Navier equations introduced

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where / , (/<. t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4> (k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/<. t) of the form14... 

For the purpose of understanding pressure filtering, attention may be restricted to the single-component, constant-property, nonreacting equations for a perfect gas. Introducing the nondimensional variables into the vector forms of the mass-continuity, constant-viscosity Navier-Stokes, and perfect-gas thermal-energy equations yields the following nondimensional system ... 

In addition to the reference scales and nondimensional variables used for the Navier-Stokes equations, new scaling parameters must be introduced to nondimensionalize the temperature and diffusive mass flux. In a mixture-averaged setting... 

Most of the models assume that neutral-species transport can be represented with either a well-mixed model or a plug flow model. The major drawback to these assumptions is that important inelastic rate processes such as molecular dissociation are usually localized in space in the reactor and are often fast compared with rates of diffusion or convection. As a result, the spatial variation of fluid flow in the reactor must be accounted for. This variation introduces a major complication in the model, because the solution of the nonisothermal Navier-Stokes equations in multidimensional geometries is expensive and difficult. 

The LGA is a variant of a cellular automaton, introduced as an alternative numerical approximation to the partial differential equation of Navier-Stokes and the continuity equations, whose analytical solution leads to the macroscopic approach of fluid dynamics. The microscopic behavior of the LGA has been shown to be very close to the Navier-Stokes (N-S) equations for incompressible fluids at the macroscopic level. 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

As was mentioned above, the Navier-Stokes equations are obtained by the appli-cation of the conservation of momentum principle to the fluid flow. The same control volume that was introduced above in the discussion of the continuity equation is considered and the conservation of momentum in each of the three coordinate directions is separately considered. The net force acting on the control volume in any of these directions is then set equal to the difference between the rate at which momentum leaves the control volume in this direction and the rate at which it enters in this direction. The net force arises from the pressure forces and the shearing forces acting on the faces of the control volume. The viscous shearing forces for two-dimensional flow (see later) are shown in Fig. 2.3. They are expressed in terms of the velocity field by assuming the fluid to be Newtonian and are then given by [4],[5] ... 

Consider the velocity field first. It is governed by the continuity and Navier-Stokes equations which, subject to the assumptions introduced above, are, as previously presented ... 

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 total number of independent variables appearing in Fq. (4.32) is thus quite large, and in fact too large for practical applications. However, as mentioned earlier, by coupling Eq. (4.32) with the Navier-Stokes equation to find the forces on the particles due to the fluid, the Ap-particle system is completely determined. Although not written out explicitly, the reader should keep in mind that the mesoscale models for the phase-space fluxes and the collision term depend on the complete set of independent variables. For example, the surface terms depend on all of the state variables A[p ( x ", ", j/p" j, V ", j/p" ). The only known way to determine these functions is to perform direct numerical simulations of the microscale fluid-particle system using all possible sets of initial conditions. Obviously, such an approach is intractable. We are thus led to reduce the number of independent variables and to introduce mesoscale models that attempt to capture the average effect of multi-particle interactions. 

Finally, we note that it is frequently convenient to introduce the concept of a dynamic pressure into (2-89). This is a consequence of the utility of having the pressure gradient that appears explicitly in the Navier Stokes equation act as a driving force for motion. In the form (2 89), however, a significant contribution to Vp is the static pressure variationVp = pg, which has nothing to do with the fluid s motion. In other words, nonzero pressure gradients in (2 89) do not necessarily imply fluid motion. Because of this it is convenient to introduce the so-called dynamic pressure P, such that VP = 0 in a static fluid. This implies... 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

If the boundaries of the flow domain are not parallel, the magnitude of the primary velocity component must vary as a function of distance in the flow direction. This not only introduces a number of new physical phenomena, as we shall see, but it also means that the Navier-Stokes equations cannot be simplified following the unidirectional flow assumptions of Chap. 3, and exact analytical solutions are no longer possible. In this chapter, we thus consider only a special limiting case, known as the thin-gap limit, in which the distance between the boundaries is small compared with the lateral gap width. In this case, we shall see that we can obtain approximate analytical solutions by using the asymptotic and scaling techniques that were introduced in the preceding chapter. 

The governing equations for arbitrary X and s are the Navier-Stokes and continuity equations (3—55)—(3—58). We assume that the cylinders are infinitely long so that allz dependence is eliminated. Further, because there is no motion in the z direction at the boundaries, it can be seen from (3 58) that u z = 0. Finally, it will be convenient for comparisons between this and later sections of this chapter if we introduce the modified radial variable... 

In the forced convection approximation, with p and p fixed, the Navier-Stokes and continuity equations can be solved (at least in principle) to determine the velocity field u, and this solution is completely independent of the temperature distribution in the fluid. Once the velocity u is known, the thermal problem, represented by (9-1) and (9-2), can then be solved (again, in principle) to determine the temperature field T. Because the boundary values of T are assumed to be constant, we may anticipate that the temperature distribution throughout the fluid will be independent of time (with the exception of some initial period after the heated body is first introduced into the moving fluid). It is the steady-state temperature field that is most often our goal. For this reason, the time derivative in (9-1) will be dropped in subsequent developments. 

Nam et al. (2000) have shown that the theoretical framework introduced in the previous sections for the description of passively ad-vected decaying scalar field also applies to the description of the small scale structure of the vorticity field in a two-dimensional turbulent flow with linear damping. The vorticity dynamics in this case is described by the Navier-Stokes equation... 

Let us nondimensionalize the Navier-Stokes equations by introducing a characteristic velocity u0 and characteristic length L and defining the dimensionless variables... 

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