Channel Boundary Layer as DAEs

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

The direction of differencing for the two first-order equations, Eqs. 7.60 and 7.61, is opposite that is, the pressure derivative involves information at points j + 1 and j, while the radial-coordinate equation involves information at points j and j — 1. This differencing is essential to the implicit-boundary condition specification that sets the radial coordinate at both boundaries. The opposite sense of the differencing permits information to be propagated from both boundaries into the interior of the domain. 

The pressure p(z) is a function of z alone. Thus it could be carried as a single scalar dependent variable, rather than defined as a variable at each mesh point. However, analogous to the reasoning used in Section 16.6.2 for one-dimensional flames, carrying the extra variables has the important benefit of maintaining a banded Jacobian structure in the differential-equation solution. 

An important issue in the boundary-layer problem, and in differential-algebraic equations generally, is the specification of consistent initial conditions. We think first of the physical problem (not in Von Mises form), since the inlet profiles of u, v, and T, as well as pressure p, must be specified. However, all the initial conditions are not independent, as they would be for a system of standard-form ordinary differential equations. So assuming that the axial velocity u and temperature T profiles are specified, the radial velocity must be required to satisfy certain constraints. 

By integrating the continuity equation in the radial direction along the inlet plane, we see that 

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