A Study of the Spatial Variation of Vehicle-Induced Turbulence on Highways Using Measurements from a Mobile Platform

Abstract

During July 2016, an on-road study was conducted in and around the Toronto, Canada region to investigate the spatial variation of vehicle-induced turbulence on highways. The power spectral density of turbulent kinetic energy (TKE) while following on-road vehicles is significantly enhanced for frequencies greater than 0.5 Hz. This increase is not present while driving isolated from traffic, demonstrating that TKE is enhanced considerably on highways in the presence of vehicles. The magnitude of normalized TKE is found to decay following a power-law relationship with increasing normalized distance behind on-road vehicles, which is most pronounced behind heavy-duty trucks. The results suggest that the TKE in the vehicle wake is maximized in the upper shear layer near the vehicle top. An extended parametrization is outlined that describes the total on-road TKE enhancement due to a composition of vehicles, which includes a vertical dependence on the magnitude of TKE.

Appendices

Appendix 1: Following Distance Calculation

Figure 11 Displays the pixel geometry and Fig. 12 shows the physical geometry. In Fig. 11, $P_{H}$ is the pixel distance between the attachment points on the hood of the sport-utility vehicle, (corresponding to the physical distance $Y_{H}$ in Fig. 12) and $P_{r}$ is the pixel width of the road (corresponding to the physical road width $Y_{r}$ in Fig. 12). Noting that the ratio between $P_{H}$ and the image width $W$ (Fig. 11) is equal to the ratio between $Y_{H}$ and the width of the field of view at this location, $Y_{W,H}$ (Fig. 12) gives

$$ \frac{{Y_{H} }}{{Y_{W,H} }} = \frac{{P_{H} }}{W}. $$

(9)

Fig. 11

A still image while vehicle chasing an HD-B truck on 12 July. Superimposed is the pixel geometry used to estimate the following distance. Automated software was developed to determine the value of $H_{f} + H_{H}$. The red circle represents the location of the vanishing point

Fig. 12

The basic geometry. Refer to Fig. 3 which displays the measured physical geometry (i.e. $X_{h}$ and $Y_{H}$)

Likewise, the ratio between $P_{r}$ (at the location of the vehicle’s shadow) and the image width $W$ (Fig. 11) is equal to the ratio between road width $Y_{r}$ and the width of the field of view at this location $Y_{W,r}$ (Fig. 12),

$$ \frac{{Y_{r} }}{{Y_{W,r} }} = \frac{{P_{r} }}{W}. $$

(10)

From Fig. 12, similar triangles give

$$ \frac{{Y_{W,H} }}{{X_{H} }} = \frac{{Y_{W,r} }}{{x_{m} }}, $$

(11)

where $x_{m}$ is the following distance and $X_{H}$ is the measured distance between the dashcam and the attachment points at the front of the vehicle (see Fig. 3). Using Eq. 9 and Eq. 10 in Eq. 11 yields

$$ \frac{{Y_{H} W}}{{P_{H} X_{H} }} = \frac{{Y_{r} W}}{{P_{r} x_{m} }}. $$

(12)

Noting that the highway intersects the image frame at X (see Fig. 11) which is also equal to image frame width, W, then

$$ \frac{Q}{W} = \frac{{H_{s} }}{{P_{r} }} \Rightarrow P_{r} = \frac{{WH_{s} }}{Q}, $$

(13)

where $H_{s}$ is the pixel difference between the vanishing point $H_{v}$ and the target distance $H_{f} + H_{H}$ (at location $P_{r}$) and Q is the distance in pixels between W (denoted $W/2$ in Fig. 11) and the vanishing point.

Substituting Eq. 13 into Eq. 12 gives

$$ \frac{{Y_{H} W}}{{P_{H} X_{H} }} = \frac{{Y_{r} Q}}{{H_{s} x_{m} }}. $$

(14)

If we assume that the calculated vanishing point is centred about W, then

$$ \tan \beta = \frac{Q}{W/2} \Rightarrow \frac{\tan \beta }{2} = \frac{Q}{W}, $$

(15)

and the following distance in physical units can be expressed as

$$ x_{m} = \frac{{Y_{r} P_{H} X_{H} \tan \beta }}{{2Y_{H} H_{s} }}. $$

(16a)

Since $H_{s} = H_{v} - H_{f} - H_{H}$, $x_{m}$ can be written as

$$ x_{m} = \frac{{Y_{r} P_{H} X_{H} \tan \beta }}{{2Y_{H} \left( {H_{v} - H_{f} - H_{H} } \right)}}. $$

(16b)

Finally, since $x_{m}$ is defined as the distance between the measurement location (i.e. the sonic anemometer) and the back-end of the target vehicle (i.e. its shadow), the measured distance between the dashcam and the anemometer (≈  $X_{H}$) needs to be removed from Eq. 16b, giving

$$ x_{m} = \frac{{Y_{r} P_{H} X_{H} \tan \beta }}{{2Y_{H} \left( {H_{v} - H_{f} - H_{H} } \right)}} - X_{H} . $$

(17)

In Eq. 17, $Y_{r}$ is estimated from a Google Earth satellite image. Since the terrain was generally flat and the dashcam was adhered to the windshield, a single calibration image is used to determine the pixel values of $H_{v}$, $H_{H}$, $P_{H}$ and $\beta$. Physically, $H_{H}$ includes the distance between the dashcam and the vehicle’s front end, and some of the highway surface immediately in front of the vehicle (it cannot be seen by the dashcam). Therefore, $H_{H}$ places a lower limit on the domain of $x_{m}$. To determine the target distance $H_{f}$, automated software was developed to locate the step change in greyscale values between the sunlit highway surface and the shadow behind the vehicle. It should be noted that Eq. 17 approaches infinite distance near the horizon (i.e. as $H_{f} + H_{H} \to H_{v}$), and therefore a small error in the pixel location near the horizon results in a large error in the following distance. Consequently, the measurement domain is limited to a maximum following distance of 100 m to limit error related to image resolution (except Sect. 5). Equation 17 was tested in a parking lot using distances measured up to 70 m. The results determined from Eq. 17 were generally within ± 1 m of the measured distances.

Appendix 2: Near-Road Spectral Details

See Table 5.

Table 5 Stationary measurements obtained near Highway 400

Appendix 3: On-Road Spectral Details

See Table 6.

Table 6 Details related to the spectra presented in Fig. 8, where x_m is the following distance, V is the vehicle speed, $u_{s}$ is the velocity component in the flow parallel to vehicle motion, $u_{s}^{\prime 2}$, $v_{r}^{\prime 2}$, and $w_{r}^{\prime 2}$ are the longitudinal, lateral and vertical velocity variances respectively and e is the turbulent kinetic energy