Consider the 2-D incompressible, invisicid Navier-Stokes equation in the horizontal plane. Recall that the momentum equations...

Transcribed Image Text:

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 z and y, as shown in Fig. 4. Use typical numbering, e.g. u.j refers to the solution for the i-th point in the x-, and j-th point in the y-direction. i-1.j+1 ij+1 i-1.j ij i-1.j-1 x i.j-1 i+1j+1 i+1j i+1j-1 Figure 4: Two-dimensional grid with equal spacing. The fluid has a density of 1000. 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 u.j = 1.1, u2.j = 1.5, U3.j = 2.5 for all j except cell 2, 2, and u,1 = 0.3, 4,2 = 0.5, 1,3 = 0.8 for all i except cell 2,2. In cell 2,2 the velocity is u2,2 = [2, 0.6]. a) Simplify the equations for the r- and y-momentum for this case. b) Calculate the x-momentum fluxes for the cell 2, 2. Use the 2-momentum convective flux as fr = uu. Number the cell faces clockwise from k = 1 representing the interface to cell 2, 3. c) Calculate the residual for the x-momentum divided by velocity, i.e. , for the cell 2, 2. du at d) Calculate the y-momentum fluxes for the cell 2, 2. Use the y-momentum convective flux as f = vu. e) Calculate the residual for the y-momentum divided by density, for cell 2, 2. 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 z and y, as shown in Fig. 4. Use typical numbering, e.g. u.j refers to the solution for the i-th point in the x-, and j-th point in the y-direction. i-1.j+1 ij+1 i-1.j ij i-1.j-1 x i.j-1 i+1j+1 i+1j i+1j-1 Figure 4: Two-dimensional grid with equal spacing. The fluid has a density of 1000. 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 u.j = 1.1, u2.j = 1.5, U3.j = 2.5 for all j except cell 2, 2, and u,1 = 0.3, 4,2 = 0.5, 1,3 = 0.8 for all i except cell 2,2. In cell 2,2 the velocity is u2,2 = [2, 0.6]. a) Simplify the equations for the r- and y-momentum for this case. b) Calculate the x-momentum fluxes for the cell 2, 2. Use the 2-momentum convective flux as fr = uu. Number the cell faces clockwise from k = 1 representing the interface to cell 2, 3. c) Calculate the residual for the x-momentum divided by velocity, i.e. , for the cell 2, 2. du at d) Calculate the y-momentum fluxes for the cell 2, 2. Use the y-momentum convective flux as f = vu. e) Calculate the residual for the y-momentum divided by density, for cell 2, 2.