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
- Simulate the flow with the steady and unsteady case and calculate the Strouhal Number for Re= 100.
- 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)
- Discuss the effect of Reynolds number on the coefficient of drag
- 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.
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,
FL = 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,
FD= Drag Force
ρ = Fluid Density
A = Projected Area of Object with respect to which drag force is calculated
v = Flow Velocity.
Why does the wake form like this ?.
Near the wall, fluid slows down.Pressure increases as the wall falls away.This pushes the fluid back,which causes boundary layer separation.The separation of the boundary layer can distinguish the wake.The further upstream separation occurs,the more drag is experienced by the cylinder.
Now consider a stream of fluid flow past a circular cylinder.As the fluid passes the top most point of the cylinder,it finds itself
unable to negotiate the rear half of the cylinder where there is an uphill pressure gradient.Hence the fluid separates from the leading surface and the flow is separated individually with highly organized and typically consists of two sequences of vertices,one from each side of the body,with circulations of opposite signs.Depending on the shape and motion of the body more complicated patterns of vortices may arise.
In the wake of the cylinder,the flow pattern consists of an alternating system of vortices or regions of rotation.This repeating pattern of swirling vortices at the wake of the cylinder is known as Karman vortex street.
As stated above vortex street can be observed only over a given range of Reynolds Number.At sufficiently large reynolds number the vortex street persists for many cylinder diameters,but,it can break down far downstream and reorganise itself into a secondary structure.
Strouhal made the first measurements of the frequencies associated with the shedding of vortices by defining a Strouhals number(S). At high Strouhal numbers oscillation dominate the flow while,at low Strouhal numbers oscillations are swept up by the fast moving flow.
S = fD/U
where,
D - Diameter
U - Speed
f - Frequency.
On the practical side ,the Karman street can lead to unwanted noise or even cause failure of structures when the vortex shedding frequency coincides with the natural frequency of the structure.This type of vortex induced oscillations can occur in a variety of situations such as,chimneys,bridges,heat exchanger tubes,overhead power cables and marine structures.Hence,understanding the physics of karman vortex street is crucial to avoide disasters in these situations.
Procedure -
We are supposed to consider 2 cases for simulation,
Case 1) At Re = 100,simulate the flow with steady and transient method and calculate the Strouhal number.
Case 2) Compairing the fluid at steady case with different Reynolds number 10,100,1000,10000 & 100000.
GEOMETRY --> MESHING --> SETUP & SOLUTION --> RESULT.
Geometry Creation -
Meshing -
The generated 2D geometry meshed with element size of 0.25 m.The mesh was generated and proper inflation layers are given around the cylinder all to capture the wall boundary with growth rate of 1.2.Number of the generated elements is 39728.
Setup and Solution-
Setup parameters :
- Viscous model - Laminar
- Solver type - Pressure bases
- Solving Scheme - SIMPLE
- Initializing method - Hybrid(for Steady case)
Standard(for Transient case) - No. of Iterations - 1000
Case A) At Re = 100,simulate the flow with steady and transient method and calculate the Strouhal number.
- Material - User defined fluid with rho = 1kg/m^3
- mu - 0.02 kg/m-s
- Inlet velocity(U) -
- Re = UL/nu
- U = Re*(mu/rho)/L
In our case,diameter of the cylinder is 2 m ,therefore our characteristics length will be 2 m. - U = 100*(0.02/1)/2
- U = 1 m/s
Steady Case
Fig - Velocity Contour For Re = 100(Steady)
Fig - Pressure Contour For Re = 100(Steady)
Transient Case
Fig - Velocity Contour For Re = 100(Transient)
Fig - Pressure Contour For Re = 100(Transient)
Now lets compare the results of both the cases -
Fig - Residual Plot For Re = 100(Steady)
Fig - Residual Plot For Re = 100(Transient)
From both the residual plots it's clear that,there are huge differences in both steady and transient cases.In steady state solver we can say that, our solution has converged after around 370 iterations.However,in transient case, it's not exactly possible to determine when the solution has converged just by looking at the residual plot.We need some other parametrs to,firmly decide whether the solution has converged or not.Hence,we have created a moniter point at 8 m in the wake of the cylinder to compute our measuring parameter (velocity),and to see whether we are getting repeated results(results with same frequency) or not.The moniter point results will help us to get confidence of converged results.
Fig - Vertex Average of Velocity Magnitude Plot For Re = 100(Steady)
Fig - Vertex Average of Velocity Magnitude Plot For Re = 100(Transient)
In the first graph,the frequency of velocity is repeating after around 450 iterations,from this we can say that solution has
converged for steady case.And for the transient simulation the velocity frequency is constant,which will be going to increase with respect to time.
Fig - Coefficient of Drag Plot For Re = 100(Steady)
Fig - Coefficient of Drag Plot For Re = 100(Transient)
If we compare steady and transient cases with respect to their drag coefficient,at steady state drag is almost constant after around 100 iterations.While in transient,it is varying in a small frequency with respect to time.Regardless both solutions are showing the same value of drag(1.33).
Fig - Coefficient of Lift Plot For Re = 100(Steady)
Fig - Coefficient of Lift Plot For Re = 100(Transient)
In lift plot,steady solution improving frequency from zero to approximately +-0.2.For transient we see a repeating sine wave.
Strouhal Number
From the above plot of Strouhals Number and ,using highest value of magnitude Strouhal number was found to be 0.1.
Compairing the fluid at steady case with different Reynolds number 10,100,1000,10000 & 100000.
Fig- Project Schematic For Steady Cases
As shown in the above image,we will be sharing geometry and mesh part for all the cases.
For Re = 10
Inlet velocity(U) -
Re = UL/nu
U = Re*(mu/rho)/L
In our case,diameter of the cylinder is 2 m ,therefore our characteristics length will be 2 m.
U = 10*(0.02/1)/2
U = 0.1 m/s
For Re = 100
Inlet velocity(U) -
Re = UL/nu
U = Re*(mu/rho)/L
U = 100*(0.02/1)/2
U = 1 m/s
B3)For Re = 1000
Inlet velocity(U) -
Re = UL/nu
U = Re*(mu/rho)/L
U = 1000*(0.02/1)/2
U = 10 m/s
For Re = 10000
Inlet velocity(U) -
Re = UL/nu
U = Re*(mu/rho)/L
U = 10000*(0.02/1)/2
U = 100 m/s
For Re = 100000
Inlet velocity(U) -
Re = UL/nu
U = Re*(mu/rho)/L
U = 100000*(0.02/1)/2
U = 1000 m/s
From the above plots and contours we can see that,ass the value of reynolds number increased further and further ,the wake region is characterized by a pair of counter rotating vortices that remain attached.At higher Re,as the wake becomes unstable,the viscous forces are no longer sufficient to squash these disturbances.Although the flow is still laminar and 2D,the instability grows further with the increasing Reynolds number,i.e icreased levels of turbulence in the boundary layer and wake region.
As per our discussion above,in the vortex street simulation we may probably never get our results fully converged.Thus we need alternate options to see our results are converged or not.If we look at the residual plots,we can't determine the solution convergence. But,from the velocity magnitude plots,we can see the repeation of velocity frequency after certain number of iterations for each run. Which validates our confidence in converged results.
Drag Coefficient -
Drag coefficient is a dimensionless factor of proportionality between the overall hydrodynamic force vector on a body in a fluid and the product of reference area of the body and velocity head.
Effect of Re number on Drag -
As it can be seen through the results,the coefficient og drag keeps decreasing whe the reynolds number is increasing.This is because, the flow becomes turbulent for reynolds number greater than 2000,which means laminar flow though present is negligible leading to a lower skin friction drag.Also,for low reynolds number(Re=10),the alternating vortices cannot be visualized due to the low velocity values.In such case only boundary layer separation occured.
@ Re = 10
At very slow flow there is no separation of flow and no wake downstream of the cylinder,as wee have neglected viscosity.Since the flow
is symmetric from upstream to downstream,there is no drag on the cylinder.But this type of flow does not occur in nature,where there
is always some small amount of viscosity is present in the fluid.
@ Re = 100
Stable vortices are formed.The flow is separated but steady.The vortices gererate a high drag on the cylinder.
@ Re = 1000
As the flow velocity increases,the down stream vortices becomes unstable,separate from the body and are alternatively shed from the downstream.The wake is wide and generate large amount of drag.
@ Re = 10000
The flow velocity is increased even more and the periodic flow breaks down into chaotic wake.The flow is laminar and orderly while on the back side of the cylinder,it is turned into a chaotic wake causing drag.
@ Re = 100000
The boundary layer transioned into turbulent flow and vortices of different scales are being shed in the wake region.But since the separation point is slightly downstream from the laminar separation point,which cause a slightly smaller wake and the drag is than the corresponding laminar drag.
Conclusion -
- The Karman Vortex Street phenomenon for flow over a cylinder is studied and observed in this project.
- Increasing velocity eventually brings the turbulent drag up to and even higher than laminardrag value,but during transition from laminar to turbulent ,there is a range of reynolds number,for which turbulent drag is less than laminar drag.
- Vortex shedding can be seen i the case of both steady and transient simulation.But transient simulation produces better and more realistic results.
- Strouhal Number calculation is only possible in the case of transient simulations as the frequency changes with respect to time.