Solution of the Navier-Stokes Equation

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. 

The flow phenomena described by the Navier-Stokes equation fall into two classes discriminated by the nature of the compressibility effects to be taken into account. For compressible flow, the Navier-Stokes equation [Eq. (1)] has to be solved in com- 

The situation is different for incompressible flow. In that case, no equation of motion for the pressure field exists and via the mass conservation equation Eq. (17) a dynamic constraint on the velocity field is defined. The pressure field entering the incompressible Navier-Stokes equation can be regarded as a parameter field to be adjusted such that the divergence of the velocity field vanishes. 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

A discretized version of the steady-state incompressible Navier-Stokes equation derived from Eq. (16) can be written as 

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Taylor, C. and Hood, P., 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73-100. 

Lee, R. L., Gresho, P. M. and Sani, R. L., 1979. Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations. Int. J. Numer. Methods Eng. 14, 1785-1804. 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

Turbulent inlet conditions for LES are difficult to obtain since a time-resolved flow description is required. The best solution is to use periodic boundary conditions when it is possible. For the remaining cases, there are algorithms for simulation of turbulent eddies that fit the theoretical turbulent energy distribution. These simulated eddies are not a solution of the Navier-Stokes equations, and the inlet boundary must be located outside the region of interest to allow the flow to adjust to the correct physical properties. 

There is an analytical solution of the Navier-Stokes equations for the flow between two rotating cylinders with laminar flow (see e.g. [37]). The following equation applies for the velocity gradient in the annular gap in the general case of rotation of the outer cylinder (index 2) and the inner cylinder (index 1) ... 

There is no general solution of the Navier-Stokes equations, which is due in part to the non-linear inertial terms. Analytical solutions are possible in cases when several of the terms vanish or are negligible. The skill in obtaining analytical solutions of the Navier-Stokes equations lies in recognizing simplifications that can be made for the particular flow being analysed. Use of the continuity equation is usually essential. 

Since velocity is a vector quantity, it is usually necessary to identify the component of the velocity, as was done for the rectangular Cartesian coordinate system in Eq. (1). The value of the integral as it differs from zero may be employed as a measure of the accuracy with which average characteristics (Kl) of the stream may be used to describe the macroscopic aspects of turbulence. Such methods do not yield results of practical significance when applied to the solution of the Navier-Stokes equations. 

Recently, a set of correlations including the effect of channel shape has been proposed by Ramanathan et al. (2003) on the basis of solution of the Navier-Stokes equations in the channel, with different solutions derived for ignited-reaction and extinct-reaction regimes. The comparison of various empirical and theoretical correlations with experimentally evaluated mass transfer coefficients is given by West et al. (2003). The correlations by Ramanathan et al. (2003) or Tronconi and Forzatti (1992) have been used in most simulations presented in this chapter. 

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

The equations that form the theoretical foundation for the whole science of fluid mechanics were derived more than one century ago by Navier (1827) and Poisson (1831) on the basis of molecular hypotheses. Later the same equations were derived by de Saint Venant (1843) and Stokes (1845) without using such hypotheses. These equations are commonly referred to as the Navier-Stokes equations. Despite the fact that these equations have been known of for more than a century, no general analytical solution of the Navier-Stokes equations is known. This state of the art is due to the complex mathematical (i.e., nonlinearity) nature of these equations. 

Let V be a steady solution of the Navier-Stokes equations (with prescribed body forces and zero boundary velocity). Note that v is not assumed to be small . Then, there exists a steady solution (Vt,Tj,pj) of any Jeffreys model with a sufficiently small Weissenberg number We, and with a sufficiently small retardation parameter e, such that (ve,rj) is close to (v,0) and close 0. (See [26].)... 

J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 3 (1959) 120-122. 

However, it must be remembered that although these schemes are comprehensive they are not perfectly accurate in the same way as are, for instance, the rigorous solution of the Navier-Stokes equations for gas motion. The information on the elementary rate constants and their temperature and pressure dependence is very imperfect for any combustion system beyond that of H2/O2. This is exemplified by the uncertainties assigned in rate constant evaluations [62], and discussed in Chapter 3. 

To find a rate, one must generally identify the driving force and the resistauice against flow. We elaborated this for a number of examples in sec. 1.6.4. All these examples involved macroscopic amounts of fluid, moving under the influence of external forces and having a resistance of a viscous nature. Under such conditions solution of the Navier-Stokes equation [1.6.1.15] or variants thereof, suffices to describe the fluid dynamics. For a droplet, spreading on a (Fresnel) surface, the situation is more complicated. Flow in the bulk of the drop obeys Navier-Stokes... 

More complicated 3D effects were studied in Refs. 6 and 7 with the help of 3D Monte Carlo digital simulation performed with a rather powerful computer (RISK System/6000). Sedimentation FFF with different breadth-to-width channel ratios and both codirected and counterdirected rotation and flow were studied. Secondary flow forming vortexes in the y-z plane is generated in the sedimentation FFF channel, both due to its curvature, and the Coriolis force caused by the centrifuge rotation. The exact structure of the secondary flow was calculated by the numerical solution of the Navier-Stokes equations and was used in the Monte Carlo simulation of the movement of solute molecules. 

A general solution of the Navier-Stokes equations has not been possible until now. The main cause of these difficulties is the non-linear character of the differential equations by the product of the inertia terms... 

Solutions of the Navier- Stokes equations for laminar flows at large Reynolds numbers only describe real flows if the solutions are stable against small disturbances. A momentarily small disturbance has to disappear again. This is no longer the case above the critical Reynolds number. A momentarily small disturbance does not disappear, rather it grows. The flow form changes from laminar to turbulent flow. 

In stochastic Lagrangian particle models, the evolution of the concentration field is computed in a two-step process. First, the Eulerian velocity field in the region of interest must be calculated, either by solution of the Navier-Stokes equations or via an approximate method that satisfies mass consistency. The solution must also provide the local statistics of the velocity field. Individual particles are then released, and their position is updated over a time increment dt using an equation of the form (Wilson and Sawford 1996)... 

In the modern theory of fluid dynamic systems the term turbulence is accepted to mean a state of spatiotemporal chaos (e.g., [155], chap 5). That is, the fluid exhibits chaos on all scales in both space and time. Chaos theory involves the behavior of non-linear dynamical systems and their response to initial and boundary conditions. Using such methods it can be shown that although the solution of the Navier-Stokes is apparently random for turbulent flows, its behavior presents some orderly structures. In addition, the numerical solution of the Navier-Stokes equations is sometimes strongly dependent on the initial conditions, thus even very small inaccuracies in the initial conditions may be fatal providing completely erroneous results. ... 

Chorin AJ (1968) Numerical solution of the Navier-Stokes equations. Math Comput 22 745-762. 

Due to its great importance in reactor simulations, a brief survey of the main steps involved in Stokes solution of the Navier-Stokes equation for creeping motion about a smooth immersed rigid sphere is provided. The details of the derivation is not repeated in this book as this task is explained very well in many textbooks [169, 14, 103, 15]. 

Our analysis shows that there is a steady, unidirectional flow solution of the Navier-Stokes equations for the Poiseuille flow problem for all flow rates. However, it says nothing about the stability of this solution. To examine this question, further analysis is necessary.7... 

In the present book, we focus our attention on the solution of the Navier-Stokes equations for laminar flows, frequently without any attempt to analyze the stability (or experimental realizability) of the resulting solutions. In using these solutions, it is therefore quite apparent that we must always reserve judgment as to the range of parameter values where they will exist in practice. We have already noted that experimental observation shows that Poiseuille flow exists for Reynolds numbers only less than a critical value. A general introduction to hydrodynamic stability theory is given in Chap. 12. It should be noted, however, that the stability of Poiseiulle flow is a very difficult problem, and only a short introductory section that is relevant to this problem is provided.7... 

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