Diffusers and rear-wheel wakes

There is a persistent notion amongst some Formula One technical analysts that the low pressure wake behind the rear wheels can be connected to the lateral extremities of the diffuser airflow, thereby enhancing the flow capacity of the underbody, and its downforce-generating potential. In particular, the notion has been repeatedly promoted by Autosport Technical Consultant Gary Anderson:

"Mercedes has worked very hard in making the low pressure area behind the rear tyres connect up to the trailing edge of the diffuser. In effect this gives the diffuser more extraction capacity," ( The key technical developments from Australia, 25th March 2017).

Now, it's certainly true that if the diffuser is expanded in a lateral direction without causing separation of the boundary layer, then the expansion ratio of the diffuser will be increased, and it'll generate more downforce. It's also true that the wakes shed by the wheels are areas of low pressure, situated as they are behind rotating bluff bodies. So surely, one might think, there will be a pressure gradient directed towards those rear-wheel wakes, and surely the airflow exiting the diffuser can be connected to them, thereby increasing its effective expansion ratio?

Unfortunately, whilst the wakes behind bluff bodies do indeed tend to be regions of low pressure, they are also regions of high turbulence, and the airflow 'sees' a region of turbulence as an obstruction. Directing the lateral extremities of diffuser airflow towards the rear wheel wakes does not therefore offer a straightforward boost in the power of the diffuser, and could even promote diffuser separation.

One illuminating way to understand this is to look at the Reynolds-averaged Navier-Stokes (RANS) equations for a flow-field containing turbulence. A solution of these equations represents the mean velocity flow field $\overline{u}$ and the mean pressure field $\overline{p}$ in a region of space. For a time-independent incompressible flow, each component $\overline{u}_i$ of the mean velocity vector field is required to satisfy the equation

$$
\rho (\mathbf{\overline{u}} \cdot \nabla) \overline{u}_i  = - \frac{\partial \overline{p}}{\partial x_i} +\frac{\overline{\tau}_{ij}}{\partial x_j} - \rho \frac{\overline{u'_i u'_j}}{\partial x_j} \;.
$$ This equation is simply a version of Newton's second law, $F=ma$, albeit with the accelerative term on the left-hand side, and the force terms on the right-hand side.

In the case of a continuous medium, the density $\rho$ is substituted in place of the mass, and $(\mathbf{\overline{u}} \cdot \nabla) \overline{u}_i$ represents the acceleration experienced by parcels of air as the velocity field changes from one spatial position to another. 

Each term on the right-hand side of the equation represents a different type of force. The first term $- \partial \overline{p}/\partial x_i$ is the familiar pressure gradient. The negative sign indicates that the force points in the opposite direction to the gradient: the fluid will be pushed away from high pressure, and sucked toward low pressure.

Pressure, however, is only the isotropic component of stress. When the isotropic component has been subtracted from the total stress, what remains is called the 'deviatoric' stress $\tau_{ij}$. This represents the stresses which occur due to viscosity $\nu$. These are the forces which occur within a continuous medium when there are shear motions. In the case of a Newtonian fluid such as air, the deviatoric stress is a function of the viscosity and the velocity shear:

$$ \tau_{ij} = \rho \nu \bigg[ \frac{\partial u_i}{\partial x_j}+ \frac{\partial u_j}{\partial x_i} \bigg] $$In general, forces are generated by spatial gradients of the stress, and the second term on the right-hand side of the RANS equation represents the force due to the spatial gradient in the mean deviatoric stress. These 'tangential' forces are crucial inside the boundary layer of a fluid, but more generally they play a role wherever one layer of fluid runs parallel to another layer travelling at a different speed. Here, the viscosity entails that momentum is transferred from the higher velocity layer to the lower velocity layer, helping to pull it along. This is a source of acceleration in the flow-field which cannot be explained by pressure gradients alone.

The third term on the right-hand side of the RANS equation represents the effective force due to spatial gradients in the turbulence. In a turbulent flow-field, the velocity at a point is decomposed into a sum $u_i = \overline{u}_i + u'_i$ of the mean-flow $\overline{u}_i $ and the turbulent fluctuations, $u'_i$. The expression $\overline{u'_i u'_j}$ represents a type of turbulent stress, hence its spatial gradient provides another source of acceleration in the mean flow-field.

This third term is crucial to understand why the rear wheel wakes behave like obstructions in the flow-field. Note the negative sign associated with the turbulent-stress term. That entails that the force vector points away from a region of turbulence. Airflow exiting a region of low turbulent intensity will effectively experience a repulsion force as it approaches a region of high turbulence.  

Hence, trying to join the diffuser-flow to the rear wheel wake is not necessarily a good idea. A better idea is to create vortices from the edges of the diffuser which push the rear wheel wake further outboard. This might enable one to increase the expansion ratio of the diffuser without provoking separation.

Why nuclear disarmament is wrong

The 2017 Nobel Peace Prize was awarded this week to the International Campaign to Abolish Nuclear Weapons (ICAN), a coalition of 468 non-governmental organisations across 101 countries. Berit Reiss-Andersen, the chair of the Nobel committee, stated that the award recognised ICAN's work “to draw attention to the catastrophic humanitarian consequences of any use of nuclear weapons and for its groundbreaking efforts to achieve a treaty-based prohibition of such weapons”. According to the BBC, ICAN's supporters “include actor Michael Sheen.”

Now, whilst one contradicts a B-list actor at one's peril, it is nevertheless a good juncture to review exactly why organisations such as ICAN are wrong, and why nuclear disarmament would be a bad thing. Let's begin with those “catastrophic humanitarian consequences of any use of nuclear weapons”, by returning to 1945 and the use of nuclear weapons to end the Second World War.

The image of the mushroom cloud, and the destruction inflicted on Hiroshima and Nagasaki dominates modern media coverage of these events. Rarely, however, does the media also recall the incendiary bombing campaign conducted by the Americans prior to the use of nuclear weapons.

Between March and June of 1945, Japan's six largest industrial centres, Tokyo, Nagoya, Kobe, Osaka, Yokohama and Kawasaki, were devastated. As military historian John Keegan wrote, “Japan's flimsy wood-and-paper cities burned far more easily than European stone and brick...by mid-June...260,000 people had been killed, 2 million buildings destroyed and between 9 and 13 million people made homeless...by July 60 per cent of the ground area of the country's sixty larger cities and towns had been burnt out,” (The Second World War, 1989, p481).

Unfortunately, this mass bombing campaign, conducted with conventional chemical munitions, and inflicted upon civilians and military alike, did not stop the war. Only the bombing of Hiroshima and Nagasaki stopped the war.

In terms of the number of deaths, “reported numbers vary, but it has been estimated that by the end of 1945, 90 000 to 120 000 out of a civilian population of about 330 000 in Hiroshima, and 60 000 to 80 000 out of 280 000 in Nagasaki, would be dead as a result of exposure to the intense heat, physical force, and ionizing radiations emitted by the bombs,” (Long-term Radiation-Related Health Effects in a Unique Human Population: Lessons Learned from the Atomic Bomb Survivors of Hiroshima and Nagasaki).

So, the first conclusion to draw from this is that conventional munitions killed more people, and didn't stop the war, while nuclear weapons killed less people, and did stop the war. In terms of “humanitarian consequences”, being burnt alive by incendiary weapons rather than the blast wave, thermal radiation or ionising radiation of a nuclear detonation, seems scant consolation. 

In the decades since the Second World War, the presence of nuclear weapon stockpiles have been justified on the basis of deterrence: as long as the use of nuclear weapons by one side will result in a retaliatory strike that guarantees their own destruction, then a nuclear war is unwinnable, hence there is no incentive to use nuclear weapons. 

Despite the logic of deterrence, many continue to argue that nuclear weapons should now be abolished by means of multi-lateral disarmament. A recent article in NewScientist by Debora Mackenzie argued that deterrence is unstable:

“The growth in US missile defence systems...undermine deterrence by, in theory, allowing a country to launch a first attack safe in the knowledge that it can intercept any retaliatory strikes...deterrence is only ever a temporary stand-off, lasting just until the enemy finds a way to neutralise your deterrent. Ultimately, the technological capacity to see, hear and otherwise detect and destroy other countries' weapons could become so good that first strikes will become winnable, and deterrence will no longer work...What else will keep the nuclear peace? Optimists are promoting a UN treaty to ban all nuclear weapons,” (Accidental Armageddon, 23rd September 2017).

Which brings us back to ICAN, who promoted the 'Nuclear Weapons Ban Treaty'. The nine recognised nuclear powers refused to sign this at the United Nations in July. And they were right not to do so, for the following reason:

A world without nuclear weapons is a world in which a nuclear war is winnable. As demonstrated in the 1940s, it only requires one nation to secretly begin the production of nuclear weapons, (breaking whatever treaty they may have signed), to gain a head-start on their enemies, and they will be able to use nuclear weapons without fear of reprisal. A world without nuclear weapons is a world in which there is an incentive to use nuclear weapons. Multi-lateral nuclear disarmament would therefore take us into the most unstable and dangerous state of all.

Once nuclear weapons have been invented, there is no going back to a world without them. It's not a question of optimism or pessimism, it's a question of logic.