Omri et al 2019a - Revision history

Rimni at 14:59, 24 February 2021

2021-02-24T14:59:44Z

Rimni at 14:55, 24 February 2021

2021-02-24T14:55:50Z

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Rimni: /* 2.2. Numerical model */

2021-02-24T13:44:30Z

‎2.2. Numerical model

Rimni at 13:43, 24 February 2021

2021-02-24T13:43:24Z

Rimni at 08:45, 8 May 2020

2020-05-08T08:45:49Z

Rimni at 08:44, 8 May 2020

2020-05-08T08:44:42Z

Rimni at 08:42, 8 May 2020

2020-05-08T08:42:50Z

Rimni at 08:41, 8 May 2020

2020-05-08T08:41:52Z

Medd at 10:33, 30 April 2020

2020-04-30T10:33:16Z

Rimni at 12:29, 15 April 2020

2020-04-15T12:29:38Z

@@ Line 426: / Line 426: @@
-[[#img-15|Figure 15]] indicates vertical variations of velocity <math>x</math>-component at different positions along <math>x</math>-axis (<math>x=0</math>, 4.75, 12.25, 19.75, 24.75) at mid-height and mid-width of the channel (Positions are <math>P_{xi}</math> as indicated in Figure 10). For the two last positions, they were in the medium of the two micro-channels.
+[[#img-15|Figure 15]] indicates vertical variations of velocity <math>x</math>-component at different positions along <math>x</math>-axis (<math>x=0</math>, 4.75, 12.25, 19.75, 24.75) at mid-height and mid-width of the channel (Positions are <math>P_{xi}</math> as indicated in [[#img-10|Figure 10]]). For the two last positions, they were in the medium of the two micro-channels.
 <div id='img-15'></div>
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 |style="padding:10px;"| [[File:Omri_et_al_2019a_5901_Fig15.png]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 15'''. Normalized velocity profiles along the flow-path predicted by SA model at different positions (<math>P_{x0}</math>, <math>P_{x1}</math>, <math>P_{x2}</math>, <math>P_{x3}</math>, <math>P_{x4-1/2}</math>, <math>P_{x5-1/2}</math> as indicated in Figure 10) in the case of the fourth tests. The red color indicated the positions <math>P_{x4-2}</math> and <math>P_{x5-2}</math>
+| colspan="1" style="padding-bottom:10px;"| '''Figure 15'''. Normalized velocity profiles along the flow-path predicted by SA model at different positions (<math>P_{x0}</math>, <math>P_{x1}</math>, <math>P_{x2}</math>, <math>P_{x3}</math>, <math>P_{x4-1/2}</math>, <math>P_{x5-1/2}</math> as indicated in [[#img-10|Figure 10]]) in the case of the fourth tests. The red color indicated the positions <math>P_{x4-2}</math> and <math>P_{x5-2}</math>
 |}
@@ Line 442: / Line 442: @@
 |style="padding:10px;"| [[File:Omri_et_al_2019a_2022_Fig16.png]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;"| '''Figure 16'''. Velocity profiles along the flow-path predicted by SA model and SST <math>k-\omega</math> model in different positions (<math>P_{x0}</math>, <math>P_{x1}</math>, <math>P_{x2}</math>, <math>P_{x3}</math>, <math>P_{x4-1/2}</math>, <math>P_{x5-1/2}</math> as indicated in Figure 10)
+| colspan="1" style="padding-bottom:10px;"| '''Figure 16'''. Velocity profiles along the flow-path predicted by SA model and SST <math>k-\omega</math> model in different positions (<math>P_{x0}</math>, <math>P_{x1}</math>, <math>P_{x2}</math>, <math>P_{x3}</math>, <math>P_{x4-1/2}</math>, <math>P_{x5-1/2}</math> as indicated in [[#img-10|Figure 10]])
 |}
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 |style="padding:10px;"| [[File:Omri_et_al_2019a_4487_Fig17.png]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding-bottom:10px;padding-left:10px;padding-right:10px;"| '''Figure 17'''. Velocity profiles along the flow-path predicted by SA model and SST <math>k-\omega</math> model at the mid-height of the channel in two different places (<math>P_{y1}</math> and  <math>P_{y2}</math> as indicated in Figure 10)
+| colspan="1" style="padding-bottom:10px;padding-left:10px;padding-right:10px;"| '''Figure 17'''. Velocity profiles along the flow-path predicted by SA model and SST <math>k-\omega</math> model at the mid-height of the channel in two different places (<math>P_{y1}</math> and  <math>P_{y2}</math> as indicated in [[#img-10|Figure 10]])
 |}
 <div id='img-18'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
-|-
+|-[[#img-10|Figure 10]]
 |style="padding:10px;"| [[File:Omri_et_al_2019a_5887_Fig18.png]]
 |- style="text-align: center; font-size: 75%;"

@@ Line 151: / Line 151: @@
 When adding radiation, the air as transparent medium was considered radiatively non-participating; surface to surface was used where all surfaces were considered gray and diffuse, consequently conduction equilibrated convection and radiation on each surface. To obtain net radiative heat fluxes over the walls, the number of zones of each surface was determined by the solving-used-mesh.
-The solid horizontal were included in the calculation domain where only the heat conduction energy conservation was resolved. The thermal conductivities of the three solid layers indicated in Figure 1 were considered constant respectively.
+The solid horizontal were included in the calculation domain where only the heat conduction energy conservation was resolved. The thermal conductivities of the three solid layers indicated in [[#img-1|Figure 1]] were considered constant respectively.
-The coupled elliptic partial differential equations (1) to (5) were discretized using second-order finite volume scheme. In the cavity, the rectangular grid was finer near the walls (boundary layer) and coarser in the central region of the cavity (Figure 1). Grid independence, as done by the same author [3], was tested by considering three non-uniform grids: <math>30\times 30</math>, <math>50\times 50</math> and <math>80\times 80</math>. The SA model verified the same result, and the grid <math>50\times 50</math> was used for all calculations of this cavity.
+The coupled elliptic partial differential equations (1) to (5) were discretized using second-order finite volume scheme. In the cavity, the rectangular grid was finer near the walls (boundary layer) and coarser in the central region of the cavity ([[#img-1|Figure 1]]). Grid independence, as done by the same author [3], was tested by considering three non-uniform grids: <math>30\times 30</math>, <math>50\times 50</math> and <math>80\times 80</math>. The SA model verified the same result, and the grid <math>50\times 50</math> was used for all calculations of this cavity.
 The following boundary conditions were applied:

@@ Line 42: / Line 42: @@
 ===2.1. Experimental data===
-The natural convection cavity used here was experimentally investigated by Tian et al.  [9] and also by Ampofo et al. [2] by considering an air-filled cavity. The dimensions and boundary conditions were indicated in Figure 1. The Rayleigh number was <math>1.5810^6</math>. The outside temperature was <math>30 \pm 0.2</math>°C. In addition, the measurements were presented in the middle of the <math>z</math>-direction (<math>z = 0.75</math>m) of the cavity, where Tian et al. [9] and Ampofo et al. [2] demonstrated that the flow fields were 2D.
+The natural convection cavity used here was experimentally investigated by Tian et al.  [9] and also by Ampofo et al. [2] by considering an air-filled cavity. The dimensions and boundary conditions were indicated in [[#img-1|Figure 1]]. The Rayleigh number was <math>1.5810^6</math>. The outside temperature was <math>30 \pm 0.2</math>°C. In addition, the measurements were presented in the middle of the <math>z</math>-direction (<math>z = 0.75</math>m) of the cavity, where Tian et al. [9] and Ampofo et al. [2] demonstrated that the flow fields were 2D.
 ===2.2. Numerical model===
@@ Line 145: / Line 152: @@
 The solid horizontal were included in the calculation domain where only the heat conduction energy conservation was resolved. The thermal conductivities of the three solid layers indicated in Figure 1 were considered constant respectively.
 The coupled elliptic partial differential equations (1) to (5) were discretized using second-order finite volume scheme. In the cavity, the rectangular grid was finer near the walls (boundary layer) and coarser in the central region of the cavity (Figure 1). Grid independence, as done by the same author [3], was tested by considering three non-uniform grids: <math>30\times 30</math>, <math>50\times 50</math> and <math>80\times 80</math>. The SA model verified the same result, and the grid <math>50\times 50</math> was used for all calculations of this cavity.
@@ Line 168: / Line 167: @@
 ===2.3. Numerical results and discussion===
-Figure 2 indicates the non-dimensional vertical velocity in the boundary layer close to the hot wall at the nine different heights presented in the experimental data. The calculations perfectly reproduce the whole core of the cavity (i.e. <math>x/H=0.1</math> to <math>x/H=0.9</math>) where the flow in this zone is quiescent, explaining identical behavior of all models. The influence of constants modification and also the influence of the radiation are seen within the boundary layers (i.e <math>x/H<0.08</math> and <math>x/H>0.92</math>). Model SA slightly underestimates the vertical velocity component; such behaviour is also noticed in the predictions of Peng and Davidson [7]. The constant modification suggested allows improving the velocity behaviour, i.e. a boundary layer thicker but the SAM model continue under estimating the experiment. The addition of radiation to SAM allows a better adjustment of the considered velocity component but not until reproducing quantitatively the experiment. Indicate that SAMR gives a satisfactory behaviour especially in the boundary layers located at <math>y/H=0.4, 0.5</math> and <math>0.6</math>. The differences are appreciable especially while approaching the zones of inversion. Remember that these predictions are satisfactory with reference to other levels of modelling (SST <math>k-w</math> By Omri [3] or LES by Peng [7]).
+[[#img-2|Figure 2]] indicates the non-dimensional vertical velocity in the boundary layer close to the hot wall at the nine different heights presented in the experimental data. The calculations perfectly reproduce the whole core of the cavity (i.e. <math>x/H=0.1</math> to <math>x/H=0.9</math>) where the flow in this zone is quiescent, explaining identical behavior of all models. The influence of constants modification and also the influence of the radiation are seen within the boundary layers (i.e <math>x/H<0.08</math> and <math>x/H>0.92</math>). Model SA slightly underestimates the vertical velocity component; such behaviour is also noticed in the predictions of Peng and Davidson [7]. The constant modification suggested allows improving the velocity behaviour, i.e. a boundary layer thicker but the SAM model continue under estimating the experiment. The addition of radiation to SAM allows a better adjustment of the considered velocity component but not until reproducing quantitatively the experiment. Indicate that SAMR gives a satisfactory behaviour especially in the boundary layers located at <math>y/H=0.4, 0.5</math> and <math>0.6</math>. The differences are appreciable especially while approaching the zones of inversion. Remember that these predictions are satisfactory with reference to other levels of modelling (SST <math>k-w</math> by Omri [3] or LES by Peng [7]).
+<div id='img-2'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 50%;"
 |-
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-Figure 3 shows the second velocity component which represents the secondary flow, for example the maximum of <math>U/Vo</math> at mid height (<math>y/H=0.5</math>) does not exceed the tenth of <math>V/Vo</math>. Consequently, we are in the presence of low velocity values; however, the model qualitatively reproduces the experiment. The difference between the 3 models is better seen especially when approaching the horizontal boundary layers. The three model versions indicate certain symmetry of the profile speed (also checked by <math>k-\varepsilon </math>). The radiation (SAMR model) tends indeed to align the model to the experiment and its contribution is clear in the regions near the horizontal boundary layers.
+[[#img-3|Figure 3]] shows the second velocity component which represents the secondary flow, for example the maximum of <math>U/Vo</math> at mid height (<math>y/H=0.5</math>) does not exceed the tenth of <math>V/Vo</math>. Consequently, we are in the presence of low velocity values; however, the model qualitatively reproduces the experiment. The difference between the 3 models is better seen especially when approaching the horizontal boundary layers. The three model versions indicate certain symmetry of the profile speed (also checked by <math>k-\varepsilon </math>). The radiation (SAMR model) tends indeed to align the model to the experiment and its contribution is clear in the regions near the horizontal boundary layers.
+<div id='img-3'></div>
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 |-
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-The air temperature profiles at the same nine different heights were indicated in Figure 4. Indicate that at the cavity mid-height (<math>y/H = 0.5</math>) the numerical calculations are confused and also in the positions <math>y/H=0.4</math> and <math>0.6</math>. The differences are increasingly seen while tending towards the top or the bottom horizontal walls. Therefore, considering only the mid height cavity does not allow seeing the impact of some phenomenon (radiation, conducting horizontal walls and grid validation) is most likely not enough.
+The air temperature profiles at the same nine different heights were indicated in [[#img-4|Figure 4]]. Indicate that at the cavity mid-height (<math>y/H = 0.5</math>) the numerical calculations are confused and also in the positions <math>y/H=0.4</math> and <math>0.6</math>. The differences are increasingly seen while tending towards the top or the bottom horizontal walls. Therefore, considering only the mid height cavity does not allow seeing the impact of some phenomenon (radiation, conducting horizontal walls and grid validation) is most likely not enough.
 By comparing all heights, the differences between SA, SAM and SAMR are clearer; the correction of the Prandtl number seems to improve the agreement with experiment but it is insufficient especially for <math>y/H=0.1,0.2,0.8</math> and <math>0.9</math>. As an example for <math>y/H=0.9</math> and <math>y/H=0.1</math> the SA and SAM models deviate one and two degrees from the experiment respectively. Indeed, the radiation seems important at these locations; even though it is still difficult to estimate its exact contribution since predictions of the model without radiation is not exact. Therefore, radiation is relevant and as important in this cavity configuration as the conduction of the horizontal walls, especially in the top and bottom regions near the horizontal walls.
+<div id='img-4'></div>
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 |-
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-Lets consider the temperature at mid-section cavity in Figure 5, the prediction of SST <math>k-w</math> model obtained by Omri [3] has also been added, which legitimates in a way the constant modification of the SA model. In the central zone (<math>0.4<y/H <0.6</math>) there are no differences between the various versions of SA model. These differences were accentuated while going towards the horizontal walls. The radiation was non-negligible, especially near the horizontal walls (<math>y/H<0.4</math> and <math>y/H>0.6</math>), where we can verify that for <math>y/H=0.02</math> and <math>y/H=0.98</math> radiation varies the SAM predictions by about 1.6°C.
+Lets consider the temperature at mid-section cavity in [[#img-5|Figure 5]], the prediction of SST <math>k-w</math> model obtained by Omri [3] has also been added, which legitimates in a way the constant modification of the SA model. In the central zone (<math>0.4<y/H <0.6</math>) there are no differences between the various versions of SA model. These differences were accentuated while going towards the horizontal walls. The radiation was non-negligible, especially near the horizontal walls (<math>y/H<0.4</math> and <math>y/H>0.6</math>), where we can verify that for <math>y/H=0.02</math> and <math>y/H=0.98</math> radiation varies the SAM predictions by about 1.6°C.
+<div id='img-5'></div>
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 |-
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-Figure 6 shows the mean horizontal velocity component at the same place as in Figure 5 compared to Tian et al. [9] experimental data. The results of the considered turbulence model compare favorably with experimental data. In particular, at the central region where <math>U/Vo = 0</math> are similar. The maximum velocity values are under estimated, but the SAMR is closer to the experience peaks and valleys are better predicted and at the right vertical position <math>y/H=0.94</math>. Moreover, it changes sign exactly like in the experiment indicating the importance of accounting for radiation in such a flow. It increases the maximum velocity of the SAM model by about 29%. Furthermore, this velocity component is essentially antisymmetric. They also show that horizontal boundary layer was thicker than the vertical one; it was approximately three times the ascending or descending flows boundary layer.
+[[#img-6|Figure 6]] shows the mean horizontal velocity component at the same place as in [[#img-5|Figure 5]] compared to Tian et al. [9] experimental data. The results of the considered turbulence model compare favorably with experimental data. In particular, at the central region where <math>U/Vo = 0</math> are similar. The maximum velocity values are under estimated, but the SAMR is closer to the experience peaks and valleys are better predicted and at the right vertical position <math>y/H=0.94</math>. Moreover, it changes sign exactly like in the experiment indicating the importance of accounting for radiation in such a flow. It increases the maximum velocity of the SAM model by about 29%. Furthermore, this velocity component is essentially antisymmetric. They also show that horizontal boundary layer was thicker than the vertical one; it was approximately three times the ascending or descending flows boundary layer.
+<div id='img-6'></div>
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 |-
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-The Nusselt Number along the four interior walls of the cavity is calculated in Figure 7. The calculation of this number strongly depends on the manner how derivative of temperature is calculated at the point near the wall. It should be noticed that Joubert et al. [23] showed that the difference in calculating the Nusselt number could reach 25% by using various computer codes. Over all, the Nusselt number is well reproduced by the three versions of the model.
+The Nusselt Number along the four interior walls of the cavity is calculated in [[#img-7|Figure 7]]. The calculation of this number strongly depends on the manner how derivative of temperature is calculated at the point near the wall. It should be noticed that Joubert et al. [23] showed that the difference in calculating the Nusselt number could reach 25% by using various computer codes. Over all, the Nusselt number is well reproduced by the three versions of the model.
+<div id='img-7'></div>
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 |-

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 <div class="center" style="font-size: 75%;">'''Table 1'''. Experimental test boundary conditions</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:45%;"
+{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:35%;"
 |-
 |  style="border-top: 1pt solid black;vertical-align: top;border-bottom: 1pt solid black;"|

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 {| style="text-align: center; margin:auto;"
 |-
-|<math>{ \mu }_{{ t}} ={ \rho }\tilde{{ \nu }}{ f}_{{ v1}}\,\,</math> <math>{ f}_{{ w}} ={ g}\left(\frac{{ 1}+{ C}_{{ w3}}^{{ 6}} }{{ g}^{{ 6}} +{ C}_{{ w3}}^{{ 6}} } \right)^{{ 1/6}} \quad</math>  , <math>\quad{ f}_{{ \nu 2}} ={ 1}-\frac{{ \chi }}{{ 1}+{ \chi }^{{ 3}} { f}_{{ \nu 1}} }\quad</math> ,  <math>\quad{ \chi }=\frac{\tilde{{ \nu }}}{{ \nu }}</math>
+|<math>{ \mu }_{{ t}} ={ \rho }\tilde{{ \nu }}\quad ,\quad { f}_{{ v1}}\quad</math> ,  <math>\quad{ f}_{{ w}} ={ g}\left(\frac{{ 1}+{ C}_{{ w3}}^{{ 6}} }{{ g}^{{ 6}} +{ C}_{{ w3}}^{{ 6}} } \right)^{{ 1/6}} \quad</math>  , <math>\quad{ f}_{{ \nu 2}} ={ 1}-\frac{{ \chi }}{{ 1}+{ \chi }^{{ 3}} { f}_{{ \nu 1}} }\quad</math> ,  <math>\quad{ \chi }=\frac{\tilde{{ \nu }}}{{ \nu }}</math>
 |-
 |<math>

@@ Line 126: / Line 126: @@
 |}
-where <math>{ \mu }_{{ t}} ={ \rho }\tilde{{ \nu }}{ f}_{{ v1}}\,\,</math> <math>{ f}_{{ w}} ={ g}\left(\frac{{ 1}+{ C}_{{ w3}}^{{ 6}} }{{ g}^{{ 6}} +{ C}_{{ w3}}^{{ 6}} } \right)^{{ 1/6}} \quad</math> , <math>\quad{ f}_{{ \nu 2}} ={ 1}-\frac{{ \chi }}{{ 1}+{ \chi }^{{ 3}} { f}_{{ \nu 1}} }\quad</math>, <math>\quad{ \chi }=\frac{\tilde{{ \nu }}}{{ \nu }}</math>
+where
-<math>
+{| class="formulaSCP" style="width: 100%; text-align: center;"
 \tilde{S}=\sqrt{{ 2\Omega }_{{ ij}} { \Omega }_{{ ij}} } +\frac{\tilde{{ \nu }}}{{ \kappa }^{{ 2}} +{ d}^{{ 2}} } { f}_{{ v2}} \quad ,\quad  { g}=\frac{\tilde{{ \nu }}}{\tilde{{ S}}{ \kappa }^{{ 2}} { d}^{{ 2}} } +{ C}_{{ w2}} \left( \left(\frac{\tilde{{ \nu }}}{\tilde{{ S}}{ \kappa }^{{ 2}} { d}^{{ 2}} } \right)^{{ 6}} -\frac{\tilde{{ \nu }}}{\tilde{{ S}}{ \kappa }^{{ 2}} { d}^{{ 2}} } \right)\quad ,\quad{ f}_{{ \nu 1}} =\frac{{ \chi }^{{ 3}} }{{ \chi }^{{ 3}} +{ C}_{{ \nu 1}}^{{ 3}} }. </math>
+|}
 <math>\hat S</math> is the mean strain rate  and <math>\Omega_{ij}=(\partial u_i/\partial u_j - \partial u_j/\partial u_i)/2</math> is the mean rate of rotation. The constants: <math>{C}_{ b1} ,\, {C}_{b2} ,\, \sigma_{\tilde{{ \nu }}} ,C_{ \nu 1} ,\, { C}_{ w1} ,\, { C}_{ w2} ,\, { C}_{ w3}</math> and  <math>\kappa</math> as indicated in Spalart-Allmaras [1] were equal to 0.1355, 0.622, 0.667, 7.1, 3.2, 0.3, 2.0 and 0.4187 respectively, and were used with an energy Prandtl number equal to 0.85. The default model was referred to as SA model. The turbulent Prandtl number  <math>\sigma _{\tilde{\nu }}</math> has also been modified from 0.667 to 0.75, which automatically modifies  <math>{C}_{w1}</math> to 2.935 with a turbulent energy Prandtl number = 0.6. This modification was referred to as SAM model. When radiation was added to the SAM model, the model is referred to as SAMR model.

@@ Line 474: / Line 474: @@
 ==Acknowledgments==
-This project was funded by the deanship of Scientific Research (DSR) at king Abdulaziz University, Jeddah, under Grant no: G-1436-305-568. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
+This project was funded by the deanship of Scientific Research (DSR) at king Abdulaziz University, Jeddah, under Grant no: G-568-305-1436. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
 ==References==

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 |-
-|style="padding:10px;"| [[Image:Draft_Omri_961141282-small-image25.png|354px]]
+|style="padding:10px;"| [[Image:Draft_Omri_961141282-small-image25.png|454px]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" style="padding-bottom:10px;"| '''Figure 8'''. Test module setup