Unsteady Theory Research Articles

Experimental investigation of instantaneous lift on NACA 0018 airfoil section due to a non-harmonic pitching motion

In this study, we explore the problem of an airfoil in a harmonic and arbitrary periodic pitching motion and seek to understand how theoretical estimation with first-order potential theory match experiments. We adopt Jones’ approximation for the Wagner function for lift computations in the time domain and Theodorsen’s model in the frequency domain. We experimentally investigate a two-dimensional symmetric NACA 0018 airfoil in pitching motion under attached flow with reduced frequencies up to 0.25 and Reynolds number varying from 1.5×105 to 3×105. To eliminate the influence of the laminar separation bubble, free-stream turbulence intensity was elevated above the baseline level by placing turbulence generating grid, thus, obtaining better compatibility between the flow conditions in the experiments and the theoretical assumptions. Time-resolved sectional lift is determined by simultaneous static pressure measurement with miniature pressure transducers. The transient response is captured with accuracy in the time domain by quantifying the non-circulatory contribution. Excellent agreement is achieved in the time and frequency domains due to the highly accurate measurement of the instantaneous magnitude and phase of the time-resolved surface pressures. We further explore the applicability of the theory for high-pitching amplitudes, which results in a temporary deviation of the measured lift from the theoretical predictions. As the frequency domain unsteady theory is more widely studied than its time domain counterpart, this study provides a unique opportunity to highlight the significance of time-domain analysis in estimating instantaneous lift in non-harmonic motion.

Aeroelastic analysis of membrane airfoils and flexible-chord airfoils with trailing-edge flaps

This paper studies the static and dynamic aeroelastic characteristics of membrane airfoils and flexible-chord airfoils (deformable airfoils) with the emphasis on the effects of a trailing-edge (TE) flap which is a novel topic. Two modeling approaches are presented; the first method is the Rayleigh–Ritz method, and the second method is the finite element method which is an efficient method to study the TE flap effects. The two models are presented in the Laplace domain which enables the transient response analysis. The models adopt the potential flow aerodynamics based on the Prandtl–Glauert thin-airfoil theory and the Theodorsen’s unsteady theory. The airfoils are assumed to have small deformations, so linear structural models are used. The effect of the airfoils’ flexibilities on the static aeroelastic characteristics and the dynamic responses due to step and harmonic TE flap inputs is presented through a parametric study.

Analytical and numerical modelling of tandem oscillating foils propulsors

The present work investigates theoretical and numerical methods to model oscillating hydrofoils, in single and tandem arrangement. The first part of this work presents a semi-analytical 2D model for a hydrofoil propulsor travelling at constant speed, in infinite domain, undergoing heaving and pitching oscillations, based on the Theodorsen unsteady theory. Following the previous work, the model is developed to estimate the thrust and propulsive efficiency attempting to incorporate the main physical contributions. The second part investigates the same foil arrangements using a high-fidelity CFD approach. In particular, a robust deforming grid method is used, enabling the simulation of cases with large body motions. Results of the semi-analytical model are compared with available published numerical and experimental data, with good agreement. The numerical model results are analysed using verification and validation procedures, allowing accurate physical interpretation of the flow and forces. Additionally, the tandem foil configuration, if properly optimized, is found to increase the single foil thrust by almost three-fold. The overall results suggest that the semi-analytical model may be quite suitable in design studies, and the CFD model showed capable of providing accurate results for the most complex cases and of contributing to a comprehensive understanding of the flow dynamics.

Mitigation of Airfoil Gust Loads Through Pitch

Wings can experience a large increase in lift when subjected to a high-amplitude transverse gust, which may adversely affect their controllability. This study analyzes the capability of airfoil pitch motions to mitigate such lift spikes. The mitigation pitch kinematics are calculated preemptively from gust velocity measurements using two approaches. First, a simple quasi-steady model of the lift force is used to counteract the gust-induced angle of attack. Second, the classical unsteady theories of Wagner and Küssner are utilized to account for the effects of trailing-edge vorticity and noncirculatory forces. Experimental force and flow measurements are analyzed for gust ratios from 0.25 to 1.5. Results show that the motions predicted by the quasi-steady model achieve a lift peak alleviation of 70%, whereas the unsteady pitch profile results in a superior 90% alleviation. This accurate mitigation for high-amplitude gusts is analyzed by comparing the flow development with the main assumptions in the models, like small angle of attack and no shedding from the leading edge. Further analysis includes experiments where the wing experiences only the gust or only the pitch motion. Ultimately, the outcome of this work can be used to improve the control mechanisms on board of flying vehicles.

State-Space Modeling of Viscous Unsteady Aerodynamic Loads

In this paper, we summarize our recently developed viscous unsteady theory, which couples potential flow with the triple-deck boundary-layer theory. This approach provides a viscous extension of potential-flow unsteady aerodynamics. As such, a Reynolds-number-dependent transfer function is determined for unsteady lift. We then use the Wiener–Hammerstein structure to develop a finite-dimensional approximation of such an infinite-dimensional theory, presenting it in a state-space model. This novel nonlinear state-space model of viscous unsteady aerodynamic loads is expected to serve aerodynamicists better than the classical Theodorsen’s model because it captures viscous effects (that is, Reynolds number dependence) as well as nonlinearity and additional lag in the lift dynamics; it also allows simulation of arbitrary time-varying airfoil motions (not necessarily harmonic). Moreover, being in a state-space form makes it quite convenient for simulation and coupling with structural dynamics to perform aeroelasticity, flight dynamics analysis, and control design. We then develop a linearization of such a model, which enables analytical results. Subsequently, we derive an analytical representation of the viscous lift frequency response function: an explicit function of both the frequency and Reynolds number. We also develop a state-space model of the linearized response. We finally simulate the nonlinear and linear models to a nonharmonic small-amplitude pitching maneuver at a Reynolds number of 100,000 and compare the resulting lift and pitching moment with those obtained from potential flow; this is in reference to relatively higher-fidelity computations of the unsteady Reynolds-averaged Navier–Stokes equations.

Feasibility of Boosting Low-Energy-Quality Steam by Using Wave Rotor

Owing to the difficult utilization of the low-pressure level in the process industry, the low-energy-quality steam is often condensed to recover the demineralized water or just discharged directly, causing a huge waste of thermal energy. A novel technology of enhancing the steam’s energy quality by using the wave rotor based on the principle of moving shockwave compression is proposed. The supercharging ability of 3-port wave rotor is studied by meaning of 1-dimension unsteady theory and computational fluid dynamic. A practical thermodynamic flowsheet of boosting the low-pressure steam driven by high-pressure steam is also proposed and analysed in detail. As an example, to boost the saturated steam of pressure 1.0 MPa to 1.953 MPa, a three-stage wave rotor solution is proposed and is verified its feasibility. The high supercharging ratio and entrainment ratio of the wave rotor are much higher than the traditional steam ejector shows the feasibility of enhancing energy-quality of low-pressure steam.

Flutter analysis of rotary laminated composite structures using higher-order kinematics

The flutter analysis of rotary laminated structures has been performed using Carrera Unified Formulation (CUF). CUF is a one-dimensional higher-order structural model, in which the displacement at the cross-section is the only unknown parameter. The cross-sectional displacement fields of the model are defined using the Lagrange- and Taylor expansion. The quasi-steady and unsteady theories (Theodorsen and Loewy) have been used to apply the aerodynamic force. Subsequently, the equation of motion is defined in ‘fundamental nucleus’ form using Hamilton’s principle. The engineering solution of the flutter condition has been obtained by using p-k method. The computational studies of flutter analysis have been done for various rotor hub, lamination sequence, and structural models. The regular, irregular plies and arbitrary fiber orientation of the orthotropic structures have also been considered. The present study suggests that the proposed CUF model can obtain an aeroelastic solution of rotary structure in a computationally and economically efficient manner.

Unsteady lift on a high-amplitude pitching aerofoil

The ability to accurately predict the forces on an aerofoil in real-time when large flow variations occur is important for a wide range of applications such as, for example, for improving the manoeuvrability and control of small aerial and underwater vehicles. Closed-form analytical formulations are only available for small flow fluctuations, which limits their applicability to gentle manoeuvres. Here we investigate large-amplitude, asymmetric pitching motions of a NACA 0018 aerofoil at a Reynolds number of 3.2 times 10^4 using time-resolved force and velocity field measurements. We adapt the linear theory of Theodorsen and unsteady thin-aerofoil theory to accurately predict the lift on the aerofoil even when the flow is massively separated and the kinematics is non-sinusoidal. The accuracy of the models is remarkably good, including when large leading-edge vortices are present, but decreases when the leading and trailing edge vortices have a strong interaction. In such scenarios, however, discrepancies between the theoretically predicted and the measured lift are shown to be due to vortex lift that is calculated using the impulse theory. Based on these results, we propose a new limiting criterion for Theodorsen’s theory for a pitching aerofoil: when a coherent trailing-edge vortex is formed and it advects at a significantly slower streamwise velocity than the freestream velocity. This result is important because it extends significantly the conditions where the forces can be confidently predicted with Theodorsen’s formulation, and paves the way to the development of low-order models for high-amplitude manoeuvres characterised by massive separation.Graphic abstract

Unsteady theory of heat transfer and fluid flow during instantaneous transition to film boiling

The paper deals with the problem of unsteady convective heat transfer during instantaneous transition to film boiling over a flat vertical surface, when heat flux is suddenly switched on. This heat flux exceeds the critical value, which corresponds to the onset of the film boiling regime. As a result, instantaneous transition to film boiling takes place. To solve the problem, two sets of the initial and boundary conditions were used. The first set was used in the classical solution of the film condensation problem derived by Nusselt. The second set was proposed by Ellion for a stationary liquid at the vapor-liquid interface. The analytical solution of the problem enabled obtaining unsteady temperature distributions, Nusselt numbers, velocity profiles and friction coefficients. Results obtained based on the Laplace transform and the self-similar solution are analyzed.

New Method for Analyzing the Flutter Stability of Hingeless Blades with Advanced Geometric Configurations in Hovering

A new method used to analyze the aeroelastic stability of a helicopter hingeless blade in hovering has been developed, which is especially suitable for a blade with advanced geometric configuration. This method uses a modified doublet-lattice method (MDLM) and a 3-D finite element (FE) model for building the aeroelastic equation of a blade in hovering. Thereafter, the flutter solution of the equation is calculated by the V-g method, assuming blade motions to be small perturbations about the steady equilibrium deflection. The MDLM, which is suitable to calculate the unsteady aerodynamic force of nonplanar rotor blade in hovering, is developed from the doublet-lattice method (DLM). The structural analysis tool is the commercial software ANSYS. The comparisons of the obtained results against those in the literatures show the capabilities of the MDLM and the method of structural analysis. The flutter stabilities of swept tip blades with different aspect ratios are analyzed using the new method developed in this work and the usual method on the basis of the unsteady strip theory and beam model. It shows that considerable differences appear in the flutter rotational velocities with the decrease of the aspect ratio. The flutter rotational velocities obtained by the present method are evidently lower than those obtained by the usual method.

Flutter Analysis of a Tapered Viscoelastic Wing Subjected to a Follower Thrust Force

The flutter instability of a wing considering many parameters is investigated. The effects of the structural damping, wing taper ratio, engine thrust, and engine mass and location on the flutter characteristics of the wing are studied. The material of the wing is modeled as a viscoelastic material where Kelvin-Voigt model is used to represent the viscoelastic behavior of the material. The unsteady Theodorsen theory is used to model the aerodynamic forces (i.e. lift and pitching moment). The governing equations of motion are developed using the extended Hamilton’s principle and solved via Galerkin’s method along with the classical flutter investigation approach. The developed model is validated against the well-known Goland wing and HALE wing. The flutter determinant method is employed to carry out non-dimensional parametric study on the Goland wing. The study revealed that a tapered wing would be more dynamically stable than a uniform wing. It is also observed that the viscoelastic damping provides wider stability region for the wing. This work shows that the engine thrust and mass have significant effects on the dynamic stability of the wing.

A Comparative Assessment of Aerodynamic Models for Buffeting and Flutter of Long-Span Bridges

Abstract Wind-induced vibrations commonly represent the leading criterion in the design of long-span bridges. The aerodynamic forces in bridge aerodynamics are mainly based on the quasi-steady and linear unsteady theory. This paper aims to investigate different formulations of self-excited and buffeting forces in the time domain by comparing the dynamic response of a multi-span cable-stayed bridge during the critical erection condition. The bridge is selected to represent a typical reference object with a bluff concrete box girder for large river crossings. The models are viewed from a perspective of model complexity, comparing the influence of the aerodynamic properties implied in the aerodynamic models, such as aerodynamic damping and stiffness, fluid memory in the buffeting and self-excited forces, aerodynamic nonlinearity, and aerodynamic coupling on the bridge response. The selected models are studied for a wind-speed range that is typical for the construction stage for two levels of turbulence intensity. Furthermore, a simplified method for the computation of buffeting forces including the aerodynamic admittance is presented, in which rational approximation is avoided. The critical flutter velocities are also compared for the selected models under laminar flow.

Solution and type curves for the seepage model of the water-bearing coalbed with leakage recharge

To analyze the effects of the leakage recharge of the aquifer on the initial dewatering of coalbed methane wells, the mathematical seepage model of water in the coalbed considering the aquifer leakage was established by using the leakage coefficient according to the unsteady seepage theory. The model was solved after Laplace transform and the Stehfest numerical reverse inversion was used to obtain the solution in right space. Then, the log-log type curves of pressure and pressure derivative were created with new combinations of parameters. Based on the natural seepage mechanism, the influence of aquifer leakage on curve shape was judged. It is found that the radial flow ends earlier as the leakage coefficient increases. Moreover, it was proposed to obtain reservoir permeability, skin factor, and leakage coefficient by using type curve matching. The type curves are useful for quantitatively evaluating the level of leakage, thereby guiding the adjustment of the following production system for CBM wells. Curvas de solución y tipo para el modelo de filtración de capas carboníferas acuíferas con recarga de fugasResumenEste estudio estableció el modelo matemático de filtración de agua en una capa carbonífera al estimar la salida acuífera con el uso del coeficiente de fuga, de acuerdo con la teoría de filtración inestable, para analizar los efectos en la recarga de pérdida de fluidos de un acuífero en el drenado inicial para pozos de gas metano. El modelo se resolvió tras usar la transformación Laplace y la inversión numérica Stehfest para encontrar la respuesta en el lugar indicado. Luego, se creó la representación algorítmica de la presión y la presión derivativa con nuevas combinaciones de parámetros. Se evaluó la influencia de la pérdida de fluido del acuífero en la forma de la curva con base al mecanismo físico de filtración. Se estableció que el flujo radial finaliza antes de que el coeficiente de pérdida de fluido se incremente. Además, se propone el uso de la curva tipo correspondiente para obtener la permeabilidad del reservorio, el factor de daño y el coeficiente de pérdida de fluido. Las curvas tipo son útiles para evaluar cuantitativamente el nivel de la pérdida de fluido, y de esta manera guiar el ajuste de un sistema de producción consecuente para pozos de gas metano de carbón.

Aerodynamic and mechanical hierarchical aeroelastic analysis of composite wings

The coupled bending-torsion flutter is here investigated through Carrera Unified Formulation (CUF). The hierarchical capabilities of CUF offer a procedure to obtain refined one-dimensional models that, by going beyond the assumptions of classical theories, accurately describe the kinematics of structures. Aerodynamic loadings have been determined according to Theodorsen theory, from which the steady formulation can be easily obtained. The displacement variables over the cross section (x-z plane) are approximated by x,z polynomials of any order, N. The finite element method is used to solve the governing equations, which are derived in a weak form through the principle of virtual displacements. The equations are written in terms of “fundamental nuclei,” which do not vary with the theory order, N. Several wing configurations have been studied, giving great attention to thin-walled box beams made of orthotropic material. The effects of sweep angle and lamination scheme on flutter conditions have been investigated, and the results have been compared with solutions obtained from two-dimensional theories, experimental tests, and aeroelastic analyses carried out with the doublet lattice method (DLM). The unsteady theory, combined with advanced beam theories, represents a computationally cheap tool for preliminary aeroelastic studies of complex wing structures.

Streamwise Oscillation of Airfoils into Reverse Flow

A NACA 0012 airfoil is oscillated in streamwise direction in a constant freestream and at a fixed incidence angle such that reverse flow occurs cyclically. Force measurements reveal that lift is close to unsteady theory while advancing into the freestream, if the angle of attack permits attached flow. Lift is augmented at large angles of attack, where the flow is separated under steady conditions, and does not become appreciatively negative in flow reversal for either attached or separated flow, contrary to one unsteady theory but supported by another. Dye flow visualization reveals a coherent vortical structure upstream of the leading edge before flow reversal, which is believed to attenuate negative lift.

Geometrically-exact unsteady model for airfoils undergoing large amplitude maneuvers

A semi-analytical, geometrically-exact, unsteady potential flow model is developed for airfoils undergoing large amplitude maneuvers. For this objective, the classical unsteady theory of Theodorsen is revisited relaxing some of the major assumptions such as (1) flat wake, (2) small angle of attack, (3) small disturbances to the mean flow components, and (4) time-invariant free-stream. The kinematics of the wake vortices is simulated numerically while the wake and bound circulation distribution and, consequently, the associated pressure distribution are determined analytically. The steady and unsteady behaviors of the developed model are validated against experimental and computational results. The model is then used to determine the lift frequency response at different mean angles of attack. Both qualitative and quantitative discrepancies are found between the obtained frequency response and that of Theodorsen at high angles of attack.

Wave Breaking in Dense Plumes

AbstractSinking dense plumes are important in many oceanographic settings, notably the polar formation of deep and bottom waters. The dense water sources feeding such plumes are commonly affected by tidal modulation, leading to plume variability on short time scales. In a simple unsteady theory of one-dimensional plumes (based on conservation equations for volume, momentum, and buoyancy), this plume variability is manifested as waves that travel down the resulting current. Using numerical techniques applied to the hyperbolic conservation equations, this study investigates the novel concept that these waves may break as they travel down the plumes, triggering intense local mixing between the dense fluid and surrounding ocean. The results demonstrate that the waves break at geophysically relevant distances from the plume source. The location of wave breaking is very sensitive to plume drag from the seabed, the properties of the dense source, and the amplitude and period of the source modulation. To the extent that the simple model represents the real world, these results suggest that wave breaking originating from the tidal modulation of dense plumes could lead to a strong and previously unexplored source of local deep-ocean mixing.

Simulation on Torpedo Unsteady Hydrodynamic with the Movement in the Longitudinal Plane

The unsteady hydrodynamic accurate calculation is the premise of submerged body trajectory design and maneuverability design. Calculation model of submerged body unsteady hydrodynamic with the movement in the longitudinal plane was established, which based on unsteady three-dimensional incompressible fluid dynamics theory. Variable speed translational and variable angular velocity of the pitching motion in the longitudinal plane of submerged body was achieved by dynamic mesh method. The unsteady hydrodynamic could be obtained by model under the premise of good quality grid by the results. Modeling methods can learn from other similar problems, which has engineering application value.

동요하는 날개를 이용한 파랑 중 선박의 추력발생

In this paper, the effects of a foil attached to a ship on the ship motion, added resistance, and thrust generation in waves are analyzed. The unsteady theory for a two-dimensional foil is introduced to determine the coupled motion responses of the ship and foil. The thrust caused by the oscillating foil is evaluated and compared to the added resistance of the ship, so that a positive net thrust can be possible in waves. A comparison of the results of unsteady, quasi-steady, and experimental analyses is performed.

Interdependence between High-Speed Mill Chatter and Unsteady Lubrication of Metal Rolling Process

A metal-rolling process with chatter is chosen as the case for analysis. An isothermal, full film unsteady lubrication model is developed. The influences of time dependent variables such as back tension stress, variation on lubrication are investigated. By utilizing this model, roll forces and torque variation with respect to time can be successfully predicted. Chatter vibration in cold rolling due to unsteady lubrication phenomena is discussed. The current unsteady lubrication theory indicates that, when the nondimensional frequency of back tension stress variation is in a specific range, the roll force and torque variations can be significantly “amplified”.