Difference between revisions of "OpenFOAM/C3/Flow-over-a-flat-plate/English-timed"

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|-
 
|-
 
| 00:30
 
| 00:30
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|'''Flow over flat plate''' is a fundamental problem in '''fluid mechanics'''.
+
| The tutorials were recorded using the versions specified in previous slide. Subsequently the tutorials were edited to latest versions. To install latest system requirements go to Installation Sheet
  
 
|-
 
|-
 
| 00:35
 
| 00:35
 +
|'''Flow over flat plate''' is a fundamental problem in '''fluid mechanics'''.
 +
 +
|-
 +
| 00:40
 
| We can visualize the growth of the '''boundary layer'''. '''Boundary layer''' is a very thin region above the body  
 
| We can visualize the growth of the '''boundary layer'''. '''Boundary layer''' is a very thin region above the body  
  
 
|-
 
|-
−
| 00:41
+
| 00:46
 
|where the velocity is 0.99 times the '''free stream velocity'''.
 
|where the velocity is 0.99 times the '''free stream velocity'''.
  
 
|-
 
|-
−
| 00:46
+
| 00:51
 
| This is a diagram of '''flow over the flat plate'''.
 
| This is a diagram of '''flow over the flat plate'''.
  
 
|-
 
|-
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| 00:49
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| 00:54
 
|The''' boundary conditions''' are given as follows.
 
|The''' boundary conditions''' are given as follows.
 
You have the '''Inlet''', the '''Plate''', '''Top''' – which is the '''Farfield''' and '''Outlet''' – which is the''' pressure outlet boundary'''.
 
You have the '''Inlet''', the '''Plate''', '''Top''' – which is the '''Farfield''' and '''Outlet''' – which is the''' pressure outlet boundary'''.
  
 
|-
 
|-
−
| 01:00
+
| 01:05
 
| The '''Free stream velocity U = 1 m/s''' and we are solving this for '''Reynolds number (Re) = 100'''.
 
| The '''Free stream velocity U = 1 m/s''' and we are solving this for '''Reynolds number (Re) = 100'''.
  
 
|-
 
|-
−
| 01:08
+
| 01:13
 
|Now let us go to the '''home''' folder. In the '''home''' folder, click on the '''OpenFoam''' folder.
 
|Now let us go to the '''home''' folder. In the '''home''' folder, click on the '''OpenFoam''' folder.
  
 
|-
 
|-
−
| 01:15
+
| 01:20
 
|Then go to the 'run' directory. You will see 'Tutorials'. Click on it. Scroll down and then click on '''Incompressible.''' Scroll down.  
 
|Then go to the 'run' directory. You will see 'Tutorials'. Click on it. Scroll down and then click on '''Incompressible.''' Scroll down.  
  
 
|-
 
|-
−
| 01:27
+
| 01:32
 
|You will see the 'simpleFoam' folder. Click on it. This '''solver''' suits our case.
 
|You will see the 'simpleFoam' folder. Click on it. This '''solver''' suits our case.
  
 
|-
 
|-
−
| 01:34
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| 01:39
 
| In this, create a folder by the name '''flatplate. '''Right click - '''Create New Folder -''' '''flatplate'''.
 
| In this, create a folder by the name '''flatplate. '''Right click - '''Create New Folder -''' '''flatplate'''.
  
 
|-
 
|-
−
| 01:44
+
| 01:53
 
| Now, let's open the '''pitzdaily''' case.  
 
| Now, let's open the '''pitzdaily''' case.  
  
 
|-
 
|-
−
| 01:47
+
| 01:56
 
| Let me zoom this. Copy the three folders - '''0, constant '''and '''system'''. Copy this.
 
| Let me zoom this. Copy the three folders - '''0, constant '''and '''system'''. Copy this.
  
 
|-
 
|-
−
| 01:56
+
| 02:04
 
| Now let us go one level back. '''Paste''' these  three folders inside the''' flatplate '''folder.
 
| Now let us go one level back. '''Paste''' these  three folders inside the''' flatplate '''folder.
  
 
|-
 
|-
−
| 02:05
+
| 02:14
 
| Open the '''constant''' folder and then the '''polyMesh''' folder.
 
| Open the '''constant''' folder and then the '''polyMesh''' folder.
  
 
|-
 
|-
−
|02:10
+
|02:19
 
| Change the geometry and '''boundary condition''' names in the '''blockMeshDict '''file.
 
| Change the geometry and '''boundary condition''' names in the '''blockMeshDict '''file.
  
 
|-
 
|-
−
|02:15
+
|02:24
 
| I have already made the changes. Let us open the '''blockMeshDict''' file . Scroll down. The geometry is in meters.
 
| I have already made the changes. Let us open the '''blockMeshDict''' file . Scroll down. The geometry is in meters.
  
 
|-
 
|-
−
|02:25
+
|02:34
 
| We have set the dimensions of the '''flatplate'''.
 
| We have set the dimensions of the '''flatplate'''.
  
 
|-
 
|-
−
| 02:29
+
| 02:38
 
| We can see the  '''simpleGrading. '''It is kept as (1 3 1) as we need a finer '''mesh''' near the plate.
 
| We can see the  '''simpleGrading. '''It is kept as (1 3 1) as we need a finer '''mesh''' near the plate.
  
 
|-
 
|-
−
|02:35
+
|02:24
 
| Now close this. Go two levels back.
 
| Now close this. Go two levels back.
  
 
|-
 
|-
−
|02:41
+
|02:50
 
| Similarly, make changes in the '''boundary condition''' names inside the files in the '0' folder.
 
| Similarly, make changes in the '''boundary condition''' names inside the files in the '0' folder.
  
 
|-
 
|-
−
| 02:48
+
| 02:57
 
|These files have '''pressure, velocity''' and '''wall''' functions.
 
|These files have '''pressure, velocity''' and '''wall''' functions.
  
 
|-
 
|-
−
|02:54
+
|03:03
 
| To calculate the values of '''wall''' functions, please refer to the earlier tutorial in the '''OpenFoam''' series. Let us go one level back.
 
| To calculate the values of '''wall''' functions, please refer to the earlier tutorial in the '''OpenFoam''' series. Let us go one level back.
  
 
|-
 
|-
−
|03:03
+
|03:12
 
| The '''system''' folder can be kept default. Let us close this.
 
| The '''system''' folder can be kept default. Let us close this.
  
 
|-
 
|-
−
|03:09
+
|03:18
 
| Now let us open the '''terminal window.''' In the terminal window, type "run" and press '''Enter.'''
 
| Now let us open the '''terminal window.''' In the terminal window, type "run" and press '''Enter.'''
  
 
|-
 
|-
−
| 03:16
+
| 03:27
 
| Type '''cd space tutorials ''' press''' Enter.'''
 
| Type '''cd space tutorials ''' press''' Enter.'''
  
 
|-
 
|-
−
| 03:21
+
| 03:30
 
| Type '''cd incompressible '''press''' Enter.'''
 
| Type '''cd incompressible '''press''' Enter.'''
  
 
|-
 
|-
−
| 03:25
+
| 03:34
 
| Type '''cd space simpleFoam '''press''' Enter.'''
 
| Type '''cd space simpleFoam '''press''' Enter.'''
  
 
|-
 
|-
−
| 03:31
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| 03:40
 
| Now type "ls" and press '''Enter.'''
 
| Now type "ls" and press '''Enter.'''
  
 
|-
 
|-
−
|03:34
+
|03:43
 
| We can see the '''flatplate''' folder.
 
| We can see the '''flatplate''' folder.
  
 
|-
 
|-
−
| 03:37
+
| 03:46
 
| Now, type '''cd space flatplate '''and press '''Enter.'''
 
| Now, type '''cd space flatplate '''and press '''Enter.'''
  
 
|-
 
|-
−
| 03:42
+
| 03:51
 
| Now type "ls" and press '''Enter.'''
 
| Now type "ls" and press '''Enter.'''
  
 
|-
 
|-
−
|03:45
+
|03:54
 
| You can see the three folders '''0, constant''' and '''system.'''
 
| You can see the three folders '''0, constant''' and '''system.'''
  
 
|-
 
|-
−
| 03:49
+
| 03:58
 
| Now, we will '''mesh''' the geometry. We are using a '''course mesh''' for this problem. '''Meshing''' can be done by typing '''blockMesh''' in the terminal.
 
| Now, we will '''mesh''' the geometry. We are using a '''course mesh''' for this problem. '''Meshing''' can be done by typing '''blockMesh''' in the terminal.
  
 
|-
 
|-
−
|03:58
+
|04:07
 
| Press '''Enter.''' '''Meshing''' has been done.
 
| Press '''Enter.''' '''Meshing''' has been done.
  
 
|-
 
|-
−
|04:01
+
|04:10
 
| Note that if there is some error in the '''blockMesh''' file, it will be shown in the terminal window.
 
| Note that if there is some error in the '''blockMesh''' file, it will be shown in the terminal window.
  
 
|-
 
|-
−
| 04:07
+
| 04:16
 
| To view the geometry, type “paraFoam”, press '''Enter.'''
 
| To view the geometry, type “paraFoam”, press '''Enter.'''
  
 
|-
 
|-
−
| 04:13
+
| 04:22
 
| After the '''ParaView''' window opens, on the left hand side of the '''object inspector''' menu, click '''Apply'''.
 
| After the '''ParaView''' window opens, on the left hand side of the '''object inspector''' menu, click '''Apply'''.
  
 
|-
 
|-
−
| 04:21
+
| 04:30
 
|We can see the '''geometry'''. Close the '''ParaView''' window. Let me switch back to the slides.
 
|We can see the '''geometry'''. Close the '''ParaView''' window. Let me switch back to the slides.
  
 
|-
 
|-
−
| 04:28
+
| 04:37
 
| The '''solver''' we are using here is '''simpleFoam.''' '''SimpleFoam''' is a '''steady state solver ''' for '''in compressible '''and''' turbulent''' flows.
 
| The '''solver''' we are using here is '''simpleFoam.''' '''SimpleFoam''' is a '''steady state solver ''' for '''in compressible '''and''' turbulent''' flows.
  
 
|-
 
|-
−
| 04:37
+
| 04:46
 
| Let me switch back to the terminal window. In the terminal window, type "simpleFoam" and press '''Enter.'''
 
| Let me switch back to the terminal window. In the terminal window, type "simpleFoam" and press '''Enter.'''
  
 
|-
 
|-
−
| 04:45
+
| 04:54
 
|You will see the '''iterations''' running in the terminal window.
 
|You will see the '''iterations''' running in the terminal window.
  
 
|-
 
|-
−
| 04:51
+
| 05:00
 
| Once the solving is done, type "paraFoam" to view the results.
 
| Once the solving is done, type "paraFoam" to view the results.
  
 
|-
 
|-
−
| 04:55
+
| 05:04
 
| On the left hand side of the '''Object Inspector''' menu, click '''Apply''' to view the geometry.  
 
| On the left hand side of the '''Object Inspector''' menu, click '''Apply''' to view the geometry.  
  
 
|-
 
|-
−
| 05:01
+
| 05:10
 
| Scroll down the '''properties''' panel of the '''Object Inspector''' menu for '''time step''', '''regions''' and '''fields'''.
 
| Scroll down the '''properties''' panel of the '''Object Inspector''' menu for '''time step''', '''regions''' and '''fields'''.
  
 
|-
 
|-
−
| 05:08
+
| 05:17
 
| To view the '''contours''' from the top drop down menu, in the '''Active Variable Control '''menu, change from '''solid color''' to capital 'U'.
 
| To view the '''contours''' from the top drop down menu, in the '''Active Variable Control '''menu, change from '''solid color''' to capital 'U'.
  
 
|-
 
|-
−
|05:19
+
|05:28
 
| You can see the initial condition of the '''velocity'''.
 
| You can see the initial condition of the '''velocity'''.
  
 
|-
 
|-
−
| 05:23
+
| 05:32
 
| Now on top of the '''ParaView''' window, you will see the '''VCR''' control.
 
| Now on top of the '''ParaView''' window, you will see the '''VCR''' control.
  
 
|-
 
|-
−
| 05:28
+
| 05:37
 
| Click on the '''Play''' button.
 
| Click on the '''Play''' button.
  
 
|-
 
|-
−
|05:33
+
|05:42
 
| You will see the '''contour''' of '''Pressure''' or '''Velocity''' on the flat plate accordingly.
 
| You will see the '''contour''' of '''Pressure''' or '''Velocity''' on the flat plate accordingly.
  
 
|-
 
|-
−
| 05:39
+
| 05:48
 
| This is the '''velocity contour'''. Toggle on the '''Color legend'''.
 
| This is the '''velocity contour'''. Toggle on the '''Color legend'''.
  
 
|-
 
|-
−
| 05:43
+
| 05:52
 
| To do this, click on the '''color legend '''icon on the '''Active Variable Control''' menu.
 
| To do this, click on the '''color legend '''icon on the '''Active Variable Control''' menu.
  
 
|-
 
|-
−
| 05:50
+
| 05:59
 
| Click '''Apply''' in the '''Object inspector '''menu.
 
| Click '''Apply''' in the '''Object inspector '''menu.
  
 
|-
 
|-
−
| 05:53
+
| 06:02
 
| In the '''Object inspector '''menu, click on '''Display'''.  
 
| In the '''Object inspector '''menu, click on '''Display'''.  
  
 
|-
 
|-
−
| 05:57
+
| 06:06
 
|'''Scroll down''' and click on '''Rescale to data range'''.
 
|'''Scroll down''' and click on '''Rescale to data range'''.
  
 
|-
 
|-
−
| 06:03
+
| 06:12
 
| Let me shift this '''Color legend''' on top to visualize the '''Vector Plot'''. Go to the '''Filters''' Menu > '''Common''' > '''Glyph'''.
 
| Let me shift this '''Color legend''' on top to visualize the '''Vector Plot'''. Go to the '''Filters''' Menu > '''Common''' > '''Glyph'''.
  
 
|-
 
|-
−
|06:15
+
|06:24
 
| Go to the '''Properties''' in '''Object Inspector menu'''.
 
| Go to the '''Properties''' in '''Object Inspector menu'''.
  
 
|-
 
|-
−
|06:20
+
|06:29
 
| Click '''Apply''' on the left hand side of '''Object Inspector''' menu.
 
| Click '''Apply''' on the left hand side of '''Object Inspector''' menu.
  
 
|-
 
|-
−
| 06:24
+
| 06:33
 
| You can change the number of '''vectors''' by changing their size at the bottom.
 
| You can change the number of '''vectors''' by changing their size at the bottom.
  
 
|-
 
|-
−
| 06:29
+
| 06:38
 
| Also, the size of the '''vectors''' can be changed by clicking on the '''Edit''' button. The '''Set Scale Factor''' can be changed to 0.1.
 
| Also, the size of the '''vectors''' can be changed by clicking on the '''Edit''' button. The '''Set Scale Factor''' can be changed to 0.1.
  
 
|-
 
|-
−
| 06:41
+
| 06:50
 
| Again, click the '''Apply''' button.
 
| Again, click the '''Apply''' button.
  
 
|-
 
|-
−
|06:44
+
|06:53
 
| Now let me zoom this.
 
| Now let me zoom this.
  
 
|-
 
|-
−
| 06:46
+
| 06:55
 
| To do this, in the '''Active Variable Control '''menu, click on '''zoom To Box '''option
 
| To do this, in the '''Active Variable Control '''menu, click on '''zoom To Box '''option
  
 
|-
 
|-
−
|06:52
+
|06:61
 
| and '''zoom''' over any area that you desire.
 
| and '''zoom''' over any area that you desire.
  
 
|-
 
|-
−
| 06:58
+
| 07:07
 
| We can see the parabolic variation of the '''vector plots '''as the '''flow''' moves over the''' plate'''.
 
| We can see the parabolic variation of the '''vector plots '''as the '''flow''' moves over the''' plate'''.
  
 
|-
 
|-
−
| 07:04
+
| 07:13
 
| Delete this. Now delete the '''vector plot.'''
 
| Delete this. Now delete the '''vector plot.'''
  
 
|-
 
|-
−
| 07:09
+
| 07:18
 
| Also, we can see that the color near to 1 corresponds to the velocity of 0.99 times the '''free stream velocity'''.
 
| Also, we can see that the color near to 1 corresponds to the velocity of 0.99 times the '''free stream velocity'''.
  
 
|-
 
|-
−
| 07:17
+
| 07:26
 
| You can also plot the variation of velocity along the X and Y axes using the '''plot over data line'''.
 
| You can also plot the variation of velocity along the X and Y axes using the '''plot over data line'''.
  
 
|-
 
|-
−
| 07:26
+
| 07:35
 
| This brings us to the end of the tutorial. In this tutorial, we learnt:
 
| This brings us to the end of the tutorial. In this tutorial, we learnt:
 
'''Geometry''' and '''meshing''' of the '''flat plate geometry''' and
 
'''Geometry''' and '''meshing''' of the '''flat plate geometry''' and
Line 299: Line 303:
  
 
|-
 
|-
−
| 07:37
+
| 07:46
 
| As an Assignment- Create a '''geometry''' of '''flow over a flat plate'''.Refine the '''grid spacing''' near the plate.
 
| As an Assignment- Create a '''geometry''' of '''flow over a flat plate'''.Refine the '''grid spacing''' near the plate.
  
 
|-
 
|-
−
| 07:45
+
| 07:54
 
| Watch the video available at this URL: http://spoken-tutorial.org/What_is_a_Spoken_Tutorial  
 
| Watch the video available at this URL: http://spoken-tutorial.org/What_is_a_Spoken_Tutorial  
 
It summarizes the Spoken Tutorial project. If you do not have good bandwidth, you can download and watch it.  
 
It summarizes the Spoken Tutorial project. If you do not have good bandwidth, you can download and watch it.  
  
 
|-
 
|-
−
| 07:55
+
| 08:04
 
| The Spoken Tutorial Project team:
 
| The Spoken Tutorial Project team:
 
Conducts workshops using spoken tutorials. Gives certificates to those who pass an online test.
 
Conducts workshops using spoken tutorials. Gives certificates to those who pass an online test.
Line 314: Line 318:
  
 
|-
 
|-
−
| 08:08
+
| 08:38
 
| '''Spoken Tutorial''' project is a part of the '''Talk to a Teacher''' project. It is supported by the National Mission on Education through ICT, MHRD, Government of India.  
 
| '''Spoken Tutorial''' project is a part of the '''Talk to a Teacher''' project. It is supported by the National Mission on Education through ICT, MHRD, Government of India.  
  
 
|-
 
|-
−
| 08:17
+
| 08:48
 
|More information on this mission is available at this URL:http://spoken-tutorial.org/NMEICT-Intro.
 
|More information on this mission is available at this URL:http://spoken-tutorial.org/NMEICT-Intro.
 
This is Rahul Joshi from '''IIT Bombay''', signing off.Thanks for joining.
 
This is Rahul Joshi from '''IIT Bombay''', signing off.Thanks for joining.
  
 
|}
 
|}

Latest revision as of 16:50, 11 April 2019

Time Narration
00:01 Hello and welcome to the spoken tutorial on Flow over a flat plate using OpenFOAM.
00:06 In this tutorial, I will teach you about:

Geometry of the flat plate Changing the grid spacing in meshing Postprocessing results in ParaView and Visualizing using Vector Plot.

00:19 To record this tutorial, I am using:

Linux Operating system Ubuntu version 12.04. OpenFOAM version 2.1.1 and ParaView version 3.12.0

00:30 The tutorials were recorded using the versions specified in previous slide. Subsequently the tutorials were edited to latest versions. To install latest system requirements go to Installation Sheet
00:35 Flow over flat plate is a fundamental problem in fluid mechanics.
00:40 We can visualize the growth of the boundary layer. Boundary layer is a very thin region above the body
00:46 where the velocity is 0.99 times the free stream velocity.
00:51 This is a diagram of flow over the flat plate.
00:54 The boundary conditions are given as follows.

You have the Inlet, the Plate, Top – which is the Farfield and Outlet – which is the pressure outlet boundary.

01:05 The Free stream velocity U = 1 m/s and we are solving this for Reynolds number (Re) = 100.
01:13 Now let us go to the home folder. In the home folder, click on the OpenFoam folder.
01:20 Then go to the 'run' directory. You will see 'Tutorials'. Click on it. Scroll down and then click on Incompressible. Scroll down.
01:32 You will see the 'simpleFoam' folder. Click on it. This solver suits our case.
01:39 In this, create a folder by the name flatplate. Right click - Create New Folder - flatplate.
01:53 Now, let's open the pitzdaily case.
01:56 Let me zoom this. Copy the three folders - 0, constant and system. Copy this.
02:04 Now let us go one level back. Paste these three folders inside the flatplate folder.
02:14 Open the constant folder and then the polyMesh folder.
02:19 Change the geometry and boundary condition names in the blockMeshDict file.
02:24 I have already made the changes. Let us open the blockMeshDict file . Scroll down. The geometry is in meters.
02:34 We have set the dimensions of the flatplate.
02:38 We can see the simpleGrading. It is kept as (1 3 1) as we need a finer mesh near the plate.
02:24 Now close this. Go two levels back.
02:50 Similarly, make changes in the boundary condition names inside the files in the '0' folder.
02:57 These files have pressure, velocity and wall functions.
03:03 To calculate the values of wall functions, please refer to the earlier tutorial in the OpenFoam series. Let us go one level back.
03:12 The system folder can be kept default. Let us close this.
03:18 Now let us open the terminal window. In the terminal window, type "run" and press Enter.
03:27 Type cd space tutorials press Enter.
03:30 Type cd incompressible press Enter.
03:34 Type cd space simpleFoam press Enter.
03:40 Now type "ls" and press Enter.
03:43 We can see the flatplate folder.
03:46 Now, type cd space flatplate and press Enter.
03:51 Now type "ls" and press Enter.
03:54 You can see the three folders 0, constant and system.
03:58 Now, we will mesh the geometry. We are using a course mesh for this problem. Meshing can be done by typing blockMesh in the terminal.
04:07 Press Enter. Meshing has been done.
04:10 Note that if there is some error in the blockMesh file, it will be shown in the terminal window.
04:16 To view the geometry, type “paraFoam”, press Enter.
04:22 After the ParaView window opens, on the left hand side of the object inspector menu, click Apply.
04:30 We can see the geometry. Close the ParaView window. Let me switch back to the slides.
04:37 The solver we are using here is simpleFoam. SimpleFoam is a steady state solver for in compressible and turbulent flows.
04:46 Let me switch back to the terminal window. In the terminal window, type "simpleFoam" and press Enter.
04:54 You will see the iterations running in the terminal window.
05:00 Once the solving is done, type "paraFoam" to view the results.
05:04 On the left hand side of the Object Inspector menu, click Apply to view the geometry.
05:10 Scroll down the properties panel of the Object Inspector menu for time step, regions and fields.
05:17 To view the contours from the top drop down menu, in the Active Variable Control menu, change from solid color to capital 'U'.
05:28 You can see the initial condition of the velocity.
05:32 Now on top of the ParaView window, you will see the VCR control.
05:37 Click on the Play button.
05:42 You will see the contour of Pressure or Velocity on the flat plate accordingly.
05:48 This is the velocity contour. Toggle on the Color legend.
05:52 To do this, click on the color legend icon on the Active Variable Control menu.
05:59 Click Apply in the Object inspector menu.
06:02 In the Object inspector menu, click on Display.
06:06 Scroll down and click on Rescale to data range.
06:12 Let me shift this Color legend on top to visualize the Vector Plot. Go to the Filters Menu > Common > Glyph.
06:24 Go to the Properties in Object Inspector menu.
06:29 Click Apply on the left hand side of Object Inspector menu.
06:33 You can change the number of vectors by changing their size at the bottom.
06:38 Also, the size of the vectors can be changed by clicking on the Edit button. The Set Scale Factor can be changed to 0.1.
06:50 Again, click the Apply button.
06:53 Now let me zoom this.
06:55 To do this, in the Active Variable Control menu, click on zoom To Box option
06:61 and zoom over any area that you desire.
07:07 We can see the parabolic variation of the vector plots as the flow moves over the plate.
07:13 Delete this. Now delete the vector plot.
07:18 Also, we can see that the color near to 1 corresponds to the velocity of 0.99 times the free stream velocity.
07:26 You can also plot the variation of velocity along the X and Y axes using the plot over data line.
07:35 This brings us to the end of the tutorial. In this tutorial, we learnt:

Geometry and meshing of the flat plate geometry and Vector plotting in ParaView.

07:46 As an Assignment- Create a geometry of flow over a flat plate.Refine the grid spacing near the plate.
07:54 Watch the video available at this URL: http://spoken-tutorial.org/What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project. If you do not have good bandwidth, you can download and watch it.

08:04 The Spoken Tutorial Project team:

Conducts workshops using spoken tutorials. Gives certificates to those who pass an online test. For more details, please write to: contact@spoken-tutorial.org

08:38 Spoken Tutorial project is a part of the Talk to a Teacher project. It is supported by the National Mission on Education through ICT, MHRD, Government of India.
08:48 More information on this mission is available at this URL:http://spoken-tutorial.org/NMEICT-Intro.

This is Rahul Joshi from IIT Bombay, signing off.Thanks for joining.

Contributors and Content Editors

DeepaVedartham, PoojaMoolya, Pratik kamble, Sandhya.np14

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