EL INFLUJO MODERNIZADOR DE LOS PLANES URBANOS

The first case considered is an incompressible laminar channel flow over a cavity. The geometry consists of a 2D channel with a cavity in the bottom wall, shown in Fig. 6.1.

The flow is at Re = 200 based on the channel height H and the average inlet velocity Uavg. At the inlet a parabolic velocity profile is prescribed:

Ux = 6Uavg · y H − ³_y H ´2¸ .

The calculation was started on the mesh with 36 cells depicted in Fig. 6.2. Second-order Central Differencing (CD) was used for discretisation of convective

Walls Outlet H H 2H 4H 8H X Y 0.25H 0.25H Parabolic velocity profile

Figure 6.1: Geometry and boundary conditions for the flow over a cavity

Figure 6.2: Starting mesh for the flow over a cavity (36 cells)

terms. Gradients of all fields are evaluated by using the Gauss Theorem, Eqn. (3.24). The required maximum velocity error in the solution is set to:

E = 1 %,

of the average inlet velocity Uavg. The solution is compared with a benchmark

solution obtained on the 501552 cells mesh with the maximum estimated velocity error of 0.2% of the inlet velocity Uavg.

The predicted velocity field on the final adapted mesh with 3257 cells can be found in Figs. 6.3(a) and 6.3(b). The flow is aligned with the x axis within the channel and the variation of velocity in the cross-stream direction is parabolic. A stagnation point at the top-right corner of the cavity, Fig. 6.3(d), is a region of high velocity and pressure gradients, Figs. 6.3(d) and 6.4. The vortex formed within the cavity has an impingement zone near the bottom-right corner where the flow parallel to the right wall meets with the bottom wall. The separation near the upper-left corner of the cavity is a region of moderate velocity and pressure gradients when compared to the zone near the stagnation point, Figs. 6.3(c) and 6.4.

The calculation stopped after 8 cycles of refinement. The final mesh has 3257 cells and is shown in Fig. 6.5. It is highly refined in the region near the stagnation point, as would be expected in a region of high gradients. Some refinement occurred

6.2 Flow Over a Cavity 139

(a) Velocity

(b) Streamlines

(c) Zoom around the top-left corner of the cavity

(d) Zoom around the top-right corner of the cavity

Figure 6.3: Velocity field for the flow over a cavity on the final adapted mesh with 3257 cells (normalised by Uavg)

Figure 6.4: Pressure coefficient field for the flow over a cavity on the final adapted mesh with 3257 cells

Figure 6.5: Mesh after 8 cycles of refinement for the flow over a cavity (3257 cells)

within the cavity to resolve the impingement zone and the shear layer. The mesh is unexpectedly fine within the channel and the main reason for such behaviour is the fact that for the uniform mesh as present here the discretisation of the diffusion term can resolve the parabolic profile within the channel exactly, Eqn. (3.54), while the error estimator is sensitive to the second gradient and reports non-zero error for such conditions, Eqn. (4.45). Fig. 6.6(a) shows that the exact error after 8 cycles of adaptive refinement is negligible within the channel while the estimated error shown in Fig. 6.6(b) is non-zero there. This undesired behaviour has caused poor agreement of the estimated error with the exact one.

Figs. 6.7(a) and 6.7(b) show the reduction of maximum and mean velocity errors with mesh refinement. The maximum error exhibits steep reduction during the final three cycles when the refinement is localised to a small number of cells with high error. The estimated mean error in velocity does not tend to the exact error because of the over-estimation within the channel.

6.2 Flow Over a Cavity 141

(a) Exact error

(b) Estimated error (cell values are the weighted average of errors on cell faces)

Figure 6.6: Errors for the flow over a cavity on the final adapted mesh with 3257 cells (given as percentage of Uavg)

Velocity Error 101 102 103 104 10-1 100 101 102 103 no of cells m ax e rr o r [% ] Exact error Estimated error

(a) Maximum error

Velocity Error 101 102 103 104 10-1 100 101 102 no of cells m ea n e rr o r [% ] Exact error Estimated error (b) Mean error

Figure 6.7: Variation of velocity errors with adaptive mesh refinement for the flow over a cavity (errors given as percentage of Uavg)

% of maximum estimated error on a given mesh

Mesh 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Starting 58.2 8.8 4.4 11 2.2 6.5 6.5 1.09 0 1.1

Final 37.1 9.9 13.1 10.2 8.5 12.4 4.2 2.2 1.5 0.8

Table 6.1: Distribution of face errors for the cavity case

The refinement procedure tends to reduce the number of faces with low estimated error and increase the number of faces with nearly average estimated error thus increasing the mesh optimality. Fig. 6.8 shows the estimated velocity error on the

Figure 6.8: Estimated velocity error on the faces of the final mesh with 3257 cells (given as percentage of the maximum estimated error on that mesh)

faces of the final mesh. The main reason for this unfavourable error distribution is the existence of many faces perpendicular to the flow which have low error due to slow variation of velocity field in that direction.

The pressure drop, defined as the difference between the average pressures at the inletPinlet and outlet boundariesPoutlet, is also used as an indicator of accuracy.

The pressure drop coefficient (Cp) defined as:

Cp =

Pinlet −Poutlet

1 2ρUavg2

can be found in Table 6.2. The refinement makes the pressure drop reach its final value quickly. It does not change during the last four cycles of refinement suggesting that the remaining faces with estimated error above the required do not influence the global accuracy of the pressure field. The pressure drop on the final adapted mesh is in good agreement (0.2% error) with the benchmark pressure drop. This suggests that the adaptive refinement is capable of achieving accurate solutions with saving

6.3 S-shaped Pipe Bend 143