## Abstract

The amplitudes induced by random excitation forces on the tubes bring continuous friction between the tube and supports, which results in gradual failure of the tubes due to fretting wear. Therefore, it is very important to determine the envelope lines of the random excitation force spectrum for the coil tube. To the authors' knowledge, there are no published studies on the normalized force spectrum of coil tubes. In this paper, a simplified three-layer experimental model was established. The robustness of the numerical method was demonstrated by comparing the experimental and simulated results, including the vibration response and the fluid excitation force spectrum. Then, a semi-empirical equation for predicting the dominant frequency of turbulent buffeting was constructed by employing the threshold envelope method. Through the observation of time-history and root-mean-square (RMS) data, it was found that the pitch diameter ratio between adjacent tube layers, a, had the greatest influence on the force coefficients. The smaller a is, the larger the force coefficients are. The pitch diameter ratio in the same layer, b, and helix angle, *α*, had little effect on the force coefficients. With the increase of *α*, the flow instability in the shell-side flow enhanced and the fluctuation of force coefficients became larger. Finally, the mechanisms of the tube position, Reynolds number (Re), and bundle structure on the normalized force spectrum were studied. The normalized envelope force spectrum for coil tubes was proposed as the guidelines to predict and evaluate the random excitation force acting on the tubes.

## 1 Introduction

The coil tube bundle is superior to the straight tube and U-tube bundles because of its compactness and greater heat transfer efficiency [1–3]. Therefore, in recent years, coil-wound heat exchangers (CWHE) have been used widely in pharmaceutical [4], chemical industries [5], and especially in the nuclear power industry. It is utilized extensively in the fourth generation reactors, such as high-temperature gas-cooled reactors [6], lead-cooled fast reactors [7], and small modular reactors [8].

The coil tube bundle is subject to the same cross flow impacted in the shell side as other structural bundles, resulting in flow-induced vibration. There are three main mechanisms that can lead to tube failure for single-phase flow: vorticity shedding excitation, turbulent buffeting, and fluid-elastic instability [9]. Among the mechanisms cited above, both vorticity shedding excitation and fluid-elastic instability can lead to large-degree vibration and destructive tube failure [10]. However, turbulent buffeting does cause continuous small-degree vibration. Thereby appear fretting wear between the tube and support. It can produce progressive damage in terms of tubes [11,12]. Due to the high maintenance and downtime costs of nuclear power plants, it is important to be able to assess the effects of random excitation to ensure the maximum safe operating life of the steam generator. At the same time, in order to design a safer and longer life steam generator, it is also necessary to accurately evaluate the random excitation force exerted by the fluid on the tube during the design process. To this end, it is meaningful to carry out the research and making contributions to the design guidelines of the coil tube bundle.

Owen obtained the semi-empirical formula of the main frequency of turbulent buffeting by using the experiments of gas-phase flow subject to the bundle and determined the relevant parameters by using the experimental data [13]. Then, previous experimental results were used to verify the formula, which proved that it had a certain reliability in engineering design [14]. As a result, Owen's method can be used for reference to explore the prediction method of the main frequency of turbulent buffeting with respect to the coil tube, as presented in the later part of this paper.

Since heat exchanger with straight tube bundle has been developed for a long time, the random excitation forces acting on straight tubes have been studied quite a lot. In order to study the fluid excitation force of vortex shedding on a bluff cylinder, Blevins and Burton established a semi-empirical dynamic model using random vibration theory, which included both the spanwise correlation effects and the amplitude dependence of the related vortex forces [15]. The model parameters were determined according to the experimental data. Finally, the model was applied to assess the force acting on the elastic cylinder, and the results were in good agreement with the experimental data [16]. This method provides a new research idea, and the random excitation of the coil tube can also be evaluated by using such a semi-empirical model. Chen and Jendrzejczyk introduced the experimental results of a square tube array with a pitch diameter ratio of 1.75 under turbulence. And the assumption that the normal component of velocity was taken as the effective velocity was also proposed for the bundle which was inconsistent with the normal direction of velocity [17]. Taniguchi and Miyakoshi conducted an experimental study on the flow around the cylinder close to a horizontal plane [18], which does not exist in the coil tube bundle. Only the flow in the bundle is considered for coil tube, and there is no direct impact of the constant flow acting on the tube. It will be discussed in this article. Taylor et al. directly measured the fluctuating forces of bundles with pitch diameter ratios of 1.5 and 3 and finally gave power spectral density (PSD) and excitation force coefficients of these structural parameters [19]. Axisa et al. systematically studied the random excitation mechanisms of tube bundles subject to single-phase cross flow and included them in the reduced force spectrum versus reduced frequency [20]. This plays an active role in predicting the excitation forces on the tube accurately in engineering design. It also provides an idea for the establishment of random excitation mechanisms in terms of the coil tube. Paїdoussis and Price revealed the physical mechanisms of flow-induced instability caused by cross flow. Under the framework of quasi-steady fluid dynamics theory, fluctuation excitation was discussed [21]. Norberg shed light on root-mean-square (RMS) lift and drag factors over time in the range of Re* *= 1 × 10^{3}–10^{4} [22,23]. The experimental results of excitation force of normal triangular bundle were given [24]. It found that there are significant forces in both directions of drag and lift. In addition to random excitations, constant frequency and quasi-periodic fluid forces were also found. These forces depend on the position of the tube within the bundle. This finding was confirmed by the experimental results of Liu et al. within our project team [25]. Sun et al. applied the improved Van der Pol model to study the vortex-induced vibration, which has a positive effect on the study of the forced vibration with respect to the tube under the elastic boundary conditions [26].

At present, there are few researches on random excitation forces of coil tubes under cross-flow impact. In Blevins's test, the tube response corresponding to the reduced velocity of 1–50 was studied. And the dimensionless force spectrum was derived, which was used to predict the tube response with respect to a single mode [27]. And then, Xu et al. proposed and implemented an RMS response estimation method for multisupported coil tubes based on the ansysapdl command object [28]. Choosing this method, Yuan et al. [29] co-developed multiphysics simulation toolkit sharp to explore random vibrations for coil tubes. It was worth noting that the simulation resulted in a higher peak frequency than the experiment. It is conservative for engineering design and evaluation.

Our research team has done some work on the flow-induced vibration mechanisms for coil tube bundles. The natural vibration characteristics and vortex shedding for coil tube bundles were explored. The details are presented in Refs. [30–32]. In this paper, a simplified three-layer experimental model was established to verify the reliability of the numerical method. In terms of the simulation data, the semi-empirical equation of the dominant buffeting frequency was obtained. The mechanism of structural parameters on the force coefficients was discussed. Finally, the normalized force spectrum for the coil tube was proposed, which provided the guidelines for the prediction and evaluation of random excitation.

## 2 Experimental Setup

*d*, was 0.008 m. The pitch diameter ratio between adjacent tube layers,

*a*, and the pitch diameter ratio in the same layer,

*b*, were quantified as

In the test section, *a* was 1.5, *b* was 2.6, and the helix angle, *α*, was set to 15 deg. The tube bundle height, *H _{c}*, and the straight tube section height,

*H*, constituted the overall height of the section, which was 0.760 m and 0.100 m, respectively. In addition,

_{s}*D*

_{in}denoted the internal diameter of the shell, which was 0.426 m.

The experimental system is indicated by Fig. 2. Pump was used to realize the water circulation. A throttle valve was used to regulate the flow, and a flowmeter was applied to monitor the flow. In this test, in order to ensure that the sensor was completely attached to the surface of the tube, resistive strain gauges with the size of 2.2 mm × 1.5 mm were selected, and the half-bridge connection was adopted as presented in Fig. 2. The sensitivity coefficient was 1.91±1%, and resistance was 120.1±0.1 Ω. In addition, support bars of the two sides that constrained the same tube were connected by welding bridges. Figure 3 shows the testing site.

## 3 Test Method

Tested coil tubes were in the middle layer of the span next to the inlet. A total of ten coil tubes were monitored, the 14th–23rd tubes, which were sequenced based on flow direction. Note that the model used in the simulation analysis below was the extraction of 25 tubes per layer of the tested span (orange tubes). If the tested tubes were ranked according to the simulation model, there were 9th–18th tubes. In order to facilitate the comparison between the test and simulated data, the tubes were numbered according to the simulation, which was T9–T18, respectively, as shown in Fig. 4. The sensors were placed in the middle of the coil tube in the single-span and monitored the time-history amplitude of the tubes in the drag and lift directions. The sampling frequency was set at 1 × 10^{4} Hz, and the collection was for a total of 30 s. The test was carried out three times based on the same Reynolds number (Re).

*F*(

*s*,

*t*) is the fluctuating force per unit tube length at location

*s*along the tube. And such function is assumed stationary Gaussian. As a result, it can be represented by its cross-correlation spectrum, which can be defined as the Fourier transform

*R*denotes function of the cross-correlation spectrum between location

_{F}*s*

_{1}and

*s*

_{2}. When uniform mass tube is subjected to an uneven cross flow, fluctuation is conveyed along the tube.

*ψ*remains a real-valued function, if the flow is perpendicular to the tube axis. Then, such function is

_{F}*γ*, describes the degree of correction of forces along the tube, approximated by

where *λ _{C}* denotes the correlation length. When

*λ*is no more than a few tube diameters and even smaller, its exact value is not acquired. The

_{C}*λ*/

_{C}*l*is needed, where

*l*is tube length, for practical applications [16,20].

*U*

_{gap}, can be approximately scaled by the uniform cross-flow velocity,

*U*. The detail is found in the previous work of Wang et al. [32]

_{H}*ε*and

_{A}*ε*are the fluid area and fluid volume porosity fractions for the shell-side. Excitation force per unit tube length is characterized by the permanent force density

_{V}where *d* represents the tube outer diameter.

*δ*(

_{n}*s*) denotes the modal displacement at

*s*,

*m*is the total mass including the added mass,

_{n}*f*is the natural frequency, and

_{n}*ζ*is the damping ratio of the

_{n}*n*th order. Here,

*L*denotes the modal joint acceptance. Referring to the generalized correlation length in the case of uniform cross flow, it is shown that

_{cn}Provided that $\lambda c/l\u2009\u226a\u20091$, such as less than 0.1. It is worth noting that *a _{n}* values depend on the mode.

where *σ _{n}* is the contribution of the

*n*th order RMS displacement.

*A*summarizes all parameters

_{n}## 4 Numerical Details

### 4.1 Computational Domains and Boundary Conditions.

Several computational simulation cases were carried out to obtain random excitation forces of fluid and tube vibration response of analyzing fluid–structure interaction mechanisms.

The adjacent layers are wound in reverse, and the influence of *α* on the flow cannot be ignored. In a word, the effect of coil tube bundle structure on the interaction mechanism cannot be determined. Therefore, the three-dimensional transient simulation in terms of the flow impinging on the bundle was carried out. The coil tube bundle is constrained by circumferential uniformly distributed support bars, and every four bars are placed in the same position of the bundle. Two bars clamp the tube with grooves and smooth edges to act as constraints. In addition, the adjacent bars are bound by welding bridges at multiple points along the axial direction as shown in zoomed figure of Fig. 2. It makes the flow between adjacent spans weak or even negligible. Therefore, the numerical model only chose the span where the tested tubes were located, that is, the span close to the inlet. In order to reduce the calculation quantities, 25 tubes in the middle of each layer of the bundle were extracted to form the coil tube bundle, and then the impinging bundle model was established by adding fluid domain, as described in Fig. 5.

Six bundles with different structures were set up in order to comprehensively consider the effect of various structural parameters on buffeting excitation, as listed in Table 2. Figure 5 also shows the boundary conditions applied to the domain. The inlet boundary was set at the uniform velocity inlet. The outlet was considered the outflow condition, where static pressure with *p*_{outlet} = 0 Pa was set. The inlet condition is set to high turbulence intensity (10%) to meet wall boundary layer requirements [34]. No-slip condition was used in the walls, which consisted of the coil tube wall, inner wall, the walls of bars, etc., in addition, to reduce computations, the domain was divided into three parts: the upper, middle, and lower domains, where the coil tube bundle was in the middle domain, and the connected surfaces of each two domains were defined as interfaces to ensure that the mesh shared nodes. The medium in the shell-side is water with density, *ρ* = 999.87 kg/m^{3} and the dynamic viscosity, *μ* = 1.01 × 10^{−3 }Pa·s. The middle 1/3 part of the detection tube was taken as the monitoring surface to ensure that the random excitation of the fluid acting on the tube was not affected by the model simplification. Monitoring tubes were named T9–T18 along the flow direction, as shown in Fig. 6.

### 4.2 Computational Method and Meshing Strategy.

where *x*, $x\u02d9$, and $x\xa8$ denote the displacement, velocity, and acceleration of the center of the coil tubes, respectively. Here, the damping and stiffness are not equal in both directions. Before the water tunnel test, the modal test was carried out by direct percussion. Because T9 was exactly the same as the restraint and structural feature of the others, one only measured vibration characteristics of T9 for three times. Figures 8 and 9 show the results of one of the times. In the end, the average values were used as the input value. The natural frequencies in the lift and drag directions are 456.21 Hz and 172.81 Hz, respectively, and the damping ratios are 0.0462 and 0.0201.

where *k* is defined as the turbulent kinetic energy, *Q*_{SAS} is set to the additional source term, and *F*_{1} is set to the blending function. More details are indicated in Refs. [42] and [43].

The ansyscfx solver was used to analyze the fluid-structure interaction problem in terms of finite volume technique. In order to reduce computation, the whole computational domain was divided into three parts. The whole fluid domain was performed in three-dimensional and nonstructured grids were applied to the region close to the bundle, and structured grids were applied to the rest, as shown in Fig. 10. Note that boundary layers were added close to the tube wall to accurately capture the tube motion. A total of 12 boundary layers are set up. Table 3 lists the grid independent results of the domain for the bundle for *a* = 1.5, *b* = 1.5, and *α* = 15 deg at Re* *=* *2426. It illustrates that *y*^{+} ≈ 1 and the accuracy of the results could be guaranteed when the thickness of the first boundary layer is set as 1/155*d*. In addition, the residual is set to 1 × 10^{−5}. The time-step is set to 1 × 10^{−4}. Furthermore, to ensure that the calculation results are not affected by the initial conditions of the cases, a stable state calculation with a certain amount of computation was carried out first, and then the transient calculation was implemented. This ensures that time-averaging of the solution for calculating the mean flow quantities can be started.

Mesh size | Total nodes/×10^{7} | y^{+} | RMS A/d (%) | Error (%) |
---|---|---|---|---|

1/135d | 2.76 | 1.13 | 0.01571 | 0.191 |

1/145d | 2.92 | 1.05 | 0.01569 | 0.064 |

1/155d | 3.09 | 0.98 | 0.01568 | — |

1/165d | 3.21 | 0.92 | 0.01568 | — |

Mesh size | Total nodes/×10^{7} | y^{+} | RMS A/d (%) | Error (%) |
---|---|---|---|---|

1/135d | 2.76 | 1.13 | 0.01571 | 0.191 |

1/145d | 2.92 | 1.05 | 0.01569 | 0.064 |

1/155d | 3.09 | 0.98 | 0.01568 | — |

1/165d | 3.21 | 0.92 | 0.01568 | — |

### 4.3 Validation Cases.

Water tunnel experiments with respect to coil tube bundle with *a *=* *1.5, *b *=* *2.6, and *α* = 15 deg were carried out to verify the reliability of the numerical model for the study. Figure 11 shows amplitude of T9 versus sampling time at Re* *=* *7905 in the lift and drag directions. Here, the amplitude is dimensionless. Simulated and experimental data fluctuate in a similar range in both directions. The amplitude in the lift direction is asymmetric with respect to the axis of *x* = 0, which is different from the response of the straight tube [44,45]. It also illustrates that the asymmetry structure in the plane affects the fluid forces acting on the coil tube. In addition, the data line in the lift direction is denser than those in the drag direction, indicating indirectly that the natural frequency in the lift direction is greater than that in the other. RMS amplitude versus Re of T9 and T12 in the drag direction are delineated in Fig. 12. The amplitude of test and simulation increases with the increase of Re, and the error is low, less than 10%, indicating that the numerical model has a certain reliability.

What's more, the normalized force PSD (obtained by Eq. (17) refer to a reference length, *L _{e}*, of 1 m) in both directions was checked (Fig. 13). The normalized force PSD for random excitation forces of the three tubes at Re

*=*

*7905 was obtained. It is found that the order of magnitude of the test PSD has a high degree of fitting with the simulation, which proves that the fluid excitation force data obtained by the simulation is applicable. Besides, PSD in the lift direction is about one decade larger than that in the drag direction. In short, the robustness of the numerical methodology is proved by comparing with the experimental data.*

## 5 Results and Discussion

### 5.1 The Dominant Buffeting Frequency.

Turbulent energy is generated at the front of the bundle and convects into it at an approximate rate $\Delta Cf\rho UH3T$. $\Delta Cf(1/2)\rho UH2$ is the pressure drop through a row of tubes embedded in the bundle. If the turbulent kinetic energy of the fluid per unit mass is 3*q*^{2}/2 (provided that the mean square component of a velocity component is *q*^{2}) and it mainly exists in eddies at scale *l*, the dissipation rate is about 3*q*^{2}/2*l*. Thereby, the rate of turbulence attenuation in the shell-side is 3*ρq*^{3}*TL*/2*l*.

*C*, of each tube is dependent on the dimensions of the tube array, based on the gap velocity, and independent of the transverse tube spacing

_{d}*l*flow through the bundle at an average velocity,

*U*, they pass through the plane containing the downstream row of tubes, and the frequency for the excitation force on these tubes is

*q*nor

*l*is constant. In addition, the above equation used for the turbulence decay rate can be accepted as merely suggestive, because the time scale for the breakdown of the high-energy eddy is similar to the time required for the fluid to flow from one row of tubes to another, so that the average flowrate changes markedly. It follows that it is not at all worth considering improvements such as the dependence of

*C*and

_{D}*U*

_{gap}/

*q*on the geometry of the tube array [13]. Then, the above equation can be written as

where *F _{b}*, which is a constant, is fit factor for various spaced in a high-Reynolds-number flow. It has been demonstrated in studies of straight tube bundles [14,46]. Resonance occurs when the dominant buffeting frequency,

*f*, fits the natural frequency for the tube. Once such a resonance is established, the tube amplitude will be greatly increased, possibly greater than the amplitude threshold of 0.02

_{t}*d. f*can be used to predict whether resonance occurs according to the relation proposed in the above equation, that is, when 1/2

_{t}*f*

_{1}≤

*f*, resonance is prone to occur [47].

_{t}Although one cannot determine the force direction, it can be noted that even if it is entirely in the drag direction, corresponding to a pure drag fluctuation, the vibration in the lift direction would be excited in the bundle due to the lack of consistency in the phase distribution along the direction of the tube elongation: similar in its effect to that of a nonuniform piston executing vibration in the drag direction within the shell-side. The PSD of eight tubes of all monitoring tubes for various structural bundles is carried out to obtain the main peak value of the spectrum of random excitation forces acting on coil tubes to determine the constant *F _{b}*, as demonstrated in Fig. 14. Here, Re

*= ρU*= 2.371 × 10

_{H}d/μ^{4}corresponding to

*U*= 3 m·s

_{H}^{−1}. Particularly, maximum three peaks of all the tubes are extracted from the figure, and if two of the peaks are equal, only two peaks are marked.

*U*

_{gap}, calculated according to the calculation correlation of

*U*

_{gap}and

*U*obtained in our own works [32], which is substituted into Eq. (25). The results of spectral analysis of various structures are plotted in the form of (

_{H}*FL*/

*U*

_{gap}) × (

*T*/

*d*) to [1 − (

*d*/

*T*)]

^{2}in Fig. 15. Considering the safety of engineering design and evaluation, a straight line is drawn above all data points to envelope all points, so as to ensure that no resonance occur when the dominant buffeting frequency calculated according to the following formula is not within 1/2

*f*

_{1}≤

*f*:

_{t}*d*/

*T*. In the structure for

*a*=

*1.1,*

*b*=

*1.1, and*

*α*= 15 deg,

*d*/

*T*=

*0.9 does not meet this condition and does not conform to Eq. (22). However, it can be found that the frequency envelope of the bundle with*

*a*=

*1.1,*

*b*=

*1.1, and*

*α*= 15 deg calculated by this equation, namely, the point in the circle, is below the line by observing Fig. 15. Although the structure does not meet the establishment condition of the equation, it can still be used for conservative evaluation. For brevity, it should be applied to engineering. Equation (25) can also be written as

### 5.2 The Excitation Force Coefficient for Different Variables.

where *C* denotes coefficient. The subscripts *f*, *l*, and *d* are related to the excitation force, lift, and drag.

Figure 16 shows the time-varying force coefficient of T9 within the tube bundle of various structures. The motion of fluid in the shell-side was still unstable for the first about 0.3 s, after that the flow developed fully. The closer the structure is, the larger the *C _{f}* is, as shown in Fig. 16(a). This phenomenon is especially evident when

*a*increases from 1.1 to 1.2 and then to 1.5. When

*a*=

*1.5, there is little difference between*

*C*of three different structures. Compared with the other two types, the shear flow on the tube surface of

_{f}*b*=

*1.5 and*

*α*= 15 deg (case 4) is more significant, resulting in a larger fluctuation of

*C*. This is because the increase of helical angle enhances the partial motion of the flow along the tube to generate a more uneven distribution of forces along the tube and a large fluctuation of excited force acting on the tube. The increase of the pitch diameter ratio between the tubes of the same layer leads to the formation of eddies behind the tube, and the interaction of eddies on both sides of the tube wall makes the weakening of excitation force fluctuation. However, the excitation force fluctuation of

_{f}*a*=

*1.2,*

*b*=

*1.5, and*

*α*= 15 deg (case 2) is greater than that of

*a*=

*1.1,*

*b*=

*1.1, and*

*α*= 15 deg (case 1). Because the tube clearance of case 1 is too low, the shear flow on the tube wall is weak and jet switching is obvious. Although the tube clearance of case 2 increases and jet switching weakens, the shear flow increases and the two modes of case 2 interact with each other, making the excitation force coefficient fluctuates significantly.

Figure 16(b) presents the variations of lift coefficient (*C _{l}*) of T9 over time. Trends for

*C*of different structures are similar to

_{l}*C*of them. The difference is that

_{f}*C*of case 2 fluctuates more obviously and shows obvious periodicity. There are periodic vortex shedding in the lift direction between the tube clearance. In addition,

_{l}*C*is asymmetric with respect to

_{l}*x*=

*0. It is caused by the asymmetry of the structures for the tube bundle and is exacerbated by the presence of*

*α*. Except for cases 1 and 2,

*C*of other structures fluctuates around 0. The gap between adjacent layers increases, and a flow channel is formed in the gap. It makes the flow pass through the bundle rather than going around it.

_{l}The variations of drag coefficient (*C _{d}*) of T9 over time are shown in Fig. 16(c). The law of

*C*over time is similar to the other two force coefficients. It is found that the smaller the gap between tubes, the greater the random excitation forces of the fluid on the tube. All in all, the influence of fluid acting on the tube is formed by the coupling of turbulence, vortex shedding, and jet switching. The tube clearance is small, the flow goes around the tube, and the streamlines are constantly changing. This phenomenon is mainly caused by jet switching. With the increase of tube clearance, the shear flow on the tube wall becomes more obvious, forming eddies behind both sides of the tube and moving backward. It enhances the fluctuation of random excitation forces. With further increase of tube clearance, the eddies formed behind the tube may interact with each other and then cancel out, so that the effect on the back tube is weakened. In this phenomenon, the vortex shedding plays an important role, while jet switching becomes weaker gradually.

_{d}Figure 17 shows the broken lines of the RMS force coefficients' distribution of all the tubes for the tube bundle with *a *=* *1.5, *b *=* *2.6, and *α* = 15 deg (case 5) at various Re. It can be seen from Fig. 17(a) that the farther back the tube is, the greater *C _{f}* is in terms of the overall trend. As the fluid continues to pass through the tube bundle, the flow disturbance is intensified, and the flow instability is further enhanced on account of the reverse wound of the tubes and the asymmetry of the structure. The excitation force on the tube not only increases the fluctuation but also the RMS force. And the

*C*of at Re = 2.371 × 10

_{f}^{4}is greater than that of other Re numbers, which indicates that when the Re number is greater than when a certain value is reached after Re = 2.371 × 10

^{4}, the flow state of the fluid in the basin or the vibration response of the bundle changes suddenly, which may be because the flowrate between the tubes is greater than the fluid elastic instability threshold, the bundle vibrates violently, and the coupling between the bundle and the fluid also changes greatly. However, the laws are not seen in Fig. 17(b). The reason is that the spacing between adjacent layers for case 5 is large, and obviously, flow channels are formed between the tube layers, which has been observed in previous studies, detailed in Ref. [32]. The fluid can easily pass through the tube bundle along the channels. Therefore, there is no regularity for

*C*in this figure. The flow develops obvious shear flow on both sides of the tube, and eddies are formed in the gap between the tubes of the same layer, which enhanced vortex shedding in the shell-side. As the flow goes deeper into the bundle,

_{l}*C*increases, as illustrated in Fig. 17(c). In addition, because of the complex and variable structure of the gaps of the bundle, the uncertainty of the flow is enhanced. The force on a certain tube may not obey the overall trend. For example,

_{d}*C*of T13 is greater than them of T12 and T14 in Fig. 17(a).

_{f}Root-mean-square force coefficients of all the tubes for various structural bundles at Re* *=* *2.371 × 10^{4} are shown in Fig. 18, where RMS *C _{f}* of case 1 is the largest, and that of

*a*=

*2.0,*

*b*=

*1.5, and*

*α*= 15 deg (case 6) is generally the smallest. And

*C*of all the tubes is uniform. There is no prominent change. In general,

_{f}*a*has a great influence on RMS

*C*(the smaller

_{f}*a*is, the larger

*C*is), but

_{f}*b*and

*α*have little influence on it. When

*α*decreases, RMS

*C*increases slightly. With the increase of

_{f}*b*, the RMS

*C*does not decrease obviously. The more compact the bundle, the greater the role of jet switching. As the gaps between the adjacent layers widen, the effect of vortex shedding becomes more and more obvious. And turbulence works all the time. Figure 18(b) shows the distribution of

_{f}*C*of all the tubes. The existence of

_{l}*α*strengthens the flow in the lift direction and makes the flow more unstable. The smaller

*a*is, the greater

*C*is. The RMS

_{l}*C*of all the tubes is shown in Fig. 18(c), and its law is the same as that of

_{d}*C*, which will not be repeated here.

_{f}### 5.3 The Envelope Spectrum for Random Excitation Forces.

The vibration amplitude caused by random excitation forces is relatively small and will not lead to short-term failure. However, these vibrations do cause constant friction between the tubes and its supports, which makes the tube gradually break down due to fretting wear. As a result, it is very necessary to determine the guidelines of random excitation force for coil bundle. As discussed earlier in Eqs. (9), (15), and (17), it is necessary to refer to a single excitation tube length, *L _{e}*, in order to properly compare the spectrum obtained by the coil tube bundle with different geometric parameters. This paper uses 1 m as the reference length. The flow into CWHE first passes through curved tube or straight tube regions (Fig. 1) and then enters the coil tube region, so there is no direct fluid impact on the tube. Therefore, only the coil tube interior the bundle is considered. In this paper, the effects of tube position, Re, and bundle structure on the normalized PSD are analyzed in order to obtain the normalized spectrum of excitation force for the bundle.

*L*= 1 m for all the tubes of case 5 at Re

_{e}*=*

*2.371 × 10*

^{4}. It is observed that PSD varies from tube to tube within the bundle. T18 is the most backward of the monitoring tubes, and its PSD is located above the rest. It can be conjectured that the more coil tubes the flow passes through, the stronger the flow instability and the larger the value of PSD. In future studies, the number of tubes for per layer can be increased to get closer to actual number of tubes. To simplify the spectrum, the narrow peaks in the PSD are ignored. The broken line that envelops PSD is plotted, and the envelope 1 is proposed as

*a*=

*1.5,*

*b*=

*2.6, and*

*α*= 15 deg. In addition, PSD of T10 and T18 under three representatives of Re is considered. The narrow peaks in PSD may be the vortex shedding frequency or the dominant buffeting frequency of flow-induced vibration. Only the universal existence of random excitation force on the tube is studied, so the narrow peaks are ignored. Draw the envelope 2 as

*L*= 1 m. Three structural parameters,

_{e}*a*,

*b*, and

*α*, are used to determine the structure of the coil tube bundle. The variation of each structural parameter affects the flow characteristics in the shell-side, thus changing the random excitation force on the tube. The selected bundle structure includes compact arrangement and transition arrangement, which are common types of bundle arrangement [48]. The helix angle,

*α*, is within the usual range. PSD of T11 and T18 is selected to determine the envelope 3, delineated as

And the final envelope of the coil tube is compared with that of the straight tube. Note that the three envelope lines are representative. Axisa et al. took into account a variety of factors to draw the envelope [20]. Taylor and Pettigrew summarized various previous research results and proposed two other envelope lines [33]. It is found that the broken line for interior the straight tube bundle intersects with the coil tube. The broken line of the straight tube bundle close to the inlet has crossing points with the coil tube at the sharp position, but most of it is enveloped. However, when *f _{r}* = 0.01, the difference in values is 1 order of magnitude, indicating a large error. Therefore, it is necessary to analyze the normalized envelope spectrum independently to predict and evaluate the force on the coil tube.

## 6 Conclusion

The coil tube bundle in the CWHE was simplified, and the test equipment was established. Flow-induced vibration of the coil tube bundle with *a *=* *1.5, *b *=* *2.6, and *α* = 15 deg was carried out through water tunnel tests. And the numerical models with respect to various structural parameters were studied in order to investigate turbulent buffeting mechanism and the influence of Re and bundle structures on the force coefficients. The main conclusions of this paper are the following.

The results show that the vibration response error is within 10%, and the order of magnitude corresponding to the normalized force spectrums and the trends are a little different for the experimental and simulated data, which proves the robustness of the numerical method.

- The dominant frequencies of turbulent buffeting corresponding to various structures were obtained by monitoring the fluid excitation of the tubes. Using the method of threshold envelope, the semi-empirical equation was set up as$ft=UgapdLT[1.89(1\u2212dT)2+0.39]$
The smaller the clearance within the tube bundle, the more obvious the jet switching, resulting in the increase of the force coefficients. As the pitch diameter ratio between adjacent tube layers (

*a*) increases, the shear flow on both sides of the tube enhances, and eddies are formed behind the tube. Lift on the tube has no obvious change law. Vortex shedding plays a major role at this moment.*a*has a great influence on the force coefficients, while the pitch diameter ratio in the same layer (*b*) and the helix angle (*α*) have little influence on it. In addition, force coefficients in the bundle*a*= 1.2,*b*= 1.5, and*α*= 15 deg (case 2) fluctuate greatly and the flow instability is obvious, which is the result of the interaction of jet switching, vortex shedding, and turbulence.The vibration amplitude caused by random excitation forces is relatively small. However, these vibrations do cause constant friction between the tube and its supports, which causes the tube to gradually break down due to fretting wear. Therefore, it is important to determine the random excitation force of the coil bundle. The mechanisms of tube position, Re, and bundle structure on the normalized force spectrum were investigated. And the envelope of each case is determined respectively. Eventually, the normalized envelope spectrum of random excitation acting on the coil tube was proposed.

## Acknowledgment

The authors would like to thank Heat Exchanger Committee, National Technical Committee for Standardization of Boiler and Pressure Vessels for technical support. Thanks also to Ami Tektronix Industrial Equipment Co., Ltd. for its help in the design and implementation of experiment.

## Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*α*=helix angle, deg

*γ*=the degree of correction of forces along the tube

*δ*=mode shape displacement constant

*ε*=porosity

*ζ*=damping ratio

*λ*=the correlation length

*ρ*=fluid density in the shell side, kg/m

^{3}- $\varphi $ =
the autocorrelation spectrum of the force per unit tube length

*ψ*=the fluctuating force per unit tube length

## References

*Bluff-Body Wakes, Dynamics and Instabilities*