Mean Flow Equations

The mean flow equations are obtained by invoking boundary layer approximation to the above conservation equations. For the two-dimensional steady incompressible flow with constant properties and Boussinesq approximation, the non-dimensional equations are written in a Cartesian coordinate system, fixed at the leading edge of the semi-infinite horizontal flat plate as, 

Following the notations and non-dimensionalization schemes of Schneider (1979), these equations are written as, 

In the above set of equations, (6.3.5) is automatically satisfied if we introduce a stream function (ip). As P indicates an excess pressure over the free-stream value, then (6.3.7) can be integrated to P = — K0dY. When these are introduced in the x-momentum and the energy equation, one gets 

It is shown in Schneider (1979), that the above formulation admits similarity solution, if oc X and for which a similarity transformations is introduced for the independent variable r] = YX / and for the dependent variables via -tp = These transformation 3ueld 

In these equations a prime indicates a derivative with respect to the independent variable, rj. These equations have to be solved subject to the boundary conditions at rj = 0 g = g = 0 and 0 = 1 and as 77 — 00 g = I and 0 = 0. The energy equation can be furthermore integrated analytically once to obtain. 

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We have also discussed the formation of spatio-temporal patterns in non-variational systems. A typical example of such systems at nano-meter scales is reaction-diffusion systems that are ubiquitous in biology, chemical catalysis, electrochemistry, etc. These systems are characterized by the energy supply from the outside and can exhibit complex nonlinear behavior like oscillations and waves. A macroscopic example of such a system is Rayleigh-Benard convection accompanied by mean flow that leads to strong distortion of periodic patterns and the formation of labyrinth patterns and spiral waves. Similar nano-meter scale patterns are observed during phase separation of diblock copolymer Aims in the presence of hydrodynamic effects. The pattern s nonlinear dynamics in both macro- and nano-systems can be described by a Swift-Hohenberg equation coupled to the non-local mean-flow equation. 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

The balanced equation for turbulent kinetic energy in a reacting turbulent flow contains the terms that represent production as a result of mean flow shear, which can be influenced by combustion, and the terms that represent mean flow dilations, which can remove turbulent energy as a result of combustion. Some of the discrepancies between turbulent flame propagation speeds might be explained in terms of the balance between these competing effects. 

For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the Vacuum Flow subsec tion) may be applied when the pore is of the same or smaller order as the mean free path, as described by Monet and Vermeulen (Chem. E/ig. Pi og., 55, Symp. Sei , 25 [1959]). 

Thus, using equation (6) to calculate (yo), equation (7) to calculate (yt), the mean flow rate (<2,) over a time period (t) can be calculated from... 

From this relatively simple test, therefore, it is possible to obtain complete flow data on the material as shown in Fig. 5.3. Note that shear rates similar to those experienced in processing equipment can be achieved. Variations in melt temperature and hypostatic pressure also have an effect on the shear and tensile viscosities of the melt. An increase in temperature causes a decrease in viscosity and an increase in hydrostatic pressure causes an increase in viscosity. Topically, for low density polyethlyene an increase in temperature of 40°C causes a vertical shift of the viscosity curve by a factor of about 3. Since the plastic will be subjected to a temperature rise when it is forced through the die, it is usually worthwhile to check (by means of Equation 5.64) whether or not this is signiflcant. Fig. 5.2 shows the effect of temperature on the viscosity of polypropylene. 

In fully developed flow, equations 12.102 and 12.117 can be used, but it is preferable to work in terms of the mean velocity of flow and the ordinary pipe Reynolds number Re. Furthermore, the heat transfer coefficient is generally expressed in terms of a driving force equal to the difference between the bulk fluid temperature and the wall temperature. If the fluid is highly turbulent, however, the bulk temperature will be quite close to the temperature 6S at the axis. 

At very high pressure, where the mean free path is short compared to the separation distance between surfaces, (a A jX)dP 1, and Eq. (101) reduces to the heat flow equation for moderate pressure gases given by Eq. (88). Although the derivation given here is not self-consistent when the mean free path is roughly the same as the separation distance, it is interesting to note that as the pressure becomes very small, (a A0/X)dP 1, and Eq. (101) reduces to the free molecular... 

The focus of RANS simulations is on the time-averaged flow behavior of turbulent flows. Yet, all turbulent eddies do contribute to redistributing momentum within the flow domain and by doing so make up the inherently transient character of a turbulent-flow field. In RANS, these effects of the full range of eddies are made visible via the so-called Reynolds decomposition of the NS equations (see, e.g., Tennekes and Lumley, 1972, or Rodi, 1984) of the flow variables into mean and fluctuating components. To this end, a clear distinction is required between the temporal and spatial scales of the mean flow on the one hand and those associated with the turbulent fluctuations on the other hand. 

It has been assumed that the flow is incompressible so that there are no fluctuations of the density. Equation 1.91 shows that the momentum flux consists of a part due to the mean flow and a part due to the velocity fluctuation. The extra momentum flux is proportional to the square of the fluctuation because the momentum is the product of the mass flow rate and the velocity, and the velocity fluctuation contributes to both. The extra momentum flux is equivalent to an extra apparent stress perpendicular to the face, ie a normal stress component. As (v x)2 is always positive it produces a compressive stress, which is positive in the negative sign convention for stress. 

In turbulent flow, there is direct viscous dissipation due to the mean flow this is given by the equivalent of equation 1.98 in terms of the mean values of the shear stress and the velocity gradient. Similarly, the Reynolds stresses do work but this represents the extraction of kinetic energy from the mean flow and its conversion into turbulent kinetic energy. Consequently this is known as the rate of turbulent energy production ... 

The final form for the scalar mean transport equation in a turbulent reacting flow is given by... 

In variable-density flow, additional estimated terms are needed for the mean energy equation (Jenny et al. 2001). 

For variable-density flow, Muradoglu etal. (2001) identify a third independent consistency condition involving the mean energy equation. 

The analysis used by these authors for constant flow conditions has been extended to constant pressure conditions also. The major change arises from the fact that now the flow is a function of the extent to which the bubble has already been formed. This has been introduced into the constant flow equation by means of an orifice equation. 

We will now show that Eq. (4.31) may be obtained by solving the atmospheric diffusion equation in which diffusion in the direction of the mean flow is neglected relative to advection ... 

The basic Gaussian plume dispersion parameters are ay and a. The essential theoretical result concerning the dependence of these parameters on travel time is for stationary, homogeneous turbulence (Taylor, 1921). Consider marked particles that are released from the origin in a stationary, homogeneous turbulent flow with a mean flow in the x direction. The y component, y, of the position of a fluid particle satisfies the equation... 

Thus, this equation has indeed the form of Eq. 1 with the exponent n = 1/4. Note that the shear velocity for river flow can be calculated either from the slope and geometry of the river bed (Eq. 20-33a) or from the mean flow velocity u (Eq. 20-3 3b). 

Although it seems natural to formulate the dynamic equations of a chemical in a river in terms of the Langrangian picture, the field data are usually made in the Eulerian reference system. In this system we consider the changes at a fixed point in space, for instance, at a fixed river cross section located atxQ. In Eq. 22-6 we adopted the Eulerian system and found that this representation combines the influence from in-situ reactions (the Langrangian picture) with the influence from transport. The latter appears in the additional advective transport term -udCJdx, where the mean flow velocity ... 

By analogy to the description of dispersion in rivers, the dispersive flux relative to the mean flow, Fdis, can be described by an equation of the First Fickian Law type (see Eq. 24-42) ... 

Moreover if we apply equation (2) to the deep interior of stars (r=0), eather the velocity or the density should become infinitely large at r=0. Therefore we cannot get any normal stellar structures. This means that equation (2) is inadequate to the interior part of stars. This difficulty comes from the steady-state approximation (1). The mass flux must reduce zero at the center of the stars or the surface of the degenerate stars. The interior flow therefore should be described by another steady states, not by equation (1). Therefore we will present a new steady-state approximation and derive mass-loss equations which is available also to the deep interior of stars. 

Figure 23 shows the dependence of CTL intensity at 400 and 500 °C on flow velocity. The CTL intensity I at 400 °C is almost independent of the mean flow velocity v of the sample gas, i.e., the catalytic oxidation is under reaction-controlled conditions. On the other hand, at 500 °C the curve plotted as log I vs. log v has a slope of 1/2. Equation 16 shows that the rate of catalytic oxidation ry is proportional to the square root of the flow velocity uo, so that the catalytic oxidation at 500 °C is under diffusion-controlled conditions. As the rate constant of diffusion k ) increases with increasing m0> the value of rx... 

FIGURE 2.15 Comparison among different views of a snapshot of a subsystem satisfying the Swift-Hohenberg equation—a simplified model of convection in the absence of mean flow. Panel (a) shows the detailed flow directions, whereas panels (b) and (c) exhibit the amplitude and phase, respectively. The latter are slowly varying and allow for easier identification of the grain boundaries of the flow. 

For a turbulence model to be useful in a general-purpose CFD code, it must be simple, accurate, economical to run, and have a wide range of applicability. Table 10-1 gives the most common turbulence models. The classical models use the Reynolds equations and form the basis of turbulence calculations in currently available commercial CFD codes. Farge eddy simulations are turbulence models where the time-dependent flow equations are solved for the mean flow and the largest eddies and where the effects of the smallest eddies are modeled. 

The mean free path, /, of the C02 molecules at the temperatures and pressures of the permeation experiment are by far smaller than the membrane pore size, d, that is, d, /.. Then, Knudsen flow is not possible since the determining process is gaseous laminar flow through the membrane pores [18]. It is therefore feasible to apply Darcy s law for gaseous laminar flow (Equations 10.19 through 10.23). 

The mean flow velocity in the capillary is given by Eq. 4.11 if we replace Ap in this equation by Eq. 4.33 and if we replace bed length L by the distance Xf that the liquid front has advanced, we get... 

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