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Navier-Stokes Equations

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Use an Excel spreadsheet to numerically solve the Navier-Stokes equations in order to find the nondimensional unsteady velocity profile for Couette flow with an impulsively started top plate, i.e., at t < 0, Vp = 0, and at t = 0, the top plate begins moving at Vp = 0.97 m/s. Use a stability number of 0.5, and at least 21 nodes. This is an individual effort project. You may not use another students’ work in any part of your project. A sketch of the problem is presented below. Include the following in your write up. All graphs must be included electronically in your Word document write up. A discussion of the graphs is expected. Equation derivations can be done neatly by hand. You must also email your instructor your Excel spreadsheet. All items must be included in one file submitted electronically (.docx or .pdf) or stapled together in order and handed in in class. a) *Derivation of the simplified, unsteady, non-dimensionalized Navier-Stokes equation for Couette flow starting with the general x-direction N-S equation and continuity equation for incompressible flow. Include assumptions! b) *Derivation of the finite-difference form of the equation developed in part a). c) *Derivation of the analytical steady state exact solution. d) One graph with five non-dimensional velocity profiles to include one profile at three intermediate non-dimensional times, the computed steady-state profile, and the exact steady-state profile (from the analytical solution). e) The corresponding elapsed real time for your computed velocity profiles, with a brief explanation of how you arrived at the number. f) A discussion on the stability of the solution, i.e., what happens if the stability number is increased/decreased. Include graphs to justify your discussion. g) A discussion on how Reynolds number affects the time variant solution, i.e., change fluids and analyze the effects. Include graphs to justify your discussion.

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