Turbulence, wall boundary

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 mean velocity and turbulent diffusivity should approach zero at solid walls. In theory, this should be enough to keep particles from crossing wall boundaries. In practice, due to the finite time step, some particles will eventually cross wall boundaries and must be accounted for. 

Wall-boundary conditions in probability density function methods and application to a turbulent channel flow. Physics of Fluids 11, 2632-2644. 

A different situation is encountered at the bottom of a water body. The sediment-water interface is characterized by, on one side, a water column which is mostly turbulent (although usually less intensive than at the water surface), and, on the other side, by the pore space of the sediment column in which transport occurs by molecular diffusion. Thus, the turbulent water body meets a wall into which transport is slow, hence the term wall boundary (Fig. 19.3b). A wall boundary is like a one-sided bottleneck boundary, that is, like a freeway leading into a narrow winding road. 

The mathematics of the wall boundary model slightly changes if the media on either side of the interface are different. As an example, consider the volatilization of a dissolved chemical into the well-mixed atmosphere from a shallow puddle of water in which advective and turbulent motion is completely suppressed. Another example is the transport between a solid phase and a turbulent water body. 

In the preceding section, the sediment surface was described as an intermedia wall boundary. Thereby we tacitly assumed that the diffusion wall, that is, the location where diffusivity drops from DB, to DA coincides with the interface between the two media. As shown in Fig. 19.86, the transition from a turbulent to a stagnant media includes a boundary layer in the former in which diffusivity drops in a characteristic manner. 

For fully developed turbulent flow of liquid metals, the Nusselt number depends on the wall boundary condition. For a constant wall temperature [Notter and Sleicher, Chem. Eng. Science, 27,2073 (1972)],... 

The random eddy motion of groups of particles resembles the random motion of molecules in a gas—colliding with each other after traveling a certain distance and exchanging momentum and licat in the process. Therefore, momentum and beat transport by eddies in turbulent boundary layers is analogous to the molecular momentum and heat diffusion. Then turbulent wall shear stress and turbulent heat transfer can be expressed in an analogous manner as... 

In addition to representing the influence of walls on turbulence, adequate boundary conditions need to be specified to solve modeled equations of k and e. At computational boundaries far from the wall, the following boundary conditions can be used (Ferziger and Peric, 1995) ... 

The authors concluded that field conditions could be reproduced by both wind tunnel and numerical models, but the numerical models were very sensitive to alternative specifications of grid resolution, wall boundary conditions, source size and turbulence... 

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. 

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. 

Accordingly, the wall boundary condition for the turbulent energy dissipation rate, e, is given by (1.429) ... 

For the velocity components parallel to the wall the calculation of the wall boundary conditions, i.e., the apparent bulk source term, for turbulent... 

It has been shown that there exists a continuous change in the physical behavior of the turbulent momentum boundary layer with the distance from the wall. The turbulent boundary layer is normally divided into several regions and sub-layers. It is noted that the most important region for heat and mass transfer is the inner region of the boundary layer, since it constitutes the major part of the resistance to the transfer rates. This inner region determines approximately 10 — 20% of the total boundary layer thickness, and the velocity distribution in this region follows simple relationships expressed in the inner variables as defined in sect 1.3.4. 

Deen et al [30] used the lift force in addition to the steady drag- and added mass forces in their dynamic 3D-model to obtain the transversal spreading of the bubble plume which is observed in experiments. A prescribed zero void wall boundary was used forcing the gas to migrate away from the wall. The continuous phase turbulence was incorporated in two different ways, using... 

Step 4 Set the boundary conditions as follows. The centerline, inlet velocity, and exit velocity/pres sure are set as in the laminar case slip/symmetry, v = 2, Normal flow/ Pressure, p = 0. The wall boundary condition, though, is set to the Logarithmic wall function. This is an analytic formula for the velocity, turbulent kinetic energy, and rate of dissipation, as determined by experiment (Deen, 1998, pp. 527-528). 

W. C. Reynolds, W. M. Kays, and S. J. Kline, Heat Transfer in the Turbulent Incompressible Boundary Layer, I—Constant Wall Temperature, NASA Mem. 12-1-58W, 1958. 

This maximum steady state turbulent flame speed depends also on the H2 concentration itself and ranges from about 100 m/sec 10% H2) to about 1800 m/sec (30% b H2 45% H2). Upon exiting the 3 m length of the obstacle (i.e., the Shchelkhin spiral) into the "smooth tube, the flame immediately decelerates. For H2 15%, the flame decelerates to a new and much lower steady state value corresponding to the smooth wall boundary condition. However, for H2 15%, the flame reaccelerates and transits to detonation after a couple of meters of flame travel (the transition distance depends on the H2 concentration). We shall elaborate more on the transition phenomenon in a later section. 

For the gas velocity u, the wall boundary conditions can be of free slip, no-slip, or turbulent law-of-the-wall type. The free-slip condition demands that the normal velocity components of the fluid and the wall coincide, while the tangential components of the stress tensor satisfy o = slip boundary conditions require the fluid velocity to coincide with the wall velocity. No-slip boundary conditions impose large velocity gradients, which, because of lack of sufficient computational resources, cannot be resolved. Therefore, wall functions are used. The following discussion reflects the exhibition given by Amsden et al. [4]. 

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