Wall functions

When a coarse grid is used, wall functions are used for imposing boundary conditions near the walls (Section 11.2.3.3). The nondimensional wall distance should be 30 < y < ]Q0, where y = u,y/p. We cannot compute the friction velocity u. before doing the CFD simulation, because the friction velocity is dependent on the flow. However, we would like to have an estimation of y" to be able to locate the first grid node near the wall at 30 < y < 100. If we can estimate the maximum velocity in the boundary layer, the friction velocity can be estimated as n, — 0.04rj, . . After the computation has been carried out, we can verify that 30 nodes adjacent to the walls. 

In ventilated rooms buoyant effects are often imported, and wall functions to take this into account are presented in ref. 5. 

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. 

Yuan, X., Moser, A., Surer, P. Wall functions for numerical simulation of turbulent natura] convection along vertical plates, /nt. J. Heat Mass Transfer, vol. 36, pp. 4477—448,5, 1993. 

The k-o) model can simulate flow close to walls while the other models require a low Re addition or wall functions to describe the flow below y+ < 30. The models are optimized to give accurate flow predictions and the model parameters kinetic... 

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). 

The standard wall function (Launder and Spalding, 1974) has been widely used for industrial flows. The wall function is based on the assumption that the velocity obeys the log law-of-the-wall... 

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. 

The heat flux to the wall and the wall temperature are related through a wall function... 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

The preferred range for the thickness of the near-wall cell layer is y+ >30. However, this is difficult to achieve in packed tubes. The cells sizes are constrained by the need to fit in between the gaps and/or narrow spaces between particles, so they cannot be too large. This can result in the y+ values being too small for proper application of wall functions. The alternative to use small enough cells to resolve the boundary layer (y+ < 1) increases the computational... 

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat flux was specified as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ x 13-14. This is somewhat low for the k- model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

Fig. 1.3 Functionalization pathways for SWNTs (a) defect-group functionalization, (b) covalent side-wall functionalization, (c) noncovalent exohedral functionalization with surfactants, (d) noncovalent exohedral functionalization with polymers, and (e) endohedral functionalization with, for example, C60. For methods (b)-(e), the tubes are drawn in idealized fashion, but defects are found in real situations. From [103] with kind permission of Wiley.

HOLLOW-FIBER MEMBRANES. A hollow-fiher membrane is a capillary having an inside diameter of - inn and an outside diameter < I mm and whose wall functions as a semipermeahlc membrane. The fibers can he employed singly or grouped into a bundle which may contain tens of thousands of fibers and up to several million libers as in reverse osmosis (Fig. 11. In most eases, hollow fibers are used as cylindrical membranes that permit selective exchange of materials across (heir walls. However, they can also he used as containers to effect the controlled release of a specific material, or as reactors to chemically modify a permeate as il diffuses through a chemically activated hollow-liher wall. e g., loaded with immobilized enzyme. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

In order to give an approximation of the velocity gradient near the wall, an empirical equation, known as the wall function, is introduced. The wall function is expressed as... 

More recent series have also documented a survival advantage (7-9). Hoye et al. in 874 consecutive patients with a CTO found a five-year survival in 93.5% of patients with successful revascularization versus 88.0% in these patients with failed revascularization (p = 0.02). In a Canadian registry of 1458 patients at seven year, successful recanalization of a chronic total occlusion was associated with improved survival as well as lower rates of PCI and/or CABG (9). In addition to survival advantage, both regional and global left ventricular function is improved in patients with successful treatment of a chronic total occlusion (10). This improvement may depend on whether the patient had a prior infarction in the distribution of the occlusion (II). If prior infarction resulted in frank myocardial necrosis, then recanalization may not improve the function however, many patients with chronic occlusion have preservation of regional wall function. 

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