Assuming two-dimensional flow, I'm trying to implement a code for a quasi DNS problem. There is no doubt that i should replace the velocities in Navier-Stokes equation (NSE) with its equivalent velocities in the form of u=u_base + uf, where u_base is the steady state solution for NSE for the channel flow and uf is the perturbed velocity which varies with time. By the new definition of the velocities, one can get new extra terms in NSE, as a function of u_base and uf. My point is to find the fluctuation behavior with time so that i can calculate the growth rate and start an instability study for the flow field.
There are two forms of the new NSE, maybe more but at least these two were the common equations which i found in most of the researches which i read. The first one is in the form of du/dt + div(uu) = - dp/dx+ iv(grad(u))/Re where u was defined earlier. The second form is: du/dt + div(uu) = - dp/dx+ div(grad(u))/Re - omega x u.
Which one should i use for the purpose of instability studies?
Note: You can check the attached files for the non-conservative form of the two previous models.
Dear Dr Ahmed: please take a look at the following useful links:
Conference Paper DNS of Incompressible Square Duct Flow and Its Receptivity t...
Article DIRECT NUMERICAL SIMULATIONS OF PLANE CHANNEL FLOW OVER A 3D...
https://arxiv.org/pdf/1703.08730.pdf
I have some comments:
1) the DNS is nothing but solving the orginal NSE equation without any turbulence model but in a fully 3d unsteady case and using a grid resolution such to resolve all the characteristc scales.
2) It is not clear the setup of the problem. Are you considering a base flow corresponding to the laminar Poiseulle solution with periodic boundary conditions or you want to solve the spatially developing boundary layer?
3) You do not need to transform the equations. After the onset of instability you can simply compute the fluctuating part of the velocity field by subtracting the base flow to the total computed velocity field.
Thank you for your answer Prof Filippo Maria Denaro
I'm considering the flow field as a base flow and fluctuating part. Considering the original NSE equations, after substituting with that assumption, we can subtract the base flow equations from NSE and the result can represent the mathematical model of the evolution of the disturbance with time, which is what my research is aiming to calculate. I want to find the response of the flow field due to these perturbations and calculate the most unstable mode and compare it with that which can be calculated using Linear Stability Theory. I'm looking forward to create a control model so that i can control the evolution of TS waves, hence, delaying the transition.
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