SIMULATE FLOW OVER A CYLINDER
SIMULATE FLOW OVER A CYLINDER
AIM: To simulate the flow over a cylinder by varying the Reynolds number and to perform grid dependency test using SOLIDWORKS flowsimulation.
DESCRIPTION:
External flows past objects have been studied extensively because of their many practical applications. For example, airfoils are made into streamline shapes in order to increase the lifts, and at the same time, reducing the aerodynamic drags exerted on the wings. On the other hand, flow past a blunt body, such as a circular cylinder, usually experiences boundary layer separation and very strong flow oscillations in the wake region behind the body.
Vortex shedding:
In certain Reynolds number range, a periodic flow motion will develop in the wake as a result of boundary layer vortice being shed alternatively from either side of the cylinder.
In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to streamlined) body at certain velocities, depending on the size and shape of the body.
In this flow, vortices are created at the back of the body and detach periodically from either side of the body forming a Von Kármán vortex street.
The fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object. The object will tend to move toward the low-pressure zone.
Vortex street created by a cylindrical object; the flow on opposite sides of the object is given different colors, showing that the vortices are shed from alternating sides of the object.
Kármán vortex street:
In fluid dynamics, 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.
A vortex street will form only at a certain range of flow velocities, specified by a range of Reynolds numbers (Re), typically above a limiting Re value of about 90.
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, vibrating with harmonic oscillations driven by the energy of the flow.
This vortex shedding can drive the structure into violent oscillations that can damage or destroy the structure.
Re = Inertial Force / Viscous Force
`Re= (ρUD)/μ = (UD)/ϑ`
Here
`ρ` is the density of the fluid (SI units: kg/m^3)
`U`is the free stream flow speed (m/s)
D is the Diameter of the Cylinder (m)
μ is the free stream fluid dynamic viscosity of the fluid (Pa·s or N·s/m^2 or kg/(m·s))
`ϑ` is the free stream kinematic viscosity of the fluid (m^2/s).
Engineering problems:
In low turbulence, tall buildings can produce a Kármán street, so long as the structure is uniform along its height. In urban areas where there are many other tall structures nearby, the turbulence produced by these prevents the formation of coherent vortices.
Periodic crosswind forces set up by vortices along object's sides can be highly undesirable, and hence it is important for engineers to account for the possible effects of vortex shedding when designing a wide range of structures, from submarine periscopes to industrial chimneys and skyscrapers. In order to prevent the unwanted vibration of such cylindrical bodies, a longitudinal fin can be fitted on the downstream side, which, provided it is longer than the diameter of the cylinder, will prevent the eddies from interacting, and consequently they remain attached.
Simulated vortex street around The same cylinder, now with a fin,
a no-slip cylindrical obstruction suppressing the vortex street
Obviously, for a tall building or mast, the relative wind could come from any direction. For this reason, helical projections resembling large screw threads are sometimes placed at the top, which effectively creates asymmetric three-dimensional flow, thereby discouraging the alternate shedding of vortices; this is also found in some car antennas.
Another countermeasure with tall buildings is using variation in the diameter with height, such as tapering - that prevents the entire building from being driven at the same frequency.
Chimneys with strakes fitted to break up vortices
OBJECTIVES:
1) To simulate the flow over a cylinder by Running a baseline simulation and then calculate the Reynolds number. Then increase the Reynolds number by a factor of 20%, 40%, and 100% and also for the case when Re =10 and Re=1000 then un the transient flow simulation.
2) To determine the Drag force, Lift force, Drag coefficient, Lift coefficient, and Strouhal number for each case.
3) Create animations for pressure-velocity for all the four Reynolds number.
USER INPUTS:
GEOMETRY:
Dimensions of the cylinder:
A circular sketch is created and the then extrusion is done to the required length of the cylinder.
Diameter of the cylinder =0.04m
Length of the cylinder = 0.05m
Dimensions of the Computational Domain/Wind tunnel:
Length of the computational domain upstream = 10D=0.4m
Length of the computational domain downstream = 20D=0.8m
Height of the computational domain= 10D=0.4m
And 2D analysis is selected along the XY plane.
General Settings:
Analysis Type: External
Transient state:
The transient state is where the time derivatives of the variables are also taken into account, and hence the magnitude of the values of the system can be monitored at any particular instant of time, but in steady-state simulation, only the initial values and steady-state values are obtained.
This will also help in monitoring the frequency of the vortex shedding with respect to various Reynolds numbers.
Since the Vortex shedding is a transient (time dependent) phenomenon so running the transient simulation, so we can see how our results change wrt time.
Case 1:
When Re=100
Vx = 0.038136 m/s
Total length of the domain is 1.2 m
Flow through time is 31.466 seconds. That is, it takes 31.466 seconds for the flow from the inlet to go to the outlet.
End time = 2 to 4 times flow through time so that we get a steady solution.
End time= 126 sec.
Time step = 0.5sec.
Case 2:
When Re=120
Vx = 0.046096 m/s
Total length of the domain is 1.2 m
Flow through time is 26.032 seconds. That is, it takes 26.032 seconds for the flow from the inlet to go to the outlet.
End time = 2 to 4 times flow through time so that we get a steady solution.
End time= 105 sec.
Time step = 0.5sec.
Case 3:
When Re=140
Vx = 0.053779 m/s
Total length of the domain is 1.2 m
Flow through time is 22.31 seconds. That is, it takes 22.31 seconds for the flow from the inlet to go to the outlet.
End time = 2 to 4 times flow through time so that we get a steady solution.
End time= 90 sec.
Time step = 0.5sec.
Case 4:
When Re=200
Vx = 0.0768 m/s
Total length of the domain is 1.2 m
Flow through time is 15.625 seconds. That is, it takes 15.625 seconds for the flow from the inlet to go to the outlet.
End time = 2 to 4 times flow through time so that we get a steady solution.
End time= 63 sec.
Time step = 0.5sec.
Case 5:
When Re=10
Vx = 0.0038136 m/s
Total length of the domain is 1.2 m
Flow through time is 314.66 seconds. That is, it takes 314.66 seconds for the flow from the inlet to go to the outlet.
End time = 2 to 4 times flow through time, so that we we get a steady solution.
End time= 1260 sec.
Case 6:
When Re=1000
Vx = 0.38136 m/s
Total length of the domain is 1.2 m
Flow-through time is 3.1466 seconds. That is, it takes 3.1466 seconds for the flow from the inlet to go to the outlet.
End time = 2 to 4 times flow through time so that we get a steady solution.
End time= 12.6 sec.
Time step = 0.5sec.
Fluids:
Gas: Air
Since the initial temperature is T=293.2K and Pressure P=101325Pa
Density=1.204 kg/m^3
Dynamic viscosity=1.85e-5 Ns/m^2 (at 300K)
Flow Type= Laminar and Turbulent.
Wall conditions: Adiabatic Wall
Roughness: 0micrometer
Initial conditions:
Pressure: P=101325Pa
the initial temperature is T=293.2K
MESH:
Global mesh > edit definition > automatic
Basic Mesh
Initial Mesh level is taken as 4
MATHEMATICAL MODEL:
1) Governing Equations
2) Boundary conditions.
1) Governing Equations:
2D steady, Navier - Stokes equations, which are nothing but the continuity equation and the momentum equations in two directions as given as
1) Conservation of mass (continuity equation):
`∇.(ρV)=0`
2) Conservation of momentum (`overlineF =moverlinea` )
`∇( ρVu)=-∇p i+∇(μ∇u)+S_u`
`∇( ρVv)=-∇pj+∇(μ∇v)+S_v`
Further in the present case viscosity (µ) and density (ρ) of the fluid are constant, and assuming that there are no body forces acting on the flow domain, the source terms and are zero.
Hit Run calculation
After solving we will get selected variables at selected points and from that we can do Post-processing.
2) Boundary conditions:
Velocity boundary condition along x-direction is applied at the inlet of the computational domain as mentioned above for different cases
And Initial conditions:
Pressure: P=101325Pa
the initial temperature is T=293.2K
GRID DEPENDENCY TEST:
This test is performed by taking the Case 1 when Re=100 and velocity is 0.038136 m/s
Project name > New Parametric study
Input variables:
Level of initial mesh is selected for variation
Output Variables:
Various cutplots and xy plots are selected.
Scenario:
RESULTS AND PLOTS FORM GRID DEPENDECY TEST:
Design point 1: Level of initial mesh = 4
Results:
Values:
Maximum values:
Minimum values:
Design point 1: Level of initial mesh = 4
Velocity cutplot:
Relative Pressure cutplot:
XY PLOTS:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
Design point 2: Level of initial mesh = 5
Velocity cutplot:
Pressure cutplot:
XY Plots:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
Design point 3: Level of initial mesh = 6
Velocity cutplot:
Pressure cutplot:
XY plots:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
Design point 4: Level of initial mesh = 7
Velocity cutplot:
Pressure cutplot:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
Comparison of Cd plots for Design point 2,3 and 4:
Comparison of Cl plots for Design point 2,3 and 4:
From the cutplots we can see that the velocity and the pressure profile for design point 1 is quite nonuniform and this mesh is unable to properly capture vortex shedding phenomenon, but in case of design point 3 where the level of mesh is 6 the velocity and pressure cutplots become more uniform and in case of design point 4 the plots improve more.
From the x-y plots of design point 1 we can see there is a large amount of the fluctuations in the values of the coefficient of drag and coefficient of lift and the values do not converge at certain number since this coarse mesh is unable to capture properly the values when compared to the case of design point 3 and 4. In the case of design point, 3 and 4 the lift coefficient values oscillate between certain maximum and minimum values.
We would use the Mesh of design point 3 for further study as it gives the best result in terms of the results and the less amount of computational time required.
VARIATION OF THE INLET VELOCITY TO GET THE FLOWS WITH DIFFERENT REYNOLDS NUMBER:
CASE 1: When Re=100, Vx = 0.038136 m/s and End time= 126 sec with time step of 0.5sec.
Velocity cutplot:
Animation:
Relative Pressure cutplot:
Animation:
XY Plots:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
CASE 2: When Re=120, Vx = 0.046096 m/s and End time= 105 sec with Time step = 0.5sec.
Velocity cutplot:
Animation:
Relative Pressure cutplot:
Animation:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
CASE 3: When Re=140, Vx = 0.053779 m/s and End time= 90 sec with Time step = 0.5sec.
Velocity cutplot:
Animation:
Relative Pressure cutplot:
Animation:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
CASE 4: When Re=200, Vx = 0.0768 m/s and End time= 63 sec with Time step = 0.5sec.
Animation:
Relative Pressure cutplot:
Animation:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
CASE 5: When Re=10, Vx = 0.0038136 m/s and End time= 1260sec with time step of 0.5sec.
Velocity cutplot:
Animation:
Relative Pressure cutplot:
Animation:
XY Plots:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
CASE 6: When Re=1000, Vx = 0.38136 m/s and End time= 12.6 sec with time step of 0.5sec.
Velocity cutplot:
Animation:
Relative Pressure cutplot:
Animation:
XY Plots:
Coefficient of Drag (Cd) Plot:
Coefficient of Lift (Cl) Plot:
OBSERVATIONS AND CONCLUSION:
1) Grid dependency test:
From the cutplots we can see that the velocity and the pressure profile for design point 1 is quite nonuniform and this mesh is unable to properly capture vortex shedding phenomenon but in case of design point 3 where the level of mesh is 6 the velocity and pressure cutplots become more uniform and in case of design point 4 the plots improve more.
From the x-y plots of design point 1 we can see there is a large amount of the fluctuations in the values of the coefficient of drag and coefficient of lift and the values do not converge at certain number since this coarse mesh is unable to capture properly the values when compared to the case of design point 3 and 4 where the lift coefficient values oscillate between certain maximum and minimum values.
We would use the Mesh of design point 3 for further study as it gives the best result in terms of the results and the less amount of computational time required
2) Variation of inlet velocity to get flows of different Reynolds number:
Velocity cutplot:
The velocity plots show the velocity of flow past the cylinder; Red color represents the magnitude of the velocity (warmer colors, faster velocity). Green in this case corresponds to the velocity far away from the cylinder, the free stream velocity which you can think of like the wind, all moving at the same velocity from left to right. As they get close enough, fluid elements approaching the cylinder straight on (at the equator) begin to slow (shifting to blue color). In fact, the fluid element on the surface of the upstream side of the cylinder is stopped -- velocity = 0. This is called a stagnation point.
Below is the velocity plot for case 1 When Re=100, free stream velocity is Vx = 0.038136 m/s
Fluid elements passing either above or below the cylinder increase their velocity magnitude (shifting to red colour). It will turn out that velocity at both the top and bottom of the cylinder (the poles) is approximate twice the free stream velocity. Velocity along the surface of the cylinder is in a tangential direction, i.e. parallel to the surface of the cylinder. Fluid elements continuing around the downstream side of the cylinder begin to slow down, reaching a second stagnation point at the downstream equator.
Pressure cutplot:
Below is the pressure plot for case 1 When Re=100, free stream velocity is Vx = 0.038136 m/s
The pressure distribution around the cylinder is shown using colour and contour lines; warmer colors correspond to higher pressure. The magnitude and direction of the (negative) pressure gradient -- the pressure force acting on the fluid elements. This is the identical flow problem as depicted above so that the pressure field here is the one corresponding to the previous velocities
Obviously enough, pressure increases as velocity decreases in accordance with the Bernoulli equation. The relationship between pressure and velocity is nonlinear, i.e. pressure is proportional to the square of the velocity. This particular mathematical relationship may not be clearly evident by simple observation of the colours, but it is clear that the pressure hotspots coincide with the stagnation points where velocity is zero. At these two locations, fluid elements have given up all of their kinetic energy and pressure is at its highest. Similarly, velocity at the poles (top and bottom) of the cylinder is at a maximum and pressure at a minimum. Note that the pressure force vectors, due to the pressure gradient, are aligned perpendicularly to the pressure contours throughout the solution and are directed from the high-pressure side of each contour to the low-pressure side. The pressure is the only force that is operative in the solution; friction due to viscosity is omitted for the ideal fluid.
Consider a fluid element approaching from the left at position 1. We know that the particle is slowing down as it approaches the equator of the cylinder. Pressure force vectors are directed exactly opposite to the direction of motion -- the force acts entirely to decrease the fluid speed (velocity magnitude). From the above image showing pressure contours, the pressure retarding force actually increases in magnitude as the element approaches the cylinder and brings the element to a halt at the surface of the cylinder. Fluid elements moving around the top side of the cylinder are subjected to a dramatically varying pressure force. Initially, it appears that the forces are directed primarily along the direction of motion, adding to the velocity of the elements there. As the elements approach the top of the cylinder, forces are directed less and less in the direction of motion and gradually become perpendicular to the motion. Forces aligned with the direction of motion change the speed of an object; forces directed perpendicular to the direction of motion change the direction of the object. BOTH constitute acceleration. Velocity is a vector and a change in EITHER magnitude or direction constitutes an acceleration. It is only the direction of the acceleration (force) vector that determines whether speed is added or subtracted to the object and whether the direction of motion is changing. As fluid elements continue around the downstream side of the cylinder, they continue to be subjected to a large pressure force which acts to keep the elements moving tangentially to the cylinder surface (direction of velocity) and also to slow them down (decelerate, magnitude of velocity).
At a certain point, the flow separates from the surface and it forms repeating pattern of swirling vortices, caused by a process known as vortex shedding, which forms the Kármán vortex street (or a von Kármán vortex street) is a which is responsible for the unsteady separation of flow of a fluid around blunt bodies. A vortex street will form only at a certain range of flow velocities, specified by a range of Reynolds numbers (Re), typically above a limiting Re value of about 90.
It can be seen that from the velocity and pressure plots that for Reynolds number less than 1 (a) the shape of every streamline is symmetrical not only around the cylinder’s upper to lower side but also around its front to rear.
As the Reynolds number increases, the front to rear symmetry disappears, and the interval of the streamlines of the cylinder’s rear flow widens. As the Reynolds number reaches close to 10, a closed region of the streamline is generated within the cylinder’s rear flow. The fluid in the upper half of this region rotates clockwise, and the fluid in the lower half rotates counterclockwise. This phenomenon is the result of the so-called generation of vortices. (b) depicts this state.
As the Reynolds number increases, the length of the vortex increases. When the Reynolds number exceeds 60, the rear flow becomes unstable as shown in Figure (c), and the vortex begins to oscillate in an up-down direction. The fully formed vortex is carried away on the surrounding main flow, and a vortex subsequently occurs on the other side. This vortex then expands before finally dissipating and departing leaving the circular cylinder. In addition to this flow process, every pair of upper and lower vortices behind the cylinder is stored in an alternative regular array to form twin rows of vortices. These rows are called the von Karman vortex street. While the main flow approaching the circular cylinder is in a uniformly steady state, a periodically varying flow characteristic of the von Karman vortex street appears in the cylinder’s rear flow. As shown in Figure (d), as the Reynolds number reaches 1,000, the vortices are mixed together, and the flow behind the cylinder (wake) behaves very irregularly in time and space scale. This behavior is the so-called state of turbulent flow. However, a free stream distant from the wake region maintains a steady-state, and the streamlines are held in a regular and smooth shape. This phenomenon is known as laminar flow.
Flow Separation:
From the velocity and pressure pots, we can observe that the presence of the fluid viscosity slows down the fluid particles very close to the solid surface and forms a thin slow-moving fluid layer called a boundary layer. The flow velocity is zero at the surface to satisfy the no-slip boundary condition. Inside the boundary layer, flow momentum is quite low since it experiences a strong viscous flow resistance. Therefore, the boundary layer flow is sensitive to the external pressure gradient (as the form of a pressure force acting upon fluid particles). If the pressure decreases in the direction of the flow, the pressure gradient is said to be favorable. In this case, the pressure force can assist the fluid movement and there is no flow retardation. However, if the pressure is increasing in the direction of the flow, an adverse pressure gradient condition as so it is called exist. In addition to the presence of a strong viscous force, the fluid particles now have to move against the increasing pressure force. Therefore, the fluid particles could be stopped or reversed, causing the neighboring particles to move away from the surface. This phenomenon is called the boundary layer separation.
Wake:
From the velocity and pressure plots, we can observe that a fluid particle flows within the boundary layer around the circular cylinder. From the pressure distribution measured in an earlier experiment, the pressure is a maximum at the stagnation point and gradually decreases along the front half of the cylinder. The flow stays attached in this favorable pressure region as expected. However, the pressure starts to increase in the rear half of the cylinder and the particle now experiences an adverse pressure gradient. Consequently, the flow separates from the surface and creating a highly turbulent region behind the cylinder called the wake. The pressure inside the wake region remains low (this can be seen in the relative pressure plot) as the flow separates and a net pressure force (pressure drag) is produced.
Vortex Shedding:
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) (this can be seen in the velocity and pressure plots above) 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.
2) From the XY Plots of the Cd and Cl:
Vortex-Induced Vibrations:
When vortices shed from the cylinder, uneven pressure distribution develops between the upper and lower surfaces of the cylinder, generating an oscillatory aerodynamic loading (lift) on the cylinder. This unsteady force can induce significant vibrations on a structure, especially if the "resonance" condition is met. The most famous example is the collapse of the Tacoma Narrow bridge in 1940 under the action of wind-induced vibrations. It is believed that natural vortex shedding frequency behind the bridge matches one of the resonant modes of the bridge and eventually lead to a catastrophic vibration that destroys the bridge.
Aerodynamic Loading:
According to Newton's second law, the time rate change of the linear momentum is equal to the sum of all external forces acting on a system. Therefore, an integration of the linear momentum inside a control volume surrounding the circular cylinder can provide information on the aerodynamic forces (lift and drag) acting on the cylinder. From the XY plot of the coefficient of lift (Cl), there are alternative upward and downward flows in the wake as the result of vortex shedding. Consequently, there must be also an oscillatory up and downforce acting periodically on the cylinder. This periodic forcing exerting on the cylinder body is responsible for the vortex-induced vibrations as described earlier.
Momentum Balance:
As stated earlier, the external force acting on an object can be determined using the momentum balance concept. In general, there is a momentum deficit in the wake profile along the streamwise direction as relative to the incoming momentum upstream of the object. Therefore, a simple balance of the momentum flow in and out of the control volume surrounding the object suggests that there is a net force acting on the object (this can be seen from the plots of the drag coefficient) (note: the pressure is considered to be relatively constant if the momentum flow is measured far away from the object.) This net force along the flow direction is called the drag. Averaged velocity profiles of the flow past a circular cylinder is provided as a general representation of the wake flow field. Selected profiles at several representative regions is also presented for your reference. Near the cylinder, the flow separates from the surface. Immediately behind the cylinder, a recirculation region exists with a strong reversing flow. The region between the cylinder and the end of the recirculation region is called the vortex formation region (this can be seen in the velocity and pressure plots). The centreline velocity becomes zero at the end of the vortex formation region. Further downstream, the two separating shear layers merge and the velocity profile presents a typical wake profile. It is clear that there is a deficit in the center of the wake. This deficit in the momentum flow is the direct result of drag force acting on the cylinder.
REFERENCES:
1) A fundamental study of the flow past a circular cylinder using Abaqus/CFD Masami Sato, and Takaya Kobayashi
2) https://en.wikipedia.org/wiki/Vortex_shedding
3) https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street