Interdisciplinary Applied Mathematics

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physics of the problem as well as by numerical stability considerations. The first incompressible version of p,Flow (I1 in Table 14.1) solves the Navier-Stokes equations as well as the energy equation. It employs the first-order slip (equation (2.19)) and temperature jump (equation (2.20b)) and thermal creep boundary conditions. It is general for two- and three-dimensional flows. Explicit (in time) implementation of the boundary conditions results in a Knudsen number limit of typically Kn < 0.1. The second version of


TABLE 14.1. Gas flow models and boundary conditions implemented in pFlow. This program is used in many examples in this book, and it can be replaced by other equivalent discretizations.


Flow


B.C.


2-D/3-D


V. Slip


T. Creep


T. Jump


Kn limit


11


0(Kn, Kn2


i 2-D/3-D


Yes


Yes


Yes


< 0.1


12


0( Kn»)


2-D


Yes


assigned


No


< 0.5


C


e>(Knn)


2-D/Axi-Sym


Yes


Yes


Yes


< 0.5

incompressible p,Flow (I2 in Table 14.1) employs the slip boundary condition (equation (2.26)). It is stable for high Kn flows, and applicability is restricted with the flow geometry and the validity of the slip flow model. It does not solve the energy equation, and therefore thermal creep effects are imposed explicitly by a prescribed tangential temperature gradient. In the compressible version of p,Flow (C in Table 14.1) the general high-order slip boundary condition (equation (2.26)) and high-order temperature jump boundary condition (equation (2.31)) are used. Their limitations are based on the limitations of the slip flow theory. This version of the program is restricted to shock-free flows; therefore, it is used for subsonic and shock-free transonic flows.

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