Turbulent boundary layer mixing

Rahman, S. Effect of Bed Roughness on Scalar Mixing in Turbulent Boundary Layers. Ph.D. Thesis, Georgia Institute of Technology, 2002. 

A numerical procedure for calculating the heat transfer rate with turbulent boundary layer flow was discussed in Chapter 5. This procedure used a mixing length-based turbulence model. Discuss the modifications that must be made to this procedure to apply it to mixed convective flow over a vertical plate. 

Since both laminar and turbulent boundary layers contain laminar or viscous layers, it would seem logical that diffusion would primarily take place across these regions. If the boundary layer thickness were known, assuming a linear decrease in concentration, Eq. 10.3 could be used to estimate diffusion current. Unfortunately, the point of uniform velocity is not necessarily the point of uniform concentration. This is because particles, with their large inertia compared to air, can be carried into laminar boundary regions by mixing as well as by diffusion. The value for 8 in Eq. 10.3 will always be less than the equivalent value for the aerodynamic boundary layer thickness, in some cases being only one-tenth or even smaller (Levich, 1962). 

In order to calculate wx/wT by solving the differential equation (3.149), the Reynolds stress w xw has to be known. The hypothesis introduced by Boussinesq (3.140) is unsuitable for this, as according to it, the Reynolds stress does not disappear at the wall. However, the condition w xw y = 0 at the wall is satisfied by Prandtl s mixing length theory, which will now be explained. In order to do this we will consider a fluid element in a turbulent boundary layer, at a distance y from the wall, Fig. 3.16. It has, at a distance y, the mean velocity wx(y) and... 

For tall structures (where H/b > 1) in turbulent boundary layers the recirculating flow field in the wake is a region where streamlines enter and leave from the external flow (Lawson et al., 1988 [361]). Although in general there is no closed streamline bubble , because of the high turbulence there is strong mixing in the horizontal... 

The existing turbulence models consist of approximate relations for the /ij-parameter in (5.246). The Prandtl mixing-length model (1.356) represents an early algebraic (zero-equation) model for the turbulent viscosity Ht in turbulent boundary layers. 

In the turbulent flow region a very efficient mixing takes place, i.e., macroscopic chunks of fluid move across stream-FIGURE 1.7 Property variation with time at some lines and transport energy and mass as well as momentum point in a turbulent boundary layer. vigorously. The most essential feature of a turbulent flow is... 

The addition of polymers has been investigated also for twenty years. Polymers reduce the drag of the turbulent boundary layer, without avoiding transition. Since the layer remains turbulent, there is no stringent requirement for surface smoothness, as there is for many of the other techniques. Although much is not known about the detailed mechanism, practical success can be achieved more or less routinely. As there would be in the application of any drag reduction technique, there are practical problems that must be solved to make polymers operational these include such problems as how and with what and when to mix the polymer and how to eject it, and how to mate the propulsion system to the altered body characteristics. 

A widely accepted hypothesis of the mechanism responsible for drag reduction is that the microscale eddies in the bufferzone of the turbulent boundary layer are suppressed. It is thought that the macromolecules (injected or pre-mixed) affect the sublayer instability which produces jet-like vortices, the so-called "bursts", which erupt into the boundary layer and lead to the production of much turbulent energy. Recent experiments have shown [l] that the polymer molecules seem to act as a "barrier" restricting communication across the boundary layer so that the fluid near the wall flows in a more laminar manner [2]. 

Lee et al. (2003) presented the effectiveness of microsynthetic jet for modification of turbulent boundary layer under adverse gradient. They showed significant mixing enhancement of the boundary layer when the foreing frequency is closer to the natural instability frequency. The... 

The aerosol model in VICTORIA is built upon the CHARM model developed by Wheatley [5]. The CHARM model treats aerosol behavior in a single computational cell, which is assumed to be well mixed. Time-varying external conditions are calculated in advance and supplied as data to the model. The implementation of CHARM in VICTORIA allows two options (1) aerosol particle composition Is independent of particle size or (2) aerosol particles are taken to be composed of a pure species, in which case a separate particle size distribution is calculated for each species that exists in the aerosol. In addition, the capability to account for the thickness of laminar or turbulent boundary layers and their effect on deposition rates has been added to the version of CHARM In VICTORIA. 

The phenomenon of concentration polarization, which is observed frequently in membrane separation processes, can be described in mathematical terms, as shown in Figure 30 (71). The usual model, which is weU founded in fluid hydrodynamics, assumes the bulk solution to be turbulent, but adjacent to the membrane surface there exists a stagnant laminar boundary layer of thickness (5) typically 50—200 p.m, in which there is no turbulent mixing. The concentration of the macromolecules in the bulk solution concentration is c,. and the concentration of macromolecules at the membrane surface is c. 

Consider the equilibrium set up when an element of fluid moves from a region at high temperature, lying outside the boundary layer, to a solid surface at a lower temperature if no mixing with the intermediate fluid takes place. Turbulence is therefore assumed to persist right up to the surface. The relationship between the rates of transfer of momentum and heat can then be deduced as follows (Figure 12.5). 

It may be assumed that the fully turbulent portion of the boundary layer starts at y+ — 30, that the ratio of the mixing length to the distance y from the surface, Af/v = 0.4, and that for a smooth surface u+ = I4 at v 30. 

Another widely used concept is that of a planetary boundary layer (PBL) in contact with the surface of the Earth above which lies the "free atmosphere." This PBL is to some degree a physically mixed layer due to the effects of shear-induced turbulence and convective overturning near the Earth s surface. 

---