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Navier-Stokes Equations - Engineering Fluid Flow - Cartesian Coordinates - Ansys Assignment Help
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Task: 1 The analytical solution of laminar flow through infinitely long ‘parallel plates’ (also called plain channel) can be derived from the full 3-D Navier-Stokes (NS) governing equations via simplification with given boundary conditions. Fig. 1a below shows an upper plate driven flow with the bottom plate fixed. Fig. 1b shows a pressure driven flow with both upper and lower plates fixed. The channel height is 2h (h=0.01m) and the flowing fluid is water at a constant temperature of 25 °C. Ignore the effects of gravity.
- Using Cartesian coordinates (x, y, z), simplify the full 3-D unsteady compressible NS equations into 1-D (one dimensional) steady incompressible laminar flow equations. You should explain and justify all steps and reasons or the simplifications.
- Considering the boundary conditions in Fig. 1a (fixed bottom plate, moving upper plate at constant velocity V), with zero pressure gradient along streamwise direction (dp/dx=0):
- Derive the analytical solution of the streamwise velocity profile u(y) and the general formula of the volumetric flow rate Q
- For V= 0.1m/s, calculate the flow Reynolds number (based on density, mean velocity, channel height, viscosity) and determine the value of the flow rate Q. Please take the water properties (density, viscosity etc...) at 25 °C from the literature.
- Considering the boundary conditions in Fig. 1.b (fixed bottom and fixed upper plate), water flow driven by a non-zero constant pressure gradient dp/dx (with dp/dx=-1 Pa/m):
- Derive the analytical solution of the streamwise velocity profile u(y) and the general formula of the volumetric flow rate Q
- Calculate the flow Reynolds number (based on density, mean/bulk, velocity, channel height, viscosity), and determine the flow rate Q
- We now consider a boundary condition which is a combination of the above conditions: fixed bottom plate, moving upper plate at constant velocity V, a non-zero constant pressure gradient dp/dx along the streamwise direction.
- Derive the analytical solution of the streamwise velocity profile u(y) and the general formula of the volumetric flow rate Q;
- Calculate the flow Reynolds number (based on mean/bulk velocity) and determine the flow rate Q for the following parameters: V= 0.1m/s, dp/dx=-1 Pa/m.
- Calculate the air flow Reynolds number (based on chord length), and Mach number
- Determine the flow status (laminar or turbulent flow, compressible or incompressible flow), select and justify the appropriate turbulence model (if required);
- Choose the right boundary layer thickness calculation formula; and estimate its value to be used in meshing procedure
- Demonstrate that inflation layers properly reflect the physical viscous layer
- Analyse the mesh quality in terms of orthogonal quality, skewness and yplus coordinate.
- Run CFD calculations on the generated unstructured mesh and analyse your results for drag and lift coefficients, and pressure distribution over the airfoil.
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