Discretization of the Momentum Equations

The unsteady term in the momentum equation has the same form as the transient term in the generic transport equation and is discretized in the same manner. 

The treatment of the convective term in the momentum equation basically follows that of the convective term in the generic equation. However, some extra linearization is required as the convective term in the momentum equation is non-linear in the velocity components. 

The body forces, like the gravity term, are integrated over the grid volume. Usually, the mean value approach is used, so that the value at the grid center is multiplied by the grid volume. The apparent forces that may occur in particular coordinate systems, are often considered as body forces and integrated in the same way as the gravity term. 

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This set of equations can be solved by a variety of approaches. Historically they were solved analytically by a separation-of-variables method, which is tedious, time-consuming, and, for most, an error-prone task. The results presented here were computed using a finite-volume discretization of the momentum equation on a 10 by 10 mesh, which was solved iteratively in a spreadsheet. The programming time was a couple of hours, and the solution is found in about a minute on a typical personal computer. The results are accurate to within one percent of the exact series solutions. The details of the spreadsheet programming for this problem are included in an appendix. 

In the coordinate discretization process one selects the node points in the domain at which the values of the unknown dependent variables are to be computed. In the finite volume method one also selects the location of the grid cell surfaces at which the property fluxes are determined. In this way the computational domain is sub-divided into a number of smaller, nonoverlapping sub-domains. There are many variants of the distribution of the computational node points and grid cell surfaces within the solution domain. The grid arrangements associated with the finite volume discretization of the momentum equation are generally more complicated than the one employed for a scalar transport equation. 

The velocity corrections in (12.247) were then substituted by the pressure corrections employing appropriately defined pressure-velocity correction relationships. The pressure-velocity correction relationships were constructed by subtracting the discrete momentum equations with the pressure at the old time level n and the preliminary estimates for velocity fields (12.242) from the semi-implicit discretization of the momentum equation with the corrected pressure and the estimates for the velocity fields that satisfy the continuity equations. The desired relationships are the given on the form ... 

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

One of the popular methods proposed by Patankar and Spalding (1972) is called SIMPLE (semi-implicit method for pressure linked equations). In this method, discretized momentum equations are solved using the guessed pressure field. The discretized form of the momentum equations can be written ... 

The discretized versions of the momentum equations and Eq. (6.38) lead to discretized equations in terms of velocity and pressure correction ... 

The discretized rr-component of the momentum equation at the surface point e, for example, can be written as ... 

The components of the momentum equation are usually solved sequentially, meaning that the components of the momentum equation are solved one by one. Since the pressure used in these iterations has been obtained from the previous outer iteration or time step, the velocities computed from (12.159) do not generally satisfy the discretized continuity equation. The predicted velocities do not satisfy the continuity equation, so the uf at iteration 1/ are not the final values of the velocity components. To enforce the continuity equation, the velocities need to be corrected. This is achieved by modifying the pressure field. 

The solution at the next time level n + 1 is then obtained from the discrete form of the momentum equation subject to the continuity constraint on the form ... 

A Poisson equation for the pressure like variable was derived from a discrete form of the momentum equation in which the volume fractions and densities were assumed independent of time ... 

The 5th and 8th terms on the RHS of the radial component of the momentum equation for the gas phase are identical and discretized in the same way as the corresponding terms in the liquid phase equation, as discussed in sect C.4.4. 

In order to properly solve (17.5), sharp changes in the properties as well as pressure forces due to surface tension effects have to be resolved. In particular, surface tension results in a jump in pressure across a curved interface. The pressure jump is discontinuous and located only at the interface. This singularity creates difficulties when deriving a continuum formulation of the momentum equation. The interfacial conditions should be embedded in the field equations as source terms. Once the equations are discretized in a finite-thickness interfacial zone, the fiow properties are allowed to change smoothly. It is therefore necessary to create a continuum surface force (CSF) equal to the surface tension at the interface, or in a transitional region, and zero elsewhere. Therefore, the surface integral term in (17.5) could be rewritten into an appropriate volume integral... 

S-3.3.5 Numerical Diffusion. Numerical diffusion is a source of error that is always present in finite volume CFD, owing to the fact that approximations are made during the process of discretization of the equations. It is so named because it presents itself as equivalent to an increase in the diffusion coefficient. Thus, in the solution of the momentum equation, the fluid will appear more viscous in the solution of the energy equation, the solution will appear to have a higher conductivity in the solution of the species equation, it will appear that the species diffusion coefficient is larger than in actual fact. These errors are most noticeable when diffusion is small in the actual problem definition. 

By use of the explicit Euler scheme, the discrete form of the momentum equations can be written as [3] ... 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

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