Briley's full Navier-Stokes solutions of a separation bubble problem and Nishioka's experimental data of a flat plate trailing edge problem have been made. In the former case, the agreement is excellent, and the pressure distribution at the wall also agrees well with Carter's inverse boundary-layer solution. In the latter case, the agreement is reasonable. It is also found that the pressure gradients in both the streamwise and normal directions are almost of the same order. This suggests that the omission of the normal momentum equation, i.e., the boundary-layer equations, is supposed to be a rough approximation for the trailing edge flow.

Intensive injection of fluid into a hypersonic stream from the surface of a finite length plate: PMM vol. 43, no. 5, 1979, pp. 829–838

Intensive fluid injection into a hypersonic stream from the surface of a finite length plate is considered. The injected fluid is assumed perfect and inviscid, and the flow in the injection region, separated from the main stream by a contact surface, is defined by approximate “thin layer” equations.A complete analytic solution of the problem is presented. It leads to the establishment of universal formulas in similarity variables that define the contact surface form and pressure distribution on the plate. These universal formulas are valid for any given pressure at the plate trailing edge.

Analysis of noise produced by jet impact near the trailing edge of a flat plate

The sound field produced by the interaction of a subsonic cold gas jet with the trailing edge of a large flat plate is analyzed. The jet issuing from a 2-in.-diameter convergent nozzle impacted the plate seven nozzle diameters downstream from the nozzle exit and one diameter from the plate's trailing edge. The plane of the plate intersected the nozzle axis at 60°. Data analysis was performed to obtain a better understanding of the dominant noise source and the mechanism governing the peak frequency of the broad-band spectra. Results indicate that noise from the plate emanates from two principal sources: first, the impact of the jet on the plate and, second, the flow over the plate's trailing edge. Above a jet Mach number of 0.5, the dominant noise as detected on the impingement side of the plate results from the jet impact (eighth power of the velocity dependence) rather than a trailing edge disturbance (fifth or sixth power of the velocity dependence), which is generally assumed dominant. Also, the frequency of the peak SPL may be governed by a feedback phenomenon, resulting in periodic formation and shedding of ring vortices from the nozzle.

An experimental investigation of the instability of an incompressible, separated shear layer

The investigation of a separated shear layer was undertaken to clarify the non-linear mechanisms associated with instability and transition to turbulence. Such an investigation is of practical importance since profiles which resemble the separated shear layer are a common occurrence.A two-dimensional free shear layer was formed by separation of a laminar boundary layer from a rearward-facing step. The free-stream speed was approximately 16 ft./sec. Hot-wire measurements were made in the region directly downstream of the plate trailing edge. The measurements included mean velocity profiles, frequency spectra of the longitudinal fluctuation, and root-mean-square amplitude and phase distributions of various spectral components of the longitudinal fluctuation. Several measurements were designed to detect the presence of periodic spanwise structure.The most important findings were: Significant non-linear distortion of the initial unstable wave occurred without periodic spanwise structure.Non-linear distortion was first manifest by the growth of a subharmonic oscillation, which was strongly intermittent. Numerous harmonics of the sub-harmonic oscillation were also present.Strong evidence suggests that secondary instabilities were present, which created still higher frequencies.