A15013950 wrote: ↑22 Jul 2018, 11:01
How do you calculate initial k and omega when using that turbulence model for external aerodynamics?
I will go over all the "major" variables that people may or may wish to calculate here - there are ways to skip to the ones you want, but this way will give you all that you need, I hope.
One equation for
turbulent kinetic energy (k), in J/kg, is:
This basically reads as "k" being equal to the time average of the velocity fluctuations in one direction squared, all divided by 2. So a rough way to calculate this would be to take your domain's inlet velocity, multiply it by your turbulence intensity (i.e. 5% = 0.05 ), and then square it. From there, you divide by 2, but since we only looked into one direction, a crude way of approximating this throughout the domain is to multiply that final result by 3; one for each axis, since there will be fluctuations in velocity in all directions.
Alternatively, you can split them out into each separate axis if you have a vector with a non-zero second or third component. If that is the case, do the same thing but use this equation below:
For Omega, its a little more involved... Assuming you know the air Pressure [Pa] and Temperature [K}, then you can use the ideal gas law equation to calculate your air density:
From there, we need to use Sutherland's Law for calculating the
fluid dynamic viscosity of air as it relates to temperature:
which can be written as:
, where
Taking "
" as being equal to 0.000001458 and "C" as being equal to 110.4 (these are commonly used values of dynamic viscosity of air at a specific temperature within CFD literature), you can then calculate back through to get your
actual at your given temperature in [ Pa . s ].
Next we need to calculate the
turbulent length scale [m] of the flow, which can be done crudely (yes, I am aware that the k-
model has it's own definition of the length scale utilizing the relationship between k, epsilon and
) by:
Note that
is the
hydraulic diameter and so depending on your shape of domain, there will be a different equation for calculating the equivalent circular section. For a rectangular domain, use this equation:
Next, we can calculate
Epsilon [J / kg . s ] by using the k-
specific coefficient
= 0.09 along with the turbulent length scale we just worked out and the turbulent kinetic energy (k) from before:
Next we need to calculate something called
nuTilda which is our turbulent viscosity [ m^2 /s ]. We can do that by
Finally, for
Omega [J / kg . s ]
Where "k" is the turbulent energy,
is the density,
is the molecular dynamic viscosity and
is the eddy viscosity ratio.
Hope this helps!!