In the second part of this article, we will further discuss important concepts of fluid mechanics and aerodynamics, such as laminar/turbulent flow, boundary layer, flows in different directions and different pressure definitions.

For reminder, you can read Part I here.

## Laminar and Turbulent Flow

Terms like laminar and turbulent flow are common phrases in F1, especially turbulent flow or turbulence. Turbulence is also very often heard in relation to airliner flights, pilot and crew may warn their passengers about “experiencing some in-flight turbulence”, meaning passengers should expect some buffeting and shaking of the cabin.

Laminar flow is a low velocity flow, where streamlines have a uniform and “proper” layout, taking up the shape defined by geometry of wall(s) surrounding them. Turbulent flow, contrary to laminar, is unsteady, chaotic, three-dimensional and unpredictable. No point in turbulent flow will have predictable fluid properties, though with experimental data collection, some of the unsteadiness (varying over time) can be averaged out.

It was Osborne Reynolds who first published the results of experiments revealing the differing nature of fluid flow at different velocities. He also established that there are other factors to this phenomenon, but also that there is a non-dimensional number that could serve as a helpful guide when determining the specific nature of a flow. This number was later called the **Reynolds number**. Reynolds number is defined as a ratio of inertial forces and viscous forces in a fluid:

where *u* is relative fluid velocity, *L* is a characteristic linear dimension (wing chord, sphere diameter, pipe internal diameter, etc) and *ν* is kinematic viscosity.

A Reynolds number, in this sense, gives value to the condition at which, depending on wall geometry, fluid pressure, temperature, etc, the flow transitions from laminar to turbulent.

*Figure 6 – Example of laminar and turbulent flow in cigarette smoke*

Laminar flow can turn to turbulent flow in free atmosphere (meaning no walls in fluid volume), as can be seen in figure 6. This transition can also occur in wall vicinity, i.e. – in boundary layer. This means that fluid adjacent to wall (e.g. on the wing) is transformed from laminar to turbulent flow at some point. You can read more about laminar and turbulent flow in our detailed dedicated article – Transforming Chaos; the Law of Nature, into Order; The Law of Man.

## The Boundary Layer

The boundary layer is related to fluid viscosity, which was discussed in Part I of this article. It is a layer of fluid where effects of viscosity are significant and there is a number of different types of it, such as laminar, turbulent, thermal, etc.

As we are discussing F1 aerodynamics, boundary layer defined by fluid velocity is of most interest. The boundary layer increases in height along the length of fluid-wall interaction, which is related to the varying fluid velocity in it. On the wall surface, fluid is static, so velocity is 0. On the edge of boundary layer, velocity is defined as 99% of free-stream velocity – i.e. surface velocity of inviscid flow.

*Figure 7 – Laminar boundary layer, graphic representation*

If fluid flow is laminar before it meets with wall, laminar boundary layer will form, vice versa with turbulent flows. However, transition between laminar and turbulent flow is almost guaranteed to happen on the wall surface at some point (on an F1 car – it happens everywhere, as all surfaces are long enough, car speeds and pressure gradients on surfaces as well).

It is very important to note that a turbulent boundary layer is not the same as free stream turbulence and is not always a bad thing. For instance, it can remain attached to curved surfaces with relatively strong adverse pressure gradients longer than laminar flow. It comes with a surface drag penalty, but the difference is minimal – take golf ball dimples as an example where keeping flow attached for longer via a turbulent boundary layer all over the ball to reduce form/pressure drag is more important than having less surface drag.

When the turbulent boundary layer does detach, fully turbulent flow also develops and everything down-stream can be (and often is) compromised. One example of this occurrence is detachment in coke-bottle area of the car, where strong pressure gradients are present and flow detachment can happen easily – as can be seen on Figure 8.

*Figure 8 – turbulent boundary layer separation on side pod and floor of Ferrari F14-T (between Brembo and Hublot logos)*

## 1D, 2D and 3D Flows

As things usually go in scientific research and transforming laws of nature into mathematical equations, simplifications are abundant. This is also the case with so called 1D and 2D flows, i.e. flows along a line and flows in a plane. No need to explain that this is simply a thing of semantics and that every fluid flow is, in fact, a flow in a volume – i.e. a 3D flow.

In general terms, 1D and 2D flows (looked at from simplified point of view) are not present in F1 car outer aerodynamics. Even flow through an S-duct can’t really be observed as 1D, as there are significant effects of 3D flow. However, there are areas of an F1 car where flow can locally be observed as a 1D flow and also as a 2D flow.

As mentioned in first part of this article, convergent slots on wings (and almost every other part of the car nowadays) are used to energize the boundary layer and to accelerate flow on the underside of a wing, while also reducing the pressure on the bottom and increasing pressure on top of the wing. Locally, slots can be observed as 1D flow, when a cross section cut-out is made, as you have one inlet and one outlet in this zone.

1D flow calculations are more important when hydraulic flow trough pipes is concerned, or air flow as well. This area of fluid dynamics is also important for calculations of nozzles, especially when Mach numbers are above 1. Wind tunnel cross sections are also largely defined with 1D flow calculations.

*Figure 9 – subsonic to supersonic flow trough de Laval nozzle*

When flow velocity at every point is parallel to a fixed plane, it can be said that this flow is a **2D flow**. Due to a large number of very strong vortices forming all over an F1 car, there really is no plane on a car where flow is a 2D flow, even though in theory in symmetry plane it should be the case. Studies concerning 2D flows are very important from another perspective – aerofoils are designed as 2D curves, tested in wind tunnels in forced 2D conditions and also tested in 2D CFD simulations. To understand better how aerofoils are looked at from analytical point of view, it is useful to mention different types of 2D flows.

A **line source** is a line from which fluid appears and flows away on planes perpendicular to the line. Same goes for a **line sink** – fluid flows towards the sink perpendicular to it. This is basically a 2D inlet/outlet in CFD simulations.

**Uniform source flow** is a radially symmetrical flow field directed outwards from a common point. The same foes for **uniform sink flow**. A uniform source flow is best depicted like a star giving away light in every direction, while the sink is a black hole – sucking in everything from every direction.

*Figure 10 – a graphic representation of uniform source flow, with streamlines pointing away from the source*

An **irrotational vortex** is a vortex where flow at every point is such that a small particle placed there undergoes pure translation and does not rotate. There is no radial flow, so at the centre the velocity is zero. An important feature of this kind of vortex is circulation (*Γ*), a line integral around a closed curve of the velocity field. In other words, circulation is a velocity flux around a closed curve, i.e. perpendicular to the surface this curve encloses.

Flow around an aerofoil can be represented as a circulation around its circumference. Needless to say, this is only true when there is some flow around the aerofoil itself in the first place, so tangent velocity on its circumference is summed with free stream velocity. Clockwise circulation provides lift, resulting in an increase in velocity on top surface of aerofoil, with a decrease on bottom surface – as is indeed the case. This is called Kutta-Zhukovsky theorem, with Magnus effect one of the best examples of correlation between circulation and lift generation.

*Figure 11 – a graphic representation of flow around an aerofoil based on KZ theorem*

**Doublet** is one last type of 2D flow which should be mentioned, it is a concept analogue to electric and magnetic dipoles (electrodynamics). It can be described as a uniform source flow and sink flow at an infinitesimally small distance apart. All other types of 2D flow can be mathematically represented as a combination of these six basic flow types and as such flow parameters can be analytically determined. Again, 2D flows are very rare and these analysis where important in early days of fluid mechanics.

## Total, Static and Dynamic Pressure

The most often used definition of pressure (*p* or *P*) is “Pressure is the amount of force applied perpendicular to the surface of an object per unit area.” Since we can define aerodynamic forces as a multiplication of pressure and surface area, fluid pressure should be defined differently – as compressive stress at some point within a fluid.

Why only compressive? As pressure acts upon a surface of an object, on its own it cannot pull this object – there has to be a pressure difference to move this surface and/or this object. This pressure difference multiplied by surface area is how aerodynamic forces are calculated.

Now, which pressure are we talking about here, exactly? Aerodynamic forces result from motion of an object trough air (or motion of air around an object), so the term **dynamic pressure** should be mentioned first. In doing so, we can define dynamic pressure as an increase over **static pressure** due to fluid motion. Dynamic pressure is often indicated as *q* or *Q* and is defined as:

This equation is analogue to kinetic energy equation and should be correlated to it – due to movement pressure rises with velocity squared.

As mentioned, dynamic pressure is an increase over static pressure and it is static pressure that is, in fact, acting upon an object. Static pressure can therefore be measured on the surface, unlike dynamic pressure. Difference in static pressure over an F1 car comes from difference in air velocity over the car. Dynamic pressure changes locally on the car, as it is related to local air velocity. This is where the term **total pressure** comes into play. Total pressure is the sum of static and dynamic pressure and gravitational head:

The term stagnation pressure is often confused with total pressure, when difference in geodesic height between two observed points is very small and can be ignored. Stagnation pressure is a sum of dynamic and static pressure and it is pressure at a point where fluid velocity is zero (e.g. leading edge of a wing).

Total pressure represents total energy of air moving around an object and is very important for overall race car aerodynamic performance. Loss of energy in air can lead to loss of aerodynamic performance (since equal dynamic pressure difference in that case yields smaller static pressure difference), and losses can happen easily. One way to lose energy is trough boundary layer, where viscous forces take their toll and the thicker the boundary layer is, the more energy is lost. Another way to lose energy is trough turbulent wakes, which is why modern F1 car designs are focused on keeping (front) tyre turbulent wake away from the car as quickly as possible.

Loss of energy in air doesn’t just represent trough smaller static pressure difference; it also affects potential flow separation, meaning it can happen more easily when total pressure is low. This is why good diffuser performance is very difficult to extract – a long floor with thick boundary layer precedes it and any intake of turbulent wake in this zone (either from front or rear tyres) can badly hurt it.

Teams counter these problems with various solutions. Increasing the number of slots in surfaces helps with keeping boundary layer thickness at the minimum, while also energizing the flow with convergent channels. These slots are also useful with another way to counter total pressure loss – induced vortices used to create flow structures that help guide the air where designers want it. Turbulent front tyre wake is forced outboard with well-known Y250 vortex and flow above side pods is forced down towards the floor with various vortices generated in barge board area and around side pod inlets.

*Figure 12 – the number of slots in barge board area of modern F1 cars seems to increase with every race (Red Bull RB14 in Canada)*

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