How to achieve an oscillatory fluid motion inside a pipe
 

I am trying to solve a FSI (fluid-structure interaction) problem in COMSOL. Fluid flowing through the pipe is deforming the solid attached inside the pipe. It has a womersley number of 7. For this model, my inlet BC is a pulsating pressure wave defined by the formula-
P(inlet)= P_mean+ P_amp*cos (wt) (time dependent analysis)
My outlet BC is zero pressure (0 Pa) condition.

With this set of BC's, I was hoping to get a oscillatory motion of fluid which will travel from inlet to outlet and reverse it's direction and start flowing from outlet to inlet. I am calculating p_mean and p_amplitude values by using two pressure values found out experimentally using the same formula as stated above : P(inlet)= P_mean+ P_amp*cos (wt)

But due to p_mean > p_amplitude, the overall value of pressure at all time steps is always positive and there is no flow reversal from outlet to inlet.

So I am guessing my pressure equation is wrong (calculation method for P-mean and P_amplitude) or my BC's are wrong (especially outlet BC to be 0 Pa). Hence I wanted to know how can I calculate p_mean and p_amplitude for such a case and how can I achieve oscillatory motion of the fluid? Your help is appreciated.

Thanks.

I think that you can obtain such effect simply using a periodical pipe and a driving force (the forcing pressure gradient) that changes in time its sign.

see for example:

https://www.researchgate.net/profile...df36000000.pdf

I personally used the setting illustrated in 5.1 here

https://www.researchgate.net/publica...ection_methods

Hi Dr.Filippo,

Thanks for your quick reply. I read the papers suggested by you. They were indeed very helpful for a clearer understanding. However, I have a few doubts related to the concepts illustrated in it. I request you to help me in this regard.

In your paper the total pressure gradient is given as,
P(tot_gradient) = P_residual+ P_base +P_oscillatory --- (1)

Looking at this equation, I could interpret it as the P_residual term to add a scalar quantity to ensure a continuity, P_base term being space dependent, it adds the vector and P_oscillatory being space and time dependent, it adds a tensor form to the equation. So adding all these terms would give a pressure equation which if enforced at the inlet, fluid flow will be pulsatory (or oscillatory?) in nature. Please tell me if I understood it correctly.

So I have a few doubts related to this,
1) How can I calculate the P_residual term (∇p)? Do I need to use the divergence principle? I am not sure what can be the right method to get the value of P_residual.
2) P_base term has a pressure difference term (Δp0). Is it the pressure difference in between the inlet and outlet in my case?
3) P_oscillatory also has Δp0 term. Is it the same as given in the P_base formula? Also how to get the value of α ? Is it safe to assume it as 1 for oscillatory flows?
4) I could imagine the equation to define a pulsatory flow having their mean as a nonzero value. Please tell me if my understanding in this regard is wrong and how does the equation needs to be altered to achieve oscillatory flow?
5) So after I calculate all the pressure component values, can I use the equation (1) as my inlet pressure boundary condition keeping the outlet boundary condition as zero pressure? (0 Pa)
Thank you for your guidance.

Hello, here my ideas about your questions









In your paper the total pressure gradient is given as,
P(tot_gradient) = P_residual+ P_base +P_oscillatory --- (1)

Looking at this equation, I could interpret it as the P_residual term to add a scalar quantity to ensure a continuity, P_base term being space dependent, it adds the vector and P_oscillatory being space and time dependent, it adds a tensor form to the equation. So adding all these terms would give a pressure equation which if enforced at the inlet, fluid flow will be pulsatory (or oscillatory?) in nature. Please tell me if I understood it correctly.

P_residual=f(x,y,z,t) is the scalar function ensuring the correct gradients that enforce the divergence-free constraint
P_base=g(x) is the linear decreasing function along the streamwise direction that force the main flow. It can be seen as the pressure for steady laminar flow.

So I have a few doubts related to this,
1) How can I calculate the P_residual term (∇p)? Do I need to use the divergence principle? I am not sure what can be the right method to get the value of P_residual.
Yes, it requires to compute the Poisson equation obtained from the continuiti equation Div v=0

2) P_base term has a pressure difference term (Δp0). Is it the pressure difference in between the inlet and outlet in my case?
Yes, Δp0 is the pressure difference along the length Lx taken along the streamwise direction. Note that using periodic condition, no velocity profile must be prescribed as inlet/outlet conditions, they simply develop from the pressure forcing.

3) P_oscillatory also has Δp0 term. Is it the same as given in the P_base formula? Also how to get the value of α ? Is it safe to assume it as 1 for oscillatory flows?
This is an input that depends on the flow physics you want to study.

4) I could imagine the equation to define a pulsatory flow having their mean as a nonzero value. Please tell me if my understanding in this regard is wrong and how does the equation needs to be altered to achieve oscillatory flow?
Have a look to eq.s(34-37) in my paper.

5) So after I calculate all the pressure component values, can I use the equation (1) as my inlet pressure boundary condition keeping the outlet boundary condition as zero pressure? (0 Pa)
The eq.(1) enters as forcing term in the momentum equation, is not a BC.s as they are periodic. Conversely, if you want to fix inflow and outflow, you need to prescribe a time-depending oscillatory velocity profile at the inlet and suitable Neumann condition at outlet. Note that you cannot fix simoultaneously velocity and pressure on the same boundary.

Hi Dr.Filippo,

Thank you for your help. I followed your advice and ran my analysis with pressure equation as the forcing term in the momentum equation. I am assigning this pressure equation at the inlet of the pipe. However, to get the oscillatory profile, I had to use same equation with sin term in it to introduce a phase difference in between inlet and outlet. See below-
P(inlet)= P_mean+ P_amp*cos (wt) and
P(outlet)= P_mean+ P_amp*sin (wt)

I used this outlet condition so that there will be pressure difference in between inlet and outlet which will govern the flow in the forward and backward direction. Although, I am not sure if the outlet pressure condition is right even when it is giving me oscillatory fluid flow. Please share your opinion on this.

Also, I am still struggling to find the P_residual term of the pressure equation given in your paper. I apologize for not understanding it correctly. Can you please tell me how can I find out P_residual?
I highly appreciate your help.

-Sajiree

the residual pressure is nothing that the solution of the elliptic pressure equation that allows to ensure the divergence-free velocity field