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August 26, 1998, 09:41 
Cell Reynolds Number

#1 
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Hi, I'm a beginner of CFD. In some paper, I read this words "cell Reynolds number". But I'm not sure what exactly this words mean. Any explanation will be very appreciated. Thanks.


August 26, 1998, 10:06 
Re: Cell Reynolds Number

#2 
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Hi. It probably refers to the local Reynolds number based on the velocity in the calculation cell.


August 26, 1998, 10:11 
Re: Cell Reynolds Number

#3 
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...and the length scale of the cell.


August 26, 1998, 10:28 
Re: Cell Reynolds Number

#4 
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Thanks for your responses. But what is the constraint of cell Reynolds number for flow with high Reynolds number ?


August 26, 1998, 11:40 
Re: Cell Reynolds Number

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The cell Reynolds number can be taught as The Peclet Number after Roache. The Reynolds Number is the ratio of Inertial to Viscous forces. While the Peclet Number is the Ratio of Convective to Diffusive Fluxes across the Cell. Therefore at low Peclet number, diffusion of a property is of higher importance than convection of that property across the cell.
In order for a discretization scheme to behave appropriately for different Peclet numbers it must be transportiveness, which is one of the fundamental properties of a discretization scheme. All this says is, that for low Peclet numbers the discretization schemes must behave as thought diffusion is the dominant characteristic, while for high Peclet number the discretization scheme must behave as thought convection of the property is the dominant characteristic. Hope this is helpful 

August 26, 1998, 11:46 
Re: Cell Reynolds Number

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There are various ways that local Reynolds numbers can be useful, including as an idea of the local stability characteristics of a discretization scheme. Some schemes have critical Re above which you get oscillations, etc... Another useful local variable is the CFL number defined as V*dt/dx where dt and dx are (local) time step and grid scale respectively. Every scheme has a theoretical maximum CFL number.


August 26, 1998, 12:51 
Re: Cell Reynolds Number

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Assuming that you are writing the finitedifference equations of the NavierStokes equations in Cartesian coordinates, ( use central difference or onesided upwinded difference scheme), you can group the coefficients of the final finitedifference equations in such way that some will appear as Re,x , Re,y , Re,z . where Re,x is defined as (rho * U * deltx)/mu, Re,y is defined as (rho * V * deltay)/mu, and Re,z is defined as (rho * W * deltaz)/mu.These are teh local cell's Reynolds number because U,V,an W are local nodal point velocity components.(U=U(i,j,k)), V=V(i,j,k), and W=w(i,j,k) ). And the deltax, deltay, and deltaz are the local cell sizes in x, y, and z directions. ( deltax= X(i+1,j,k) X(i1,j,k), deltay = Y(i,j+1,k)Y(i,j1,k), deltaz = Z(i,j,k+1)  Z(i,j,k1) ). There is nothing magic about the definition of these cell Reynolds number at this stage because you will get these in the process of deriving the finitedifference equations. The problem is when you try to solve this set of finitedifference equations, you have to use some kind of finite size mesh. Based on the mesh size you use, you can change the values of the local cell Reynolds numbers, because they are directly related to the deltax, deltay, and deltaz. When you increase the cell size ( using coarse mesh), you automatically increase the cell Reynolds numbers ( the coefficients of your finitedifference equations ). In your equations, you have terms coming from the lefthand side of the momentum equation ( convection term) , you also have terms coming from the right hand side of the momentum equation. As a result, in your final finitedifference equations, some terms will have positive coefficient, and some terms will have negative coefficient. Because not all terms have local cell Reynolds number attached to them, when you use a coarse mesh, some terms will become larger because the cell Reynolds number will increase with cell size. Sometimes, this will change the nature of the finitedifference equation. ( negative terms can be enlarged to an extent that they become larger than the rest of the terms simply because you are using coarse mesh. ). Numerically, when you try to solve the equation, you can see the solution wiggles because you have just changed the nature of the equations. In early days, when people use central difference for both the firstoder convection term and the secondorder diffusion terms, there were positive and negatice coefficients in the final equation. As the cell Reynolds number becomes greater then two, the nature of the finite difference equation was changed because of the cell size used. The result was instability in the solution ( sometimes the solution diverged). By using the "upwind" onesided difference , you can obtain the finite difference equation with all the coefficients remain the same sign regardless of the cell size you select. At least using this method, you can avoid the stability problem related to the cell size. In other words, when you use certain finitedifference methods, you can easily change the nature of the equation. ( you don't have this problem with the original partial differential equations because the cell size is approaching zero.) This is a big step in the CFD development. Once you can handle the stability problem, you would like to get more accurate solutions by using more accurate schemes ( first order upwind scheme tends to smear the solution profiles. more diffusive. ) The best way to get the feeling is to use the 1D model equation with convection and diffusion terms, and write a simple program with various difference schemes to check the sign of the coefficients and the final solutions. ( you don't get this part by just running a commercial code.)


September 8, 1998, 07:01 
Re: Cell Reynolds Number

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John,
I thought the cell Peclet number was used, not the cell Reynolds number. What's the difference? Is there a difference? I suppose one looks at fluxes the other considers absolutes. 

September 8, 1998, 07:47 
Re: Cell Reynolds Number

#9 
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Oops, of course Pe = RePr. Doh...


March 19, 2015, 12:39 
About Matleb Programe for reynold's equation solution by finite diffrence method

#10 
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Suchit
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you have any matleb programme in which Reynold's equation is solved by finite difference method???????
i am doing work on this bus not getting any reference work for complete convergent and divergent shape for parabolic shape.... My mail id : shahsuchit007@gmail.com 

March 19, 2015, 12:40 

#11 
New Member
Suchit
Join Date: Mar 2015
Posts: 8
Rep Power: 2 
you have any matleb programme in which Reynold's equation is solved by finite difference method???????
i am doing work on this bus not getting any reference work for complete convergent and divergent shape for parabolic shape.... My mail id : shahsuchit007@gmail.com 

March 19, 2015, 12:42 
About Matleb Programe for reynold's equation solution by finite diffrence method

#12 
New Member
Suchit
Join Date: Mar 2015
Posts: 8
Rep Power: 2 
you have any matleb programme in which Reynold's equation is solved by finite difference method???????
i am doing work on this bus not getting any reference work for complete convergent and divergent shape for parabolic shape.... My mail id : shahsuchit007@gmail.com 

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