Boundary layers functions, defined

Fig. 6.10 Nondimensional axial-velocity gradients and scaled radial velocities at the viscous boundary-layer edge as a function of Reynolds number in a finite-gap stagnation flow. The Prandtl number is Pr = 0.7 and the flow is isothermal in all cases. The outer edge of the boundary layer is defined in two ways. One is the z position of maximum V velocity and the other is the z. at which T — 0.01. As Re - oo, du/dz - —2 and V — 1, which are the values in the inviscid semi-infinite stagnation flow regions.

The method of Vishik-Lyusternik is well developed not only for elliptic but also for parabolic and hyperbolic partial differential equations. Some nonlinear problems can be solved using this method as well. The main advantage of this method is its simplicity. Usually the boundary layer functions are defined as solutions of ordinary differential equations in which the independent variable is the stretched distance along the normal to the boundary (the variable p in our case). However, there are some problems when the boundary layer functions are described by more complicated equations, for example, by parabolic equations. A survey of results obtained by the Vishik-Lyusternik method with many references can be found in [28]. 

The initial and boundary conditions for 6o(x, t) are defined during the construction of the boundary layer functions. 

Let us now define the boundary layer functions of zeroth order. For IloU, we have the problem... 

The initial and boundary conditions for the function ao(x, r) are defined, as before, during the construction of the boundary layer functions. For example, for QoH, y, z, t) we obtain the problem... 

The initial condition at f = 0 and the second boundary condition at Jt = 1 are defined analogously. Thus, the process of constructing the boundary layer functions is similar to that described in Section X.B. A distinction is that we use now expansions in a Fourier series in terms of the eigenfunctions F (y,z) instead of cos(irny). 

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

The slope of the lines presented in Figure 5 is defined as k(q/v). The q/v term defines the turnover of the tank contents or what is commonly referred to as the retention time. When q is increased, the liquid contacts the carbon more often and the removal of pesticides should increase, however, the efficiency term, k, can be a function of q. As the waste flow rate is increased, the fluid velocity around each carbon particle increases, thereby increasing system turbulence and compressing the liquid boundary layer. The residence time within the carbon bed is also decreased at higher liquid flow rates, which will reduce the time available for the pesticides to diffuse from the bulk liquid into the liquid boundary layer and into the carbon pores. From inspection of Table II, the pesticide concentration also effects the efficiency factor, k can only be determined experimentally and is valid only for the equipment and conditions tested. 

The boundary-layer thickness is a function of the rotation rate and can be derived from the nondimensional velocity profiles. Boundary-layer thickness can be defined in different ways, but generally it represents the thickness of the viscous layer. Defining the boundary-layer thickness as the point at which the circumferential velocity is 1% of its surface value gives zi% = 5.45. 

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, <5H. The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 5H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7] takes a figure of 10% (see below). It has been shown that... 

Because they do not contain the pressure as a variable. Eqs. (2.76) and (2.77) have been used quite extensively in solving problems for which the boundary layer equations (see later) cannot be used. For this purpose, instead of solving the Navier-Stokes and energy, simultaneously with the continuity equation, it is convenient to introduce the stream function, ip, which is defined such that... 

The two-dimensional Navier-Stokes equation is solved in stream function-vorticity formulation, as reported variously in Sengupta et al. (2001, 2003), Sengupta Dipankar (2005). Brinckman Walker (2001) also simulated the burst sequence of turbulent boundary layer excited by streamwise vortices (in X- direction) using the same formulation for which a stream function was defined in the y — z) -plane only. To resolve various small scale events inside the shear layer, the vorticity transport equation (VTE) and the stream function equation (SFE) are solved in the transformed — rj) —... 

Many boundary-layer problems possess similarity solutions—solutions in which appropriately identified dependent variables may be expressed as functions of a single independent variable, t], the nondimensional similarity variable, through functions defined by ordinary differential equations in f/. To facilitate derivations of similarity solutions, after employing the Moore-Stewartson transformation, we shall further transform the equations to a general similarity variable and introduce a nondimensional stream function F(x, t], t), according to... 

Various different types of boundary-layer flows may be addressed by selecting g so that one of the four functions defined in equation (31) is a specified, nonzero constant. In many problems, a number of the functions in equation (31) vanish for example, for steady flows G2 = — 0, while... 

According to the assumptions in Section 6.2.1, the liquid phase concentration changes only in axial direction and is constant in a cross section. Therefore, mass transfer between liquid and solid phase is not defined by a local concentration gradient around the particles. Instead, a general mass transfer resistance is postulated. A common method describes the (external) mass transfer mmt i as a linear function of the concentration difference between the concentration in the bulk phase and on the adsorbent surface, which are separated by a film of stagnant liquid (boundary layer). This so-called linear driving force model (LDF model) has proven to be sufficient in... 

In other words, the pressure distribution in the boundary-layer is completely determined at this level of approximation by the limiting form of the pressure distribution impressed at its outer edge by the potential flow. It is convenient to express this distribution in terms of the potential-flow velocity distribution. In particular, let us define the tangential velocity function ue(x) as... 

The potential-flow solution for streaming motion past a circular cylinder was obtained earlier and given in terms ofthe streamfunctionin(10-17). To calculate the pressure gradient in the boundary layer, we first determine the tangential velocity function, ue, as defined in (10-37) ... 

While this critical flux phenomena is generally accepted in MF and UF, some authors also mention limiting fluxes in NF (Levenstein et al. (1996)). Cohen and Probstein (1986) determined a characteristic permeation velocity below which no fouling was observed in the RO of colloids. Bacchin et al. (1995) defined a critical flux for the MF, UF, and RO of large colloids. The critical flux Jcot is a function of diffusion and the potential barrier between particles Vb as shown in equation (3.36), where 5 is the boundary layer thickness and D the particle diffusivity. 

---