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A computational study of the aerodynamic forces acting on a tractor-trailer vehicle on a bridge in cross-wind
Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 573–592
A computational study of the aerodynamic forces acting on a tractor-trailer vehicle on a bridge in cross-wind J. Bettle, A.G.L. Holloway*, J.E.S. Venart Department of Mechanical Engineering, University of New Brunswick, Box 4400, Fredericton NB, Canada E3B 5A3 Received 23 April 2001; received in revised form 6 September 2002; accepted 11 November 2002
Abstract This paper examines the effect of truck speed on the aerodynamic forces acting on a standard sized, North American transport truck travelling across a bridge under conditions of cross-wind. The objective is to establish a relationship between wind speed, truck speed and propensity for truck rollover that may be used to devise strategies for accident avoidance. Conditions of a moving truck travelling in both windward and leeward lanes were considered with a cross-wind speed of 120 km/h and truck speeds of 0–120 km/h. Using the calculated pressure distributions on the surface of the truck; the aerodynamic lift, drag and moment coefficients were determined for relative wind directions. The results show that the aerodynamic moment tending to overturn a truck in the windward lane of the bridge rises from approximately 120 kN m at low speeds (0–40 km/h) to 217 kN m at a truck speed of 120 km/h. For a truck in the leeward lane, the corresponding moments are substantially less, at 82 and 154 kN m, respectively. The 1.1 m barrier wall along the side of the bridge is a contributing factor to the aerodynamic difference between windward and leeward lanes. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bridge and tractor-trailer vehicle aerodynamics; Computational fluid dynamics
1. Introduction Bridges and open highways often are exposed to crosswinds that can cause driveability problems for high-sided vehicles and in some cases result in serious *Corresponding author. Tel.: 506-447-3108; fax: 506-453-5025. E-mail address: [email protected] (A.G.L. Holloway). 0167-6105/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0167-6105(02)00461-0
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accidents [1,2]. Baker [2] has described a comprehensive approach to the prediction of such accident conditions that considers wind variation, vehicle geometry and speed, suspension dynamics and driver reaction. An essential element of this method is the relationship between wind speed, truck speed, and aerodynamic forces for a particular vehicle and site. Traditionally these aerodynamic characteristics have been studied by scale model wind tunnel testing [2]. Alternatively, the present paper uses the method of computational fluid dynamics (CFD) [3] for this purpose. Specifically CFD is used to evaluate the aerodynamic forces on a tractor-trailer vehicle crossing a bridge under windy conditions. This appears to be a novel application of the method. The bridge geometry chosen for this study is similar to the Confederation Bridge, which opened in 1997, and crosses the Northumberland Strait, joining Cape Jourimaine, New Brunswick to Borden, Prince Edward Island, Canada. It is 12.9 km in length, 11 m across the deck, and has an average height above the water of 40 m. The highest point is located at the navigational span, 60 m above sea level, where the magnitude and direction of the wind is continually measured by bridge operators using a stationary anemometer. The direction of the wind has been found to be predominantly perpendicular to the bridge with 6 min averaged speeds as high as 148.6 km/h and gusting as high as 170.3 km/h [4]. It should be noted that this bridge has the distinct feature of a 1.1 m high barrier, which runs the full length of the bridge on both sides of the road deck. The vehicle is a standard North American tractor-trailer with a 2.92 m high, 14.63 m long, and 2.60 m wide trailer that is raised 1.19 m above the road surface. The model geometry with a few overall dimensions can be seen in Fig. 1. The effect of cross-wind on the aerodynamic forces is very significant. Travelling through still air the drag force acts on the relatively small frontal area of the truck and is directed nearly parallel to its longitudinal axis. However in the presence of a cross-wind the magnitude of the relative velocity between the truck and the air is increased, and its direction is skewed to the direction of motion so that the large side area of the trailer is exposed. The resulting drag force is much larger and it has components both parallel and normal to the direction of the truck’s motion. Substantial lift can also be generated in cases of separated flow over the top of the trailer. The component of drag normal to the longitudinal axis of the truck, referred
8.5 m 1.1m
11 m Fig. 1. Model geometry and overall dimensions.
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to as side force, and the lift force, are particularly important to drivability because they generate a moment about the longitudinal axis of the vehicle, reducing the wheel loading on the windward side. An experimental study of wind forces on trucks on the Confederation Bridge was conducted by King et al. [5]. The wind tunnel testing was performed on a (1/60)th scale with pure cross-wind conditions. The only measurements taken were of the wind speed required to cause rollover of the stationary truck models. Nevertheless, they provided coefficients for the rolling moment based on the model weight. Experimental studies of the aerodynamic forces acting on trucks subjected to winds on an open roadway have been summarized by Baker [2]. Typically these were conducted on (1/50)th scale models for wind directions ranging from 01 to 901 of yaw and for various wind turbulence levels. Among the measurements reported were the time average side force, lift and drag, and the pitching, rolling and yawing moments acting on the model trucks.
2. Method The present analysis has been performed using a commercial CFD program called Flovent [6], a product of Flomerics. The computational domain has three space dimensions and time. Spatial discretization is done using the control volume approach on a staggered, structured, non-uniform grid with convective terms treated using a weighted average of first order accurate upwinding and second order accurate central differencing according to the limits of numerical stability. The method is robust but does introduce numerical errors of a diffusive nature where the flow is strongly convective and skewed to the coordinate direction. Temporal discretization is uniformly spaced and uses an implicit first-order differencing method. Details of the numerical method can be found in Refs. [6,3]. The present flow problem is inherently unsteady, not only because of the flow turbulence, but also because of the large scale vortex shedding present in the separated wake behind the truck and bridge. The turbulence away from the boundaries was modelled by a constant eddy viscosity of 1000 times the molecular value for air under standard conditions. Near the solid boundaries a log law was used to calculate the wall shear stress from the velocity at the closest node to the boundary. A more complex model of turbulence, the k e model [7], was also tested on this problem with the result of increasing the number of steps required for convergence several fold and changing aerodynamic drag by only a few percent. Consequently, all solutions presented were calculated using the constant eddy viscosity model of turbulence. More recently formulated versions of the standard k e model (for example RNG) and Reynolds Stress Models [7] were not available for testing but it should be expected that such methods would make incremental improvements to the turbulent viscosity and k e model results under the present circumstances. A transient solution method was used; however, such solutions were found to develop into a steady form for both turbulence models and for a range of time step sizes as small as 100th of the vortex-shedding period. That a steady
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solution is achieved under such conditions is undoubtedly the result of the diffusive nature of the turbulence model and numerical scheme; both of which are imprecise in this regard. In fact, turbulence models such as the ones employed here are calibrated to match the time average velocity field; even in wake flows. The aerodynamic forces presented are based on these steady solutions and are therefore regarded as time average values. Unfortunately, they do not provide information about the important force fluctuations due to vortex shedding or wind gusting. The computational domain is physically very large and the number of grid nodes available to fill it was limited to 550 000 for the computer available. The mesh was more concentrated around the truck and bridge but still the smallest computational cell had a volume of 0.027 m3. The sides of which were many times the boundary layer thickness and finer features of the truck geometry. In effect, the truck and trailer are modelled as a series of solid ‘‘blocks’’ with sharp edges. As a consequence, flow separation can be expected to occur at these edges independent of the Reynolds number or boundary layer development of the flow. This is not too far from reality for this type of vehicle, which have little streamlining. A study describing the effects of domain size and mesh size is provided in the discussion. Atmospheric conditions vary widely in temperature and humidity with corresponding variations in density and viscosity. The air properties used in the present simulation were for standard conditions of 101 kPa and 201C with a density of 1.16 kg/m3 and a kinematic viscosity of 15 106 m2/s. The present calculations were all performed in a frame of reference moving with the truck, so that the boundary conditions are steady. From this viewpoint, riding in the truck, it is the angle of yaw, a; and the magnitude of the relative wind, which changes with the truck speed as shown in Fig. 2. For example, with a constant crosswind of 120 km/h and zero truck speed, the relative wind is normal to the longitudinal axis of the truck, which will in the present context will be taken as the 901 angle of yaw. For a truck speed of 120 km/h the relative wind is at 451 and 41% greater in magnitude. Further increases in truck speed swings the relative wind direction towards the line of motion of the truck. The bridge deck would also be moving relative to the truck in this frame of reference. However, to be consistent with wind tunnel testing and to simplify the present simulation, the relative velocity
Wind Velocity, VW
Truck Velocity VT
α
Relative Velocity VR
Fig. 2. Wind velocity relative to the truck.
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between the truck and road was set to zero. This approximation should not cause large errors in the present computations because of the two-dimensional nature of the bridge geometry used in the model. This is not an inherent limitation of CFD but is a limitation of the commercial package used. The aerodynamic forces acting on the truck consist of a side force, S; a lift force, L; and a longitudinal drag force, D: The simulations provide estimates of both the magnitude and the line of action of each of these forces by simply summing the pressure forces and moments acting on the vertical and horizontal surfaces of the truck and trailer. The skin friction, which is poorly resolved, is not included in the wind force calculations. The aerodynamic forces may be generalized by rendering them non-dimensional with the geometric and fluid properties of the model to give CL ; CS ; and CD ; which are defined as L CL ¼ 1 2 ; ð1Þ 2rVR AL S CS ¼ 1 2 ; 2rVR AS D : 2 rV R AD 2
CD ¼ 1
ð2Þ ð3Þ
r is the density of the air, 1.16 kg/m3, VR is the speed relative to the truck, AL is the top surface area of the trailer, 38.0 m2, AS is the side area of the trailer, 42.7 m2 , and AD is the frontal area, 9.386 m2. An additional aerodynamic property of the tractortrailer is the moment tending to roll the truck over. This is calculated using the lift and drag forces described above and by summing moments about the point of contact between the leeward tire and the road (point O in Fig. 3). This aerodynamic moment is calculated using M0 ¼ Sdy þ Ldx;
ð4Þ
where dy and dx are defined in Fig. 3 as the position of the centre of pressure of the respective surfaces. This is strictly the moment due to wind forces and it does not take into account the weight of the vehicle or its inertia. The non-dimensional form of the moment is M0 CM ¼ ; ð5Þ 1 2 rVR AS hv 2 where hv represents the overall vertical height of the trailer, 4.11 m. A total of 14 simulations were run to determine a relationship between truck speed and the aerodynamic forces and moments acting on the truck as it travels across the bridge during high winds. Simulations were run with a constant wind speed of 120 km/ h and truck speeds of 0, 20, 40, 60, 80, 100 and 120 km/h. This varied a (see Fig. 2) between 901 and 451. Separate simulations were run with the truck in the windward and leeward lanes. A summary of the simulation conditions is shown in Table 1.
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L
δx
x
S
hv
δy
O Road Fig. 3. Definition of the rollover moment due to wind forces.
Table 1 Simulation conditions Wind speed, VW (km/h)
Truck speed, VT (km/h)
Relative speed, VR (km/h)
a (deg.)
120 120 120 120 120 120 120
120 100 80 60 40 20 0
170 156 144 134 126 122 120
45.0 50.2 56.3 63.4 71.6 80.5 90
3. Geometry and boundary conditions The three-dimensional computational model of a transport truck on a section of the Confederation Bridge, created for this study, can be seen in Fig. 4. The bridge was modelled as a 60 m long section with a constant depth. All the truck dimensions fall within the Canadian standard geometric design limits for an 18-wheeled tractor-trailer with a 14.63 m (480 ) dual axle trailer [8]. The overall length and the total height of the model truck are 20.15 m and 4.11 m, respectively. The boundary conditions used in the simulations are shown in Fig. 5. The domain boundaries above, below, to the right, and behind the model truck and bridge are ‘‘free’’ outflow boundaries; that is, they are taken as surfaces of constant pressure to
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30 20 10 0
20
40 60 X (m)
80
100
60
40 20 Z (m)
0
Y (m)
40
0
Fig. 4. Three principal views of the model geometry used in the simulations.
Front - Uniform In Flow
LeftUniform In Flow
Right Free Boundary
Back - Free Boundary Fig. 5. The flow boundary condition used in the simulations.
permit wake formation. Note that the water surface is not included in the simulation and that the bridge extends fully across the domain so that no end leakage is permitted where it intersects the boundary surface. The remaining domain boundaries, the one to the left and the one in front of the truck, are upwind of the model and are set to uniform wind profiles with the correct relative magnitude and direction.
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Y
Z
X
Fig. 6. Computational grid in the vicinity of the truck and bridge.
4. Computational details The overall computational domain used for the simulations, as depicted in Fig. 4, is 100 m 40 m 60 m. A non-uniform grid was created within this domain. The grid used contained 550,000 nodes (100 55 100). A section of the grid in the vicinity of the truck, which is the region of the smallest cells, can be seen in Fig. 5. All the cells in this area are about 0.027 m3 (approx. 1 ft3) (Fig. 6).
5. Results A sample of the computational results with a ¼ 451 is shown in Figs. 7–12. Figs. 7 and 8 show contours of speed in the horizontal and vertical planes, which pass through the mid-plane of the truck for the windward and leeward simulations. The speed contours clearly show that the air is accelerated as it passes over and under the bridge (and truck) and around the ends of the truck (red regions). Peak speeds reach more than 60 m/s. The blue regions depict the low speed wake directly behind the bridge and truck where the air speed is close to the wind speed of 30 m/s. The 1.1 m barrier on the sides of the bridge deck reduced the airflow under the truck compared to what may occur on an open road. The gauge pressure contours (Figs. 9 and 10) show that a high pressure (red) region is created on the windward side of the truck and bridge, while a low pressure (blue) region is created on all other sides of the truck. Clearly the wake is larger for the truck in the windward lane and the relatively low pressure on the leeward side and the top of the truck in both cases produce a very substantial drag and lift forces.
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S P EED:
0
10
S P EED:
20
0
10
30
40
20
30
581
50
40
60
50
60
Fig. 7. Contours of speed from the simulation results corresponding to a windward solution with a ¼ 451: The dashed lines show the location of the Bridge. Speed is in m/s.
Fig. 11 gives a complete description of the surface pressure on all sides of the truck. The extreme pressures (which cannot be shown in Figs. 9 and 10) were calculated to be as high as 1000 Pa on the windward side of the truck and as
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S P EED:
0
10
S P EED:
20
0
10
30
40
20
30
50
40
60
50
60
Fig. 8. Contours of speed from the simulation results corresponding to a leeward solution with a ¼ 451: The dashed lines show the location of the Bridge. Speed is in m/s.
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583
Fig. 9. Contours of gauge pressure calculated from the windward simulation results for a ¼ 451: The dashed lines show the location of the Bridge. Pressure is in Pa.
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Fig. 10. Contours of gauge pressure calculated from the leeward simulation results for a ¼ 451: The dashed lines show the location of the Bridge. Pressure is in Pa.
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585 P 1000 500
Left (Windward) Surface
0 -500 -1000
Top Surface
Back Surface
Right (Leeward) Surface
Front Surface
Bottom Surface P 1000 500
Left (Windward) Surface
0 -500 -1000
Top Surface
Back Surface
Right (Leeward) Surface
Front Surface
Bottom Surface
Fig. 11. Distribution of pressure on the surface of the tractor and trailer as calculated from the windward and leeward simulations with a ¼ 451: Pressure is in Pa.
low as 1000 Pa on the top of the truck. These pressure distributions were used to calculate the wind forces for each simulation case. A summary of these forces and their corresponding rollover moment are provided in Table 2 for all the simulation results. The magnitudes of moments listed would be very significant to the dynamics of an unloaded trailer. Fig. 12 shows the pressure and the velocity fields in the immediate vicinity of the truck and bridge. The regions of low pressure are centred on vortices and the regions
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S P EED: 10
20
30
40
50
60
P : -1000
-500
0
500
1000
SPEED: 10
20
30
40
50
60
P: -1000
-500
0
500
1000
Fig. 12. Pressure and velocity fields on a vertical plane which passes through the mid-plane of the trailer for both the windward and leeward simulations with a ¼ 451:
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Table 2 Aerodynamic forces and moments acting on a tractor-trailer for various truck speeds. Wind speed is held constant at 120 km/h Truck speed (km/h)
S (kN)
D (kN)
L (kN)
M (kN m)
Truck in windward lane 120 100 80 60 40 20 0
73.7 67.4 53.1 50.7 39.9 39.8 42.9
6.78 4.52 3.00 0.996 0.103 0.230 0.141
20.1 20.2 17.4 14.7 9.99 10.6 9.78
217 202 165 153 120 121 127
Truck in leeward lane 120 100 80 60 40 20 0
52.5 48.2 40.7 31.0 25.4 24.9 24.9
9.91 7.13 3.32 1.53 0.440 0.434 0.315
7.52 8.50 8.29 7.43 6.63 4.33 5.10
154 146 127 102 86.2 81.5 82.4
of high pressure on stagnation points. A vortex on the top of the trailer, at the point of separation, is visible though not well resolved. The differences between the speed and pressure contours surrounding a truck travelling in the windward lane and leeward lanes is largely due to the wind being deflected upward by the side barriers. When in the leeward lane the wind strikes the side of the truck at a higher location and this has the effect of reducing the side load on the truck and consequently the rollover moment.
6. Discussion The present computational method uses a finite number of grid nodes and a finite computational domain to approximate a continuous and practically unbounded flow field. To determine the accuracy of these approximations parametric studies were performed and the primary results may be summarized as follows. Fig. 13 shows the effect of varying the number of nodes from 124 806 to 550 000 on the aerodynamic forces and rollover moment for the case where the wind is at a yaw angle of 451 to the longitudinal axis of the truck. A computational domain height of 40 m was used. Clearly the solution for rollover moment is approaching an asymptote for a grid of 550 000 nodes (as was used in all the results presented) and could be expected to be within a few percent of the continuous solution using the present level of eddy viscosity. Fig. 13b shows that the lack of convergence of rollover moment is entirely due to the lift force, which makes a small contribution to the rollover moment.
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588 250
80 (a)
(b)
200
60 50
150
Force
Rollover Moment (kN*m)
70
100
40 30 20
50
10 0
0 0
100
200 300 400 Grid nodes (Thousands)
500
600 Lift
0
100
Side-Force
200 300 400 Grid Nodes (Thousands)
500
600
Drag
Fig. 13. Effect of grid size on (a) rollover movement (b) forces acting on truck.
250
Rollover Moment (kN*m)
200
150
100
50
0 0
20
40
60 80 Domain height (m.)
100
120
Fig. 14. Effect of increasing domain height on the rollover moment.
The effect of varying the computational domain height from 40 to 100 m on the calculated rollover moment (bridge is always centred in the domain) is shown in Fig. 14. The convergence of the solution (here for a fixed number of 124 806 nodes) requires a domain height of more than 80 m. The results presented in this article use a computational domain height of 40 m and might be expected to underestimate the results for an infinite domain by approximately 5%. The results presented in Table 2 can be reduced to non-dimensional form as shown in Table 3. These non-dimensional results apply specifically to the present geometry but depend only on the angle of the relative wind, a; and the Reynolds number, Re ¼ rVr L=m; and not on the air properties or wind speed independently. They therefore have considerable generality, which may be further extended if the requirement of fixed Re is relaxed somewhat.
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Table 3 Non-dimensional lift, drag and moment coefficients for various relative wind directions CD
CL
CM
Truck in windward lane 45.0 1.34 50.2 1.44 56.3 1.33 63.4 1.47 71.6 1.30 80.5 1.41 90 1.56
0.56 0.44 0.34 0.13 0.02 0.04 0.02
0.41 0.49 0.49 0.48 0.37 0.42 0.40
0.96 1.05 1.01 1.08 0.95 1.04 1.12
Truck in leeward lane 45.0 50.2 56.3 63.4 71.6 80.5 90
0.82 0.69 0.38 0.20 0.07 0.07 0.05
0.15 0.20 0.23 0.24 0.24 0.17 0.21
0.68 0.76 0.77 0.72 0.68 0.70 0.73
CS
a
2
0.95 1.03 1.02 0.90 0.83 0.88 0.90
2
(a)
(b)
1
1
0
50
60
α
70
80
90
0
50
60
α
70
80
90
Fig. 15. Variation of CL (m), CS (), CD (.), and CM (’) with relative wind direction: (a) Truck is in windward lane. (b) Truck is in leeward lane.
In order to illustrate the relationship between truck speed and the aerodynamic forces and moment acting on the truck, the dimensionless coefficients have been plotted against a for both the windward and leeward cases in Fig. 15. It shows that the coefficient of side force, lift and rollover moment are fairly constant within the range of a ¼ 4512901: Naturally the coefficient of drag force tends to zero at 901. The most significant variation amongst force and moment coefficients lies between those for the windward and leeward lanes. In order to illustrate the generalization of the non-dimensional results, Fig. 16 was created for winter conditions with a density of 1.40 kg/m3. It shows contours of the aerodynamic moment on a plane of wind speed and the ratio of truck speed to wind
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590
25 0
140 20 0
120
30 0
25 0 15 0
20 0
100 Wind Speed (km/hr)
15 0
100
80 100 50
60 50 25
40
25 10
10
20
0 0.0
0.1
0.2
0.3
0.4 0.5 0.6 (Truck Speed)/(Wind Speed)
0.7
0.8
0.9
1.0
Fig. 16. Contours of aerodynamic moment in kN m; r ¼ 1:40 kg/m3. Solid lines are for the windward case. Dashed lines are for the leeward case.
speed. One can see that at relatively low truck speeds the aerodynamic moment depends only on wind speed whereas at higher truck speeds it depends strongly on wind speed and truck speed. Furthermore the moments for a truck travelling in the windward lane are substantially higher than when it is travelling in the leeward lane. The mass of an empty 18-wheeled tractor semi-trailer with a dual axle trailer, similar to the model used in the present study, is 14 810 kg. The critical moment, which would allow a static rollover of the tractor-trailer, is therefore about 190 kN m. It can be seen in the figures that a moment of this magnitude is certainly obtainable for this range of wind speeds and truck speeds. A fully loaded truck has a maximum mass of 39 500 kg and a static rollover moment of 504 kN m. It appears that for all practical conditions static rollover is not possible in this case. A complete dynamic rollover analysis would be required to obtain accurate safety limits. Overall, the present simulations might be considered as fairly crude because of their poor resolution of space and time features of the flow and the simplistic treatment of turbulence. Nevertheless a favourable comparison can be made between the scale model wind tunnel testing reported by King et al. [5] who used the present bridge geometry and measured the rollover moment for the case of zero truck speed. They found CM ¼ 0:720:9 for the windward case and 0.3–0.9 for the leeward case, with the higher values corresponding to low wind turbulence levels and the lower values to high wind turbulence. The comparable values from the present work are
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taken from Table 3 to be for the windward case, CM ¼ 1:12 and for the leeward case, 0.73. These are comparable to the measurements but the effect of turbulence, which was not accounted for, is profound, especially in the leeward case. While this comparison provides a partial validation of the present method, it also exposes a rather severe limitation. Baker [2] reported a summary of side force coefficients for scale model experimental studies of open road conditions under a variety of experimental conditions. When adjusted to the present definition, these give an average value of CS ¼ 0:75 for a ¼ 4512901: The present calculated values of CS for both the windward lane and leeward lanes are higher. This discrepancy might be expected due to the acceleration of the air around the bridge and side barrier prior to its impact on the vehicle. The corresponding values of CL reported by Baker [2] were found to strongly depend on Reynolds number, free-stream turbulence level and yaw angle making a comparison difficult. However, a particular finding of Coleman and Baker [9] was that the lift force was a maximum for a ¼ 451 and this was associated with a stable vortex, similar to that found on a delta wing at a high angle of incidence, present on the top of the trailer. Fig. 11 shows such a vortex, and the associated lowpressure region, on top of the trailer in the windward case.
7. Conclusion Estimates of the aerodynamic forces acting on a tractor-trailer vehicle crossing a bridge under cross-wind conditions were determined by methods of CFD. In particular, the dependence of these forces on the truck speed and lane of travel were explored. The results show qualitative agreement with the scale model wind tunnel testing but they should be regarded with some caution considering the crude approximations employed. Of these, it appears that the most serious limitation of the model is the inadequate representation of free stream wind turbulence, which was shown to have a strong effect on the aerodynamic forces measured in the scale model wind tunnel testing. The computational results show that for a fixed cross-wind speed the rollover moment increased with truck speed to nearly double its value for a yaw angle of 451. This is true for a truck travelling in either the windward or leeward bridge lanes, although the rolling moment experienced in the leeward lane is 30% less than is experienced in the windward lane.
Acknowledgements The authors gratefully acknowledge Flomerics, which provided the Flovent program [6] and Strait Crossing Inc., which provided the study by J.P.C. King, et al. [5]. Financial support for this study was provided by NSERC OGP’s 46192 and 8859. Mr. Mark Bettle performed the calculations required to evaluate convergence of the solution with respect to grid and domain size.
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References [1] R. Sigbjornsson, J.T. Snaebjornsson, Probabilistic assessment of wind related accidents of road vehicles: a reliability approach, J. Wind Eng. Ind. Aerodyn. 74-76 (1998) 1079–1090. [2] C.J. Baker, The effects of high winds on vehicle behaviour, A. Larsen, S. Esdahl (Eds.), Proceedings of the International Symposium on Advances in Bridge Aerodynamics, Copenhagen, Denmark, Balkema, Rotterdam, May 10–13, 1998. [3] J.C. Tannehill, D.A. Anderson, R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, 2nd Edition, Taylor and Francis, Philadelphia, PA, 1997, pp. 267-282. [4] D. Quesnel, Evaluating the Impact of Wind on the Confederation Bridge, MScE Thesis, Department of Civil Engineering, University of New Brunswick, 1999. [5] J.P.C. King, M.J. Mikitiuk, A.G. Davenport, N. Isyumov, A Study of Wind Effects for The Northumberland Straits Crossing, BLWT-SS8-1994, Faculty of Engineering Science, The University of Western Ontario, Canada, May 1994. [6] Flomerics Limited, How to Use FLOVENT, Document No. FLOVENT/LC/1091 Issue 1.0, 1991. [7] S.B. Pope, Turbulent Flow, Cambridge University Press, Cambridge UK, 2000. [8] Interjurisdictional committee on Vehicle Weights and Dimensions HeavyTruck Weight Dimension Regulations for Interprovincial Operations in Canada. The Federal-Provincial-Territorial Memorandum of Understanding on Interprovincial Weights, Dimensions, Ottawa, ON, 1995. [9] S.A. Coleman, C.J. Baker, An experimental study of the aerodynamic behaviour of high sided lorries in cross winds, J. Wind Eng. Ind. Aerodyn. 53 (1994) 401–429.
A computational study of the aerodynamic forces acting on a tractor-trailer vehicle on a bridge in cross-wind J. Bettle, A.G.L. Holloway*, J.E.S. Venart Department of Mechanical Engineering, University of New Brunswick, Box 4400, Fredericton NB, Canada E3B 5A3 Received 23 April 2001; received in revised form 6 September 2002; accepted 11 November 2002
Abstract This paper examines the effect of truck speed on the aerodynamic forces acting on a standard sized, North American transport truck travelling across a bridge under conditions of cross-wind. The objective is to establish a relationship between wind speed, truck speed and propensity for truck rollover that may be used to devise strategies for accident avoidance. Conditions of a moving truck travelling in both windward and leeward lanes were considered with a cross-wind speed of 120 km/h and truck speeds of 0–120 km/h. Using the calculated pressure distributions on the surface of the truck; the aerodynamic lift, drag and moment coefficients were determined for relative wind directions. The results show that the aerodynamic moment tending to overturn a truck in the windward lane of the bridge rises from approximately 120 kN m at low speeds (0–40 km/h) to 217 kN m at a truck speed of 120 km/h. For a truck in the leeward lane, the corresponding moments are substantially less, at 82 and 154 kN m, respectively. The 1.1 m barrier wall along the side of the bridge is a contributing factor to the aerodynamic difference between windward and leeward lanes. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Bridge and tractor-trailer vehicle aerodynamics; Computational fluid dynamics
1. Introduction Bridges and open highways often are exposed to crosswinds that can cause driveability problems for high-sided vehicles and in some cases result in serious *Corresponding author. Tel.: 506-447-3108; fax: 506-453-5025. E-mail address: [email protected] (A.G.L. Holloway). 0167-6105/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0167-6105(02)00461-0
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accidents [1,2]. Baker [2] has described a comprehensive approach to the prediction of such accident conditions that considers wind variation, vehicle geometry and speed, suspension dynamics and driver reaction. An essential element of this method is the relationship between wind speed, truck speed, and aerodynamic forces for a particular vehicle and site. Traditionally these aerodynamic characteristics have been studied by scale model wind tunnel testing [2]. Alternatively, the present paper uses the method of computational fluid dynamics (CFD) [3] for this purpose. Specifically CFD is used to evaluate the aerodynamic forces on a tractor-trailer vehicle crossing a bridge under windy conditions. This appears to be a novel application of the method. The bridge geometry chosen for this study is similar to the Confederation Bridge, which opened in 1997, and crosses the Northumberland Strait, joining Cape Jourimaine, New Brunswick to Borden, Prince Edward Island, Canada. It is 12.9 km in length, 11 m across the deck, and has an average height above the water of 40 m. The highest point is located at the navigational span, 60 m above sea level, where the magnitude and direction of the wind is continually measured by bridge operators using a stationary anemometer. The direction of the wind has been found to be predominantly perpendicular to the bridge with 6 min averaged speeds as high as 148.6 km/h and gusting as high as 170.3 km/h [4]. It should be noted that this bridge has the distinct feature of a 1.1 m high barrier, which runs the full length of the bridge on both sides of the road deck. The vehicle is a standard North American tractor-trailer with a 2.92 m high, 14.63 m long, and 2.60 m wide trailer that is raised 1.19 m above the road surface. The model geometry with a few overall dimensions can be seen in Fig. 1. The effect of cross-wind on the aerodynamic forces is very significant. Travelling through still air the drag force acts on the relatively small frontal area of the truck and is directed nearly parallel to its longitudinal axis. However in the presence of a cross-wind the magnitude of the relative velocity between the truck and the air is increased, and its direction is skewed to the direction of motion so that the large side area of the trailer is exposed. The resulting drag force is much larger and it has components both parallel and normal to the direction of the truck’s motion. Substantial lift can also be generated in cases of separated flow over the top of the trailer. The component of drag normal to the longitudinal axis of the truck, referred
8.5 m 1.1m
11 m Fig. 1. Model geometry and overall dimensions.
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to as side force, and the lift force, are particularly important to drivability because they generate a moment about the longitudinal axis of the vehicle, reducing the wheel loading on the windward side. An experimental study of wind forces on trucks on the Confederation Bridge was conducted by King et al. [5]. The wind tunnel testing was performed on a (1/60)th scale with pure cross-wind conditions. The only measurements taken were of the wind speed required to cause rollover of the stationary truck models. Nevertheless, they provided coefficients for the rolling moment based on the model weight. Experimental studies of the aerodynamic forces acting on trucks subjected to winds on an open roadway have been summarized by Baker [2]. Typically these were conducted on (1/50)th scale models for wind directions ranging from 01 to 901 of yaw and for various wind turbulence levels. Among the measurements reported were the time average side force, lift and drag, and the pitching, rolling and yawing moments acting on the model trucks.
2. Method The present analysis has been performed using a commercial CFD program called Flovent [6], a product of Flomerics. The computational domain has three space dimensions and time. Spatial discretization is done using the control volume approach on a staggered, structured, non-uniform grid with convective terms treated using a weighted average of first order accurate upwinding and second order accurate central differencing according to the limits of numerical stability. The method is robust but does introduce numerical errors of a diffusive nature where the flow is strongly convective and skewed to the coordinate direction. Temporal discretization is uniformly spaced and uses an implicit first-order differencing method. Details of the numerical method can be found in Refs. [6,3]. The present flow problem is inherently unsteady, not only because of the flow turbulence, but also because of the large scale vortex shedding present in the separated wake behind the truck and bridge. The turbulence away from the boundaries was modelled by a constant eddy viscosity of 1000 times the molecular value for air under standard conditions. Near the solid boundaries a log law was used to calculate the wall shear stress from the velocity at the closest node to the boundary. A more complex model of turbulence, the k e model [7], was also tested on this problem with the result of increasing the number of steps required for convergence several fold and changing aerodynamic drag by only a few percent. Consequently, all solutions presented were calculated using the constant eddy viscosity model of turbulence. More recently formulated versions of the standard k e model (for example RNG) and Reynolds Stress Models [7] were not available for testing but it should be expected that such methods would make incremental improvements to the turbulent viscosity and k e model results under the present circumstances. A transient solution method was used; however, such solutions were found to develop into a steady form for both turbulence models and for a range of time step sizes as small as 100th of the vortex-shedding period. That a steady
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solution is achieved under such conditions is undoubtedly the result of the diffusive nature of the turbulence model and numerical scheme; both of which are imprecise in this regard. In fact, turbulence models such as the ones employed here are calibrated to match the time average velocity field; even in wake flows. The aerodynamic forces presented are based on these steady solutions and are therefore regarded as time average values. Unfortunately, they do not provide information about the important force fluctuations due to vortex shedding or wind gusting. The computational domain is physically very large and the number of grid nodes available to fill it was limited to 550 000 for the computer available. The mesh was more concentrated around the truck and bridge but still the smallest computational cell had a volume of 0.027 m3. The sides of which were many times the boundary layer thickness and finer features of the truck geometry. In effect, the truck and trailer are modelled as a series of solid ‘‘blocks’’ with sharp edges. As a consequence, flow separation can be expected to occur at these edges independent of the Reynolds number or boundary layer development of the flow. This is not too far from reality for this type of vehicle, which have little streamlining. A study describing the effects of domain size and mesh size is provided in the discussion. Atmospheric conditions vary widely in temperature and humidity with corresponding variations in density and viscosity. The air properties used in the present simulation were for standard conditions of 101 kPa and 201C with a density of 1.16 kg/m3 and a kinematic viscosity of 15 106 m2/s. The present calculations were all performed in a frame of reference moving with the truck, so that the boundary conditions are steady. From this viewpoint, riding in the truck, it is the angle of yaw, a; and the magnitude of the relative wind, which changes with the truck speed as shown in Fig. 2. For example, with a constant crosswind of 120 km/h and zero truck speed, the relative wind is normal to the longitudinal axis of the truck, which will in the present context will be taken as the 901 angle of yaw. For a truck speed of 120 km/h the relative wind is at 451 and 41% greater in magnitude. Further increases in truck speed swings the relative wind direction towards the line of motion of the truck. The bridge deck would also be moving relative to the truck in this frame of reference. However, to be consistent with wind tunnel testing and to simplify the present simulation, the relative velocity
Wind Velocity, VW
Truck Velocity VT
α
Relative Velocity VR
Fig. 2. Wind velocity relative to the truck.
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between the truck and road was set to zero. This approximation should not cause large errors in the present computations because of the two-dimensional nature of the bridge geometry used in the model. This is not an inherent limitation of CFD but is a limitation of the commercial package used. The aerodynamic forces acting on the truck consist of a side force, S; a lift force, L; and a longitudinal drag force, D: The simulations provide estimates of both the magnitude and the line of action of each of these forces by simply summing the pressure forces and moments acting on the vertical and horizontal surfaces of the truck and trailer. The skin friction, which is poorly resolved, is not included in the wind force calculations. The aerodynamic forces may be generalized by rendering them non-dimensional with the geometric and fluid properties of the model to give CL ; CS ; and CD ; which are defined as L CL ¼ 1 2 ; ð1Þ 2rVR AL S CS ¼ 1 2 ; 2rVR AS D : 2 rV R AD 2
CD ¼ 1
ð2Þ ð3Þ
r is the density of the air, 1.16 kg/m3, VR is the speed relative to the truck, AL is the top surface area of the trailer, 38.0 m2, AS is the side area of the trailer, 42.7 m2 , and AD is the frontal area, 9.386 m2. An additional aerodynamic property of the tractortrailer is the moment tending to roll the truck over. This is calculated using the lift and drag forces described above and by summing moments about the point of contact between the leeward tire and the road (point O in Fig. 3). This aerodynamic moment is calculated using M0 ¼ Sdy þ Ldx;
ð4Þ
where dy and dx are defined in Fig. 3 as the position of the centre of pressure of the respective surfaces. This is strictly the moment due to wind forces and it does not take into account the weight of the vehicle or its inertia. The non-dimensional form of the moment is M0 CM ¼ ; ð5Þ 1 2 rVR AS hv 2 where hv represents the overall vertical height of the trailer, 4.11 m. A total of 14 simulations were run to determine a relationship between truck speed and the aerodynamic forces and moments acting on the truck as it travels across the bridge during high winds. Simulations were run with a constant wind speed of 120 km/ h and truck speeds of 0, 20, 40, 60, 80, 100 and 120 km/h. This varied a (see Fig. 2) between 901 and 451. Separate simulations were run with the truck in the windward and leeward lanes. A summary of the simulation conditions is shown in Table 1.
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L
δx
x
S
hv
δy
O Road Fig. 3. Definition of the rollover moment due to wind forces.
Table 1 Simulation conditions Wind speed, VW (km/h)
Truck speed, VT (km/h)
Relative speed, VR (km/h)
a (deg.)
120 120 120 120 120 120 120
120 100 80 60 40 20 0
170 156 144 134 126 122 120
45.0 50.2 56.3 63.4 71.6 80.5 90
3. Geometry and boundary conditions The three-dimensional computational model of a transport truck on a section of the Confederation Bridge, created for this study, can be seen in Fig. 4. The bridge was modelled as a 60 m long section with a constant depth. All the truck dimensions fall within the Canadian standard geometric design limits for an 18-wheeled tractor-trailer with a 14.63 m (480 ) dual axle trailer [8]. The overall length and the total height of the model truck are 20.15 m and 4.11 m, respectively. The boundary conditions used in the simulations are shown in Fig. 5. The domain boundaries above, below, to the right, and behind the model truck and bridge are ‘‘free’’ outflow boundaries; that is, they are taken as surfaces of constant pressure to
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30 20 10 0
20
40 60 X (m)
80
100
60
40 20 Z (m)
0
Y (m)
40
0
Fig. 4. Three principal views of the model geometry used in the simulations.
Front - Uniform In Flow
LeftUniform In Flow
Right Free Boundary
Back - Free Boundary Fig. 5. The flow boundary condition used in the simulations.
permit wake formation. Note that the water surface is not included in the simulation and that the bridge extends fully across the domain so that no end leakage is permitted where it intersects the boundary surface. The remaining domain boundaries, the one to the left and the one in front of the truck, are upwind of the model and are set to uniform wind profiles with the correct relative magnitude and direction.
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Y
Z
X
Fig. 6. Computational grid in the vicinity of the truck and bridge.
4. Computational details The overall computational domain used for the simulations, as depicted in Fig. 4, is 100 m 40 m 60 m. A non-uniform grid was created within this domain. The grid used contained 550,000 nodes (100 55 100). A section of the grid in the vicinity of the truck, which is the region of the smallest cells, can be seen in Fig. 5. All the cells in this area are about 0.027 m3 (approx. 1 ft3) (Fig. 6).
5. Results A sample of the computational results with a ¼ 451 is shown in Figs. 7–12. Figs. 7 and 8 show contours of speed in the horizontal and vertical planes, which pass through the mid-plane of the truck for the windward and leeward simulations. The speed contours clearly show that the air is accelerated as it passes over and under the bridge (and truck) and around the ends of the truck (red regions). Peak speeds reach more than 60 m/s. The blue regions depict the low speed wake directly behind the bridge and truck where the air speed is close to the wind speed of 30 m/s. The 1.1 m barrier on the sides of the bridge deck reduced the airflow under the truck compared to what may occur on an open road. The gauge pressure contours (Figs. 9 and 10) show that a high pressure (red) region is created on the windward side of the truck and bridge, while a low pressure (blue) region is created on all other sides of the truck. Clearly the wake is larger for the truck in the windward lane and the relatively low pressure on the leeward side and the top of the truck in both cases produce a very substantial drag and lift forces.
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S P EED:
0
10
S P EED:
20
0
10
30
40
20
30
581
50
40
60
50
60
Fig. 7. Contours of speed from the simulation results corresponding to a windward solution with a ¼ 451: The dashed lines show the location of the Bridge. Speed is in m/s.
Fig. 11 gives a complete description of the surface pressure on all sides of the truck. The extreme pressures (which cannot be shown in Figs. 9 and 10) were calculated to be as high as 1000 Pa on the windward side of the truck and as
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S P EED:
0
10
S P EED:
20
0
10
30
40
20
30
50
40
60
50
60
Fig. 8. Contours of speed from the simulation results corresponding to a leeward solution with a ¼ 451: The dashed lines show the location of the Bridge. Speed is in m/s.
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583
Fig. 9. Contours of gauge pressure calculated from the windward simulation results for a ¼ 451: The dashed lines show the location of the Bridge. Pressure is in Pa.
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Fig. 10. Contours of gauge pressure calculated from the leeward simulation results for a ¼ 451: The dashed lines show the location of the Bridge. Pressure is in Pa.
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585 P 1000 500
Left (Windward) Surface
0 -500 -1000
Top Surface
Back Surface
Right (Leeward) Surface
Front Surface
Bottom Surface P 1000 500
Left (Windward) Surface
0 -500 -1000
Top Surface
Back Surface
Right (Leeward) Surface
Front Surface
Bottom Surface
Fig. 11. Distribution of pressure on the surface of the tractor and trailer as calculated from the windward and leeward simulations with a ¼ 451: Pressure is in Pa.
low as 1000 Pa on the top of the truck. These pressure distributions were used to calculate the wind forces for each simulation case. A summary of these forces and their corresponding rollover moment are provided in Table 2 for all the simulation results. The magnitudes of moments listed would be very significant to the dynamics of an unloaded trailer. Fig. 12 shows the pressure and the velocity fields in the immediate vicinity of the truck and bridge. The regions of low pressure are centred on vortices and the regions
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S P EED: 10
20
30
40
50
60
P : -1000
-500
0
500
1000
SPEED: 10
20
30
40
50
60
P: -1000
-500
0
500
1000
Fig. 12. Pressure and velocity fields on a vertical plane which passes through the mid-plane of the trailer for both the windward and leeward simulations with a ¼ 451:
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Table 2 Aerodynamic forces and moments acting on a tractor-trailer for various truck speeds. Wind speed is held constant at 120 km/h Truck speed (km/h)
S (kN)
D (kN)
L (kN)
M (kN m)
Truck in windward lane 120 100 80 60 40 20 0
73.7 67.4 53.1 50.7 39.9 39.8 42.9
6.78 4.52 3.00 0.996 0.103 0.230 0.141
20.1 20.2 17.4 14.7 9.99 10.6 9.78
217 202 165 153 120 121 127
Truck in leeward lane 120 100 80 60 40 20 0
52.5 48.2 40.7 31.0 25.4 24.9 24.9
9.91 7.13 3.32 1.53 0.440 0.434 0.315
7.52 8.50 8.29 7.43 6.63 4.33 5.10
154 146 127 102 86.2 81.5 82.4
of high pressure on stagnation points. A vortex on the top of the trailer, at the point of separation, is visible though not well resolved. The differences between the speed and pressure contours surrounding a truck travelling in the windward lane and leeward lanes is largely due to the wind being deflected upward by the side barriers. When in the leeward lane the wind strikes the side of the truck at a higher location and this has the effect of reducing the side load on the truck and consequently the rollover moment.
6. Discussion The present computational method uses a finite number of grid nodes and a finite computational domain to approximate a continuous and practically unbounded flow field. To determine the accuracy of these approximations parametric studies were performed and the primary results may be summarized as follows. Fig. 13 shows the effect of varying the number of nodes from 124 806 to 550 000 on the aerodynamic forces and rollover moment for the case where the wind is at a yaw angle of 451 to the longitudinal axis of the truck. A computational domain height of 40 m was used. Clearly the solution for rollover moment is approaching an asymptote for a grid of 550 000 nodes (as was used in all the results presented) and could be expected to be within a few percent of the continuous solution using the present level of eddy viscosity. Fig. 13b shows that the lack of convergence of rollover moment is entirely due to the lift force, which makes a small contribution to the rollover moment.
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588 250
80 (a)
(b)
200
60 50
150
Force
Rollover Moment (kN*m)
70
100
40 30 20
50
10 0
0 0
100
200 300 400 Grid nodes (Thousands)
500
600 Lift
0
100
Side-Force
200 300 400 Grid Nodes (Thousands)
500
600
Drag
Fig. 13. Effect of grid size on (a) rollover movement (b) forces acting on truck.
250
Rollover Moment (kN*m)
200
150
100
50
0 0
20
40
60 80 Domain height (m.)
100
120
Fig. 14. Effect of increasing domain height on the rollover moment.
The effect of varying the computational domain height from 40 to 100 m on the calculated rollover moment (bridge is always centred in the domain) is shown in Fig. 14. The convergence of the solution (here for a fixed number of 124 806 nodes) requires a domain height of more than 80 m. The results presented in this article use a computational domain height of 40 m and might be expected to underestimate the results for an infinite domain by approximately 5%. The results presented in Table 2 can be reduced to non-dimensional form as shown in Table 3. These non-dimensional results apply specifically to the present geometry but depend only on the angle of the relative wind, a; and the Reynolds number, Re ¼ rVr L=m; and not on the air properties or wind speed independently. They therefore have considerable generality, which may be further extended if the requirement of fixed Re is relaxed somewhat.
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Table 3 Non-dimensional lift, drag and moment coefficients for various relative wind directions CD
CL
CM
Truck in windward lane 45.0 1.34 50.2 1.44 56.3 1.33 63.4 1.47 71.6 1.30 80.5 1.41 90 1.56
0.56 0.44 0.34 0.13 0.02 0.04 0.02
0.41 0.49 0.49 0.48 0.37 0.42 0.40
0.96 1.05 1.01 1.08 0.95 1.04 1.12
Truck in leeward lane 45.0 50.2 56.3 63.4 71.6 80.5 90
0.82 0.69 0.38 0.20 0.07 0.07 0.05
0.15 0.20 0.23 0.24 0.24 0.17 0.21
0.68 0.76 0.77 0.72 0.68 0.70 0.73
CS
a
2
0.95 1.03 1.02 0.90 0.83 0.88 0.90
2
(a)
(b)
1
1
0
50
60
α
70
80
90
0
50
60
α
70
80
90
Fig. 15. Variation of CL (m), CS (), CD (.), and CM (’) with relative wind direction: (a) Truck is in windward lane. (b) Truck is in leeward lane.
In order to illustrate the relationship between truck speed and the aerodynamic forces and moment acting on the truck, the dimensionless coefficients have been plotted against a for both the windward and leeward cases in Fig. 15. It shows that the coefficient of side force, lift and rollover moment are fairly constant within the range of a ¼ 4512901: Naturally the coefficient of drag force tends to zero at 901. The most significant variation amongst force and moment coefficients lies between those for the windward and leeward lanes. In order to illustrate the generalization of the non-dimensional results, Fig. 16 was created for winter conditions with a density of 1.40 kg/m3. It shows contours of the aerodynamic moment on a plane of wind speed and the ratio of truck speed to wind
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590
25 0
140 20 0
120
30 0
25 0 15 0
20 0
100 Wind Speed (km/hr)
15 0
100
80 100 50
60 50 25
40
25 10
10
20
0 0.0
0.1
0.2
0.3
0.4 0.5 0.6 (Truck Speed)/(Wind Speed)
0.7
0.8
0.9
1.0
Fig. 16. Contours of aerodynamic moment in kN m; r ¼ 1:40 kg/m3. Solid lines are for the windward case. Dashed lines are for the leeward case.
speed. One can see that at relatively low truck speeds the aerodynamic moment depends only on wind speed whereas at higher truck speeds it depends strongly on wind speed and truck speed. Furthermore the moments for a truck travelling in the windward lane are substantially higher than when it is travelling in the leeward lane. The mass of an empty 18-wheeled tractor semi-trailer with a dual axle trailer, similar to the model used in the present study, is 14 810 kg. The critical moment, which would allow a static rollover of the tractor-trailer, is therefore about 190 kN m. It can be seen in the figures that a moment of this magnitude is certainly obtainable for this range of wind speeds and truck speeds. A fully loaded truck has a maximum mass of 39 500 kg and a static rollover moment of 504 kN m. It appears that for all practical conditions static rollover is not possible in this case. A complete dynamic rollover analysis would be required to obtain accurate safety limits. Overall, the present simulations might be considered as fairly crude because of their poor resolution of space and time features of the flow and the simplistic treatment of turbulence. Nevertheless a favourable comparison can be made between the scale model wind tunnel testing reported by King et al. [5] who used the present bridge geometry and measured the rollover moment for the case of zero truck speed. They found CM ¼ 0:720:9 for the windward case and 0.3–0.9 for the leeward case, with the higher values corresponding to low wind turbulence levels and the lower values to high wind turbulence. The comparable values from the present work are
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taken from Table 3 to be for the windward case, CM ¼ 1:12 and for the leeward case, 0.73. These are comparable to the measurements but the effect of turbulence, which was not accounted for, is profound, especially in the leeward case. While this comparison provides a partial validation of the present method, it also exposes a rather severe limitation. Baker [2] reported a summary of side force coefficients for scale model experimental studies of open road conditions under a variety of experimental conditions. When adjusted to the present definition, these give an average value of CS ¼ 0:75 for a ¼ 4512901: The present calculated values of CS for both the windward lane and leeward lanes are higher. This discrepancy might be expected due to the acceleration of the air around the bridge and side barrier prior to its impact on the vehicle. The corresponding values of CL reported by Baker [2] were found to strongly depend on Reynolds number, free-stream turbulence level and yaw angle making a comparison difficult. However, a particular finding of Coleman and Baker [9] was that the lift force was a maximum for a ¼ 451 and this was associated with a stable vortex, similar to that found on a delta wing at a high angle of incidence, present on the top of the trailer. Fig. 11 shows such a vortex, and the associated lowpressure region, on top of the trailer in the windward case.
7. Conclusion Estimates of the aerodynamic forces acting on a tractor-trailer vehicle crossing a bridge under cross-wind conditions were determined by methods of CFD. In particular, the dependence of these forces on the truck speed and lane of travel were explored. The results show qualitative agreement with the scale model wind tunnel testing but they should be regarded with some caution considering the crude approximations employed. Of these, it appears that the most serious limitation of the model is the inadequate representation of free stream wind turbulence, which was shown to have a strong effect on the aerodynamic forces measured in the scale model wind tunnel testing. The computational results show that for a fixed cross-wind speed the rollover moment increased with truck speed to nearly double its value for a yaw angle of 451. This is true for a truck travelling in either the windward or leeward bridge lanes, although the rolling moment experienced in the leeward lane is 30% less than is experienced in the windward lane.
Acknowledgements The authors gratefully acknowledge Flomerics, which provided the Flovent program [6] and Strait Crossing Inc., which provided the study by J.P.C. King, et al. [5]. Financial support for this study was provided by NSERC OGP’s 46192 and 8859. Mr. Mark Bettle performed the calculations required to evaluate convergence of the solution with respect to grid and domain size.
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