Influence of fluid-structure interaction on vortex induced vibration and lock-in phenomena in long span bridges

Abstract

In this paper, deck models of a cable stayed bridge are generated in ABAQUS-finite element program once using only CFD model (one-way fluid-structure interaction) and another by using both the CFD model and the CSD model together (two-way fluid-structure interaction) in a co-simulation. Shedding frequencies for the associated wind velocities in the lock-in region are calculated in both approaches. The results are validated with Simiu and Scanlan results. The lift and drag coefficients are determined for the two approaches and the latter results are validated with the flat plate theory results by Munson and coauthors. A decrease in the critical wind velocity and the shedding frequencies considering two-way approach was determined compared to those obtained in the one-way approach. The results of the lift and drag forces in the two-way approach showed appreciable decrease in their values. It was concluded that the two-way approach predicts earlier vortex induced vibration for lower critical wind velocities and lock-in phenomena will appear at lower natural frequencies of the long span bridges. This helps the designers to efficiently plan and consider for the design and safety of the long span bridge against this type of vibration.

Keywords

vortex-induced vibration / fluid-structure interaction / Strouhal number / lock-in / kinetic energy

Introduction

Vortex-induced vibration is a strong fluid-structure interaction phenomenon. The application of fluid-structure interaction concept in the vibration of long span Bridges is a sensitive and important step in understanding the actual behavior of the structure during vibration resulted from a wind excitation. The vortex induced vibration of the deck is a type of vibration results from the fluid-structure interaction between the wind and the deck of the Bridge. The application of fluid-structure coupling using numerical simulations is a complicated problem. It does result in arising difficulties related to the fluid and the structure simulations in addition to coupling of these two systems which is a hard process. These difficulties resulted from the coupling process depends highly on the physical properties of the problem which is under simulation. The complexity of fluid-structure interaction cases is due to the structural nonlinear boundary conditions being imposed on the boundaries of the moving fluid where the position is a part of the solution. Eulerian fluid equation integration is necessary because the moving position of the structure is prescribing a part of the fluid boundary. These cases are usually considered a two-field coupled case, while the moving mesh can be considered different structural case, as a result the complete coupled case can be constructed as a three-field system: the structure, the fluid and the moving mesh. Lagrangian description is used for the deformation of the structure for the case of fluid–structure interaction [ 1– 51]. The fluid flow is taking place in the domain with the boundary resulted by the deformation of the structure that is exposed to change during time and it is influenced back by the flow of the fluid, so the Arbitrary Lagrangian Eulerian ALE description should be used in this situation. Noh (1964) introduced the concept of the moving and deforming frame ALE. This concept was implemented in finite element analysis by (Donea et al., 1977) [ 52– 60].

Cases of fluid-structure interaction comprise a single or multiple solid structures interacting with an internal or external surrounding fluid flow. Fluid structure interaction cases have prominent roles in wide scientific and engineering areas, up today a comprehensive study of such cases remains a challenge because of the strong nonlinear and multidisciplinary nature (Chakrabarti 2005, Dowell and Hall 2001, Morand and Ohayon 1995). In the most fluid structure interaction cases, the model equations do not accept analytical solutions and it is impossible to determine, whereas the experiments in the laboratory are limited in the scope; as a result in order to investigate the fundamental physics involving in the complex interaction between the solids and fluids, many numerical simulations may be utilized. Fluid structure interaction within finite element models is implemented using a node to surface contact algorithm. This fluid structure interaction algorithm is standard and it consists of three steps; in the first step, the surfaces and the their normal are being computed, and in the second step, a contact search for the nodes of the fluid that may contact with the surface is proceeded, in the third step, penalty forces are being applied to the fluid nodes to prevent the fluid from being penetrated at the surfaces [ 1– 10, 61– 69].

Multi-physics cases are very difficult to being solved using analytical methods. They mostly should be solved by using numerical simulations or using experiments. The most advanced techniques and the reputed commercial software in CFD and CSM have made the numerical simulation possible. Two approaches are available to solve fluid structure interaction cases by the use of software, the monolithic approach and the partitioned approach. In the monolithic approach, formulation as one combined problem is being constructed for the sub problems (the fluid and the structure). The discretization of the governing equation resulting in a system of algebraic equations, are being solved as a whole. The process of interaction between the fluid and the structure at the interface is dealt with synchronously. This will lead to possible conservation of the properties at the interface which increases the stability of the solution. This approach is more robust compared to the partitioned approach. One of the properties of this approach is that it is computationally expensive and haven't the ability to make advantage from the software modularity while the partitioned approach is able to make (see Fig. 1).

In the partitioned approach, sub problems are solved each in a separate way which indicates that the flow is not changing while the structural solution is solved. The governing equations of the flow and the structural displacement are solved alternatively in time parallel with two distinct solvers. Prescription of the intermediate fluid solution as a boundary condition for the structure is being done and vice versa, and continuous iteration are utilized until the criterion of the convergence is satisfactory. At the boundary between the solid and the fluid, information is exchanged according to the application of coupling technique type. The time lag between the integration of the fluid and the structure domains results in implementing the interface conditions asynchronously, which leads to a possibility of properties conservation loss. This problem can be polished by utilizing mapping algorithms. Therewith, this allows t preserve the modularity of the software (see Fig. 2).

Exchange of the information is done at the interface between the two solvers, this is called the coupling which is of two types, the first is the one-way coupling and the second is the two-way coupling. In one-way coupling, the motion of the fluid flow influences the solid structure but the reaction of the solid structure on the fluid flow is neglected. Also the two-way coupling can be possible. The calculation of the fluid flow is performed until reaching convergence. The resulted forces at the interface from the fluid calculation are being interpolated to the structural mesh. Then the calculations of the structural dynamic are performed until meeting convergence criterion. The process is repeated until reaching the end time. In two-way coupling the motion of the fluid flow influences the solid structure and in the same time the fluid flow is influenced by the reaction of the structural solid. During the beginning of the time step, convergence solutions of the fluid calculation produce forces acting on the solid structure. Next, the forces are interpolated to the structural mesh the same as in the one-way coupling and the solutions from the structural solver is determined with the fluid forces as boundary conditions. As a result the mesh is deformed according to the response of the structure. The values of the displacements are interpolated to the fluid mesh which results in deforming the fluid domain. The process is repeated until reaching the convergence between the force and displacement values below the limit that was previously determined [ 11– 23, 70].

The accuracy of the two-way coupling is obvious, especially in large deflection cases where the structural deformation influences the fluid field strongly. In strong two-way coupling, solutions are of second order time accuracy. And they are more stable (Vaassen et al., 2010). In one-way coupling conservation of energy at the interface is not guaranteed, but in two-way coupling is. There is a benefit of one-way coupling simulations that ends with significant lower computational time. The second benefit is that the fluid mesh has deformations without need to be calculated, which result in producing a constant quality of the mesh. Fluid structure interaction effects for the cases of totally or partially submerged body in a fluid flow with a free surface cannot be accurate predicted especially in cases related to civil engineering, offshore engineering and naval architecture. Accurate solving of fluid structure interaction coupling problem faces difficulties that can be originated due to three reasons: first, the prediction of motion of the submerged solid generated due to the interaction forces are difficult when the distortion of the finite elements discretizing the fluid domain is minimized, as a result this will reduce the need for re meshing. Second, unknown obstacles that are related to the process of solving the constraint equation that is responsible of identifying the position of fluid particles on the free surface boundary in turn. Finally, the incompressible fluid dynamic equations which are solved numerically have difficulties and typically include essential nonlinearities excluding simple and limited flow model [ 22– 34, 70, 71].

The application of fluid-structure interaction concept in the vibration of long span Bridges is a sensitive and important step in understanding the actual behavior of the structure during vibration resulted from a wind excitation. The vortex induced vibration of the deck is a type of vibration results from the fluid-structure interaction between the wind and the deck of the Bridge. When a bridge deck is excited by a wind, it starts to oscillate in the in-line and the cross-flow directions. The in-line oscillation often takes place at twice the frequency of the cross-flow oscillation, and it is very small compared to the cross-flow oscillation. Therefore it is not important in the majority of the engineering applications. When the cross-flow oscillation amplitude of the deck is large enough, the fluid-structure interaction enhances and increases the strength of the vortices or the mean drag forces on the deck. In the same time, the motion of the deck will alter the phase, sequence and the vortices pattern in the wake region. If the wind is with an intermediate and high Reynolds number, the wind around the deck starts to oscillate due to vortex shedding, and these shedding vortices exert oscillating forces on the deck in the direction perpendicular to the deck and the wind. The shedding frequency of the vortices is controlled by the deck movement. If the first natural frequency of the structural system is near enough to the shedding frequency of the vortices, the structural frequency controls the vortex shedding frequency even if the wind velocity, and consequently the nominal Strouhal frequency are varied within a certain range, so the vortices will shed at the natural frequency instead of the frequency determined by the Strouhal number and this is called lock-in (Simiu and Scanlan, 1996). This situation is a result of nonlinear interaction between the deck oscillation and the wind action. The Strouhal number is a dimensionless parameter which describes the shedding of the vortices of the wind flow in the wake region, and it is a function of the structural shape and the Reynolds number, where the latter is a function of the velocity of the wind velocity, structural diameter and the kinematic viscosity coefficient of the wind. The value of the Strouhal number is 0.2 which is constant for a wide Reynolds number range approximately, where this range is called subcritical range and is 300-2×10⁶. The lock-in range depends on the change in the rate of wind velocity and the vibration amplitude of the deck, where the larger vibrations amplitude will hold lock-in for a larger range of wind velocity compared with smaller vibrations amplitudes [ 72– 83].

Munson et al. (1998) studied the character of the drag coefficient as a function of Reynolds number for objects with various degrees of streamlining, from a flat plate normal to the upstream flow to a flat plate parallel to the flow (two-dimensional), where this value is related to the cross flow oscillation of the body. They calculated the value of the drag coefficient for a flat plate parallel to the flow which is simulating the bridge deck subjected to a wind flow, this value's range was (0.08–0.0075) for the Reynolds number range (0.2×10⁶–2.3×10⁶) [ 84] .

To understand the influence of the fluid-structure interaction on the generation of vortex induced vibration in the deck of long span bridges in addition to its role in thoroughly predicting the lock-in phenomena which is related to high amplitudes that might be the cause of a structural failure of the long span bridge, in this paper numerical simulations of 2D deck models of a long span bridge are generated in ABAQUS using two approaches, once using CFD model or one-way fluid-structure interaction approach and another using both CFD and CSD models or two-way fluid-structure interaction approach. These models are excited by multiple wind velocities to the duration of 100 s. The vortex shedding and lock-in phenomena with the associated shedding frequencies are simulated and determined for each approach to detect their effect on the vibration of the deck of the long span bridge. The effect of Reynolds number and the Strouhal number in each approach is studied and the lift and drag forces generated due to this type of vibration are determined with their coefficients, these results are compared between the two approaches and validated supporting on benchmarks from the literature.

Vortex-induced vibration

Vortex-induced vibration can be known as the aerodynamic vibration which is caused by the alternate vortices generated from the interaction between the objects and wind flow. Normally, the dimensionless constant called Strouhal number is used to evaluate the characteristic property of each bluff-body in wind flow by means of the phenomenon called the vortex shedding behind the object as shown in Eq. (1) below [ 85– 96].

(1) S={\displaystyle \frac{N_{\mathrm{s}}D}{U}},

where the Strouhal number S depends on body geometry and the Reynolds number Re,D is the across-wind dimension, U is the mean velocity and N_s is the primary frequency of the vortex shedding. Reynolds numberis one of the non dimensionless hydrodynamic numbers that is used to describe the flow around a body. By the definition, the Reynolds number is the ratio of the inertia forces to viscous forces and formulated as:

(2) Re={\displaystyle \frac{DU}{v}},

where v is the kinematic viscosity of the wind flow.

The changes of the Reynolds number create separation flows in the wake region, which are called vortices. At low values of Re ( Re<5), there no separation occurs. When Re is further increased, the separation starts to occur and becomes unstable and initiates vortex shedding at certain frequency. The vortex shedding process is defined by the Reynolds number and the shedding frequency by the Strouhal number [ 97].

When the vortices shed in the wake region is exciting the driving an elastically supported body periodically, it will undergo a small response unless the Strouhal frequency reaches close to the across-flow frequency of the structure. The surface pressure generates in-line and across-flow forces with frequencies 2N_s and N_s simultaneously, where a pitching moment has a frequency N_s. Here, the body interacts with the flow strongly. This frequency controls the vortex shedding even if the flow velocity changes, which results in change of the Strouhal frequency away by a few percent from the natural frequency, this phenomena is called lock-in.

It is known that the across-wind force increases until limited cycle of oscillation amplitude reached during synchronization. Also the along-structure correlation of force increases with oscillation amplitude. The most qualified models of the mathematical modeling of lock-in phenomena are basing on coupled oscillators representing the structure and the wake (Diana et al., 2006). Practically, the models of the single degree of freedom are usually considered in the structural analysis, where a classical one is proposed in (Simiu and Scanlan, 1996), by expressing the across-wind force as:

(3) L_{\mathrm{S}}=q[C_{\mathrm{L}\mathrm{S}}(K)\sin (\omega t+\phi )+Y_{\mathrm{1}}(K)(\mathrm{1}-\epsilon (K){\displaystyle \frac{y^{\mathrm{2}}}{D^{\mathrm{2}}}}){\displaystyle \frac{\dot{y}}{U}}+Y_{\mathrm{2}}(K){\displaystyle \frac{y}{D}}],

where C_LS, Y₁, Y₂, and \epsilon are experimental parameters, functions of the Strouhal reduced frequency K_S=f_{S}D/U . In Eq. (3), three terms are distinguished: harmonic forcing term, a nonlinear aerodynamic damping term, and an aerodynamic stiffness term. The harmonic term represents the alternating lift where lock-in does not occur due to vortex-shedding in the velocity ranges. The forcing term becomes negligible (C_LS \simeq 0) in the region close to lock-in, and the other terms are controlling the synchronized oscillations. Using Van der Pol oscillator, the nonlinear damping term represents the self-limiting oscillations and the stiffness term holds the frequencies synchronized in the lock-in region [ 90, 98].

Avoiding synchronization is suggested by Eurocode 1, but this phenomena has been observed where the wind velocity can range between wide limits, especially because a wide range of wind velocities result in the development of the phenomena. Modifying the cross-section geometry of the bridge deck changes the vortex shedding behavior as a result changing the amplitude and the frequency of the oscillations but it does not development of the phenomenon [ 99, 100].

One-way fluid-structure interaction

Interaction between fluids and structures problems can be considered uncoupled problems within both separate domains. In these problems assumption is formulated that a domain is driven by another, but the driven domain has no feedback effect on the driving domain. CFD model simulation provides the pressure on the structure by the fluid, and they are applied like load condition or as a boundary for the finite element analysis simulation of the configuration, where no feedback is done by the deflection into the CFD, which is called one-way fluid structure interaction. Many civil structures are excited by dynamic wind loading which is an example of one-way fluid structure interaction. In these cases, the structural motion and displacement have no appreciable effect on the driving wind loads, where the structure can be separately analyzed, or by treating the structure as uncoupled from the driving wind under certain loads in the range of interest [ 71– 75, 101– 104].

Two-way fluid-structure interaction

When both the fluid and the structure are interacting in a system of feedback in some practical applications in the engineering area, the structural displacements generated by the fluid motion enhance the fluid forces, and this is called two-way fluid structure interaction. In this case often the amplitude of the deflection of the structure is large. The wind power plant is one example of this type of analysis [ 70, 73– 77].

In the general form, the fluid structure interaction can be represented by the coupling both the equation of the motion of the structure and the fluid. The equation of the motion of the structure can be expressed as:

(4) M_{\mathrm{S}}\ddot{u}+D_{\mathrm{S}}\dot{u}+K_{\mathrm{S}}u=F_{\mathrm{S}},

where M_S is the structural mass matrix, D_S the structural damping matrix, K_S the structural stiffness matrix, F_S the applied load vector, and u the nodal displacement vector where the dot denotes the time derivative.

When the stagnant fluid is surrounding the oscillating structure, the fluid effects on the structure should be accounted for. While the structure is oscillating, the close fluid to the surface of the structure is starting to undergo motion in such a way that the structure becomes affected by additional fluid force F_S. As a result Eq. (4) represents the forced oscillation form, where F_S is the fluid reaction on the structural movement, and the decomposition of the force into three parts acting in-phase with the displacement u, the velocity \dot{u} and the acceleration \ddot{u} of the structure is possible [ 78– 80].

Finite element models

The models of the segmental bridge deck are generated in ABAQUS once in CFD and another in CSD. The CFD model is with a dimension 2.6 m height and total width of 22 m as shown in Fig. 3(a). The thickness is 0.01 m (should be very small so as to be treated as a 2D model). The flow domain size is 140 m length and 40 m height, the position of the deck model in the flow domain should be in a position so that to facilitate a proper area to show the vortex shedding in the downstream in addition to an appropriate area above and under the deck model to show the boundary layers around the deck with the separation points (see Fig. 3(b)). Material properties for the CFD model are assigned where the air density is assigned 1.29 kg/m³ and the dynamic viscosity of the air is assigned 1.8E‒05 Pa.s.

The CFD model part is meshed using CFD element fluid family with FC3D8: A-8 node linear fluid brick. The deck wall assigned with 0.4 element size and the flow domain with 1 element size.

The CSD model have the following material properties concrete density is assigned 2643 kg/m³, Youngs modulus 200E+08 and the Poissons ratio 0.2 (Fig.3(c)). The role of the steel reinforcement is neglected for simplicity because the thickness of the CSD model is too small 0.01m and the reinforcement has no direct contact with the air.

A flow step with 100 s duration is created for the CFD model in addition to assigning predefined Spalarat-Allmaras model turbulence due to very large number of Reynolds number and or due to random oscillation of the time history curve of the kinetic energy dissipated, and creating another dynamic implicit step with 100 s for the CSD model. The step time in each model must be the same, in addition to adding interaction with the same name in each model for the surfaces of interaction, also the important issue related to mesh generation is equality of the mesh size in each model.

The job is created for the simulation of CFD model apart or one-way fluid-structure interaction approach, and another co-simulation job is generated for both the CFD and the CSD models together or two-way fluid-structure interaction approach.

Boundary conditions

Four boundary conditions are defined for the CFD and CSD models (Fig. 4), fluid B.C for the inflow and far fields assigning the air velocity value in the horizontal direction only (zero attack angle). The other two directions with zero magnitudes, fluid B.C for the outflow assigning zero pressure and fluid B.C for the front and back of the flow model with zero velocity magnitude for the third direction perpendicular to the model (z-direction), and no-slip fluid B.C for the wall condition of the deck.

Mesh size

The vortex shedding in the flow domain at the downstream of the deck in the velocity field depends on the mesh size around the deck and the downstream so that to facilitate a better simulation for the boundary layers and separation regions, in addition to the vortex shedding periodically. Figure 5 shows the velocity field of the result with mesh size 28074 elements where the vortex shedding is not simulated correctly compared to the actual phenomenon, the vortices shedding is not in an asymmetric style, this is due to very fine mesh size around the deck model.

Where the vortex shedding simulation in Fig. 6 with mesh size 9943 elements is the best simulation of the actual phenomena, this is because the vortices shed is in an asymmetric form and the shedding is near the trail of the deck at the wake region, and it worthy that the mesh size in this case is appropriate, while the vortex shedding simulated in the model in Fig. 7 with mesh size 7724 elements is simulated in a weak style because the shedding of vortices is relatively far from the trail of the deck despite the asymmetric style, and this indicates that the mesh size is not suitable. The models in Figs. 8 and 9 with mesh sizes 6202 and 5317 elements simultaneously are not showing the vortex shedding at all due to the coarse mesh sizes of the models. They look like to have a very large Reynolds number especially high velocity magnitude of the wind flow.

As a result the mesh size of 9943 elements would be utilized to mesh both CFD and CSD models to run the co-simulations considering fluid-structure interaction influence on the vortex induced vibration and the lock-in phenomena in the deck.

The mesh size of the deck in the structural domain should be the same as the mesh size of it in the flow domain at the interface so that to match or coincide. Depending on the desirable mesh size for the CFD model, the related mesh size of the CSD model was 357 elements (see Fig. 10).

Results and discussion

Vortex shedding simulation

Generation of vortex shedding from the deck models exists with different patterns, and the screen shots are considered at the time 100 s. When considering one-way fluid-structure interaction approach the vortex shedding for wind velocity 1 m/s (see Fig. 11) is generated and it is regular in periodic shedding from the tail of the deck, but for two-way fluid-structure interaction approach and for the same wind velocity a simple vortex shedding exists with irregular shedding because the wind flow is affected by the feedback of the deck which disturbs the periodic shedding style of the vortices shed from the deck tail where the wind velocity is not high (see Fig. 12).

For wind velocity 10 m/s a very good and efficient regular vortex shedding style from the deck exists for one-way fluid-structure interaction approach (see Fig. 13) but for two-way fluid-structure interaction approach and for the same wind velocity, the vortex shedding is generated with a different style which is an evidence of effect of the deck feedback (CSD simulation) on the surrounding wind flow, where increasing the wind velocity increases the lift and drag forces as a result the disturbance in the vortices will be detected (see Fig. 14).

While for wind velocity 15 m/s another pattern of vortex shedding exists for one-way fluid-structure interaction approach which is affected by increasing the wind velocity where it changes the shedding style and shape of the vortices (see Fig. 15), but for two-way fluid-structure interaction approach and for the same wind velocity the vortex shedding is generated but without a vanishing pattern, this happens because for higher wind velocities higher than the associated lock-in velocities leads to vanishing of the vortex shedding for a range velocities because the amplitudes of vibrations of the deck are increasing due to an increase in the lift and drag forces, hence the feedback of the deck on the wind flow increases (see Fig. 16) and it is possible that the lock-in phenomena and the vortex shedding generation restarts at higher wind velocities where this needs a very long duration simulation process and computationally is much costly.

Lift forces

The lift forces are generated in the deck of the cable stayed bridge models with different values when assigning wind flow of multiple velocities (1, 5, 10 and 15 m/s) first by considering one-way fluid-structure interaction approach in the simulation and the other by considering two-way fluid-structure interaction approach. Increasing the wind velocity leads to increase in the lift forces generated in the deck models commonly in both approaches in a direct proportion. When the wind velocity is 1 m/s, the time history of the lift forces in one-way fluid-structure interaction is showing a certain value with a stable situation without vertical periodic oscillation of the deck after 40 s, while for the same wind velocity in two-way fluid-structure interaction, the lift forces have smaller values but showing a primary simple vertical periodic oscillation after 55 s (see Fig. 17).

When the wind velocity increases to 5 m/s, the lift forces in one-way fluid-structure interaction increase in a high rate with a stable situation after 20 s without vertical periodic oscillation. For the same wind velocity the lift forces are increasing with a high rate in two-way fluid-structure interaction with a better vertical periodic oscillation starting after 10 s (see Fig. 18).

Increasing the wind velocity to 10 m/s, the lift forces in one-way fluid-structure interaction are increasing with a high rate and the stability of the lift forces are obvious starting from 3 s without any vertical periodic oscillation, while for the same wind velocity the lift forces in two–way fluid-structure interaction are increasing in a high rate too, but now the vertical periodic oscillatory behavior is much better seen starting from 5 s (see Fig. 19).

Now for the wind velocity of 15 m/s, the lift forces for one-way fluid-structure interaction are increasing with a high rate and show stability without showing vertical periodic oscillation after 2 s, but in two-way fluid-structure interaction and for the same wind velocity, the lift forces are increasing with a high rate exhibiting significant vertical periodic oscillation with bigger frequency after 3 s (see Fig. 20).

The difference in the lift forces between one-way and two-way approaches is due to kinetic energy dissipation in the latter approach which results in smaller values for lift forces. The results of the lift forces in one-way fluid-structure interaction is an evidence that one-way fluid-structure interaction is not sufficient in detecting the actual vertical periodic oscillation of the deck which is called vortex induced vibration, but the two-way fluid-structure interaction have a perfect ability in detecting it. This means that the amplitudes of the lift displacement in the deck are predicted with lower values of wind velocities, and thus the lock-in phenomena is better predicted in earlier stages.

Drag forces

For Drag forces, when one-way fluid-structure interaction approach is considered in the simulation process, the drag forces increase in a direct proportion but the pattern of the horizontal periodic vibration response of the deck models is not regular when the wind flow velocity increases between (1 to 15 m/s). When the wind velocity value is 1 m/s, considering one-way fluid-structure interaction, the drag forces have a certain value with a stable horizontal periodic vibration starting after 5 s, but for two-way fluid-structure interaction and for the same wind velocity, the drag forces have smaller values compared to previous approach, and the horizontal periodic vibration have a semi stable situation starting after 12 s and by time it is decreasing in a small rate (see Fig. 21).

When the wind velocity is increasing to 5 m/s, in one-way fluid-structure interaction, the drag forces are increasing in a large rate with a stable situation of horizontal periodic vibration but with a larger frequency for the amplitudes starting from the beginning of the wind excitation directly. While for two-way fluid-structure interaction and for the same wind velocity, the drag forces are increasing with a high rate but still smaller than its values for the previous approach, in addition to creation of sudden changes in the amplitudes of the horizontal periodic vibration coinciding with the start of changes of vertical periodic vibrations after 45 s (see Fig. 22) which means that the changes in the frequency of the lift amplitudes leads to the change in the drag amplitudes in the same time.

Increasing the wind velocity to 10 m/s, in one-way fluid-structure interaction, the drag forces are increasing in a large rate again with a stable situation of horizontal periodic vibration but with a smaller frequency for the amplitudes starting from the beginning of the wind excitation directly too. While for two-way fluid-structure interaction and for the same wind velocity, the drag forces are increasing with a high rate but still smaller than its values for the one-way fluid-structure interaction approach, furthermore a creation of sudden changes in the amplitudes of the horizontal periodic vibration coinciding with the start of changes of vertical periodic vibrations after 25, 55 and 85 s (see Fig. 23) which means that the changes in the frequency of the lift amplitudes leads to the change in the drag amplitudes in the same time, where in this case three times are obvious.

Continuous increase of wind velocity to 15 m/s, and considering one-way fluid-structure interaction, the drag forces are increasing in a large rate again with a stable situation of horizontal periodic vibration but frequency for the amplitudes becomes bigger. While in two-way fluid-structure interaction and for the same wind velocity, the drag forces are increasing with a high rate but still smaller than its values in one-way fluid-structure interaction approach, and it is worthy to mention that a creation of sudden changes in the amplitudes of the horizontal periodic vibration are coinciding with the start of changes of vertical periodic vibrations after 20, 40, 60, 80 and 100 s (see Fig. 24) which means that the changes in the frequency of the lift amplitudes leads to the change in the drag amplitudes in the same time for five times along duration of 100 s of wind excitation.

The difference in the drag forces between one-way and two-way approaches is due to kinetic energy dissipation in the latter approach which results in smaller values for drag forces. It is obvious that the lift forces value in one-way fluid-structure interaction approach is three times the drag forces value approximately for all the wind velocity cases, compared to two-way fluid-structure interaction approach the lift forces value is twice the drag forces value, the reason is the same which is related to the kinetic energy dissipation.

Kinetic energy

The vibrational kinetic energy of the system is calculated from the simulation of vortex-induced vibration of the deck models of a long-span bridge due to a wind excitation in ABAQUS for both the on-way and two-way fluid-structure interaction approaches. The results are for a duration of 50 s. It is obvious that kinetic energy for the one-way approach is increasing from 4850 N•m to 4975 N•m and after this stage it becomes stable approximately till t = 20 s, while for the two-way approach this value is not stable in the beginning 20 s with a decreasing value from 4800 N•m and after that it becomes stable for the remain 30 s with a value of 4525 N•m (see Fig. 25). This means that the kinetic energy of the vibrating system in one-way approach is still the same value approximately because the effect of the structural model of the deck on the surrounding wind has not been considered and no exchange of energy exists from the deck to the wind, so the overall kinetic energy remains high and stable compared to that in two-way approach, but due to exchange of kinetic energy between the deck and the wind together in two-way approach and large dissipation of the kinetic energy in this process, the overall kinetic energy is less than of it in the on-way approach.

Lock-in phenomena

At lock-in region, the vortex shedding is generated more efficiently. For one-way fluid-structure interaction approach this phenomena starts to appear clearly at wind velocity of 11.5 m/s with regular periodic and stable shape of the vortices shedding from the tail of the deck. The screen shots of the simulations are captured at the time histories 11.5, 11.7 and 11.9 m/s. When the two-way fluid-structure interaction approach is considered, the vortex shedding generates with irregular style due to the movement of the deck and its feedback on the surrounding wind flow for all wind velocity cases. The similarity in vortex shedding style with respect to the shape of the vortices and the periodic pattern for all wind velocity values for both the one-way and two-way approaches is an indication of the lock-in phenomena where the frequency of vortex shedding remains constant approximately (see Fig. 26).

When only the one-way fluid-structure interaction approach is considered in the simulation, the lock-in phenomena exists at the wind velocity of 11.5 m/s approximately but it is not quite regular and the associated vortex shedding frequencies are near to 1.5 Hz till wind velocity magnitude of 12 m/s, and after that range the vortex shedding frequency begins to increase in magnitude in a linear proportion with the wind velocity as shown in Fig. 27.

When considering the two-way fluid-structure interaction approach, the same phenomena starts to appear at the wind velocity magnitude of 10 m/s and ends at 11.5 m/s in a regular path compared to the previous approach, while the associated vortex shedding frequencies are 1.18 Hz. The vortex shedding frequency after the lock-in region increases rapidly to 1.45 Hz and starts to undertake a linear proportion approximately. This indicates that the fluid-structure interaction has an important and effective role in detecting the actual region where the lock-in phenomena starts to generate and determines the associated shedding frequencies for lower natural frequencies of the system.

Lift coefficient

The results of the relation between the Reynolds number Re and the lift coefficient C_l for the deck model when the wind velocity changes from 1 m/s to 15 m/s when considering one-way fluid-structure interaction approach, the lift coefficient reaches a maximum value of 0.4 at Reynolds number value of 0.75E+ 06, and the value of the lift coefficient is unstable until the Reynolds number magnitude 0.9E+ 06, after that the relation becomes stable approximately for a lift coefficient value −0.4 without appreciable changes in the value of the lift coefficient (see Fig. 28).

The results of the relation between the Reynolds number and the lift coefficient when taking in account the two-way fluid-structure interaction approach, show that the maximum value of the lift coefficient is -0.3 begins from the Reynolds number value 0.9E+ 06 and in the same way as previous approach it is unstable until the latter value of the Reynolds number and becomes stable after that point.

Drag coefficient

The results of the relation between the Reynolds number Reand the drag coefficient C_d for the deck model when the wind velocity changes from 1 to 15 m/s when considering one-way fluid-structure interaction approach, the drag coefficient reaches a maximum value of 0.023 at Reynolds number value of 1.5E+ 06, and the value of the drag coefficient is stable at 0.02 approximately (see Fig. 29).

While the relation between the Reynolds number and the drag coefficient when the two-way fluid-structure interaction approach is used, show that the maximum value of the lift coefficient is 0.005 begins from the Reynolds number value 4.0E+ 05 and it becomes stable starting from Reynolds number value 0.8E+ 06 approximately.

Reynolds number

The Reynolds number Re calculations in each approach have been determined for multiple wind velocities supporting on the results of the simulations in ABAQUS in addition to lift and drag coefficients (see Table 1).

Strouhal number

The calculations of the Strouhal number S value for the deck models is between 0.322 and 0.324 for the one-way fluid-structure interaction approach, while for the two-way fluid-structure interaction approach this value is between 0.285 and 0.308 (see Table 2). The relation between the Reynolds number and the Strouhal number for the deck model when considering many cases of wind velocities for both approaches is stable and linear approximately for one-way fluid-structure interaction approach but it is not stable and nonlinear when adopting two-way fluid-structure interaction approach (see Fig. 30).

This result proofs the fact that the Strouhal number is depending on the Reynolds number and it is nearly with a value of 0.2 for a large range of Reynolds number, and the fluid structure interaction concept is better supporting this fact, this by comparing the results when considering two-way fluid-structure interaction approach, where the Strouhal number is between (0.28-0.29) approximately until Reynolds number value of (1E+ 06) but the Strouhal number value is constant at 0.32 until Reynolds number value of 2.3E+ 06 when considering one-way fluid-structure interaction approach.

Validation of the results

It is essential to accurately simulate the FEA models and this depends on verification of the numerical models through a comparison process against analytical solutions and or if it is possible to compare it against experimental data.

Validation of lock-in models

The results of the lock-in phenomena determined from ABAQUS simulations in both one-way fluid-structure interaction approach and two-way fluid-structure interaction approach are compared with the compiled numerical results obtained by Simiu and Scanlan. The lock in phenomena simulation using two-way fluid-structure interaction approach is better represents the actual phenomena this by validation with the results obtained by Simiu and Scanlan (see Fig. 31).

Where the period of equal vortex shedding frequencies is greatly similar compared to the region of equal vortex shedding frequencies in the one-way fluid-structure interaction approach, where the start of the phenomena in one-way approach is at wind velocity 11.5 m/s till 12 m/s, but for two-way approach is starting from wind velocity of 10 m/s till 11.5 m/s. This is indicates the importance of the two-way fluid-structure interaction approach in simulating the generation of lock-in phenomena as a result the vortex shedding induced vibration in the deck of the long span bridges can be determined more efficiently and predicted for earlier and wider range of wind velocities in addition to considering lower natural frequencies of the system which enhances the safety of the structure against the vortex induced vibration.

Validation of drag coefficient

Another validation parameter is the drag coefficient C_d for the deck model for different Reynolds number values, where the main effective parameter is the wind velocity changes from 1 m/s to 15 m/s. The benchmark of validation for C_d is the results of a flat plate model used by Munson and coauthors [Munson]. The C_d results obtained in the flat plate model show that this value is unstable until Reynolds number of 7.5E+ 05 where C_d value is 0.08 and becomes stable in the region between 0.002 and 0.085 approximately but the C_d value is 0.02 for one-way fluid-structure interaction approach and it is 0.005 for two-way fluid-structure interaction approach, which means that there is a very good approximation between the results specifically between the benchmark and the two-way fluid-structure interaction approach (see Fig. 32).

Conclusions

The following conclusions have been formulated:

1) The results of the vortex shedding from the deck of the long span bridge show that the two-way fluid-structure interaction approach is more efficient than the one-way fluid-structure interaction approach in detecting the generation of earlier vortex induced vibration and predicting lower critical wind velocities as a result detecting earlier lock-in phenomena, which are indications for designers to avoid the vortex induced vibration in the design stage by ensuring that the frequencies of the vortices generated are sufficiently separated from the natural frequencies of the long span bridge by altering the geometrical design of the deck or changing the natural frequencies of the system.

2) The results of lift forces generated in the deck of the long span bridge in the two-way fluid-structure interaction approach is smaller than the same values for the one-way fluid-structure interaction approach, which means that the two-way approach determines the actual magnitude of the energy dissipation for the vortex induced vibration of the deck and the vertical periodic oscillation of the deck is detected in lower wind velocities only by utilizing two-way fluid-structure interaction approach. Supporting on these results, the cable stayed bridge is designed safely through avoiding wrong estimation of late vortex induced vibration where in some cases lead to resonant and dangerous events.

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