The Low Reduced Frequency Limit of Vibrating Airfoils—Part I: Theoretical Analysis

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Stokes Equations ◽  
Order Approximation ◽  
Unsteady Aerodynamics ◽  
Mean Value ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Unsteady Pressure ◽  
Reduced Frequency ◽  
Influence Coefficients ◽  
Frequency Regime

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis on its impact on the scaling of the work-per-cycle curves, using an asymptotic approach. A perturbation analysis of the linearized Navier–Stokes equations for real modes at low reduced frequency is presented and some conclusions are drawn. The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the vibration of the airfoil. For lightly loaded airfoils, the unsteady pressure and the influence coefficients (ICs) scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence, the work-per-cycle scales linearly with the reduced frequency for any interblade phase angle (IBPA), and it is independent of its sign. For highly loaded airfoils, the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero and varies linearly with it. In this case, only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and the mode shape in first-order approximation in the reduced frequency. For symmetric cascades, the work-per-cycle scales linearly with the reduced frequency irrespective of whether the airfoil is loaded or not. These conclusions have been numerically confirmed in Part II of the paper.

The Low Reduced Frequency Limit of Vibrating Airfoils: Part II — Numerical Experiments

2015 ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Mean Value ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Navier Stokes Equations ◽  
Unsteady Pressure ◽  
Reduced Frequency ◽  
Influence Coefficients ◽  
Frequency Regime

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis in its impact on the scaling of the work per cycle curves using an asymptotic approach (Part I) and numerical simulations. A perturbation analysis of the linearized Navier-Stokes equations at low reduced frequency is presented and some conclusions are drawn (Part I of the corresponding paper). The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure field caused by the vibration of the airfoil. For lightly loaded airfoils the unsteady pressure and the influence coefficients scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence the work-per-cycle is proportional to the reduced frequency for any inter-blade phase angle and it is independent of its sign. For highly loaded airfoils the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero varies linearly with it. In this case only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and in first approximation of the mode-shape. For symmetric cascades the work-per-cycle scales linearly with the reduced frequency irrespectively of whether the airfoil is loaded or not. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on a low-pressure turbine airfoil section, the NACA0012 and NACA65 profiles and a flat plate operating at different flow conditions to show the generality and correctness of the analytical conclusions. Both the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding and explore the nature of the unsteady pressure perturbations.

The Low Reduced Frequency Limit of Vibrating Airfoils: Part I — Theoretical Analysis

2015 ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Mean Value ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Navier Stokes Equations ◽  
Unsteady Pressure ◽  
Reduced Frequency ◽  
Influence Coefficients ◽  
Frequency Regime

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis in its impact on the scaling of the work per cycle curves using an asymptotic approach (Part I) and numerical simulations (Part II). A perturbation analysis of the linearized Navier-Stokes equations for real modes at low reduced frequency is presented and some conclusions are drawn. The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the vibration of the airfoil. For lightly loaded airfoils the unsteady pressure and the influence coefficients scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence the work-per-cycle scales linearly with the reduced frequency for any inter-blade phase angle and it is independent of its sign. For highly loaded airfoils the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero and varies linearly with it. In this case only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and the modeshape in first approximation. For symmetric cascades the work-per-cycle scales linearly with the reduced frequency irrespectively of whether the airfoil is loaded or not. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on a low-pressure turbine airfoil section, the NACA0012 and NACA65 profiles and a flat plate to show the generality and correctness of the analytical conclusions (Part II of the corresponding paper). Both, the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding and explore the nature of the unsteady pressure perturbations.

The Low Reduced Frequency Limit of Vibrating Airfoils: Part IIIA — Theoretical Quantification and Influence of Unsteady Loading

2016 ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Vibration Mode ◽  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Airfoil Surface ◽  
Navier Stokes Equations ◽  
Reduced Frequency ◽  
Frequency Regime ◽  
Unsteady Loading

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis on its impact on the work per cycle curves. In Part I of the corresponding paper, a perturbation analysis of the linearized Navier-Stokes equations for real modes at low reduced frequency was presented and some conclusions were drawn. It was discovered that a new parameter, the unsteady loading, plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the oscillation of the airfoil. In this third (a) part, the theory is extended in order to quantify the new parameter. It is shown that this parameter depends solely on the steady flowfield on the airfoil surface and the vibration mode-shape. As a consequence, the effect of changing the design operating conditions or the vibration mode onto the work-per-cycle curves (and therefore in the stability) can be easily predicted and, what is more important, quantified without conducting additional flutter analysis. The relevance of the parameter has been numerically confirmed in the Part IIIb of the paper.

Quantification of the Influence of Unsteady Aerodynamic Loading on the Damping Characteristics of Airfoils Oscillating at Low-Reduced Frequency—Part I: Theoretical Support

2016 ◽  
Vol 139 (3) ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Vibration Mode ◽  
Stokes Equations ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Airfoil Surface ◽  
Navier Stokes Equations ◽  
Aerodynamic Loading ◽  
Unsteady Aerodynamic ◽  
Reduced Frequency ◽  
Frequency Regime

The effect of the unsteady aerodynamic loading of oscillating airfoils in the low-reduced frequency regime on the work per cycle curves is investigated. The theoretical analysis is based on a perturbation analysis of the linearized Navier–Stokes equations for real modes at low-reduced frequency. It was discovered that a new parameter, the unsteady loading, plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the airfoil oscillation. Here, the theory is extended in order to quantify this new parameter. It is shown that this parameter depends solely on the steady flow-field on the airfoil surface and the vibration mode-shape. As a consequence, the effect of changing the design operating conditions or the vibration mode onto the work-per-cycle curves (and therefore in the stability) can be easily predicted and, what is more important, quantified without conducting additional flutter analysis. The relevance of the parameter has been numerically confirmed in the Part II of the paper.

Simulation of Unsteady Flow Around a Cylinder

2015 ◽  
Vol 3 (2) ◽  
pp. 28-49
Author(s):  
Ridha Alwan Ahmed
Keyword(s):  
Stokes Equations ◽  
Lift Coefficient ◽  
Reynolds Numbers ◽  
Mean Value ◽  
Navier Stokes ◽  
Cylinder Surface ◽  
Navier Stokes Equations ◽  
Flow Around A Cylinder ◽  
Numerical Grid ◽  
Good Agreement

       In this paper, the phenomena of vortex shedding from the circular cylinder surface has been studied at several Reynolds Numbers (40≤Re≤ 300).The 2D, unsteady, incompressible, Laminar flow, continuity and Navier Stokes equations have been solved numerically by using CFD Package FLUENT. In this package PISO algorithm is used in the pressure-velocity coupling.        The numerical grid is generated by using Gambit program. The velocity and pressure fields are obtained upstream and downstream of the cylinder at each time and it is also calculated the mean value of drag coefficient and value of lift coefficient .The results showed that the flow is strongly unsteady and unsymmetrical at Re>60. The results have been compared with the available experiments and a good agreement has been found between them

scholarly journals On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yat Tin Chow ◽  
Ali Pakzad
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Energy Dissipation Rate ◽  
Mean Value ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Additional Hypothesis ◽  
The Mean ◽  
Deterministic Force ◽  
Martingale Solutions

<p style='text-indent:20px;'>We consider the three-dimensional stochastically forced Navier–Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper bounds of the mean value of the time-averaged energy dissipation rate are derived directly from the equations for weak (martingale) solutions. This estimate is consistent with the Kolmogorov dissipation law. Moreover, an additional hypothesis of energy balance implies the zeroth law of turbulence in the absence of a deterministic force.</p>

The Low Reduced Frequency Limit of Vibrating Airfoils: Part IIIB — Numerical Quantification and Influence of Unsteady Loading

2016 ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Unsteady Aerodynamics ◽  
Operating Conditions ◽  
Mode Shapes ◽  
Navier Stokes ◽  
Influence Coefficient ◽  
Low Pressure Turbine ◽  
Reduced Frequency ◽  
Frequency Regime ◽  
Unsteady Loading ◽  
Loading Parameter

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis on its impact on the work per cycle curves. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on rows of a low-pressure turbine airfoil section, the NACA65 section and a flat plate, to show the correlation between the actual value of the unsteady loading parameter (ULP), theoretically derived in Part IIIa, and the flutter characteristics, for different airfoils, operating conditions and mode-shapes. Both, the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding of the ULP influence in different aspects of the unsteady flowfield. It is concluded that, for a blade vibrating in a prescribed motion at design conditions, the ULP can quantitatively predict the effect of loading variations due to changes in the incidence, and also in the mode shape. It is also proved that the unsteady loading parameter can be used to compare the flutter characteristics of different airfoils.

Unsteady aerodynamics computation and investigation of magnus effect on computed trajectory of spinning projectile from subsonic to supersonic speeds

2019 ◽  
Vol 123 (1264) ◽  
pp. 863-889
Author(s):  
F.A. Chughtai ◽  
J. Masud ◽  
S. Akhtar
Keyword(s):  
Stokes Equations ◽  
Unsteady Aerodynamics ◽  
Navier Stokes ◽  
Magnus Force ◽  
Accurate Analysis ◽  
Navier Stokes Equations ◽  
Eddy Simulation ◽  
Magnus Effect ◽  
Computationally Intensive ◽  
The Stability

AbstractThis paper describes the extensive numerical investigation carried out on a 203-mm spin-stabilised projectile to study the effects of Magnus force at high angles of attack on the stability and flight-trajectory parameters, for further validation and incorporation in a 6-DOF trajectory solver for flight-stability analysis. Magnus force typically influences the course of flight by causing the projectile to drift from its intended path in addition to generation of inbuilt dynamic instabilities in pitch and yaw orientation and is a function of AoA and spin rate. This study is a consolidation of the authors’ previous research on the same caliber projectile but with time-accurate analysis. It has been found that typically, the Magnus force and moment calculation requires time-accurate Navier Stokes equations to be solved numerically for accurate prediction(1,2). Hence, to complete the extraction of static and dynamic coefficients derivatives, unstructured time-accurate CFD analysis on multiple configurations, ranging from subsonic to supersonic Mach regimes, has been evaluated using Large Eddy Simulation (LES) and found to be suitable for capturing the desired effects. However, the LES simulation requires non-dimensional wall distance (y+) of the order of 0.5 – 1, with LES_IQ > 75% thus, is computationally-intensive. In addition, to cover the entire flight envelope from Charge 1 (249 m/s) to Charge 7 (595 m/s), at spin rate from 500 rad/s to 750 rad/s, 30 cases have been evaluated to generate the time-accurate coefficient library for integration with 6-DOF solver analysis. The results obtained have been compared with the available experimental data and found to be in reasonable agreement. The results of 6-DOF solver, incorporating the extracted coefficients, were compared with firing-table results, which further validated the computational methodology. This study provides an insight on how opposite flow interacts with the attached boundary layer due to spin rate and generates a turbulent interacting flow with variation in vortical structures for Q-Criterion vortex-flow visualization.

Numerical Investigation Into the Unsteady Aerodynamics of a Ducted Helicopter Tail Rotor Under Side-Wind Conditions

2010 ◽  
Author(s):  
Marc Kainz ◽  
Florian Danner ◽  
Hans-Peter Kau ◽  
Fre´de´ric Le Chuiton
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Unsteady Aerodynamics ◽  
Drive Shaft ◽  
Navier Stokes ◽  
Wind Effect ◽  
Navier Stokes Equations ◽  
Power Characteristics ◽  
Computational Fluid Dynamics Analysis

A numerical study into the unsteady aerodynamics of a ducted helicopter tail rotor is presented. Computations were carried out for ideal hover flight conditions and under the influence of side-wind. The results are validated against existing experimental performance data. The investigated numerical model incorporates the full annulus turbomachinery components as well as the entire tail boom including non-axis-symmetric casing contour, drive shaft fairing, fin and stabilisers. The rotor is characterised by an uneven azimuth-wise blade distribution, while the stator vanes complemented by the drive shaft fairing are distributed evenly. For the computational fluid dynamics analysis the fully turbulent three-dimensional Favre-averaged Navier-Stokes equations were solved on a Chimera grid system with 24.7 million cells. Besides thrust and shaft power characteristics, the varying blade loading due to the azimuth-wise position and the side-wind effect is analysed.

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