Solution methods and test accuracy

Once boundary conditions were set, the solution methods can be specified. Fluent provides a list of solver formulations. From these methods, a pressure-velocity value is required to predict the pressure distribution in the flow domain with reasonable accuracy. In the present study, SIMPLE algorithm (Semi-Implicit Method for the Pressure-Linked Equation) was employed for pressure-velocity coupling. This algorithm has the ability to converge the solution faster and is often quite accurate for flows around bluff bodies such as tractor semi-trailer vehicles. Additionally, in SIMPLE algorithm, velocities are corrected and a new set of conservative fluxes is calculated [135].

Green-Gauss Node-based gradient evaluation has been selected for computing secondary diffusion terms and velocity derivatives at the cell faces. This scheme is more suitable than the cell-based gradient option for unstructured meshes[135], as it reconstructs exact values of a linear function at a node from surrounding cell-centred values on arbitrary unstructured meshes. Furthermore, the 2nd _{order implicit method was used for performing the time} integration. This method was blended with a second-order upwind scheme for interpolating the variables on the surface of the control volumes. It should be noted here that use of the first- order upwind should be avoided, whenever possible, in LES due to the excessive amount of numerical dissipation introduced [79].

The Fluent solutions are provided in the ‘Results’ task page, where the user can set up and display the results of the CFD simulation. The graphical results that let the users visually inspect the results are also generated using Fluent. Those graphical results include contours, vectors, path lines, particle tracks, animations and plots. In addition, the user also has the

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ability to get the numerical solutions for the drag, side and lift forces from the ‘Reports’ task page.

4.4.1 Simulation time step and Convergence Criteria

The CFL number (Courant-Friedrichs-Lewy condition) is a mathematical convergence condition used when solving partial differential equations. In Fluent, the CFL number relates velocity with time and length of the computational cell size and is given as [140]:

𝑣 ∆𝑡

∆𝑥 ≤ 𝐶𝐹𝐿 (4-11)

where Δx is minimum length side of the mesh in the domain and Δt is time step.

In order to acquire a correct and steady solution, the CFL number should be smaller than 1 [140]. To achieve this condition, side aerodynamic coefficient (Cs) of the vehicle is obtained using three time steps based on different lengths of the computational cell, and constant CFL of 0.92 are compared in the Table 4-2. It is observed that there are almost no differences between the results from the first and second tests. Therefore, a constant time step of t = 14 ×10-3 sec was used in the following numerical simulation. Also, 20 iterations were performed for each time step to have good convergence.

(Note: all calculations have been done for a flow angle of 45o)

Table 4-2: Time step calculation CFL At iteration number Time step

(sec) Minimum x (mm) Cs (-) 0.92 8000 10 ×10-3 _{130 mm -1.08} 8000 14 ×10-3 _{182 mm -1.1} 8000 19 ×10-3 _{250 mm -0.67}

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In addition, in numerical CFD method, only a converged solution can be treated as the solution of the flow problem. The converged solution indicates that the solution has reached a stable state and the variations in the flow parameters and iterative process of the solver have died out.The default convergence criterion for the continuity, velocities in three dimensions and the turbulence parameters in ANSYS 17 is 0.001. This means that when the continuity changes, velocities and turbulence parameters drop down to the fourth place after the decimal, and then the solution is treated as a converged solution. However, in many practical applications, the default criterion does not necessarily indicate that the changes in the solution parameters have died out. Hence, it is often better to monitor the convergence rather than relying on the default convergence criteria [135].

In this simulation, static pressure on the leeward and windward surfaces of the trailer unit has been monitored throughout the iterative process. The solution has been considered converged once the static pressure at both these surfaces has become stable. Here a stable solution can be either one in which the pressure fluctuations have died out completely or have become cyclic, having the same amplitude in each cycle.

4.4.2 Mesh independent study

A grid sensitivity analysis for the model was performed to confirm the precision of the results, and to identify the most effective mesh sizing in order to achieve an appropriate mesh discretisation. This type of analysis must be performed to reduce the influence of the number of nodes on the computational results, since the solution must be independent of the mesh resolution in the computational domain. For this purpose, the numerical simulation is run using three different mesh sizes [141]. The first mesh comprises of 3,346,646 elements, the second mesh of 6,172,308 elements, and the third mesh of 7,709,906 elements. The mesh distributions over the trailer surfaces and the height of first layer grid over the walls were kept

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constant. The computed average drag force from these three simulations is listed in Table 4- 3. It has been observed that the time-average drag force is well predicted by both the second and the third mesh schemes, and the obtained results do not show significant changes. Therefore, this study employs the third mesh (comprising of 7,709,906 mesh elements) to investigate the aerodynamic forces acting on the tractor semi-trailer vehicle. Furthermore, the mesh independence test indicates that five layers of inflation are sufficient for this study.

(Note: all calculations have been carried out for a flow angle of 45o)

Table 4-3: Mesh dependency

Min size Number of iterations Elements FD (N) 

70 mm 2500 3,346,646 3761 6.6%

100 mm 2500 6,172,308 3898 2.2%

182 mm 2500 7,709,906 3903 0.01%