A Perard Lecomte

G Fokoua

A Mehel

A Taniere

Impact of Downscaling on the Wake Flow of a Heavy Truck : Experiments and Simulations

Keywords: Computational Fluid Dynamics, Heavy Truck, Particle Image Velocimetry, Similitude Parameter, Wake Airflow

Pollutants emitted from ground vehicles are the main contributor in urban areas. Wind tunnels are used to understand their dynamics and interactions with the flow topology. As reduced scale models are often used in such wind tunnels, some similitudes between real cases and reduced models must be respected. In this paper, the influence of similitude parameter is investigated on flow characteristics in the wake of a heavy truck for typical urban speeds, below 15 m.s -1 . Aerodynamics through drag coefficient and flow topology are compared when Reynolds number or upstream velocity are kept constant between a heavy truck model at different scale sizes. For this purpose, Computational Fluid Dynamics (CFD) simulations have been performed to model the airflow around full-and reduced scaled heavy trucks (full-, 1/4-, 1/28-and 1/46-scales). The Reynolds Average Navier-Stokes (RANS) approach combined to the turbulence Reynolds Stress Model (RSM) has been used. Particle Image Velocimetry (PIV) measurements have been achieved with the aim of validating the simulation model. Flow topology obtained numerically shows a good agreement with experimental results. For the four investigated scales, simulations give similar drag coefficients regardless of the similitude parameter. Furthermore, a different flow structure containing two counter-rotating vortices is observed for the two similitude parameters when the scale size is reduced. However, this low impact of scale size reduction is limited for the relatively low airflow velocities investigated here.

Introduction

Gaseous and particulate pollutants are emitted in urban area mainly by ground vehicles with concentrations that are very high. These pollutants, in particular fine and ultrafine particles, are transported from regions with very high concentrations to all over the surrounding local environments; resulting in the exposure for the population. It has been shown that the dispersion of particles highly depends on flow topology ( [START_REF] Kanda | A wind-tunnel study on exhaust gas dispersion from road vehicles -Part I: Velocity and concentration fields behind single vehicles[END_REF] [2] [START_REF] Mehel | Effect of air velocity on nanoparticles dispersion in the wake of a vehicle model: Wind tunnel experiments[END_REF]). It is then important to characterize flow characteristics before studying particles that are emitted by ground vehicles. To investigate the transport of such pollutants, an original way is to use reduced scale models in wind tunnels ( [START_REF] Kanda | A wind-tunnel study on exhaust gas dispersion from road vehicles -Part I: Velocity and concentration fields behind single vehicles[END_REF] [2] [START_REF] Mehel | Effect of air velocity on nanoparticles dispersion in the wake of a vehicle model: Wind tunnel experiments[END_REF]).

In automotive industry, reduced scale models are often used for wind tunnel studies. Indeed, studying aerodynamics of a full-scale vehicle would demand a full-scale wind tunnel, which is rare and very expensive. Working on bulky ground vehicles is also problematic, in particular on heavy trucks, that are frequently investigated using reduced scale models. Indeed, most experimental studies on heavy trucks have been conducted on a large range of reduced scale models. According to the study, a physical parameter, called here the similitude parameter, is kept constant between the real case and the reduced scale model investigated in a wind tunnel. In some cases, the Reynolds number is the similitude parameter ( [START_REF] Storms | A Study of Reynolds Numver Effects and Drag Reduction Concepts on a Generic Tractor-Trailer[END_REF]). [START_REF] Storms | A Study of Reynolds Numver Effects and Drag Reduction Concepts on a Generic Tractor-Trailer[END_REF] have conducted an experimental study on a 1/8 scale truck and highlighted that drag force varies from less than 1%, for Reynolds numbers lower than 3 million ( [START_REF] Storms | A Study of Reynolds Numver Effects and Drag Reduction Concepts on a Generic Tractor-Trailer[END_REF]). Other researchers chose the upstream velocity to be identical between the real heavy truck and the reduced scale model ( [START_REF] Cooper | Model and Full-Scale Wind Tunnel Tests of Second-Generation Aerodynamics Fuel Saving Devices for Tractor-Trailers[END_REF] [6] [START_REF] Lo | Flow Characteristics over a Tractor-Trailer Model with and without Vane-Type Vortex Generator Installed[END_REF]). In study [START_REF] Cooper | Model and Full-Scale Wind Tunnel Tests of Second-Generation Aerodynamics Fuel Saving Devices for Tractor-Trailers[END_REF], drag coefficients have been measured on full-and 1/10 scales, using the same velocities for the two scales. A maximal gap of drag coefficient about 11% has been noted between full-and the reduced scales ( [START_REF] Cooper | Model and Full-Scale Wind Tunnel Tests of Second-Generation Aerodynamics Fuel Saving Devices for Tractor-Trailers[END_REF]). Typical highway speeds going from 90 to 180 kilometers per hour have been used to study drag reduction on a 1/4 scale heavy truck ( [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF]) and on a 1/20 scale of tractor-trailer ( [START_REF] Lo | Flow Characteristics over a Tractor-Trailer Model with and without Vane-Type Vortex Generator Installed[END_REF]). A relative gap of drag coefficient fewer than 4% has been obtained for two different upstream velocities ( [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF]). Wake flow topology has also been investigated, highlighting the presence of a large vortex in the rear of the truck ( [START_REF] Lo | Flow Characteristics over a Tractor-Trailer Model with and without Vane-Type Vortex Generator Installed[END_REF]). However, most of mentioned experimental studies do not provide information on flow topology around a heavy truck. Numerical investigations on heavy vehicles, fewer in number, have also been achieved and afford a better knowledge of the flow topology around heavy trucks. A Large Eddy Simulation investigation on a simplified truck, modelled by a parallelepiped, showed a recirculation region composed of two contra-rotating vortices in the wake of the vehicle [START_REF] Rao | An LES Investigation of the Near-Wake Flow Topology of a Simplified Heavy Vehicle[END_REF]. Both numerical and experimental researches mentioned earlier contribute to knowledge on heavy truck's flow characteristics, especially drag coefficients and flow topology.

However, researches on the scale influence on flow characteristics are missing; in particular on heavy utility vehicles travelling at typical urban speeds, below 15 m.s -1 . This paper aims to characterize numerically the influence of scale on two flow characteristics: drag coefficient and wake flow topology through velocity profiles. Four scale size are investigated, and flow characteristics are compared using two parameters that are kept identical between different scales: Reynolds number and upstream velocity. This numerical study draws on experimental measurements, using Particle Image Velocimetry on a reduced scale heavy truck model. It should be mentioned that this study is part of a wider research project on particle dispersion in the wake of a heavy truck. The present study is the first step of this research project, it consists in characterizing numerically and experimentally the single-phase flow around the truck without particles.

After presenting numerical and experimental methodologies, validation approach is introduced. Results on flow topology and aerodynamics through drag coefficient are then presented for both mentioned parameters: Reynolds number and upstream velocity.

Methodology

Studied parameters a. Recirculation length

The first studied parameter is the recirculation length noted 𝐿 𝑟 . This length is the characteristic length of the recirculation region, which is present in the rear of any obstacle and where velocities are relatively low (fig. 1), often used for wake flows studies of the Ahmed Body [START_REF] King | Active Flow Control II[END_REF]. This length is obtained in drawing x-velocity profile in the wake of the heavy truck, starting from the mid-height of the trailer (fig. 2). The recirculation length is the highest length for which the x-velocity component equals zero (fig. 2).

b. Drag coefficient

Scale effects have been characterized also in term of drag coefficient of the heavy truck. Drag coefficient, noted 𝐶 𝑑 , is defined in the following equation [START_REF] Kanda | A wind-tunnel study on exhaust gas dispersion from road vehicles -Part I: Velocity and concentration fields behind single vehicles[END_REF]:

𝐶 𝑑 = 𝐹 𝑑 1 2 𝜌𝑈 0 2 𝑆 𝑥 ( 1
)
𝐹 𝑑 is the drag force, ρ is the fluid density, 𝑈 0 is the upstream velocity and 𝑆 𝑥 is the projected surface on

x-axis.

Heavy truck geometry

CFD-simulations and experimental measurements have been conducted on a simplified heavy truck model type class 8. The heavy truck model used in the present study is shown in figure 3. This Figure 1 Recirculation area and length on an Ahmed Body [START_REF] King | Active Flow Control II[END_REF] Figure 2 Streamwise x-velocity profile in the wake of the truck geometry is a simplified model of the heavy truck used in the experimental study [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF]. Indeed, small elements like mirrors and under-trailer have been removed. Wheels and trailer axles have also been simplified to ease the meshing for numerical simulations. Four different scales of the heavy truck have been investigated numerically: the full-scale, the 1/4 scale, the 1/28 scale and the 1/46 scale. Only the 1/28 scale has been used for experimental study. The trailer's height H, given in table I, has been chosen to be the characteristic length of the vehicle (fig. 3 andtable I).

Numerical methodology a. Numerical set up

The calculation domain has been dimensioned with two parameters: the height of the truck's trailer H and an aspect ratio X, defined by Eq. 2. 𝐿 𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚 and 𝐿 𝑑𝑜𝑤𝑛𝑠𝑡𝑟𝑒𝑎𝑚 are respectively lengths between the inlet of the domain and the front of the truck, and between the outlet of the domain and the back of the vehicle. 𝐿 𝑡𝑜𝑝 , 𝐿 𝑏𝑜𝑡𝑡𝑜𝑚 and 𝐿 𝑠𝑖𝑑𝑒 are respectively distances from the truck to the top, to the bottom and to each side of the domain (fig. 4). Edges effects must be negligible at the domain's borders. Previous simulations have shown this condition is observed when the ratio X is higher than 9.5; the parameter X has also been set at 9.5.

Tetrahedral elements have been used to compose the grid. A coarse mesh has been set in Zone 1 (fig. 4). A finer mesh has been created in Zone 2 (fig. 4), near the heavy truck. The mesh size has also been reduced in complex geometry areas, like staircases, deflector and trailer's axles. In order to model the boundary layer, an inflation mesh has been applied on the truck's faces and on the domain's bottom. The first layer's height has been set to ensure that, for the four scales, the non-dimensional parameter y + , given by ( 3) is around 5. 𝑢 𝜏 is the friction velocity and 𝜈 the kinematic viscosity. For each scale, three grids have been created : a coarse, a medium and a fine mesh. Recirculation length obtained from these three meshes have been compared to investigate the mesh sensitivity on the wake flow and to fix the final grid. Grids characteristics and recirculation lengths are presented in table II, for the 1/4 scale. The mesh size in the full domain (fig. 4, zone 1) and in the body of influence (fig. II). Inflation mesh and local grid refinements remain unchanged for the three grids.

𝑦 + = 𝑦 𝑢 𝜏 𝜈 (3)
For medium and fine meshes, recirculation lengths are very close to each other, with a deviation of 0.6%. Results obtained with the medium grid are thus very similar to results given by the fine grid, whereas it counts more than two times less elements. Whereas recirculation length obtained with the coarse grid is 9.3% shorter. For these reasons, medium grid (fig. 5) has been chosen as the final 1/4 scale grid for simulations, providing a good compromise between precision and calculation duration.

This mesh sensitivity study has been realized for the four studied scale models. A summary table of the final grid characteristics for the four scales is given in table III.

b. Simulation model

The CFD software Ansys-Fluent has been used for the simulations. Inlet velocity and outlet pressure have been set as boundary conditions. Stationary wall conditions have been applied to the heavy truck's faces and to the bottom of the domain to be comparable to experimental results. Both sides and top of the domain have been set to symmetry condition. Transient simulations have been conducted, using a time step of 10 -2 s.

The Reynolds-Averaged Navier-Stokes (RANS) approach has been used for numerical simulations. In this method, variables are decomposed in the sum of mean and fluctuating components. This operation called Reynolds decomposition is then applied to Navier-Stokes equations and equations of mass conservation, given in (4) and momentum conservation (5) are then obtained.

𝜕𝑢 𝑖 𝜕𝑥 𝑖 = 0 (4)
𝜕𝑢 𝑖 𝜕𝑡 + 𝜕𝑢 𝑖 𝑢 𝑗 𝜕𝑥 𝑗 = - 1 𝜌 𝜕𝑝 𝜕𝑥 𝑖 + 1 𝜌 𝜕 𝜕𝑥 𝑗 [𝜇 ( 𝜕𝑢 𝑖 𝜕𝑥 𝑗 + 𝜕𝑢 𝑗 𝜕𝑥 𝑖 ) -𝜌𝑢 𝑖 ′𝑢 𝑗 ′ ] (5)
𝑢 𝑖 and 𝑢 𝑖 ′ are respectively the mean and the fluctuating velocity components, 𝑝 is the mean pressure.

The dynamic viscosity and the density of the fluid are respectively given by 𝜇 and 𝜌. 𝑢 𝑖 ′𝑢 𝑗 ′ represents the Reynolds tensor components. The turbulence model called Reynolds Stress Model (RSM) has been chosen to close the system of equations. In this closure model, the transport equation for each Reynolds tensor component is solved; thus, the turbulence anisotropy is considered.

The enhanced wall treatment has been used in order to model the near-wall turbulence. This method consists in combining a two-layer model with enhanced wall functions based on the boundary layer non-dimensional distance to the wall y+, defined earlier in equation (3).

Experimental methodology a. Wind tunnel set up

Experimental measurements have been conducted in an open-circuit tunnel, composed of a test section of 0.60m x 0.60m x 1.50m width x height x length respectively (fig. 6). The wind tunnel can reach a velocity of 35 m.s -1 in this section. In the present study, the flow velocity has been set to 12 m.s - 1 . This value corresponds to a velocity around 42 km/h, which is the mean urban speed in French cities [START_REF] Onisr | Observatoire des Vitesses[END_REF]. A raised floor with a rounded leading edge is installed 10 cm above the test section bottom. The mean turbulence intensity outside the boundary layer has been measured in the test section with the raised floor by Particle Image Velocimetry technique and equals 1.55 % at the studied velocity (12 m.s - 1 ). At the end of the PIV-measurable test section (x=1050mm), the boundary layer thickness is 19 mm.

The heavy truck model in scale 1/28 is placed on the raised floor at 10 cm from the leading edge (fig. 6). In this configuration, the blockage ratio is around 4.3 %. Reynolds number based on the trailer's height, defined by ( 6) is 85,000.

𝑅𝑒 = 𝜌𝑈 0 𝐻 𝜇 (6)

b. Measurement technique

2D Particle Image Velocimetry measurements have been performed to characterize velocity/turbulence fields in the wake of the heavy truck. The Safex S195G fog generator is used with a mix of water and diethylene glycol to seed the test section. Particles are illuminated by a double pulsed laser, Litron LPU450 NanoPIV, from Dantec. The maximum thickness of this laser sheet is around 3 mm. Pictures of illuminated particles are taken using a 4 Mpx camera, FlowSenseEO 4M-32. These two elements are both bound to a traverse positioning system, which allows the laser and the camera to move simultaneously in x-and y-directions.

3

Results and discussion

Simulation model validation

Before analyzing scale effects using numerical methods, the simulation model has been validated based on two aspects using experimental results. The first one is the aerodynamics of the heavy truck in particular drag coefficient and the second one is the flow topology in the wake of the truck.

a. Aerodynamics validation

In order to validate the CFD-model regarding aerodynamics, a first simulation has been completed and compared to an existing experimental study, conducted by Landman et al [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF]. This first simulation is therefore set in the same conditions as in the experimental study, that is on the 1/4 scale and with an inlet velocity of 29 m.s -1 . Results of drag coefficients are given in table IV. Relative gap between the two drag coefficients is relatively low, equal to 3.6% thus the present CFD-model is valid, regarding the truck's aerodynamics.

TABLE IV EXPERIMENTAL AND NUMERICAL PARAMETERS

Inlet velocity

Drag coefficient Numerical present study 29 m.s -1 0.515 Experimental study [START_REF] Mehel | Effect of air velocity on nanoparticles dispersion in the wake of a vehicle model: Wind tunnel experiments[END_REF] 29 m.s -1 0.534 In the near wake of the heavy truck, vortical structures are forming. A large vortex (A) coming from the under-trailer is observed experimentally and numerically. Locations of vertices A centers for experimental and numerical results are given in table V. In the top of the recirculation area, a second vortex (B) appears clearly on numerical results but is less visible experimentally, forming an angle (α) with the horizontal plane (fig. 8). This difference between experimental and numerical results is still further investigated. Angles α obtained experimentally and numerically are very close to each other, as shown in table V. A third structure (C) emerges further in the recirculation area, at respectively x = 1.33*H and x = 1.51*H for experimental and numerical cases. Flow topologies observed experimentally and numerically are similar to the wake flow of a different heavy truck, studied experimentally by Lo and Kontis (2016) ( [START_REF] Lo | Flow Characteristics over a Tractor-Trailer Model with and without Vane-Type Vortex Generator Installed[END_REF]). In their study, the recirculation region is composed of a large vortex and an additional structure comparable to vortex A and structure C of the present study (fig. 8). The first vortex center locates around x=0.18*H in study [START_REF] Lo | Flow Characteristics over a Tractor-Trailer Model with and without Vane-Type Vortex Generator Installed[END_REF] against x=0.16*H for numerical simulation and x=0.22*H for experimental measurements of the present study.

Scale effects

Two similitude parameters have been investigated. In this paper, the term of "similitude parameter" refers to the parameter that is kept constant between different scales. In the first case, the Reynolds number is the similitude parameter and has been fixed at 85,000. For the second case, an upstream or inlet velocity 𝑈 0 = 12 𝑚. 𝑠 -1 is kept identical for the four scales. This velocity corresponds to a speed around 42 kilometers per hour, which is the mean velocity in a French city [START_REF] Onisr | Observatoire des Vitesses[END_REF]. For the four scales, corresponding velocities and Reynolds numbers are given in table VI. 2,400,000 600,000 85,000 51,000

Recirculation lengths 𝐿 𝑟 and drag coefficients 𝐶 𝑑 have been quantified and relative gaps between reduced scales and full-scale have been calculated using equations ( 7) and [START_REF] Rao | An LES Investigation of the Near-Wake Flow Topology of a Simplified Heavy Vehicle[END_REF]. Relative gaps of drag coefficients between present numerical study and the experimental one obtained by Landman et al [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF] have also been calculated with equation [START_REF] King | Active Flow Control II[END_REF]. 𝐶 𝑑 𝑚𝑒𝑎𝑛 and 𝐿 𝑟 𝑚𝑒𝑎𝑛 are respectively drag coefficients and recirculation lengths mean values obtained for the four scales 𝑖: 1, 1/4, 1/28 and 1/46. As mentioned earlier drag coefficient obtained experimentally by [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF] [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF] is 𝐶 𝑑 𝑒𝑥𝑝 = 0.534.

𝛥𝐶 𝑑 𝑖 = | 𝐶 𝑑 𝑚𝑒𝑎𝑛 -𝐶 𝑑 𝑖 𝐶 𝑑 𝑚𝑒𝑎𝑛 | * 100 (7)
𝛥𝐿 𝑟 𝑖 = | 𝐿 𝑟 𝑚𝑒𝑎𝑛 -𝐿 𝑟 𝑖 𝐿 𝑟 𝑚𝑒𝑎𝑛 | * 100 (8)
𝛥𝐶 𝑑 𝑒𝑥𝑝 = | 𝐶 𝑑 𝑒𝑥𝑝 -𝐶 𝑑 𝑖 𝐶 𝑑 𝑒𝑥𝑝 | * 100 (9)

a. Reynolds number similitude

The four scales of heavy truck models have been simulated for the same Reynolds number (Re=85,000). Results concerning flow topology are observed to the downstream of the truck's trailer, in the recirculation region. Velocity magnitude contours and streamlines in this area, in the central vertical longitudinal plane y=0 of the heavy truck, are given in fig. 9 for the four scales.

A large vortex (A) coming from the under-trailer appears for the four scales (fig. 9.a, b, c andd). This vortex center is located between x=0.13H and x=0.18H; z=-0.34H and z=-0.26H. The A vortex's length and height respectively range from 0.59H to 0.74H, and from 0.52H to 0.69H. For the three reduced scales (1/4 (fig. A general observation is that for a constant Reynolds number, as the scale size decreases, a second vortex (B) appears. A different flow structure is thus developing, containing two contra-rotating vertices: the first one (A) is stuck to the trailer's rear, with a counterclockwise rotation direction and the second one (B) locates further and higher in the recirculation area, with a clockwise direction of rotation. Recirculation length and drag coefficient values for each scale obtained with the Reynolds number similitude are given in table VII. A graph of velocity profiles in the wake of the heavy truck is also given in fig. 10. For the four scales, recirculation lengths are quite close one from the other, with a gap lower than 2.2%. Velocity profiles given in figure 10 are very similar for all scales outside the recirculation area, for 𝑥 > 𝐿 𝑟 𝐻. Drag coefficients are also quite close for the four scales, with a maximal relative gap around 5%. Relative gaps between numerical and experimental drag coefficients, obtained by [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF] [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF], are between 0.9% and 7.7%.

b. Inlet velocity similitude

Flow topologies in the wake of the four scales are given in fig. 11 As in case of the Reynolds number similitude, the flow topology changes as the scale size is reduced. Indeed, a second vortex appears for the two smallest reduced scale models, forming a flow structure with two counter-rotating vortices. Recirculation lengths and drag coefficients obtained for the four scales simulated with the same inlet velocity 𝑈 0 = 12 𝑚. 𝑠 -1 are given in table VIII. It appears that recirculation lengths are more scattered than in the first case (table VII). Indeed, recirculation length is much lower for the full-scale than for the three reduced scales, where the gap with the mean value of recirculation length exceeds 15%. Indeed, figure 12 shows a difference between velocity profiles obtained for the full-scale and reduced scales. However, the trend of velocity profiles (fig. 12) remains unchanged for the four scales. When focusing on drag coefficients, the highest gap observed equals 3.4 %. Moreover, drag coefficient values are close to experimental [START_REF] Landman | Understanding Practical Limits of Heavy Truck Drag Reduction[END_REF] value, with a maximal relative gap of 3.4% too.

c. Conclusion

In the present study, the question of the influence of scale on the flow characteristics is raised, when dealing with airflow in a limited velocity range, below 15 m.s -1 . The question that arises is whether the similitude parameter, kept constant between full-and reduced scales, actually influences flow characteristics. To help answer this question, simulation results obtained for four scales were compared: full size, 1/4, 1/28 and 1/46 models. In particular, the influence of two similitude parameters (Reynolds number and upstream velocity) has been investigated on aerodynamics through drag coefficient and on flow topology, through velocity profiles and recirculation length.

Flow topology especially the main vortex in the recirculation region looks similar for the four scales and for the two similitude parameters. However, some vortical structures do not appear on each studied case. From a certain reduced scale size, a change in flow structure is observed regardless of the similitude parameter. Indeed, two counter-rotating vortices appear when scale size is reduced: a first vortex with a rotational direction which is counterclockwise and located in the truck's rear and a second vortex turning clockwise positioned at the top of the recirculation region.

Drag coefficient values and flow topology follow the same trend regardless of the similitude parameter. Indeed, drag coefficients are very close to each other for the four scales, for the two similitude parameters, with a maximal gap of 5.3%. Velocity profiles are quite similar for the two similitude parameters too, outside of the recirculation area, even if recirculation region seems to be bulkier at reduced scale for simulations where the inlet velocity is kept constant between different scales.

In conclusion, when studying airflow around reduced scale models in wind tunnels for limited velocity range (below 15 m.s -1 ), both Reynolds number and upstream velocity could be chosen as Figure 12 Horizontal x-velocity profiles in the wake of the heavy truck, at mid-height of the trailer, for 𝑈 0 = 12 𝑚. 𝑠 -1 for the four scales. similitude parameter between full-and reduced scale. For feasibility reasons, as most wind tunnels are limited in terms of velocity, the upstream velocity can therefore be used as the similitude parameter.

Figure 3 Figure 4

 34 Figure 3 Heavy truck model used in the present study

Figure 5

 5 Figure 5 Final medium grid for the 1/4 scale

Figure 6

 6 Figure 6 Heavy truck model, scale 1/28 in the test section

Figure 7

 7 Figure 7 Wind tunnel and PIV-measurement system

Figure 8

 8 Figure 8 2D-velocity magnitude and streamlines in the recirculation region, longitudinal plane y=0, for the 1/28 scale model obtained for 𝑈 0 = 12 𝑚. 𝑠 -1 (a) experimentally, (b) numerically.

  9.b), 1/28 (fig. 9.c) and 1/46 (fig. 9.d)), a second vortex (B) is visible, at respectively (x;z)=(0.97H;0.30H), (x;z)=(1H;0.28H) and (x;z)=(0.96H;0.29H). This vortex rotates in the opposed direction to the A vortex rotation direction. An additional structure (C) appears in the rear part of the recirculation area, where velocities are notably low, at x=1.52H for 1/4 and 1/28 scales; x=1.43H for 1/46 scale x=1.23H for the full-scale.

Figure 9

 9 Figure 9 Velocity magnitude and streamlines in the recirculation region, longitudinal plane y=0, for Re=85,000 for scale (a) 1, (b) 1/4, (c) 1/28 and (d) 1/46.

  , for the simulation conducted setting the same inlet velocity, 𝑈 0 = 12 𝑚. 𝑠 -1 . Flow topologies are similar to those observed in the case of the Reynolds number similitude. Indeed, the large vortex (A) appears for the four scales, formed with the flow coming from the under-trailer. The center of this vortex is located around x=0.21H and z=-0.25H for the 1/4-and full-scale (fig. 11.a and b). For the two smallest scale models (fig. 11.c and d), it is located lower and closer to the heavy truck, around x=0.15H and z=-0.34H. The vortex length is quite constant for the four scales, around x=0.71H. The second counter-rotating vortex (B), centered respectively at x=H, z=0.28H and x=0.80H, z=0.29H only appears for the 1/28 scale (fig. 11.c) and the 1/46 scale (fig. 11.d). The additional structure (C) is visible at x=1.23H for the full-scale (fig. 11.a) against x=1.51H for 1/28 scale (fig. 11.c) and x=1.45H for the two other scales (fig. 11.b and 11.d).

Figure 10

 10 Figure 10 Horizontal x-velocity profiles in the wake of the heavy truck, at mid-height of the trailer, for Re=85000 for the four scales.

Figure 11

 11 Figure 11 Velocity magnitude and streamlines in the recirculation region, longitudinal plane y=0, for 𝑈 0 = 12 𝑚. 𝑠 -1 for scale (a) 1, (b) 1/4, (c) 1/28 and (d) 1/46.

TABLE II GRIDS

 II CHARACTERISTICS FOR THE GRID SENSITIVITY STUDY, 1/4 SCALE

	Grid	Coarse	Medium	Fine
	Domain mesh size (m)	0.30	0.25	0.20
	Body of influence mesh size (m)	0.05	0.04	0.025
	Number of elements (million)	5.3	8.1	21
	Recirculation length 𝐿 𝑟	1.62*H	1.77*H	1.78*H

TABLE V

 V

	LOCATIONS OF VORTICAL STRUCTURE CENTERS FOR EXPERIMENTAL AND NUMERICAL RESULTS
		Vortex A, x	Vortex A, z	Structure C, x	Angle α
	Experimental (fig. 8.a)	0.22 H	-0.23 H	1.33 H	11 °
	Numerical (fig. 8.b)	0.16 H	-0.30 H	1.51 H	12 °

TABLE VII

 VII

			RESULTS FOR RE=85000
	Scale	Mean value	1	1/4	1/28	1/46
	𝐿 𝑟	1.63*H	1.60*H	1.66*H	1.59*H	1.65*H
	𝛥𝐿 𝑟 𝑖 (%)	-	1.5	2.2	2.2	1.5
	𝐶 𝑑	0.546	0.575	0.539	0.552	0.519
	𝛥𝐶 𝑑 𝑖 (%)	-	5.3	1.3	1.1	5.0
	𝛥𝐶 𝑑 𝑒𝑥𝑝 (%)	-	7.7	0.9	3.4	2.8

TABLE VIII RESULTS

 VIII FOR 𝑈 0 = 12 𝑚. 𝑠-1

	Scale	Mean value	1	1/4	1/28	1/46
	𝐿 𝑟	1.52*H	1.29*H	1.69*H	1.59*H	1.52*H
	𝛥𝐿 𝑟 𝑖 (%)	-	15.3	11	4.4	0.2
	𝐶 𝑑	0.534	0.524	0.517	0.552	0.542
	𝛥𝐶 𝑑 𝑖 (%)	-	1.8	3.1	3.4	1.5
	𝛥𝐶 𝑑 𝑒𝑥𝑝 (%)	-	1.9	3.2	3.4	1.5