Assessment of the upwind differencing scheme Conservativeness: The upwind differencing scheme utilises consistent
5.6.1 Assessment of the upwind differencing scheme Conservativeness: The upwind differencing scheme utilises consistent
expressions to calculate fluxes through cell faces: therefore it can be easily shown that the formulation is conservative.
Boundedness: The coefficients of the discretised equation are always posi- tive and satisfy the requirements for boundedness. When the flow satisfies
continuity the term (F e −F w ) in a P (see (5.31)) is zero and gives a P =a W +a E , which is desirable for stable iterative solutions. All the coefficients are positive and the coefficient matrix is diagonally dominant, hence no ‘wiggles’ occur in the solution.
Transportiveness: The scheme accounts for the direction of the flow so transportiveness is built into the formulation.
Accuracy: The scheme is based on the backward differencing formula so the accuracy is only first-order on the basis of the Taylor series truncation error (see Appendix A).
Because of its simplicity the upwind differencing scheme has been widely applied in early CFD calculations. It can be easily extended to
150 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS
multi-dimensional problems by repeated application of the upwind strategy embodied in the coefficients of (5.31) in each co-ordinate direction. A major drawback of the scheme is that it produces erroneous results when the flow is not aligned with the grid lines. The upwind differencing scheme causes the distributions of the transported properties to become smeared in such problems. The resulting error has a diffusion-like appearance and is referred to as false diffusion. The effect can be illustrated by calculating the trans- port of scalar property φ using upwind differencing in a domain where the flow is at an angle to a Cartesian grid.
In Figure 5.14 we have a domain where u = v = 2 m/s everywhere so the velocity field is uniform and parallel to the diagonal (solid line) across the grid. The boundary conditions for the scalar are φ = 0 along the south and east boundaries, and φ = 100 on the west and north boundaries. At the first and the last nodes where the diagonal intersects the boundary a value of 50 is assigned to property φ.
Figure 5.14 Flow domain for the illustration of false diffusion
To identify the false diffusion due to the upwind scheme, a pure convec- tion process is considered without physical diffusion. There are no source terms for φ and a steady state solution is sought. The correct solution is known in this case. As the flow is parallel to the solid diagonal the value of φ at all nodes above the diagonal should be 100 and below the diagonal it
should be zero. The degree of false diffusion can be illustrated by calculating the distribution of φ and plotting the results along the diagonal (X–X). Since
there is no physical diffusion the exact solution exhibits a step change of φ from 100 to zero when the diagonal X–X crosses the solid diagonal. The cal- culated results for different grids are shown in Figure 5.15 together with the exact solution. The numerical results show badly smeared profiles.
The error is largest for the coarsest grid, and the figure shows that refinement of the grid can, in principle, overcome the problem of false diffusion. The results for 50 × 50 and 100 × 100 grids show profiles that are closer to the exact solution. In practical flow calculations, however, the degree of grid refinement required to eliminate false diffusion can be prohibitively expensive. Trials have shown that, in high Reynolds number
5.7 THE HYBRID DIFFERENCING SCHEME
Figure 5.15
flows, false diffusion can be large enough to give physically incorrect results (Leschziner, 1980; Huang et al., 1985). Therefore, the upwind differencing
scheme is not entirely suitable for accurate flow calculations and considerable research has been directed towards finding improved discretisation schemes.
5.7 The hybrid differencing
The hybrid differencing scheme of Spalding (1972) is based on a combina- scheme
tion of central and upwind differencing schemes. The central differencing scheme, which is second-order accurate, is employed for small Peclet num- bers (Pe < 2) and the upwind scheme, which is first-order accurate but
accounts for transportiveness, is employed for large Peclet numbers (Pe ≥ 2). As before, we develop the discretisation of the one-dimensional convection– diffusion equation without source terms. This equation can be interpreted as a flux balance equation. The hybrid differencing scheme uses piecewise formulae based on the local Peclet number to evaluate the net flux through each control volume face. The Peclet number is evaluated at the face of the control volume. For example, for a west face,
The hybrid differencing formula for the net flux per unit area through the west face is as follows:
for Pe w ≥2 (5.36)
q w =F w φ P
for Pe w ≤ −2
152 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS
It can be easily seen that for low Peclet numbers this is equivalent to using central differencing for the convection and diffusion terms, but when |Pe| > 2
it is equivalent to upwinding for convection and setting the diffusion to zero. The general form of the discretised equation is
a P φ P =a W φ W +a E φ E (5.37)
The central coefficient is given by
a P =a W +a E + (F e −F w )
After some rearrangement it is easy to verify that the neighbour coeffi- cients for the hybrid differencing scheme for steady one-dimensional convection–diffusion can be written as follows:
I C 2 F L Example 5.3
Solve the problem considered in Case 2 of Example 5.1 using the hybrid scheme for u = 2.5 m/s. Compare a 5-node solution with a 25-node solution.
Solution If we use the 5-node grid and the data of Case 2 of Example 5.1 and u =
2.5 m/s we have: F =F e =F w = ρu = 2.5 and D = D e =D w = Γ/ δx = 0.5 and hence a Peclet number Pe w = Pe e = ρuδx/Γ = 5. Since the cell Peclet number Pe is greater than 2 the hybrid scheme uses the upwind expression for the convective terms and sets the diffusion to zero.
The discretised equation at internal nodes 2, 3 and 4 is defined by (5.37) and its coefficients. We also need to introduce boundary conditions at nodes
1 and 5, which need special treatment. At the boundary node 1 we write
F e φ P −F A φ A =0−D A ( φ P − φ A )
and at node 5
(5.39) It can be seen that the diffusive flux at the boundary is entered on the right
F B φ P −F w φ W =D B ( φ B − φ P ) −0
hand side and the convective fluxes are given by means of the upwind method. We note that F A =F B = F and D B = 2Γ/ δx = 2D so the discretised
equation can be written as
a P φ P =a W φ W +a E φ E +S u
with
a P =a W +a E + (F e −F w ) −S P
and
5.7 THE HYBRID DIFFERENCING SCHEME 153
Node a W
1 0 0 −(2D + F ) (2D +F) φ A 2,3,4
5 F 0 −2D
2D φ B
Substitution of numerical values gives the coefficients summarised in Table 5.8.
Table 5.8
Node a W
The matrix form of the equation set is
G 3.5
0 0 0 0 J Gφ 1 J G3.5J H−2.5 2.5 0 0 0 K Hφ 2 K
H 0K
H 0 −2.5 2.5 0 0 K Hφ 3 K = H 0K (5.41) H0 0 −2.5
2.5 0 K Hφ 4 K
H 0K
I 0L The solution to the above system is
I0 0 0 −2.5 3.5 L Iφ 5 L
Gφ 1 J G 1.0 J Hφ 2 K H 1.0 K Hφ 3 K=H 1.0 K
(5.42) Hφ 4 K H 1.0 K
Iφ 5 L I0.7143L Comparison with the analytical solution
The numerical results are compared with the analytical solution in Table 5.9 and, since the cell Peclet number is high, they are the same as those for pure
Table 5.9
Node Distance Finite volume Analytical Difference Percentage
solution
solution
error
154 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS
upwind differencing. When the grid is refined to an extent that the cell Pe < 2, the scheme reverts to central differencing and produces an accurate
solution. This illustrated by using a 25-node grid with δx = 0.04 m so F = D = 2.5. The results computed on both the coarse and the fine grids are shown in Figure 5.16 together with the analytical solution. Now Pe = 1, the hybrid scheme reverts to central differencing, and it can be seen that the solution obtained with the fine grid is remarkably good.
Figure 5.16