Wednesday, September 13, 2017

F1 1980 - Nozzles and streamtubes

Let’s delve a little more deeply into the nature of ground-effect downforce. The underbody of a ground-effect car can be treated as a type of (subsonic) converging-diverging nozzle. Such a nozzle consists of a mouth, a throat, and a diffuser. The mouth consists of a duct with a contracting cross-section, which accelerates air into the narrowest section, the throat. In accordance with the Bernoulli effect, the pressure is at its lowest in the throat, and the airflow velocity is at its highest. The air then flows from the throat into the diffuser, a duct with an expanding cross-section, which decelerates the air, and thereby returns it towards the freestream pressure, a process referred to as ‘pressure recovery’.

To give an illustration of the relative proportions here, the MKIV underbody on the Williams FW07B had a throat about 30 inches (762mm) in length, compared with a mouth only about 10 inches (254mm) long. The diffuser was about 45 inches (1143mm) in longitudinal extent.

Pressure recovery is a delicate process because it creates an ‘adverse pressure gradient’. The pressure increases in the direction of flow, hence there is a force pushing against the flow in the diffuser. Such an adverse pressure gradient tends to promote separation of the boundary layer. When separation occurs, the boundary layer is released into the interior of the fluid, where it breaks up into turbulence. This reduces the effective cross-sectional area and flow capacity of the diffuser, which in turn reduces the low pressure upstream at the throat. Separation also transforms a portion of the mean-flow kinetic energy into turbulent kinetic energy, which eventually dissipates as heat energy. To avoid separation, the diffuser tends to be much longer than the mouth and throat, with a more gradual slope than that between mouth and throat.

At a fixed freestream velocity (determined by the car-speed), the steady-state mass-flow rate through this nozzle is determined by the area of the diffuser outlet (assuming there is no separation), and by the ‘base pressure’* at the diffuser exit. The latter will be lower than the freestream pressure due largely to the low pressure created by the suction surface of the rear-wing, but also due to the low-pressure wake behind the car.

To understand this further, it’s useful to introduce the concept of a ‘streamtube’. This is defined by taking a closed loop in the flowfield, identifying the streamline which passes through each point of the loop, and extruding the loop along those streamlines. This defines the surface of the streamtube. By definition, because the surface of a streamtube is constructed from streamlines, the velocity field is tangent to the surface of the tube, hence no mass can flow through the surface. Moreover, in a steady flow the mass flow-rate is the same through any cross-section of the streamtube.

Now, whilst the underbody of a ground-effect car has a solid mouth, (defined in 1980 by the geometry of the sidepod inlets), the flow upstream of the mouth is not confined by solid walls. Instead, it is defined by the streamtube of the flow which enters each venturi tunnel.

At a fixed car-speed, the greater the exit area of the diffuser, and/or the lower the base pressure created by the rear-wing, the greater the cross-sectional area of the streamtubes feeding the sidepod inlets. The greater the cross-sectional area of the streamtube feeding the mouth of each venturi tunnel, the greater the contraction as the air enters the throat of the tunnel, hence the greater the acceleration of the air and the greater the pressure drop. Therefore, “the degree of expansion of the air in the diffuser rather than the physical dimensions of the mouth determines the effective contraction of air into the throat, hence the maximum airspeed that will be obtained,” (Ian Bamsey, The Anatomy and Development of the Sports Prototype Racing Car, Haynes, 1991, p63).

A principal concern in the design of the underbody mouth is the avoidance of separation. Depending upon the car-speed and the base-pressure, the stream-tubes entering the venturi tunnels may either expand or contract as they approach the mouth. There will be a stagnation line somewhere around the upper-lip of each mouth: flow below this line will enter the venturi duct, while flow above it will pass over the top of the sidepod. If the stream-tubes expand approaching the mouth of each tunnel, (as they might do at high car speeds), then the stagnation line might lie just inside the upper lip of the tunnel, and the external flow might separate as it accelerates over and around the upper lip. Conversely, if the stream-tubes contract approaching each mouth, the stagnation line might exist just outside the upper lip, and the flow might separate as it accelerates under that lip into the tunnel. The latter condition would inject turbulence into the throat of the underbody tunnel, leading to a significant loss of downforce.

*Note that whilst the ‘base pressure’ is lower than the static pressure of the freestream, it is not the point of lowest pressure, the latter being located in the throat of the venturi. The air doesn't flow towards the rear of the car because of a pressure gradient; it flows to the rear because the car is in motion with respect to the air!

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