Steady Vs Unsteady flow over a cylinder

Aim:

• To simulate flow over cylinder and observe phenomena of Von Karman Vortex
• Compare steady and unsteady flow over cylinder for various Reynolds number

Introduction:

Flow around cylinder represents many practical applications such as flow around high-rise buildings, bridges, towers, chimneys, heat exchangers etc. A lot of numerical and experimental research is going on related to flow over cylinder to understand the complex unsteady dynamics of cylinder wake flow. Cross flow normal to axis of cylinder is frequently encountered flow in engineering applications.

Governing Equations:

The equations given below are solved in solution domain to capture the flow physics. The equations are for 3D geometry, but here we will model 2D geometry for study.

Continuity: `∂ρ/∂t+(∂(ρu))/∂x+(∂(ρv))/∂y+(∂(ρw))/∂z=0`

X-momentum: `(∂(ρu))/∂t+(∂(ρuu))/∂x+(∂(ρuv))/∂y+(∂(ρuw))/∂z=-∂P/∂x+(∂τ_xx)/∂x+(∂τ_xy)/∂y+(∂τ_xz)/∂z+ρg_x`

y-momentum: `(∂(ρv))/∂t+(∂(ρvu))/∂x+(∂(ρvv))/∂y+(∂(ρvw))/∂z=-∂P/∂y+(∂τ_yx)/∂x+(∂τ_yy)/∂y+(∂τ_yz)/∂z+ρg_y`

z-momentum: `(∂(ρw))/∂t+(∂(ρwu))/∂x+(∂(ρwv))/∂y+(∂(ρww))/∂z=-∂P/∂z+(∂τ_zx)/∂x+(∂τ_zy)/∂y+(∂τ_zz)/∂z+ρg_z`

While simulating flow over cylinder vortex shedding will appear in flow field. Vortex shedding is responsible for vibrations in body, if it is not properly designed then it may lead to damage to body. There are number of parameters which on vortex shedding depends flow parameters like velocity and type of fluid and geometrical parameters like shape and size of object.

In this study steady and unsteady state simulations are performed for flow over cylinder (Re =100) and vortex shedding effect is observed for both cases.

Some dimensionless numbers required for this study are:

Reynolds Number:

`Re=ρVD/μ`

Where,

- Density of fluid

V- Velocity of fluid

D- Characteristic length of object

- Dynamic viscosity of object

Strouhal’s Number: it is dimensionless number used to detect vortex shedding.

`S= fD/u`

Where,

f- Frequency of shedding

D- Characteristic length of object

u- Velocity of fluid

Work Flow of Project

Problem Definition:

Diameter of Cylinder: 2 m

Upstream distance of cylinder: 20 m

Downstream distance of cylinder: 40 m

Reynolds number of flow: 100

The geometry used for simulation is as shown in above figure.

Meshing:

Meshing is needed to divide computational domain into number of small volumes on which we will solve the above stated governing equations. Sufficiently fine mesh is generated to capture the rapid change in variables i.e. velocity, pressure etc. due to vortex shedding. An unstructured triangular mesh is used.

As you move towards the wall of cylinder from freestream, the velocity decreases non-linearly up to zero velocity at wall due to no-slip condition. And we can observe large gradient in velocity near cylinder wall. Hence, in order to capture this large gradient correctly an inflation layers are added near wall of cylinder with first layer thickness 1e-3 m.

Mesh Details:

Element type: Triangular

Element size: 0.25 m

Number of inflation layers: 6

First layer thickness: 5e-3 m

Growth size: 1.2

Mesh quality: as shown in figure below.

PART I:

Setting and Results:

User defined fluid is defined in Materials section. With parameters as:

Density of fluid: 1 kg/m³.

Viscosity: 0.02 N-s/m².

This parameters will give Reynolds number as 100. Hence, viscous model is selected as laminar.

• Steady State Analysis:

ANSYS Fluent solver is used for simulation. The double precision solver setting with pressure based solver and steady state analysis is chosen. And continuity and momentum equations are used for solution.

• Transient Analysis: The double precision solver with pressure based and transient state analysis is better to approach such problem which are naturally transient in nature.

for transient analysis

time step size:0.1 sec.

number of time steps: 1800

number of iterations in each time step: 20

Boundary conditions:

Inlet: Velocity Inlet, V= 1 m/s

Outlet: Pressure Outlet, Gauge pressure =0 Pa

Operating Pressure: 1Bar

Wall: No-slip condition.

Report definition:

The additional quantities of interest which are required while post-processing results are included using report definition tab.

Initialization: the hybrid initialization method is used to initialize the flow field. In this initialization the solver solves the potential flow equations i.e. Euler’s equation.

Solution Monitoring:

In order to monitor the flow of fluid over cylinder the run-time animation is created.

Results:

Results of both the cases i.e. steady and unsteady state solvers are compared here.

• Residuals:

• Velocity fluctuations at monitor point:

• Lift and Drag Coefficient Plots:

• Velocity Contour:

Case 2: calculate drag and lift for flow over cylinder with Re=10, 100,1000,10000 and 100000

to change the Reynolds number the velocity is varied accordingly.

• Re = 10, V=0.1 m/s, viscosity = 0.02 kg-m/s

Residual Plot:

Lift coefficient:

Drag Coefficient:

Lift and drag coefficient values:

Monitor point velocity:

Velocity and pressure contour:

• Re = 100 ,V=1 m/s, viscosity = 0.02 kg-m/s

Residual Plot:

Lift coefficient:

Drag coefficient:

Lift and Drag coefficient values:

Velocity and pressure contour:

• Re = 1000, V=10 m/s, viscosity = 0.02 kg-m/s

Residual Plot

Lift coefficient plot

Drag Coefficient Plot

Lift and drag coefficient values

Velocity and pressure contour

• Re = 10000, V=100 m/s, viscosity = 0.02 kg-m/s

Residual Plot

Lift coefficient plot

Drag Coefficient Plot

Lift and drag coefficient value:

Velocity and pressure contour

• Re = 100000, V=1000 m/s, viscosity = 0.02 kg-m/s

Residual Plot

Lift coefficient plot

Drag Coefficient Plot

Lift and drag coefficient values

Velocity and pressure contour

Re	Cd	Cl
10	3.31897	-0.004028
100	1.299	-0.064
1000	0.6946	0.3141
10000	0.5892	0.2398
100000	0.5301	0.2673

• Strauhal Number Calculatation:

• Steady state simulation:

As we know from the definition of the strauhal number that it is directly proportional to the frequency of the vortex shedding.

`S= fD/u`

and we cannot calculate the frequency of vortex shedding for staedy state simulation, hence we cannot calculate the strauhal number.

• Transient Simulation:

in case of transient (unsteady) simualtion the strauhal number is calculated from the FFT plot.

with the help of MS Excel the median of Strauhal Number values are calculated and which comes out to be near to 0.2657.

Conclusion:

following conclusions can be made from the study:

1. Karman Vortex street is generated in both steady and unsteady state solver
2. the frequency of vortices depends on the reynoldsnumber of flow
3. the strauhal number is calculated in transient simuation only as it gives the property variation with respect to flow time and from that we cam calculate the frequency of oscillation of vortices.