Convective heat transfer optimization with rib's deployment in a turbulent offset jet flow

During this work, we inspired the investigated geometry from Kim et al's (1996) experimental work, represented in (figure 1). The air was ejected from a two-dimensional nozzle, having dimensions of 800 mm × 20 mm (L × d) which gives an aspect ratio, A = L/d, of 40 and proves the two-dimensionality criteria of the flow. Besides the lower wall is subjected to a constant heat flux, 1951 W m⁻², and mounted with a rib, having a square section (20 mm × 20 mm). Hence, the chosen domain dimensions proved no effect on the flow development and behavior.

Figure 1 represents the 'no-rib' case, which is used to validate our numerical code, in terms of the mesh type and size and the turbulence model choices. h is defined as the offset distance and d as the nozzle width.

The RANS equations are written and solved in the Cartesian (Ox, Oy) coordinate system, with the adoption of these following working hypothesis:

• The flow is considered turbulent and fully developed.
• The air is the working fluid.
• The heat flux is assumed to be constant all along the bottom wall.
• The flow is stationary and two-dimensional.
• The air is ejected in the longitudinal direction.

The averaged mass conservation, momentum and energy equations, of an incompressible fluid, are expressed in their general form in a Cartesian coordinate as follows:

$$\begin{equation}\frac{\partial }{{\partial {x_i}}}\left( {{u_i}} \right) = 0\end{equation} \tag{ 1 }$$

$$\begin{equation}\frac{\partial }{{\partial {x_j}}}\left( {{u_i}{u_j}} \right) = - \frac{1}{\rho }\frac{{\partial p}}{{\partial {x_i}}} + \frac{1}{\rho }\frac{\partial }{{\partial {x_j}}}\left[ {\mu \left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}} - \frac{2}{3}{\delta _{ij}}\frac{{\partial {u_l}}}{{\partial {x_l}}}} \right)} \right] + \frac{\partial }{{\partial {x_j}}}\left( { - \overline {{{u^\prime}_i}{{u^\prime}_j}} } \right)\end{equation} \tag{ 2 }$$

$$\begin{equation}\frac{{\partial ({C_p}T)}}{{\partial t}} + \frac{{\partial ({C_p}{U_i}T)}}{{\partial {x_j}}} = \frac{\partial }{{\partial {x_j}}}\frac{1}{\rho }\left( {\frac{{\lambda \partial T}}{{\partial {x_j}}}} \right) + S\end{equation} \tag{ 3 }$$

Equation (2), presents an additional term that reflects the effect of turbulence: It is the Reynolds stress constraint. This constraint must be modeled in order to close equation (2). By exploiting the Boussinesq approximation, a relationship between the shear stress and the mean velocity gradient appears as follow:

$$\begin{equation} - \overline {{{u^\prime}_i}{{u^\prime}_j}} = \frac{{{\mu _t}}}{\rho }\left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}}} \right) - \frac{1}{\rho }\frac{2}{3}\left( {\rho k + {\mu _t}\frac{{\partial {u_k}}}{{\partial {x_k}}}} \right){\delta _{ij}}\end{equation} \tag{ 4 }$$

Thus, for the closure of the whole equations system, four turbulence models, standard k-ω, SST k-ω, RNG k- and Realizable k- were tested. Only the SST k-ω turbulence model proved its accuracy in resolving this flow. A comparison test will be presented in the results section in order to highlight that the SST k-ω model presented the best results compared to the other models, (figure 4). Actually, the SST k-ω combines between the standard k- model in the near-wall region and the standard k-ω model in the far field region, which gives more consistent results and overcome the limits of each model used separately. In fact, to blend these two models, two blending function F1 and F2 were used. In this case the transport equations of k, the turbulent kinetic energy, and ω, the specific dissipation rate, are given as follows:

$$\begin{equation}\frac{\partial }{{\partial {x_i}}}\left( {\rho k{u_i}} \right) = \frac{\partial }{{\partial {x_j}}}\left( {{\Gamma _k}\frac{{\partial k}}{{\partial {x_j}}}} \right) + {\mathop G\limits^ \sim _k} - {Y_k} + {S_k}\end{equation} \tag{ 5 }$$

$$\begin{equation}\frac{\partial }{{\partial {x_i}}}\left( {\rho \omega {u_i}} \right) = \frac{\partial }{{\partial {x_j}}}\left( {{\Gamma _\omega }\frac{{\partial \omega }}{{\partial {x_j}}}} \right) + {G_\omega } - {Y_\omega } + {D_\omega } + {S_\omega }\end{equation} \tag{ 6 }$$

The generation of turbulence kinetic energy is modeled with the term ${\mathop G\limits^ \sim _k}$, while G_ω represents the generation of ω. ${\Gamma _k}$ and ${\Gamma _\omega }$ represent the effective diffusivity of k and ω, respectively, which are calculated as described below. Y_k and Y_ω represent the dissipation of k and ω due to turbulence. D_ω term represents the cross-diffusion term while S_k and S_ω are the user-defined source terms.

Returning to the effective diffusivity, it is modeled as follows:

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

$$\begin{equation}{\Gamma _\omega } = \mu + \frac{{{\mu _t}}}{{{\sigma _\omega }}}\end{equation} \tag{ 8 }$$

The SST turbulence model details are found in Ansys Fluent user guide (2013).

On the first hand, the boundary conditions, figure 2 gathers all the different used boundary conditions with their physical significations. As for the emission conditions, a uniform velocity profile was applied at the nozzle inlet, U0 = 28.47 m s⁻¹, with a constant turbulence intensity of 0.01% and a Reynolds number of 39 000, (Kim et al 1996), which was calculated based on the nozzle width. Meanwhile, the turbulence dissipation rate, ω, was estimated basing on the hydraulic diameter as a length scale parameter. With regards heat flux, as it was assumed to be constant and applied uniformly on at the channel's base.

On the other hand, the choice of the mesh size and form is a mandatory phase to valid in order to guarantee a credible resolution of our problem's governing equation. As the geometry is not complicated, we adopted a quadratic non-uniform mesh. We ensured that the grid is enough fine in the vicinity of the jet nozzle, in the near rib-region and the heated bottom wall, and a quasi-loose elsewhere. Figure 3 represents a caption of the mesh distribution near the rib. As it is shown in figure 3, the mesh is well refined near the rib and gets coarser moving downstream and upstream.

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As the work was purely numerical, it was mandatory to validate the numerical CFD software we used, in terms of its reliability of the resolution of such flows. There so, comparisons between numerical results and experimental ones, of (Kim et al 1996), will be presented in this section.

The first step is to choose the adequate mesh type and size that gives an accurate resolution of the flow behavior. There so, figure 4 demonstrates the comparison between the numerical results of three different meshes against the experimental data of (Kim et al 1996), at three OR values. These meshes were assigned respectively as Mesh1, Mesh2 and Mesh3 each having 11 875, 33 600 and 52 332 non- uniform and quadratic cells. Refereeing to figure 4, we may see that Mesh1 presents huge error compared to the experimental values, especially for OR equal to 6.5 and 7. Meanwhile, Mesh2 and Mesh3 give approximately the same Nusselt pattern, that nearly coincides with the Nusselt experimental profile. There so, Mesh2 is adopted in the rest of the work, as it offers enough accurate resolution of this problem, and any increase of the mesh size would not have a significant amelioration effect on the predicted results. In fact, Mesh2 precision was found to be up to 97%.

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In addition to the mesh size choice, knowing the most reliable turbulence model was equally important. In this context, four different models were tested. This second validation phase was conducted to prove the efficiency of one particular turbulence model in the resolution of such a flow. In fact, with the adoption of the same emission conditions of (Kim et al 1996)'s experimental work and for a constant OR, at 7,5, we tested four different turbulence models: RNG k-, Realizable k-, Standard k-ω and SST k-w.

On the first hand, figure 5 highlights the precision of the SST k-ω in the detecting of the local Nusselt values, at the bottom wall, compared to all the other three models. Actually, as it was previously mentioned, in the 'Governing Equation' section, this particular model blends between the Standard k-ω and the Standard k- turbulence models, which makes it more reliable. On the other hand, this figure demonstrates Fluent's capability to solve this problem, as the numerical results error does not even reach 3%. Basing on these results, the SST k-ω is used to conduct the rest of the simulations.

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Overall, the SST k-ω turbulence model and a non-uniform mesh, of 33 600 quadratic cells, will be used to solve numerically the dynamic and thermal behavior of this flow.

In this section, a detailed analysis of the rib position effect on the flow development and on the mean flow structures is presented. As it is previously mentioned, the investigated configuration is demonstrated in figure 3, in which stands a rectangular rib on the heated bottom wall. For all the numerical simulations, the OR was maintained at 5 while the Reynolds number was fixed at 39 000, similarly to Kim et al (1996) experimental work. However, the rib's position was the only varying parameter.

Actually, 9 different rib positions were tested and investigated. The rib position, X_rib, is expressed as follow:

$$\begin{equation}{X_{{\text{rib}}}} = \frac{{{x_{{\text{rib}}}}}}{{{x_{{\text{rp0}}}}}}\end{equation} \tag{ 9 }$$

Having x_rib as the obstacle first edge position and x_rp0 as the reattachment point for the no-rib case. In addition, the dimensionless reattachment point position, X_rp0, was found to be 11.3. Table 1 gathers all the tested cases.

In order to visualize the flow pattern for these different cases, the downstream velocity (U) contours and stream traces were plotted. First, figure 6 demonstrates the flow development for the four first cases. At a first sight, we may say that the rib presence changes significantly the dynamic behavior of this flow.

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For X_rib = 0, stand two different sized and counter-rotating vortices. The first one lies just above the top of the rib, having a counter-clockwise rotation direction. Meanwhile, throughout the rest of the recirculation region stands the main clockwise rotating vortex that extends to almost X_rp0, the same reattachment point as the 'non-rib' case.

However, for X_rib = 0.25, three vortices were found to be present. It is like that the obstacle divided the recirculation region into two regions with comparable sized counter-rotating vortices, that lie in the upstream and downstream rib areas. Further an additional horse-shoe vortex, the smallest one, stands on the top of the obstacle. It is important to mention that the final reattachment point position moves further downstream when X_rib = 0.25 and reaches a value of X_rp = 12.3, which makes the downstream vortex more elongated compared to the upstream one.

Figure 6(c) demonstrates a totally different behavior of the flow essentially in the recirculation region, for X_rib = 0.5. This figure shows the presence of two counter-rotating vortices, having different pattern. The first vortex occupies almost one third of the recirculation region, having a counter-clockwise rotation direction and lies from the nozzle-wall to the rib's upper surface with a high of around H/2. The second vortex is a double-centered, having a deflection point at the rib's right edge position. Even the huge effect of the rib presence, the reattachment point position returns to coincide, almost, at Xrp0. For this case, it was found that the negative velocity values and region are minimum compared to the previous cases.

Regarding X_rib = 0.75, the flow pattern changes again and presents three distinguishable vortices. The main one is situated between the nozzle wall and the first edge of the rib. This vortex is the most elongated and characterized with the most negative velocity values and with a counter- clockwise direction. Further the fluid attaches the upper edge of the rib at X_rp inferior to X_rp0. The other two vortices bounded the rib from both sides, left and right. The flow development downstream the rib is similar to a flow behavior facing a step. Hence the vortex attaches the heated wall at a position of X_rp = 11.1, which is slightly less than X_rp0. Figure 7 represents also the velocity contours and stream-traces corresponding to the rest of the rib location cases, _Xrib in the range from 1 to 1.75. First for X_rib = 1 and X_rib = 1.25, the flow development behavior is similar to a flow facing step behavior, in which two distinguishable vortices upstream and downstream the rib is present. Actually, for these two cases, the fluid attaches the heated wall at the first facing edge of the rib. However, the recirculation region for X_rib = 1.25 is more elongated compared to X_rib = 1. Regarding the downstream vortices, they have almost the same dimensions.

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Regarding the last two rib positions, the flow development is significantly different from the other cases. Hence, figures 7(c) and (d) highlight the existence of, considerably large, two co-rotating vortices. The first one stands in the recirculation region, having a reattachment length equal to X_rp0 which means that the rib presence has no longer effect on the development of the pre-attachment region. The second one extends from the upper rib edge to X = 30 and X = 37, respectively for X_rib = 1.5 and X_rib = 1.75, which gives dimensionless attachment lengths, Le, of 12 and 16 respectively. There so, we may say that Le increases the downstream movement of the rib when X_rib ⩾ 1. Because it was found to be 2 and 2.8 for X_rib = 1 and X_rib = 1.25, respectively.

In this section, a qualitative and detailed presentation of the turbulent kinetic energy for all the range of the rib positions is communicated in figures 8 and 9. The first four captions show the TKE distributions for X_rib going from 0 to 0.75. At a first sight, we see that the TKE contours change significantly with the variation of the rib position, we adopted the same scaling system for all the cases.

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Figures 8(a)–(c) represent qualitatively almost the same phenomenon and distributions. In fact, for these cases, the upper and lower shear layers correspond to high TKE, which is predictable because of the presence of important friction forces that bound the potential core. Further, the counter-clockwise rotating vortices are characterized with the lowest TKE, unlike the main vortex, which is resulted from the fluid recirculation, that is characterized with higher TKE that increases in the downstream direction till the reattachment point. From which, it gets on decreasing.

However, for X_rib = 0.75, the maximum TKE appear to coincide with the rib position because the fluid attaches the wall just at the left upper limit of the rib. Elsewhere, in the recirculation region a progressive increase of TKE is seemed, which is contrary its behavior starting from the attachment position of the downstream rib vortex, that presents a sudden reduce in the TKE due to the fluid blocking.

Hence, figure 9 represents the TKE contours for the rest of the examined cases. The first two captions show nearly the same TKE distribution, that gets in increasing throughout the recirculation region till the reattachment point, from which it returns to decline due to the presence of the downstream rib vortex.

Meanwhile, figures 9(c) and (d) demonstrate the rib presence effect on the TKE's cartography when it is placed in the impingement region. All along the recirculation region the same phenomenon was seen, compared to figures 9(a) and (b). However, downstream the rib the minimum TKE's values' area gets elongated

Examining the mean flow structures and the TKE distributions can be related to the heat transfer behavior of this particular flow. So, we may relate the thermal characteristics ultimately to the dynamic features and expressions of the flow. Hence, after passing the first phase that deals with the overall flow hydrodynamic response, an investigation of the heat exchange phenomenon comes after in the next sections.

The main purpose of this present work is the optimization of the convective heat transfer of a turbulent offset jet, with the use of a rib mounted on the heated-bottom wall. There so, 9 different rib positions were tested in order to figure out the optimum one, in terms of the amelioration of the heat exchange between the heated wall and the cold fluid. Hence, a detailed thermal analysis is presented, in this section, throughout the exploration of the local Nusselt number, Nu_x, paces and the temperature contours.

As a first step, the local Nu distributions are presented in figures 10–12, from which comparisons of the heat transfer performance are demonstrated for all the range of X_rib. In fact, the Nu_x were plotted against the 'no rib' case Nu_x, to see if the used X_rib enhances the heat transfer or do not. These figures illustrate how the Nu paces change significantly with the variation of X_rib. Actually, it is obviously seen that for each three successive rib positions the Nu pace presents a different shape. Regarding the temperature contours, the same scaling system is adopted for all the captions.

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First, for X_rib between 0 and 0.5, figure 10(a) shows that the local distributions of Nu, calculated on the heated bottom wall, reflect almost a common shape with the no-rib case Nu pace, essentially in the post-reattachment region and they peak nearly at the same position, which is found to be the reattachment point position where stands the largest TKE value, figure 8, that intensifies the heat transfer phenomenon. Despite this similarity, a noticeable difference is detected near the rib area, where Nu was found to increase slightly and reach a certain local peak, from which it drops till a local minimum that corresponds to the first facing edge. In fact, at both left and right sides of the rib, different Nu values were detected that correspond to different temperatures presented in figures 10(b)–(d). This fact is owing to the near-rib vortices that prevent hot fluid, at the wall lowest limit, from exchanging heat with the colder fluid by blocking it. So, every increase of Nu, in figure 10(a), is equivalent to a decrease of the superficial detected temperature in figures 10(b)–(d).

Moreover, figure 10(a) highlights that only X_rib = 0, provides a slight enhancement in the heat transfer coefficient. Consequently, when a rib is placed at X_rib = 0.25 or X_rib = 0.5, the convective heat transfer gets less intensified and this is owing to the existence of considerably large counter- clockwise rotating vortices, in the recirculation region, which blocks the hot fluid from communicating with the colder fluid. However, throughout the rest of the recirculation region, Nu increases in the downstream direction till the reattachment point, from which it changes the variation trend.

On the other hand, for 0.75 < X_rib < 1.25, the Nu distributions, in figure 11(a), have shown a dramatic change compared to the no-rib case, in terms of their variation trends. For all these cases the flow attaches the wall around the rib positions, X_rib, as it was described above. Overall, from the nozzle wall up to X_rib, Nu presents an upward trend that decreases in intensity with the downstream movement of the rib. After that, Nu decreases, due to the presence of a small vortex standing just upstream the rib, which reduces the amount of the exchanged heat flux to a local minimum situated at the lowest part of the first facing edge. However, on the same edge appear different values of Nu, that correspond to different temperature values in figures 11(b)–(d), including the global maximum value, as the flow attaches the wall around X_rib with a greater TKE that enhances the heat transfer phenomenon. All along the rib's upper and right edges, Nu declines significantly and hit a local minimum value and then it returns to rise once again up to the attachment point of the rib's downstream vortex. In fact, when the fluid passed the rib, it is like the behavior of an offset jet flow, which explains this behavior. Finally, in the impingement and wall jet region the heat transfer gets in decreasing in the (OX) direction.

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Regarding now the effect of the rib presence, figure 11 provides all the information about the performance of each case separately. Starting from the nozzle wall to X_rib, only when X_rib = 0.75 the convective heat transfer is enhanced, meanwhile for the remaining cases the local Nu distributions were found to be underneath the local Nu distribution for the no-rib case, which is not the case in the downstream rib area. Furthermore, the global maximum convective heat transfer coefficient climbs with X_rib. In the impingement and wall jet regions, no considerable amelioration is noticed.

Concerning the last three positions, in which the rib is placed far away downstream the recirculation region, figure 12 represents similarly the local Nu distributions in addition to the temperature contours corresponding to investigated X_rib. Once again, the Nusselt patterns have remarkably changed, compared to the other X_rib and the no-rib case. However, the Nu variations seem to have the same paces, qualitatively, in all the range of X_rib between 1.25 and 2, in other words when the rib is located in the development region.

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A similar behavior of all the local Nusselt distributions appears between the nozzle wall up to the reattachment points, which is owing to the rib's emplacement that has no effect on the flow development in the recirculation region, as described above. Furthermore, in this region, no heat transfer amelioration has resulted from the rib use. In fact, when the cold air passes the recirculation region, a dramatic fall of the Nu looms up having a slope that decreases in intensity with the downstream movement of the obstacle. This increase makes it up to the first rib facing edge, on which different Nu and temperature values were detected for the same reasons as the previous cases. Just after passing the rib, and all along the clockwise rotating downstream rib vortex, the heat transfer coefficient gets in increasing and peaks around the attachment point, Le, just as the behavior of a typical offset jet. Even though, according to the comparison with the no-rib case, facts have shown that this range of X_rib provides no heat transfer amelioration, with the exception of the attachment position at which the local Nu distributions were found to be slightly above the Nusselt (Nu) distribution for the no-rib case.

So far, the convective heat transfer has been found enhanced in some cases and in some local positions, there so a call to averaged and maximum parameters would be useful in order to judge which is the optimum rib position that offers the heights amelioration rate. Hence, averaged and maximum Nusselt numbers are question of comparison in this section. As for the maximum Nu, figure 13 demonstrates comparisons between the no-rib case and the other investigated cases. The examination of this characteristics would give information about the maximum exchanged convective heat transfer in each case and its location, see figure 14.

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The maximum Nusselt value, in other words the heat transfer coefficient h, is found to be enhanced considerably for X_rib = 0.75, X_rib = 1 and X_rib = 1.25. Moreover, this amelioration rises with the downstream movement of the rib and gets 29%, 45% and 52% of the maximum no-rib case coefficient respectively. The determination of the position of these maximum values is equally important. That is why the dimensionless Numax positions, X_Numax' = X_Numax/X_rp0, were plotted against the rib position in figure 14. This figure highlights the fact that Numax appears around X_rp0, except for X_rib = 0.75 and X_rib = 1.25, where Numax position coincides with X_rib. In fact, these locations are characterized with the maximum TKE that intensifies the heat transfer and makes it maximum.

However, in order to get the appropriate choice of which X_rib gives the best heat transfer, the averaged heat transfer performance was examined. Therefore, figure 15 reveals comparisons between the averaged Nu values obtained for each case. Obviously, only the fourth and the fifth X_rib guaranteed a heat transfer enhancement with just above 9% and around 4%, respectively, compared to the averaged heat transfer coefficient of the no-rib case.

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Accordingly, the adequate choice of the optimum X_rib depends on the application's aim. If the target is the get a maximum local heat transfer, then X_rib = 1.25 would be the appropriate choice as it provides the maximum heat transfer coefficient. Otherwise, X_rib = 0.75 would be the chosen rib position as it offers the optimal overall averaged convective heat transfer performance.