Convection term, computing

Lagranglan codes are characterized by moving the mesh with the material motion, u = y, in (9.1)-(9.4), [24]. The convection terms drop out of (9.1)-(9.4) simplifying all the equations. The convection terms are the first terms on the right-hand side of the conservation equations that give rise to fluxes between the elements. Equations (9.1)-(9.2) are satisfied automatically, since the computational mesh moves with the material and, hence, no volume or mass flux occurs across element boundaries. Momentum and energy still flow through the mesh and, therefore, (9.3)-(9.4) must be solved. 

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first-order schemes (e.g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e.g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accurate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can continue the calculations with a second-order scheme. 

The boundary conditions provide a tight coupling between the vorticity and stream-function fields. Also velocities still appear in the convective terms. Given the stream-function field, velocity is evaluated from the definition of stream function. That is, velocity is computed from stream-function derivatives. 

M 44a] [P 40] Numerical errors which are due to discretization of the convective terms in the transport equation of the concentration fields introduce an additional, unphysical diffusion mechanism [37]. Especially for liquid-liquid mixing with characteristic diffusion constants of the order of 1CT9 m2 s 1 this so-called numerical diffusion (ND) is likely to dominate diffusive mass transfer on computational grids. 

Incorporation of this term is straightforward, since the velocities u are known after the first iteration. The convective terms are unsymmetric, however, which requires using less efficient storage and solution schemes than are possible for symmetric systems. Even more seriously, inclusion of convective transport effects tends to produce numerical instability in the computed results, as will be illustrated below. 

The application of CFD in the modeling of solid-liquid mixing is fairly recent. In 1994, Bakker et al. developed a two-dimensional computational approach to predict the particle concentration distribution in stirred vessels. In their model, the velocity field of the liquid phase is first simulated taking into account the flow turbulence. Then, using a finite volume approach, the diffusion-sedimentation equation along with the convective terms is solved, which includes Ds, a... 

Flatt (F3) computes bubble growth rates due to radiolytic gas formation in a power excursion in a homogeneous reactor. For very small bubbles the growth is assumed to be quasi-stationary so that the convective term can be dropped from the diffusion equation. A distributed exponential volume source is assumed. An integral equation for the radius is then obtained, from which upper and lower bounds for the radius of the bubble can be deduced. 

Figure 3 presents results for a 0.05 step change in YAOn (similar results were obtained for the other inputs). The expected, realistic response is a gradual increase of outlet concentration occurring after a certain dead time. The computed response showed oscillations, which is attributed to numerical approximation of the convection term. This effect is discussed by Lefevre et. al (2000), in the context of tubular reactors. 

Particular care must be paid to the choice of the differencing scheme used for the solution of the governing equations. The choice is not univocal, since a tradeoff between accuracy and computational cost exists. A first-order approximation for the convection term is the most stable approach, however if the target is the prediction of the mixing efficiency between ammonia and exhaust gas stream, then the solution will be affected by a significant amount of artificial viscosity, comparable to the turbulent one [33]. This issue can be overcome resorting to more... 

It is noted that the convection, growth and breakage terms are linear. Hence, it is not necessary to repeate the computation of (12.515H 12.519) within the iteration loop. The convective term in physical space is the simplest term to implement. Pseudo code 4 shows the computation of the convective term, [Li] (12.515). The algorithm for convection in the property coordinate is presented in pseudo code 5. The algorithms of the breakage death and birth terms are presented in pseudo codes 6 and 7, respectively. 

Pseudo code 4 Computation of the problem matrix for the convective term in physical... 

Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete eqnalions associated with a finite set of discrete points within the domain and specified by the user, directly or through an antomated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD). 

To set against the drawback of possible uncertainty about effects of the numerical diffusion associated with the convective terms, the Eulerian approach has very considerable operational advantages if the computational objective is solely the determination of steady-state properties. Because the properties at each grid point are time-invariant in the steady state, Eulerian time steps are in themselves successive approximations to the end result. They... 

All governing equations are all solved using a finite volume discretization, see [7]. All vectors quantities, e.g. position vector, velocity and moment of momentum, are expressed in Cartesian coordinates. Non-staggered variable arrangement is used to define dependent variables all physical quantities are stored and computed at cell centers. An interpolation practice of second order accuracy is adopted to calculate the physical quantities at cell-face center [8]. The deferred correction approach [9] is used to compute the convection term appearing in the governing equations by blending the upwind difference and the centi difference scheme. 

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