# Vorticity evolution in laminar flows
**1**. **A universal time scale for vortex ring formation -by Gharib(1997)**
### Key points
1.
**1**. **Circulation and formation number of laminar vortex rings - by Gharib(1998)**

### Key points

1. As the duration of piston stroke(discharge time) is increased, GRS showed evidence for the existence of a limiting process that imposes an upper bound on the circulation a vortex ring can acquire for a given set of flow parameters. They suggested that this limiting process is the outcome of the Kelvin-Benjamin variational principle for steady axis touching vorterx rings.
2. It state that the kinetic energy of impulse-preserving rearrangements of the vorticity field by an arbitrary divergence-free velocity field is maximum for a steady vortex ring.
3. Formation time/Non-dimensional time($t^*$)
$$t^* = \frac{U_{avg}*t}{D}$$
where $U_{avg}$ is the mean velocity discharged from a tube of diameter $D$ and $t$ is the time.
4. Formation time when the discharge of fluid stops is thus given by $T^* = \frac{L_m}{D}$ , wher $L_m$ is the maximal stroke.
5. Formation number is defined as that formation time at which vortex ring have achieved their maximum circulation. Any further discharge of fluid does not increase the vortex ring circulation. Instead the excess vorticity is accumulated in the tail of the vortex.
6. Based on their experimental study, they have concluded that there is a maximum circulation that a vortex ring can acquire as the maximal piston stroke increases. This maximum is reached at a piston stroke ratio of $\frac{L_m}{D}$ ~ 4.
7. The total circulation was calculated by integrating the vorticity in the entire domain of computation. The circulation of the vortex is more subjective, especially before a clear pinch-off is observed(i.e for $t^*$ < 12 ). It was estimated by calculating the circulation inside a polygon that included the vortex ring to the best of our judgement.
The above figure shows the variation of total and vortex circulation. An examination of the vorticity field indicates that the total circulation is constant as long as the tail of the vortex ring is away from the axis of symmetry. As the tail reaches to the axis of symmetry significant vorticity cancellation is observed and the total circulation decays at an increase rate.
8. The formation number is the formation time when the total circulation imparted by the discharging flow is equal to the circulation of the pinched-off vortex ring. In the calculation of formation numbers of of all cases a vortex circulation $\Gamma_{vortex}$ = 2.3 is used and it is found out to be 3.6.
9. They defined the pinch-off to occurwhen the contour line of 5% of the maximal vorticity in the core of the vortex encircles the vortex for the first time
The above figure shows the variation of $\Gamma_{vortex}$ vs $T^*$ at $t^*$ ~ 10
10. They reveals that the total circulation depends on both the velocity programme and velocity profile.
11. Velocity profile which they have used mainly are uniform (impulse, trapezoidal and linear) parabolic(impulse).
12. For parabolic velocity profile vortex centre is closer to the axis of symmetry and its shape is more slender because of the larger axial velocity near the axis of symmetry, and the total circulation increase rate is four times larger than that of the uniform velocity case and due to this vortex ring disconnects itself later.
13. The formation number depends weakly on the velocity programme but strongly on the velocity profile of the discharged flow.It does not depend, however on the discharge time beyond a threshold value of $T^*$~4.
14. Vortex circulation is relatively insensitive to the formation conditions, once its asymptotic state ($T^*$>4) has been reached. Total circulation depends strongly on both velocity programme and profile.
15. The circulation is scaled by $U_p$*$D$. It is also possible that other scaling velocities can be determined such that even smaller variation will be obtained in the scaled vortex circulation.

## Results

## Computational Domain
The axisymmetry model used in our simulation is shown below:-

Inlet Diameter ($D_i$)=20mm
Outlet Diameter ($D_o$)=80mm
Upstream Length ($L_u$)=120mm
Downstream Length ($L_d$)=300mm

Non-Dimensional time $$T^* = \frac{\bar{U}*t}{D_i}$$
where $t$ is time and $\bar{U}$ is average velocity.

Non-Dimensional vorticity $$\omega^* = \frac{\omega*D_i}{\bar{U}}$$
where $\omega$ is vorticity.

Non-Dimensional Circulation $$\Gamma^* =\frac{\Gamma}{D_i*\bar{U}}$$
where $\Gamma$ is Circulation

Reynolds Number $$ Re = \frac{\bar{U}*D_i}{\nu}$$
where $\nu$ is kinematic viscosity and it's value used in all cases is fixed i.e $3*10^{-6}$ $m^2/s$

## Grid Independence

## Validation
We have validated our case with Rosenfeld.et.al paper in which there $Re$ $No.$ is 2500 and no forcing is present.The velocity profile is uniform at inlet but it will be time dependent means fluid flow will be stoped after certain time. The Non-dimensional time $T^*$ at which discharge will be stoped is also known as discharge time. In their paper they have considered the discharge time of 6.

## Note
All our results will be based on $Re$ $No.$ of 2000 and discharge time of 6 but there will be axial forcing
The inlet velocity profile is given below:-
$$U = \bar{U}(1+A\sin(2*\pi*f*t))$$
where $A$ is amplitude and $f$ is Frequency in Hz and $\bar{U}$ is 0.3m/s

## Frequency vs CIrculation
### Amplitude (10% of $\bar{U}$)
### Amplitude (30% of $\bar{U}$)
### Amplitude (50% of $\bar{U}$)
### Amplitude (70% of $\bar{U}$)
### Amplitude (90% of $\bar{U}$)

## Amplitude vs circulation
### Frequency (10 Hz)
### Frequency (50 Hz)
### Frequency (90 Hz)
### Frequency (150 Hz)

## Vortex centre distance vs time at fixed frequency and varying amplitude
### Frequency (10 Hz)

### Frequency (50 Hz)

### Frequency (90 Hz)

### Frequency (150 Hz)

## Vortex centre distance vs time at fixed amplitude and varying frequency

### Amplitude (10% of $\bar{U}$)

### Amplitude (30% of $\bar{U}$)

### Amplitude (50% of $\bar{U}$)

### Amplitude (70% of $\bar{U}$)

### Amplitude (90% of $\bar{U}$)

## Vorticity contour plots
### Case 1:- Amplitude (10% of $\bar{U}$) and frequency (10 Hz)


### Case 2:- Amplitude (30% of $\bar{U}$) and frequency (10 Hz)

### Case 3:- Amplitude (50% of $\bar{U}$) and frequency (10 Hz)

### Case 4:- Amplitude (70% of $\bar{U}$) and frequency (10 Hz)

### Case 5:- Amplitude (90% of $\bar{U}$) and frequency (10 Hz)

### Case 6:- Amplitude (10% of $\bar{U}$) and frequency (50 Hz)

### Case 7:- Amplitude (30% of $\bar{U}$) and frequency (50 Hz)

### Case 8:- Amplitude (50% of $\bar{U}$) and frequency (50 Hz)

### Case 9:- Amplitude (70% of $\bar{U}$) and frequency (50 Hz)

### Case 10:- Amplitude (90% of $\bar{U}$) and frequency (50 Hz)

### Case 11:- Amplitude (10% of $\bar{U}$) and frequency (90 Hz)

### Case 12:- Amplitude (30% of $\bar{U}$) and frequency (90 Hz)

### Case 13:- Amplitude (50% of $\bar{U}$) and frequency (90 Hz)

### Case 14:- Amplitude (70% of $\bar{U}$) and frequency (90 Hz)

### Case 15:- Amplitude (90% of $\bar{U}$) and frequency (90 Hz)

### Case 16:- Amplitude (10% of $\bar{U}$) and frequency (150 Hz)

### Case 17:- Amplitude (30% of $\bar{U}$) and frequency (150 Hz)

### Case 18:- Amplitude (50% of $\bar{U}$) and frequency (150 Hz)

### Case 19:- Amplitude (70% of $\bar{U}$) and frequency (150 Hz)

### Case 20:- Amplitude (90% of $\bar{U}$) and frequency (150 Hz)