L 2 Norm – CFD Temporal and Spatial Convergence Study

The CFD model of the airport terminal space simulated in FLUENT is the more important part of the coupled simulation, because the temperature data obtained from the FLUENT building model controls the entire simulation. As a result, it is important to ensure that the data obtained from FLUENT is accurate enough, so that it represents the actual behaviour of the airport terminal space.

The validity of CFD in building simulations has already been confirmed in a wide range of studies, described in section 4.1, as well as the validation study conducted in section 4.3. However, because of the influence of the mesh size and time-step on the final results of a particular geometry (Gowreesunker et al, 2013 a, b), it is important to design a CFD model whereby the errors associated with these aforementioned parameters are justified within the limits of accuracy and adequate computing power.

This error analysis is performed following an L2 norm study for the temperature and velocities in the model. The L2 norm study is essentially a sensitivity analysis of the mesh size and time-step, with the temperature and velocities variables as the monitored parameters. It quantifies the errors in the model based on the difference between the exact solution of the governing differential equations and the solution of the discrete governing equations. However as the exact solution of the governing equations is not known, the results of a simulation with a uniformly very fine mesh and small time-step are taken as the benchmark (Gowreesunker and Tassou, 2013a; Alauzet et al, 2007).

The L2 norm errors are calculated from Eq. (6.6) (Veluri et al, 2012):

L2 norm =

(6.6) The inputs to the L2 norm simulation are described in Figs. 6.5, representing typical values for airports (Parker et al, 2011; Simmonds et al, 2000).

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Fig. 6.5(a). External temperatures schedule input for L2 norm study

Fig. 6.5(b). Heat gains schedule input for L2 norm study

Fig. 6.5(c). Ventilation input schedule for L2 norm study

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In this case, the results of a simulation with very fine-uniform mesh (307,000) and time-step (10 s) are taken as the benchmark. This benchmark model employs second-order discretisation schemes (as proposed by Ramponi and Blocken (2012)), has a mean y⁺ value of 3.2 and a global Courant number of 1.55. Furthermore, the scaled residuals are 10^-4 for continuity; 10^-5 for x/y velocities, k and ε; and 10^-7 for energy and DO intensity, as shown in Fig. 6.6.

Three meshes: coarse (≈26,000 cells), medium (≈40,000 cells) and fine (≈61,000 cells);

and four time-steps: 60s, 120s, 360s and 720s were evaluated in this model independence study. Mesh refinement was performed by varying all mesh sizes by the same ratio. Apart from the benchmark model simulation, the default FLUENT residual convergence criteria and first-order discretisation schemes were employed for all other investigated models. Furthermore, air is modelled as an ideal gas to account for buoyancy effects, the body-force weighted scheme was used for the pressure discretisation, the RNG k-ε model non-slip wall conditions were used for turbulence and the SIMPLE algorithm was used for the pressure-velocity coupling. The enhanced wall treatment is used to model the near-wall flows, because the enhanced-wall function in FLUENT allows an adequate representation of the velocity and thermal profiles for different flow regimes, including in the buffer region (3 < y⁺ < 10) (ANSYS FLUENT theory guide, 2010).

Fig. 6.6. Scaled residuals for the benchmark model only, over the course of the simulation

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The mean errors and standard deviations from the L2 norm study shown in Figs. 6.7 are determined from 36 uniformly distributed points in the comfort zone at heights of 0.3 m, 1.2 m, and 2 m, and time intervals of 1 hour.

Fig. 6.7(a). L2 norm for Temperature

Fig. 6.7(b). L2 norm for x-velocity

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Fig. 6.7(c). L2 norm for y-velocity

In order to enhance the practicality and realism of this study, the maximum errors are taken to be similar to the uncertainty of real-life sensors. Hence, the error requirements are taken to be 0.5K (similar to errors in K-type thermocouples) and 0.15 m/s (similar to errors in the TSI VelociCalc 8386® Pitot-tube velocity meter). Figs. 6.7 show that although the scaled residuals convergence criteria is relaxed for the investigated models, relative to the benchmark model, the errors for the different mesh sizes and time-steps are still within reasonable margins. The use of first order discretisation schemes in the investigated models improved the residual convergence stability in this case.

For the case of the simulated models, the velocity error requirement is satisfied with the Coarse-60s/120s, and all the medium and fine configurations. However, the temperature error requirement is only satisfied with the Coarse-60s/120s, Medium-60s/120s/360s and Fine-60s/120s/360s configurations. As a result, the medium mesh with 360s time-step was finally chosen for the CFD model, on the basis of a temperature error level lower than 0.5K and the relatively lower computing times. The resulting CFD grid is shown in Fig. 6.8 (Gowreesunker et al, 2013b).

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Fig. 6.8. Airport Terminal Space Model Grid

The mesh shown in Fig. 6.8 is designed using the in-built ANSYS® design modeller meshing algorithm, and consists of mainly hexahedral cells. The air domain is made of unstructured mesh, with a face cell size of 5 cm and growth rate of 1.1 at the internal gains; a face cell size of 10 cm and growth rate of 1.05 at the envelope surfaces, producing a first inflated layer of 4 cm; and a face cell size at the inlet and outlet of 6 cm. The mesh gradually increases towards the bulk of the air domain producing a maximum cell size of 75 cm. The roof, glazing and floor domains are made of structured hexahedral cells ranging from 3 cm to 10 cm, with 2-5 mesh intervals, with the finest meshes in the floor domain to appropriately account for the solar heat fluxes.

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