**2.2 Computational model**

In this study, the effects of the installation of RWP and CRWP VGs in the rectangular channel on thermal–hydraulic performance were compared. The geometry of the VG used in this study can be seen in **Figure 2**. In this simulation, VGs were made from an aluminum plate with a thickness of 1 mm with/without holes

**Figure 1.** *Schematic of experimental set-up.*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular… DOI: http://dx.doi.org/10.5772/intechopen.96117*

with a diameter of 5 mm. **Table 1** shows the geometric parameters of the CRWP and RWP VGs. **Figure 3** is a top view of the RWP and CRWP VGs. VG with the angle of attack (α) 45° arranged in-line in common flow-down orientation with a longitudinal pitch of 125 mm. The distance between the first row and the inlet channel is 125 mm. Meanwhile, the leading-edge transverse distance between winglet pairs VG is 20 mm. The rectangular channel modeled in this simulation has dimensions of length (P), width (L), height (H) of channels of 500 mm, 75.5 mm, 65 mm, respectively.

**Figure 4** shows the computational domain used in this modeling. This domain consists of an inlet extended region and an outlet extended region. An inlet extended region was provided to ensure that the airflow entering the channel is a fully developed flow. Meanwhile, an extended region outlet was added so that the air does not experience reverse flow in the channel.

#### **2.3 Governing equations**

length of 370 cm, a width of 8 cm, and a height of 18 cm, as shown in **Figure 1**. The blower sucks air into the channel from the inlet side through a straightener composed of pipes with a diameter of 5 mm and wire mesh to equalize the flow velocity. The flow velocity in the channel was varied in the range of 0.4 m/s to 2.0 m/s with an interval of 0.2 m/s using a motor regulator controlled by an inverter (Mitsubishi Electric-type FR-D700 with an accuracy of 0.01 Hz) and measured with a hotwire anemometer (Lutron type AM-4204 with an accuracy of 0.05). In this study, the airflow flowed through VGs with variations in the number of rows (one, two, and three rows) as well as variations with/without holes to investigate the effect on heat transfer rate and pressure drop. The VGs were mounted on a flat plate that was heated at a constant rate of 35 W using a heater that was regulated by a heater regulator and monitored by a wattmeter (Lutron DW-6060 with an accuracy of 0.01). Thermocouples K type was used to measure surface temperature, inlet and outlet temperatures, which were connected to data acquisition (Advantech type USB-4718 with accuracy 0.01) and were monitored and stored in the CPU. In the pressure drop test, two pitot tubes were installed at the inlet and outlet of the test section and connected to a micro manometer (Fluke 922 with accuracy 0.01) to monitor the pressure drop due to the installation of VGs. Flow visualization tests were also carried out to observe the longitudinal vortex formed as a result of VGs insertion. The longitudinal vortex was formed when the smoke resulting from the evaporation of oil in the heater was flowed through VGs and captured by the transverse plane formed by the luminescence of the laser beam. The camera was

*Heat Transfer - Design, Experimentation and Applications*

used to record the longitudinal vortex structure that was formed.

In this study, the effects of the installation of RWP and CRWP VGs in the rectangular channel on thermal–hydraulic performance were compared. The geometry of the VG used in this study can be seen in **Figure 2**. In this simulation, VGs were made from an aluminum plate with a thickness of 1 mm with/without holes

**2.2 Computational model**

**Figure 1.**

**56**

*Schematic of experimental set-up.*

In this 3-D flow modeling, air was assumed to be steady state, incompressible and has constant physical properties. Flow can be laminar or turbulent based on its

**Figure 2.** *Geometry of RWP and CRWP VG with and without holes.*


#### **Table 1.**

*Geometry parameters of vortex generator (VG).*

Energy equation

*DOI: http://dx.doi.org/10.5772/intechopen.96117*

stated as follows:

**2.4 Boundary conditions**

Inlet upstream extended region

Outlet downstream extended region

*∂u <sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup><sup>v</sup>*

follows:

Wall

**59**

*∂ ∂xi*

*∂ ∂xi* ð Þ¼ *ρkui*

ð Þ¼ *ρωui*

*∂ ∂xi* ð Þ¼ *<sup>ρ</sup>uiT <sup>∂</sup>*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular…*

dynamic viscosity, respectively. Meanwhile, Γ is the diffusion coefficient

where *λ* is the thermal conductivity, and *cp* is the specific heat of air.

*∂ ∂x <sup>j</sup>*

*∂ ∂x <sup>j</sup>*

turbulence kinetic energy. The *Γω* and *Γ<sup>k</sup>* equations are stated as follows:

*Γk ∂k ∂x <sup>j</sup>* 

*Γω ∂ω ∂x <sup>j</sup>* 

where *Γω* is the specific dissipation rate and *Γ<sup>k</sup>* is the diffusion effectiveness of

*Γω* <sup>¼</sup> *<sup>μ</sup>* <sup>þ</sup> *<sup>μ</sup><sup>t</sup>*

*<sup>Γ</sup><sup>k</sup>* <sup>¼</sup> *<sup>μ</sup>* <sup>þ</sup> *<sup>μ</sup><sup>t</sup>*

*σ* and *μ<sup>t</sup>* are the turbulent Prandtl number and turbulent viscosity, respectively. In this governing equation, the turbulent intensity can be formulated as follows:

*<sup>I</sup>* <sup>¼</sup> <sup>0</sup>*:*16R�1*=*<sup>8</sup>

The boundary conditions used in this computational domain are described as

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup><sup>w</sup>*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup><sup>T</sup>*

*u* ¼ *u*, *v* ¼ *w* ¼ 0, *T* ¼ *T* ¼ *Const:* (9)

*u* ¼ *v* ¼ *w* ¼ 0, *T* ¼ *Tw* (11)

*σω*

*σk*

*∂xi Γ ∂T ∂xi*

where *ρ*, *p*, *ui*, and *μ* are the density, pressure, mean velocity on the x-axis, and

*Γ* ¼ *λ=cp*

The turbulent flow modeling used in this simulation is the standard *k-ω* model. The transport equation for the standard *k-ω* model consists of the turbulent kinetic energy (*k*) and specific dissipation rate (*ω*) equations, respectively, which are

(3)

þ *Gk* � *Yk* þ *Sk* (4)

þ *G<sup>ω</sup>* � *Y<sup>ω</sup>* þ *S<sup>ω</sup>* (5)

*Dh* (8)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup> (10)

(6)

(7)

**Figure 3.** *Top view of (a) RWP VG, (b) CRWP VG.*

**Figure 4.** *Computational domain.*

Reynolds number value. Flow velocities were set in the range of 0.4–2 m/s with 0.2 m/s intervals. The Reynolds number is determined from R ¼ *ρumDh=μ* in the range of 1800 < Re <9100. Therefore, the flow was assumed to be laminar at a velocity of 0.4 m/s with Re = 1800 and the others were turbulent. Based on these assumptions, the governing equations used to solve this case are:

Continuity equation

$$\frac{\partial \boldsymbol{u}\_{j}}{\partial \boldsymbol{\omega}\_{j}} = \mathbf{0} \tag{1}$$

Momentum equation

$$\frac{\partial}{\partial \mathbf{x}\_j} \left( \rho u\_i u\_j \right) = \frac{-\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left( \mu \frac{\partial u\_k}{\partial \mathbf{x}\_i} \right) \tag{2}$$

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular… DOI: http://dx.doi.org/10.5772/intechopen.96117*

Energy equation

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho u\_i T) = \frac{\partial}{\partial \mathbf{x}\_i} \left( \Gamma \frac{\partial T}{\partial \mathbf{x}\_i} \right) \tag{3}$$

where *ρ*, *p*, *ui*, and *μ* are the density, pressure, mean velocity on the x-axis, and dynamic viscosity, respectively. Meanwhile, Γ is the diffusion coefficient

*Γ* ¼ *λ=cp*

where *λ* is the thermal conductivity, and *cp* is the specific heat of air.

The turbulent flow modeling used in this simulation is the standard *k-ω* model. The transport equation for the standard *k-ω* model consists of the turbulent kinetic energy (*k*) and specific dissipation rate (*ω*) equations, respectively, which are stated as follows:

$$\frac{\partial}{\partial \mathbf{x}\_i} \left( \rho k u\_i \right) = \frac{\partial}{\partial \mathbf{x}\_j} \left( \Gamma\_k \frac{\partial k}{\partial \mathbf{x}\_j} \right) + G\_k - Y\_k + \mathbf{S}\_k \tag{4}$$

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho a u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left( \Gamma\_{a} \frac{\partial \alpha}{\partial \mathbf{x}\_j} \right) + \mathbf{G}\_{a\nu} - \mathbf{Y}\_{a\nu} + \mathbf{S}\_{a\nu} \tag{5}$$

where *Γω* is the specific dissipation rate and *Γ<sup>k</sup>* is the diffusion effectiveness of turbulence kinetic energy. The *Γω* and *Γ<sup>k</sup>* equations are stated as follows:

$$
\Gamma\_w = \mu + \frac{\mu\_t}{\sigma\_w} \tag{6}
$$

$$
\Gamma\_k = \mu + \frac{\mu\_t}{\sigma\_k} \tag{7}
$$

*σ* and *μ<sup>t</sup>* are the turbulent Prandtl number and turbulent viscosity, respectively. In this governing equation, the turbulent intensity can be formulated as follows:

$$I = \mathbf{0}.\mathbf{1}6\Re\_{\mathrm{D}\_{\mathrm{h}}}^{-1/8} \tag{8}$$

#### **2.4 Boundary conditions**

The boundary conditions used in this computational domain are described as follows:

Inlet upstream extended region

$$u = u, v = w = 0, T = T = \text{Const.}\tag{9}$$

Outlet downstream extended region

$$\frac{\partial u}{\partial \mathbf{x}} = \frac{\partial v}{\partial \mathbf{x}} = \frac{\partial w}{\partial \mathbf{x}} = \frac{\partial T}{\partial \mathbf{x}} = \mathbf{0} \tag{10}$$

Wall

$$
\mu = \upsilon = \omega = \mathbf{0}, T = T\_w \tag{11}
$$

Reynolds number value. Flow velocities were set in the range of 0.4–2 m/s with 0.2 m/s intervals. The Reynolds number is determined from R ¼ *ρumDh=μ* in the range of 1800 < Re <9100. Therefore, the flow was assumed to be laminar at a velocity of 0.4 m/s with Re = 1800 and the others were turbulent. Based on these

> *∂u <sup>j</sup> ∂x <sup>j</sup>*

> > *∂xi* þ *∂ ∂x <sup>j</sup> μ ∂uk ∂xi*

¼ 0 (1)

(2)

assumptions, the governing equations used to solve this case are:

*ρuiu <sup>j</sup>* <sup>¼</sup> �*∂<sup>p</sup>*

*∂ ∂x <sup>j</sup>*

Continuity equation

**Figure 3.**

**Figure 4.**

**58**

*Computational domain.*

*Top view of (a) RWP VG, (b) CRWP VG.*

*Heat Transfer - Design, Experimentation and Applications*

Momentum equation

*Heat Transfer - Design, Experimentation and Applications*

Symmetry

$$v = 0, \frac{\partial u}{\partial \mathbf{y}} = \frac{\partial w}{\partial \mathbf{y}} = \frac{\partial T}{\partial \mathbf{y}} = \mathbf{0} \tag{12}$$

Nusselt number

*DOI: http://dx.doi.org/10.5772/intechopen.96117*

is defined as follows:

**2.7 Validation**

**61**

determined by the following equation:

*T* ¼ ÐÐ

ÐÐ

*Tout* ¼

which *Pin* and *Pout* can be described as follows:

*Nu* <sup>¼</sup> *hDh*

where *ρ*, *um*, *μ*, *Dh*, and *λ* are the density, average fluid velocity, dynamic viscosity, hydraulic diameter, and thermal conductivity of the fluid, respectively. *h* is the convection heat transfer coefficient obtained from the following equation:

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular…*

*<sup>h</sup>* <sup>¼</sup> *<sup>q</sup>*

*AT Tw* � *T <sup>f</sup>*

*q*, *AT*, and *Tw* are the convection heat transfer rate, heat transfer surface area, and hot wall temperature, respectively, while *Tf* is the bulk fluid temperature which

*<sup>T</sup> <sup>f</sup>* <sup>¼</sup> *<sup>T</sup>* <sup>þ</sup> *Tout*

ÐÐ

ÐÐ

*P* ¼ ÐÐ *<sup>A</sup>pdA* ÐÐ

> ÐÐ *AoutpdA* ÐÐ

An independent grid test was performed to ensure that the number of grids does not affect the numerical simulation results. Four different grid numbers were used for grid-independent testing. The test was carried out on the computational domain with three CRWP pairs at a velocity of 0.4 m/s. **Table 2** shows the simulation results of the variation in the number of different grids on the convection heat transfer coefficient. Because the convection heat transfer coefficient of the simulation results shows a slight difference, the optimum number of grids is determined by comparing the heat transfer coefficient from the modeling results and the results from the experiment. The smallest error from the simulation results and experimental results is used as an independent grid. Based on the comparison of the simulation results for the various numbers of grids with the experimental results, it is found that the grid with the number of elements close to 1,600,000 was chosen for use in this numerical simulation because it has the lowest error, namely 0.337%. Validation was also carried out by comparing the

*Pout* ¼

*Tin* is the inlet temperature and *Tout* is the temperature at the outlet side which is

*<sup>A</sup>u x*ð Þ , *y*, *z T x*ð Þ , *y*, *z dA*

*Aoutu x*ð Þ , *y*, *z T x*ð Þ , *y*, *z dA*

*ΔP* is the pressure drop of fluid flow which can be formulated as *ΔP* = *Pin*-*Pout* in

*<sup>λ</sup>* (14)

� � (15)

<sup>2</sup> (16)

*<sup>A</sup>u x*ð Þ , *<sup>y</sup>*, *<sup>z</sup> dA* (17)

*Aoutu x*ð Þ , *<sup>y</sup>*, *<sup>z</sup> dA* (18)

*<sup>A</sup>dA* (19)

*AoutdA* (20)

#### **2.5 Numerical method**

The finite volume method (FVM) was used to analyze the thermo-hydraulic characteristics of the rectangular channel installed with VGs. Laminar flow was simulated using a laminar model, while the turbulent flow was simulated using the *k-ω* model. The turbulent *k-ω* model was used in this simulation because this model is suitable for modeling fluid flow in the viscous region [21]. The SIMPLE algorithm was chosen to obtain a numerical solution of the continuity and momentum equations. The governing equations for momentum, turbulent kinetic energy, specific dissipation rate and energy were discretized with a second-order upwind scheme. The convergence criterion assigned to the continuity, momentum, and energy equations was 10�<sup>5</sup> , 10�<sup>6</sup> , 10�<sup>8</sup> , respectively.

In this numerical simulation, the mesh type was differentiated between the upstream extended and downstream extended regions with the computational domain, as shown in **Figure 5**. The hexagonal mesh was used in both parts of the extended region because it has a simple geometric shape. Meanwhile, the part of the computational domain, namely the fluid and plate, uses a tetrahedral mesh because it has a more complex geometry due to the presence of VGs. The tetrahedral mesh was also used to obtain more accurate results in this area so that it can show flow separation and secondary flow in the test section.

#### **2.6 Parameter definitions**

The parameters used in this study are as follows: Reynolds number

$$\Re = \frac{\rho u\_m D\_h}{\mu} \tag{13}$$

**Figure 5.** *Mesh generated.*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular… DOI: http://dx.doi.org/10.5772/intechopen.96117*

Nusselt number

Symmetry

**2.5 Numerical method**

equations was 10�<sup>5</sup>

**2.6 Parameter definitions**

Reynolds number

**Figure 5.** *Mesh generated.*

**60**

, 10�<sup>6</sup>

separation and secondary flow in the test section.

The parameters used in this study are as follows:

, 10�<sup>8</sup>

*<sup>v</sup>* <sup>¼</sup> 0, *<sup>∂</sup><sup>u</sup>*

*Heat Transfer - Design, Experimentation and Applications*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup><sup>w</sup>*

The finite volume method (FVM) was used to analyze the thermo-hydraulic characteristics of the rectangular channel installed with VGs. Laminar flow was simulated using a laminar model, while the turbulent flow was simulated using the *k-ω* model. The turbulent *k-ω* model was used in this simulation because this model is suitable for modeling fluid flow in the viscous region [21]. The SIMPLE algorithm was chosen to obtain a numerical solution of the continuity and momentum equations. The governing equations for momentum, turbulent kinetic energy, specific dissipation rate and energy were discretized with a second-order upwind scheme. The convergence criterion assigned to the continuity, momentum, and energy

, respectively. In this numerical simulation, the mesh type was differentiated between the upstream extended and downstream extended regions with the computational domain, as shown in **Figure 5**. The hexagonal mesh was used in both parts of the extended region because it has a simple geometric shape. Meanwhile, the part of the computational domain, namely the fluid and plate, uses a tetrahedral mesh because it has a more complex geometry due to the presence of VGs. The tetrahedral mesh was also used to obtain more accurate results in this area so that it can show flow

> <sup>R</sup> <sup>¼</sup> *<sup>ρ</sup>umDh μ*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup><sup>T</sup>*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>0</sup> (12)

(13)

$$Nu = \frac{hD\_h}{\lambda} \tag{14}$$

where *ρ*, *um*, *μ*, *Dh*, and *λ* are the density, average fluid velocity, dynamic viscosity, hydraulic diameter, and thermal conductivity of the fluid, respectively. *h* is the convection heat transfer coefficient obtained from the following equation:

$$h = \frac{q}{A\_T (T\_w - T\_f)}\tag{15}$$

*q*, *AT*, and *Tw* are the convection heat transfer rate, heat transfer surface area, and hot wall temperature, respectively, while *Tf* is the bulk fluid temperature which is defined as follows:

$$T\_f = \frac{T + T\_{out}}{2} \tag{16}$$

*Tin* is the inlet temperature and *Tout* is the temperature at the outlet side which is determined by the following equation:

$$T = \frac{\iint\_{A} \mu(\mathbf{x}, \mathbf{y}, z) T(\mathbf{x}, \mathbf{y}, z) dA}{\iint\_{A} \mu(\mathbf{x}, \mathbf{y}, z) dA} \tag{17}$$

$$T\_{out} = \frac{\iint\_{A\_{out}} \mu(\mathbf{x}, \mathbf{y}, z) T(\mathbf{x}, \mathbf{y}, z) dA}{\iint\_{A\_{out}} \mu(\mathbf{x}, \mathbf{y}, z) dA} \tag{18}$$

*ΔP* is the pressure drop of fluid flow which can be formulated as *ΔP* = *Pin*-*Pout* in which *Pin* and *Pout* can be described as follows:

$$P = \frac{\iint\_{A} p dA}{\iint\_{A} dA} \tag{19}$$

$$P\_{out} = \frac{\iint\_{A\_{sw}} pdA}{\iint\_{A\_{out}} dA} \tag{20}$$

#### **2.7 Validation**

An independent grid test was performed to ensure that the number of grids does not affect the numerical simulation results. Four different grid numbers were used for grid-independent testing. The test was carried out on the computational domain with three CRWP pairs at a velocity of 0.4 m/s. **Table 2** shows the simulation results of the variation in the number of different grids on the convection heat transfer coefficient. Because the convection heat transfer coefficient of the simulation results shows a slight difference, the optimum number of grids is determined by comparing the heat transfer coefficient from the modeling results and the results from the experiment. The smallest error from the simulation results and experimental results is used as an independent grid. Based on the comparison of the simulation results for the various numbers of grids with the experimental results, it is found that the grid with the number of elements close to 1,600,000 was chosen for use in this numerical simulation because it has the lowest error, namely 0.337%. Validation was also carried out by comparing the


**Table 2.**

*Grid independent test.*

experimental results of Wu et al. (2008) and current experimental results with slightly different conditions, see Ref. [22].

## **3. Results and discussion**

This study aims to investigate the effect of holes on VGs and the number of pairs of VGs on airflow and heat transfer characteristics. The installation of VG generates vortices and forms swirl flow so that the convection heat transfer rate on the airside increases [7–9].

#### **3.1 Flow field**

To determine the difference in flow structure in the test section, simulations were carried out on a channel with VGs and without VGs (baseline). **Figure 6(a)** is a flow in the baseline case where vortices and swirl flows are not observed. Whereas in **Figure 6(b)**, the simulation results show that the installation of VGs on the channel results in the formation of swirl flow [7], which results in longitudinal vortices due to flow separation along the VGs caused by pressure differences on the upstream and downstream VGs [10]. **Figure 7** illustrates the counter-rotating pairs of longitudinal vortices due to the installation of RWPs and CRWPs VGs with a 45° angle of attack. A strong counter-rotating longitudinal vortex forms behind the VG with the left rotating clockwise and the right rotating counterclockwise [23]. These two longitudinal vortices result in the formation of downwash flow in the center of the channel towards the lower wall of the channel and upwash flow on both sides of the channel to the upper wall of the channel. This longitudinal vortex configuration is also called common-flow-down.

**Figure 8** is a comparison of tangential velocity vectors in the cross-plane X1 with

*Tangential velocity vector on a channel with three pairs of VG: (a) perforated CRWP and (b) perforated RWP.*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular…*

*DOI: http://dx.doi.org/10.5772/intechopen.96117*

three pairs of RWP and CRWP VGs for with and without holes at 2.0 m/s. The tangential velocity vector in the use of RWP and CRWP VGs is high in the downwash region, which results in improved heat transfer [7]. In the case of CRWP VGs, the longitudinal vortex radius formed is larger than that of the RWP VGs. This is because the frontal area of the CRWP is larger, which results in a better heat transfer rate increase than that of the RWP VGs [24, 25]. The hole in VG causes a jet flow, which removes stagnant fluid in the back region of VG and increases the kinetic energy in this area so that the pressure difference before and after passing VG can be reduced [26]. Because of this decrease in the pressure difference, the longitudinal vortex strength decreases. The main vortex, induced vortex, and corner vortex are observed on CRWP VGs installation, as shown in **Figure 9**. The structure of the longitudinal vortex is formed due to several factors. The main vortex is formed due to flow separation when the flow passes through the VG wall due to the pressure difference [27]. Induced vortex is formed due to the interaction between the main vortex. Meanwhile, the corner vortex is formed as a result of the

**Figures 10** and **11** show the counter-rotating longitudinal vortex as the flow passes through the VGs. Counter-rotating longitudinal vortices are observed in the

interaction between the VG wall and the main vortex.

**Figure 7.**

**Figure 8.**

**63**

*Tangential velocity vector in the cross-section X1.*

**Figure 6.** *Velocity streamline in a channel; (a) without VG (baseline), (b) with VG.*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular… DOI: http://dx.doi.org/10.5772/intechopen.96117*

#### **Figure 7.**

experimental results of Wu et al. (2008) and current experimental results with

**Number of element h(simulation) h(experiment) Error (%)** 1,262,840 18.27726 18.18571 0.503 1,478,060 18.34781 18.18571 0.891 1,661,610 18.24699 18.18571 0.337 1,868,587 18.29429 18.18571 0.597

This study aims to investigate the effect of holes on VGs and the number of pairs of VGs on airflow and heat transfer characteristics. The installation of VG generates vortices and forms swirl flow so that the convection heat transfer rate on the airside

To determine the difference in flow structure in the test section, simulations were carried out on a channel with VGs and without VGs (baseline). **Figure 6(a)** is a flow in the baseline case where vortices and swirl flows are not observed. Whereas in **Figure 6(b)**, the simulation results show that the installation of VGs on the channel results in the formation of swirl flow [7], which results in longitudinal vortices due to flow separation along the VGs caused by pressure differences on the upstream and downstream VGs [10]. **Figure 7** illustrates the counter-rotating pairs of longitudinal vortices due to the installation of RWPs and CRWPs VGs with a 45° angle of attack. A strong counter-rotating longitudinal vortex forms behind the VG with the left rotating clockwise and the right rotating counterclockwise [23]. These two longitudinal vortices result in the formation of downwash flow in the center of the channel towards the lower wall of the channel and upwash flow on both sides of the channel to the upper wall of the channel. This longitudinal vortex configuration

slightly different conditions, see Ref. [22].

*Heat Transfer - Design, Experimentation and Applications*

**3. Results and discussion**

is also called common-flow-down.

*Velocity streamline in a channel; (a) without VG (baseline), (b) with VG.*

increases [7–9].

**Table 2.**

*Grid independent test.*

**3.1 Flow field**

**Figure 6.**

**62**

*Tangential velocity vector on a channel with three pairs of VG: (a) perforated CRWP and (b) perforated RWP.*

#### **Figure 8.**

*Tangential velocity vector in the cross-section X1.*

**Figure 8** is a comparison of tangential velocity vectors in the cross-plane X1 with three pairs of RWP and CRWP VGs for with and without holes at 2.0 m/s. The tangential velocity vector in the use of RWP and CRWP VGs is high in the downwash region, which results in improved heat transfer [7]. In the case of CRWP VGs, the longitudinal vortex radius formed is larger than that of the RWP VGs. This is because the frontal area of the CRWP is larger, which results in a better heat transfer rate increase than that of the RWP VGs [24, 25]. The hole in VG causes a jet flow, which removes stagnant fluid in the back region of VG and increases the kinetic energy in this area so that the pressure difference before and after passing VG can be reduced [26]. Because of this decrease in the pressure difference, the longitudinal vortex strength decreases. The main vortex, induced vortex, and corner vortex are observed on CRWP VGs installation, as shown in **Figure 9**. The structure of the longitudinal vortex is formed due to several factors. The main vortex is formed due to flow separation when the flow passes through the VG wall due to the pressure difference [27]. Induced vortex is formed due to the interaction between the main vortex. Meanwhile, the corner vortex is formed as a result of the interaction between the VG wall and the main vortex.

**Figures 10** and **11** show the counter-rotating longitudinal vortex as the flow passes through the VGs. Counter-rotating longitudinal vortices are observed in the

cross-sectional plane at positions X1 to X6 and move spirally downstream to a certain distance and sweep towards the lower wall of the channel [26]. The strength of the longitudinal vortex is observed to be greater in CRWP than in RWP. CRWP has greater longitudinal vortex strength because CRWP has a larger frontal area than that of RWP, which results in a larger longitudinal vortex radius causing in better heat transfer performance [19]. From **Figures 10** and **11**, it is observed that the longitudinal vortex in the X1 plane is stronger than that in the X2 plane for all types of VG with/without holes. This is due to viscous dissipation, which causes the longitudinal vortex to gradually weaken as the flow away from VG [28]. In the X3 plane, the longitudinal vortex strength increases compared to the X2 plane due to the addition of VGs, which results in an increase in fluid velocity in the downwash region [29]. The hole in the VG results in the weakening of the longitudinal vortex

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular…*

*Comparison of the tangential velocity distribution in the channel installed CRWP at several cross-section*

strength due to jet flow formation [26].

*DOI: http://dx.doi.org/10.5772/intechopen.96117*

**Figure 11.**

**65**

*positions at a velocity of 2.0 m/sec.*

**Figure 9.**

*Tangential velocity vector in the cross-section X1 in the channel installed VG: (a) perforated RWP, (b) perforated CRWP.*

**Figure 10.**

*Comparison of the tangential velocity distribution in the channel installed RWP at several cross-section positions at a velocity of 2.0 m/sec.*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular… DOI: http://dx.doi.org/10.5772/intechopen.96117*

cross-sectional plane at positions X1 to X6 and move spirally downstream to a certain distance and sweep towards the lower wall of the channel [26]. The strength of the longitudinal vortex is observed to be greater in CRWP than in RWP. CRWP has greater longitudinal vortex strength because CRWP has a larger frontal area than that of RWP, which results in a larger longitudinal vortex radius causing in better heat transfer performance [19]. From **Figures 10** and **11**, it is observed that the longitudinal vortex in the X1 plane is stronger than that in the X2 plane for all types of VG with/without holes. This is due to viscous dissipation, which causes the longitudinal vortex to gradually weaken as the flow away from VG [28]. In the X3 plane, the longitudinal vortex strength increases compared to the X2 plane due to the addition of VGs, which results in an increase in fluid velocity in the downwash region [29]. The hole in the VG results in the weakening of the longitudinal vortex strength due to jet flow formation [26].

#### **Figure 11.**

*Comparison of the tangential velocity distribution in the channel installed CRWP at several cross-section positions at a velocity of 2.0 m/sec.*

**Figure 9.**

**Figure 10.**

**64**

*at a velocity of 2.0 m/sec.*

*perforated CRWP.*

*Tangential velocity vector in the cross-section X1 in the channel installed VG: (a) perforated RWP, (b)*

*Heat Transfer - Design, Experimentation and Applications*

*Comparison of the tangential velocity distribution in the channel installed RWP at several cross-section positions*

### **3.2 Longitudinal vortex intensity**

The longitudinal vortex intensity is a dimensionless number studied by K Song et al. [30] and represents the magnitude of the inertia force induced by secondary flow to the viscous force. In this study, the longitudinal vortex intensity is defined in Eq. (22)

$$\text{Se} = \frac{\rho D\_h U}{\mu} \tag{21}$$

*<sup>U</sup>* <sup>¼</sup> *Dh <sup>ω</sup><sup>n</sup>* j j¼ *Dh*

*DOI: http://dx.doi.org/10.5772/intechopen.96117*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular…*

*Sex* <sup>¼</sup> *<sup>ρ</sup>D*<sup>2</sup>

*h A x*ð Þ*μ*

Eq. (24)

**Figure 13.**

**67**

*∂w <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup><sup>v</sup> ∂z*

� � � �

*<sup>ω</sup><sup>n</sup>* j j*dA* (23)

(22)

� � � �

where *ω<sup>n</sup>* is the vortices about the normal axis of the spanwise plane. The mean longitudinal vortex intensity in the spanwise plane at position x (*Se*x) is defined by

ðð

**Figures 12** and **13** show the ratio of *Se*<sup>x</sup> in RWP and CRWP cases at a velocity of 0.4 m/s and 2.0 m/s. In general, CRWP insertion produces a greater longitudinal

*The mean spanwise longitudinal vortex intensity at a velocity of 2.0 m/s for the case of (a) one-pair RWP; (b) one-pair of CRWP; (c) two pairs RWP; (d) two-pairs CRWP; (e) three pairs RWP; (f) three-pairs CRWP.*

*A x*ð Þ

where *Se* is the longitudinal vortex intensity, and U is the secondary flow velocity characteristic, which can be formulated in the following equation:

**Figure 12.**

*The mean spanwise longitudinal vortex intensity at a velocity of 0.4 m/s for the case of (a) one-pair RWP; (b) one-pair of CRWP; (c) two pairs RWP; (d) two-pairs CRWP; (e) three pairs RWP; (f) three-pairs CRWP.*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular… DOI: http://dx.doi.org/10.5772/intechopen.96117*

$$U = D\_h |\boldsymbol{\alpha}^n| = D\_h \left| \frac{\partial \boldsymbol{w}}{\partial \mathbf{y}} - \frac{\partial \boldsymbol{v}}{\partial \mathbf{z}} \right| \tag{22}$$

where *ω<sup>n</sup>* is the vortices about the normal axis of the spanwise plane. The mean longitudinal vortex intensity in the spanwise plane at position x (*Se*x) is defined by Eq. (24)

$$\text{Se}\_{\mathfrak{x}} = \frac{\rho D\_h^2}{A(\mathfrak{x})\mu} \iint\_{A(\mathfrak{x})} |w^n| dA \tag{23}$$

**Figures 12** and **13** show the ratio of *Se*<sup>x</sup> in RWP and CRWP cases at a velocity of 0.4 m/s and 2.0 m/s. In general, CRWP insertion produces a greater longitudinal

**Figure 13.**

*The mean spanwise longitudinal vortex intensity at a velocity of 2.0 m/s for the case of (a) one-pair RWP; (b) one-pair of CRWP; (c) two pairs RWP; (d) two-pairs CRWP; (e) three pairs RWP; (f) three-pairs CRWP.*

**3.2 Longitudinal vortex intensity**

*Heat Transfer - Design, Experimentation and Applications*

in Eq. (22)

**Figure 12.**

**66**

The longitudinal vortex intensity is a dimensionless number studied by K Song et al. [30] and represents the magnitude of the inertia force induced by secondary flow to the viscous force. In this study, the longitudinal vortex intensity is defined

> *Se* <sup>¼</sup> *<sup>ρ</sup>DhU μ*

*The mean spanwise longitudinal vortex intensity at a velocity of 0.4 m/s for the case of (a) one-pair RWP; (b) one-pair of CRWP; (c) two pairs RWP; (d) two-pairs CRWP; (e) three pairs RWP; (f) three-pairs CRWP.*

where *Se* is the longitudinal vortex intensity, and U is the secondary flow velocity characteristic, which can be formulated in the following equation:

(21)

vortex intensity than that of RWP because the frontal area of the CRWP is larger than that of the RWP and due to the instability of centrifugal force when the flow passes over the CRWP surface [19, 31]. The longitudinal distribution of the vortex intensity is shown in **Figure 14** for a velocity of 0.4 m/s and **Figure 15** for a velocity of 2.0 m/s. In the case of CRWP and RWP, the longitudinal vortex intensity tends to dissipate after passing VGs due to viscous effects [2, 26, 28]. Therefore, the installation of the second and third rows of VG reinforces the intensity of the longitudinal vortex as illustrated in **Figures 12(c)–(f)** and **Figures 13(c)–(f)** for velocities of 0.4 m/s and 2 m/s, respectively.

**3.3 Temperature distribution**

*DOI: http://dx.doi.org/10.5772/intechopen.96117*

**Figure 16.**

**Figure 17.**

**69**

The temperature distribution for the RWP and CRWP cases with/without holes and the baseline in the spanwise plane at a certain position with a velocity of 2.0 m/s is shown in **Figures 16** and **17**. Visually, the temperature distribution in the channel in the presence of VG is better than the baseline. The placement of VG in the channel increases the temperature distribution due to the counter-rotating pairs of longitudinal vortices, which result in increased fluid mixing [32]. Counter-rotating pairs of longitudinal vortices produce a downwash that pushes the fluid towards the surface of the heated plate resulting in increased local heat transfer coefficients and thinning of the thickness of the thermal and dynamic boundary layers [32, 33].

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular…*

Meanwhile, counter-rotating pairs of longitudinal vortices also generate upwash on the outer side of the vortex and push the hot fluid on the plate wall towards the flow-stream resulting in a decrease in the local heat transfer coefficient and a

*Temperature distribution in channel with: (a) RWP without holes; (b) RWP with holes.*

*Temperature distribution in channel with: (a) CRWP without holes; (b) CRWP with holes.*

The hole in the VG results in a decrease in the intensity of the longitudinal vortex, as shown in **Figures 12**–**15**. The hole in VG causes jet flow formation, which can interfere with the generation of the longitudinal vortex [26]. For RWP VGs with a velocity of 2.0 m/s, the intensity of the longitudinal vortex experiences the highest decrease, namely 17% at x/L = 0.48 for the case of one pair with holes, 11% at x/L = 0.4 for the case of two pairs with holes and 13% at x/L = 0.48 for the case of three pairs with holes of ones without holes. Meanwhile, in the case of CRWP VGs with a velocity of 2.0 m/s, the intensity of the longitudinal vortex experiences the highest decrease, namely 35% at x/L = 0.48 for the case of one pair with holes, 14% at x/L = 0.68 for the case of two pairs with holes and 22% at x/L = 0.68 for the case of three pairs with holes compared to ones without holes.

#### **Figure 14.**

*The longitudinal vortex intensity for the case of three pairs of RWP and CRWP at locations x / L = 0.34 and x/L = 0.32 at a velocity of 0.4 m/s, respectively.*

#### **Figure 15.**

*The longitudinal vortex intensity for the case of three pairs of RWP and CRWP at locations x / L = 0.34 and x/L = 0.32 at a velocity of 2.0 m/s, respectively.*

*Numerical Investigation of Heat Transfer and Fluid Flow Characteristics in a Rectangular… DOI: http://dx.doi.org/10.5772/intechopen.96117*
