Sublayer viscous

In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between. 

The parameters fcg and k are the transfer velocities for chemically unreactive gases through the viscous sublayers in the air and water, respectively. They relate the flux density F to the concentration gradients across the viscous sublayers through expressions similar to Equation (42) ... 

Fig. 4-16 A simplified model of flux resistances and concentration gradients in the viscous sublayers at the air-sea interface.

A review on drag-reducing polymers is given in the literature [1359]. It has been suggested that drag reduction occurs by the interactions between elastic macromolecules and turbulent-flow macrostructures. In turbulent pipe flow, the region near the wall, composed of a viscous sublayer and a buffer layer, plays a major role in drag reduction. 

When electrically insulated strip or spot electrodes are embedded in a large electrode, and turbulent flow is fully developed, the steady mass-transfer rate gives information about the eddy diffusivity in the viscous sublayer very close to the electrode (see Section VI,C below). The fluctuating rate does not give information about velocity variations, and is markedly affected by the size of the electrode. The longitudinal, circumferential, and time scales of the mass-transfer fluctuations led Hanratty (H2) to postulate a surface renewal model with fixed time intervals based on the median energy frequency. 

The near-wall region is conceptually subdivided into three layers, based on experimental evidence. The innermost layer is the viscous sublayer in which the flow is almost laminar, and the molecular viscosity plays a dominant role. The outer layer is considered to be fully turbulent. The buffer layer lies between... 

To save computational effort, high-Reynolds number models, such as k s and its variants, are coupled with an approach in which the viscosity-affected inner region (viscous sublayer and buffer layer) are not resolved. Instead, semiempiri-cal formulas called wall functions are used to bridge the viscosity-affected region between the wall and the fully turbulent region. The two approaches to the sublayer problem are depicted schematically in Fig. 2 (Fluent, 2003). 

It is important to place the first near-wall grid node far enough away from the wall at yP to be in the fully turbulent inner region, where the log law-of-the-wall is valid. This usually means that we need y > 30-60 for the wall-adjacent cells, for the use of wall functions to be valid. If the first mesh point is unavoidably located in the viscous sublayer, then one simple approach (Fluent, 2003) is to extend the log-law region down to y — 11.225 and to apply the laminar stress-strain relationship U — y for y < 11.225. Results from near-wall meshes that are very fine using wall functions are not reliable. 

At the wall, e O, but this behaviour cannot be calculated from the l/7th power law, which is not valid near the wall (ie in the viscous sublayer and buffer zone). The equation is also slightly in error at the centre-line where it does not predict the required zero velocity gradient, e tends to a non-zero value at the centre-line. Although the shear stress and velocity gradient both tend to zero at the centre-line and e is therefore indeterminate from equation 1.95, it can be determined by applying L Hopital s rule [Longwell (1966)]. 

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

Equation 2.40 is an empirical equation known as the one-seventh power velocity distribution equation for turbulent flow. It fits the experimentally determined velocity distribution data with a fair degree of accuracy. In fact the value of the power decreases with increasing Re and at very high values of Re it falls as low as 1/10 [Schlichting (1968)]. Equation 2.40 is not valid in the viscous sublayer or in the buffer zone of the turbulent boundary layer and does not give the required zero velocity gradient at the centre-line. The l/7th power law is commonly written in the form... 

Consider a fully developed turbulent flow through a pipe of circular cross section. A turbulent boundary layer will exist with a thin viscous sublayer immediately adjacent to the wall, beyond which is the buffer or generation layer and finally the fully turbulent outer part of the boundary layer. 

In the viscous sublayer, the magnitude of the time-averaged value of the shear stress f is given by Newton s law of viscosity which can be written in this case as... 

The dimensionless distance y+ has the form of a Reynolds number. Equation 2.58 fits the experimental data in the range 0 y+ 5. In the viscous sublayer, the velocity increases linearly with distance from the wall. 

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... 

The changing character of the flow in the different regions of the turbulent boundary layer explains certain aspects of the friction factor chart. If the absolute roughness of the pipe wall is smaller than the thickness of the viscous sublayer, flow disturbances caused by the roughness will be damped out by viscosity. The wall is subject to a viscous shear stress. Under these conditions, the line on the friction factor chart... 

As the wall becomes rougher, the velocity profile in the turbulent zone changes as shown in Figure 2.5, and the viscous sublayer and generation zone eventually disappear. 

For Newtonian fluids the velocity profile in the viscous sublayer adjacent to the wall is... 

While the viscous sublayer may be important for momentum transport, it is everything for mass and heat transport through liquids. Virtually the entire concentration boundary layer is within the viscous sublayer This difference is important in our assumptions related to interfacial transport, the topic of Chapter 8, where mass is transported through an interfacial boundary layer. 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

More recent experiments [62] concerning the viscous sublayer have shown a three-dimensional structure for turbulence near the wall. In a plane normal to the mean flow, counterrotating eddy pairs are involved (Fig. 6c), whereas in the direction of the mean flow, the motion is quasi-periodic (as described earlier). Since the wavelength along the mean flow is much larger than along the perimeter of the tube, a simplified bidimensional model may account only... 

Fortuna G, Hanratty TJ (1972) The influence of drag reducing polymers on the viscous sublayer J Fluid Mech 53 575... 

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