Investigation of aerodynamic instability of a thin plate

This paper is devoted to methods and principles of investigation of aerodynamic instability of structures. Object of interest is a thin plate. This is an important aspect of the design of large-span structures, bridges and other structures sensitive to wind loads. As Russian standards and Eurocode obliges to check the conditions for the occurrence of galloping, divergence and flutter for a certain class of structures. Modern computing facilities allow to calculate aerodynamic coefficients with high accuracy.


Introduction
The relevance of this work is determined by the need to take into account the sensitivity of structures to wind loads that produces aeroelastic vibrations, which can lead to collapse of the structure.This type of structures includes the large-span structures, large-span bridges, thin-walled cooling towers, high-rise buildings, roofs of long-span stadiums and so on.
Exciting analytical, experimental and numerical techniques that allow to assess the possibility of the occurrence of such aerodynamic instability effects as vortex shedding, galloping or divergence need to be improved [1,2].Nowadays the modern numerical methods and computational tools let to perform detailed simulations of the aeroelastic behavior of structures and fluid flow around it, but it is necessary to verify and validate them.

Problem statement
The object of interest is a thin plate (fig. 1) placed in a viscous incompressible turbulent flow (Re= 7•10 5 ).The purpose of this study is an identification of the occurrence of such aerodynamic instability effects for the interesting object, such as vortex shedding, galloping or divergence.We consider two-dimensional problem (at the plane OXY).Therefore, in the third dimension (along the axis OZ) the domain has one element and size equal 0.1 m.For simulation of the turbulent flow RANS/URANS SST turbulence models are used (for steady state and unsteady simulations respectively).Simulation time for unsteady simulations is 40 s with Time step equal 0.005 s.Following boundary conditions are set (Fig. 2): x velocity at the domain inlet is constant and equal 21.7 m/s, turbulence kinetic energy 42.9 m 2 /s 2 and turbulence eddy dissipation 0.94 m 2 /s 3 ; x «Opening» boundary conditions are defined at domain outlet with zero average relevance pressure, turbulence kinetic energy equal 42.9 m 2 /s 2 and turbulence eddy dissipation equal 0.94 m 2 /s 3 ; x «No slip wall» boundary conditions on the surface of the plate are set; x «Symmetry» boundary conditions are set at the left and right side of the domain.

Results
Computational domain was meshed by ANSYS Mechanical preprocessor.We considered 5 different computational grids (table 1).As controlled parameters following characteristics are chosen: aerodynamic coefficients of the drag (Cd) and the lift (Cl) forces.Model 2 (fig.3) was chosen as a basic model for steady state simulations, Model4 (fig.3) was chosen as a basic model for unsteady simulations.In Ref. [3] the author shows the dependence of the aerodynamic coefficients of the drag (Cd) and the lift (Cl) forces on angle of attack α for the infinite thin plate with a rectangular cross-section.Graphs of the dependence of the obtained aerodynamic coefficients on the angle of attack α in comparison with the coefficients from [3] are presented below.

Galloping
Galloping is a self-excited oscillations of flexible structures in the form of bending vibrations along the normal to the direction of the wind As a result of researches by scientists such as Den-Hartog and Glowrt, the necessary condition for the occurrence of an aerodynamic instability was obtained: According to the Eurocode [4], aerodynamic instability with galloping occurs when the wind speed m V reaches a value CG V at which the oscillating process begins with an increasing amplitude.
The critical wind velocity of galloping, CG V , is given in Expression (3): where: G -structural damping expressed by the logarithmic decrement; U -air density under vortex shedding conditions; m -equivalent mass; b -reference width of the cross-section at which resonant vortex shedding occurs.
Values of the aerodynamic coefficients of the drag (Cd) and the lift (Cl) forces at various angles of attack were determined by performing steady state simulations using specialized software ANSYS CFX.Then, according to formula (1), the Glowrt -Den-Gartog criterion H was calculated, and the critical velocity of the beginning of the oscillatory process CG V was determined using formula (3) (at f1 = 0.4 Hz, Sc = 551; ag = ks•H, where ks = 0.9).For all angles, the critical velocity exceeded the value of 1 .27 25 . 1 m V m / s, despite the fact that for some angles of attack the Glowrt -Den-Gartog criterion H is fulfilled (i.e.there is a possibility of the occurrence of the galoping).In the table 2, the angels of attack for which the Glowrt -Den-Gartog criterion H is fulfilled are marked in red.

Vortex shedding
Vortex shedding occurs if the vortices periodic break from the opposite edges of the structure and, as the result, variable load perpendicular to the direction of the wind occur.
If the frequency of the vortex excitations coincides with the natural frequency of the structure, large amplitude oscillations may arise.This occurs at the so-called critical wind speed: where: b -reference width of the cross-section at which resonant vortex shedding occurs and where the modal deflection is maximum for the structure or structural part considered;  According to [4], the effect of vortex shedding should be investigated when the ratio of the largest to the smallest crosswind dimension of the structure, both taken in the plane perpendicular to the wind, exceeds 6.
The effect of vortex shedding need not to be investigated when where: i cr V , -is the critical wind velocity for mode i; m V -is the characteristic 10 minutes mean wind velocity.
Below there are the results of unsteady simulations, performed using the specialized software ANSYS CFX.The main frequencies of vortex shedding at various angles of attack were determined (Fig. 4).Then, for two lowest (main) frequencies the Strouhal numbers Sh were calculated and the critical wind speed (at f 1 = 0.4 Hz) was determined using formula (5).For all angles of attack the critical wind speed did not exceed the value 1 .27 25 . 1 m V m/s; therefore, according to the Eurocode, vortex shedding can occur at the considered angles of the wind attack (table 3).

Divergence and Flutter
According to [4], such types of dynamic instability as divergence and flutter, which occur if structural deformations cause an aerodynamic loads changing, should be excluded in principle.To be prone to either divergence or flutter, the structure satisfies all of the three criteria specified in the Eurocode.If all the conditions are met, it is necessary to check the structure under study for the possibility of a flutter or divergence arising from the condition: The critical wind velocity for divergence div V is given in Expression (8): Results of the unsteady simulations are shown below.Values of the aerodynamic moment Cmz at different angles of attack were determined.Since all three conditions specified in the Eurocode are satisfied for the present structure, the critical wind speed for divergence was calculated (for k Θ = 14.23•10 4 Nm 2 , U = 1.185 kg / m 3 ) for all angels of attack according to formula (8).In the table 4, those angles for which the critical wind speed div V does not exceed 4 .43 2 m V m/s are marked in red.For these angles, when a corresponding wind speed is reached, a divergence may occur (Table 4).

Conclusions
Present paper is devoted to the computational investigations of aerodynamic instability of thin flat plate placed in a viscous incompressible turbulent flow.For this structure a check of the conditions for the occurrence of galloping, divergence and flutter was conducted.
Obtained results can be used as a reference for the investigation of the sensitivity of structures like thin roofs of the large-span stadiums or bridges to wind loads.

Fig. 3 .
Fig. 3. Aerodynamic coefficients of the drag (Cd) and the lift (Cl) forces depending on the angle of attack α.Comparison of the results of the simulations (Model 2) with the reference [3].
i f -cross-wind fundamental frequency of the structure; b -the width; g a -factor of galloping instability; Sc -Scruton number: i f -natural frequency of the considered flexural mode i of cross-wind vibration; Sh -Strouhal number.

Fig. 4 .
Fig.4.Results of the unsteady simulations (Model 4) for angel of attack α=0° and α=5°from left to right.Field of instantaneous velocity [m/s], aerodynamic coefficient of the lift force (Cl), PSD for Cl -from top to bottom.

4 k
-torsional stiffness; M c -aerodynamic moment coefficient; 4 d dc M -rate of change of aerodynamic moment coefficient with respect to rotation about the torsional centre, 4 is expressed in radians; M -aerodynamic moment of a unit length of the structure; U -density of air; d -in wind depth (chord); b -width.

Table 1 .
Variants of the computational grids and the corresponding values of the aerodynamic coefficients of the drag (Cd) and the lift (Cl) forces.Angle of the wind attack α=1°.

Table 2 .
Aerodynamic coefficients of the drag (Cd) and the lift (Cl) forces, the Glowrt -Den-Gartog criterion H, the critical wind velocity CG V on angle of the wind attack α.

Table 3 .
Strouhal numbers Sh and critical wind velocity

Table 4 .
Aerodynamic moment coefficients Cmz and critical wind velocity div