ABSTRACT
The calculation procedure described in this book is aimed at solving coupled nonlinear equations by an iterative scheme. At this point, we shall take an overall look at the iterative process.
The iteration technique plays two different roles:
Our equations are, in general, nonlinear and interlinked. We cast them into nominally linear form and calculate the coefficients from the previous-iteration values of the variables.
The nominally linear algebraic equations for one dependent variable at a time are solved by an iterative method (such as the line-by-line method) rather than by a direct method.
The iterative solution of the algebraic equations need not be taken to complete convergence, because we are, at any intermediate stage, working with only tentative coefficients. After the discretization equations have been iterated to a certain extent, one must return to the recalculation of the coefficients. A sense of proportion is appropriate here. After having spent a certain amount of effort on calculating the coefficients, we must extract a fairly good solution of the algebraic equations, but refrain from doing an excessive amount of work with coefficient values that we know well to be only tentative. A direct solution method used for multidimensional problems 140usually results in a disproportionately large amount of work spent in the equation-solving activity.
A similar consideration has been used in Chapter 6 in choosing a sequential, rather than simultaneous, procedure for calculating fluid flow. The momentum equations and the pressure-correction equation are solved sequentially. The alternative, which is commonly adopted in most finite-element methods for fluid flow, is to obtain a simultaneous solution of the linearized forms of the continuity equation and all the momentum equations. Such a simultaneous solution by a direct method requires large amounts of computer time and storage. Since the momentum equations are nonlinear, these large amounts of effort must be spent at every iteration. Further, the continuity and momentum equations may not be the only equations governing the situation. These equations are often coupled with the energy equation (through fluid properties and buoyancy forces), with the equations for turbulence parameters (through the turbulent viscosity), with the equations for chemical-species concentration, and so on. Obviously, it would not be practicable to attempt a simultaneous solution of all these equations; these additional equations would normally be solved in a sequential manner. Under these circumstances, the expenditure of large amounts of computing effort for the simultaneous solution of the continuity and momentum equations seems out of proportion.
In the numerical method presented in this book, there is no fundamental difference between solving a steady-state problem and performing one time step in an unsteady problem. In a steady problem, we start with guessed values for the variables ϕ and proceed to obtain the steady-state solution. For an unsteady situation, the problem is this: Given the values of ϕ at time t and a guess for ϕ at t + Δt, find the values of ϕ at t + Δt. As in the steady-state problem, we must perform a number of iterations at each time step for an unsteady problem. Further, many such time steps must be sequentially executed to cover the desired time period.
Thus, the solution of an unsteady problem seems to involve an effort that is equivalent to the task of solving a succession of steady-state problems. This is partially true, but there is one consolation. For reasonable values of Δt, the known ϕ values at time t can be used as a guess for the unknown ϕ values at time t + Δt. Since this is a relatively good guess (compared with a rather arbitrary guess, which one must make in a steady-state situation), only a few iterations are normally needed to obtain a converged solution for the time step. Sometimes, the number of iterations per time step can be as small as one. Thus, when a method for a nonlinear unsteady problem is claimed to be noniterative, it is, in fact, accepting the solution at the end of one iteration as a sufficiently converged solution for that time step. Such methods must employ rather small time steps, whereas the use of multiple iterations for a time step would allow larger values of Δt.
Such a one-iteration-per-time-step method is sometimes used to obtain 141the steady-state solution at the end of many time steps. Such time steps are truly iterations, with the unsteady term in the equations providing a kind of underrelaxation.
A computer program that employs iteration within a time step should provide storage for the values of ϕ at time t and for the ϕ values at t + Δt. A steady-state program, on the other hand, requires storage for only one set of ϕ values, which are continually overwritten until convergence is attained.
The iterative technique greatly simplifies the construction of the numerical method and provides a way in which, at least in principle, one can handle any nonlinearity and interlinkage. Of course, the technique is of no value if a converged solution cannot be reached. It is useful at this stage to examine the prospects of convergence.
The four basic rules (introduced in Section 3.4) have enabled us to obtain such discretization equations as would, for fixed values of the coefficients, ensure convergence of the point-by-point or line-by-line solution procedure.
If the coefficients do not remain fixed but change rather slowly, it seems reasonable that we shall still obtain convergence. A proper linearization of the source term and an appropriate underrelaxation of the dependent variables would, in general, slow down the changes in the variables and hence in the coefficients.
In addition to the dependent variables, other quantities can be under-relaxed with advantage. For example, the density ρ is often the main link between the flow equations and the equations for temperature, concentration, etc. An underrelaxation of ρ via () ρ = α ρ new + ( 1 − α ) ρ old https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275130/9c5aefb9-a272-4bc2-b902-60b84d4db7b5/content/eq524.tif"/>
would cause the velocity field to respond rather slowly to the changes in temperature and concentration. A diffusion coefficient Γ can be under-relaxed to restrain, for example, the influence of the turbulence quantities on the velocity field. The present value of Γ is then calculated from () Γ = α Γ new + ( 1 − α ) Γ old . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275130/9c5aefb9-a272-4bc2-b902-60b84d4db7b5/content/eq525.tif"/>
Here, as in Eq. (7.1), α stands for the relaxation factor. Underrelaxation requires α to be positive but less than 1. The interlinkage between different variables often comes through the source term (for example, the buoyancy force in a momentum equation depends on temperature). We may decide to underrelax the source term via () S C = α S C , new + ( 1 − α ) S C , old . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275130/9c5aefb9-a272-4bc2-b902-60b84d4db7b5/content/eq526.tif"/>
Even the boundary conditions can be underrelaxed. For example, a hot 142wall or a rotating disc need not assume its final temperature or rotational speed right from the first iteration; the boundary value may be slowly adjusted, during the course of the iterations, to ultimately achieve the desired value. Thus, () ϕ B = α ϕ B , given + ( 1 − α ) ϕ B , old . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275130/9c5aefb9-a272-4bc2-b902-60b84d4db7b5/content/eq527.tif"/>
Of course, the value of α appearing in Eqs. (7.1)-(7.4) need not be the same, nor is it necessary to use the same value for a for every grid point.
It must be remembered that there is no general guarantee that, for all nonlinearities and interlinkages, we will always get a converged solution. The underrelaxation procedures that are introduced here have been found to be helpful in many cases, but special underrelaxation practices may be needed for special problems. In the absence of an unconditional guarantee, we can nevertheless derive hope from the fact that, for a large number of rather complex problems, it has been possible to get converged solutions. A sample of such solutions will be presented in Chapter 9, but many other problems have also been solved and published.
As we have noted, an iterative process is said to have converged when further iterations will not produce any change in the values of the dependent variables. In practice, the iterative process is terminated when some arbitrary convergence criterion is satisfied. An appropriate convergence criterion depends on the nature of the problem and on the objectives of the computation. A common procedure is to examine the most significant quantities given by the solution (such as the maximum velocity, total shear force, a certain pressure drop, or overall heat flux) and to require that the iterations be continued only until the relative change in these quantities between two successive iterations is greater than a certain small number. Often the relative change in the grid-point values of all the dependent variables is used to formulate the convergence criterion. This type of criterion can sometimes be misleading. When heavy underrelaxation is used, the change in the dependent variables between successive iterations is intentionally slowed down; this may create an illusion of convergence although the computed solution may be far from being converged. A more meaningful method of monitoring convergence is to examine how perfectly the discretization equations are satisfied by the current values of the dependent variables. For each grid point, a residual R can be calculated from () R = ∑ a nb ϕ nb + b − a P ϕ P . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275130/9c5aefb9-a272-4bc2-b902-60b84d4db7b5/content/eq528.tif"/>
Obviously, when the discretization equation is satisfied, R will be zero. A suitable convergence criterion is to require that the largest value of |R| be less than a certain small number. Incidentally, as mentioned in Section 6.7.2, the 143quantity b in Eq. (6.22), which is the residual of the continuity equation, can be used as one of the indicators of the convergence of the iterative process.
