### Table 3 gives the available boundary conditions together with their codes and the geometrical cases where they may be formally applied (in terms of the integer parameter ICASE describing the geometrical cases).

"... In PAGE 27: ... Table3 : Boundary condition codes of pde OSH Data to be furnished by the user in the routine BVALUE: { At an in ow boundary, the boundary state (uB; vB; cB; zB) must be fully prescribed in BVALRE(1:4). { At an out ow boundary, the boundary pressure must be given in BVALRE(1).... ..."

### Table 2. Boundary Conditions for

"... In PAGE 11: ... Boundary conditions for the n expansion coe cients are required. Consistent with the boundary conditions for the tensor function f ij , the corresponding n bound- ary conditions are listed in Table2 as functions of ij (see Appendix A for details). The equivalence of the elliptic relaxation of the expansion coe cients n given by Eq.... ..."

### Table 1: Boundary conditions

in SUMMARY

"... In PAGE 5: ... The narrowness of the Gaussian pro#0Cle has a relevant in#0Duence on the calculated #0Dame length, so that its parameters have to be determined appropriately #5B19#5D. The boundary conditions are summarized in Table1 . Finally, we note that the use of the de#0Cnition of the vorticity #281#29 for the vorticity outlet boundary condition does not yield any relevantchanges in the computed solution.... In PAGE 5: ... 3. GENERAL SOLUTION ALGORITHM The partial di#0Berential equations #282#29 together with the boundary conditions #28see Table1 #29 are discretized on a two dimensional tensor product grid. A solution is #0Crst obtained on an initial coarse grid.... In PAGE 6: ...i#0Berence expressions. Di#0Busion and source terms are evaluated using centered di#0Berences. We adopt a monotonicity preserving upwind scheme for the convective terms #28see #5B20, p. 304#5D#29, for instance, v r @S @r = maxf#28v r #29 i, 1 2 ; 0g S i , S i,1 r i , r i,1 , maxf,#28v r #29 i+ 1 2 ; 0g S i+1 , S i r i+1 , r i : #283#29 The boundary conditions given in Table1 involve only zero or #0Crst order derivatives. For the latter terms, #0Crst order back or forward di#0Berences can be used, except for two boundary conditions which require a more accurate treatment.... In PAGE 7: ... By comparing our numerical solutions with a primitivevariable solution of the same problem #5B19#5D, we found that these two boundary conditions exerted a strong in#0Duence on the overall accuracy of the numerical solution. The discretization of the partial di#0Berential equations #282#29 together with the boundary conditions #28 Table1 #29 yields a set of algebraic equations of the form F #28U#29 = 0, which is solved using a damped Newton method J#28U n #29#01U n = ,#15 n F #28U n #29; n =0;1;:::; #285#29 with convergence tolerance k#01U n k S #3C 10 ,5 . The Jacobian matrix J#28U n #29 is computed numerically using vector function evaluations and the grid nodes are split into nine independent groups which are perturbed simultaneously #28see #5B2#5D for more details#29.... ..."

### Table 4: Boundary condition codes of pde EUL Data to be furnished by the user in the routine BVALUE: { At a far{ eld boundary, the boundary state in terms of primitive variables ( B; uB; vB; pB) must be fully prescribed in BVALRE(1:4). { At a solid boundary, no data have to be given. 4.4.3 Initial solution The initial solution has to be given in terms of conservative variables ( ; u; v; E). We usually initialize the solution in the domain with the far{ eld state. 4.4.4 Remarks It has not been possible to nd a convenient discretization using Dick apos;s ux{splitting at solid walls with a special shape (corner, l-corner, split points). That is the reason why we use, in these cases, the Osher apos;s discretization.If NSTOAD 2 is selected, the Mach number and the Entropy, respectively, are also written to the grid function le GRFDAT.

"... In PAGE 29: ...4.2 Boundary conditions Table4 gives the available boundary conditions together with their codes and the geometrical cases where they may be formally applied (in terms of the integer parameter ICASE describing the geometrical cases).... ..."

### TABLE 1. Boundary Conditions

2008

### TABLE I BOUNDARY CONDITIONS

### Table 1 Boundary Condition Types

2004

"... In PAGE 39: ...Six types of boundary conditions are implemented in M2D, and these can be distinguished as specifying forcing and non-forcing boundaries. Table1 lists the boundary-condition types and contains a short description of each. Each boundary-condition type is described next.... ..."

### Table 1. Boundary Conditions for the f

"... In PAGE 8: ... Note that since the linear form of the pressure-strain rate model is used here, the value for C L di ers from that used previously (C L =0:2, see Manceau and Hanjali c 2000) for the form of the elliptic relaxation equation given in (11). Boundary conditions are needed for the f ij and are determined, in the vicinity of the wall, by the balance of the redistributivetermby the viscous di usion of the Reynolds stresses resulting in Table1 . Only the 22- and 12-components of f have determinate solutions to the near-wall balance of the stress transport equations.... In PAGE 19: ... n Boundary Conditions The expressions for the n boundary conditions are derived from the basis tensors T (n) ij used in the representation of f ij f ij = 3 X n=1 n ^ T (n) ij ! 8 gt; gt; gt; gt; gt; lt; gt; gt; gt; gt; gt; : f 11 = 2 T (2) 11 + 3 T (3) 11 f 22 = 2 T (2) 22 + 3 T (3) 22 f 33 = 3 T (3) 33 f 12 = 1 T (1) 12 (A1) Table1 gives the corresponding boundary conditions for these f ij components. The boundary condition for 1 is directly proportional to the f 12 boundary condition and is given by 1;;w = f 12;;w T (1) 12 = p 2f 12;;w = ;20 p 2 2 12 quot; 2 w y 4 (1) (A2) The coe cient 3 appears in all three expansions of the diagonal terms of f ij .... ..."

### Table 1: ATMI boundary conditions

2006

"... In PAGE 2: ... The conductance h2 between the copper layer and the ambient medium is computed as h2 = 1=(RhsL2), where Rhs is the heat sink thermal resistance and L is the heat sink width. Boundary conditions are listed in Table1 , where T1 and T2 are the temperatures in layers 1 and 2 respectively, and q(x;y; t) is the surface power density. The ff3d model is depicted on Figure 2.... ..."

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