Channel flow
In this challenge we are simulating Channel flow, in which we have inlet at which we set Total (Stable) pressure to be 100001Pa and 100000Pa at the Outlet.Airflow is incompressible in our case. According to thermodynamics because of pressure difference fluid flow will accure from Inlet towards Outlet, so my goal is to simulate and describe the results which I already have after the process is finished.
I want to represent two chapters. First, conserning Theory of channel flow and the Second, post processing the simulation done in converge cfd.
CHAPTER 1
Channel flow is a problem which could be described by poiseuille equations, describes how velocity will be indused in respect of length of the channel. This equation is derived from Navier-stockes equation.
As we know in Navier-stockes equations we have continuity and momentum equations.
Continuity eqaution 1)
In our case flow is incompressible, thus density will be constant, also we assume that velocity will be induced only in one direction (x-direction along to the length of the channel)
According to these assumptions our continuity equation looks like this: `nabla*u=0` divergent of velocity remains to be zero, meaning that if we choose any point in our domain scalar difference between inflow and outflow at that point should be zero.
When it comes to momentum equation in the x-direction. because of we have steady state simulation time dependent variables will be zero, furthermore body forces are neglected and radial momentum eqaution also goes to zero. According to that our momentum eqaution looks like this:
2)
We assume that rate of change of pressure in respect to space (x-dimension) is constant, therefore:
`3)DeltaP=(P_(out)-P_(flow))/L` - where `P_(out)` is pressure at Outlet
`P_(flow)` is pressure at Inlet\'
L is the length of the channel
`4)d/(dr)(r*(du)/(dr)) = r/mu((P_(out)-P_(flow))/L)`
`5)r*(du)/(dr) = r^2/(2mu)((P_(out)-P_(flow))/L)+C_1`
After integrating eqaution No.4 we get equation No.5
after dividing equation No.5 by \"r\" and than integrating it again we get eqaution No.6
`6)u(r) = r^2/(4mu)((P_(out)-P_(flow))/L)+C_1ln(r)+C_2`
Now it\'s time to apply the boundary conditions
1)When r=0 velocity is finite, `ln(r)` is undefined, thus meaning `C_1` equals to zero
2) According to No-slip condition at the channel wall when r=R, u=0.
According to the second BC `6) C_2=-R^2/(4mu)((P_(out)-P_(flow))/L)`
`7)u(r) = (DeltaP)/(4muL)(R^2-r^2)`
According to the equation No.7, which is the function of velocity in respect of channel height when r=0 at the middle of the height of the channel we get maximum velocity.
CHAPTER 2
2.1) all 3 base-size Meshes
Top one is 2e-4mm, middle one is 1.5e-4mm and the last (bottom) one is 1e-4mm base-size mesh.
2.2) Animations of Velocity and pressure distribution for all 3 base-size meshes
2.3) post processing simulations
2.3.1) Mesh size 2e-4mm
Mass Flow
Its clear that continuity converged perfectly
Total Pressure
The plot above shows Total pressure at inlet and outlet. According to bernoulli\'s equations energy is conserved, meaning that total pressure should be constant in out case when we don\'t have power input to the system (Heat or mechanical work). However, in our case we can see little deviation between total pressure at the Inlet and Outlet which could be the result of convergence or friction loss due to boundary layer phenomenon
Static Pressure
Static pressure At the Inlet and at the Outlet is not the same and it\'s intuitive because of energy conservation. As we know total pressure shoud be constant, thus static pressure plus dynamic pressure should be constant. if we have difference in static pressure it means that we should also have difference in dynamic pressure in order total pressure to remain constant, if our static pressure is less at the Inlet than at the Outlet, means that dynamic pressure is higher at the Outlet causing velocity induction towards the Outlet boundary.
Average Velocity
we see that velocity at the outlet is higher than at the inlet, it\'s because what I have already described Dynamic pressure, which is represented as it follows: `rho*V^2/2`, as we discussed dynamic pressure should be greater at the outlet, thus we have higher value of velocity at that Boundary.
Total cell count
Plot over line
According to this plot we bravely can say that simulation is logical, because the highest value of velocity is located at the middle of the height of the channel (what we predicted in CHAPTER 1).
In CHAPTER 1 we also assumed rate of change of pressure to be constant in respect to x-direction.
At the beggining of the flow it\'s not linear but afterwards we should admit that it is, meaning that our assumption was partially, but correct.
2.3.2) Mesh size 1.5e-4mm
Mass Flow
Total Pressure
Static Pressure
Average Velocity
Total cell count
Plot over line
2.3.3) Mesh size 1e-4mm
Mass Flow
Total Pressure
Static Pressure
Average Velocity
Total cell count
Plot over line
According to plots of 1.5e-4mm and 1e-4mm base-size meshes the simulation is not converged, because of some reasons, first total pressure at Inlet and Outlet does not match and dynamic pressure at these boundaries don\'t vary, which is a contradiction to the what we call enegry conservation accordhing to Bernouilli\'s equations. As we decrease the size of base mesh this means that simulation must take more iterations to converge. i didn\'t run simulations with these base-size meshes with more iterations in order to show the difference between the meshes when it comes to converges, On top of that you may ask how i know that my first simulation in which mesh size is 2e-4mm converged within 15k iterations. to make sure that my simulation conveged i run 2e-4mm size mesh simulation once again with the number of iterations 30K with the steady state monitoring.
Instead of increasing the number of iterations i got the same results, meaning that my simulation converged, however it does not mean that this results should meet real results perfectly, but only one reason i did not run simulations with lower base-size mesh is the computational time which is always problem when increasing the number of mesh elements.
After all as you remembered i assumed that deviation of Total pressure was due to convergence or friction loss, according to the last simulation with 30K iterations i should say that this little deviation between Total pressures at the Inlet and Outlet is due to boundary layer, as we transfer fluid further and further along the channel it experience drop in Total pressure (in other words energy loss).