Turbulence, wall boundary conditions

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

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)],... 

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... 

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... 

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). 

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]. 

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 ... 

Darrieus and Landau established that a planar laminar premixed flame is intrinsically unstable, and many studies have been devoted to this phenomenon, theoretically, numerically, and experimentally. The question is then whether a turbulent flame is the final state, saturated but continuously fluctuating, of an unstable laminar flame, similar to a turbulent inert flow, which is the product of loss of stability of a laminar flow. Indeed, should it exist, this kind of flame does constitute a clearly and simply well-posed problem, eventually free from any boundary conditions when the flame has been initiated in one point far from the walls. 

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. 

For homogeneous turbulent flows (no walls, periodic boundary conditions, zero mean velocity), pseudo-spectral methods are usually employed due to their relatively high accuracy. In order to simulate the Navier-Stokes equation,... 

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). 

Furthermore, it is assumed that there is no particle fluctuation on the wall so that the boundary condition of particle turbulent kinetic energy at the wall is given by... 

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).

The standard k-e model simulates the turbulence in the reactor. For flow within the porous catalyst bed, however, we suppress the turbulence. We enter the appropriate physical properties of the system, and employ standard boundary conditions at the impermeable walls and the reactor outlet. To represent the turbulence of the feed stream at the inlet, we treat it as pipe-flow turbulence. These model equations can then be solved for instance, via the well-known Simple algorithm [3]. To facilitate fast convergence, it is useful to make a reasonable initial guess of the pressure drop across the catalyst bed. 

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