A study of the time-resolved structure of the vortices shed into the wake of an isolated F1 car wheel

Abstract

This paper describes the flow about a smooth isolated wheel rolling steadily along a plane surface. The shape chosen is typical of those used for Formula 1 racing cars, and it is fitted with an inflatable tyre to ensure that an appropriate contact patch geometry is realised. Single-pass PIV is used to obtain instantaneous records of the velocity field around the wheel, and examples of the detailed wake structure are presented. The evolution of the vortices generated by the wheel and shed into the wake is studied for a range of yaw angles up to 6°. It is shown that the structure of the wake for the unyawed wheel fluctuates considerably, switching between distinct states where a dominant lower ground vortex is observed on either the left or the right side of the wake or a third state where a symmetrical pair of vortices forms. Yawing the wheel, even by as little as 2°, which will frequently arise in practical situations due to cross winds or steering the vehicle, biases the flow towards the leeside which inhibits the switching. In such a case, a stable dominant leeside lower ground vortex is formed and the wake structure changes significantly. The origin of the ground vortex formation ahead of the wheel is identified as vorticity coming from the boundary layer generated on the front face of the tyre and not the “jetting phenomenon” as is widely assumed.

Graphical abstract

Appendices

Appendix 1: The Γ₂ variable

Within a flow, Γ₂ indicates the degree of rotation about a point P by considering all the velocity vectors in a region S surrounding P containing N grid points after first subtracting the average drift velocity UP within the region. Γ₂ was devised by Graftieaux et al. (2001) and is a non-local variable that does not involve differentiation. As such it is generally better at identifying vortices from an array of velocity vectors than the vorticity which is susceptible to noise within the data. It is directly based on the velocity field topology without using velocity derivatives. A visual interpretation of the function is depicted in Fig. 

Fig. 22

Gamma Criterion calculation

22.

Γ₂ is the mean of the sine of the angles between the radial vector PM and the flow velocity vectors after the removal of the average drift velocity U_P. As such it can vary between − 1 and 1, with 1 representing all vectors laying at 90° to the radial vector and − 1 representing 270°. Zero indicates that there is no swirl in this plane.

Simpson et al. (2018) discusses the effectiveness of applying the Γ₂ method to PIV measurements and demonstrated its effectiveness in analysing single-shot data. For the present data, an interrogation window 11 by 11 data points (that is 0.09 D square) has been selected giving the best balance between in-plane averaging and resolution, taking into account the scale of typical vortices observed in these tests. Contours of |Γ₂ | > 0.3 only are shown since any structure below this value are unlikely to be part of any coherent vortices.

There are several alternative ways to identify vortical feature (De Gregorio and Visingardi 2020), a popular one being the method based on Q proposed by Hunt et al. (1988) which is the residual of the vorticity tensor norm squared subtracted from the strain-rate tensor norm squared,

$$Q = \frac{1}{2}\left( {\left\| {\overline{\overline{\Omega }} } \right\|_{F}^{2} - \left\| {\overline{\overline{S}} } \right\|_{F}^{2} } \right) > 0$$

where S and Ω are the symmetrical and asymmetrical parts of the velocity gradient tensor. This method can lead to ambiguities as strictly it also requires knowledge that the pressure is below the ambient value to ensure that it detects vortices unambiguously. Alternative to the Γ₂ criterion have not proved as effective in this study.

Simpson et al. (2018) showed that |Γ₂| values above 0.6 were capable of unambiguously detecting a simulated Lamb Oseen vortex when significant random noise was added to the vectors with a standard deviation of 30% of peak velocity. This noise rejection property of the Gamma Criterion is of great value when analysing PIV data, particularly that from single-pass tests which are inevitably more prone to noise than results from averaged multiple runs.

Appendix 2: Streamline segments

Short streamline segments that indicate the local crossflow direction are constructed as an aid in interpreting the Γ₂ contour plots. Figure 

Fig. 23

The streamline segment construction

23 illustrates this construction for a y – z plane located at a streamwise location x_Γ2. A streamwise interrogation length d_x is selected together with a grid of points in a y – z plane at $x={x}_{\gamma }-0.5{d}_{x}$ from which the streamline to be constructed originate. The velocity at each grid point is evaluated by interpolation from the measured PIV data. The streamline segments are then constructed incrementally from the starting point for a short distance δ, at which point the velocities are re-evaluated by interpolation and the process repeated until $x\; = \;x_{\gamma } + 0.5d_{x} .$ It should be noted that these are not true instantaneous streamlines; the model wheel moves past a stationary laser plane and thus data for incremental values of x are captured at slightly increasing values of time.