
[Sponsors] 
November 21, 2011, 09:29 
Nondimensionalizing Navier Stokes

#1 
New Member
Vincent
Join Date: Jul 2011
Posts: 29
Rep Power: 7 
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

#3 
New Member
wuzj
Join Date: Mar 2011
Posts: 6
Rep Power: 8 
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 

#4 
New Member
Vincent
Join Date: Jul 2011
Posts: 29
Rep Power: 7 
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 

#5 
New Member
Vincent
Join Date: Jul 2011
Posts: 29
Rep Power: 7 
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 

#6 
Senior Member
Astio Lamar
Join Date: May 2012
Location: Pipe
Posts: 175
Rep Power: 6 
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 

#7 
Senior Member
Astio Lamar
Join Date: May 2012
Location: Pipe
Posts: 175
Rep Power: 6 
for thetacomponent, we have:
and finally for zcomponent, we have: 

January 24, 2013, 23:03 

#8 
Member
Osman
Join Date: Oct 2012
Location: Japan
Posts: 51
Rep Power: 6 
Hi VincentD, could you tell me how to choose V and H ???


November 25, 2013, 07:00 

#9 
New Member
teja
Join Date: Nov 2013
Posts: 2
Rep Power: 0 
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

#10 
New Member
Ahmed Aql
Join Date: Aug 2015
Location: Kuwait
Posts: 4
Rep Power: 3 
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 

#11 
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 3,060
Rep Power: 35 
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 
New Member
Ahmed Aql
Join Date: Aug 2015
Location: Kuwait
Posts: 4
Rep Power: 3 
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  
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 3,060
Rep Power: 35 
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  
New Member
Ahmed Aql
Join Date: Aug 2015
Location: Kuwait
Posts: 4
Rep Power: 3 
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  
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 3,060
Rep Power: 35 
Quote:


August 21, 2015, 17:40 

#16 
New Member
Ahmed Aql
Join Date: Aug 2015
Location: Kuwait
Posts: 4
Rep Power: 3 

September 14, 2015, 08:45 

#17 
New Member
Fábio Mallaco Moreira
Join Date: Sep 2015
Posts: 1
Rep Power: 0 
It should be noted this seems to be a derivation for an incompressible flow.


Thread Tools  
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Filtered navier stokes equation..LES::  Palani Velladurai  Main CFD Forum  7  September 6, 2013 02:51 
Adding a term to Navier Stokes Equation  ashtonJ  CFX  3  January 15, 2011 07:32 
Navier stokes compresible viscid flow fea, somebody can help?  Jose Choy  Main CFD Forum  3  October 24, 2003 02:28 
Newbie:Viscoelasticity and Navier stokes equation  Rajil Saraswat  Main CFD Forum  2  June 9, 2003 07:21 
help: I am trying to solve Navier Stokes compressible and viscid flow  Jose Choy  Main CFD Forum  2  May 18, 2000 05:45 