Turbulent boundary layer eddies

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

Conditions in the fully turbulent outer part of the turbulent boundary layer are quite different. In a turbulent fluid, the shear stress f is given by equation 1.95. As illustrated in Example 1.10, outside the viscous sublayer and buffer zone the eddy kinematic viscosity e is much greater than the molecular kinematic viscosity v. Consequently equation 1.95 can be written as... 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

It is interesting to compare equations (6.32) and (6.33) with those for a fully developed laminar flow, equations (6.29) and (6.30). In Example 5.1, we showed that eddy diffusion coefficient in a turbulent boundary layer was linearly dependent on distance from the wall and on the wall shear velocity. If we replace the diffusion coefficient in equation (6.30) with an eddy diffusion coefficient that is proportional to hu, we get... 

The random eddy motion of groups of particles resembles the random motion of molecules in a gas—colliding with each other after traveling a certain distance and exchanging momentum and licat in the process. Therefore, momentum and beat transport by eddies in turbulent boundary layers is analogous to the molecular momentum and heat diffusion. Then turbulent wall shear stress and turbulent heat transfer can be expressed in an analogous manner as... 

Note the similarity with the intermittent behaviour of the turbulent boundary layer with polymers. This may be relevant to the flow over very flexible plant canopies (Ptasinski et al., 2003 [511]). There is a finite jump in velocity across this fluctuating layer. The external turbulence is blocked by the vorticity of the layer (Hunt and Durbin, 1999 [292]). This is why there is only a very weak effect of large eddies moving above the canopy. The layer is analogous to that at the outer edge of jets, wakes and boundary layers (Bisset et al., 2002 [63]). 

Figure 6.1 Three regimes of canopy flow. Three scales of turbulence are present. The smallest scale (black circles) is set by the canopy morphology, specifically the diameter of and spacing between individual canopy elements, such as stems and branches. Drag discontinuity at the canopy interface generates a shear-layer that produces vortices via Kelvin-Helmholtz (K-H) instability (shown as solid, black ovals). Boundary layer vortices are present above the canopy (dashed gray). When H/h is small the water surface constrains the boundary layer eddy scale.

Turbulent Boundary Layer Theory - Eddy Viscosity Consept... 

The turbulent boundary layer model accounts for the transfer of a solute molecule A from a turbulent stream to a fixed surface. Eddy diffusion is rapid in the turbulent stream and molecular diffusion is relatively insignificant. It is supposed that the turbulence is damped out in the immediate vicinity of the surface. In the intermediate neighborhood between the turbulent stream and the fixed surface, it is supposed that transport is by both molecular and eddy diffusion which take place in parallel. The total rate of transfer (moles of A transferred per unit time per unit area) is given by an extended form of Fick s law... 

Farther away from the surface the fluid velocities, though less than the velocity of the undisturbed fluid, may be fairly large, and flow in this part of the boundary layer may become turbulent. Between the zone of fully developed turbulence and the region of laminar flow is a transition, or buffer, layer of intermediate character. Thus a turbulent boundary layer is considered to consist of three zones the viscous sublayer, the buffer layer, and the turbulent zone. The existence of a completely viscous sublayer is questioned by some, since mass transfer studies suggest that some eddies penetrate all the way through the boundary layer and reach the wall. 

Eddy Diffusivity Models. The mean velocity data described in the previous section provide the bases for evaluating the eddy diffusivity for momentum (eddy viscosity) in heat transfer analyses of turbulent boundary layers. These analyses also require values of the turbulent Prandtl number for use with the eddy viscosity to define the eddy diffusivity of heat. The turbulent Prandtl number is usually treated as a constant that is determined from comparisons of predicted results with experimental heat transfer data. 

The presence of wall and wake regions in the turbulent boundary layer is reflected in the distribution of the eddy viscosity. In the outer wake region, Clauser s empirical form (Eq. 6.158), has been adopted for finite-difference boundary layer computations, although Ref. 87 suggests that the constant in Eq. 6.158 be altered to 0.0168. In analyses, however, the need for defining the wake region eddy viscosity has not been critical, largely because of the nearness of the value of the turbulent Prandtl number to unity and the use of the Crocco transformation, as demonstrated in the next section. 

As mentioned previously, even when the flow becomes turbulent in the boundary layer, there exists a thin sub-layer close to the surface in which the flow is laminar. This layer and the fully turbulent regions are separated by a buffer layer, as shown schematically in Figure 7.1. In the simplified treatments of flow within the turbulent boundary layer, however, the existence of the buffer layer is neglected. In the laminar sub-layer, momentum transfer occurs by molecular means, whereas in the turbulent region eddy transport dominates. In principle, the methods of calculating the local values of the boundary layer thickness and shear stress acting on an immersed surface are similar to those used above for laminar flow. However, the main difficulty stems from the fact that the viscosity models, such as equations (7.13) or (7.27),... 

The border diffusion layer model was introduced as an amendment to the film model to present a more realistic description. It accounts for an undefined film thickness, turbulence effects, and the role of molecular diffusion. When the flow is turbulent, the flow around the bubble is split into four sections the main turbulent stream, the turbulent boundary layer, the viscous sublayer, and the diffusion sublayer. Eddy turbulence accounts for mass transfer in the main turbulent stream and the turbulent boundary layer. The viscous sublayer limits eddy turbulence effects so that the flow is laminar and mass transfer is controlled by both molecular diffusion and eddy turbulence. Microscale eddy turbulence is assumed to be dominant in the viscous sublayer. Mass transfer in the diffusion sublayer is controlled almost completely by molecular diffusion (Azbel, 1981). 

From this experimental work it is found out that the addition of drag reducing polymer stabilizes the Gortler vortices. If one accept that the Gortler vortices and sublayer eddies in turbulent boundary layer are analogous structure, it is inferred that the addition of polymer in turbulent boundary layer flows would also increase the size of the sublayer eddies and suppress the production of energy dissipating eddies. 

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]. 

Pamies et al. [19] expanded the method of Jarrin et al. [10] by dividing the inflow plane of an incompressible flat plate boundary layer into several zones depending on the wall distance. At each zone turbulent eddy shapes are prescribed in the sense of Marusic [17], i.e., these shapes are representative for t3q> ical coherent structures of the turbulent boundary layer. This resulted in a good approximation for the low-order statistics of wall-bounded flows and reduced the... 

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