Near-wall

If the user wants higher accuracy in the predictions, and if he/she is willing to pay for the increase in accuracy in terms of increased computing cost, more grid nodes should be used. In general, more grid nodes should be located where the flow is complex. For an empty room without furniture or persons, this normally means that more grid nodes should be placed near walls and in... 

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. 

Chen, H. C, Patel, V. C. Near-wall turbulence models for complex flows including separation, AIAA J., vol. 26, pp. 641-648, 1988. 

R. Dickman, C. K. Hall. High density Monte Carlo simulations of chain molecules Bulk equation of state and density profile near walls. J Chem Phys 59 3168-3174, 1988. 

The paper by Davies et al. (2006) reports results of a numerical investigation of the laminar, periodically repeating flow in a parallel-plate micro-channel with superhydrophobic walls. In particular, the influence of the Reynolds number and the vapor cavity size on the overall flow dynamics was explored. A schematic of the near-wall and cavity regions is shown in Fig. 3.18. 

Fig. 3.18 Schematic of the near-wall and cavity regions for liquid flow over a superhydrophobic surface exhibiting micro-rib structures and flow perpendicular to the ribs...

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

Use A- = 0.25. The stability criterion at the near-wall position is obtained from Equation (8.36) with otr replacing or from Equation (8.59) evaluated at - = 1 — A. The result is... 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

B) The DNB in a medium- or low-subcooling bubbly flow is caused by near-wall bubble crowding and vapor blanketing. 

Near-wall bubble crowding and vapor blanketing... 

Figure 5.17 Physical model of heat balance near wall. (From Tong, 1972. Reprinted with permission of U.S. Department of Energy, subject to the disclaimer of liability for inaccuracy and lack of usefulness printed in the cited reference.)...

Lints, M. C., and Glicksman, L. R., Structure of Particle Clusters Near Wall of a Circulating Fluidized Bed, AIChE Symp. Series, 89(296) 35 47 (1993)... 

The differential form of this equation is used in calculating the effective viscosity in the RNG k-s model. This method allows varying the effective viscosity with the effective Reynolds number to accurately extend the model to low-Reynolds-number and near-wall flows. 

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... 

The viscosity-affected region is not The near-wall region is resolved... 

The turbulence models ought to be valid throughout the near-wall region. 

Fig. 2. Near-wall treatments (reproduced from Fluent Inc., Version 6.1 Manual, 2003, by permission).

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 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... 

With increased computer power, our next step was to construct full beds of particles N —2 for validation studies as described above in Section II.D.2, and N = 4 for further investigation of the temperature fields and near-wall transport processes as described above in Section II.B.2. Some early flow maps and path... 

From this illustration we can see that the added detail of the radial temperature profile near the wall that could be provided by CFD simulations does not help in obtaining better estimates for the standard heat transfer parameters. It also implies that experimental efforts to measure temperatures closer to the wall are, in fact, counter-productive. Finally, it is clear that the standard model with plug flow and constant effective transport parameters does not fit satisfactorily to temperature profiles in low-Abeds. These considerations have led us to look for improved approaches to near-wall heat transfer. 

At this stage it seemed clear that to improve near-wall heat transfer modeling would require better representation of the near-wall flow field, and how it was connected to bed structure and wall heat transfer rates. Our early models of full beds of spheres at N — 4 were too large for our computational capacity when meshed at the refinement that we anticipated to be necessary for the detailed flow fields that we wanted. We therefore developed the WS approach described above in Section II.B.3. 

In Fig. 18, flow path lines are shown in a perspective view of the 3D WS. By displaying the path lines in a perspective view, the 3D structure of the field, and of the path lines, becomes more apparent. To create a better view of the flow field, some particles were removed. For Fig. 18, the particles were released in the bottom plane of the geometry, and the flow paths are calculated from the release point. From the path line plot, we see that the diverging flow around the particle-wall contact points is part of a larger undulating flow through the pores in the near-wall bed structure. Another flow feature is the wake flow behind the middle particle in the bottom near-wall layer. It can also be seen that the fluid is transported radially toward the wall in this wake flow. 

The second picture in Fig. 18 shows a temperature map for a vertical plane in the middle of the WS. The tube wall is to the right of the picture, and the scale has been chosen to emphasize the temperature gradients in the near-wall region. 

For relating the wall heat flux and the near-wall flow patterns quantitatively the separate pieces of information had to be linked. Detailed information on the... 

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