y i-1,j+1 i,j+1 i+1, j+1 i-1,j i,j i-1,j-1 i,j-1 X i+1,j i+1,j-1 Figure 4: Two-dimensional grid with equal spacing.
Consider the 2-D incompressible, invisicid Navier-Stokes equation in the horizontal plane. Recall that the momentum equations are simply solving the transport of the velocity on a frozen velocity field. Use a finite volume method on a structured grid numbered i, j with uniform h = 0.3 in x and y, as shown in Fig. 4. Use typical numbering, e.g. ui,j refers to the solution for the i-th point in the x-, and j-th point in the y-direction. The fluid has a density of 1000 kgm3. Use first-order upwinding for the fluxes.
The pressure field of the initial solution is taken as uniform pi,j = 0.
Assume that you have computed the first step of the SIMPLE scheme from an initial solution, and the resulting velocity field u* is given by the components u = [u, v] ^T with u1,j = 1.1, u2,j= 1.5, u3,j = 2.5
for all j except cell 2, 2, and ui,1 = 0.3, ui,2 = 0.5, ui,3 = 0.8 for all i except cell 2, 2. In cell 2,2 the velocity is u2,2 = [2, 0.6]^T.
a) Simplify the equations for the x− and y-momentum for this case.
[10 marks]
b) Calculate the x-momentum fluxes for the cell 2, 2. Use the x-momentum convective flux as fx = uu.
Number the cell faces clockwise from k = 1 representing the interface to cell 2, 3.
[5 marks]
c) Calculate the residual for the x-momentum divided by velocity, i.e. ∂u/∂t , for the cell 2, 2.
[4 marks]
d) Calculate the y-momentum fluxes for the cell 2, 2. Use the y-momentum convective flux as fx = vu.
[5 marks]
e) Calculate the residual for the y-momentum divided by density, ∂v/∂t for cell 2, 2. [4 marks]
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