Nodes near-boundary

When a coarse grid is used, wall functions are used for imposing boundary conditions near the walls (Section 11.2.3.3). The nondimensional wall distance should be 30 < y < ]Q0, where y = u,y/p. We cannot compute the friction velocity u. before doing the CFD simulation, because the friction velocity is dependent on the flow. However, we would like to have an estimation of y" to be able to locate the first grid node near the wall at 30 < y < 100. If we can estimate the maximum velocity in the boundary layer, the friction velocity can be estimated as n, — 0.04rj, . . After the computation has been carried out, we can verify that 30 nodes adjacent to the walls. 

The approximation of the type (12) applies equally well to a non-equidistant grid and at the near-boundary nodes in the case of an arbitrary... 

In spite of the fact that this scheme generates no approximation at the near-boundary nodes t = 1 and i = IV, scheme (17) is of second-order accuracy in the space C z = 0 h ). In order to obtain this estimate at the points a = Xj, Xn, we rewrite equation (18) as... 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

Other ideas are connected with the operator A, which coincides with A at the near-boundary nodes and with A at the remaining inner nodes, leading to an alternative form of writing... 

After that, we look at equation (8) at a near-boundary node x ... 

If a node x is near-boundary not only with respect to but also in other directions, then sum (12) contains no other terms for < 2> >... 

Let X G be a near-boundary and irregular node only in some direction and = (+ ) G g From the equation... 

If a is a regular near-boundary node only with respect to a , then... 

This provides enough reason to conclude that in the case of any complex domain the number of the iterations depends only on the main step h of the grid regardless of near-boundary nodes. 

We offer below more a detailed classification of inner nodes. With this aim, let us draw up a straight line parallel to the axis Oxa through an inner node x u>h. Its intersection with the domain G is an interval (or several intervals), whose ends are boundary nodes along the direction Oxa. The nodes from this interval can be distinguished by the approved rules. We call the nearest to the end of the interval node a near-boundary node along the direction Oxa (with respect to z0). If the distance to the boundary jh is h ha, any such node is said to be irregular with respect to xa. Let... 

Let u>/t(G) be a grid in a p-dimensional domain and x 6 ui h be a near-boundary irregular node. Then... 

Further development with y on the grid coh is due to a special choice of the near-boundary node ij = 1, i2 = 1 at the left lower corner of the domain of interest so that two other nodes ((fj — 1) h1, i2 h2) and (ix hx, ((i2 — 1) h2) should belong to the boundary on which the values j/,- j and j/, j are already known. Formula (42) gives the value y with further contingency either along rows or along columns. 

Let now H = Q be a space of all grid functions given on the grid uik and vanishing on the boundary 7ft. Having involved the operators Ay = —A y and Ry — —Ay specified for any y H, we deduce instead of (46) that Ay = [Pg.692]

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