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.

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.

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 ◽  
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 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.

Assessing the Dynamic Stability of an Offshore Supply Vessel

2011 ◽  
Author(s):  
Vladimir Shigunov ◽  
Ould el Moctar ◽  
Thomas E. Schellin ◽  
Jan Kaufmann ◽  
Rasmus Stute
Keyword(s):  
Dynamic Stability ◽  
Stokes Equations ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Ship Motions ◽  
Navier Stokes Equations ◽  
Counter Measures ◽  
Seakeeping Analysis ◽  
Design Speed ◽  
Run Up

The dynamic stability was investigated of a typical offshore service vessel operating under stability critical operating conditions. Excessive roll motions and relative motions at the stern were studied for two loading conditions for ship speeds ranging from zero to the design speed. A linear frequency-domain seakeeping analysis was followed by nonlinear time-domain simulations of ship motions in waves. Based on results from these methods, critical scenarios were selected and simulated using finite-volume solvers of the Reynolds-averaged Navier-Stokes equations to understand the phenomena related to dynamically unstable ship motions as well as to confirm the results of the simpler analysis methods. Results revealed the possibility of excessive roll motions and water run-up on deck; counter measures such as a ship-specific operational guidance are discussed.

Numerical Characterisation of Jet-Vane based Thrust Vector Control Systems

2015 ◽  
Vol 65 (4) ◽  
pp. 261 ◽  
Author(s):  
M.S.R. Chandra Murthy ◽  
Debasis Chakraborty
Keyword(s):  
Vector Control ◽  
Control Systems ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Static Tests ◽  
Thrust Vector Control ◽  
Thrust Vector

<p>Computational fluid dynamics methodology was used in characterising jet vane based thrust vector control systems of tactical missiles. Three-dimensional Reynolds Averaged Navier-Stokes equations were solved along with two-equation turbulence model for different operating conditions. Nonlinear regression analysis was applied to the detailed CFD database to evolve a mathematical model for the thrust vector control system. The developed model was validated with series of ground based 6-Component static tests. The proven methodology is applied toa new configuration.</p><p><strong>Defence Science Journal, Vol. 65, No. 4, July 2015, pp. 261-264, DOI: http://dx.doi.org/10.14429/dsj.65.7960</strong></p>

3-D Numerical Simulation for Fuel Cell Performance

2002 ◽  
Author(s):  
Tien-Chien Jen ◽  
S. H. Chan ◽  
T. Z. Yan
Keyword(s):  
Fluid Flow ◽  
Fuel Cell ◽  
Stokes Equations ◽  
Current Density Distribution ◽  
Mass Transfer Process ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fuel Cell Performance ◽  
Gas Channel

A 3-D mathematical model for the PEM fuel cell including gas channel has been developed to simulate fluid flow, current density distribution, and multi-component transport. In order to understand the developing fluid flow and mass transfer process inside the fuel cell channels, the conventional Navier-Stokes equations for gas channel, and volume-averaged Navier-Stokes equations for porous gas diffusers and catalyst layer are adopted individually in this study. A set of conservation equations and species concentration equations are solved numerically in a coupled gas channel and porous media domain using the vorticity-velocity method with power law scheme. Detailed development axial velocity and secondary flow fields at various axial positions in the entrance region are presented. Polarization curves under various operating conditions are demonstrated by solving the equations for electrochemical reactions and the membrane phase potential. Compared with experimental data from published literatures, numerical results of this model agree closely with experimental results. Finally, mass transport equations are solved at a preset condition of electrochemical reaction, and oxygen and hydrogen mole fraction distribution fields are displayed.

A Multiscale Method Modeling Surface Texture Effects

2006 ◽  
Vol 129 (2) ◽  
pp. 221-230 ◽  
Author(s):  
Alex de Kraker ◽  
Ron A. J. van Ostayen ◽  
A. van Beek ◽  
Daniel J. Rixen
Keyword(s):  
Surface Texture ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Stokes Equations ◽  
Multiscale Method ◽  
Operating Conditions ◽  
Contact Load ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Inertia Effects

In this paper a multiscale method is presented that includes surface texture in a mixed lubrication journal bearing model. Recent publications have shown that the pressure generating effect of surface texture in bearings that operate in full film conditions may be the result of micro-cavitation and/or convective inertia. To include inertia effects, the Navier–Stokes equations have to be used instead of the Reynolds equation. It has been shown in earlier work (de Kraker et al., 2006, Tribol. Trans., in press) that the coupled two-dimensional (2D) Reynolds and 3D structure deformation problem with partial contact resulting from the soft EHL journal bearing model is not easy to solve due to the strong nonlinear coupling, especially for soft surfaces. Therefore, replacing the 2D Reynolds equation by the 3D Navier–Stokes equations in this coupled problem will need an enormous amount of computing power that is not readily available nowadays. In this paper, the development of a micro–macro multiscale method is described. The local (micro) flow effects for a single surface pocket are analyzed using the Navier–Stokes equations and compared to the Reynolds solution for a similar smooth piece of surface. It is shown how flow factors can be derived and added to the macroscopic smooth flow problem, that is modeled by the 2D Reynolds equation. The flow factors are a function of the operating conditions such as the ratio between the film height and the pocket dimensions, the surface velocity, and the pressure gradient over a surface texture unit cell. To account for an additional pressure buildup in the texture cell due to inertia effects, a pressure gain is introduced at macroscopic level. The method also allows for microcavitation. Microcavitation occurs when the pressure variation due to surface texture is larger than the average pressure level at that particular bearing location. In contrast with the work of Patir and Cheng (1978, J. Lubrication Technol., 78, pp. 1–10), where the microlevel is solved by the Reynolds equation, and the Navier–Stokes equations are used at the microlevel. Depending on the texture geometry and film height, the Reynolds equation may become invalid. A second pocket effect occurs when the pocket is located in the moving surface. In mixed lubrication, fluid can become trapped inside a pocket and squeezed out when the pocket is running into an area with higher contact load. To include this effect, an additional source term that represents the average fluid inflow due to the deformation of the surface around the pocket is added to the Reynolds equation at macrolevel. The additional inflow is computed at microlevel by numerical solution of the surface deformation for a single pocket that is subject to a contact load. The pocket volume is a function of the contact pressure. It must be emphasized that before ready-to-use results can be presented, a large number of simulations to determine the flow factors and pressure gain as a function of the texture parameters and operating conditions have yet to be done. Before conclusions can be drawn, regarding the dominanant mechanism(s), the flow factors and pressure gain have to be added to the macrobearing model. In this paper, only a limited number of preliminary illustrative simulation results, calculating the flow factors for a single 2D texture geometry, are shown to give insight into the method.

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.

The Validity of the Compressible Reynolds Equation for Gas Lubricated Textured Parallel Slider Bearings

2012 ◽  
Author(s):  
Mingfeng Qiu ◽  
Brian Bailey ◽  
Rob Stoll ◽  
Bart Raeymaekers
Keyword(s):  
Gas Flow ◽  
Reynolds Equation ◽  
Stokes Equations ◽  
Hydrodynamic Lubrication ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slider Bearings ◽  
Simplifying Assumptions ◽  
Simulation Results

The Navier-Stokes and compressible Reynolds equations are solved for gas lubricated textured parallel slider bearings under hydrodynamic lubrication for a range of realistic texture geometry parameters and operating conditions. The simplifying assumptions inherent in the Reynolds equation are validated by comparing simulation results to the solution of the Navier-Stokes equations. Using the Reynolds equation to describe shear driven gas flow in textured parallel slider bearings is justified for the range of parameters considered.

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