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November 21, 2011, 09:29 
Nondimensionalizing Navier Stokes

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Vincent
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This post is in response to a message I recieved of someone that wanted to have a more detailed explanation on how to obtain the nondimensional NS equations. Since others might be interested (and the fact that I like this tex compiler) I posted on this forum.
For starters we introduce the Navier Stokes equations and look at the xdirection component: We assume no additional body forces. The components of the velocity field v are denoted by subscripts (x,y,z). The pressure is given by p and is the density. First we divide by the density in order to simplify the equation. The reader should check that indeed each terms has a dimension equal to . Now in order to obtaint the nondimensional equation we make the following substitutions. Quantities with a superscript star are dimensionless quantities and the capital letters V and H have dimensions and respectively. Please check that these substitutions are indeed valid. For instance the physical length is 4 m. We decide to take the length H equal to 1 m, giving a nondimensionalised length of 4. Using the above substitutions we end up with the following equation: We simplify this equation and use that T = H/V. Here is our inverse reynolds number. Where Re is defined as . Now we have nondimensionalised the NS equations. Please rememeber that if you solve this equation you will end up with the nondimensionalised quantities. In order to revert them to physical quantities you will need to use the equation we proposed above when we nondimensionalised the quantities. Good luck! Regards, Vincent 

November 21, 2011, 22:08 
the force term problem

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wuzj
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if there are force term in the NS equation, how to deal with it??
for example, when use the immersed boundary method to solve the flow past fixed circular, what should i do with the force term nondimension??? 

November 22, 2011, 07:03 

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Vincent
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In the case of an additional body force (like gravity) the equation changes to include an additional term.
Again dividing by yields this term indeed as an acceleration . If we then substitute in the following way: We end up with the following equation: So for instance if we want to simulate gravity where . Then we first determine by multiplying by H and dividing by . For instance, let's say we took H = 0.4 m and V = 2 m/s, this gives that is equal to 9.8*0.4/4 is 0.98. Good luck! Regards, Vincent 

November 22, 2011, 07:40 

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Vincent
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A simple immersed boundary method can be implemented in the following way. The philosophy behind the idea is that we will determine the fluid flow without the obstacle and the in a next step force the fluid flow to zero using a body force.
Introducing a phase indicator function that can either be one (solidphase) or zero (fluidphase). By multiplying our body force with this phase indicator function we only imply a force where our obstacle is. However we can't use a simple constant for the forcing term since we want this opposing force to set the velocity exactly to zero and not a positive or negative value. So we need a parametrization for our body force. The forcing in a certain cell can be expressed in the following form: Here is our phase indicator function as defined above. is our numerical timestep the velocity in a certain cell at the current timestep before forcing. Our desired velocity is which is equal to zero in the case of a solid. Combining these elements gives that our full equation: Now we need to supply a geometry field () and the desired velocity field (which for simple cases will be zero everywhere). Good luck! Regards, Vincent 

May 3, 2012, 08:28 

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Astio Lamar
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Dimensionless NaviorStoks in cylindrical coordinate :
nondimensional parameters which we use here: for rcomponent we have: Last edited by asal; May 3, 2012 at 10:35. 

May 3, 2012, 10:39 

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Astio Lamar
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for thetacomponent, we have:
and finally for zcomponent, we have: 

January 24, 2013, 23:03 

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Osman
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Hi VincentD, could you tell me how to choose V and H ???


November 25, 2013, 07:00 

#9 
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teja
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Hi,
I need to non dimensionalize energy equation of navier stokes. Can you help me out 

August 21, 2015, 12:39 
Nondimensionalization of NV equations in cylindrical coordinates

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Ahmed Aql
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Please, Thank you for the great explanation. I have a question, how didn't you nondimensionalized theta?! and is it okay to express the azimuthal (tangential) velocity with the same reference parameter as the axial and radial velocities? thank you!


August 21, 2015, 13:14 

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Filippo Maria Denaro
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theta is not dimensionally homogeneous to a lenght that needs to be nondimensionalized but it is an angle position (rad).


August 21, 2015, 17:08 

#12 
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Ahmed Aql
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Thank You! but what will be wrong with the below dimensionless parameters?
I considered theta as a reference variable to dimensionalize the tangential coordinate I also expressed the tangential dimensionless velocity considering the angular velocity as a reference variable. 

August 21, 2015, 17:18 

#13  
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Filippo Maria Denaro
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Quote:
No, theta ranges from 0 to 2*pi is already a nondimensional number, furthermore, the velocity reference is unique 

August 21, 2015, 17:26 

#14  
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Ahmed Aql
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Quote:
I can see now how theta is already dimensionless. I am still a little bet confused about how the tangential velocity can be expressed in terms of a the unique reference velocity (Vr) which has [ L / T ] dimensions while omega r have a dimension [ T ^1] ? Thank you so much 

August 21, 2015, 17:33 

#15  
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Filippo Maria Denaro
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Quote:


August 21, 2015, 17:40 

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Ahmed Aql
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September 14, 2015, 08:45 

#17 
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Fábio Mallaco Moreira
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It should be noted this seems to be a derivation for an incompressible flow.


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