Numerical Analysis of a Transient Flow over a Cylinder

Aim:

To simulate a transient flow over a Circular Cylinder with varying Reynolds number (Re) and determining its Pressure and Velocity animations over changing Reynolds number in the factor of 1.2(20%), 1.4(40%), 2(100%).

Abstract:

Flow over circular cylinders is a simple and useful model for many applications concerning obstacles to flow. This classic case is fairly simple in geometry but it encapsulates all the complexity and variety of the topology of the flow by preserving its interactions and other potentially unstable regions where turbulence is generated at higher Reynold’s number. Generally, the solutions found are of very limited scope and range in validity because of the basic assumptions taken into consideration before evaluation, various large factors such as Basic differencing method, Mesh sizing, Boundary conditions, and Stability & convergence criteria decide a lot on the output values of the solution.

Theory:

Reynold's number (Re) is of fundamental importance in fluid mechanics as it’s essentially the ratio of inertial to viscous forces and is a dimensionless number i.e., it has no units, it is just a number.

`Re = frac{VD}{v}`

or

`Re = frac{rho*V*D}{mu}`

Where,

D = Diameter of the Cylinder

V = Free Stream Velocity

`v` = Kinematic viscosity = `frac{mu(Dynamic Viscosity)}{rho(Density)}`

A Simple illustration below shows different regions of flow conditions over a cylinder.

• Flow Separation: In the presence of the fluid viscosity there forms a layer of fluid particles in the vicinity to the solid surface (or system) where fluid slows down (due to fluid viscous forces) and forms a thin slow-moving fluid layer called a boundary layer.

Case – 1(Flow Separation – Laminar Boundary Layer): The flow velocity is zero at the surface to satisfy the no-slip boundary condition. Inside the boundary layer, flow momentum is quite low since it experiences a strong viscous flow resistance. As we know the Velocity and Pressure are inversely proportional to each other. `frac{partialP}{partialx}`(Pressure Gradient) and `frac{partialV}{partialx}` (Velocity Gradient) is seen to be negative and positive in this regime. Which is favorable for the fluid to flow over the surface.

Note: Boundary Layer separation takes when the `frac{partialP}{partialx}`(Pressure Gradient) or `frac{partialV}{partialx}` (Velocity Gradient) is equal to zero. In this condition, the fluid is too fatigued to flow over the surface anymore.

Case – 2 (Flow Separation – Turbulent Boundary Layer):

In this region of flow over the cylinder (rear-end of the cylinder) mostly the pressure is increasing in the direction of the flow, an adverse pressure gradient condition as so it is called exist. In addition to the presence of a strong viscous force, the fluid particles now have to move against the increasing pressure force. Therefore, the fluid particles could be stopped or reversed (separation bubble), causing the neighboring particles to move away from the surface. If the direction of flow particles is reversed this phenomenon is called the turbulent boundary layer separation.

• Wake: Consider a fluid particle flows within the boundary layer around the circular cylinder. From the pressure distribution, `Pmax` is at the stagnation point and gradually decreases along the front half of the cylinder. The flow stays attached in this favorable pressure region as expected. However, the pressure starts to increase in the rear half of the cylinder and the particle now experiences an adverse pressure gradient. Consequently, the flow separates from the surface and creating a highly turbulent region behind the cylinder called the wake. The pressure inside the wake region remains low as the flow separates and a net pressure force (pressure drag) is produced.
• Vortex Shedding: The boundary layer separates from the surface forms a free shear layer and is highly unstable. This shear layer will eventually roll into a discrete vortex and detach from the surface (a phenomenon called vortex shedding). Another type of flow instability emerges as the shear layer vortices shed from both the top and bottom surfaces interact with one another. They shed alternatively from the cylinder and generates a regular vortex pattern (the Karman vortex street) in the wake. The vortex shedding occurs at a discrete frequency and is a function of the Reynolds number. The dimensionless frequency of the vortex shedding, the shedding Strouhal number, `St = frac{fD}{V}`, is approximately equal to 0.21 when the Reynolds number is greater than 1,000.
• Vortex-Induced Vibrations: When vortices shed from the cylinder, uneven pressure distribution develops between the upper and lower surfaces of the cylinder, generating an oscillatory aerodynamic loading (lift) on the cylinder. This unsteady force can induce significant vibrations on a structure, especially if the "resonance" condition is met. The major reason for a colossal failure at Tacoma bridge, USA.
• Aerodynamic Loading: According to Newton's second law, a time rate change of the linear momentum is equal to the sum of all external forces acting on a system. Therefore, an integration of the linear momentum inside a control volume surrounding the circular cylinder can provide information on the aerodynamic forces (lift and drag) acting on the cylinder. Consequently, there must be also an oscillatory up and downforce acting periodically on the cylinder. This periodic forcing exerting on the cylinder body is responsible for the vortex-induced vibrations as described earlier.

So, the flow around a circular cylinder under the classic case of flow of superfluid (i.e., inviscid in nature), incompressible is generally transverse flow. In such a case the flow after the cylinder body is unidirectional and uniform. With a change in Reynold’s number, different regimes of flows are seen after the cylinder body as shown in the picture.

· Flow Regime – 1: (Re < 5)

The overall flow has no vorticity (i.e., the velocity field is irrotational.) Such a kind of system has total net-drag as zero on the body. As shown in the figure below

In the given figure,

(Stream function around Cylinder without swirl) `Psi = Ursintheta - frac{Ksintheta}{r}`

(Velocity Potential around Cylinder without swirl) `phi = Ursintheta - frac{kcostheta}{r}`

· Flow Regime – 2: (5<Re < 40)

At low Re (less than about 20) the flow is steady (its pattern does not change with time), with the fluid moving in orderly layers (lamina or sheets, so-called laminar flow). After that, a pair of symmetric counter-rotating vortex can be seen. In this flow regime.

· Flow Regime – 3: (40<Re<200)

At the Re = 100, one vortex detaches and gets carried downstream, a new one grows in its place whilst the other vortex detaches - the vortices detach in a periodic and alternating fashion. The flow is no longer steady since it changes periodically over time.

Until the Re becomes 190 laminar periodic vortex shedding is observed (also known as Karman Vortex Street)

· Flow Regime – 4: (200<Re<300)

This regime of flow is called as Williamson regime of Mode A instability(200<Re<260)

At the onset of this regime, it is characterized by dominant spanwise scales having a wavelength of approximately four-cylinder diameters.

Then there is another flow regime generated called as Williamson regime of Mode B instability. At the onset of this regime, it is characterized by a spanwise characteristic length of around one-cylinder diameter. With increasing Reynolds number, the three-dimensional cylinder wake becomes more chaotic(turbulent), which is believed to lead to the reduction in the base suction coefficient defined as the negative of the base pressure coefficient.

· Flow Regime – 5: (300<Re<3*10⁵)

In this flow regime, at Re = 1200 the shear layer separation happens. (But in literature it varies from 300 – 3000). This is a subcritical flow region. It is believed that the separating shear layers are very sensitive to various experimental factors such as free-stream turbulence level, acoustic noise, cylinder vibrations, end boundary conditions, and aspect ratio.

· Flow Regime – 6: (3*10⁵<Re<3.5*10⁵)

In this flow regime, there is a dramatic decrease in the base suction coefficient. This is a complicated flow regime in which there is a transitioning of flow from Laminar to Turbulent. So it is also the critical (Lower transition) flow region.

· Flow Regime – 7: (3.5*10⁵<Re<1.5*10⁶)

This flow regime marks the advent of post-critical Reynold’s number. Boundary-Layer of Cylinder in this region is partly turbulent and partly laminar before separation, it has lower base suction coefficient & lower drag due to later separation of the turbulent boundary layer. So, it is also the Supercritical flow region.

· Flow Regime – 8: (1.5*10⁶<Re<4*10⁶)

In this flow regime. The boundary Layer around Cylinder in this region is completely turbulent at one side before separation, it has a lower base suction coefficient & lower drag due to later separation of the turbulent boundary layer. So, it is also the (Upper transition) flow region.

· Flow Regime – 9: (4*10⁶<Re)

This is the final flow regime in this the boundary layer around Cylinder is completely turbulent at both sides before separation. This region is called a Trans-critical flow region.

Note:

As noticed from the literature reference [1], After the Flow regime – 1

The Vorticity Transport Equations are `frac{partialomega}{partialt} + frac{partialuomega}{partialt} +frac{partialvomega}{partialt} = nu(frac{partial^2omega}{partialx^2} +frac{partial^2omega}{partialy^2})`

Where, (vorticity) `omega = [frac{partialv}{partialx} -frac{partialu}{partialy}]`

t = time,

and u and v are components of velocity in (X – direction and Y – direction) terms of stream function. So, `u = frac{partial Psi}{partialy}` & `v= - frac{partial Psi}{partialx}`

Objective: To measure the Pressure and Velocity animations over changing Reynolds number in the factor of 1.2(20%), 1.4(40%), 2(100%) from the reference baseline Reynold’s number.

Basic Calculations: `Re = frac{rho*V*D}{mu}`

Given Data: (Dynamic vicosity)`mu`= 1.81*10^-5 kg/(m.s)

`rho`(Density) = 1.225 kg/m³

D (diameter of Cylinder) = 0.04m,

Length (Length of Cylinder) = 0.05m

Case - 1: Reference Reynolds number setup

At V(Velocity of fluid flow) = 20m/s

`Re = frac{1.225*20*0.04}{1.81*10^-5}` `approx` 54143.6838

Case - 2: (1.2Re)

1.2Re `approx` 64972.420

`V = frac{Re*mu}{rho*D}` `approx`24m/s

Case - 3: (1.4Re)

1.4Re `approx` 75801.1573

`V = frac{Re*mu}{rho*D}``approx` 28m/s

Case - 4: (2Re)

2Re `approx`108287.3676

`V = frac{Re*mu}{rho*D}` `approx` 40m/s

Case - 1	Re `approx`54,143.6838	V = 20m/s
Case - 2	1.2Re `approx` 64972.420	V = 24m/s
Case - 3	1.4Re `approx`75801.1573	V = 28m/s
Case - 4	2Re `approx` 108287.3676	V = 40m/s

Simulation Setup:

• Create a Cylinder body of length – 0.05m by using the extrude command on a circle sketch of Diameter – 0.04m.
• Use Flow Simulation wizard for presetting the conditions of type of fluid, fluid flow type, reference flow velocity for the first case, and respectively derive another velocity iterations for predefined incremental Reynold’s conditions (1.2Re, 1.4Re, 2Re). Also, Set a time-dependent flow for the study of this model in flow wizard.
• So the reference line velocity is set to be 20m/s in the X-direction. The Reynolds number to this setup condition is Re = 54143.6838
• Now while setting up Computational domain make sure the dimensions are as follows of the domain as shown in the figure
• • Now calculate the Simulation time by Start time = `frac{Total Computaiona Distance}{Fluid Velocity}`and End time = 2*Start time.

  In this equation the total computational domain distance = 10D + 5D + D(Diameter of Cylinder) = 16D = 16*0.04 = 0.64m

Results & Analysis:

Case – 1: Reference Velocity = 20 m/s

Pressure Cut Plot

Velocity Cut Plot

Animations of the Cut Plot are as follows:

Case – 2: Fluid Velocity at 1.2Re = 24 m/s

Pressure Cut Plot

Velocity Cut Plot

Animations of the Cut Plot are as follows:

Case - 3: Fluid Velocity at 1.4Re = 28m/s

Pressure Cut Plot

Velocity Cut Plot

Animations of the Cut Plot are as follows:

Case - 4: Fluid Velocity at 2Re = 40m/s

Pressure Cut Plot

Velocity Cut Plot

Animations of the Cut Plot are as follows:

In these cases, as already discussed in the theory section it is observed there is a maximum pressure found at the front stagnation point which is indicated by the red color in the Pressure plots in each case. This Pressure reduces until it reaches a minimum before the separation point as seen by the green color layer. This creates a negative pressure gradient favorable for the flow around the cylinder body.

After the pressure gradient becomes zero or positive Boundary layer separation takes place these form a wake region behind the cylinder body creating pressure drag as seen by greenish-blue color at the back of the body.

In the Velocity Curves, it can be seen that velocity was in its free stream velocity speed but before it hits the front stagnation point the velocity is reduced and is slow, which gradually increases as it moves across the obstacle(Cylinder body). It is found maximum at the upper and lower points of the cylinder radius seen by red color on the top and tip in velocity Cut-plots. This creates a positive velocity gradient favorable for the flow around the cylinder. After this, the body has a separation point from which there is a turbulent boundary layer in which reverse direction flow and vortices are seen forming at the back of the cylinder body.

Conclusion: All the Flows examined in this study are subcritical flows around the cylinder. Therefore, at the macroscopic level of the research. The Results observed in each case is more or less the same with the arbitrary values of velocity and pressure changing.

Bibliography

[1]	D. Thoman and A. Szewczyk, "Time-Dependent Viscous Flow over a Circular Cylinder," Physics of fluids (AIP), 1969.
[2]	[Online]. Available: http://www.eng.fsu.edu/~shih/succeed/cylinder/cylinder.htm.
[3]	A. Kravchenko and P. Moin, "Numerical studies of flow over a circular cylinder at ReD = 3900," Physics of Fluids (AIP), 2000.
[4]	R. Stringer, J. Zang, and A. Hillis, "Unsteady RANS computations of flow around a circular cylinder for a wide range of Reynolds number," Ocean Engineering (Elsevier), 2014.