Transversal pitches of VGs
1. Introduction
Developing and innovating new techniques to enhance the heat transfer of a new compact heat exchanger is not only useful but also necessary for energy saving. The flow with longitudinal velocity components is an important phenomenon in fluid dynamics and heat transfer. Longitudinal vortices are generated by flow separation along the side edges of the vortex generators (VGs) due to the pressure differences between the upstream and the downstream sides and are perpendicular to the main flow direction. A description of the typical vortices structure formed by a delta winglet VG is given in some publications [1, 2]. There is a main vortex that is formed as a result of the flow separating in the tip of the halfdelta wing and rolling up due to the lower pressure in the back side of the VG. Then there is a corner vortex that is horseshoelike vortex formed in the corner between the front side of the VG and the fin. Finally, there is an induced secondary vortex formed in the corner between the back side of the wing and the fin as a result of the redirection of the near wall flow caused by the lower pressure behind the generator.
The longitudinal vortices can potentially enhance heat transfer with small pressure loss penalty and a better heat transfer effect than that of latitudinal vortices. The longitudinal vortices can cause bulk fluid mixing, boundary–layer modification, flow destabilization, and thereby enhance convective heat transfer. Setting protrusions that can generate longitudinal vortices on the fin surface is a promising technique to enhance the airside heat transfer. There are many protrusions that can generate longitudinal vortices. Vortex generators (VGs) are among the most popular actuators for the finside heat transfer enhancement. The winglet VG is capable of enhancing heat transfer with less increase in pressure penalty compared to other type of protrusions. A thorough review of the progress made in the application of longitudinal VGs is performed in reference [3].
In order to obtain a better heat transfer performance, researchers always try to punch lots of VGs out of the fin surface. However, the increasing number of VGs is not necessarily linked with the rise in heat transfer performance augmentation. This is because the vortices not only change the boundary layer structure but also interact with each other when they meet in the flow channel and the interaction of vortices affects the intensity of vortices and their effect on heat transfer enhancement. Experimental and numerical studies focusing on the interactions between vortices and boundary layers have been carried out in references [4–6]. The experimental investigation about the interaction between vortices and the boundary layers indicated that in the region where two neighboring vortices induced flow toward the heat transfer surface, local heat transfer was locally enhanced. Conversely, in the regions where neighboring vortices induced outflow departs the heat transfer surface, the local heat transfer was decreased. Close proximity of other vortices strongly affects the spreading of the vorticity. The heat transfer modification produced by the vortex was strongly dependent on vortex interaction. These previous works have shown that the strength of the vortices interaction with the wall is strongly dependent on the arrangement of vortices in the array. However, seldom works consider the interaction of longitudinal vortices and their effect on heat transfer. The effect of interaction of longitudinal vortices generated by winglet VGs on heat transfer enhancement of a flat tube bank fin heat exchanger was qualitatively analyzed in reference [7]. The quantitative study of the interaction of longitudinal vortices was seldom reported due to the lack of parameter that can define the intensity of longitudinal vortices. A nondimensional parameter Se was defined for the intensity of secondary flow in reference [8]. The parameter Se provides a powerful tool for the quantitative study of the interaction between longitudinal vortices. By using Se, quantitative studies about the interaction of longitudinal vortices generated by VGs mounted on the fin surfaces of flat tube bank fin heat exchanger were carried out in references [9, 10].
In this chapter, the nondimensional parameter Se that can be used for the description of the intensity of the longitudinal vortices is introduced first, then the interaction of counterrotating longitudinal vortices generated by VGs is quantitatively studied, and the effect of interaction on the intensity of vortices and heat transfer are discussed in detail by using the nondimensional parameter Se.
2. Physical model
As stated above, the longitudinal vortices can provide good performance for fluid flow and heat transfer enhancement. As the intensity of the longitudinal vortices decreases along the main flow direction, in order to obtain a high intensity of longitudinal vortices in the flow field, lots of VGs are always protruded into the flow field. Different arrangements of VGs will generate longitudinal vortices with different intensity and different rotating directions. These vortices with different rotating directions will inevitably meet and interact with each other when they are flowing downward. The interaction between these vortices affects not only the intensity of the vortices but also the heat transfer enhancement of the longitudinal vortices. For the platefin heat exchangers, there are many rows of VGs, and interaction between these vortices generated by different VGs will be a common physics phenomenon. This chapter focuses on the interaction between two counterrotating longitudinal vortices with different transversal pitches.
The physical model is shown in Figure 1. The flow channel is formed by two plain fins. Two winglet VGs with a certain longitudinal pitch are mounted on the bottom fin surface. The VGs are mounted around the longitudinal center line of the channel and the longitudinal pitch










4.818  3.212  1.606  0.0  1.606  3.212  4.818  6.424 

3  2  1  0  1  2  3  4 
3. Governing equations and boundary condition
In the case of the heat exchangers, the fin spacing is small and the flow in gas side is usually laminar. With this application in mind, the delta winglet VG in a laminar developing flow is considered in the present investigation. In such case, the compact forms of governing equations in physical space are as follows:
Continuity equation
Momentum equations
Energy equation
For developing flow, the boundary conditions at the inlet surface are given as follows:
At the outlet surface
At the symmetric surfaces
At the solid surfaces, constant temperature and noslip condition for velocity are applied as follows:
The Reynolds number is
The local Nusselt number is determined by
The spanaveraged Nu_{s} is obtained by averaging Nu_{local} over the span strip fin surfaces at position
The overall average Nu is obtained by averaging Nu_{local} over the entire fin surfaces:
4. Parameter for the intensity of longitudinal vortices
If the main flow direction is along the
where
Based on the study of
Se has the same form as the definition of Re, but the physical meaning is quite different. Se represents the ratio of inertial force to viscous force, which are induced by the secondary flow. Re represents the ratio of inertial force to viscous force, which are induced by the main flow. The crosssectional average value of Se at position
The volumeaveraged value of Se is obtained by integrating the local value of Se over the total flow field:
5. Numerical method
The simulation domain in physical space (
A typical structured grid system used in the present study is shown in Figure 3. Figure 3(a) shows the schematic view of the threedimensional grid system, and Figure 3(b) is the grid in



134 × 98 × 22  6.6418  0.0580 
194 × 142 × 32  6.6073  0.0578 
258 × 190 × 44  6.6291  0.0582 
6. Results and discussion
The generation of longitudinal vortices and the subsequent disruption of thermal boundary layers are the prime movers of heat transfer augmentation. Thus, the flow field attracts special attention. In order to show the development of the flow field, eight cross sections are selected as shown in Figure 4, the locations of these selected sections are summarized in Table 3.










0.291  0.368  0.448  0.525  0.602  0.682  0.836  0.993 
6.1. Effect of interaction of longitudinal vortices on the flow field
The velocity vectors on the cross sections for transversal pitch c2 and c6 with Re
Figures 5 and 6 correspond to the arrangements of Figure 1(b) and (c), respectively. By comparing the flow field structure in Figures 5 and 6, one can find that the flow field structures in the common region between the vortices are different. In Figure 5, the fluid in the common region is directed away from the bottom fin surface and forms a common flow–up flow field structure. However, in Figure 6, the fluid in the common region is directed toward the bottom fin surface and forms a common flow–down flow field structure.
The velocity vectors on the cross section s6 with Re
6.2. Effect of interaction of longitudinal vortices on Se_{s}
The parameter Se relates to the intensity of the longitudinal vortices generated by the VGs, and the intensity of these vortices is affected by the interaction between them. Thus, the distribution curve of the spanaveraged value of Se along the flow direction can reflect the interaction between the counterrotating longitudinal vortices for different transversal pitches of the VGs.
The distribution of the spanaveraged values of Se_{s} for the range of
6.3. Effect of interaction on Nu_{s}
Figure 9 shows the distributions of the spanaveraged values of Nu_{s} at Re
The local spanaveraged Nu_{s} for the case c6 is the largest in the region around the second VG. However, the intensity of the longitudinal vortices for the case c6 is not the largest. The reason maybe that a common flow region formed between the VGs and the fluid in the common flow region is directed downwash toward the bottom surface on which the VGs are mounted on, as shown in Figure 6. The heat transfer is locally enhanced benefiting from the local thinning of the thermal boundary layer in the common flow down region. Experimental investigation in reference [4] presents the same conclusion that the heat transfer is locally enhanced in the region where two neighboring vortices impose a flow toward the surface. Therefore, the interaction of longitudinal vortices does not necessarily decrease the heat transfer performance. The heat transfer performance depends not only on the intensity of the vortices but also on the flow field structure of the vortices.
6.4. Effect of interaction on average values of Se, Nu, and f
The average value of Se over the entire flow and the average value of Nu over the entire area included in the heat transfer are of great interest as they are directly linked to the intensity of longitudinal vortices in the channel and to the amount of total heat transfer. The distributions of Se and Nu for the range of
6.5. Effect of interaction on Se/Se_{ref} and Nu/Nu_{ref}
In order to study the effect (in percentage) of the interaction of the counterrotating longitudinal vortices on Se and Nu, the values of Se and Nu for c1 are selected as the reference values, then the ratios of Se/Se_{ref} and Nu/Nu_{ref} mean the percentage of Se and Nu compared with the reference values. The distributions of Se/Se_{ref} and Nu/Nu_{ref} as a function of the distance between the VGs are presented in Figure 11 for Re ranging from 200 to 1800. As expected, when values of transversal pitch between the VGs are
The distributions of Nu/Nu_{ref} are much similar to the distributions of Se/Se_{ref}; the values of Nu/Nu_{ref} also reach the minimum values at
6.6. Effect of interaction on JF and JF/JF_{ref}
The surface goodness factor JF under same pump power is more suitable for engineering application and is commonly used as the criteria for evaluating the good performance of heat transfer exchangers or heat transfer surfaces. Figure 12(a) shows the distribution of JF for different values of
7. Conclusions
The interaction of two counterrotating longitudinal vortices generated by VGs mounted on the bottom of a channel formed by two neighboring fins and the effect of interaction of counterrotating longitudinal vortices on the intensity of vortices and heat transfer are quantitatively studied using the numerical method. The following conclusions were derived:
The strength of the interaction between the counterrotating vortices is strongly dependent on the transversal pitches between the vortices. The distribution of Se_{s} does not only reflect the changing of the intensity of the longitudinal vortices in the flow channel but also reflect the interaction between the longitudinal vortices.
The interaction between the counterrotating longitudinal vortices does not necessarily decrease the intensity of the vortices. When the counterrotating vortices partially interact with each other, the intensity of the vortices can also be increased. When the counterrotating vortices fully interact with each other, the intensity of the vortices decreases seriously.
The interaction between counterrotating vortices does not necessarily decrease the heat transfer performance of the longitudinal vortices. The heat transfer performance depends on not only the intensity of the vortices but also the structure of the vortices. The common flow region formed between the counterrotating longitudinal vortices is beneficial for the heat transfer enhancement.
Due to the interactions of counterrotating longitudinal vortices and their effect on heat transfer enhancement, an optimum arrangement of VGs exists for obtaining a better heat transfer performance. When the distance between the VGs is twice the projected length of the base of VGs, the best heat transfer performance can be obtained.
Nomenclature
Nu; Nusselt number: Nu
Re; Reynolds number: Re
Se; secondary flow intensity
u, v, w; components of velocity vector (m/s)
x, y, z; coordinates
Greeks
θ; attack angle of VG (°)
λ; heat conductivity (W/(m K))
μ; viscosity (kg/(m s))
ρ; density (kg/m^{3})
ω; vorticity (1/s)
Subscripts
ABS; absolute value
bulk; bulk temperature on the cross section
local; local value
s; spanaveraged or crosssectional average value
w; fin surface
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant nos. 51376086 and 51366008) and the Gansu Provincial Foundation for Distinguished Young Scholars (grant no. 145RJDA324).
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