Channel flow simulation using CONVERGE CFD
Aim: To create a channel flow simulation in CONVERGE for different base mesh sizes:-
- dx = 2e-4m, dy = 2e-4m, dz = 2e-4m
- dx = 1.5e-4m, dy = 1.5e-4m, dz = 1.5e-4m
- dx = 1.0e-4m, dy = 1.0e-4m, dz = 1.0e-4m
Objectives:
1. Velocity and pressure contours for all the 3 base mesh sizes
2. Show mesh (i.e surface with edges) for the 3 base mesh sizes
3. Plots for velocity, pressure, mass flow rate and total cell count for all 3 base mesh sizes.
4. Explain the plots and write a detailed report on your project.
5. Upload an animation to youtube and attach the link here using the add media option. Explain the development of the flow using the animation.
Note: Please view the project in Full Screen
Since the dimensions of the channel were not mentioned, therefore I used the same dimensions of the channel as shown in the video lectures
Dimensions of the channel:-
dx = 0.1m, dy = 0.01m, dz = 0.01m
Results:
Mesh 1:-
Velocity Contour
Pressure Contour
Mesh(Surface with edges)
Mesh(Surface with edges) - closeup
Mesh 2:
Velocity Contour
Pressure Contour
Mesh(Surface with edges)
Mesh(Surface with edges) - closeup
Mesh3:
Velocity Contour
Pressure Contour
Mesh(Surface with edges)
Mesh(Surface with edges) - closeup
Plots:-
RED = Mesh1 , BLUE = Mesh2 , YELLOW = Mesh3
Velocity at inlet
Velocity at Outlet
Total Pressure at inlet
Total Pressure at outlet
Mass flow rate at the inlet
Mass flow rate at the outlet
Total Cell count
Explanation:
Velocity Profile inside the channel:-
Assuming:
1 Newtonian Fluid
2 Neglecting effect of gravity(g=0)
3 Steady Flow(all-time derivatives tend to zero)
4 Neglecting velocities in Y and Z directions(V= 0,W=0){since,the pressure difference is only in the axial direction i.e X}
Therefore the equations reduce to:
`frac(delp)(delx) = mu(frac(del^2U)(dely^2))`
The pressure gradient can be assumed as: `(P_o - P_i)/L`
Therefore integrating the equation `1/mu(P_o-P_i) = (frac(del^2U)(dely^2))` twice with respect to y
`U = −frac( ∆P )(2µL) y ^2+ C_3y + C_4`
Boundary Conditions:
No slip conditions at either extreme ends:
Therefore,at `y=d/2`, U = 0
at `y =-d/2`,U =0
hence we get two simultaneous equations to solve for `C_3` and `C_4`
`0 = −frac(∆P )(8µ L )d ^2 + C_3 d/2 + C_4`
`0 = −frac(∆P)( 8µ L )d ^2 − C_3 d /2 + C4`
On sollving the above equations we get:
`C_3 = 0` , `c_4 = frac(∆P d^2 )(8µL)`
Therefore the function of velocity is:
`U = frac(∆P )(2µl) [d^ 2 /4 − y 2 ]` which is a parabolic profile
[The same was observed in the velocity contour of our simulation after converge was reached]
[This analysis can be taken further to find the mass flow rate:
`dot m = rho(intUdA)`
`dot m = rho(int_-{w/2}^{w/2} int_-{d/2}^{d/2}Udzdy)` , where d is height of the cross-section and w is the width of the cross-section
on integration we get
`dot m =rho frac{∆P d^3w }{12µL}`, This is the \'Haigen Poiseuille Equation\' for a Rectangular Tube`
for air as the fluid of density 1.225 kg/m^3 and viscosity 18.1 micro pascal-second, L =0.1m,w =0.01m,d=0.01m,∆P = 1 Pa
we get mass flow rate at both inlet and outlet as 5.63*10^-4 kg/s which is very small, as indicated in the mass flow rate plots.]
[The Pressure Plots and contours obtained are in accordance with our pre-set conditions of 1Pa pressure difference between the inlet and outlet]
Animation:
Development of flow in a Pipe:
At the entrance, the fluid has uniform velocity. As it enters into the pipe the fluid at the pipe surface interface sticks to the surface and assumes zero velocity(No-slip condition),As the shear stresses are a function of the velocity gradient(Newton\'s Law of Viscosity `tau = mu frac[delu][dely]`) Therefore the shear stresses at the centre are zero and increase linearly up to a max value at walls. Hence, the velocity of the fluid is maximum at the centre and decreases in a parabolic fashion to zero at the walls`.The fluid takes a certain length to develop completely into a parabolic profile of velocity, this length is called the 'Hydrodynamic entrance length' and after this length, it maintains this parabolic profile till the outlet which is called the fully developed region.
This can be observed in the above animation as the fluid near the exit side has more pronounced of a parabolic velocity than near the entrance side
As far as the mass fraction simulation of N2 goes, it indicates that the fluid conditions inside the channel initially do not hinder the flow from the inlet to the outlet side. The fluid inside gets washed away due to the pressure gradient between inlet and outlet and the flow resumes as usual.