Wednesday, September 13, 2017

F1 1980 - Separation and curvature

As noted in the previous post, the airflow in the aft section of a venturi duct has a propensity to separate. Whilst the primary cause of boundary layer separation is the severity of the adverse pressure gradient experienced during pressure recovery, curvature upstream of the pressure recovery region can also exert a significant influence. In this context, a useful rule-of-thumb to remember is that the thicker the boundary layer at the start of the pressure recovery region, the earlier separation will occur. The rate at which the thickness of the boundary layer on a flat surface increases with distance from the leading edge is generally used as a baseline, with respect to which the effects of curvature can be compared.

To understand the influence of curvature, let’s first introduce a distinction between 2-dimensional and 3-dimensional boundary layers. In a 2-dimensional boundary layer, the velocity profile and thickness of the boundary later vary only in a longitudinal direction, along the direction of streamwise flow. The boundary-layer velocity is a function only of height above the solid surface and longitudinal distance; it is therefore 2-dimensional. In contrast, in a 3-dimensional boundary layer the velocity profile and thickness vary in both a longitudinal and a lateral direction. 

Consider first a 2-dimensional boundary layer on a surface with either convex or concave curvature. Concave curvature increases the rate at which a boundary layer thickens (compared to a flat surface), whilst convex curvature either thins a boundary layer, or reduces the rate at which the thickness would otherwise increase.

One way to understand this is in terms of radial pressure gradients. For a flowfield to negotiate a curve, a pressure gradient develops which is directed towards the centre of the radius of curvature, balancing the centrifugal force associated with the curved flow.

A flowfield bounded by a concave curve is such that the centre of curvature is located inside the fluid itself, hence a pressure gradient develops which points upwards from the solid surface into the fluid, effectively trying to peel the boundary layer off the surface

In contrast, a flowfield bounded by a convex surface is such that the centre of curvature is located the ‘other side’ of the solid surface, hence a pressure gradient develops which points downwards onto the surface, effectively pushing the boundary layer onto it.

Hence, concave curvature is liable to trigger boundary layer separation, while convex curvature promotes boundary layer adhesion.

So much for the influence of curvature on a 2-dimensional boundary layer. Most actual flowfields tend to possess ‘crossflow’ velocity components in addition to streamwise components. Crossflow components point in a lateral direction. In the context of wings, this is often referred to as ‘spanwise flow’. The representation of separation under these circumstances requires the introduction of the aforementioned 3-dimensional boundary layers.

The crossflow velocity components correspond to the existence of crossflow pressure gradients. These pressure gradients will induce streamline curvature both inside the boundary layer attached to the solid surface, and in the adjacent outer-flow streamlines. The streamline curvature, however, will be greater inside the boundary layer. Hence, the skin-friction lines on the solid surface (otherwise known as the shear stress at the wall), have greater curvature than the streamlines just outside the boundary layer. (Understanding Aerodynamics, Doug McLean, Wiley, 2013, p88).

Inserting a bend or kink into the wall of a venturi tunnel will generate a radial crossflow pressure gradient, pointing towards the centre of the radius of curvature. The outer-flow streamlines will turn the corner due to this radial pressure gradient. The skin-friction lines on the ceiling of the tunnel, however, will turn the corner at a tighter angle.

The curvature of a surface will itself generate streamline curvature, but this effect is distinct from the streamline curvature generated by a crossflow pressure gradient. If an outer-flow streamline is projected onto a curve in the solid surface, the curvature at each point of that curve can be decomposed into a component which is parallel to the tangent plane of the surface at that point, and a component which is perpendicular to the tangent plane. The perpendicular component represents the part of the curvature which is due to the streamline simply following the extrinsic curvature of the surface in 3-dimensional space. In contrast, the parallel component represents the intrinsic curvature of the projected streamline due to a crossflow pressure gradient. If there is no crossflow, then the projected streamlines are geodesics of the surface, with zero intrinsic curvature. (McLean, p306-307).   

A similar but distinct type of curvature effect occurs when a solid is bounded by an axisymmetric surface, whose radius varies in a longitudinal direction. If the lateral extent of a surface tapers in a longitudinal direction, then successive lateral slices through the surface possess an increasingly smaller diameter. For example, in the special case of a cone-shaped surface, oriented with the tip of the cone pointing downstream, successive lateral slices through the surface of the cone have a smaller diameter. A boundary layer attached to such a surface will thicken at a faster rate than it would over a flat surface with the same streamwise pressure gradient, (McLean p124). This occurs as a consequence of the preservation of mass and the relative incompressibility of the air: the boundary layer air is forced to thicken as its lateral dimensions contract. This makes such a boundary layer more liable to detach.

Conversely, consider a surface which flares outwards with longitudinal direction, an extreme case of which would be a cone-shaped surface with its tip pointing upstream. The boundary layer on such a surface will either get thinner as the lateral extent of the surface increases, or its thickness will increase at a slower rate than it would on a flat surface in the same streamwise pressure gradient. Hence, a surface which spreads outwards promotes boundary layer adhesion.

In both cases the outer-flow streamlines are following longitudinal geodesics of the surface, and there is no pressure-driven crossflow, (ibid). A Formula 1 car, however, is rarely equipped with axisymmetric appendages. Rather, it exhibits reflection symmetry in a longitudinal plane, and as a consequence the flow around the nose and engine cover are special cases of ‘plane of symmetry’ flows (ibid., p125-126). In such flows, the boundary layer along the plane of symmetry resembles a 2-dimensional boundary layer, with no crossflow component, but either side of the symmetry plane there are crossflow components which either induce divergence or convergence.

In the case of a Formula 1 car, the flow over the nose will be a divergent plane-of-symmetry flow, and that over the engine cover will tend to be a convergent plane-of-symmetry flow. 

So, equipped with this understanding of the effects of curvature, let’s consider an example of its impact on F1 ground-effect aerodynamics. In 1980, some of the teams created vertical surfaces at the rear of the sidepods to partially seal the venturi tunnels from the effects of the rotating rear wheel. The motive for this may have been twofold: to enhance underbody performance, and also to reduce rear wheel lift and drag. However, these plates, when considered in horizontal cross-section, traced a sinuous curve which started with concave curvature, passed through a point of inflection, and ended with convex curvature. Hence, whilst such plates may have prevented the flow in the venturi tunnels from directly interacting with the rotating wheel, the geometrical restriction imposed by the presence of the wheel was in no way eliminated.

If a venturi tunnel entered a constriction towards the rear of the sidepod, then the reduced cross-sectional area would have a tendency to thicken the boundary layer. Moreover, at just this point, the initial concave curvature on the outer wall of the tunnel would also contribute towards thickening the boundary layer. Exacerbating matters yet further, the turbulent jet from the inner contact patch of the rotating rear wheel would be injected into this region of the underbody. All three factors, in conjunction, would have tended to promote boundary layer separation in this part of the underbody. The only mitigation here is that the cross-sectional constriction would have weakened the adverse pressure gradient.

As a specific example of the challenges in this region of the underbody, the Williams FW07B MKIV underwing, as specified in a design drawing from April 1980, contained a dashed outline of an alternative profile for the sinuous section of the outer wall as it passes inside the rear wheel. The rationale behind this is alluded to in a briefing note written by Patrick Head, dated 1st April 1980, (just in advance of the introduction of the MKIV underwing at the Belgian Grand Prix). Here, he notes that Williams would be “running the wide rear track with new rear plates and engine fairings plus a wheel fairing which will reduce leakage into the rear of the side wing and increase the velocities. A new side wing profile is also to be made with an altered profile in the defuser (sic) section to reduce proneness to separation.”

The alternative profile reduced the concave curvature, but it did so at the expense of beginning the transition further upstream, therefore sacrificing channel width. Hence, there was a trade-off here: concave curvature or convergence; both would have thickened the boundary layer.

Frank Dernie has since testified that “most people’s diffusers stopped at the rear suspension. It was very difficult to keep the flow attached any further back…I am told the Brabham BT49 never had attached flow rearward of the chassis because they never found a solution to keeping the flow attached after the sudden change of section.” (Motorsport Magazine, November 2004, X-ray Spec: Williams FW07, p77).

In fact, the initial underbody profile on the Williams FW07B in 1980 did attempt to extend the diffuser tunnels beyond the leading edge of the rear suspension. These gearbox enclosures and sidepod extensions appeared on the car during practice in Argentina, but serious porpoising problems were experienced, and the sidepods and underbodies were returned to 1979 MKIII specification for the race. The porpoising was attributed to the skirts jamming, hence the extensions were tried again in conjunction with the MKIII sidepods and underwing during practice in South Africa. They were, however, notable by their absence when the MKIV underwing made its debut in Belgium. 

FW07B venturi extensions, as seen at Kyalami. (Grand Prix International magazine)

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!