1 Introduction

Helical tube has a broad range of application in various industry devices such as heat exchangers, chemical reactors, steam generators and mixing vessels [1]. It has the advantages of compact structure and high heat exchange efficiency [2, 3]. The study on fluid flow characteristic and heat transfer mechanism in helical tubes is always a hot issue in the last several decades. A large number of literatures have reported the researches on fluid flow characteristic, heat transfer performance and compound enhanced heat transfer in helical tubes by using experimental or numerical methods. The main parameters affecting the heat transfer efficiency of helical tube include curvature and torsion of the tube, flow parameter characterized by Reynolds number or Dean number as well as properties of fluids. The high heat transfer efficiency of the helical tube is mainly due to the self-generated secondary flow under the action of centrifugal force. Yang et al. [4, 5] discussed the effects of the Dean number, torsion, and Prandtl number of fluid on the convective heat transfer in helical tubes. It was concluded that the secondary flow would become stronger when the Dean number increases. As the torsion increased, the temperature distribution in the cross-section was asymmetrical. Khoshvaght-Aliabadi [6] analyzed the thermal–hydraulic performance in the helical tube for different working fluids. They found that the Nusselt number increased with Prandtl number increasing, but the friction coefficient changed little. Based on different helical tube structural parameters and different fluids, El-Genk [7] proposed correlations of flow resistance coefficient and Nusselt number by handling the experimental data reported in the literatures. Hardik et al. [8] analyzed the local heat transfer performance in a helical tube. The results showed that values of Nusselt number of the outer wall were greater than those of the inner wall. Li et al. [9] and Zhang et al. [10] analyzed the field synergy of the velocity and temperature fields in helical tubes with semicircular and rectangular cross sections. They proposed that heat transfer efficiency of helical tubes could be further enhanced by improving the secondary flow fields near the inner wall.

At present, some passive and active heat transfer enhancement methods have been applied to helical tubes. The mechanism of heat transfer enhancement is to increase the intensity of fluid turbulence and improve the characteristics of secondary flow fields. The passive method includes changing the tube wall characteristics or installing disturbance elements. Li et al. [11] and Zachár et al. [12] numerically studied the heat transfer enhancing performance by adding spiral corrugation on the inner wall of helical tube, respectively. Conclusions were drawn that heat transfer performance could be improved to a certain extent, however, the flow resistance would increase significantly at the same time. And Rainieri et al. [13, 14] pointed out that this method was more suitable for the high Reynolds number. In recent literatures [15,16,17,18], researches on installing disturbance element in helical tube to enhance heat transfer have also been reported, such as spiral coils or vortex generators. The disadvantage of placing a spiral coil in a helical tube to achieve heat transfer enhancement is that the flow resistance increases significantly. The vortex generator can enhance heat transfer by changing the flow field characteristics near the wall, but the number and location of installation are easy to be limited.

In addition to the above passive methods, active enhanced heat transfer method is also applied to helical tubes. Pulsating flow is one of the effective active methods. Some experimental and numerical studies have been carried out to study pulsating flow in helical tubes. Sinusoidal and square wave are two common used patterns of pulsation. Many researches on heat transfer enhancement of sinusoidal pulsating flow in helical tubes have been reported in the literatures. The main affecting factors are Reynolds number, pulsating frequency and pulsating amplitude [19, 20]. Rabadi et al. [21] pointed out that heat transfer efficiency might be even reduced at some specific frequency and amplitude of sinusoidal wave pulsating flow in the curved tube. Pan et al. [22, 23] revealed the mechanism of sinusoidal pulsating flow enhancing heat transfer in helical tubes based on the field synergy principle. The results showed that volume average field synergy angle under the pulsating flow was less 2.45% than that under the steady flow. Kharvani et al. [24, 25] considered that the upstream pulsation heat transfer coefficient was larger than the downstream pulsation in helical tubes. For larger mean Reynolds numbers, the relative average heat transfer coefficient was increased by 16–26%. Guo et al. [26] simulated nanofluids heat transfer in a helical coil under pulsation condition. They found the secondary flow generated in the cross section and the counter-rotating vortex formed in the axial direction both devoted to the heat transfer augmentation under pulsating state.

Compared with the sinusoidal wave, there is a drastic change of velocity in square wave. According to reference [27], many scholars pointed out that square wave pulsating flow significantly enhanced heat transfer. Zhang et al. [28] pointed out that square wave pulsating flow caused greater disturbance to the fluid and promoted the mixing of fluids more easily. Thus it might be more virtue for heat transfer enhancement. In addition, square wave pulsating flow is easy to achieve. However, the study on square wave pulsating flow in helical tube is rare, except that Hamed et al. [29] carried out the experimental study. They investigated heat transfer and pressure drop of square wave pulsating flow in helical coiled tube at the condition of constant heat flux. The results revealed that increment in pressure drop was nearly 3–7% in pulsating flow compared to the steady one, while convective heat transfer was enhanced up to 39%.

According to the literature reviews, it can be found that the research on the flow field and heat transfer characteristics of the square wave pulsating flow in the helical tube is not deep enough. Its mechanism of heat transfer enhancement is not clear. In addition, the research of the fluids properties affecting on pulsating flow enhancing heat transfer in helical tubes has not been found. For the above reasons, the paper has studied the heat transfer enhancing characteristics of the square wave pulsating flow in the helical tube using the numerical method. The influences of the dimensionless frequency and dimensionless amplitude of the square wave pulsating flow on heat transfer enhancement have been investigated in detail based on the boundary condition of constant wall temperature. The distribution of flow field, temperature field and local heat transfer characteristics in a pulsating period is analyzed in detail. Based on these, heat transfer enhancing mechanism of square wave pulsating flow in helical tube has been revealed. In addition, the influence of the Prandtl number of working fluids on the heat transfer enhancing for pulsating flow in the helical tube has also been studied.

2 Numerical simulation

2.1 Physical and mathematical models

In the present study, the helical tube with a circular cross section is vertically oriented, seen in Fig. 1. The number of turns of the helical tube is set as 4, namely the angle α = 1440°. The helical diameter and the pitch of the helical tube are represented by D and Pc respectively. The inner diameter of the circular tube is characterized by d. The geometric parameters of the helical tube have been listed in Table 1.

Fig. 1
figure 1

Basic geometry of the helical tube

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Table 1 Geometric parameters of helical tube
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The fluid domain is three-dimensional, incompressible and unsteady. Water is used as working fluid. The governing equations including continuity, momentum and energy equations for the fluid domain can be expressed as follows [26]:

Continuity equation:

$$\nabla \cdot \left(\rho \overrightarrow{v}\right)\text{=}0$$
(1)

Momentum equation:

$$\frac{\rho \partial \overrightarrow{v}}{\partial t}+\rho (\overrightarrow{v}\cdot \nabla \overrightarrow{v})=-\nabla p+\mu {\nabla }^{2}\overrightarrow{v}$$
(2)

Energy equation:

$$\rho c_p\frac{\partial T}{\partial t}+\rho c_p\overrightarrow{v\cdot }\nabla T=\nabla \cdot \left[k\left(\nabla T\right)\right]$$
(3)

where ρ is density, p is pressure, μ is viscosity, cp is specific heat, T is temperature, t is the time and k is thermal conductivity.

Details of the boundary conditions are given as follows: At the inlet, the inlet velocity vin and inlet temperature Tin are set to be uniformly distributed with Tin = 293 K. A square wave pulsation displayed in Fig. 2 is used to describe the changing of vin and its mathematical expression is given in Eq. (4).

Fig. 2
figure 2

The pattern of pulsation inlet velocity

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$$\begin{array}{cc}{v}_{\text{in}}=\left\{\begin{array}{c}\begin{array}{cc}v_{\text{m}}+A& n\Gamma <t\le (n+1-\beta )\end{array}\\ \begin{array}{cc}v_{\text{m}}-A& (n+\beta )\Gamma <t\le (n+1)\Gamma \end{array}\end{array}\right.& n=\mathrm{0,1},2,..\end{array}$$
(4)

Here, Γ is the period of the square wave pulsation and n is the natural number. vm is the average velocity in a pulsating period. A is the amplitude of square wave pulsating flow. It is the velocity difference between extreme velocity (vmax or vmin) and average velocity vm. vmax and vmin are respectively the maximum and minimum velocity in the helical tube at pulsating state. β = Δt/Γ is defined as the duty ratio of the square wave and used to characterize the proportion of the maximum velocity time length Δt in a pulsating period. β = 0.5 is selected in the present study. The inlet velocity is guided by the user-defined functions (UDF) of the CFD software in the present study.

At the outlet, the pressure outlet boundary condition is adopted and the relative pressure of outlet is set as zero. The no-slip and uniform wall temperature boundary conditions are imposed on the wall, where the value of wall temperature Tw is set as Tw = 343 K. The initial condition is that the initial fluid velocity and temperature is taken as the mean velocity and mean temperature at the inlet. The inlet thermo-physical properties of water are displayed in Table 2 at room temperature of 293 K [6].

Table 2 Thermo-physical properties of working fluids
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2.2 Main parameter definition

Dimensionless parameter Dean number De is used to characterize the flow of fluids in helical tube and is expressed as [30]:

$$De=Re\sqrt{d/D}$$
(5)

where, Re is the Reynolds number and is defined as:

$$Re=\frac{\rho v_{\text{m}}d}{\mu }$$
(6)

The dimensionless amplitude \(\overline{A}\) and dimensionless frequency Wo are used to characterize the pulsating flow and are defined respectively as follows:

$$\overline{A }=\frac{A}{v_m}$$
(7)
$$Wo=\frac{d}{2}\sqrt{\frac{2\pi f}{\upsilon }}$$
(8)

where, f = 1/Γ is pulsating frequency and υ is the kinematic viscosity of the fluid. In the present study, the value of f ranges from 0.125 Hz to 8 Hz to obtain good heat transfer enhancement effect. At the same time, the value of f should be taken into account that the value of Γ is an integer.

The dimensionless time \(\overline{t}\) is expressed as:

$$\overline{t }=t/\Gamma$$
(9)

The average Nusselt number Nu and flow resistance coefficient fD are defined as follows:

$$Nu=\frac{hd}{k}$$
(10)
$$f_D=\frac{2d\Delta p}{\rho {v}_{\text{m}}^{2}L}$$
(11)

where, L is the length of the helical tube. h is average heat transfer coefficient and △p is average pressure drop calculated by [22]:

$$h=\frac{1/t{\int }_{0}^{t}qdt}{1/t{\int }_{0}^{t}(T_{\text{w}}-T_{\text{f}})dt}$$
(12)
$$\Delta p=1/t{\int }_{0}^{t}\Delta pdt$$
(13)

where, q is heat flux, Tf is the temperature of working fluid, Tw is the wall temperature.

2.3 Numerical simulation method

The commercial CFD code ANSYS Fluent 16.2 is used for the numerical solution. Unsteady segregated solver is used to handle Navier–Stokes equations and energy equations by finite volume method. The realizable k-ε turbulent model [31] is employed for it has the superior performance in deal with the flows involving rotation, great reverse pressure gradient and back flow. The PISO algorithm [23] is utilized to deal with the coupling of pressure and velocity. The convective term in momentum and energy equations are discretized by a second-order upwind scheme. And the time discretization adopts a second order accurate fully implicit scheme. For all simulations, the computations are considered to be converged when the residuals of continuity equations are less than 10–4 and the momentum and energy equation are less than 10–6. Computers with 3.3 GHz processor and 8.0 GB RAM are used. It takes about 12 h to obtain the calculation results at the specific amplitude, frequency and velocity.

2.4 Grid and time step validation

In order to ensure the accuracy of the calculation results, the independence of the calculation grid and time step has been verified. The calculation zones are discretized with structured hexahedral grid, and local grid refinement is applied in the boundary layer. Figure 3 presents grids of the cross section. The standard wall function method is used for the near wall region. The wall y plus is evaluated to check the requirement of standard wall function. Four sets of grid system with gradually increasing grid number are generated to valid the influence of grid on the numerical results and they are Grid 1 (1,111,808), Grid 2 (1,389,760), Grid 3 (1,516,352) and Grid 4 (1,683,856). The values of fD and Nu calculated based on the four sets of grid system are displayed in Table 3. It can be found that relative deviation of fD and Nu values computed based on the Grid 3 and the Grid 4 is very small. For saving computer resource and keeping a balance between computational economics and accuracy, Grid 3 is adopted in the present simulation.

Fig. 3
figure 3

Grids in the cross-section of the helical tubes

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Table 3 Variation of fD and Nu with cell number for the helical tube at Re = 2982
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The time step independence test has been performed based on three time step sizes and they are Γ/50, Γ/100 and Γ/200, respectively. After the flow reaches periodic stability, the results of temperature difference (ΔT) between inlet and outlet with three different time step sizes are shown in Fig. 4. It can be seen that the relative deviation of ΔT in the cases of Γ/100 and Γ/200 is less than 0.01%, thus the time step size is set as Γ/100.

Fig. 4
figure 4

Time step △T independence

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3 Experimental validation

3.1 Experimental setup

In order to test the accuracy of the simulation model and numerical method adopted in the present study, an experiment has been carried out on the helical tube. Figure 5 illustrates schematic diagram of experimental setup. The experimental system is mainly composed of pump, helical tube, pulsating device, heating device, measuring instruments, connecting pipes and valves. The helical tube is made of copper. The pulsating flow is generated by a solenoid valve. The time of opening and closing of the solenoid valve is controlled by the time controller to achieve the desired frequency. The heating device is a constant temperature water tank with three electrical heaters. The constant temperature water tank is made of stainless steel. Three electrical heaters and three thermocouples are installed at the bottom of the tank. In order to maintain the ambient hot water temperature at a constant value, the thermocouple is connected with the temperature control switch to control the working state of the heater. The constant temperature tank is also equipped with a circulating pump, which starts automatically when the temperature difference of the three thermocouples exceeds 0.5 K and closes automatically when the temperature difference of the three thermocouples is less than 0.1 K, so that the overall water temperature in the tank is kept at (343 ± 2) K.

Fig. 5
figure 5

Schematic diagram of experimental setup

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Cold water at 20 °C is pumped out of the storage tank and discharged after heat exchange in the test section. In order to ensure the stability of the mainstream flow, a pipeline is designed in parallel with the pulsating flow. This can also avoid damage to the pipeline in practice. The flowmeter is installed on mainstream pipe to measure flow rate. The valve is used for controlling the flow rate which is positioned at the pulsating device and the parallel pipe. The flow rate is measured by volume flow rate method. Namely, the measuring cylinder measures the liquid collected in a period, and the time is measured by stopwatch. The total flow rates of water ranges from 0.03 to 0.3 cubic meter per hour and the corresponding Re number is between 1400 and 8950. The fluid temperature at the inlet and outlet of test section is measured by type-K thermocouples. The temperature is collected by HIOKI (LR8432-30) heat flow meter and saved automatically within 0.5 s intervals, and then the data are transferred to the computer for processing. In order to obtain the pressure drop, two intelligent manometers (JC-80XB) are installed at the inlet and outlet of the helical tube. All parameters are collected and recorded when the system reaches the steady condition. After that, the flow rate changed to the new desired velocity and above procedure is repeated. The same experimental procedure has been independently repeated for three times to assess the reproducibility and reliability.

3.2 Experimental uncertainty

The uncertainty of the above experiments is analyzed, based on the equations presented by Kline and McClintock [32]:

$$R=R(x_1,x_2,...,x_J)\cdots U_R=\sqrt{\sum_{i=1}^{J}{\left(\frac{\partial R}{\partial x_i}U_i\right)}^{2}}$$
(14)

where, UR and Ui are the uncertainties for the results and the independent variables, respectively. The accuracy of measuring instruments and uncertainty of calculated variables are presented in Table 4. For all experimental runs, the maximum uncertainties of Nu and fD are about 8.8% and 9.3%, respectively.

Table 4 Measuring instruments accuracy
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3.3 Comparison of simulated and experimental results

Figure 6 shows the comparison of simulated and experimental values of Nu and fD. The uncertainty bars are indicated on the experimental data. Based on experimental values, the average values of relative deviation of Nu and fD are 8.51% and 8.97%, respectively for the pulsating flow. The agreements between the numerical and experimental results are acceptable. This illustrates that the calculation model and numerical method in the present study are feasible and the simulated results are reliable.

Fig. 6
figure 6

Comparisons between the numerical results and the experimental results at Wo = 9 and \(\overline{A}\) = 0.5

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4 Results and discussions

4.1 Effects of pulsating parameters

Figure 7 presents the effects of Wo number on Nu and fD at \(\overline{A}\) = 0.5. In the present study, the pulsating frequency f of the square wave pulsating flow is in the range of 0.125 Hz-8 Hz and the corresponding calculated Wo number is between 3–27. The average flow rate of helical tube in pulsating flow state is the same as that in steady flow state for comparison. It can be seen from Fig. 7 that both Nu and fD values under pulsating flow are greater than those under steady flow (Wo = 0) at the same De number. This illustrates that the square wave pulsating flow can further improve the heat transfer performance of the helical tube, however, it will increase the flow resistance. Computed results show that compared with the steady flow, Nu is increased by 11.0% on average, and fD is increased by 9.22% on average in the pulsating state.

Fig. 7
figure 7

The variations of Nu and fD with Wo at \(\overline{A}\) = 0.5

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It is worth noting that in pulsating state, the effect of Wo on Nu is more significant at high De number. This is mainly because the heat transfer enhancing of helical tubes mainly depend on the action of secondary flow. The existing research results [16] show that the secondary flow plays a primary role in heat transfer enhancement at low Re number or De number. With the increase of Re number, the location of secondary flow gradually approaches the outer wall, which reduces its contribution to enhanced heat transfer. It is pointed out in the literature [33] that when the Reynolds number increases to a certain value, the heat transfer effect of the helical tube is equivalent to that of the straight tube. At this time, the heat transfer performance mainly depends on the turbulence intensity of the fluid. In the present study, the curvature of the helical tube remains unchanged and the increases of De number mean increases of Re number. In the case of small De number, increasing the frequency of pulsating flow cannot significantly improve the intensity of secondary flow, so the effect of heat transfer enhancement is not obvious. However, the flow resistance coefficient increases significantly due to the increase of disturbance. Under the condition of high De number, the larger the pulsating frequency Wo is, the easier it is to enhance the turbulence intensity of the fluid and improve the heat transfer performance.

In order to evaluate the comprehensive heat transfer enhancing performance of square wave pulsating flow in the helical tube, comprehensive enhancement heat transfer factor TP is adopted and is defined as [34]:.

$$TP=(\frac{Nu_{\text{p}}}{Nu_s})\cdot {(\frac{f_\text{D,p}}{f_\text{D,s}})}^{-1/3}$$
(15)

where, Nup and fD,p is the values in the pulsating state. Nus and fD,s is the values in the steady flow state with the same average flow rate.

Figure 8 displays the variations of TP values against Wo. It can be seen that TP values are all greater than unity in the studied scope, which illustrate that it is significant to use square wave pulsating flow to enhance fluid heat transfer in helical tubes. It can also be observed that case of Wo = 9 is optimal in the studied range. When Wo is greater than 9, the TP value decreases gradually. This is for the reason that the improvement and development of the secondary flow not only contribute to heat transfer enhancement but increase the flow resistance under the pulsating flow condition. Only the optimal pulsating frequency can appropriately balance the relation between the generation, the growth and the expansion of the vortex so as to obtain the sufficient mixing of the fluid as well as the best heat transfer enhancement effect [26]. The results shows that TP values is in the range of 1.03–1.11 at the case of Wo = 9.

Fig. 8
figure 8

The variations of TP with Wo at \(\overline{A}\) = 0.5

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Figure 9 presents the effects of \(\overline{A}\) on Nu and fD at Wo = 9. It can be found that similar to the effect of Wo, an increase in \(\overline{A}\) values also leads to the increase in Nu and fD. This is for the reason of the larger the pulsating amplitude, the stronger ability is of disturbing and mixing the fluid. This benefits to the heat transfer enhancement but makes the flow resistance increase at the same time. Figure 10 displayed the variations of TP values against \(\overline{A}\). It can be found that TP increases at first and then decreases with the increase of \(\overline{A}\). When \(\overline{A}\) is greater than 0.25, the TP value decreases gradually, and especially TP value decreases more obviously under the condition of low De numbers than that of high De numbers. This illustrates that the increase amplitude of flow resistance is greater than that of heat transfer with the increase of \(\overline{A}\). There is an optimum De number De = 1106 at which the TP value obtains maximum. The optimal value of dimensionless amplitude is \(\overline{A}\) = 0.25. Computed results show that TP values is in the range of 1.04–1.12 at \(\overline{A}\) = 0.25 and Wo = 9.

Fig. 9
figure 9

The variations of Nu and fD with \(\overline{A}\) at Wo = 9

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Fig. 10
figure 10

The variations of TP with \(\overline{A}\) at Wo = 9

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4.2 Heat transfer enhancement mechanism

4.2.1 Flow and temperature fields

The characteristic of square wave pulsating flow different from the steady flow is that there are two times of abrupt velocity changes in one pulsating period. One is at the beginning of the period and the other is in the middle of the period. The sudden increase or decrease of velocity will change the original flow fields in helical tubes, and then affect the convection heat transfer performance. In order to reveal the heat transfer enhancement mechanism of square wave pulsating flow in helical tube, flow and temperature fields in one pulsating period have been discussed in detailed. The cross section of α = 720° is selected and the flow parameters are De = 1106, Wo = 9 and \(\overline{A}\) = 0.25.

Figure 11 gives the contour of instantaneous dimensionless temperature \(\overline{T}\) = (Tm-Tw)/(T-Tw) and dimensionless tangential velocity \(\overline{v}_\mathrm{\theta}\) = vθ/vm in the cross section of one pulsating period. Tm is the average temperature of cross section. Here, the right of the cross section is the outer wall of the helical tube. Figure 12 gives the change of dimensionless tangential velocity \(\overline{v}_\mathrm{\theta}\) distribution on the centerline of the cross section during one pulsating period. From the two figures it can be found that the location of the maximum values of \(\overline{v}_\mathrm{\theta}\) is near the outer wall in the cross section. This phenomenon occurs due to the effect centrifugal force. The tangential velocity gradient and the temperature gradient near the outer wall of the cross section are both greater than those near the inner wall at any moment. Therefore, the heat transfer capacity of the outer wall is better than that of the inner in helical tubes under the pulsating flow condition, which is similar with that under the steady flow condition. It can also be found from the two figures that the velocity field is more uneven around the time when the velocity changes suddenly. On the contrary, the temperature distribution in the cross section tends to be more uniform except the location near the inner wall. This indicates that the sudden change of velocity of the pulsating flow is beneficial to promote the mixing of the fluid in the helical tube and enhance the heat transfer performance. It is worth noting from Fig. 12 that the values of \(\overline{v}_\mathrm{\theta}\) near the inner wall is negative at the time of \(\overline{t}\) = 0.5 and \(\overline{t}\) = 0.6. This illustrates that there is a backflow at the end of half period due to the sharp reduction of flow rate. The appearance of reflux near the inner wall also contributes to the enhancement of heat transfer.

Fig. 11
figure 11

The distribution of \(\overline{T}\) and \(\overline{v}_\mathrm{\theta}\) in a pulsating period at α = 720° for De = 1106, Wo = 9 and \(\overline{A}\) = 0.25

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Fig. 12
figure 12

The distribution of \(\overline{v}_\mathrm{\theta}\) on the centerline of the cross section of α = 720° during one pulsating period at De = 1106, Wo = 9 and \(\overline{A}\) = 0.25

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It is well known that the intensity and scope of secondary flow directly affect the heat transfer performance of the helical tube. The distributions of the time-average dimensionless vorticity magnitude of secondary flow during one pulsating period have been shown in Fig. 13. It can be deduced that there is mainly one pair of symmetrical vortices of secondary flow in the cross section. Define the time segment between \(\overline{t}\) = 0.1–0.5 as high flow rate time period and that between \(\overline{t}\) = 0.6–1.0 as low flow rate time period. It can be seen that the vorticity of the former period is higher than that of the latter due to the great flow rate. Moreover, the intensity of secondary flow increases when the flow changes suddenly. This shows that the square wave pulsating flow can further enhance the strength of secondary flow in the helical tube.

Fig. 13
figure 13

The distribution of the time-average dimensionless vorticity magnitude on the cross section of α = 720° during one pulsating period at De = 1106,Wo = 9 and \(\overline{A}\) = 0.25

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4.2.2 Local enhanced heat transfer characteristics

Figure 14 has displayed the distribution of local Nusselt number Nulocal along the circumference of wall of the cross section both in steady and pulsating state. It can be seen that under the influence of the flow field, the Nulocal values near the outer wall of the helical tubes is significantly higher than those near the inner wall in the two flow states. It shows that the heat transfer near the inner wall of the helical tube is inefficient. The Nulocal values in the pulsating state are greater than the corresponding values in the steady state. This is mainly because the pulsating flow not only increases the turbulent ability of the fluid, but also enhances the intensity of the secondary flow. It can also be observed that values of Nulocal near the outer wall at \(\overline{t}\) = 0.1 are higher than those at \(\overline{t}\) = 0.6 due to the greater flow rate. However, Nulocal values near the inner wall at \(\overline{t}\) = 0.6 is greater than the corresponding values at \(\overline{t}\) = 0.1 although the flow rate at this time is smaller. This may be due to the backflow at the inner wall when the flow suddenly decreases.

Fig. 14
figure 14

Distribution of Nulocal on the circular wall at α = 720° for De = 1106, Wo = 9 and \(\overline{A}\) = 0.25

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Define the average value of Nulocal on the heated wall is as (Nulocal)m. Figure 15 gives the change of (Nulocal)m with time in one pulsating period at α = 720°. It can be found that in the pulsating state, the (Nulocal)m values in the high flow period are higher than those values in the steady state which are computed based on the maximum velocity vmax. Similarly, during the low flow period, the (Nulocal)m values are higher than those values in the steady state which are computed based on the minimum velocity vmin. In particular, at the initial time of the low flow rate period, the (Nulocal)m value is even higher than the steady state value calculated based on the average velocity vm. The above results show that the square wave pulsating flow can enhance the fluid heat transfer in the helical tube mainly due to two points: one is the enhancement of the intensity of the secondary flow and turbulence, and the other is the emergence of back flow at the inner wall when the velocity suddenly decreases.

Fig. 15
figure 15

The average Nu number of the cross section wall during one pulsating period at α=720º, De = 1106, Wo = 9 and \(\overline{A}\) = 0.25

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4.2.3 Field synergy mechanism analysis

Field synergy principle is used to further reveal the mechanism of square wave pulsating flow enhancing heat transfer in helical tubes. It has been pointed out by Tao and co-workers [35,36,37] that the reduction of the intersection angle between velocity vector and temperature gradient can effectively enhance convective heat transfer. The intersection level of the synergy between the temperature gradient and the velocity vector can be expressed by synergy angle θ and it is defined as [38]:

$$\theta =\mathrm{arccos}\ \left(\frac{\overline{U }\cdot \nabla \overline{T} }{\left.\left|\overline{U }\right.\right|\left|\left.\nabla \overline{T }\right|\right.}\right)$$
(16)

Here \(\overline{\varvec{U}}\) and \(\nabla \overline{T}\) are the dimensionless secondary flow velocity vector and temperature gradient, respectively. Figure 16 displayed the distribution of θ in the cross section of α = 720° at De = 1106, Wo = 9 and \(\overline{A}\) = 0.25. It can be seen that θ values decrease obviously when the velocity changes suddenly in one pulsating period, which indicates convection heat transfer enhancing.

Fig. 16
figure 16

The average field synergy angle of the cross section of α = 720° during one pulsating period at De = 1106, Wo = 9 and \(\overline{A}\) = 0.25

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4.3 Effects of fluids properties

Some studies have reported the effect of Prandtl Number Pr of fluid on heat transfer characteristic in helical tubes under the steady flow condition, but there is no relevant research on the pulsating flow state. Considering this, the oil of Pr = 50.41 and ethylene–glycol of Pr = 150.46 have been selected as the medium for comparison with water. Figure 17 gives the effect of Pr number on Nu and fD values in the helical tubes under pulsating flow condition. It can be found that in steady flow state, Nu is enhanced significantly as Pr increases, but the increase amplitude of fD is small. The similar conclusions can be found in the literature [6]. In the pulsating flow state, both Nu and fD values are enhanced as Re number increases. Computed results show that compared with the steady flow, Nu of three type of medium in the pulsating flow is increased by 10.3%, 5.6% and 2.3%, however, fD increased by 8.6%, 6.0% and 31.0% on average, respectively. This lead to TP values decreases with the increase of Pr number, seen in Fig. 18. Thus the conclusion can be drawn that the fluid of low Pr number is more suitable for heat transfer enhancement of square wave pulsating in helical tubes.

Fig. 17
figure 17

The variation of Nu and fD for helical tubes with different working fluids at Wo = 13 and \(\overline{A}\) = 0.5

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Fig. 18
figure 18

The variation of TP for helical tubes with different working fluids at Wo = 13 and \(\overline{A}\) = 0.5

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5 Conclusions

The square wave pulsating flow is used to enhance convection heat transfer in a helical tube. The influences of dimensionless frequency Wo, dimensionless amplitude \(\overline{A}\) of the pulsating flow and the Prandtl Number Pr of fluid have been discussed in detailed. The heat transfer enhancement mechanism has been revealed. The major conclusions can be drawn as follows.

  • Square wave pulsating flow can significantly improve the heat transfer performance of the helical tube. Both Nu and f values increase as Wo and \(\overline{A}\) increase.

  • The optimum condition of square wave pulsating flow is Wo = 9 and \(\overline{A}\) = 0.25 based which the comprehensive heat transfer enhancement factor TP is within the scope of 1.03–1.12 in the studied range.

  • Both increase in secondary flow intensity and turbulence intensity as well as the emergence of the backflow near the inner wall contribute to the heat transfer enhancement of square wave pulsating flow in the helical tube.

  • Square wave pulsating flow is more suitable for enhancing heat transfer of fluid with small Pr number in the helical tube.