# Turbulent Flow Around Obstacles: Simulation and Study with Variable Roughness

Turbulent Flow Around Obstacles: Simulation and Study with Variable Roughness
MECACOMP Laboratory at the University of Tlemcen, P.O. Box 230, Chetouane, Tlemcen 13000, Algeria

Corresponding Author Email:
kh_aliane@yahoo.fr
Page:
1659-1666
|
DOI:
https://doi.org/10.18280/ijht.390530
12 May 2021
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Accepted:
20 July 2021
|
Published:
31 October 2021
| Citation

OPEN ACCESS

Abstract:

The present work aims to investigate the recirculation and incipient mixing zones in a channel flow supplied with obstacles. The main objective is to develop a new technique to control these recirculation zones by setting a variable roughness. For the purpose of varying that roughness, 4 small bars of heights 0.25H, 0.5H, 0.75H and H were placed downstream of the obstacle; H is the height of the obstacle. For this, a three-dimensional numerical approach was carried out using the ANSYS CFX computer code. In addition, the governing equations were solved using the finite volume method. The K-ω shear-stress transport (SST) turbulence model was utilized to model the turbulent stresses. In the end, we presented the time-averaged simulation results of the contours of the current lines (3D time-averaged streamlines, trace-lines), three components of the velocities: <u> (velocity u contour), <v> (velocity v contour) and <w> (velocity w contour), trace-lines, stream ribbons and mean Q-criterion iso-surface.

Keywords:

turbulent flow, variable roughness, obstacle, finite volume, ANSYS-CFX

1. Introduction

2. Problematic

The present work aims to investigate the influence of the variable roughness that is placed downstream of an obstacle on the occurrence of mixing and circulation zones in the region that is closest to the obstacle.

3. Mathematical Formulas

The averaged mass and momentum conservation equations are expressed as:

• Mass conservation:

$\frac{\partial \rho }{\partial t}+\frac{\partial \left( \rho {{U}_{i}} \right)}{\partial {{x}_{i}}}=0$      (1)

• Momentum conservation:

$\frac{\partial {{U}_{i}}}{\partial t}+{{U}_{j}}\frac{\partial {{U}_{i}}}{\partial {{x}_{j}}}=-\frac{1}{\rho }\frac{\partial P}{\partial {{x}_{i}}}+v\frac{{{\partial }^{2}}{{U}_{i}}}{\partial {{x}_{j}}\partial {{x}_{j}}}$   (2)

In addition, the Reynolds equation is expressed as:

$\rho \frac{\partial {{{\bar{U}}}_{i}}}{\partial t}+\rho {{\bar{U}}_{j}}\frac{\partial {{{\bar{U}}}_{i}}}{\partial {{x}_{j}}}=-\frac{\partial \bar{P}}{\partial {{x}_{i}}}+\frac{\partial }{\partial {{x}_{j}}}\left( \mu \frac{\partial \overline{{{U}_{i}}}}{\partial {{x}_{j}}}-\rho \overline{{{u}^{\prime }}iu_{j}^{\prime }} \right)$     (3)

The SST (Shear Stress Transport) k- ω turbulence model of Menter  was used in this study; it was derived from the Standard k-ω model . This model combines the robustness and accuracy of the k-ω model in the near-wall region of the channel with the k-ε model  and all free-flow related types, away from the wall. The SST k-w model is mainly recommended for applications where the fluids undergo sudden stress changes and flow around curved surfaces, or in the case of boundary layer separation. Therefore, the SST k-omega turbulence model may be considered as ideal for our simulation.

The turbulent viscosity was modified so as to take into account the transport of turbulent shear flow.

The formulation of the two-equation turbulence model is:

$\rho \frac{\partial k}{\partial t}+\rho {{\bar{U}}_{j}}\frac{\partial k}{\partial {{x}_{j}}}={{\tilde{P}}_{k}}-\rho {{C}_{\mu }}\omega k+\frac{\partial }{\partial {{x}_{j}}}\left[ \left( \mu +{{\mu }_{t}}/{{\sigma }_{k}} \right)\frac{\partial k}{\partial {{x}_{j}}} \right]$     (4)

The specific dissipation rate is given as:

\begin{align} & \rho \frac{\partial \omega }{\partial t}+\rho {{{\bar{U}}}_{j}}\frac{\partial \omega }{\partial {{x}_{i}}}=2\alpha \rho {{S}_{ij}}{{S}_{ij}}-\beta \rho {{\omega }^{2}}+ \\& \frac{\partial }{\partial {{x}_{j}}}\left[ \left( {{\mu }_{t}}+{{\sigma }_{\omega }}{{\mu }_{t}} \right)\frac{\partial \omega }{\partial {{x}_{j}}} \right]+2\left( 1-{{F}_{1}} \right)\rho {{\sigma }_{\omega 2}}\frac{1}{\omega }\frac{\partial k}{\partial {{x}_{j}}}\cdot \frac{\partial \omega }{\partial {{x}_{j}}} \\\end{align}      (5)

As for the blending function F1, it is defined using the following equation:

${{F}_{1}}=\tanh \left\{ {{\left\{ \min \left[ \max \left( \frac{\sqrt{k}}{{{C}_{\mu }}\omega L},\frac{500v}{{{L}^{2}}\omega } \right),\frac{4\rho {{\sigma }_{\omega 2}}k}{C{{D}_{k\omega }}{{L}^{2}}} \right] \right\}}^{4}} \right\}$       (6)

In the near-wall region, F1=1; however it goes to zero in the outer region $C D_{k \omega}$ that is given as:

$\mathop{CD}_{k\omega }=\max \left( 2\rho \mathop{\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\mathop{\partial x}_{j}}.\frac{\partial \omega }{\mathop{\partial x}_{j}},\mathop{10}^{-10} \right)$     (7)

Moreover, the eddy viscosity is expressed as:

${{v}_{t}}=\frac{{{\alpha }_{1}}k}{\max \left( {{\alpha }_{1}}\omega ,\sqrt{2}{{S}_{ij}}{{F}_{2}} \right)}$    (8)

The second blending function F2 is defined by the following equation:

${{F}_{2}}=\tanh \left[ {{\left[ \max \left( \frac{2\sqrt{k}}{{{C}_{\mu }}\omega L},\frac{500v}{L{{\omega }^{2}}} \right) \right]}^{2}} \right]$     (9)

To prevent the accumulation of turbulence in the stagnation regions, it was decided to limit the production of turbulence:

${{\tilde{P}}_{k}}=\min \left( {{P}_{k}},10.{{C}_{\mu }}\rho k\omega \right)$     (10)

${{P}_{k}}={{\mu }_{t}}\frac{\partial {{U}_{i}}}{\partial {{x}_{j}}}\left( \frac{\partial {{U}_{i}}}{\partial {{x}_{j}}}+\frac{\partial {{U}_{j}}}{\partial {{x}_{i}}} \right)$     (11)

The model constants are calculated using the blending function below:

$F1:\phi ={{F}_{1}}{{\phi }_{1}}+\left( 1-{{F}_{1}} \right){{\phi }_{2}}$     (12)

The values of the model constants are: $C_{\mu}=0.09, \alpha_{1}=5 / 9$, $\alpha_{2}=0.44, \beta_{2}=0.0828, \sigma_{\mathrm{k}_{1}}=0.85, \sigma_{\mathrm{k}_{2}}=1.0, \sigma_{\omega_{1}}=0.5$ and $\sigma_{\omega_{2}}=0.856$.

4. Model Description and Computational Domain

The geometric model used in this work is represented by a cubic-shaped obstacle of height (H), as shown in Figure 1. A variable roughness was added downstream of the obstacle in order to disturb the boundary layer at that position and enhance the dynamic and thermal exchanges. The obstacle and the variable roughness were placed in a horizontal channel of length (11H), height (2H) and width L = 7H. The variable roughness consisted of four small bars. Note that the length of each bar is equal to the channel width (7H). These bars all have the same width (0.5H). Their heights are successively equal to (0.25H), (0.5H), (0.75H) and (H). The distance between two successive bars is equal to (0.5H), as depicted in Figure 2.

## 1.png Figure 1. Three-dimensional view of the computational domain of the geometry

## 2.png Figure 2. Geometry of computational domain and boundary conditions

5. Boundary Conditions

As the flow is turbulent, the SST k-ω turbulence model was chosen to analyse the problem. In this case, all walls were taken as adiabatic, with no-slip conditions. According to the previously selected model, the equations should be solved using the following parameters: The incoming flow velocity U0 corresponds to the Reynolds number 8.104 (Re = U0.h/) and the channel height (h). The height of the obstacle was H = 25 mm, and the channel height was h = 2H. The velocity was zero (u = 0 m/s) near the lower and upper walls of the channel and above the obstacle. Moreover, the roughness was varied. A constant pressure was imposed at the outlet of the channel such that Pout = 0.

For the purpose of achieving highly accurate results, it was necessary to generate a well refined mesh. For this, it was deemed interesting to opt for a structured hexahedral mesh. The meshing of the domain was performed using ANSYS CFX. The general practice consists of using a fine mesh size in regions where the domain geometries undergo small changes. Figure 3 shows the mesh grid used.

## 3.png Figure 3. The mesh grid configuration

The grid independence was investigated using a series of simulation tests within the calculation domain corresponding to the velocity under consideration.

Table 1. Different meshing sizes

 L/H h/H Grid Configuration 1 7 2 685020 Configuration 2 7 2 865274 Configuration 3 7 2 987100

## 4.png Figure 4. Testing the grid sensitivity

For this, several mesh sizes were tested (Figure 4) in order to attain results that are independent of the number of meshes. Consequently, three meshes with hexahedral elements and sizes 685020, 865274 and 987100 were used. The values reported in Table 1 correspond to the situation where x /H= 0.5. The findings of this study showed that there are comparatively small differences between the three grids. Eventually, the mesh including 865274 elements was chosen as the best solution with regard to precision and calculation time.

6. Flow Structures

The flow above the wall-mounted cube (without roughness) engenders complex phenomena such as horseshoe vortices and recirculation zones on the top and back of the cube, as illustrated in Figure 5. It is worth noting that the presence of the obstacle causes the fluid flow to separate near the upstream face of the obstacle. The flow forms a recirculation zone near this face, which causes the turbulence intensity to augment. In addition to the separation of the fluid flow in the three zones (X1, X2 and X3), the fluid velocity also increases in this region because the cross-sectional area of the fluid passage gets narrower at that point. Note that the length of the recirculation zone on the upper face of the cube is represented by X2 in Figure 5. The flow separation takes place on the upstream and downstream edges of the cube. A large separation region develops downstream (X2) and upstream (X1) of the cube. The effect of the separation (X1) on the flow adds up to that of the horseshoe vortex. The length of this region is represented by X1.

## 5.png Figure 5. Flow pattern around an inclined cube attached to the wall with three vortex lengths

7. Results and Discussion

In this study, the experimental flow model carried out experimentally by Martinuzzi and Tropea (1993) was considered for the purpose of validating the characteristics of the flow over the surface-mounted cube (without roughness) placed in a channel of height h = 2H; the height of the cube was H = 25 mm. The structure of the flow around the obstacle was validated by the work of Hussein and Martinuzzi (1995) for a Reynolds number Re = 8.0×104 (Figure 6a). This part of the fluid, which is blocked between the lower wall of the channel and the front face of the obstacle, leads to zero speed in that region, which explains the appearance of this small recirculation zone. Upstream of the obstacle, part of the fluid remained blocked, forming a small recirculation zone (point (a) in the case of Hussein and Martinuzzi (1995) and point (a') in our case Figure 6b). Downstream of the obstacle, a large vortex appeared, as is clearly shown in the two figures (point (b) in the case of Hussein and Martunizzi and point (b') in our case). This was due to the strong depression of the flow in that region, which engendered a return current (vortex). The separation occurred above the obstacle; this was caused by the stopping point upstream of the obstacle (point of separation) (point (c) in the case of Hussein and Martunizzi and point (c') in our case). At that point, the flow accelerates due to the presence of the obstacle which leads to the constriction of the flow passage section. A simple comparison between the two simulations allowed concluding that the results obtained are satisfactory and encouraging.

## 6a.png (a) Hussein and Martinuzzi (1995)

## 6b.png (b) SST k-ω

Figure 6. Streamlines velocity on the symmetry plane for Re= 8.104

In this study, the influence of surface roughness on the flow structure was subsequently examined. For this, a three-dimensional study was carried out using the ANSYS CFX calculation code. The size of the computational domain is 11H × 7H × 2H. The inlet of the computational field, which is located at a distance of 3H upstream of the cube; note that the fully developed velocity profiles were used, with a Reynolds number (Reh=Ubh/ν) equal to 8.0×104. At the outlet of the canal, a constant pressure pout = pref was maintained. In addition, no-slip conditions were imposed at the solid walls (upper surface, lower surface, and cube). In this case, the side boundaries were considered as slip surfaces, while considering the symmetry conditions. The hexahedral structured meshes were employed for solving the fluid dynamics equations (Figure 3).

It is worth indicating that the SST K-ω turbulence model was used to study the characteristics of the flow around the cube with a rough surface for a Reynolds number Re = 8.104. The normalized time-averaged results of the longitudinal velocity <u>, and the transversal velocity components <v> and <w> on the symmetry plane (z = 0) around the obstacle are all presented in Figure 7.

The stream-wise velocity is clearly depicted in Figure 7. This figure allows observing that the velocity is low around the cube and along the roughness bars. This velocity increased and reached its maximum value downstream of the fourth and last bar in the region close to the upper wall. It is minimal at the exit of the channel in the lower zone of the channel. The cross-stream velocity component <v > for Re = 8.104 is explicitly shown in Figure 7. The maximum values of the cross-stream velocity were reached upstream of the cube where part of the fluid remained blocked due to the presence of the obstacle, which caused the velocity to increase. Likewise, it was observed that the velocity increased also above the first three roughness bars. The minimum values were more predominant downstream of the last bar and at the exit of the canal. This same Figure 7 also indicates that the velocity component <w> was very negligible, i.e. nearly equal to zero inside the major part of the canal, except downstream of the obstacle where the velocity was quite low.

Figure 8 was used to conduct an accurate analysis of the flow. Moreover, the flow separation and reattachment phenomena on the top of, at the lateral sides of, and behind the obstacle, as well as the time-averaged streamlines near the channel floor for the Reynolds number Re = 8.104 are clearly illustrated as well.

## 7a.png ## 7b.png ## 7c.png Figure 7. Components of velocity vectors <u>, <v> and <w> on the symmetry plane (z=0) at Re = 8.104

## 8.png Figure 8. Time-averaged streamlines at the floor of the channel at the Reynolds number of Re=8.104

It is worth indicating that the blocking effect due to the obstacle created an unfavorable pressure gradient that was responsible for the separation of the flow, and then moved away from the cube, forming a horseshoe vortex that appeared upstream and bypassed the obstacle; it is represented by the point (VHS). On the other hand, two reattachment points (R1, R2) were observed on the last roughness bar, as well as a separation point (S) upstream of the cube.

7.1 Trace-lines

Figure 9 shows trace-lines on the surface of the cube as well as on the surfaces of the four bars representing the roughness on the bottom wall of the channel. These trace-lines suggested that there was a regular flow on the upper faces of both the cube and the bars. A flow separation was observed at the middle of the front face of the obstacle, thus forming a node and a recirculation zone on the lateral faces of this cube. Similarly, a flow circulation zone was observed on the front faces of the four roughness bars.

7.2 Stream-ribbons

Figure 10 shows the stream-ribbons as well as a along with a large vortex and a recirculation zone downstream of the last roughness bar. A flow separation was also observed upstream of the cube due to the presence of the obstacle. It can therefore be concluded that the separation region around a three-dimensional body cannot be closed.

7.3 Mean Q-criterion iso-surface

The Q-criterion is a new way to visualize turbulent flows.

This criterion is a scalar invariant that is defined by the expression: $\mathrm{Q}=-\frac{1}{2} \mathrm{Ui}, \mathrm{j} \mathrm{Uj}, \mathrm{i}=-\left(\left\|\mathrm{S}^{2}\right\|-\left\|\Omega^{2}\right\|\right)$. In addition, the Q-criterion can be dimensioned using the expression: $\bar{Q}=$ $\mathrm{Q} /(\mathrm{Ub} / \mathrm{H}) 2 .$

## 9.png Figure 9. Trace-lines on the surface of the cube and the four roughness bars

## 10a.png (a) Front view

## 10b.png (b) Lateral view

## 10c.png (c) Rear view

Figure 10. Stream-ribbons

Figure 11 clearly depicts the three-dimensional structure of the flow, in particular the presence of a horseshoe vortex upstream of the cube, noted VHS, in addition to marginal vortices noted VM and wake vortex noted VS. One may also observe that the fluid was reflected immediately after the last roughness bar.

## 11.png Figure 11. Iso-outline of mean Q-criterion

8. Conclusion

This work was an attempt to analyze a three dimensional turbulent flow around an obstacle with a variable roughness that was located downstream of that obstacle. For this, an interactive three-dimensional approach was applied using the ANSYS CFX Calculation Code. In addition, the governing equations were solved using the finite-volume method. Moreover, the shear-stress transport (SST) K- ω turbulence model was utilized to study the characteristics of the flow around the cube with rough surfaces, for the Reynolds number Re = 8.104. This work sought to study the recirculation zones as well as the incipient mixing in a flow, in the presence of a cubic obstacle. The main purpose was to develop a new technique for controlling these recirculation zones by adding a variable roughness, represented by four bars of different heights, i.e. 0.25H, 0.5H, 0.75H and H, placed downstream of the obstacle on the bottom wall of the channel. The velocity was found quite low around the cube and the roughness bars. Moreover, it turned out that this velocity increased and reached its maximum value downstream of the fourth and last bar in the region close to the upper wall. Its minimal value was found at the exit of the channel, and in the lower zone of the channel. The maximum values of the cross-stream velocity were achieved upstream of the cube where part of the fluid remained blocked due to the presence of the obstacle. The blocking effect of the obstacle created an unfavorable pressure gradient which separated the flow and moved away from the cube; this allowed the formation of a horseshoe vortex upstream of the obstacle and then bypassed the obstacle; it is represented by the point (VHS). On the other hand, two reattachment points (R1, R2) were detected on the last roughness bar and a separation point (S) was observed upstream of the cube. Furthermore, a flow separation was noticed in the middle of the front face of the obstacle, which contributed to the formation of a node and a recirculation zone on the lateral faces of the cube. In addition, a circulation zone was observed on the front faces of the four roughness bars. Flow separation was noticed upstream of the cube due to the presence of the obstacle. Therefore, it can be concluded that the separation region around a three-dimensional body cannot be closed.

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