Vortex shedding occurs when the periodic excitation frequency coincides with the mechanical resonant frequency of the tubes. Turbulent closure is the excitation caused due to the change of velocity at the periphery of the tubes [2]. The impact of the heat exchanger tubes with the loose support causes structural nonlinearities [4].
But in the case of the cylindrical array without loose support, the effect of nonlinearity due to average axial stress plays an important role in predicting the flutter amplitude of the cylindrical tubes. In a recent work by Xia and Wang [12], the effect of nonlinearity associated with the average axial elongation of the cylindrical tubes was taken into account. This initial axial load in the system is either due to externally imposed force or due to thermal expansion of the pipes.
The onset of fluidelastic instability leads to relatively large vibration of the heat exchanger tubes, which increases with increasing speed. Several experimental works were carried out to study the various mechanisms involved in the fluidelastic instability of tube bundles. The UFM has a detailed picture of fluid forces, but is highly dependent on empirical data.
Fluidoelastic excitation was considered independent of wake phenomena, and free flow was considered along the sides of the tube.
Problem definition
Semianalytical approach was proposed by Lever and Weaver [11], where analytical expression coupling fluid forces and pipe motions simplifies the fluid forces. The flow through the arrays was divided into wake and channel regions and was assumed to be incompressible and one-dimensional. Inviscid fluid flow model is discussed in the work of Dalton and Helfinstine [8], where the problem of an accelerating potential flow past a series of cylindrical tubes using image method.
Thesis structure
In this approach, the cylinder under consideration is assumed to be flexible while the other cylinders in the group are rigid. It is assumed that the cylinder is provided with a flexibility of the system by an orthogonal spring system supporting the cylinder (see Fig. 2.1). 2.3) where ρ is the density of the liquid, a is the ratio between the flow rate falling in the cylinder and the free flow rate (U∞); a= T / T−12d.
The flow approaches the cylinder at a slowed speed rather than at a constant speed (i.e. U < U∞) and at some point impinges on the cylinder. If the fluid flow had been steady, it would have collided with the cylinder sooner (t−∆t). If the fluid flow rate (U) impinging on the cylinder is constant, then the time required for the fluid to flow from x = x1 to x=R+ is ∆Ris:. where R is the radius of the tube.
If the cylinder motion is in damped harmonic form, apparent displacement ∆y=y(t)−y(t+ ∆t) occurred due to flow retardation, it can be expressed as:. The velocity at which fluid impinges on the cylinder except those in the upstream row is U =aU∞. Consider the retardation effect in Eq. 2.13). P0 represents the initial axial load, F is the cross-flow induced force acting on the cylinder and f is the force due to the loose support constraint, δ(x−xb) represents the dirac delta function with xb as the loose support constraint location.
Here, D denotes the diameter of the cylinder, and ρ and U are the liquid density and velocity, respectively. Cma is the virtual or added mass coefficient of the fluid around the cylinder, µ is a parameter related to the array pattern and ∆ is the time delay that occurs due to phase lag between cylinder motion and fluid dynamic forces. We use an orthogonal set of basis functions to take advantage of the properties of orthogonal functions.
In this work, we first modeled the loose support as a cubic spring as follows: where κ is the stiffness of the cubic spring. Using the above force models (equation 3.8) using Matlab ode solvers we obtain an approximation of the response of the PDE equation. In this section we analyze the stability and branching of the system with different types of loose supports.
On the other hand, the cylinder may also undergo Hopf branching due to the flutter instability due to crossflow. The load pos which is also known as the critical axial load, which is obtained by ωi= 0 in Eq. 4.7) tells the critical axial load position for different values of the flow velocity.
Bifurcation Analysis
Cubic Spring model
We therefore include the loose constraint and the axial deformation nonlinearities and study the associated bifurcation diagrams. In particular, it should be noted that, apart from the flutter instability, the effect of buckling due to axial loading leads to pitchfork branching. 4.2, we approximately determine the critical flow velocity at instability, at post-instability bifurcation, and at chaotic motions.
We found that for the load condition p0 = 15, the critical current velocity for instability (U1) is much smaller compared to the velocity of the linearized model. In region (d), the system gravitates toward a curved equilibrium position, and in regions (b) and (c) it undergoes stable limit cycle motion. 4.3 (without loose support), we note that the rate at which the system loses stability is the same in both cases.
This indicates that the presence of chaotic motions in the system is due to the impact of the loosely supported cylinder.
Trilinear Spring model
Bifurcation diagrams of the system response for different values of initial axial loads, p0=−15, p0= 0,p0= 9 and p0= 15 are shown in Fig. 4.8(a), bifurcation diagram for p0=−15, until the flow velocity shown reaches a value U = 2.785, the system is stable and gravitates towards the equilibrium point. It should be noted that the amplitude of the cylinder during this phase is less than d, i.e. the cylinder will vibrate within the gap.
Time trace plots and phase diagrams for the loosely supported system modeled as a trilinear spring for flow rate U = 1, at different axial loads sp0=−15, p0= 0, p0 = 9, and p0= 15, are shown in Fig. Similar to the system modeled with cubic springs, at the lowest value of initial axial load (P0) and low fluid flow rate, the motion of the system gradually gravitates to the origin due to structural damping 4.9(a) and 4.9(b) ). The different critical speeds for the onset of instability and chaotic motion of the system at different values of axial loads are given in Table 4.2.
In this work we studied the fluidelastic instability of a series of cylindrical tubes with initial axial loading and with loose support. A quasi-static approach was used to analyze the fluidelastic instability of the cylinder, considering a single flexible cylinder among an array of rigid cylinders. The flow delay was taken into account by introducing a time delay in the tube motion when calculating the fluid forces.
The effects of loose support on the stability of the cylindrical array were analyzed by comparing it with a similar system without loose support. The critical flow rate at which the system starts amplifying the oscillation at the onset of chaotic motions was obtained from the bifurcation diagrams. The regions of instability were located using a linearized model of the governing equation and were plotted in a (U, p0) plane.
Stability graphs of the system are generated by considering the loose support as cubic and trilinear spring constraints. It was found that the stability plots for the two non-linear models differ from the stability plot of the linearized model. At higher values of the initial axial load, when the fluid flow rate exceeds the critical value, the system loses its stability due to buckling, leading to a subcritical fork bifurcation, and at lower values of the axial loads, the system becomes unstable due to flutter instability, which leads to hopf. split.
From the bifurcation diagram of the system in which the loose support is modeled as a three-line spring, it can be seen that as soon as the amplitude reaches the "gap distance", chaotic movements begin and they are no further increase in the amplitude. Large eddy simulation of the fluctuating forces on a square tube array and comparison with experiment.
