In case of a flow with variable density, Equation 10-4 and Equation 10-5 must be solved together with the equation of state that expresses the density as a function of the temperature and pressure (for instance the ideal gas law).

For incompressible flow, the density stays constant in any fluid particle, which can be expressed as

+ u = 0t p

Equation 10-4 together with the above condition simplifies into the following form of the continuity equation

u = Qbr

References for the Brinkman Equations Interface

1.D. Nield and A. Bejan, Convection in Porous Media, 3rd ed., Springer, 2006.

2.M. Le Bars and M.G. Worster, “Interfacial Conditions Between a Pure Fluid and a Porous Medium: Implications for Binary Alloy Solidification,” Journal of Fluid Mechanics, vol. 550, pp. 149–173, 2006.

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T h e o r y f o r t h e F r e e a n d P o r o u s M e d i a F l o w I n t e r f a c e

The Free and Porous Media Flow Interface uses the Navier-Stokes equations for describing the flow in open regions, and the Brinkman equations for the flow in porous regions.

The same fields, u and p, are solved for in both the free flow domains and in the porous domain. This means that the pressure in the free fluid and the pressure in the pores is continuous over the interface between a free flow domain and a porous domain. It also means that continuity is enforced between the fluid velocity in the free flow and the Darcy velocity in the porous domain. This treatment is one of several possible approximations of the physics at the interface. Example of other approximations can be found in Ref. 1.

The continuity in u and p implies a stress discontinuity at the interface between a free flow domain and a porous domain. The difference corresponds to the stress absorbed by the rigid porous matrix which is implicitly treated as a consequence of the formulations of the Navier-Stokes and Brinkman equations.

Reference for the Free and Porous Media Flow Interface

1. M. L. Bars and M. G. Worster, “Interfacial Conditions Between a Pure Fluid and a Porous Medium: Implications for Binary Alloy Solidification”, Journal of Fluid Mechanics, vol. 550, pp. 149–173, 2006.

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T h e o r y f o r t h e T w o - P h a s e D a r c y ’ s L a w I n t e r f a c e

The The Two-Phase Darcy’s Law Interface theory describing the Darcy’s Law— Equation Formulation is described here.

Darcy’s Law—Equation Formulation

Darcy’s law states that the velocity field is determined by the pressure gradient, the fluid viscosity, and the structure of the porous medium. According to Darcy’s law, the net flux across a face of porous surface is

In this equation, u is the Darcy velocity o (SI unit: m/s); is the permeability of the porous medium (SI unit: m2); is the fluid’s dynamic viscosity (SI unit: Pa·s); p is the fluid’s pressure (SI unit: Pa) and is its density (SI unit: kg/m3). Here the permeability, , represents the resistance to flow over a representative volume consisting of many solid grains and pores.

The average density and average viscosity are calculated from the fluids properties and the saturation of each fluid

The Two-Phase Darcy’s Law interface combines Darcy’s law with the continuity equation

with the transport of the fluid content c1 = s1 1

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here, p is the porosity, defined as the fraction of the control volume that is occupied by pores, and Dc is the capillary diffusion (SI unit: m2/s).

Inserting Darcy’s law (Equation 10-6) into the continuity equation (Equation 10-10) produces the generalized governing equation

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11

H i g h M a c h N u m b e r F l o w B r a n c h

There are several fluid-flow physics interfaces available with the CFD Module. The fluid-flow is grouped by type under the Fluid Flow main branch. This chapter discusses applications involving the High Mach Number Flow branch () in the

Model Wizard.

In this chapter:

•The High Mach Number Flow Interfaces

•Theory for the High Mach Number Interfaces

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T h e H i g h M a c h N u m b e r F l o w I n t e r f a c e s

There are three variations of the same predefined multiphysics interface (all with the interface identifier hmnf) that combine the heat equation with either a laminar or a turbulent flow. The advantage of using the multiphysics interfaces—compared to adding the individual interfaces separately—is that predefined couplings are found in both directions. In particular, interfaces use the same definition of the density, which can therefore be a function of both pressure and temperature. Solving this coupled system of equations usually requires numerical stabilization, which the predefined multiphysics interface also sets up.

These interfaces vary only by one or two default settings (see Table 11-1), which are selected during Model Wizard selection, or from a check box or drop-down list in the

Physical Model section on the Settings window.

TABLE 11-1: THE HIGH MACH NUMBER FLOW PHYSICAL MODEL DEFAULT SETTINGS

Most of the other features share the same setting options as described in this section and in Shared Interface Features. See also The Heat Transfer

Interfaces for Settings window details for the Heat Transfer in Solids

Note

feature.

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