1 Introduction

So far, heat exchangers used in domestic and foreign are in the form of rectangular, circular or other regular equal cross section heat exchange surface runner. However, while the cooling medium flows from the inlet to the outlet through the fins and the substrate of the equal cross section heat exchange surface runner, the temperature difference between inlet and the cooling medium has the largest value. The cooling medium is continuously heated while it flows to the outlet that resulting in a smaller temperature difference between the outlet and the cooling medium, which suppressing the ability of heat transfer from the substrate to the cooling medium in the downstream region of the runner [1]. In addition, the cooling medium generates a certain pressure loss during flowing, and the loss increases in the form of the quadratic with the increase of the flow rate, which greatly increased the consumption of the pump power and exacerbates the energy loss [2].

In view of the defects of the above-mentioned conventional equal cross section heat exchange surface runner, this paper presents a kind of cosine style heat exchange surface runner. The cosine style runner is a cosine-shaped structure with a gradually expanding shape on a section parallel to the flow direction. However, in the direction perpendicular to the flow, the shape of the cross-sectional area is rectangular, and the size of the cross-sectional area changes. The height and the heat transfer area of the fins gradually increase in the flowing process, so that more heat can be transferred. Thus the temperature difference between the fins and the cooling medium is relatively uniform, which enhances the capacity of convection heat transfer. Since the flow path is a diverged structure with big inlet and small outlet in the direction of flow, the cooling medium is also pressurized during the flow. Thus, the pressurizing process can greatly reduce the power consumption.

At present, the research on the runner of the cosine style heat exchanger has not been reported. But, there are many studies on trapezoidal tapered runner structure. Wu et al. [3] measured the friction coefficient in the trapezoidal cross section of smooth silicon-based microchannel with different hydraulic diameters. The Navier-Strokes equation is still valid when the hydraulic diameters range from 25.9 to 291.0 μm. Kuo et al. [4] numerically simulated the periodic trapezoidal structure and the method was used to evaluate the heat transfer performance and flow characteristics of the trapezoidal runner. They also found that the trapezoidal runner can reduce the field synergy angle between velocity vector and the temperature gradient, which can improve the field synergy in the runner. Amrahmadi et al. [5] used a vortex generator to analyze the forced convection heat transfer performance in a trapezoidal runner, and they solved the mass conservation, momentum and energy conservation equations by using finite volume method. For the trapezoidal runner structure in the above-mentioned references, the cross section perpendicular to the flow direction is a structure including rectangle, circle and trapezoid, while the equal section area remains unchanged. As for diverged flow path, there is little research on the trapezoidal tapered structure. For this kind of structure, Farhanieh et al. [6] studied the cross section of trapezoidal runner with different geometries by means of numerical analysis, and analyzed its flow and heat transfer performance in laminar flow. Li et al. [7] experimentally investigated the heat transfer performance of the trapezoidal runner inside the turbine blade under both rotating and static conditions. Zhang et al. [1] studied the flow and heat transfer characteristics in the trapezoidal heat exchange surface channel in laminar runner. The results showed that the temperature difference distribution in the trapezoidal runner was more uniform, and the pressure drop loss was lower than that of the equal cross section runner with the Reynolds numbers ranged from 800 to 3000. In contrast, with the same heat transfer area, the heat transfer factor and friction factor ratio of the trapezoidal runner increased by 8 and 22.6% respectively. At the same time, when the inclination angle of the trapezoidal runner is less than 40°, the overall heat transfer performance in the trapezoidal runner was better than the equal cross section runner.

In the application of trapezoidal flow path, Cur and Anselmino [8] first proposed an Accelerated Flow Evaporator which can replace common evaporator and reduce the volume, weight and manufacturing cost of evaporator. In the evaporator, the air was gradually accelerated by the contracted structure, thereby increased the local convection heat transfer performance of the downstream region. As a consequence, the increment of convection heat transfer can compensate the influence caused by the reduction of heat transfer area. However, the accelerated flow evaporator also increased reversible and irreversible pressure drop losses. The reversible pressure drop loss caused by the decreased heat transfer area in the downstream, but the irreversible pressure drop losses caused by the increased average velocity in downstream zone. Waltrich et al. [9,10,11] had also experimentally studied for this accelerated flow evaporator.

In addition to the trapezoidal runner structure, Rush et al. [12] studied the local heat transfer and flow characteristics in the sinusoidal corrugated runner under laminar and transitional flow conditions. They found that relatively significant flow instability exists at the outlet with small Reynolds numbers, and the unstable state gradually goes to the inlet as the Reynolds number increases. Akbarzadeh et al. [13,14,15,16] analyzed the sensitivity of heat transfer and pump work of nanofluids in corrugated runners. The mass, momentum and energy conservation equations of two-dimensional steady state were solved by finite volume method. It was found that the mixability of the fluid can be improved in the corrugated runner, and the temperature gradient at the wall surface increased with the increase of the amplitude. Ramgodia et al. [17] simulated three-dimensional numerical simulation of three corrugated runners with different phase angles, and analyzed the heat transfer and flow characteristics of fluid at different phase angles during the complete development phase. Bahaidarah et al. [18] studied the volumetric entropy production rate in the corrugated runner. The results showed that the total entropy increases with the increase of Re. However, when the Reynolds numbers range from 25 to 400, the total entropy increase in the direction of the flow path is gradually reduced. Akbarzadeh et al. [19] also used entropy-increasing method to study three different types of special-shaped runners including sinusoidal, trapezoidal and triangular, and numerical simulation was carried out. The results for the three different structures showed that the sinusoidal structure has the best heat transfer performance. Sarkar et al. [20] investigated two-dimensional flow characteristics in corrugated runners at different Reynolds numbers (100 < Re < 2123). The heat transfer and flow properties of six structures under different amplitudes (0.05, 0.075 and 0.1 mm) and wavelength (0.5 and 1 mm) were investigated in the state of laminar flow and transition flow.

As can be seen from the analysis of the above mentioned references, the flow and heat transfer characteristics of the cosine type heat exchange surface runner structure are studied for the first time in this paper. The temperature field, flow field and synergy field of different cosine type structural parameters are experimentally and numerically studied. The cosine style runner can obviously improve the efficiency of the heat transfer and reduce the pressure drop.

2 Test study

2.1 Test sample

The effect of different cosine style runner parameters on heat transfer and pressure drop characteristics was studied by using orthogonal design method. The material used in the experiment was 6063 aluminum alloy, and the meridian cutting process was processed. In the process of experiment and simulation, the cosine curve with half wavelength ([0, π]) as the heat exchange surface of the runner. The divergent curved surface is also more conducive to discharge condensate [21].

The diagram of the cosine style runner structure is shown in Fig. 1. The equation of the cosine curve is y = A·cos(Bx), where B = π/ Lf. Detailed dimension parameters of the cosine runner structure is shown in Table 1 including nine different experimental samples. As a reference runner of equal cross section, the picture of real product and schematic diagram of structural parameters are shown in Fig. 2a and b respectively. Tables 1 and 2 are specific structural parameters of the cosine style runner and equal cross section runner respectively.

Fig. 1
figure 1

Cosine style runner. a Isometric drawing; b The experimental physical drawing (View point A); c Schematic diagram of structural parameters (View point A)

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Table 1 Structural parameters list of cosine style runner
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Fig. 2
figure 2

Equal cross section runner. a Picture of real product; b Schematic diagram of structural parameters

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Table 2 Structural parameters list of equal cross section runner
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2.2 Test bed

The schematic diagram of the test bed is shown in Fig. 3, which including an air circulation system, a water circulation system, a control system, a data acquisition system and other auxiliary systems. The measuring range and precision of air were 0.05–30 m/s and ± 5% respectively. A pulsed flow transducer was used to measure the water flow rate, which had an accuracy of 0.1 g. The JM9 compensation micro-manometer with higher precision is selected, which had the accuracy of 0.1 Pa. T-type thermocouple was used to measure temperature ranges from −200 to 300 °C, and its accuracy was ±0.1 °C.

Fig. 3
figure 3

Schematic diagram of wind tunnel test bed

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The uncertainty in the experiment was analyzed according to the method proposed by Moffat [22]. The maximum uncertainty of convective heat transfer coefficient calculated by this method was 2.9%, and the maximum uncertainty of pressure drop ΔP was 5.1%. Therefore, the error of the system can meet the requirements of the experiment, and the experimental data had certain reliability. At the same time, each data point was obtained through three repeated measurements, and each measurement was maintained at a thermal equilibrium state for a minimum of 1 h.

2.3 Data processing method

The heat transfer coefficient (h) of air side can be expressed as:

$$ h=\frac{Q}{A\varDelta {T}_{\mathrm{ln}}} $$
(1)

where Q is the average value of the heat flux of air side and water side, ΔTln is the logarithmic average temperature difference which can be described by [23]:

$$ \varDelta {T}_{\mathrm{ln}}=\frac{\left({T}_C-{T}_B\right)-\left({T}_D-{T}_A\right)}{\ln \left(\frac{T_C-{T}_B}{T_D-{T}_A}\right)} $$
(2)

The flow state of the fluid can be expressed in two Reynolds numbers according to a given fin and flow path structure. Two Reynolds numbers are used in the two types of runners, which are Rem and Remax respectively.

Reynolds number in the rectangular flow path is Reynolds number Rem:

$$ R{e}_m=\frac{u_m{d}_h}{\mu /\rho } $$
(3)

The flow condition of cosine runner can be evaluated by the maximum Reynolds number (Remax):

$$ R{e}_{\mathrm{max}}=\frac{u_{in}{d}_{h, in}}{\mu /\rho } $$
(4)

Where dh and dh,in are hydraulic diameters in rectangular and cosine ducts respectively.

The Colburn factor j and the Fanning friction factor f are respectively solved as follows [24]:

$$ j=\frac{Nu}{P{r}^{1/3}.\mathit{\operatorname{Re}}} $$
(5)
$$ f=\frac{D_h}{4{L}_f}\cdot \frac{\varDelta p}{\left(1/2\right)\rho {u}^2} $$
(6)

Where Re and Dh in the rectangular runner are calculated by using mean velocity (um) and hydraulic diameter (dh) respectively; Re and Dh in the cosine runner are calculated by using maximum velocity (uin) and hydraulic diameter at the inlet (dh,in). The difference between dh and dh,in lie in dh is hydraulic diameter of rectandular runner with the same form and dh,in is hydraulic diameter at the inlet of the cosine runner.

3 Numerical simulation analysis

3.1 Basic hypothesis and mathematical model

3.1.1 Basic assumption

Before adopting numerical method to analyze the cosine style runner, the following assumptions were made:

  1. (1)

    The Mach number of air less than 0.3, so air can be considered as incompressible fluid;

  2. (2)

    The physical model was steady state heat transfer and flow problem.

  3. (3)

    Ignoring the influence of gravity on numerical results;

  4. (4)

    Ignoring the effect of radiation heat transfer;

  5. (5)

    The heat exchange surface was an ideal smooth surface.

3.1.2 Calculation area and boundary conditions

In order to maintain uniform air velocity distribution at the inlet, the computation domain was extended upstream 1.5 times of the flow path length. Also computation domain was extended downstream 5 times in order to prevent a backflow phenomenon at the outlet. The boundary conditions employed in the simulation are “velocity-inlet”, “pressure-outlet”, “symmetry”, which is shown in Fig. 4. The model of the heat exchange surface of the cosine style runner was symmetric plane where no heat and mass exchange on it, so the plane can be set as “symmetry” condition. All the wall surfaces except the inlet, the outlet and the symmetrical plane were set as “Wall” conditions, and the heat flow rate on wall surface was set as “0”. The initial temperature at the inlet was 293.15 K, and on the fins and cosine surface were 353 K. The static pressure at the outlet is 0 Pa. The air flow rate of the cosine style runner were 0.28, 0.39, 0.5, 0.61, 0.72, 0.83 kg/h respectively.

Fig. 4
figure 4

Three-dimensional calculation model and boundary condition of fin

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3.2 Grid division and discretization

The mesh needed to be refined in the vicinity of the wall and fin surfaces in calculation area, and the area was divided by adopting a structured grid partitioning method by using Software ANSYS ICEM. The overall grid structure and the refinement mesh structure near the wall surface are shown in Fig. 5a and b respectively. ANASYS FLUENT software was used as a computing platform to discretize and solve the mass, momentum, energy conservation equations and other control equations. In this paper, the SIMPLE method was adopted for the velocity and pressure coupling equations, and the momentum, energy and turbulence equations were solved by the second-order windward format. Residual and energy residuals for continuity, velocity components, turbulent kinetic energy, turbulent dissipation, and energy residuals were controlled within 10−6.

Fig. 5
figure 5

Cosine runner grid division. a Overall grid; b Local grid

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3.3 Control equation

In this paper, the RNG k-ε model modified in turbulent state was used to solve the calculation region. The three control equations were as follows:

  • Continuity equation:

$$ \frac{\partial }{\partial {x}_j}\left(\rho {u}_j\right)=0 $$
(7)

  • Momentum equation:

$$ \frac{\partial }{\partial {x}_i}\left(\rho {u}_i{u}_k\right)=\frac{\partial }{\partial {x}_i}\left(\mu \frac{\partial {u}_k}{\partial {x}_i}\right)-\frac{\partial p}{\partial {x}_k} $$
(8)

  • The energy equation is:

$$ \frac{\partial }{\partial {x}_j}\left(\rho {u}_jT\right)=\frac{\partial }{\partial {x}_j}\left(\frac{\lambda }{c_p}\cdot \frac{\partial T}{\partial {x}_j}\right) $$
(9)

3.4 Grid independence and mathematical model verification

3.4.1 Grid independence

In the case of a heat exchanger unit, tf = 0.4 mm, Lf = 60 mm, Fh = 10 mm, Sf = 4 mm, 2A = 5 mm, the independence verification of the grid was carried out with Re = 3140. The number of seven different grids was studied, which are 246,000, 400,000, 546,000, 699,000, 852,000, 997,000, 110,100 grids respectively. As shown in Fig. 6, when the number of grids was divided into 997,000 and 1.101 million, the difference between the heat transfer coefficient and the pressure drop of the two calculation models was less than 0.6%. Synthesizes convergence time, convergence precision and economy, the grid number of 997,000 can meet the simulation needs.

Fig. 6
figure 6

Grid independence verification

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3.4.2 Mathematical model verification

Taking tf = 0.4 mm, Lf = 60 mm, Fh = 10 mm, Sf = 4 mm and 2A = 5 mm of cosine style runner as an example, the numerical solution was carried out under the condition of 500 ≤ Re ≤ 5000. In order to verify the simulation method used in this paper, the simulation data was compared with the experimental data of the cosine duct, as shown in Fig. 7a and b. The numerical calculation of j and f factors and the curve of experimental results had good agreement and consistency. Therefore, it was proved that the mathematical model and calculation method used in this paper had certain accuracy and reliability.

Fig. 7
figure 7

Comparison of simulation values and experimental data. aj factor; bf factor

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4 Results and discussion

4.1 Experimental study on effect of different structural parameters on heat transfer and fluidity

4.1.1 Heat transfer and pressure drop performance

Figure 8 is the relationship between convective heat transfer coefficient and pressure drop of a cosine style runner at different inlet heights Fh (9.5, 10.5 and 11.5 mm) and different amplitudes 2A (5, 7 and 9 mm).

Fig. 8
figure 8

Analysis of performance in different Cosine runners. a Heat transfer performance comparison; b Pressure drop performance comparison

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We can see that with the increase of air velocity, both h and ΔP increase. In addition, the trend of heat transfer coefficient increases gradually, while the pressure drop is increased in the form of parabolic growth. # 1 and # 9 have maximum and minimum convective heat transfer coefficients, respectively; While # 2 and # 8 have the maximum and minimum pressure drop respectively. By making comparison among # 1, # 2, and # 3, the convective heat transfer performance is increased as the inlet height decreases, that is, the smaller the inlet height the better the convective heat transfer performance. By making comparison among # 2, # 5, and # 8, the convective heat transfer performance is gradually reduced as the amplitude increases, that is, larger amplitude weakens the convective heat transfer performance.

Through multiple regression analysis and F significance test of experimental data, the experimental correlation formulas of j factor and f factor in cosine style runner are obtained, as shown in Eqs. (10) and (11) respectively.

The relative accurate prediction of j and f factors can be accomplished by two test correlations, and the predictive error of 90% of the experimental data is within ±10%. Comparing with the experimental values, the arithmetic mean error between j factor and f factor are 0.35 and 4.1%, and the mean deviation are 2.5 and 4.9%, respectively.

$$ j=0.1136R{e}^{-0.199}{\left({S}_f/{F}_h\right)}^{0.229}{\left({S}_f/2A\right)}^{-0.143}{\left({L}_f/{t}_f\right)}^{-0.426} $$
(10)
$$ f=1.48R{e}^{-0.25}{\left({S}_f/{F}_h\right)}^{0.527}{\left({S}_f/2A\right)}^{-0.215}{\left({L}_f/{t}_f\right)}^{-0.243} $$
(11)

4.1.2 Contrast analysis of integrated heat transfer performance of cosine and equal cross section flow path

The Core Volume Goodness Factor is used to evaluate the overall heat transfer performance of the two different ducts including cosine style runner and equal cross section runner. The significance of using the Core Volume Goodness Factor is that it can compare the overall heat transfer performance of different ducts with different structures. The Core Volume Goodness Factor was proposed by Shah and Sekulić [25] and its expression is

$$ {\eta}_o{h}_{std}\alpha =\frac{c_p\mu }{P{r}^{2/3}}{\eta}_o\frac{4\sigma }{D_h^2} jRe $$
(12)
$$ {E}_{std}\alpha =\frac{\mu^3}{2{g}_c{\rho}^2}\frac{4\sigma }{D_h^4} fR{e}^3 $$
(13)

In the above equation, ηohstdα is energy consumption per unit volume while the temperature changes by 1 K in a heat exchanger; Estdα is the frictional power consumed per unit volume. The subscript “std” is arbitrarily selected standard temperature and pressure conditions. From the perspective of the volume of the heat exchanger, in the case of the same Estdα, the larger the ηohstdα, the smaller the volume of the heat exchanger used at the same heat transfer capacity. That is, with the same “Estdα”, a larger value of ηohstdα means that only a small heat exchanger volume is required at the same heat capacity. Rem is the mean Reynolds number calculated in the rectangular runner; Remax is the maximum Reynolds number calculated in the cosine runner at the inlet (dh,in).

The variation curve of “ηohstdα” value with “Estdα” is given in Fig. 9. As can be seen from the figure, the “ηohstdα” values for both structures are increased as the “Estdα” increases. From the results of the comparative analysis, under the same “Estdα”, the cosine style runner has the best comprehensive enhanced heat transfer effect. In the whole Estdα interval, the ηohstdα value in the cosine style runner is about 7–25% greater than the ηohstdα value in the equal cross section runner. The value of ηohstdα for the cosine style runner is about 25% greater than that of equal cross section runner when Estdα is the largest. Therefore, the overall heat transfer performance of the cosine style runner mentioned in this paper is better than that of the equal cross section runner.

Fig. 9
figure 9

Comparison of core volume goodness factors under different special-shaped runner structures

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4.2 Effect of amplitude on heat transfer and resistance performance

Figure 10 shows the j and f factors of the cosine style runners with different amplitudes. The runners have the same Lf (53 mm), Sf (2.5 mm) and Fh (8.0 mm), but have different amplitudes. As can be seen from the figure, the j factor and the f factor decrease with the increase of Re. With the same Re, the increment of j and f factors is substantially consistent. That is to say, the increase of amplitude does not result in dramatic changes of the j factor and f factor. Under the same Re, the j factor and f factor increase as the amplitude decreases. So the small amplitude plays an important role in heat transfer, but the small amplitude also brings a relatively large pressure drop. And the amplitude 2A = 5 mm has the best j factor and the maximum f factor. When 1210 ≤ Re ≤ 5080, the j factors of the runners with amplitude 2A = 7 mm and 2A = 9 mm were reduced by 6.96 and 12.6% respectively. Meanwhile, the f factors decreased by 29.8 and 50% respectively.

Fig. 10
figure 10

Comparison of heat transfer and resistance performance of different amplitude. aj factor; bf factor

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In this paper, the influence of j factor and f factor must be comprehensively considered. JF factor is used as an evaluation index for the comprehensive enhanced heat transfer performance.

The expression of the JF factor is [26]:

$$ JF=\frac{j_i/{j}_0}{{\left({f}_i/{f}_0\right)}^{1/3}} $$
(14)

Where the subscript “i” represents different runners and the “0” represents a reference runner with equal cross sections. The higher the JF factor is, the better thermal performance is.

In Fig. 11, the effect of the JF factor on the comprehensive enhanced heat transfer in three different cosine style runners is compared. As can be seen from the figure, with the same Re, the flow path of amplitude 2A = 5 mm has the largest JF factor value. That is, the smaller the amplitude, the better the heat transfer performance of the cosine runner.

Fig. 11
figure 11

Comparison of enhanced heat transfer effects of three different amplitudes

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4.3 Temperature field and flow field with different amplitudes

Taking the case of Re = 3140 as an example, the cosine style runners with amplitude of 5, 7 and 9 mm were numerically analyzed. Figure 12a and b are distribution images of the temperature field and velocity field under different amplitudes respectively, and the cloud picture section is the middle plane of two adjacent fins.

Fig. 12
figure 12

Effect of different amplitude on temperature field and velocity field. a Temperature field; b Velocity field

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The temperature field images corresponding to different amplitudes are shown in Fig. 12a. For three different amplitudes, the temperature in the mainstream region near the intermediate fluid domain is relatively low and the temperature value near the wall surface is relatively high. Therefore, the process of convection heat transfer mainly depends on the convection and heat conduction of the wall surface or fin surface. The comparison of the three sub-graphs in the figure shows that the thermal boundary layer thickness gradually increases as the amplitude increases from 5 to 9 mm. This explains the larger the amplitude, the thicker the thickness of the thermal boundary layer, and the weaker the heat transfer performance.

The velocity field images corresponding to different amplitudes are shown in Fig. 12b. It can be seen from the figure that the flow rate of the main stream is relatively high and the flow rate near the wall is relatively low. This is because the air is affected by the effect of the viscous force on the wall when the air flows through the wall. In addition, the thickness of the velocity boundary layer is gradually thickened as the amplitude increases. With the increase of the amplitude, there has no backflow or boundary layer separation in the flow domain. It is because the maximum slope angle in the domain does not reach the degree of boundary layer separation. As the amplitude decreases, the high speed region of the intermediate fluid region is pushed forward. This indicates that the smaller amplitude has better heat transfer performance.

4.4 Temperature field and velocity field with different Reynolds numbers

The cosine style runner are numerically analyzed when Reynolds numbers are 1210, 2170, 3140, 4110 and 5080. Figure 13a and b are distribution images of the temperature field and velocity field under different Re conditions respectively.

Fig. 13
figure 13

Effect of different Re on temperature field and velocity field. a Temperature field; b Velocity field

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The selected plane for exhibition is the middle position of two adjacent fins. The temperature field contours corresponding to different Reynolds number are shown in Fig. 13a. As can be seen from the figure, for the five different Reynolds numbers, the temperature in the main flow zone near the intermediate fluid domain is relatively low, and the temperature near the wall surface is higher. Therefore, in the cosine style runner, heat transfer near the wall surface plays a leading role. Comparing the five subgraphs in the figure, it can be found that the thickness of the thermal boundary layer gradually decreases during the increase of Reynolds number. This means that the larger the Reynolds number, the smaller the thickness of the thermal boundary layer and the stronger the heat transfer performance. Therefore, the greater the Re is, the stronger the convective heat exchange enhancement.

As shown in Fig. 13b, a velocity field cloud image corresponding to different Re is displayed. Wherein red represents high speed region, blue represents low speed region. It can be seen from the figure that the flow rate in the main stream is relatively high but the flow rate near the wall is relatively lower with different Re. This is because the air is affected by the effect of the viscous force on the wall when the air flows through the wall. In addition, there is also a velocity boundary layer at the proximal wall. By contrast of the five sub-graphs in the graph, it is found that the thickness of the velocity boundary layer is gradually reduced as Re increases. Another significant phenomenon that can be found from velocity field clouds is that with the increase of Re, the high speed region of the intermediate fluid region is pushed forward. The higher velocity in the runner also increases convective heat transfer performance.

4.5 Strengthening heat transfer mechanism of cosine style runner based on field synergy theory

The field synergy theory is mainly to determine the synergy between the temperature field and the velocity field by calculating the angle between the velocity vector and the temperature gradient. The field synergy can be evaluated by local field synergy angle and average field synergy angle. The smaller the average field synergy angle, the more favorable the heat transfer is facilitated. The relationship between velocity vector and temperature gradient, local field synergy angle θ’ and the average field synergy angle (FSA) θ is defined as follows [27]:

$$ \overrightarrow{U}\cdot \nabla T=\mid \overrightarrow{U}\mid \cdot \mid gradT\mid \mathit{\cos}{\theta}^{\prime } $$
(15)
$$ {\theta}^{\prime }= co{s}^{-1}\frac{u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}}{\mid \overrightarrow{U}\mid \cdot \mid gradT\mid } $$
(16)
$$ \theta =\frac{\int {\theta}^{\prime } dV}{\int dV} $$
(17)

In above equations, u, v, w are represented velocities in x, y, z coordinate axes, \( \overrightarrow{U} \) is the velocity vector, ∇T is the temperature gradient, and V is the minute control body in the discretized fluid region.

Figure 14 shows a local field synergy profile under the same inlet air flow rate u = 4 m/s (Re = 3140). It can be seen from the figure that there is a certain similarity to the distribution of the local field synergy angles in the cosine style runners of three different amplitudes. The local field synergy angle distribution in the cosine runner and the equal cross section runner is different. The inner wall of the equal cross section runner has a small local field synergy angle because the wall surface or the fin surface has large temperature gradient and velocity gradient [28]. A large local field synergy in the cosine style flow path occurs at the wall surface, while the smaller local field synergy is mainly distributed in the inlet and the central main region. When the amplitude is 5 mm, the field synergy angle distribution at the center region is good. However, the field synergy angle inside the central region is mixed with larger field synergy angle when the amplitude is 7 and 9 mm. Therefore, the magnitude of amplitude has a certain effect on the degree of field synergy.

Fig. 14
figure 14

Local field synergy angle distribution

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Figure 15 shows a plot of the average field synergy with different Re numbers and different amplitudes. As can be seen from the figure, under the same Re condition, with the increase of the amplitude, the average field synergy is increased, which is not conducive to enhanced heat transfer. Therefore, the smaller the amplitude is, the smaller the average field synergy angle, the better the field synergy. When Re > 3000, the average field synergy angle of three different amplitudes is increased with a smaller amplitude. But when Re < 3000, the average field synergy angle of three different amplitudes is increased with a larger amplitude. Therefore, increasing the Reynolds number to a certain extent does not have a great influence on the average field synergy.

Fig. 15
figure 15

Comparison of average field synergy at different amplitudes

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5 Conclusions

This paper presents the steady state heat transfer and friction characteristics and performance analysis of convective heat transfer through a cosine style runner with different amplitudes 2A (5, 7 and 9 mm). We concluded that:

  1. (1)

    At the same amplitude, the convective heat transfer performance gradually increases as the inlet height (Fh) decreases. At the same inlet height (Fh), the convective heat transfer performance is gradually reduced as the amplitude increases, that is, the larger amplitude weakens the convective heat transfer performance.

  2. (2)

    Through the regression analysis method and the F significance test method, the experimental data of heat transfer and resistance performance of various structures is fitted, and the obtained j and f factors correlation formula can predict 90% data, and the error range is within ±10%.

  3. (3)

    In the range of 5 mm ≤ 2A ≤ 9 mm, with the decrease of amplitude, the overall heat transfer performance is better and the cosine style runner at 2A = 5 mm has the largest JF factor. At the same amplitude, the thickness of the thermal boundary layer and velocity boundary layer is decreasing as the Reynolds number increases. Under the same Re condition, with the decrease of amplitude, the average field gradually decreases, and therefore, the smaller amplitude has a better field synergy.

  4. (4)

    In the cosine style runner of 5 mm ≤ 2A ≤ 9 mm, with the decrease of amplitude, the smaller the average field synergy angle (FSA), the better the field synergy. When compared with the runner structure of 2A = 5 mm, the j factors of 7 and 9 mm decreased by 6.96 and 12.6% respectively, and the f factors decreased by 29.8 and 50% respectively. That is, the small amplitude increases the pressure drop while facilitating heat transfer. And with the increase of amplitude, the thickness of the boundary layer increases gradually.

  5. (5)

    By using the Core Volume Goodness Factor to compare the overall enhanced heat transfer performance of the cosine and other equal cross section runners, it is concluded that the value of ηohstdα in the cosine style runner is 25% larger than that of the equal cross section runner, and the heat transfer performance of the cosine type runner structure is greatly improved.