# Properties of discretization schemes

In computational fluid dynamics, finite volume analysis introduces inaccuracies based on the discretization of the inherently continuous variables. These errors can be minimised by properly choosing the finite volume elements.

## Introduction

Numerical results that are obtained in theory are indistinguishable from the 'exact' solution of the transport equation when the number of computational cells is infinitely large irrespective of the differencing method used. However, in practical calculations we can only use a finite number of cells (quite small) and our numerical results will only be physically realistic when the discretization discretization scheme has certain fundamental properties.

The most important ones of these properties are:

• Conservativeness
• Boundedness
• Transportiveness
• Accuracy

## Conservativeness

To ensure conservation of $\phi$ (any property) for the whole solution domain, the flux of $\phi$ leaving a control volume across a certain face must be equal to the flux of $\phi$ entering the adjacent control volume through the same face. To achieve above condition the flux through a common face must be represented in a consistent manner by one and the same expression in adjacent control volumes.The fluxes across the domain boundaries are denoted by qA and qB. Four control volumes are considered and central differencing method is applied to calculate the diffusive flux across the cell faces.

The expression for the flux leaving the element around node 2 across its west face is

$\Gamma_{w2} \left (\phi_2 - \phi_1 \right )$

and the flux entering across its east face is

$\Gamma_{e2} \left (\phi_3 - \phi_2 \right )$

An overall flux balance may be obtained by summing the net flux through each control volume taking into account the boundary fluxes for the control volumes around nodes 1 and 4.

Example of consistent specification of diffusive fluxes

Boundary fluxes for the control volumes around nodes 1 and 4 is given by

$\left [ \Gamma_{e1} \frac{\phi_2-\phi_2}{\delta x} - q_A \right ] + \left [ \Gamma_{e2} \frac{\phi_3-\phi_2}{\delta x} - \Gamma_{w2}\frac{\phi_2 - \phi_1}{\delta x} \right ] + \left [ \Gamma_{e3} \frac{\phi_4 - \phi_3}{\delta x} - \Gamma_{w3} \frac{\phi_3 - \phi_2}{\delta x} \right ] + \left [ q_B - \Gamma_{w4}\frac{\phi_4 - \phi_3}{\delta x} \right ] = q_b - q_A$

Since Γe1 = Γw2, Γe2 = Γw3 and Γe3 = Γw4, the fluxes across control volume faces are expressed in a consistent manner and cancel out in pairs when summed over the entire domain. Only the two boundary fluxes qA and qB remain in the overall balance, so above equation expresses overall conservation of property ф. Flux consistency ensures conservation of ф over the entire domain for the central difference formulation of the diffusion flux.

## Boundedness

The discretised equations at each nodal point Nodal points represent a set of algebraic equations that needs to be solved. Iterative numerical techniques are used to solve large equation sets. These methods start the solution process from a guessed distribution of the variable ф and perform successive updates until a converged solution is obtained. Scarborough (1958) has shown that a sufficient condition for a convergent iterative method can be expressed in terms of the values of the coefficients of the discretised equations:

$\frac{\sum |a_{nb}|}{|a'_{p}|} \begin{cases} \leq 1, & \text{at all nodes}\\ <1, & \text{at all nodes at least} \end{cases}$

Here a'p is the net coefficient of the central node P and the summation in the numerator is taken over all the neighbouring nodes.

If the differencing scheme produces coefficients that satisfy the above criterion the resulting matrix of coefficients is diagonally dominant. To achieve diagonal dominance we need large values of net coefficient so the linearisation practice of source terms should ensure that SP is always negative. If this is the case -SP is always positive and adds to aP. Diagonal dominance is a desirable feature for satisfying the boundedness criterion. This states that in the absence of sources the internal nodal values of the property ф should be bounded by its boundary values. Hence in a steady state conduction problem without sources and with boundary temperatures of 500 °C and 200 °C all interior values of T should be less than 500 °C and greater than 200 °C.

Another essential requirement for boundedness is that all coefficients of the discretised equations should have the same sign (usually all positive). Physically this implies that an increase in the variable ф at one node should result in an increase in ф at neighbouring nodes. If the discretisation scheme does not satisfy the boundedness requirements it is possible that the solution does not converge at all or, if it does, that it contains 'wiggles'.

## Transportiveness

The non-dimensional cell peclet number is defined as a measure of the relative strengths of convection and diffusion.

$Pe = \frac{F}{D}=\frac{\rho u}{\Gamma/\delta x}$

where δx = characteristic length (cell width).

Distribution of ф in the vicinity of a source at different Peclet numbers

Consider two extreme cases to identify the extent of the influence of the upstream node P at the downstream node E:

In the case of pure diffusion the fluid is stagnant (Pe = 0) and the contours of constant ф will be concentric circles with P at their center since the diffusion process tends to spread ф equally in all directions. Conditions at the east node E will be influenced by those upstream at P and also by conditions further downstream.

As Pe increases the contours change shape from circular to elliptical and are shifted in the direction of the flow. Influencing becomes increasingly biased towards the upstream direction at large values of Pe so that the node E is strongly influenced by conditions at P, but conditions at P will experience weak influence or no influence at all from E. In the case of pure convection (Pe → ∞) the elliptical contours are completely stretched out in the flow direction. All of property ф emanating from the source at P is immediately transported downstream towards E. Thus the value of ф at E is affected only by upstream conditions and since there is no diffusion фE is equal to фP. It is very important that the relationship between the magnitude of the Péclet number and the directionality of influencing, known as the transportiveness, is borne out in the discretisation scheme.

## References

• Versteeg, Henk Kaarle; Malalasekera, Weeratunge (2007). An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson. ISBN 9780131274983.