| OCR Text |
Show 53 The next step is to compute the values of the normal stresses r~~ and ryy at z = 0. To do that, one needs to know the w component of displacement at z = -1, i.e., one grid point above the free surface. These are fictitious values and will only serve to calculate the normal stresses at the surface. To get the fictitious displacement component w, Equation (2.10) can be evaluated at z = 0 while taking advantage of the fact that Tzz is zero at that point, i.e., ( . 112 . O) _ (2 ')ow(i + 1/2,j,O) , (ou(i + 1/2,j,O) ov(i + 1/2,j,O)) Tzz z + '), - Jl +A OZ +A OX + Oy = 0. Expressing the above equation in second-order finite differences, one gets : (2p +A) ( [w(i + 1/2, j, 1/2)- w(i + 1/2, j, -1/2)]) +A ([u(i + l,j, 0)- u(i,j, 0)] + [v(i + 1/2,j + 1/2, 0) - v(i + l/2,j- 1/2, O)J) 0. (B.5) Equation (B.5) is used to evaluate w(i + 1/2, j, -1/2) which will in turn be used to compute r~~ and ryy at z = 0, and that concludes the free surface treatment. |
