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Computational Engineering: Analytical Solutions of Laminar Flow and Turbulence Modelling
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## Simplifying the 3-D unsteady compressible Navier-Stokes equations

(1.a) The analytical solutions of laminar flow through a circular pipe (Figure Q1a below) can be derived by solving the simplified 3-D Navier-Stokes governing equations in cylindrical form available in your lecture notes or in the general literature. All steps and explanations should be clearly detailed and shown in your work. In this question, you are required to undertake the followings three main tasks.

(1.a.i). Simplify the 3-D unsteady compressible Navier-Stokes equations into axisymmetric 2-D steady incompressible laminar flow equations. Ignore the effects of gravity.

(1.a.ii.) From the simplified equations in Question1.a.i., formulate the analytical solution of the streamwise velocity profile u(r) using the appropriate boundary conditions. Also formulate the equation for the volumetric flow rate.

(1.a.iii) In this section, you will calculate the volumetric flow rate and plot the velocity graphs at given conditions. Assume the fluid is water. Take the water properties at a temperature between 20°C and 25 °C from the literature (chose one single temperature value in this range, then obtain the necessary data from the literature, providing your references/sources).

For the radius of the pipe, choose one single value in the range between R=20 cm and R=90 cm. Assume a single value of dp/dx= - 0.5 Pa/m OR dp/dx= - 2 Pa/m. Calculate the volumetric flow rate in the pipe. Create the velocity plots/graphs for u(r) in Excel and present them in your report. Discuss and analyse in detail the graphs.

(1.b) Turbulence modelling remains one of the main challenges in CFD modelling and simulations. Questions in this section of the assignment are related to turbulence modelling. In your answers you could use the lecture notes and also literature materials. All references/sources used in your work should be provided in your submission in the Reference section to avoid plagiarism.

(1.b.i). Clearly discuss the main differences between the Reynolds Averaged Navier Stokes (RANS) turbulence approach and the Large Eddy Simulation (LES) turbulence approach. Provide some examples where RANS and LES would be suitable to use in the simulations.

(1.b.ii). Near wall treatment is important in CFD modelling because the flow features in near wall region are different from the freestream region far from the wall. Dimensionless velocity analysis from a wide variety of turbulent duct and boundary layer flows lead to

(1.b.ii.a). Clearly explain the different layers/zones inside the turbulent velocity boundary layer and the thickness of each zone for a flat plate. Discuss and analyse the details of the velocity distribution profiles in the boundary layer over a flat plate. Use referenced literature plots in your explanations and discussion.

(1.b.ii.b). In real world CFD simulations, discuss and analyse few examples/cases where wall functions can or cannot be used.

Use the available meshing tools in ANSYS to generate mesh and perform CFD calculations using ANSYS-CFX for the three-dimensional air flow over NACA0012 airfoil at the following given conditions: Angle of attack: 5 degrees Free stream flow velocity: chose any value between 20 m/s 40 m/s Free stream flow temperature: 20 °C Chord length: 1 m The generated unstructured mesh should reflect the key physical flow features, e.g. viscous boundary layer, rapid changes in flow field, etc. Also discuss and comment on mesh quality. To create the 3D slice of the airfoil, extrude the 2D profile with 0.1 m depth.

(a) Calculate the air flow Reynolds number (based on chord length), and Mach number and determine the flow status (laminar or turbulent, compressible or incompressible), select and justify the appropriate turbulence model (if required);

(b) Choose the right boundary layer thickness calculation formula; and estimate its value to be used in meshing procedure, calculate and design the inflation layers and demonstrate that inflation layers properly reflect the physical viscous layer

(c) Generate and present the mesh and report the mesh quality and resolution.

(d) Run CFD calculations on the generated mesh using ANSYS-CFX and analyse your results for drag and lift coefficients, and pressure distribution over the airfoil.

In your report, you should include supporting formula, domain, mesh parameters, graphs (including near wall) and also discuss your findings/results.