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 1

^{st}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) |

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