Indicial Functions Research Articles

Unsteady aerodynamics of a flapped airfoil in subsonic flow by indicial concepts

An approach based on indicial concepts is described to model the unsteady airloads on a thin airfoil in subsonic compressible flow caused by the arbitrary motion of a trailing-edge flap. Exact indicial aerodynamic responses at small values of time as a result of flap deflection and angular deflection rate about the flap hinge are obtained from linear unsteady subsonic theory in conjunction with the aerodynamic reverse flow theorems. Using the known exact initial (piston theory) and asymptotic values of the airloads, along with an assumed analytic form for the indicial functions, these exact results are used to help obtain complete approximations for the respective indicial responses. The airloads from arbitrary flap motion in subsonic flow are subsequently obtained in state - space form. Validation of the method is conducted with experimental data for time-dependent flap motions.

NUMERICAL CALCULATION OF FLUTTER DERIVATIVES VIA INDICIAL FUNCTIONS

The central focus of the present study is the numerical calculation of flutter derivatives. These aeroelastic coefficients play an important role in determining stability, or rather, instability of long, flexible structures under ambient wind loading. In recent years, the Finite Element Method (FEM) has gained considerable acceptance in the solution of problems governed by the viscous, incompressible flow equations. FIDAP is one such general fluid dynamics analysis package that makes use of Finite Element methodology. The most direct way of obtaining flutter derivatives would be to simulate the full oscillatory object motion in a flow and use the unsteady lift and moment forces thus determined. This would essentially entail an internal boundary motion in the flow, which requires considerable computational effort in the case of most Eulerian non-adaptive grid-based numerical methods. However, the alternative indicial function approach investigated in this study limits itself to step changes in motional variables. Inherent to using this approach, then, is the assumption of reduced computational complexity.

Indicial functions of arterial remodeling in response to locally altered blood pressure.

We investigated the effect of locally altered blood pressure on the remodeling processes of the cells and extracellular matrices of the splenic and ileal arteries and used an indicial function approach to quantitatively analyze the relationship between the altered blood pressure and the remodeling processes. Blood pressure in these arteries was locally modulated by constricting the aorta at a location between the celiac and mesenteric bifurcations, resulting in a higher blood pressure at the splenic arteries then at the ileal arteries, After the pressure changes, the cross-sectional areas and the fractions of the cells and extracellular matrices of the splenic and ileal arteries were examined by electron microscopy at 2, 6, 10, 20, and 30 days. We found that both arteries remodeled, but the splenic arteries (higher blood pressure) remodeled more rapidly and to a larger degree than the ileal arteries (lower pressure compared with the splenic arteries) of the same animal. To verify whether an identical change in the blood pressure at the splenic and ileal arteries leads to the same remodeling process in these arteries, we created another model by constricting the aorta at a location between the mesenteric and renal bifurcations, resulting in hypertension of the same level at both splenic and ileal arteries. We found that the remodeling processes of the cells and matrices were almost identical in the arteries with similar changes in blood pressure. Thus we conclude that the remodeling processes of cells and matrices of the splenic and ileal arteries are dependent on the local blood pressure in aorta constriction-induced hypertension, and the indicial analysis is a useful approach in the description of the relationship between the blood pressure and the arterial remodeling processes.

Stability investigation of long-span bridges using indicial functions with oscillatory terms

Stability of long-span suspension bridges is shown to be dependent upon turbulence in the wind flow. Turbulence is represented as a stationary wide band random excitation. Incorporating the turbulence as random velocity fluctuations in the equations of motion, renders them time-variant with random coefficients, which requires time domain analysis, where indicial functions represent the aerodynamic loads. A model of indicial functions with oscillatory terms is introduced, that agrees very well with measured flutter derivatives for some bridge decks. Using this model leads to a lower estimate of the mean critical wind speed at low levels of turbulence, compared to estimates when using the nonoscillatory model. It also predicts a greater stabilizing effect of turbulence or a lower destabilizing effect, at higher levels of turbulence.

State-space representation of unsteady airfoil behavior

A method is presented to model the unsteady lift, pitching moment, and drag acting on a two-dimensional airfoil operating under attached-flow conditions in a compressible flow. Starting from suitable generalizations and approximations to aerodynamic indicial functions, the unsteady airloads due to an artibrary forcing are represented in a state-space (differential equation) form. This model is in a form compatible with the aeroelastic analyses of both fixed-wing and rotary-wing systems

Validation of approximate indicial aerodynamic functions for two-dimensional subsonic flow

Approximations for two-dimensional indicial (step) aerodynamic responses due to angle of attack and pitch rate are obtained and generalized to account for compressibility effects up to a Mach number of 0.8. Using the Laplace transform method, these indicial functions are manipulated to produce explicit solutions for idealized harmonic forcings. These explicit solutions are subsequently compared with experimentally obtained pitch and plunge aerodynamic data in the reduced frequency domain. The results of this comparison are used to relate back and substantiate the generalization of the compressible indicial lift and moment functions.

On flutter and buffeting mechanisms in long-span bridges

Using linear formulations throughout, the present paper reexamines the problem of the three-dimensional dynamic response of long, flexible bridge decks to wind, particularly with the modern cable-stayed bridge in view. There are some particular motivations for the approach presently used. First, it has been observed that the bridge deck, the principal attractor of aerodynamic forces, undergoes three-dimensional motion having not only vertical and twist components, but lateral sway as well. This is a pronounced feature of many cable-stayed bridge vibration modes; and, since lateral sway can play a strong, contributory role in total aeroelastic action, it is important that its effects be included. Second, under buffeting by turbulent wind, total bridge response is random in the time domain, precluding the direct expression of aeroelastic forces via the well-known flutter derivatives which are applicable only to harmonic motion. In principle their role must be supplanted for general motions by the integrated effects of those motions as weighted by the corresponding aerodynamic indicial functions. However, to date such indicial functions have themselves been derived only through inverse Fourier transformation of experimentally obtained flutter derivatives; hence there exists an impetus to carry out the entire analysis in the frequency domain, where the original flutter derivatives can be used directly. The details of this procedure and its consequences regarding both the stability and level of bridge deck response are pursued in the body of the paper.

Motion Stability of Long‐Span Bridges under Gusty Winds

The effects of spanwise wind-velocity correlation and turbulence intensity on the coupled vertical-torsional response of long-span suspension bridges are studied. Turbulence is accounted for in the self-excited loads that are expressed in terms of the indicial aerodynamic functions. Buffeting forces enter the stability calculations because they are related to the self-excited loads through the common turbulence field. Assuming the wind-turulence terms to be Gaussian white-noise processes, the coupled parametric-type equations of motion are transformed into an Ito stochastic differential equation. Stationary response moments are obtained for both vertical and torsional displacemnts for different degrees of spanwise correlation and turbulence intensity. Motion instability is determined when responses become asymptotically unbounded. Higher turbulence leads to lower critical wind velocity for constant correlation, but less correlation results in higher critical velocity for constant turbulence. Higher turbulence leads to a more gradual transition from a stable to an unstable state.

Airfoil gust response and the sound produced by airfoil-vortex interaction

This paper contributes to the understanding of the noise generation process of an airfoil encountering an unsteady upwash. By using a fast Fourier transform together with accurate airfoil response functions, the lift-time waveform for an airfoil encountering a delta function gust (the indicial function) is calculated for a flat plate airfoil in a compressible flow. This shows the interesting property that the lift is constant until the generated acoustic wave reaches the trailing edge. Expressions are given for the magnitude of this constant and for the pressure distribution on the airfoil during this time interval. The case of an airfoil cutting through a line vortex is also analyzed. The pressure-time waveform in the far field is closely related to the left-time waveform for the above problem of an airfoil entering a delta function gust. The effects of varying the relevant parameters in the problem are studied, including the observed position, the core diameter of the vortex, the vortex orientation and the airfoil span. The far field sound varies significantly with observer position, illustrating the importance of non-compactness effects. Increasing the viscous core diameter tends to smooth the pressure-time waveform. For small viscous core radius and infinite span, changing the vortex orientation changes only the amplitude of the pressure-time waveform, and not the shape.

Role of Indicial Functions in Buffeting Analysis of Bridges

The problem of the wind buffeting of bluff bodies, in this instance bridge decks, is reexamined in the present paper from the standpoint of linearized theory. The links of flutter derivatives to ae...

Unsteady Newton-Busemamn Flow Theory Part III. Frequency Dependence and Indicial Response

SummaryA complete unsteady Newton-Busemann flow theory, including centrifugal force corrections, was recently given by Hui and Tobak and successfully applied to study the dynamical stability of oscillating aerofoils (Part I) and bodies of revolution (Part II). The present paper extends these results, which are restricted to the first order in frequency, to general frequencies that may be applicable to flutter analysis as well. Furthermore, the new results are used to investigate the behaviour of the indicial response functions in unsteady flow at very high Mach numbers. In particular, it is found that for a family of body shapes in the Newtonian flow, including the cone, the wedge, the delta wing and slender aerofoils, the aerodynamic response to a step change in angle of attack or pitching velocity contains an impulse at the initial instant. This is followed by a rapid adjustment to the new steady-flow conditions, the adjustment time being equal to that required for a fluid particle to travel a distance of the body length. The impulse component appearing at the infinite Mach number limit (Newtonian flow) is in effect an apparent mass term, analogous to that which occurs initially in the aerodynamic indicial response at the zero Mach number limit (incompressible flow). In the latter case the initial impulse is followed by an asymptotic adjustment to the new steady-flow conditions, theoretically taking infinite time, in direct contrast to the rapid adjustment in Newtonian flow.

Unsteady Transonic Flows over an Airfoil

Inviscid, transonic flows over the NACA 64A410 airfoil were calculated using a finite-difference analog of the Euler equations. Steady flow solutions were obtained for angles of attack of 0°, 2° and 4° at Mach 0.72. Unsteady calculations were made at the same Mach number and in the same general angle-of-attack range for a step change of angle of attack, a step change in pitching angular velocity, and for sinusoidal pitching oscillations about the midchord at reduced frequencies 0.2,1.0, and 5.0. (Reduced frequency, K^uC/U^.) A summary of the resulting surface pressures, shock locations, and the forces and moments is presented. The forces and moments obtained for the cases with harmonic oscillations were compared to the results from a linearized analysis which used, to approximate the indicial functions, the responses calculated for the step changes.

Indicial Aerodynamic Functions for Bridge Decks

With certain classical results for airfoils as a guide, lift and moment functions of the indicial type are derived for suspension bridge deck sections of either the open truss or streamlined variety. Bluff deck sections susceptible to coherent vortex shedding are not treated. The indicial functions are obtained analytically from the experimentally-derived linear-theory flutter derivatives of the bridge deck sections in question. The method used is nonlinear least squares, wherein a prescribed exponential form with unknown coefficients for the indicial function is determined by fitting transformed counterparts of it to the corresponding experimental flutter derivative curves. Several examples of indicial moment functions of bridge deck sections are presented. These offer rather conclusive proof that Wagner or thin airfoil theory alone is inadequate for application to bridge buffeting flutter and buffering analyses.

Electric Circuit Theory and the Operational Calculus

So far in our discussions of wave propagation in lines and wave-filters, we have confined attention to the case where the impressed voltage is applied directly to the infinitely long line. We have found that, by virtue of this restriction, the indicial admittance functions of the important types of transmission systems are rather easily derived and expressible in terms of well known functions, and the essential phenomena of wave propagation clearly exhibited. In practice, however, we are concerned with lines of finite length with the voltage impressed on the line through a terminal impedance Z1 and the distant end closed by a second terminal impedance Z2. We now take up the problem presented by such a system.