NUMERICAL SIMULATION OF LAMINAR FLOW PAST A CIRCULAR CYLINDER

AIM:

SIMULATION OF FLOW OVER A CYLINDER TO CAPTURE THE PHENOMENON OF KARMAN VORTEX STREET

PART-I

• Simulation of the flow with the steady and unsteady case and calculation of the Strouhal Number for Re= 100.

PART-II

• Calculation of the coefficient of drag and lift over a cylinder by setting the Reynolds number to 10, 50, 100, 150, 200, 250, 300. (steady solver)
• Discussing the effect of Reynolds number on the coefficient of drag. [ Results validated with standard literature (https://www.sciencedirect.com/science/article/pii/S0307904X08000243) and error within 5 %]

OBJECTIVES:

• In both approaches, showing the flow has converged. With quantities or plots need to be looked at to determine that the flow has converged.
• Plots of the coefficient of drag and lift.
• Plots to show vortex shedding behind the cylinder.

GOVERNNG EQUATONS:

Naiver Stokes Equation:

Reynolds Number (Re_d):

Re_d = `(ρ × V × d )/ μ`

ρ = Density of fluid

V = Inlet velocity of fluid

d = Diameter of the cylinder

μ = Dynamic viscosity

KÁRMÁN VORTEX STREET

A Kármán vortex street (or a von Kármán vortex street) is a repeating pattern of swirling vortices, caused by a process known as vortex shedding, which is responsible for the unsteady separation of flow of a fluid around blunt bodies.

It is named after the engineer and fluid dynamicist Theodore von Kármán and is responsible for such phenomena as the "singing" of suspended telephone or power lines and the vibration of a car antenna at certain speeds. Re_d range (47<Re_d<105 for circular cylinders; reference length is d: diameter of the circular cylinder) eddies are shed continuously from each side of the circle boundary, forming rows of vortices in its wake.

This formula will generally hold true for the range 40 < Re_d < 150:

where:

• f= vortex shedding frequency.
• d= diameter of the cylinder
• U= flow velocity.

FLOW DOMAIN:

2D geometry constructed using ANSYS SpaceClaim.

Cylinder: Diameter(D) = 2m

BASELINE MESH:

REFINED MESH:

Method: All triangles method

Edge sizing:

Given to the cylinder to get accurate simulation results.

Type: Number of Divisions

Number of Divisions = 36

Inflation Layer:

BOUNDARIES:

MONITOR POINT:

Monitor point created at x=4D downstream. In this wake, flow over the cylinder would help us to analyze and check the occurrence of vortex shedding by capturing velocity or pressure variation at this point.

SOLVER AND SETUP (FOR ALL SIMULATIONS):

Material for all simulation:

Material Assigned to cell Zone (surface): User Defined Material (properties below)

Re_d = `(ρ × V × d )/ μ`

ρ = Density of fluid = 1 kg/m³

V = Inlet velocity of fluid

d = Diameter of the cylinder = 2m

μ = Dynamic viscosity = 0.02kg/m-s

Therefore, Re_d = `(1 × V × 2 )/ 0.02`

Re_d = 100 x V

Varying ‘V’ to get the desired Reynolds number

Solution Method:

The solver used is ANSYS Fluent (Double precision).

Type: Pressure-based

Velocity Formulation: Absolute

Physics Model: Laminar and standard wall functions. (will be used for the entirety of this project since the desired Reynolds numbers are all in the laminar range)

Solution Initialization: Hybrid Initialization

• For convergence we look at both residuals and the velocity variation at monitor point.

PART 1 (Re_d = 100):

Steady Simulation

Time: Steady

Boundary Conditions:

Inlet: velocity magnitude (normal to boundary) = V = 1m/s

Outlet: Gauge Pressure (Pascal) = 0

Wall: Stationary wall

Shear Condition = No slip

Convergence Plot: Number of Iterations given to run = 1500

Number of Iterations to converge = 390

Velocity Variation at Monitor Point:

Velocity Contour:

Drag Coefficient Variation: Drag Coefficient = 1.3352895

Here, Strouhal number cannot be calculated since we can’t determine the Vortex Shedding Frequency.

Unsteady Simulation

Time: Transient

Boundary Conditions:

Inlet: velocity magnitude (normal to boundary) = V = 1m/s

Outlet: Gauge Pressure (Pascal) = 0

Wall: Stationary wall

Shear Condition = No slip

Convergence Plot: Number of Time steps = 1500

Time Step Size = 0.1 s

Velocity Variation at Monitor Point:

Velocity Contour:

Drag Coefficient variation: Drag Coefficient = 1.45

From above velocity and lift coefficient variation we can observe that the flow simulation has converged at flow time 110 s.

From the Lift Coefficient Plot, Vortex Shedding Frequency = `(1.5)/(140-120)` = 0.075

To calculate Strouhal Number:

From Hand Calculation: For Re_d = 100, this formula will generally hold true for the range 40 < Re_d < 150

S_t = `0.198*(1-(19.7/100))`

S_t = 0.158994

From Simulation:

• f= vortex shedding frequency = 0.075
• d= diameter of the cylinder = 2m
• U= flow velocity = 1m/s

S_t = `(0.075 *2)/1`

S_t = 0.15 (equal to the hand calculated value)

PART 2 (Re_d = 10, 50, 150, 200, 250, 300):

Steady Simulation

Time: Steady

Boundary Conditions:

Inlet: velocity magnitude (normal to boundary) = V = 0.1m/s (Re_d = 10), 0.5m/s (Re_d = 50), 1m/s (Re_d = 100), 1.5m/s (Re_d = 150),

2m/s (Re_d = 200), 2.5m/s (Re_d = 250), 3m/s (Re_d = 300)

Outlet: Gauge Pressure (Pascal) = 0

Wall: Stationary wall

Shear Condition = No slip

Convergence Plot: Number of Iterations given to run = 1500

Re_d = 10, V = 0.1m/s

Convergence Plot: Number of Iterations given to run = 600

Number of Iterations to converge = 500

Drag Coefficient = 3.3264739

Velocity Contour:

Re_d = 50, V = 0.5m/s

Convergence Plot: Number of Iterations given to run = 700

Number of Iterations to converge = 50

Drag Coefficient = 1.5944976

Velocity Contour:

Re_d = 150, V = 1.5m/s

Convergence Plot: Number of Iterations given to run = 700

Number of Iterations to converge = 400

Drag Coefficient = 1.1992105

Velocity Contour:

Re_d = 200, V = 2m/s

Convergence Plot: Number of Iterations given to run = 700

Number of Iterations to converge = 350

Drag Coeffcient = 1.09292

Velocity Contour:

Re_d = 250, V = 2.5m/s

Convergence Plot: Number of Iterations given to run = 700

Number of Iterations to converge = 300

Drag Coeffcient = 1.0325197

Velocity Contour:

Re_d = 300, V = 3m/s

Convergence Plot: Number of Iterations given to run = 700

Number of Iterations to converge = 350

Drag Coeffcient = 0.94617954

Velocity Contour:

OBSERVATIONS:

Red (Reynolds Number)	Drag Coefficient (CD)
10	3.3264739
50	1.5944976
100	1.3352895
150	1.1992105
200	1.09292
250	1.0325197
300	0.94617954

INFERENCE:

Values of Reference Material:(https://www.sciencedirect.com/science/article/pii/S0307904X08000243)

Recently Zdravkovich, in an excellent monograph, has compiled almost all the experimental, analytical and numerical simulation data on flow past cylinders, available since 1938 and systematically classified this challenging flow phenomenon into five different flow regimes based on the Reynolds number. In the present study, the computation is restricted only to the first few regimes designated by Zdravkovich as (1) creeping laminar state (L1) of flow ð0 < Re < 4Þ, (2) laminar flow (L2) with steady separation ð4 < Re < 48) forming a symmetric contra-rotating pair of vortices in the near wake, (3) laminar flow (L3) with periodic vortex shedding ð48 < Re < 180Þ and finally (4) part of the transition-in-wake (TrW) regime ð180 < Re < 400Þ when the three-dimensional instabilities lead to the formation of streamwise vortex structure. Computations are carried out in the present work in the range of 1 < Re < 400 for comparison with available measurement data or other computation results for both steady and unsteady flow situations. All the computations use an implicit pressure-based finite volume Navier–Stokes algorithm (RANS3D) for time-accurate prediction of the flow.

As we can see the trend of variation is same for both materials. The values of drag coefficient keeps decreasing as the value of Reynolds number increases. As the value of drag coefficient also converges at high enough velocities the phenomenon of Karman vortex street also disappears. The case where Karman vortex street was captured at ts entirety was at Reynolds number 100. But this also depends on the size of the cylinder also, if we would have increased the size of cylinder as we increased the Reynolds number, vortex shedding could have been observed at those high values.

CONCLUSION:

1. From the graphs of variation, it is observed that drag coefficient is reduced as we increase the Reynolds number.
2. For the Re_d = 10 and 50, the flow is very slow so the inertial forces are much smaller than the viscous forces. That is why there is less perpendicular force (lift), but a higher backward force (drag).
3. At Re_d = 100, the boundary layer seperates just behind the leading edge of the cylinder and a toroidal recirculating region forms. The pressure in the recirculating region is same as at the point of seperation and is lower than the pressure on the corresponding region at the front of the cylinder. This causes "Form Drag".
4. The negative coefficient of lift indicates that the force is downwards.
5. When the fluid material blows across the cylinder, vortices are shed alternately from one side to the other, and where alternating low-pressure zones are generated on the downstream of the cylinder giving rise to a fluctuating force (Lift force) acting at right angles to the flow direction.