Performing CFD Analysis Over A 2D Cylinder To Visualize Von Kármán Vortex Shedding Using ANSYS Fluent.

Abstract:- Flow past a cylinder is one of the classical problems of fluid mechanics & the primary objective of this simulation is to perform external flow analysis for various Reynolds no (Re) over a 2-dimensional cylinder using both steady-state & transient solver. The flow is assumed to be 2D therefore the cylinder can be represented by a circle.  For lower values of Reynolds number, the flow is steady and symmetric but as the Reynolds number is increased, the disturbance at the upstream flow can not be damped. This leads to a very important periodic phenomenon downstream of the cylinder, known as ‘vortex shedding’. CFD Software Ansys Fluent (Student version) has been selected for solving the governing equations using a finite volume method (FVM). A schematic representation of the computational domain has been shown below. The project is divided into two parts. For Part 1, we will be performing 3 case study for simulating flow over a cylinder using Steady state & transient solver & calculate Strouhal number for Reynolds no of 100. The next part of the project i.e. Part 2, is based on calculating the lift & drag coefficients (`C_L  &  C_D`) for Reynolds number of 10, 100, 1000, 10000, 100000 using a steady-state solver, plot the velocity contours, residual plots, `C_L  &  C_D`plots & discuss the effects of Reynolds no on `C_D`.   

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          1. Introduction

Two-dimensional computations of flow around a circular cylinder were carried out by integrating the incompressible Navier-Stokes equations. The Finite-Volume Method was utilized to discretize the fundamental equations. The flow was kept laminar & no explicit turbulence model was used. From the fluid mechanics, hydraulics and wind engineering points of view, the flow past a circular cylinder have been the subject of numerous experimental and numerical studies because:(i) this type of flow exhibits fundamental mechanisms, and (ii) it's numerous industrial applications; for example, when designing the shape of an aerodynamic body, it is of great importance to understand how the dynamic force is generated. To achieve the goal of obtaining detailed information on the flow field being investigated, computational fluid dynamics(CFD) has emerged as an attractive, powerful tool in many design processes. In fluid mechanics, vortex shedding is an oscillating flow pattern 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. During such flows, 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. Von Kármán vortex street is a repeating pattern of swirling vortices, caused by a process known as a vortex shedding, which is responsible for the unsteady flow separation of a fluid around blunt bodies as shown in Fig 1.1. 

Fig 1.1:- Animation of 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.

          2. Formulas Used

               A.  `L = 1/ 2 ρV ^2SC_L`   Where,  C_L: lift coefficient, L: lift force, S: relevant surface, q: fluid dynamic pressure, ⍴: fluid density, u: flow speed.

               B.  `D = 1/2 ρV^2AC_D`    Where,  D: drag force, ρ: density of the air, V: velocity of the air, A: body area, and `C_D`is the drag coefficient.

               C.  `Re = (rho V L)/mu`      Where,  ρ: density of the air,  V: velocity of the air, L: length, `mu:` viscosity,  Re: Reynolds number.

               D.   `S_t = (fd)/V`              Where, f: vortex shedding frequency, d: diameter of the cylinder, V: velocity of the air,  `S_t:` Strrouhal number.

          3. Designing The Computational Domain

Ansys geometry editing software "Spaceclaim" was used to design the computational domain. Fig 2.1 shows the various dimensions of the computational domain using cylinder diameter (D) as the characteristic length scale. We start by designing the 2-D cylinder which is essentially a circle with a diameter of 2m or radius of 1m with the center located on the origin of the plane. The upstream and downstream lengths of the domain are 10D and 20D respectively & this distance is computed from the center of the circle. the downstream distance has been kept longer to capture the flow physics, in our case we are interested in capturing the vortex formation. Once the design is done we use the "Pull-tool" feature to convert the design into a solid surface. Once done, we move to the next part, which is generating the mesh & assigning boundaries. 

                                                                                                     Fig 2.1:- Dimensions of the Computational domain & converting it into a solid surface. 

          4. Mesh Generation & Assigning Boundaries

In the Ansys mesh module, we start by generating a base mesh upon which we will use different mesh refinement strategies at different locations of the geometric model to capture the flow physics accurately. Fig 4.1 shows the base mesh that's automatically generated by Ansys. We can observe that the automated mesh generated is quad-dominated i.e. the entire geometry surface is discretized using quadrilateral mesh. A domain dominated by quad based mesh occupies a fewer number of cells compared to the triangular meshes which simply means the reduced memory usage. Hence, we convert the domain from quad dominated to triangular mesh as shown in the fig below.  

                                                             Mesh -> Insert -> Method -> Select face -> Method (Triangles) -> Generate Mesh

                                                                                                   Fig 4.1:- Transforming domain from Quad-dominated mesh to Triangular mesh.

After transforming the mesh type we generate the mesh again & observed a total cell count of just 824 cells & 449 nodes. This can be considered as a "coarse mesh". Although a coarse mesh requires less computational resources to solve but gives a very inaccurate solution. Hence, we will be converting it into a "Fine mesh". But before doing so we need to edit the circle edge, if looked closely we can observe the edges of the circle have sharp corners & depicts the shape of a polygon rather than a circle as shown in Fig 4.2. These sharp corners can disturb the flow physics around the cylinder.

                                            Sizing -> Select Edge of cylinder -> Type (Number of Divivsions) -> 36 divisions -> Generate Mesh

                                                                                                 Fig 4.2:- Transforming sharp edges or corners into smooth edges

The next phase of the mesh generation process is to add "inflation layers" near the circle as shown in Fig 4.3 where the fluid entering the domain will interact. Inflation layers are used in certain areas of the geometry to correctly capture the velocity and temperature gradients near no-slip walls. For our case, we used 8 inflation layers with a growth rate of 1.2 & the first layer height is 5e-3 or 5 mm. These parameters indicate that the first layer near the cylinder will have the smallest size of 0.05 & the subsequent layers will keep increasing at a rate of 15% [ `(1.2/8)*100`]. 

                                                                                                                                 Fig 4.3:- Adding Inflation Layers

The final few steps in the meshing module involve specifying the boundaries & refining the mesh. For our geometry, we have 4 boundaries, inlet, outlet, symmetric walls & stationary walls as shown in Fig 4.4. Initially, our geometry had just 824 cells which is extremely low, hence for our final mesh, we used an element size of 0.12m which generated 0.14 million cells.

                                                                                                                  Fig 4.4:- Specifying the boundaries & final mesh after refinement.

          5. setting up the Flow Physics

In Ansys Fluent after creating the mesh & assigning the boundaries, the next stage of the process is to set up the simulation parameters. This step involves specifying certain properties to capture the physics of the problem perfectly, such as materials used, simulation time parameters, solver parameters, initial & boundary conditions to solve the Navier Stokes equation which are PDEs, defining body forces, type of fluid flow & species involved. We start by setting up the Domain, here we will be using both pressure-based steady-state & transient solver with absolute velocity formation. We can justify using a pressure-based solver since the velocity or Mach number or Reynold's no at the inlet from the inlet is extremely small. The next parameter that we need to set up is the Flow Physics, here we will be navigating to the viscous feature & use a Laminar flow since our velocity at inlet is small. In the flow physics parameters itself, we will be defining the materials, our choice of fluid for the experiment is a user-defined fluid with properties such as density (`rho`) = 1 & viscosity (`mu`) varies as we vary the Reynolds no. We then navigate to the boundary zone, these are user-defined parameters where we assign the inlet velocity & we define the pressure near the outlet area. The walls of the pipe are stationary walls with a "no-slip" condition. These parameters will vary for all 3 cases as shown below. The final step includes performing a hybrid initialization & running the simulation for 1200 iterations till the solutions converge.

          6. Results & Observations for Part 1

                                                                                                                                                  A. Case 1

                                                                                                                                                   B. Case 2

                                                                                                                                                   C. Case 3

          7. Results & Observations for Part 2

                    A. Velocity Contours For Varying Reynolds Number

                    B. Velocity Contour Animations For Varying Reynolds Number

                    C. `C_L` Plots For Varying Reynolds Number

                    D. `C_D` Plots For Varying Reynolds Number

                    E. Residual Plots For Varying Reynolds Number

          8.  Conclusion

After simulating the flow over a 2D cylinder for varying Reynolds number (Re) using a Steady-state solver with no turbulence model, the data collected gives us the relationship between Reynolds number & Lift & Drag Coefficient (`C_L  &  C_D`) near the wall of the cylinder. Since the solutions have been run for several iterations, we observed oscillatory behavior for all cases above Re=10, this is caused due to vortex shedding downstream of the cylinder. The values for `C_L  &  C_D`have been averaged for all cases to obtain a single value. From the simulation results, we can observe that with an increase in Reynolds number the `C_L  &  C_D`also increases & is maximum for Re=100000. For the smaller values of Re we observe a negative `C_L' value which indicates downforce i.e. the lift force is acting downwards. In each of the cases, we can also observe a boundary layer separation region also known as the wake region. The pressure value in these wake regions can reach negative values. 

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