Turbulence has several characteristics:

• Presence of "non-deterministic" fluctuations in flow properties. This means that one cannot precisely predict the value of a flow property at a future instant by any known means, since the precise relationship is not known.
• Examples: In the boundary layer over a large wing in flight, the velocity at a given point cannot be predicted for every instant, because it keeps fluctuating. We can watch the fluctuations for as long as we care – we will not be able to predict the exact value for a future instant.
• Counter-example: Consider the flow in the wake of a helicopter rotor. Every time the blade comes around, the flow velocity undergoes sharp fluctuations. It looks "chaotic", but only until one realizes that exactly the same thing is happening every time the blade comes by. Once this realization occurs, we see that we can indeed predict the value of , say, velocity associated with a particular position of the rotor blade during its cycle: it’s a "periodic" process, with a predictable period and therefore a frequency.
• Turbulent fluctuations occur over many frequencies. "Broadband", as opposed to the single frequency seen when vortices are shed from a circular cylinder, or the processes in the wake of a flapping wing or a rotor or propeller, or a sound wave.
• Turbulence is 3-dimensional: When turbulent fluctuations occur, they occur in all 3 dimensions. There is no such thing as a "2-d turbulence". Fluctuations occur in properties along all directions.
• Cascade of scales. If one asks: "how big are the turbulent eddies?" the answer is: "all kinds of sizes, ranging from eddies as large as the stream width (or boundary layer thickness), to very small eddies. The lower limit on eddy size can be estimated (as we’ll see presently).

Even turbulent flows must obey the laws of physics. Therefore, the momentum and continuity equations must hold, on an instantaneous basis, in relating the properties of the flow. The trouble is, all the properties keep fluctuating from instant to instant. How do we deal with this?

We develop approximate means. We ask what we really want to know about turbulent flows, for a given problem. One of the most urgent needs for an aerodynamics engineer is to be able to predict the friction due to a turbulent bounday layer. Although there are fluctuations from instant to instant, the aircraft is too big to bounce around in response to each of these: what matters is a "time-averaged" value of properties like the skin friction.

We then consider that in most turbulent flows, the fluctuations in velocity, pressure etc. are in fact quite small compared to the mean values of the flow properties. Thus, the u-component of velocity might be 100 m/s with a fluctuation of plus/minus 2 meters per second, so that that value of u ranges from about 98 to about 102 m/s.

We use this fact: assume that each flow property is comprised of a time-mean value,.which of course does not vary with time, and a small fluctuating quantity added to that. We substitute this into the Navier-Stokes relation, and find out what happens when we take the time average of the resulting terms. This yields the "Reynolds –averaged Navier-Stokes Equations".

Below, we see the Reynolds-averaged Boundary layer equations: the simplifications have already been made to the Navier-Stokes equations to deal with thin boundary layers.

Derivation of the Momentum Equation for Turbulent Boundary Layer

If each flow variable such as u is broken up into a "ubar", its mean value, and "u-prime", its fluctuation, the continuity equation becomes:

(1)

Expanding this, we get:

Now, the mean values themselves must satisfy the continuity equation, so:

(1a)

This leaves an equation for the fluctuating quantities in the continuity equation:

(1b)

The u-momentum equation for the boundary layer becomes:

………(2)

Again, expanding this gives:

(2a)

Taking the time average: If we take the time average of a quantity like u’, the average will be zero. However, the average of a quantity like u’v’ may not be zero, unless "u’ and v’ are totally uncorrelated". Perhaps there may be some relation, like "when u goes up, v goes up as well" or "when u goes up, about half the time v goes down". In these cases, the time average of a product of two fluctuating quantities need not be zero. Applying this logic, we time-average each term of the above equation. We are left with:

(3)

We now take the fluctuating portion of the continuity equation (1b) above, and multiply each term by the mass flow rate per unit area: (rho)*(ubar + uprime)

(4)

Taking the time average of each term,

(4a)

Add equations (4a) and (3):
 
 

According to our boundary layer assumptions, the last term, which is a streamwise gradient of a square of a fluctuation, should be much smaller than the transverse gradient of the product of two fluctuations. We neglect the last term. This gives:

Examine the last 2 terms:

The first term is the laminar shear stress. The second term is the "turbulent shear stress". One way to think about it is the momentum lost because the flow which was moving along the u-direction is now wasting some momentum mving along the y-direction, a motion which it cannot sustain (obviously, since it is just a fluctuation). When the flow changes momentum, a net force is exerted on the surface. Thus, the wall shear stress in turbulent flow is given by:

The value of the second term is difficult to predict: it depends on the "correlation" between fluctuations in u and v.

Note that the "turbulent shear stress" is not due to viscosity: it is due to the inertial forces in the flow. In turbulent boundary layers, this part of the stress may be many times higher than the laminar shear stress.

People develop empirical correlations for the Reynolds stresses for specific flows. Attempts to develop more general predictions of the Reynolds stresses take various forms, ranging from "large-eddy simulation" to "models" of turbulence at various levels of complexity. There are no closed-form analytical expressions.

The Cascade of Turbulent Kinetic Energy

Consider a turbulent flow. We first ask:where does the flow get energy to put into fluctuations? The answer is obvious: the energy comes from the mean flow. If there were no flow, there would be no fluctuations sustained. So, we see the first concept of the turbulent cascade: the mean flow drives the fluctuations. For example, a large vortex starts up: a lot of energy is taken up there. So, the energy cascades from the mean flow to the largest "eddies".

Next we ask: "what happens to the energy finally? Where does it go?" Well, energy cannot just disappear. So, this energy is finally transferred as heat to the bounding surfaces, or goes to heat up the flow itself.

How does the conversion from kinetic energy to heat occur? It occurs by viscous friction between layers of fluid moving at different velicities, at the smallest level. This occurs at the edges of the smallest eddies.

In between these sizes, there is a "cascade" of eddy sizes. The bid eddies spin around. At their edges, smaller eddies are generated, and at their edges, still smaller ones are generated, until they get so small that their energy is dissipated as heat due to viscous friction. So, "dissipation" of turbulence occurs at the smallest scales.

In a steady turbulent flow, there must be a balance: kinetic energy is being transferred to the eddies just as fast as it is being dissipated as heat in the smallest eddies.

How small are the smallest eddies? Well, so small that they encounter laminar flow, where viscous forces are as big as inertial forces. So small that if you construct a Reynolds number using the size of the eddy and a velocity characteristic of the fluctuation at that level you will get a Reynolds number of approximately 1.

How big are the largest eddies? As big as the mean flow itself.