Boundary conditions turbulence near wall

The Presumed Probability Density Function method is developed and implemented to study turbulent flame stabilization and combustion control in subsonic combustors with flame holders. The method considers turbulence-chemistry interaction, multiple thermo-chemical variables, variable pressure, near-wall effects, and provides the efficient research tool for studying flame stabilization and blow-off in practical ramjet burners. Nonreflecting multidimensional boundary conditions at open boundaries are derived, and implemented into the current research. The boundary conditions provide transparency to acoustic waves generated in bluff-body stabilized combustion zones, thus avoiding numerically induced oscillations and instabilities. It is shown that predicted flow patterns in a combustor are essentially affected by the boundary conditions. The derived nonreflecting boundary conditions provide the solutions corresponding to experimental findings. 

It is noted that turbulence may also affect the boundary conditions of velocity and heat flux near the wall. A detailed discussion using the k-c model for these boundary conditions is given by Launder and Spalding (1974). 

Figure 13 plots an example of the processed PIV frame. The turbulent velocity field and its boundaries, solid wall, and liquid-free surface are simultaneously shown in Figure 13. The turbulence structures such as the coherent vortical structure near the bottom wall and its modification after release from the no-slip boundary condition near the free surface of the open-channel flow, and the evolvement of the free-surface wave can be seen in Figure 13. This simultaneous measurement technique for free-surface level and velocity field of the liquid phase using PIV has been successfully applied to the investigation of wave-turbulence interaction of a low-speed plane liquid wall-jet flow (Li et al., 2005d), and the characteristics of a swirling flow of viscoelastic fluid with deformed free surface in a cylindrical container driven by the constantly rotating bottom wall (Li et al., 2006c).

In most high Reynolds number flows, the wall function approach gives reasonable results without excessive demands on computational resources. It is especially useful for modeling turbulent flows in complex industrial reactors. This approach is, however, inadequate in situations where low Reynolds number effects are pervasive and the hypotheses underlying the wall functions are not valid. Such situations require the application of a low Reynolds number model to resolve near-wall flows. For the low Reynolds number version of k-s models, the following boundary conditions are used at the walls ... 

At the outlet, extrapolation of the velocity to the boundary (zero gradient at the outlet boundary) can usually be used. At impermeable walls, the normal velocity is set to zero. The wall shear stress is then included in the source terms. In the case of turbulent flows, wall functions are used near walls instead of resolving gradients near the wall (refer to the discussion in Chapter 3). Careful linearization of source terms arising due to these wall functions is necessary for efficient numerical implementation. Other boundary conditions such as symmetry, periodic or cyclic can be implemented by combining the formulations discussed in Chapter 2 with the ideas of finite volume method discussed here. More details on numerical implementation of boundary conditions may be found in Patankar (1980) and Versteeg and Malalasekara (1995). 

Both equilibrium and non-equilibrium wall boundary implementations are considered. For equilibrium flows the local production rate of turbulence equals the dissipation rate in the near wall grid node. The first set of wall function boundary conditions reported was apparently used for equilibrium flows by Gosman et al. [59]. Denoting the dependent variables in the first point near the wall by a subscript P, an approximate sketch of their approach is given next. 

Design of a turbulent reactor requires consideration of V and ijf since both will affect reaction yields. Eor turbulent flow in long, empty pipes, the time-average velocities in the radial and tangential directions are zero since there is no net flow in these directions. The axial velocity component will have a nonzero time-average profile y (r). This profile is considerably flatter than the parabolic profile of laminar flow, but a profile nevertheless exists. The zero-slip boundary condition still applies and forces V (R) = 0. The time average-velocity changes very rapidly near the tube wall. 

Analogous to circular ducts, the fully developed turbulent Nusselt numbers for uniform wall temperature and uniform wall heat flux boundary conditions in parallel plate ducts are nearly identical for Pr > 0.7 and Re > 105. This is also true for the Nusselt number of turbulent thermally developing flow in a parallel plate duct [147]. 

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]. 

For the velocity components parallel to the wall the calculation of the wall boundary conditions, i.e., the apparent bulk source term, for turbulent flows starts with the estimation of yp, the dimensionless distance of the near wall node, P, to the solid surface. For turbulent flows where yp < 11.63, the value of the laminar wall shear stress is determined from ... 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

The K-E turbulence model discussed above only applies when e v. This will not be true near the wall. The most common way of dealing with this problem is to assume that there is a universal velocity distribution adjacent to the wall and the K-E turbulence mo del is then only applied outside of the region in which this wall region velocity distribution applies. Alternatively, more refined versions of the K-E turbulence model have been developed that apply under all conditions, i.e., across the entire boundary layer. 

The eddy diffusitives for momentum and heat, and Ejj, respectively, are not properties of the fluid but depend on the conditions of flow, especially on all factors that affect turbulence. For simple analogies, it is sometimes assumed that and jf are both constants and equal, but when determined by actual velocity and temperature measurements, both are found to be functions of the Reynolds number, the Prandtl number, and position in the tube cross section. Precise measurement of the eddy diffusivities is diflScult, and not all reported measurements agree. Results are given in standard treatises. The ratio Sh/sm also varies but is more nearly constant than the individual quantities. The ratio is denoted by i/f. For ordinary liquids, where Np > 0.6, is close to 1 at the tube wall and in boundary layers generally and approaches 2 in turbulent wakes. For liquid metals is low near the wall, passes through a maximum of about unity at j/r X 0.2, and decreases toward the center of the pipe. ... 

Drag reduction can be achieved by direct injection of microbubbles through slots or porous skin (193-196) or the generation of hydrogen by electrolysis at the wall (197). The primary parameters, independent of gas type and Reynolds number, appear to be the actual gas flow rate referenced to injector conditions of temperature and pressure (198-200) and the location of the bubbles in the turbulent boundary layer (198,199,201-203). Merkle and Deutsch (196) have provided a comprehensive review on skin friction reduction by microbubble injection. Mahadevan and co-workers (204) postulated that microbubbles like polymer solution destroy turbulence production by selectively increasing the viscosity near the buffer region. They increase the local dynamic viscosity. Pal and co-workers (205) demonstrated that microbubble and polymer solution shear stress statistics as measured by flush moimted hot film sensors are similar at equivalent value of drag reduction. 

---