Quadratic upwind differencing scheme: the QUICK scheme The quadratic upstream interpolation for convective kinetics (QUICK)

5.9.1 Quadratic upwind differencing scheme: the QUICK scheme The quadratic upstream interpolation for convective kinetics (QUICK)

scheme of Leonard (1979) uses a three-point upstream-weighted quadratic interpolation for cell face values. The face value of φ is obtained from a quadratic function passing through two bracketing nodes (on each side of the face) and a node on the upstream side (Figure 5.17).

5.9 HIGHER-ORDER DIFFERENCING SCHEMES

Figure 5.17 Quadratic profiles used in the QUICK scheme

For example, when u w > 0 and u e > 0 a quadratic fit through WW, W and P is used to evaluate φ w , and a further quadratic fit through W, P and E to calculate φ e . For u w < 0 and u e < 0 values of φ at W, P and E are used for φ w , and values at P, E and EE for φ e . It can be shown that for a uniform grid the value of φ at the cell face between two bracketing nodes i and i − 1 and upstream node i − 2 is given by the following formula:

φ face = φ i −1 + φ i − φ i −2

When u w > 0, the bracketing nodes for the west face w are W and P, the

upstream node is WW (Figure 5.17) and

φ w = φ W + φ P − φ WW

When u e > 0, the bracketing nodes for the east face e are P and E, the upstream node is W, so

The diffusion terms may be evaluated using the gradient of the approximat- ing parabola. It is interesting to note that on a uniform grid this practice gives the same expressions as central differencing for diffusion, since the slope of the chord between two points on a parabola is equal to the slope of the

tangent to the parabola at its midpoint. If F w > 0 and F e > 0, and if we use equations (5.46) – (5.47) for the convective terms and central differencing for the diffusion terms, the discretised form of the one-dimensional convection– diffusion transport equation (5.9) may be written as

H F e B φ P + φ E − φ W E −F w B φ W + φ P − φ 8 WW 8 8 8 8 8 E K

=D e ( φ E − φ P ) −D w ( φ P − φ W ) which can be rearranged to give

H D w −F w +D e +F e K φ P =D H w +F w +F e K φ W

1 +D H e −F e K φ E −F w φ WW

I (5.48)

158 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS

This is now written in the standard form for discretised equations:

For F w < 0 and F e < 0 the flux across the west and east boundaries is given

by the expressions

φ w = φ P + φ W − φ E (5.50)

6 3 1 φ e = φ E + φ P − φ EE

Substitution of these two formulae for the convective terms in the discretised convection–diffusion equation (5.9) together with central differencing for the diffusion terms leads, after rearrangement as above, to the following coefficients:

General expressions, valid for positive and negative flow directions, can be obtained by combining the two sets of coefficients above.

The QUICK scheme for one-dimensional convection–diffusion problems can be summarised as follows:

a P φ P =a W φ W +a E φ E +a WW φ WW +a EE φ EE (5.51)

with central coefficient

a P =a W +a E +a WW +a EE + (F e −F w ) and neighbour coefficients

a WW

a E a EE

D w + α w F w + α e F e − α w F w D e − α e F e − (1 − α e )F e (1 − α e )F e

+ (1 − α w )F w

− (1 − α w )F w

5.9 HIGHER-ORDER DIFFERENCING SCHEMES

where α w = 1 for F w > 0 and α e = 1 for F e >0

α w = 0 for F w < 0 and α e = 0 for F e <0

Example 5.4 Using the QUICK scheme solve the problem considered in Example 5.1 for u = 0.2 m/s on a five-point grid. Compare the QUICK solution with the

exact and the central differencing solution.

Solution As before, the five-node grid introduced in Example 5.1 is used for the discretisation. With the data of this example and u = 0.2 m/s we have F = F e

=F w = 0.2 and D = D e =D w = 0.5 everywhere so that the cell Peclet number becomes Pe w = Pe e = ρuδx/Γ = 0.4. The discretisation equation with the QUICK scheme at internal nodes 3 and 4 is given by (5.51) together with its coefficients.

In the QUICK scheme the φ-value at cell boundaries is calculated with formulae (5.46) – (5.47) that use three nodal values. Nodes 1, 2 and 5 are all affected by the proximity of domain boundaries and need to be treated

separately. At the boundary node 1, φ is given at the west (w) face (φ w = φ A ), but there is no west (W ) node to evaluate φ e at the east face by (5.47). To overcome this problem Leonard (1979) suggested a linear extrapolation to create a ‘mirror’ node at a distance δx/2 to the west of the physical bound- ary. This is illustrated in Figure 5.18.

Figure 5.18 Mirror node treatment at the boundary

It can be easily shown that the linearly extrapolated value at the mirror node is given by

(5.52) The extrapolation to the ‘mirror’ node has given us the required W node for

φ 0 =2 φ A − φ P

the formula (5.47) that calculates φ e at the east face of control volume 1:

6 3 1 φ e = φ P + φ E − (2 φ A − φ P )

= φ P + φ E − φ A (5.53)

At the boundary nodes the gradients must be evaluated using an expression consistent with formula (5.53). It can be shown that the diffusive flux through the west boundary is given by

160 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS

where D * A = δx

The superscript * is used to indicate that, in the QUICK scheme, the diffu- sion conductances at boundary nodes and interior nodes have the same value,

* A i.e. D = D = Γ/ δx. This aspect is different from the discretisation schemes we have discussed thus far. These used the half-cell approximation, so the diffusive conductance at the boundary cell was always D A = 2D = 2Γ/ δx.

The discretised equation at node 1 is

At control volume 5, the φ-value at the east face is known (φ e = φ B ) and the diffusive flux of φ through the east boundary is given by

where D B * = δx

At node 5 the discretised equation becomes

F B φ B −F w H φ W + φ P − φ WW K

(8 φ B −9 φ P + φ W ) −D w ( φ P − φ W ) (5.57)

Since a special expression is used to evaluate φ at the east face of control volume 1 we must use the same expression for φ to calculate the convective

flux through the west face of control volume 2 to ensure flux consistency. So at node 2 we have

F e H φ P + φ E − φ W K −F w H φ W + φ P − φ A K

(5.58) The discretised equations for nodes 1, 2 and 5 are now written to fit into the

=D e ( φ E − φ P ) −D w ( φ P − φ W )

standard form to give

a P φ P =a WW φ WW +a W φ W +a E φ E +S u

with

a P =a WW +a W +a E + (F e −F w ) −S P and

5.9 HIGHER-ORDER DIFFERENCING SCHEMES

Node a WW a W

Substitution of numerical values gives the coefficients summarised in Table 5.10.

The matrix form of the equation set is

G 2.175 −0.592 0 0 0 J Gφ 1 J G 1.583J

H−0.7

1.075 −0.425 0 0 K Hφ 2 K H−0.05 K

H 0.025 −0.675 1.075 −0.425 0 K Hφ 3 K=H0 K (5.60) H0 0.025 −0.675 1.075 −0.425 K Hφ 4 KH0

K I0 0 0.025 −0.817 1.925 L Iφ 5 LI0

The solution to the above system is Gφ 1 J G0.9648J

Hφ 2 K H0.8707K Hφ 3 K = H0.7309K

Hφ 4 K H0.5226K Iφ 5 L I0.2123L

Comparison with the analytical solution Figure 5.19 shows that the QUICK solution is almost indistinguishable from

the exact solution. Table 5.11 confirms that the errors are very small even with this coarse mesh. Following the steps outlined in Example 5.1 the cen- tral differencing solution is computed with the data given above. The sum of absolute errors in Table 5.11 indicates that the QUICK scheme gives a more accurate solution than the central differencing scheme.

162 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS

Figure 5.19 Comparison of QUICK solution with the analytical solution

Table 5.11

Node Distance Analytical QUICK Difference CD Difference

∑ Absolute error