In Chapter 6 we discussed the fundamentals of turbulence and reviewed the mixing length and eddy diffusivity models. As was mentioned there, these classical models do not treat turbulence as a transported property, and as a result they are best applicable to equilibrium turbulent fields. In an equilibrium turbulent field at any particular location there is a balance among the generation, dissipation, and transported turbulent energy for the entire eddy size spectrum, and as a result turbulence characteristics at each point only depend on the local parameters at that point.

Our daily experience, however, shows that turbulence is in general a transported property, and turbulence generated at one location in a flow field affects the flow field downstream from that location. One can see this by simply disturbing the surface of a stream and noting that the vortices resulting from the disturbance move downstream.

In this chapter, turbulence models that treat turbulence as a transported property are discussed. Turbulence models based on Reynolds–averaged Navier–Stokes [(RANS)-type] models are first discussed. These models, as their title suggests, avoid the difficulty of dealing with turbulent fluctuations entirely. We then discuss two methods that actually attempt to resolve these turbulent fluctuations, either over the entire range of eddy sizes [direct numerical simulation (DNS) method] or over the range of eddies that are large enough to have nonuniversal behavior [largeeddy simulation (LES) method].

An integral method is a powerful and flexible technique for the approximate solution of boundary-layer problems. It is based on the integration of the boundary-layer conservation equations over the boundary-layer thickness and the assumption of approximate and well-defined velocity, temperature, and mass-fraction profiles in the boundary layer. In this way, the partial differential conservation equations are replaced with ODEs in which the dependent variable is the boundary-layer thickness. The solution of the ODE derived in this way then provides the thickness of the boundary layer. Knowing the boundary-layer thickness, along with the aforementioned approximate velocity and temperature profiles, we can then easily find the transport rates through the boundary layer. The integral technique is quite flexible and, unlike the similarity solution method, can be applied to relatively complicated flow configurations.

Integral Momentum Equations

Let us first consider the velocity boundary layer on a flat plate that is subject to the steady and uniform parallel flow of a fluid, as shown in Fig. 5.1. We define a control volume composed of a slice of the flow field that has a thickness dx and height Y. We choose Y to be large enough so that it will be larger than the boundary-layer thickness throughout the range of interest. The inflow and outflow parameters relevant to momentum and energy are also depicted in Fig. 5.1.

Miniature flow passages, defined here as passages with hydraulic diameters smaller than about 1 mm, have numerous applications. Some current applications include monolith chemical reactors, inkjet print-heads, bioengineering and biochemistry (lab-on-the-chip; drug delivery with ultrathin needles, etc.), microflow devices (micropumps, micro heat exchangers, etc.), and cooling systems for microelectronic and high-power magnets, to name a few. Miniature flow passages are an essential part of microfluidic devices, in which can be broadly defined as devices in which minute quantities of fluid are applied. Cooling systems based on microchannels can provide very large volumetric heat disposal rates that are unfeasible with virtually any other cooling technology. Their widespread future applications may in fact revolutionize some branches of medicine and industry.

The serious study of flow in capillaries (tubes with D ≈ 1 mm) goes back to at least the 1960s. The application of microchannels for cooling of high-power systems is relatively new, however (Tuckerman and Pease, 1981). The literature dealing with flow in microtubes is extensive. Useful reviews include those of Papautsky et al. (2001), Morini (2004), Krishnamoorthy et al. (2007), and Fan and Luo (2008). The field of flow in miniature channels, in particular with respect to very small channels (microfluidics and nanofluidics) is a rapidly developing one. In this chapter we review the flow regimes and size-based miniature flow passage categories, and we discuss the limitations of the classical convection heat and mass transfer theory with respect to its application to miniature flow passages.