Fully turbulent sublayer

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. 

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. 

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... 

In reality the laminar sublayer is continuously transformed into the fully turbulent region. A transition region exists between the two, known as the buffer layer, so that the wall law of velocity can be split into three areas, whose boundaries are set by experimentation. The laminar sublayer extends over the region... 

In the immediate vicinity of the solid wall, the turbulent fluctuations will be damped even in fully turbulent flow. In this thin layer adjacent to the wall, also known as the viscous sublayer, the viscous effect of the fluid outweighs that of its turbulent viscosity. 

We will now assume that the fully turbulent core adjoins directly the laminar sublayer, as in Fig. 3.21, and that Boussinesq s laws (3.141), (3.143) and (3.145) are valid, which accordingly give... 

In this paragraph the wall function concept is outlined. The wall functions are empirical parameterizations of the mean flow variable profiles within the inner part of the wall boundary layers, bridging the fully developed turbulent log-law flow quantities with the wall through the viscous and buffer sublayers where the two-equation turbulence model is strictly not valid. These empirical parameterizations thus allow the numerical flow simulation to be carried out with a finite resolution within the wall boundary layers, and one avoids accounting for viscous effects in the model equations. Therefore, in the numerical implementation of the k-e model one anticipates that the boundary layer flow is not fully resolved by the model resolution. The first grid point or node used at a wall boundary is thus placed within the fully turbulent log-law sub-layer, rather than on the wall itself [95]. In effect, the wall functions amount to a synthetic boundary condition for the k-e model. In addition, the limited boundary layer resolution required also provides savings on computer time and storage. 

The behavior of the alternate forms of eM/v in the near-wall region of a turbulent boundary layer is shown in Fig. 6.35. The classical Prandtl-Taylor model assumes a sudden change from laminar flow (eM/v = 0) to fully turbulent flow (Eq. 6.173) at y = 10.8. The von Kftrman model [88] allows for the buffer region and interposes Eq. 6.174 between these two regions. The continuous models depart from the fully laminar conditions of the sublayer around y+ = 5 and asymptotically approach limiting values represented by Eq. 6.173. In finite difference calculations, eM/v is allowed to increase until it reaches the value given by Eq. 6.158 and then is either kept constant at this value or diminished by an intermittency factor found experimentally by Klebanoff [92]. 

Early theories for transpiration of air into air [114, 115] were based on the Couette flow approximation. Reference 114 extended the Reynolds analogy to include mass transfer by defining a two-part boundary layer consisting of a laminar sublayer and a fully turbulent core. Here, t = 0 in the sublayer (y < y ), and t = OAy and (i = 0 in the fully turbulent region. The density was permitted to vary with temperature. The effect of foreign gas injection in a low-speed boundary layer was studied in Ref. 116, and all these theories were improved upon in Ref. 117. 

The large values indicate a fully turbulent region. We expect that du fdx2 should be independent of V in this region, since the thickness of the laminar sublayer, proportional to v/m, is on the order of 0.01 cm. Thus we can set F2(x3U. /v) = constant = a, and... 

The P functions are preferred to presume pdf because they are able to change continuously from pdf shapes to delta shapes [20]. For a < 1 a singularity is developed at Z = 0 for 8 < 1 a singularity is developed at Z = 1. For such shapes, also found in jets, a composite model of three parts (a fully turbulent, a sublayer, and an outer flow) was developed by Effelsberg and Peters [19]. 

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. 

A.M. Mollinger, F.T.M. Nieuwstadt, Measurement of the lift force on a particle fixed to the wall in the sublayer of a fully developed turbulent boundary layer, J. Fluid. Mech. 316 (1996) 285-306. 

Water flows at a mean velocity of 1 m/s through a 1-cm diameter pipe, the mean water temperature being 20°C. The flow can be assumed to be fully developed and turbulent. If the thickness of the sublayer is given by y = 12, find the actual thickness of the sublayer in mm. 

Figure 10.1. Time-averaged velocity profiles in fully developed turbulent flow identifying the presence of a laminar sublayer, a buffer layer, and the turbulent core of the flow. Velocity profile calculated from Eqs. 10.2.14-10.2.16.

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. 

As flow rate was increased, the thin stream of colored water in the center of the pipe would begin to waver and oscillate back and forth across the pipe. The flow rate at which this occurred would vary according to the pipe diameter. Eventually, as flow continued to be increased, the colored filament would break up altogether (see Fig. 48.3b). The final stage of this flow development where the colored filament breaks up altogether and disappears we nowadays understand to be an illustration of fully developed turbulent flow. The existence of eddy currents randomly distributed across the whole cross-section of the pipe causes effective mixing throughout the fluid as it flows in all but a very narrow layer of stationary fluid, termed "the laminar sublayer," which persists next to the wall of the pipe itself (see Fig. 48.4). 

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