Steady Vs Unsteady flow over a cylinder

Objective:

Simulating fluid flow over a cylinder to understand the vortex formation.

When driving along a road it is a common safety measure to slow down at sharp corner turns in order to avoid being thrown off the road. This phenomenon can be observed as “separation of cars” of the road. A similar process occurs in case of fluid flowing over a curved surface at higher velocities. A fluid climbs the uphill portion of the curved body very easily but finds it difficult to remain attached to the surface at the downhill. At sufficiently high speeds the fluid detaches from the surface of the body. This is termed as flow separation. Flow can separate from a surface even if is fully or partially submerged in a fluid. The region of the flow trailing the body where effect of body on velocity can be felt is called as wake region

An important consequence of flow separation is formation of vortices.

Vortex Shedding: When a fluid flows past a body at certain velocities, depending upon the size and shape of the body, an oscillating flow takes place. The flow vortices which are created at the back of the body detach periodically from the body from either side. This phenomenon is termed as vortex shedding.

The frequency at which vortex shedding takes place for an infinite cylinder is related to the Strouhal Number by following equation:

`St = frac{fD}{u}`

where, `f` is the vortex shedding frequency

`D` is diameter of the cylinder

`u` is the free stream velocity

As Strouhal Number is affected by velocity or Reynolds number, hence change in velocity not only changes the Strouhal number but also affects the physics of flow. In case of flow over cylinder when the Reynolds number is increased beyond 40 the wake behind the cylinder becomes unstable. Experiments have shown that the wake develop a slow oscillation in which velocity is periodic in time and downstream distance. The amplitude of the oscillation increases downstream. The oscillating wake rolls up into two staggered rows of vortices with opposite sense of rotation.

Von-Karman found this phenomenon and concluded that a non-staggered row of vortices is unstable and a staggered row is stable only if the ratio of lateral distance between the vortices to their longitude distance is 0.28. Because of similarity of the wake with footprints in street, the staggered row of vortices behind the body is called von Karman Vortex Street. The vortices move downstream with a speed less than the upstream velocity. In our current simulation the downstream velocity is observed by using a monitor point behind the body.

Reynolds Number in range 40 to 80, the vortex street does not interact with the pair of attached vortices. As `Re` is increased the vortex street forms closer to the cylinder and the attached eddies themselves begin to oscillate. Finally, the attached eddies periodically break off alternately from the two sides of cylinder. While an eddy on one side is shed, that on the other side forms resulting in an unsteady flow near the cylinder. As vortices of opposite circulations are shed off alternately from the two sides, the circulation around the cylinder changes the sign, resulting in an oscillation lift or lateral force. If the frequency of vortex shedding is close to the natural frequency of cylinder then a substantial amount of lateral vibration culminates in the body.

Drag Coefficient: It is a dimensionless quantity used to measure the amount of resistance or drag faced by a body when it is moving in fluid with certain velocity. Mathematically it is given as

`C_D = frac{2F_D}{rhou^2A}`

Where, `F_D` is the Drag Force in the direction of movement in fluid

`rho` is the density of fluid

`u` is the velocity of object relative to fluid

`A` is the frontal area

Lift Coefficient: It is a dimensionless quantity that yields lift generated by a body when moving in fluid of density rho with certain velocity. Mathematically it is given as

`C_l = frac{2F_l}{rhou^2A}`

where, `F_l` is the Lift Force in the direction normal to the direction of fluid

`rho` is the density of fluid

`u` is the velocity of object relative to fluid

`A` is the frontal area

Solving and Modelling:

Geometry: The geometry is created in Ansys Space Claim. Bounding Box method is implemented for the same. Details of geometry and bounding box:

Diameter of cylinder = `2m`

Length, Breadth of Bounding Box = `60m, 20m`

The centre of cylinder is placed `20m` from the inlet of bounding box as shown in figure below.

A monitor probe point is placed `8m` behind the centre of the cylinder to analyse the physical properties like velocity.

Meshing: The element size used for generating mesh is 0.25. 6 layers of inflation layer is added to refine the mesh with a growth factor of 1.2

Solution Configuration:

The present study is divided into two parts:

In both the parts:

1. Lift coefficient plot is used to determine the convergence of the flow.
2. A material with following values of density is preferred as fluid `rho = 1 kg/m^3`, `mu = 0.05 kg/ m.s`

Hence, Reynolds Number `Re = frac{rhouD}{mu}` => `Re = 40u`

3. Diameter of cylindrical body , `D=2m`

PART - I Running simulation for `Re=100` in steady state and unsteady state

Case	1. Steady State	2. Unsteady State
Flow	Laminar	Laminar
Reynold's Number	100	100
Velocity	2.5	2.5
Iterations/Time Step	500	800
Step Size	-	0.1
Solver	Pressure Based	Pressure Based
Scheme	SIMPLE	SIMPLE
Strouhal Number	0.008	0.16

In both the steady and transient state we observe Vortex shedding in the wake region behind the body.

However, the transient state calculation is more preferred as it accounts the time factor. This can be observed while calculating the Strouhal Number for each case. The only downside of transient calculation is that they tend to be computationally heavy.

Steady State: For calculating the Strouhal Number in case of Steady State, FFT plot was used.

The peak value observed here is for St = 0.008 which is very low owing to the fact that steady state calculation does not account for time factor.

Transient State: For Transient State, frequency was calculated using cycles completed in a given time period in lift coefficient plot and following formula was applied to calculate the Strouhal Number.

`St = frac{fD}{u}`

It can be observed from the below lift coefficient plot that after steady state is reached , the frequency of oscillations of vortices is 2 cycles every 10 seconds.

Hence, `f = 2 / 10 = 0.2`

Hence, `St = frac{0.2 * 2}{2.5} = 0.16` which can be considered more accurate as it is close to the value obtained by experiments.[1]

PART - II Running simulation in steady state for varying Reynolds number 10 ~ 100000.

Case : Steady State solver with varying Reynold's Number	Velocity (m/s)	Iterations	Drag Coefficients	Lift Coefficients	Monitor Point Velocity (m/s)
10	0.25	500	3.330	0.00091691	0.10800854
100	2.5	500	1.33418160	-0.099518576	0.97447091
1000	25	500	0.69397210	-0.12076091	17.021389
10000	250	500	0.74596599	-0.21348441	173.88028
100000	2500	500	0.57420545	0.79366316	1936.4769

In case of `Re = 10` it can be observed there is no vortex formation. The monitor point which is behind the body in wake region did not see any change or fluctuation in velocity.

Monitor Point Velocity and Velocity Contour At Re = 10

As soon as we increase the velocity which in turn increases the Reynolds Number to 100, even though the flow remains laminar, fluctuation in monitor velocity can be observed.

Monitor Point Velocity and Velocity Contour At Re = 100

As we further increase the Reynolds Number the Vortex Shedding becomes more prominent as seen below.

Monitor Point Velocity and Velocity Contour At Re = 1000

Monitor Point Velocity and Velocity Contour At Re = 10000

Monitor Point Velocity and Velocity Contour At Re = 100000

Effect of Reynold's Number in Drag and Lift coefficient:

The drag friction has two components:

`C_D` = `C_[D, Friction] = frac{F_[D, Friction]}{1/2 rhoV^2A} + C_[D, Pressure] = frac{F_[D, Pressure]}{1/2 rhoV^2A}`

1. Pressure Drag: The pressure drag is proportional to the frontal area and to difference between pressure between front and rear of the body. Hence, this value is dominant for the blunt shape bodies while it is almost negligible for the thin flat plates parallel to the fluid flow and streamlined bodies. The pressure drag attains high value when the fluid velocity (Reynolds Number) is high and as a result of this the fluid is unable to follow the curvature of the body and thus separates from the body which creates a low pressure zone at the back of the body.
2. Friction Drag: The friction drag is component of wall shear force in the direction of flow, and invariantly it depends on the orientation of the body as well as the wall shear stress. Friction drag is a strong function of viscosity and hence increases with increase in viscosity. However, as Reynold's Number is inversely proportional to viscosity the contribution of friction drag to total drag is very negligible at higher Reynold's Number and in such cases the total drag is mostly due to pressure drag only.

In this set up the Reynold's Number value is controlled by increasing the velocity of fluid, hence, pressure drag component dominates the total drag value. Moreover, the viscosity is kept constant over the range of Reynold's number, hence friction drag contributes very less to the overall total drag value.

Consequently, from plot below we observe that there is negative trend in drag coefficient with respect to Reynold's Number and positive trend in magnitude of lift coefficient with respect to Reynold's number.

Reynolds Number	Velocity	Drag Coefficient	Lift Coefficient	Monitor Point Velocity
10	0.25	3.330	0.00091691	0.10800854
100	2.5	1.33418160	-0.0995186	0.97447091
1000	25	0.69397210	-0.1207609	17.021389
10000	250	0.74596599	-0.2134844	173.88028
100000	2500	0.57420545	0.79366316	1936.4769

Similary the velocity of fluid particles in the wake region of cylinderical body increases as we increase the free stream velocity of fluid(Reynold's Number) however, it remains less than that of the free stream velocity.

Plots of drag and lift coefficient at various values of Reynolds Number.

Re = 10

Re = 100

Re = 1000

Re = 10000

Re = 100000

Link to files : https://drive.google.com/file/d/1A-WAV_WAQtkqF2wreCllMsXX_F-g5bGs/view?usp=sharing