“If you want to be a Millionaire, start with a billion dollars and launch a new airline…” – Richard Branson

*The intention of this set of posts is to embark on a journey of connecting the dots between CFD and turbulence modeling with the phenomenological and practical concepts of engineering aerodynamics. **Even though it will by no means be all encompassing nor highly accurate I hope for the essence to be captured, and even more so communicated…*

Nothing is as beneficiary for the flight industry as the ability to cut substantially on fuel budget. From a CFD perspective, such a goal could potentially be achieved by high fidelity simulation for a detailed design full airborne vehicle.

A naive approach for the highest physical fidelity could be a Direct Numerical Simulation (DNS) of Navier-Stokes equations. But the plea for a direct numerical description of the equations is a mixed blessing as it seems the availability of such a description is directly matched to the power of a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to the body – The *Reynolds Number.*It is found that the computational effort in Direct Numerical Simulation (DNS) of the Navier-Stokes equations rises as Reynolds number in the power of 9/4 which renders such calculations as prohibitive for most engineering applications of practical interest and it shall remain so for the foreseeable future, its use confined to simple geometries and a limited range of Reynolds numbers in the aim of supplying significant insight into turbulence physics that can not be attained in the laboratory.

Nevertheless, for the kinds of effects we generally wish to model, we find that we’re more interested in the local statistical properties of the turbulence than in the organized structure, and in fact, that our increasing knowledge of organized structures has not contributed much to our quantitative prediction capabilities of day-to-day analysis.

It is important to note though, that in essence, turbulence **is** unsteady. Examples for aerodynamic applications for which unsteadiness is paramount and must be dealt with include (among others…):

- Flow separation associated unsteady phenomena, such as Buffeting which is a high-frequency instability caused by airflow separation flow separation associated unsteady phenomena, such as Buffeting which is a high-frequency instability from one object striking another downstream.

- Aerodynamic noise in general – and specifically “airfraim noise” from deployed landing gear and flaps or cabin noise resulting from surface-pressure fluctuations of an external to cabin boundary layer.

I have intentionally included these two examples, the first of a wake related phenomenon and the latter of a boundary-layer related phenomenon to emphasize that the turbulence in boundary layers and wakes is of course unsteady. Even still, in most aerodynamic applications this unsteadiness doesn’t produce additional valuable insight, a consequence of the length scales associated with the turbulent motions typically being small in comparison with the dimensions of the body, subsequently leading to integrated forces for which the unsteady fluctuations are also small, allowing us to take interest only in **time- averaged properties of the turbulent flowfield**.

## The Basis for Day-to-Day Turbulence Modeling: Mixing Length Theory

It was the German engineer (arguably the most prominent figure in leading to what is our current understanding of aerodynamic flows phenomenology) Ludwig Prandt’l, who first saw this connection and eloquently explained the analogy between turbulent motions and molecular mixing. Although molecules and turbulent eddies are fundamentally VERY different, this amazing analogy is directly responsible for billions of dollars in fuel budget savings in the past 50 years…

Prandt’l first hypothesized that fluid flow as consisting of collections of fluid parcels moving about randomly with some characteristic speed over some characteristic length scale, would essentially retain their momentum. This hypothesis is based on a similar one from the kinetic theory of (rare) gasses of which molecules moving about randomly (Brownian motion) following the maxwellian distribution, such that all directions are equally possible, with some characteristic speed (the average molecular velocity being the thermal velocity), over some characteristic length scale (being the mean-free-path) are holding their characteristic momentum from the velocity layer they where coming from:

In the molecular level a decomposition we may propose a decomposition of the following kind:

u=U+u”

(While U is defined by U(y) and u” is molecular random movement).

The sudden flux of every property through y=0 is proportional to the normal to plane velocity normal to plane. Concerning the description above it is v”. Hence the sudden change in momentum through a differential element dS may be described as:

After conducting an ensemble average this becomes:

**By definition** the stress acting on y=0 may be written as:

Breaking the stress into hydrostatic pressure and viscous stresses (shall be proven extremely useful later in breaking aerodynamic drag to its distinguished constituents):

Will allow for the following relation between the momentum transfer of colliding molecules and the earlier defined viscous stresses.

## Connecting RANS to the Phenomenology of Aerodynamics

Returning back to Prandt’l, we now postulate the following:

Momentum transfer by molecule collisions—>Momentum transfer by turbulent motion

Continuum———————->Mean Flow

molecule————————->Fluid parcel

Mean free path —————>Mixing length

Thermal velocity————–>Mixing velocity

Random (Brownian motion)—————->Turbulent motion

Molecular transport of momentum—–>Turbulent transport of momentum

Such an analogy between turbulent motions in a mean flow and molecular motions to a continuum may be very valuable to the level of phenomenological description for which (small scale) unsteadiness is not of direct engineering interest.

We need therefore a framework for which we can deploy the strategy, and that framework is Reynolds-Averaged Navier-Stokes (RANS). RANS is based on the Reynolds decomposition according to which a flow variable is decomposed into mean and fluctuating quantities. When the decomposition is applied to Navier-Stokes equation an extra term known as the *Reynolds Stress Tensor *arises.

In essence, we’ve seen that molecular motions relative to the continuum fluid transports momentum, and averaging these small scale momentum transfers defines the static pressure and in-plane viscous stresses. On the same token, turbulent motions also transfer momentum, and the averaging process in this case gives rise to what we may define as turbulent stress or Reynolds Stress:

Transport of momentum by turbulent motions has, all in all, a similar effect **on the time-averaged flow** as additional viscous stresses would, hence referred to as turbulent stresses or Reynolds stresses, and may be regarded as an additional dissipation effect.

This turbulent dissipation process essentially involves two steps:

**Step 1:**

Production of turbulent kinetic energy) The work done locally against the Reynolds stresses is transferred directly into the kinetic energy of the turbulent motions, a turbulent production which is almost always positive, much like molecular viscous dissipation.**Step 2:**

dissipation of turbulent kinetic energy) These turbulent motions, containing local velocity gradients eventually dissipate the turbulent kinetic energy into heat. So in essence the Reynolds stresses result in (turbulent) dissipation into heat much like the molecular viscous stresses do.

There are a few takes to bare in mind which will prove valuable in later connecting practical aerodynamic concepts with turbulence based phenomenology :

- It is important to note that although Prandt’l mxing length theory is the basis to modeling approaches which have been proven to have a far reaching, day-to-day predictive power (interestingly even beyond their theoretical range of applicability), turbulent time scales are many orders of magnitude longer than the that of molecular collisions, hence we should always remember that the scale-separation requirement is much more limiting for turbulence averaging than for molecular.
- Attention should be payed to the fact that Prandt’l mixing length hypothesis and the above continuation of the analogy to RANS and Reynolds stresses serves as the basis for the Boussinesq hypothesis. Yet, the latter is only a possible route. The Reynolds stresses may be estimated by any means, and there is no necessity at this stage that it would be through the eddy viscosity concept. Nonetheless, it is should be very clear by now how much the eddy-viscosity concept is phenomenologically important if for all practical (time-averaged flows) purposes we shall treat turbulence as as additional dissipation mechanism (especially in drag calculations), indeed, we may continue the analogy as explained in Understanding the Boussinesq Hypothesis and the Eddy-Viscosity Concept, and extend the derivation to include the constitutive relation between molecular stress and the strain rate Tenzor (haha… 🤓) through the definition of molecular viscosity and complete the analogy with the eddy viscosity.
- Although the production of turbulence kinetic energy and its dissipation are roughly in equilibrium over most parts of a flow, this is not always the case, especially when the flow goes into separation, a feature which obviously would like to predict with high accuracy. Streamlined bodies such as wings often have adverse pressure gradient over their aft portions or whenever there is an increase in the angle of attack. In such scenarios the turbulent production may be larger than its dissipation (20% larger in some frequently encountered scenarios). Many turbulence models relying on the equilibrium assumption (such as most standard two equation models), are bound in such scenarios to over-predict the shear stress (predicting shear to be higher than should), therefore delaying separation or even preventing it in some cases.

Below there is a comparison between the k-ε and k-ω SST for a plane diffuser scenario. The k-ε, intrinsically reliant on the equilibrium between the production of turbulent kinetic energy and its dissipation to correctly calculate the turbulent shear stresses model, failed (due to excess of turbulence kinetic energy production) to capture the separation as the flow is completely attached while the k-ω SST predicts a strong separation and a re-circulation zone and is in close agreement with data by K. Gersten et al.

*This concludes Part I of the “Engineering Turbulence” series. In Part II: “Lifting The Wright Way!”, the aerodynamic concept of Lift shall be somewhat explored and explain, followed by Part III: “What A Drag!” , which is all about Drag, and its reduction by flow control methodologies which are based on certain fundamental aspects of transition and turbulence coherent structures.*

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