Week 2 - Flow over a Cylinder.

AIM

            Simulate the steady and unsteady  flow of air over a cylinder and explain the phenomenon of Karman vortex street, lift and drag coefficients.

 

OBJECTIVE

1. Simulate the flow with the steady and unsteady case and calculate the Strouhal Number for Re= 100.
2. Calculate the coefficient of drag and lift over a cylinder by setting the Reynolds number to 10,100,1000,10000 & 100000. (Run with steady solver)
3. Discuss the effect of Reynolds number on the coefficient of drag
4. So The total number of cases is as below

• Case 1 – Re=10, Steady Flow
• Case 2 – Re=100, Steady flow
• Case 3 – Re=100, Laminar Flow
• Case 4 – Re=1000, Steady Flow
• Case 5 – Re=10000, Steady Flow
• Case 6 – Re=100000, Steady Flow

 

INTRODUCTION

             Flow around objects exists from commonly observed movement of animals in nature such as the flight of birds ranging from the smallest in size to the largest to man-made applications viz., in heat exchangers, chimney stacks, and commercial and cargo flight aircrafts respectively and a growing significance to understand the different fluid flow phenomena like lift and drag coefficients, boundary layer separation, vortex shedding, etc., has been extensively exploited in recent years and in many cases to solve existing issues with the previously mentioned engineering applications. One such peculiar interest that gained importance over time was the flow around a circular cylinder and more specifically to comprehend the phenomenon of Karman vortex street which is when the wake behind the circular cylinder becomes unstable beyond a certain value of Reynolds number

              In this project, we are going to simulate the flow over a cylinder inorder to capture the vortex shedding that is formed behind the cylinder surface. This is done for various cases by varying the velocity of the fluid flow for each case inorder to change the reynolds number and so by taking the user defined fluid with constant density of 1 kg/m^3 and viscosity of 0.05 kg/m-s. This simulation is done with steady state and transient state for Reynolds number is 100 and so as to analyse the vortex formation for both the cases and also to calculate the Strouhal number.

 

THEORY

 

Karman vortex 

             In Fluid Dynamics, a Karman 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 Fluid around blunt bodies. It is named after the engineer and fluid dynamicist Theodore Von Karman.

             Vortex shedding is a phenomenon when the wind blows across a structural member, vortices are shed alternately from one side to the other, and where alternating low-pressure zones are generated on the downwind side of the structure giving rise to a fluctuating force acting at right angles to the wind direction.

             Vortices are a major component of turbulent flow. The distribution of velocity, vorticity (the curl of the flow velocity), as well as the concept of circulation is used to characterize vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis.

                  For common flows (the ones which can usually be considered as incompressible or isothermal), the kinematic viscosity is everywhere uniform over all the flow field and constant in time, so there is no choice on the viscosity parameter, which becomes naturally the kinematic viscosity of the fluid being considered. On the other hand, the reference length is always an arbitrary parameter, so particular attention should be put when comparing flows around different obstacles or in channels of different shapes: the global Reynolds numbers should be referred to the same reference length. This is actually the reason for which most precise sources for air foil and channel flow data specify the reference length at a prefix to the Reynolds number. The reference length can vary depending on the analysis to be performed: for a body with circle sections such as circular cylinders or spheres, one usually chooses the diameter; for an air foil, a generic non-circular cylinder or a bluff body or a revolution body like a fuselage or a submarine, it is usually the profile chord or the profile thickness, or some other given widths that are in fact stable design inputs; for flow channels usually, the hydraulic diameter about which the fluid is flowing.

                    For an aerodynamic profile, the reference length depends on the analysis. In fact, the profile chord is usually chosen as the reference length also for the aerodynamic coefficient for wing sections and thin profiles in which the primary target is to maximize the lift coefficient or the lift/drag ratio (i.e. as usual in thin air foil theory, one would employ the chord, Reynolds, as the flow speed parameter for comparing different profiles). On the other hand, for fairings and struts the given parameter is usually the dimension of internal structure to be streamlined (let us think for simplicity it is a beam with circular section), and the main target is to minimize the drag coefficient or the drag/lift ratio. The main design parameter which becomes naturally also a reference length is therefore the profile thickness (the profile dimension or area perpendicular to the flow direction), rather than the profile chord. 

                   The range of Re values will vary with the size and shape of the body from which the eddies are being shed, as well as with the kinematic viscosity of the fluid. Over a large Re range (475 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. The alternation leads to the core of a vortex in one row being opposite the point midway between two vortex cores in the other row, giving rise to the distinctive pattern shown in the picture. Ultimately, the energy of the vortices is consumed by viscosity as they move further downstream, and the regular pattern disappears.

                   When a single vortex is shed, an asymmetrical flow pattern forms around the body and changes the pressure distribution. This means that the alternate shedding of vortices can create periodical lateral (sideways) forces on the body in question, causing it to vibrate. If the vortex shedding frequency is similar to the natural frequency of a body or structure, it causes resonance. 

             Some of the real-life examples of Vortex shedding are

     Physics behind Vortex formation:

                           As a Fluid particle flows towards the leading edge of a cylinder, the pressure on the particle rises from the free stream pressure to stagnation pressure. The high fluid pressure near the leading edge impels flow about the cylinder as a boundary layer develops about both sides. The high pressure is not sufficient to force the flow about the back of the cylinder at high Reynolds number. Near the widest section of the cylinder, the boundary layer separates from each side of the cylinder surface and form 2 shear layers. Since the innermost portion of the shear layers, which is in contact with the cylinder, moves much more slowly than the outermost portions of the shear layers which is in contact with the free flow, the shear layers roll near the wake.    

Understanding the physics behind vortex shedding is quite important from an engineering point of view. For example, if the bluff structure is not mounted rigidly and the frequency of vortex shedding matches the resonance frequency of the structure, then the structure can begin to resonate and starts vibrating. These vibrations can cause severe damage to the structure.

                        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 Karaman 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 = f D/V, is approximately equal to 0.21 when the Reynolds number is greater than 1,000.

Wake

     Wake is the region of recirculating flow immediately behind a moving or stationary blunt body, caused by viscosity, which may be accompanied by flow separation and turbulence. It is basically the region of disturbed flow downstream of a solid body moving through a fluid.  

Wake pattern created by a small boat

Reynold’s Number

            Reynold’s number is a dimensionless quantity that is used to determine the type of flow pattern as laminar or turbulent while flowing through a pipe. Reynolds number is defined by the ratio of inertial forces to that of viscous forces.

  The formula is given by,

                                             

  Where,

ρ = Density of fluid Kg/m³

D = Diameter of Cylinder m

v = Velocity of Fluid m/s

µ= Kinematic Viscosity of Fluid m²/s

 

Strouhal Number 

             Strouhal number is a dimensionless number describing oscillating flow mechanisms. The Strouhal Number represents a measure of the ratio of the inertial forces due to the unsteadiness of the flow or local acceleration to the inertial forces due to changes in velocity from one point to another, in the flow field.

            The frequency at which vortex shedding takes place for an infinite cylinder is related to the Strouhal number.

 The Strouhal number is often given as,

                            

 

Where, 

f = Frequency of Vortex Shedding 

L = Characteristic Length (for example, hydraulic diameter or the airfoil thickness)

U = Flow Velocity.

 

Lift Co-efficient

             The lift coefficient (CL) is a dimensionless coefficient that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. CLCL is a function of the angle of the body to the flow, its Reynolds number and its Mach number.

 

Where,

F_L = Lift Force

ρ = Fluid Density

A = Projected Area of Object with respect to which lift force is calculated

v = Flow Velocity.

 

 

Drag Co-efficient

           Drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. A lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.

 

Where,

F_D= Drag Force

ρ = Fluid Density

A = Projected Area of Object with respect to which drag force is calculated

v = Flow Velocity.

 

SOLVING AND MODELLING APPROACH

 

            The computational domain for the flow over a cylinder is constructed in space claim

Calculation of velocity for different Reynold’s Number

Density ρ = 1

Velocity v

Diameter D = 2m

Dynamic viscosity µ = 0.05

    

With is equation different velocities are calculated for different Reynold’s Number

 

 

 

PRE PROCESSING AND SOLVER SETTING

            In our challenge we will create a flow simulation of air over a cylinder and observe lift drag and vortex shedding. In general I will be explaining only one case and posting the results of the other cases.

 

Case 1

                        Reynold’s Number      - 10     

Flow                            - Steady State

 

            Step 1 : Open the geometry Space Claim. Create the fluid domain for the problem as below in space claim. 

Once this process is done. We will close the space claim and open the mesh module.

 

            Step 2 : Open the mesh module and under the mesh details give CFD Fluent as preference and Element size of 5mm.

                        After the meshing is done check for the quality criterion. Check whether the mesh quality is above 5%. Once this is done name the faces of the volume as Inlet, Outlet, Cylinder wall & Symmetry Wall

 

Step 3 : After the meshing and face naming are done move on to Fluent Solver. In the fluent launcher select double precision, display mesh after reading and give the appropriate solver processors and GPUs.

              In the ANSYS CFD we need to give all the conditions, parameters and models. To start with go to physics menu and click general settings. In that select Pressure based type solver, Absolute velocity formulation and steady state flow. Next we need to give the model. For this case Laminar is selected.

Next is the flow material selection. By default air is selected. Density and viscosity are given

Next the boundary conditions of the inlet and other are set as per the problem statement. For the first case the velocity of the inlet is set to 0.25m/sec.

              After the physics part is done move to the solution part. Initialise the simulation first. click Initialize to initialize the boundary conditions. Hybrid method is selected before initialization. Click on Autosave to obtain a animation of the flow in the post results.

              Then run the calculation for a given number of iterations till the convergence is obtained. Depending upon the number of elements and model selected the time required for convergence varies. In the CFD module itself we can compute, measure, plot, animate, etc., if needed.

 

Step 4 : After the solutions and calculations move to the result module to get different graphs, plots, contours, animations etc.,. Sectional views can be created if required.

 

 

RESULT

 

Case 1

Reynold’s Number      - 10     

Flow                            - Steady State

Residual Plot

 

Velocity contour

 

Pressure contour

 

velocity Vector

 

Vertex Average

 

Lift Coefficient Cl

 

Drag Coefficient Cd

 

Strouhal Number St

 

Animation

 

 

Case 2

Reynold’s Number      - 100   

Flow                            - Steady State

Residual Plot

 

Velocity contour

 

Pressure contour

 

velocity Vector

 

Vertex Average

 

Lift Coefficient Cl

 

Drag Coefficient Cd

 

Strouhal Number St

 

Animation

 

Case 3

Reynold’s Number      - 100   

Flow                            - Transient State

Residual Plot

 

Velocity contour

 

Pressure contour

 

velocity Vector

 

Vertex Average

 

Lift Coefficient Cl

 

Drag Coefficient Cd

 

Strouhal Number St

 

Animation

 

 

Case 4

Reynold’s Number      - 1000

Flow                            - Steady State

Residual Plot

 

Velocity contour

 

Pressure contour

 

velocity Vector

 

Vertex Average

 

Lift Coefficient Cl

 

Drag Coefficient Cd

 

Strouhal Number St

 

Animation

 

 

Case 5

Reynold’s Number      - 10000           

Flow                            - Steady State

Residual Plot

 

Velocity contour

 

Pressure contour

 

velocity Vector

 

Vertex Average

 

Lift Coefficient Cl

 

Drag Coefficient Cd

 

Strouhal Number St

 

Animation

 

Case 6

Reynold’s Number      - 100000         

Flow                            - Steady State

 

Residual Plot

 

Velocity contour

 

Pressure contour

 

velocity Vector

 

Vertex Average

 

Lift Coefficient Cl

 

Drag Coefficient Cd

 

Strouhal Number St

Animation

 

For all the cases lift coefficient, Drag coefficient , Vertex average velocity and strouhal number are plotted using ANSYS. The below table shows the values corresponding to the Reynold’s number.

 

This graph plots the Reynolds number against Strouhal Number. From the graph it is evident that the Strouhal number increases as the Reynold’s number increases. Which means that oscillating flow mechanism increases as the Reynold’s number increases. Vortex shedding is directly proportional to the Strouhal number.

 

The graphs plots the Reynolds number against lift and drag coefficients. From the graph we can conclude that the Drag coefficient decreases as the Reynold’s number increases and lift coefficient increase as the Reynold’s number increases.

 

This graph plots the Reynolds number with the vertex average velocity. From the graph it can be understood that as the Reynolds number increases the vertex average velocity increases. Initially gradual increase in the Reynolds number doesn’t affect the vertex velocity. But when the flow is turbulent with a high Reynolds number, the vertex velocity increases drastically. This means that the vortex shedding phenomenon happens highly at high Reynold’s number.Vortex Shedding, which is responsible for the unsteady separation of Fluid around blunt bodies.

 

 

CONCLUSION

 

1. The fluid flow over a cylinder has successfully been simulated for the Reynold’s Numbers – 10 , 100 , 1000 , 10000 and 100000 and the respective plots and contours have been obtained.

 

2. From the plots and contours, we can observe that no significant vortex shedding is observed in very low Reynold’s Numbers. Therefore, to observe the vortex shedding phenomenon, we need to simulate the flow at higher Reynold’s numbers by increasing the fluid velocity at the inlet.

 

3. Also, from the simulations run, we can observe that the transient state simulation takes a very high simulation time to compute the results and the number of iterations or timesteps it takes to reach a steady state is also very high. Therefore, transient state simulation requires much higher simulation time and the time taken to reach a steady state is very high.  

 

4. From the numerical results obtained from the simulation, we can observe that as the Reynold’s Number increases, the drag co-efficient is substantially decreasing. This means that, at high Reynold’s numbers, the effect of drag due to the boundary layer is less.

 

1.  As the Reynold’s number increases Drag coefficient Decreases and lift coefficient increases.
2. Transient state capture more accurate vortex shedding than steady state solving.