Week 2 - Flow over a Cylinder.

FLOW OVER A CYLINDER

 

Objectives:

• To simulate the two-dimensional laminar flow of a fluid over a cylinder. using steady and transient simulations.
• To visualize vortex shedding in laminar flows.
• To study the effect of Reynolds number on coefficient of drag and lift.

 

 

Introduction:

The flow of a fluid over a cylinder is a very important topic of study in the field of fluid dynamics. The nature of this flow depends to a large extent, on the Reynolds’ number of the flow. The Reynolds number for a given flow changes with variations in the flow velocity and the value of kinematic viscosity. As the fluid passes over the circular cylinder, a kind of disturbance is seen in the flow which arises due to the uphill pressure gradient. The flow is typically separated into two channels on either side of the body, arising from 2 different vortices. The direction of these two channels is opposite to each other. However, with the increase in Reynolds number, a complex flow pattern is generated.

It is found that beyond a critical Reynolds number of approximately ‘40’ a system of alternating vortices can be observed during flow over the cylinder. This swirling pattern of flow arising due to the alternating vortices near the cylinder is known as Von Karman Vortex Street. It is caused by a phenomenon called vortex shedding, responsible for the unsteady separation of fluid flowing around bluff bodies.

Karman Vortex Street is a kind of periodic flow which can be identified by definite frequencies. The vortices are formed on either side of the symmetry plane. However, the point of flow separation is determined by the upstream boundary layer. It is typically observed after a critical Reynolds number Re_cr ≈ 40.

It has been observed that at Re < 5, the flow remains steady and is attached to the cylinder. At Re=5, the flow starts separating into two symmetric channels with circulation zones having direction opposite to each other. This happens on account of the symmetric counter rotating vortices formed behind the cylinder.

After Re_cr ≈ 45, the flow becomes unsteady and Von Karman vortex street becomes visible. Transverse oscillations are set due to the shedding vortices and the symmetry of the flow breaks down. This vortex fluctuation can break down far downstream for sufficiently large values of Re, at a sufficient distance from the cylinder to again develop a uniform flow.

It has also been found that for a Reynolds number in the range of 180-400, three dimensional instabilities are introduced which lead to a streamwise vortex structure.

The phenomenon of vortex shedding has great importance in engineering applications. As the oscillating flow exerts an oscillating force on the body over which the fluid flows, it is responsible for the ‘singing’ of bodies like suspended telephone and overhead power lines, antenna of a car, etc.

If the frequency of oscillation of the flow coincides with the natural frequency of the structure, it may cause resonance and may further lead to failure of structures such as chimneys, skyscrapers, bridges, overhead cables, submarine periscopes, marine structures etc.

Thus, it is important to study vortex shedding to understand its significance in engineering applications.

 

 

Approach:

The main aim of this project is to study the phenomenon of vortex shedding and observe Von Karman Vortex Street for laminar flows.

Part 1: Observing Von Karman Vortex Street using steady state and transient simulation and determining the Strouhal’s number.
A flow having Reynolds number of 100 was simulated using both, transient and steady state simulations, to make accurate observations of the Karman Vortex Street. Strouhal’s number was determined using the transient simulation. This helped to find out the frequency of oscillation of the vortices.

Part 2: Observing the effect of Reynolds number on the coefficient of drag.
This was done using a steady state simulation.
Different values of Re (10, 100, 1000, 10000, 100000) were setup by varying the flow velocity (0.25, 2.5, 25, 250, 2500) respectively. This helped to find the relationship between coefficient of drag (C_D) and Reynolds number.

Geometry and mesh for both the parts of this project were kept the same.

 

Relevant terms:

 

Co-efficient of drag (C_D)

Drag co-efficient is a dimensionless number that is used to quantify the drag on a body moving through a fluid. It relates the drag force to the fluid flow velocity, fluid density and associated area of cross section. A lower value of drag coefficient signifies that a lesser amount of aerodynamic drag will act on the body.

C_D = 2F_D/ρv²A

F_D: Drag force

ρ: Density of fluid

v: Flow velocity

A: Associated area

 

Co-efficient of lift (C_L)
Drag co-efficient is a dimensionless number that relates the lift force to the fluid flow velocity, fluid density and associated area of cross section. It helps to determine the lift acting on the body moving through a fluid.
It also depends on the angle of attack, Reynolds number and Mach number.

C_L = 2F_L/ρv²A

F_L: Lift force

ρ: Density of fluid

v: Flow velocity

A: Associated area

 

Strouhal number (St)
It is a dimensionless number that describes oscillating flow mechanism. Quantitatively speaking it is the ratio of frequency to the mean speed of flow.

St = (f D)/v    i.e. (frequency x diameter)/flow velocity

Since it relates to the frequency of oscillations, Strouhal number for a given flow can only be obtained using transient simulations.

It typically takes a value between 0.1 to 0.3.

 

Geometry:

A planar geometry is created to simulate a 2D flow of fluid over the cylinder.

Diameter of cylinder= 2m

Effective length of flow= 60m

Effective width of flow= 20m

The boundary along the flow direction is symmetrical with respect to the cylinder.

 

 

Meshing:

To capture the geometry and physics accurately, the following methods were employed in meshing:

Element size- 250mm       Cell count- 40502

Triangles – Triangle dominated unstructured mesh

Edge sizing – A total of 36 divisions for the cylinder fluid interface

Inflation – 6 inflation layers with growth rate 1.2 and base layer thickness 5mm

            

      

Average mesh quality: 96%

Named selections: Inlet, Outlet, Wall, Symmetry

 

 

PART 1: Re=100

Setup

   

    

Viscous model: Laminar

Solution method scheme: SIMPLE

Materials:

 

Re= (ρ v D)/µ

Re: Reynolds number

ρ: Density=1kg/m³

v: Velocity of flow = 1m/s

µ: Dynamic viscosity = 0.02ks/m-s

Thus, Re = (1x1x2)/0.02 = 100

 

Steady State

Initialization method: Hybrid

No. of iterations: 1400

Residuals-

 

Velocity Contour-

Pressure Contour-

Velocity at monitor point vs iteration-

Pressure at monitor point vs iteration-

Lift coefficient-

Drag coefficient-

 

Since the Strouhal’s number is related to frequency, we need to conduct a transient simulation to calculate St.

 

 

Transient

Time step: 0.1     No. of timesteps: 500

Density: 1kg/m³    Dynamic viscosity: 0.05kg/m-s

Flow velocity: 2.5m/s

 

Residuals

Velocity contour

Pressure contour

Lift coefficient

Drag coefficient

Strouhal’s number

 

By observing the graph, the Strouhal Number is determined to be around 0.17

 

 

 

Part 2: Re and C_D- Steady State

Density = 1kg/m³         Dynamic viscosity ()= 0.05kg/m-s

 

Re = 10 (v= 0.25m/s)

The flow velocity is very low. The coefficient of drag is nearly 0. Viscosity can be neglected at this value of Re. Since there is no unsteady separation of flow, the wake length is very short and the boundary layer sticks to the surface of the cylinder.
The flow is symmetric and no oscillations are seen at this Reynolds number. There is no drag on the cylinder.

 

Re = 100 (v= 2.5m/s)

At Re=100, the flow separates. However, it is steady and transverse periodic oscillations can be seen due to the alternating vortices.
The drag on the cylinder is high due to the vortices.

 

Re = 1000 (v= 25m/s)

The flow is still laminar at this value of Re. Vortex shedding is still visible and due to the high flow velocity, and the wake length is large. Very high drag is produced on the cylinder.

 

Re = 10000 (v= 250m/s)

This flow was in the turbulent region and thus, Realizable K-epsilon turbulent model was use for this case and the next case.
The velocity was increased further. The oscillations are no longer prominent. A chaotic nature of flow is observed.
The boundary layer on windward side of the cylinder was laminar. However, chaotic fluctuations in the flow were observed on the leeward side. The wake for Re=10000 was lesser than for the previous case. Thus, drag was lesser.

 

Re = 100000 (v= 2500m/s)

This is the case with highest flow velocity and Reynolds number. The flow is of chaotic and turbulent nature. The separation point is slightly downstream as compared to other cases. So the wake is smaller than the previous case and thus, a lower drag acts on the cylinder.

 

Conclusion:

As the Reynolds number stays below a value of around 40, the flow is steady, laminar and boundary layer is attached to the cylinder. No fluctuations take place and there is no drag on the cylinder.

However, beyond the critical Reynolds number, vortex shedding can be observed and the drag on the cylinder increases with increase in Re. This is the region when the Von Karman Vortex Street is visible.

For a range of Reynolds number where the flow is transitioning from laminar to turbulent, there is a decrease in the drag, to a value below the laminar drag. Beyond this range in the turbulent region, the drag increases with increase in Reynolds number.

 

References:

https://www.sciencedirect.com/science/article/pii/S0307904X08000243

http://www2.eng.cam.ac.uk/~mpj1001/learnfluidmechanics.org/LFM_blank_notes/handout_8_v3.pdf

http://www2.eng.cam.ac.uk/~mpj1001/learnfluidmechanics.org/LFM_blank_notes/handout_8_v3.pdf