Natural convection, laminar boundary layer equations

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

In this section we derive the equation of motion that governs the natural convection flow in laminar boundary layer. The conservation of mass and energy equations derived in Chapter 6 for forced convection are also applicable for natural convection, but tlie momentum equation needs to be modified to incorporate buoyancy. 

In Chapter 5, we learned the foundations of convection. Integrating the governing equations for laminar boundary layers, we obtained expressions for the heat transfer associated with forced convection over a horizontal plate and natural convection about a vertical plate. We also found analytically, as well as by the analogy between heat and momentum, that the thermal and momentum characteristics of laminar flow over a flat plate are related by... 

Some of the more commonly used methods of obtaining solutions to problems involving natural convective flow have been discussed in this chapter. Attention has been given to laminar natural convective flows over the outside of bodies, to laminar natural convection through vertical open-ended channels, to laminar natural convection in a rectangular enclosure, and to turbulent natural convective boundary layer flow. Solutions to the boundary layer forms of the governing equations and to the full governing equations have been discussed. 

The equation for the laminar Nusselt number Nut is obtained in a two-step procedure. In the first step, not only is the flow idealized as everywhere laminar, but the boundary layer is treated as thin. There results from this idealization the equation for the laminar thin-layer Nusselt number Nur. As already explained, natural convection boundary layers are generally not thin, so the second step is to correct Nur to account for thick boundary layers. This correction uses the method of Langmuir [175]. The corrected Nusselt number is the laminar Nusselt number Nuc. 

The fluid dynamic structure within the boundary layers adjacent to natural solid surfaces such as soil, sediment, snow, and ice, is complex. Typically, the flows fields have both laminar and turbulent regions. The flows magnitudes and directions respond according to the angle of incidence to the surface and the overall shape of the object as well as thermal-induced fluid density differences (i.e., stratification) and so on. All these factors operate into shaping the mass transfer boundary layer which controls the chemical flux. Ironically, the traditional approach to handling such complex flow situations has been to use a simple flux equation. The so-called convective mass flux equation is... 

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