Characteristics of Flow Symmetry and Heat Transfer of Winglet Pair in Common Flow Down Configuration

Abstract

The effect of transverse pitch between a pair of delta-winglet vortex generators arranged in a common flow down configuration on the symmetrical flow structure and heat-transfer performance was numerically investigated. The results showed that symmetrical longitudinal vortices form a common flow down region between the vortices. The fluid is induced to flow from the top towards the bottom of the channel in the common flow region, which is advantageous to the heat transfer of the bottom fin. The vortex interaction increases and the vortex intensity decreases along with the decrease in transverse pitch of vortex generators. Vortex interaction has a slight influence on pressure penalty. The Nusselt number decreases with increasing vortex interaction. The vortices gradually attenuate and depart from each other during the process of flowing downward. A reasonable transverse pitch of delta-winglet vortex generators in a common-flow-down configuration is recommended for high thermal performance.

1. Introduction

Improvement of the heat-transfer efficiency can decrease the size, weight, and cost of heat exchangers. The thermal performance of fins can be improved by a longitudinal vortex combined with low pressure loss. Vortex generators (VGs) are widely applied as the longitudinal vortex producer in heat exchangers and computational fluid dynamics is the most efficient method for the design and optimization of complex heat exchangers [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Meng et al. [5] numerically analyzed the influence of multiple longitudinal vortices on thermal performance. The results showed that the heat transfer is significantly enhanced with a similar increase in flow resistance. Wu and Tao [6] stated that improvement in thermal performance can be obtained through optimization of VG parameters. Song et al. [7,8] reported that the VGs arranged on both fin surfaces of the flow channel can simultaneously increase heat transfer and decrease pressure loss compared with the plane VG under the same VG area. Lei et al. [9] studied the in-line and staggered tube-fin heat exchangers with VGs and found that the heat transfer increment is greater than the increase in resistance. Samadifar and Toghraie [10] studied several new-type VGs and found that the best VG attack angle was 45 degrees. Luo et al. [11] proposed a new combination of VGs and a wavy fin. A considerable increase in thermal performance was reported compared with the normal wavy fin. Li et al. [12] experimentally studied a fin with radiantly arranged VGs and showed that the fin with radiantly arranged VGs had a better comprehensive performance. Song et al. [14] summarized different geometric shapes of VGs in the open literature and found that the concave-curved VG had better thermal performance than the normal-plane VG. Yang et al. [15] reported that wedge-shaped VGs can lead to an increase in volume-goodness factor of up to 30% compared with the dimple channel without VGs. Silva et al. [13] studied the application of VGs in solar collectors and found that the heat transfer was effectively enhanced by the VGs.

VGs are usually arranged in pairs in real applications due to the common flow which is beneficial to heat-transfer improvement. Gupta et al. [16] reported a heat transfer increase of 34% by the VGs with holes arranged in common-flow-up configuration compared with the corresponding case without VGs. Skullong et al. [17] studied the rectangular and trapezoidal VG pairs punched with holes and great heat transfer enhancement was obtained by the trapezoidal VGs due to the downward common vortex flow to the solar absorber plate. Sinha et al. [18] studied the effect of combined VG arrays with different configurations on the performance of a plate-fin heat exchanger. Tian et al. [19] analyzed the influence of different VG arrangements on heat transfer. The results showed that the heat transfer and the flow resistance of the channel with VGs arranged in common flow down are greater than that in the common flow up configuration. Yang et al. [20] studied the influence of VGs in common flow down on the thermal hydraulic performance. Their research showed that the heat transfer is enhanced due to the longitudinal vortex. Sarangi and Mishra [21] studied the location of VGs in common flow up configuration and reported that the VGs near the central tube were effective at heat-transfer enhancement. Lu and Zhai [22] numerically studied the performance of tear-drop VGs in common flow up. They found that tear-drop VGs can enhance the heat transfer with a negligible increase in the pressure drop.

Although there are many studies about VG pairs in the literature, there is no paper considering the vortex interaction between the pair of longitudinal vortices. The pair of longitudinal vortices will inevitably interact with each other and affect the vortex intensity and heat-transfer performance when the VGs move close to each other. Song et al. [23] quantitatively analyzed the intensity of vortices by a secondary flow intensity parameter Se and found that the vortex intensity increased when an obvious common flow region formed between counter-rotating longitudinal vortices. The effect of vortex interaction on the heat transfer of a flat-tube-fin heat exchanger was reported in [24,25]. The results showed that an optimal arrangement of VGs exists for the best heat-transfer enhancement by considering the vortex interaction.

As the vortex interaction significantly affects the heat transfer, it is meaningful to study the vortex interaction to find the optimal transverse pitch of a pair of VGs for the highest thermal performance. In this paper, the vortex interaction between a pair of VGs in a common flow down configuration was numerically studied. The effect of the vortex interaction on the flow symmetry and thermal performance was discussed in detail.

2. Physical Model, Methods, and Formulations

The schematic view of the studied physical model is shown in Figure 1. Two winglet VGs were arranged in a common flow down configuration with a transverse pitch of c. The vortex generator had a height of H = 1.4 mm. The VG baseline equaled 2H. The VG attack angle was θ = 35°. The studied transverse pitches between VGs, as shown in Figure 2 and

Table 1. Transverse pitches of VGs.

Transverse pitch	c1	c2	c3	c4	c5	c6
c = C/(2Hsinθ)	2.5	2	1.5	1	0.5	0

, were named c₁ to c₆ with the ratio of c/(2Hsinθ) = 2.5, 2.0, 1.5, 1.0, 0.5, and 0.0, respectively. The channel width and length were B = 10H and L = 31.5H, respectively. The net height of the channel was F_p = 2 mm. The VGs were symmetrical around the center with a distance of D = 10 mm from the inlet. The wall temperature was kept at 80 degrees centigrade and the fluid inlet temperature was 40 degrees centigrade. The properties of the fluid of air were assumed as constant under the mean temperature of 60 degrees centigrade.

Steady and incompressible laminar flow was considered without considering the volume force and viscosity dissipation. The governing equations were as follows [14].

$$\frac{\partial }{\partial x_i}(\rho u_i)=0$$

(1)

$$\frac{\partial }{\partial x_i}(\rho u_i{u}_k)=\frac{\partial }{\partial x_i}(\mu \frac{\partial u_k}{\partial x_i})-\frac{\partial p}{\partial x_k}(k=1,2,3)$$

(2)

$$\frac{\partial }{\partial x_i}(\rho c_p{u}_{i}T)=\frac{\partial }{\partial x_i}(\lambda \frac{\partial T}{\partial x_i})$$

(3)

Boundary conditions:

At the inlet:

$$u_{\mathrm{in}}\left(x,y,z\right)=u_0,\begin{array}{c}\end{array}{v}_{\mathrm{in}}\left(x,y,z\right)=0,\begin{array}{c}\end{array}{w}_{\mathrm{in}}\left(x,y,z\right)=0,\begin{array}{c}\end{array}{T}_{\mathrm{in}}\left(x,y,z\right)=T_0$$

(4)

At the outlet:

$$\frac{\partial }{\partial x}{u}_{\mathrm{out}}(x,y,z)=0,\begin{array}{c}\end{array}\frac{\partial }{\partial x}{v}_{\mathrm{out}}(x,y,z)=0,\begin{array}{c}\end{array}\frac{\partial }{\partial x}{w}_{\mathrm{out}}(x,y,z)=0,\begin{array}{c}\end{array}\frac{\partial }{\partial x}{T}_{\mathrm{out}}(x,y,z)=0$$

(5)

At the symmetric side surfaces:

$$\frac{\partial }{\partial y}u(x,y,z)=0,\begin{array}{c}\end{array}v(x,y,z)=0,\begin{array}{c}\end{array}\frac{\partial }{\partial y}w(x,y,z)=0,\begin{array}{c}\end{array}\frac{\partial }{\partial y}T(x,y,z)=0$$

(6)

At the solid surfaces:

$$u(x,y,z)=0,\begin{array}{c}\end{array}v(x,y,z)=0,\begin{array}{c}\end{array}w(x,y,z)=0,\begin{array}{c}\end{array}T=T_{\mathrm{w}}$$

(7)

The hydraulic diameter was selected as the fin spacing. The parameters were defined as follows:

$$Re=\frac{\rho .u_{\mathrm{m}}.d_{\mathrm{h}}}{\mu }$$

(8)

$$f=\frac{\Delta p{d}_{\mathrm{h}}}{(L\rho u_{\mathrm{m}}^2/2)}$$

(9)

$$N{u}_{\mathrm{local}}=-\frac{d_{\mathrm{h}}}{T_{\mathrm{w}}-T_{\mathrm{s}}(x)}\frac{\partial T}{\partial n}$$

(10)

The average temperature on the cross section was:

$$T_s(x)=\frac{{\displaystyle {\int }_{A}u(x,y,z)T(x,y,z)dA}}{{\displaystyle {\int }_{A}u(x,y,z)dA}}$$

(11)

Span average Nusselt number (Nu_s) on the fin surfaces:

$$N{u}_{\mathrm{s}}(x)={\displaystyle {\int }_{\delta S}N{u}_{\mathrm{local}}dS}/{\displaystyle {\int }_{\delta S}dS}$$

(12)

The secondary flow intensity [7,23,24,25]:

$$Se=\frac{\rho d_{\mathrm{h}}{U}_s}{\mu }$$

(13)

The secondary flow characteristic speed [7,23]:

$$U_s=d_{\mathrm{h}}\left|{\mathit{\omega }}^{{}_{\mathrm{n}}}\right|$$

(14)

where ωⁿ was the vortex flux along the main flow direction.

The thermal performance factor [14,23] was defined as:

$$JF=\frac{Nu/f^{1/3}}{N{u}_0/f_0^{1/3}}$$

(15)

The above governing equations were discretized by the second-order central difference scheme in the control volume and solved by the code written by FORTRAN. The velocity and pressure were coupled using the SIMPLE algorithm [26]. The relaxation factors for velocity, pressure, and temperature were 0.6, 0.5, and 0.4, respectively. The iteration of the governing equations was first run for a thousand steps and then convergence was judged with a residual of 10⁻⁶ for the equations when the relative errors of Nu, f, and T between every 200 iterations were less than 0.01%.

The grid independence was tested out between three different structured grid numbers, 138 × 114 × 24, 194 × 142 × 32, and 234 × 166 × 38 at Re = 1000 and c₃ = 2, as shown in

Table 2. Grid independence test.

No.	Grid (x × y × z)	Nu	Relative Error	f	Relative Error
1	138 × 114 × 24	6.649	0.47%	5.854 × 10−2	0.67%
2	194 × 142 × 32	6.618	-	5.815 × 10−2	-
3	234 × 166 × 38	6.629	0.17%	5.832 × 10−2	0.29%

. The differences in Nu and f between the neighboring grids were smaller than 1%. Thus, the numerical results are not dependent on the grid number. The medium-size grid 194 × 142 × 32 was adopted to obtain the numerical results in the present paper. The mesh of the model is shown in Figure 3.

The numerical method and code were validated using a comparison with the results of Tian et al. [27]. The model for comparison was formed by two parallel fins mounted with a pair of winglet vortex generators in a common flow down configuration. The comparisons of Nu and f are presented in Figure 4. The numerical results agreed with the results reported in the literature. The largest difference in Nu was less than 2.9% and the largest difference in f was less than 1.8% in the studied range of Re. Thus, the numerical method and code were reliable.

3. Results and Discussions

3.1. Longitudinal Vortices on the Cross Sections

Eight cross sections marked as s1 to s8, as shown in Figure 1e, were selected to show the development of longitudinal vortices in the channel. The distributions of longitudinal vortices are shown in Figure 5. The transverse pitch of the VGs is c₅ and Re is 1800 in Figure 5. The vortices were located near the bottom fin because the VGs were arranged on the bottom fin. The longitudinal vortices rotated in counter-rotating directions and there was a symmetrical vortex-flow structure on the cross sections. The fluid was induced to flow from the top fin towards the bottom fin of the channel in the common-flow region. Thus, the arrangement of VGs was always called a common flow down configuration. The vortex intensity was the strongest in the region around the VG and the vortex intensity gradually decreased along the flow direction. There was an obvious vortex interaction on cross section s1 owing to the small distance between the strong vortices. The longitudinal vortices attenuated and the distance between the symmetrical vortices gradually increased when the vortices flowed downstream. Thus, the induced flow in the common region became weak and the vortex interaction also decreased.

Figure 6 shows the vortices on s4 for different transverse pitches when Re = 1800. The distance between the symmetrical vortices decreased with a decreasing transverse pitch of VGs. There was a marginal difference between the symmetrical vortices and the vortices were nearly the same when the transverse pitch changed between c₁ and c₃. This is because the distance between the symmetrical vortices was large and there was no vortex interaction between the vortices. The intensity of the vortices decreased when the transverse pitch decreased from c₄ to c₆ due to the increase in vortex interaction. The vortex intensity of c₆ is apparently much weaker than that of c₅ due to the strongest vortex interaction between the symmetrical vortices.

3.2. Contour Plot of Vortex Intensity

The distributions of vortex intensity Se corresponding to the same cross sections in Figure 5 are presented in Figure 7 when the transverse pitch between the VGs was c₅. The distribution of Se reflected the distribution of longitudinal vortices perfectly. A contour plot of Se corresponding to the symmetrical vortex-flow structure was also symmetrical. A large value of Se was attained in the vortex zone and was located nearer to the bottom fin. The small zones of Se corresponding to the corner vortex were located on the outer sides of the large zones of the main vortices. The intensity of the contour plot of Se gradually decreased from s1 to s8 due to the attenuation of longitudinal vortices and the zones departing from each other. The distribution of Se for different transverse pitches corresponding to Figure 6 is shown in Figure 8. The distance between the main zones of Se decreased with decreasing transverse pitch. Se was almost the same for large transverse pitches between c₁ and c₃ due to a weak vortex interaction. The contour plot of Se decreased slightly when the transverse pitch decreased from c₄ to c₅, and Se for c₆ was apparently the smallest due to the strongest vortex interaction.

3.3. Distribution of Nu_s and Se_s

The longitudinal vortices increased the secondary flow intensity and hence the heat-transfer enhancement. Figure 9 shows the distribution of cross-sectional averaged vortex intensity Se_s along the flow direction for different transverse pitches. Se_s was zero at the inlet because the fluid was uniform. Se_s then gradually increased due to the increase in secondary flow in the channel. In the region near the VG, Se_s increased sharply due to the flow separation and the generation of longitudinal vortices. Se_s reached the largest value around the trail end and then started to decrease quickly due to the attenuation of longitudinal vortices. Se_s then decreased smoothly to the outlet. The differences in Se_s were marginal in the region between the inlet and the VGs for different transverse pitches because there was no vortex interaction. In the region behind the VGs, the values of Se_s for c₁ to c₄ were nearly the same while there was an obvious difference for c₄ to c₆. Se_s decreased and attained the smallest value for c₆ due to the vortex interaction, as has been discussed above. The distribution of Se_s for Re = 1800 was similar with that for Re = 600 and the value of Se_s for Re = 1800 was obviously larger than that for Re = 600. The difference in Se_s between c₅ and c₆ increased when Re increased from 600 to 1800. Se_s of c₆ decreased by up to 23.1% and 26.5% compared with c₁ for Re = 600 and 1800, respectively.

The distributions of span-averaged Nu_s for different transverse pitches are shown in Figure 10. The largest value of Nu_s was obtained at the entrance of the channel due to the forming of a boundary layer, then Nu_s gradually decreased because of the development of the thermal boundary layer. Just as the values of Se_s were nearly the same, the values of Nu_s were also nearly the same for different transverse pitches in the region between the inlet and the VGs. An obvious difference existed in the region between the VGs and the outlet owing to the vortex interaction. Nu_s increased quickly in the location of VGs and showed peak values at the trail end of the VGs. Nu_s decreased from the peak point to the outlet due to the attenuation of the vortices. In the region behind the VGs for a short distance, there was a slight difference in Nu_s and the value of Nu_s for c₅ was the largest due to the common flow formed between the symmetrical vortices. The difference in Nu_s between different values of c then increased and Nu_s generally decreased with the decrease of c. The differences in Nu_s between the cases between c₁ and c₄ were slight. Obvious differences existed between the transverse pitches c₅ and c₆, and Nu_s of c₆ was the minimum due to the smallest vortex intensity. The value of Nu_s and the difference in Nu_s increased with the increase of Re from 600 to 1800. The value of Nu_s of c₆ decreased by 3.4% and 7.2% compared with c₁ at the outlet for Re = 600 and 1800, respectively.

3.4. Distributions of Nu, Se, f, and JF

The distributions of Se and Nu are presented in Figure 11. When Re was small, the vortex intensity was weak and the values of Se were nearly the same. Se increased with increasing Re and the difference in Se between different cases also increased due to the vortex interaction. The difference in Se was quite slight when the transverse pitch was large between c₁ and c₄. Obvious differences in Se can be attained for c₄ to c₆ due to the increase in vortex interaction. Se of c₆ decreased by 29.2% compared with c₁ at Re = 1800. The values of Nu were nearly the same when Re was small at Re = 200, as shown in Figure 11a. Nu was enhanced by the VGs and the difference in Nu between the model with VGs and the plain fin increased with increasing Re. The difference in Nu between different c was slight and an obvious difference in Nu only existed between c₅ and c₆ due to the decrease in vortex intensity caused by the vortex interaction. Nu of c₆ was the smallest and Nu decreased by about 2.0% compared with c₁ at Re = 1800. Although there was also an obvious vortex interaction for c₅, the formation of common flow down flow structure was beneficial for heat transfer, and as a result the value of Nu for c₅ was slightly different to other cases, except c₆.

Figure 12 shows the distributions of f and JF in the Re range between 200 and 1800. f of the model with VGs was obviously larger than that of the plain fin. The values of f for different transverse pitches were nearly the same. Thus, the change of c had no influence on f. The values of JF were less than 1.0 for Re = 200. This is because the vortices were too weak when Re was small at 200, while the pressure loss was increased compared with the plain fin. JF increased quickly from Re = 200 to Re = 600 and then gradually increased from Re = 600 to Re = 1800. The difference in JF was slight for c₁ to c₄. There was an obvious difference in JF for c₄ to c₆. JF of c₆ was the smallest and was apparently smaller than that of c₅. Thus, VG pairs in a common flow down configuration should be arranged with the transverse pitch greater than c₅ = 0.5 in order to obtain high thermal performance.

4. Conclusions

The symmetrical flow structure of the longitudinal vortices generated by VG pairs in a common flow down configuration and the influence of vortex interaction on thermal performance were numerically reported. The main conclusions can be summarized as follows:

• The symmetrical longitudinal vortices form a common-flow region between the vortices and the fluid is induced to flow from the top towards the bottom of the channel, which is beneficial for the heat transfer of the bottom fin;
• The vortex interaction in the symmetrical common-flow region increases with decreasing transverse pitch of the VG pair. The vortex intensity is obviously affected by the vortex interaction, while the friction factor is not influenced by the transverse pitch of the VG pair and the values of f for different transverse pitches are nearly the same;
• The vortex intensity, heat transfer, and thermal performance factor are obviously decreased when the transverse pitch of the studied VGs is smaller than a certain value of c₅. VG pairs in a common flow down configuration should be arranged with the transverse pitch greater than c₅ = 0.5 in order to obtain high thermal performance.