DNS 1-6 Quantification of Resolution: Difference between revisions
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=Wing-body junction= | |||
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<math>\Delta=\sqrt[3]{\dfrac{V}{N_{DoF}}}</math> | <math>\Delta=\sqrt[3]{\dfrac{V}{N_{DoF}}}</math> | ||
To assess the mesh resolution, five planes have been extracted within the highly resolved region of the turbulent flow, see [[ | To assess the mesh resolution, five planes have been extracted within the highly resolved region of the turbulent flow, see [[DNS_1-6_Quantification_of_Resolution_#figure7|Fig. 7]]. | ||
Planes A and B are parallel to the horizontal solid wall (<math>y/T=0</math>) and are placed at <math>y/T=0.0025</math> and <math>y/T=0.1</math>, respectively. | Planes A and B are parallel to the horizontal solid wall (<math>y/T=0</math>) and are placed at <math>y/T=0.0025</math> and <math>y/T=0.1</math>, respectively. | ||
Planes C and D are perpendicular to the streamwise direction and are placed at <math>x/T=0.75</math> (location of maximum wing thickness) and <math>x/T=4.5</math> (behind the wing trailing edge), respectively, where <math>x/T=0</math> is the streamwise coordinate of the wing leading edge. | Planes C and D are perpendicular to the streamwise direction and are placed at <math>x/T=0.75</math> (location of maximum wing thickness) and <math>x/T=4.5</math> (behind the wing trailing edge), respectively, where <math>x/T=0</math> is the streamwise coordinate of the wing leading edge. | ||
Plane E is the test case geometric symmetry plane (<math>z/T=0</math>). | Plane E is the test case geometric symmetry plane (<math>z/T=0</math>). | ||
The comparison with respect to the Taylor microscale is shown in [[ | The comparison with respect to the Taylor microscale is shown in [[DNS_1-6_Quantification_of_Resolution_#figure8|Fig. 8]]. | ||
For all the planes extracted, the ratio <math>{\Delta}/{\eta_{T}}</math> is lower than 0.6. | For all the planes extracted, the ratio <math>{\Delta}/{\eta_{T}}</math> is lower than 0.6. | ||
Accordingly, the current space resolution is sufficient to capture the turbulence scales in the intertial range. | Accordingly, the current space resolution is sufficient to capture the turbulence scales in the intertial range. | ||
In [[ | In [[DNS_1-6_Quantification_of_Resolution_#figure9|Fig. 9]] is reported the comparison with respect to the Kolmogorov length scale. | ||
It is commonly accepted that DNS requirements are satisfied when <math>{\Delta}/{\eta_{K}}\leq 5</math>. | It is commonly accepted that DNS requirements are satisfied when <math>{\Delta}/{\eta_{K}}\leq 5</math>. | ||
In all planes considered there is clearly visible a region around the wing in which the ratio <math>{\Delta}/{\eta_{K}}</math> is greater than 8, but no larger than 10. | In all planes considered there is clearly visible a region around the wing in which the ratio <math>{\Delta}/{\eta_{K}}</math> is greater than 8, but no larger than 10. | ||
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Notice that the accurate simulation of the wake behind the wing away from the floor was not within the scope of the current computational campaign. The low mesh resolution in this region was indeed expected as the computational grid had to be coarsened along the vertical (normal to the floor) direction due to computational cost constraints. | Notice that the accurate simulation of the wake behind the wing away from the floor was not within the scope of the current computational campaign. The low mesh resolution in this region was indeed expected as the computational grid had to be coarsened along the vertical (normal to the floor) direction due to computational cost constraints. | ||
<div id=" | <div id="figure7"></div> | ||
{|align="center" width=600 | {|align="center" width=600 | ||
|[[Image:DNS1-6 Wing-body junction scale planes.png|600px]] | |[[Image:DNS1-6 Wing-body junction scale planes.png|600px]] | ||
|- | |- | ||
|align="center"|'''Figure | |align="center"|'''Figure 7:''' Wing-body junction. Extracted planes for mesh resolution analisys. | ||
|} | |} | ||
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<div id=" | <div id="figure8"></div> | ||
{|align="center" width=800 | {|align="center" width=800 | ||
|align="center"|[[Image:DNS1-6 Wing-body junction Taylor scale plane A.png|800px]] | |align="center"|[[Image:DNS1-6 Wing-body junction Taylor scale plane A.png|800px]] | ||
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|align="center"|[[Image:DNS1-6 Wing-body junction Taylor scale plane E.png|800px]] | |align="center"|[[Image:DNS1-6 Wing-body junction Taylor scale plane E.png|800px]] | ||
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|align="center"|'''Figure | |align="center"|'''Figure 8:''' Wing-body junction. Relation between the mesh size and the Taylor microscale. | ||
|} | |} | ||
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<div id=" | <div id="figure9"></div> | ||
{|align="center" width=800 | {|align="center" width=800 | ||
|align="center"|[[Image:DNS1-6 Wing-body junction Kolmogorov scale plane A.png|800px]] | |align="center"|[[Image:DNS1-6 Wing-body junction Kolmogorov scale plane A.png|800px]] | ||
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|align="center"|[[Image:DNS1-6 Wing-body junction Kolmogorov scale plane E.png|800px]] | |align="center"|[[Image:DNS1-6 Wing-body junction Kolmogorov scale plane E.png|800px]] | ||
|- | |- | ||
|align="center"|'''Figure | |align="center"|'''Figure 9:''' Wing-body junction. Relation between the mesh size and the Kolmogorov length scale. | ||
|} | |} | ||
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The average near-wall resolution in streamwise (<math>{\Delta x^{+}}</math>), spanwise (<math>{\Delta z^{+}}</math>) and wall-normal (<math>{\Delta y^{+}}</math>) directions at different locations on the horizontal solid wall is reported in [[ | The average near-wall resolution in streamwise (<math>{\Delta x^{+}}</math>), spanwise (<math>{\Delta z^{+}}</math>) and wall-normal (<math>{\Delta y^{+}}</math>) directions at different locations on the horizontal solid wall is reported in [[DNS_1-6_Quantification_of_Resolution#table3|Tab. 3]]. | ||
<div id=" | <div id="table3"></div> | ||
{|align="center" border="1" cellpadding="10" | {|align="center" border="1" cellpadding="10" | ||
|- | |- | ||
|align="center"|'''<math>{x/ | |align="center"|'''<math>{x/T}</math>'''||align="center"|'''<math>{z/T}</math>'''||align="center"|'''<math>{\Delta x^{+}}</math>'''||align="center"|'''<math>{\Delta z^{+}}</math>||align="center"|'''<math>{\Delta y_{1}^{+}}</math>''' | ||
|- | |- | ||
|align="center"|'''<math>{-2.15}</math>'''||align="center"|'''<math>{0}</math>'''||align="center"|'''<math>{23.3}</math>'''||align="center"|'''<math>{21.6}</math>'''||align="center"|'''<math>{0.46}</math>''' | |align="center"|'''<math>{-2.15}</math>'''||align="center"|'''<math>{0}</math>'''||align="center"|'''<math>{23.3}</math>'''||align="center"|'''<math>{21.6}</math>'''||align="center"|'''<math>{0.46}</math>''' | ||
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|align="center"|'''<math>{8.5}</math>'''||align="center"|'''<math>{1}</math>'''||align="center"|'''<math>{24.6}</math>'''||align="center"|'''<math>{19.1}</math>'''||align="center"|'''<math>{0.43}</math>''' | |align="center"|'''<math>{8.5}</math>'''||align="center"|'''<math>{1}</math>'''||align="center"|'''<math>{24.6}</math>'''||align="center"|'''<math>{19.1}</math>'''||align="center"|'''<math>{0.43}</math>''' | ||
|} | |} | ||
<center>'''Table | <center>'''Table 3:''' Near-wall space resolution at different locations. <math>(x/T,z/T)=(0,0)</math> is the wing leading edge - horizontal solid wall intersection point </center> | ||
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As a first step, an assessment of code MIGALE in closing the budgets is performed. | As a first step, an assessment of code MIGALE in closing the budgets is performed. | ||
[[ | [[DNS_1-6_Quantification_of_Resolution#figure10|Fig. 10]] reports the budgets of streamwise normal Reynolds-stress and TKE in a channel flow at <math>{Re_\tau = 180}</math> using a DG polynomial approximation of degree 5 on a mesh of <math>91\times 43\times 48</math> hexahedral elements (10.5 million DoF/eqn.). | ||
Domain | Domain dimensions are those of the DNS of [[DNS_1-6_Quantification_of_Resolution#1|Moser ''et al.'' (1999)]] and their results are included as reference. | ||
As the maximum value of the residual is <math>0.22\%</math> and <math>0.64\%</math> of the production peak for the Reynolds-stress xx and the TKE budgets, respectively, the results show that MIGALE code can close well the budgets when sufficient spatial and time resolution is | As the maximum value of the residual is <math>0.22\%</math> and <math>0.64\%</math> of the production peak for the Reynolds-stress xx and the TKE budgets, respectively, the results show that the MIGALE code can close well the budgets when sufficient spatial and time resolution is used. | ||
<div id=" | <div id="figure10"></div> | ||
{|align="center" width=500 | {|align="center" width=500 | ||
|[[Image:Channel UniBG MIGALE Re xx budget.png|500px]]||[[Image:Channel UniBG MIGALE TKE budget.png|500px]] | |[[Image:Channel UniBG MIGALE Re xx budget.png|500px]]||[[Image:Channel UniBG MIGALE TKE budget.png|500px]] | ||
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!align="center" colspan="2"|'''Figure | !align="center" colspan="2"|'''Figure 10:''' Channel flow at <math>Re_{\tau}=180</math> Reynolds-stress xx and TKE budgets: dissipation <math>\varepsilon</math>, production <math>P</math>, turbulent diffusion <math>D^{1}</math>, pressure diffusion <math>D^{2}</math>, viscous diffusion <math>D^{3}</math> and pressure strain <math>\Phi</math>. Solid lines from the DG P5 computation and symbols from [[DNS_1-6_Quantification_of_Resolution#1|Moser ''et al.'' (1999)]]. | ||
|} | |} | ||
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Considering now the budgets for the wing-body junction simulation, the conclusion obtained comparing the mesh | Considering now the budgets for the wing-body junction simulation, the conclusion obtained comparing the characteristic mesh length and the Kolmogorov length scale (see [[DNS_1-6_Quantification_of_Resolution_#figure9|Fig. 9]]) still holds. | ||
Indeed, on [[ | Indeed, on [[DNS_1-6_Quantification_of_Resolution_#figure11|Fig. 11]] and [[DNS_1-6_Quantification_of_Resolution_#figure12|Fig. 12]] are reported the TKE budgets for two different locations, i.e., <math>(x/T,z/T)=(-2.15,0)</math> (the chekpoint streamwise location) and <math>(x/T,z/T)=(2.125,0.75)</math> (on the side of the wing, in the region of the horse-shoe vortex), respectively. | ||
In both locations the budget closure has not been achieved. | In both locations the budget closure has not been achieved. | ||
The oscillating behaviour of the profiles of the different terms can be ascribed to a lack of spatial resolution. | The oscillating behaviour of the profiles of the different terms can be ascribed to a lack of spatial resolution. | ||
This result suggests that for future campaigns it is recommended to increase further the grid density. | This result suggests that for future campaigns it is recommended to increase further the grid density. | ||
<div id=" | <div id="figure11"></div> | ||
{|align="center" width=1000 | {|align="center" width=1000 | ||
|align="center"|[[Image:DNS1-6 Wing-body junction TKE budget P1.png|500px]] | |align="center"|[[Image:DNS1-6 Wing-body junction TKE budget P1.png|500px]] | ||
|- | |- | ||
!align="center" colspan="1"|'''Figure | !align="center" colspan="1"|'''Figure 11:''' Wing-body junction. TKE budgets at location <math>x/T=-2.15</math>, <math>z/T=0</math>: convection <math>C</math>, production <math>P</math>, turbulent diffusion <math>D^{1}</math>, pressure diffusion <math>D^{2}</math>, viscous diffusion <math>D^{3}</math>, pressure strain <math>\Phi</math>, pressure dilatation <math>\Phi'</math>, dissipation <math>\varepsilon</math> and density fluctuation <math>K</math>. | ||
|} | |} | ||
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<div id=" | <div id="figure12"></div> | ||
{|align="center" width=1000 | {|align="center" width=1000 | ||
|align="center"|[[Image:DNS1-6 Wing-body junction TKE budget P4.png|500px]] | |align="center"|[[Image:DNS1-6 Wing-body junction TKE budget P4.png|500px]] | ||
|- | |- | ||
!align="center" colspan="1"|'''Figure | !align="center" colspan="1"|'''Figure 12:''' Wing-body junction. TKE budgets at location <math>x/T=2.125</math>, <math>z/T=0.75</math>: convection <math>C</math>, production <math>P</math>, turbulent diffusion <math>D^{1}</math>, pressure diffusion <math>D^{2}</math>, viscous diffusion <math>D^{3}</math>, pressure strain <math>\Phi</math>, pressure dilatation <math>\Phi'</math>, dissipation <math>\varepsilon</math> and density fluctuation <math>K</math>. | ||
|} | |} | ||
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| organisation=University of Bergamo (UNIBG), ICL (Imperial College London), ONERA | | organisation=University of Bergamo (UNIBG), ICL (Imperial College London), ONERA | ||
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Latest revision as of 11:47, 27 February 2023
Wing-body junction
Quantification of resolution
This section provides details of the solution accuracy obtained by tackling the wing-body junction DNS with MIGALE. After providing details of the mesh resolution in comparison with spatial turbulent scales, a discussion on the closure of the Reynolds stress budgets is given.
Mesh resolution
The mesh resolution is quantified by comparing the characteristic mesh length () with the characteristic lengths of the turbulence, i.e., the Taylor microscale () and the Kolmogorov length scale (). Here, the characteristic mesh length takes into account the degree of the DG polynomial approximation. In particular, it is defined as the cubic root of the ratio between the mesh element volume and the number of DoFs within the mesh element per equation
To assess the mesh resolution, five planes have been extracted within the highly resolved region of the turbulent flow, see Fig. 7. Planes A and B are parallel to the horizontal solid wall () and are placed at and , respectively. Planes C and D are perpendicular to the streamwise direction and are placed at (location of maximum wing thickness) and (behind the wing trailing edge), respectively, where is the streamwise coordinate of the wing leading edge. Plane E is the test case geometric symmetry plane ().
The comparison with respect to the Taylor microscale is shown in Fig. 8. For all the planes extracted, the ratio is lower than 0.6. Accordingly, the current space resolution is sufficient to capture the turbulence scales in the intertial range.
In Fig. 9 is reported the comparison with respect to the Kolmogorov length scale. It is commonly accepted that DNS requirements are satisfied when . In all planes considered there is clearly visible a region around the wing in which the ratio is greater than 8, but no larger than 10. This region is characterized by the presence of the horse-shoe vortex. Besides, for planes D and E an additional region of high ratio can be seen downstream of the wing trailing edge, close to the symmetry plane. This is the region where the turbulent boundary layer that develops along the wing solid wall moves downstream and generates a wake. As outcome of this analysis, it can be stated that the DNS requirements are not fulfilled for the current simulation. For this reason the present study was referred to as under-resolved DNS (uDNS). For future highly resolved simulations, further mesh refinement is strongly advised in these regions. Notice that the accurate simulation of the wake behind the wing away from the floor was not within the scope of the current computational campaign. The low mesh resolution in this region was indeed expected as the computational grid had to be coarsened along the vertical (normal to the floor) direction due to computational cost constraints.
Figure 7: Wing-body junction. Extracted planes for mesh resolution analisys. |
Figure 8: Wing-body junction. Relation between the mesh size and the Taylor microscale. |
Figure 9: Wing-body junction. Relation between the mesh size and the Kolmogorov length scale. |
The average near-wall resolution in streamwise (), spanwise () and wall-normal () directions at different locations on the horizontal solid wall is reported in Tab. 3.
Solution verification
One way to verify that the DNS are properly resolved is to examine the terms in the Reynolds-stress and turbulent kinetic energy (TKE) equation (the budgets).
As a first step, an assessment of code MIGALE in closing the budgets is performed. Fig. 10 reports the budgets of streamwise normal Reynolds-stress and TKE in a channel flow at using a DG polynomial approximation of degree 5 on a mesh of hexahedral elements (10.5 million DoF/eqn.). Domain dimensions are those of the DNS of Moser et al. (1999) and their results are included as reference. As the maximum value of the residual is and of the production peak for the Reynolds-stress xx and the TKE budgets, respectively, the results show that the MIGALE code can close well the budgets when sufficient spatial and time resolution is used.
Figure 10: Channel flow at Reynolds-stress xx and TKE budgets: dissipation , production , turbulent diffusion , pressure diffusion , viscous diffusion and pressure strain . Solid lines from the DG P5 computation and symbols from Moser et al. (1999). |
---|
Considering now the budgets for the wing-body junction simulation, the conclusion obtained comparing the characteristic mesh length and the Kolmogorov length scale (see Fig. 9) still holds. Indeed, on Fig. 11 and Fig. 12 are reported the TKE budgets for two different locations, i.e., (the chekpoint streamwise location) and (on the side of the wing, in the region of the horse-shoe vortex), respectively. In both locations the budget closure has not been achieved. The oscillating behaviour of the profiles of the different terms can be ascribed to a lack of spatial resolution. This result suggests that for future campaigns it is recommended to increase further the grid density.
References
- Moser, R. D., Kim, J., Mansour, N. N. (1999): Direct numerical simulation of turbulent channel flow up to Re_tau 590. Physics of Fluids, Vol. 11(4), pp.943-945.
Contributed by: Francesco Bassi (UNIBG), Alessandro Colombo (UNIBG), Francesco Carlo Massa (UNIBG), Michael Leschziner (ICL/ERCOFTAC), Jean-Baptiste Chapelier (ONERA) — University of Bergamo (UNIBG), ICL (Imperial College London), ONERA
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