# Aerodynamics: Drag & Bernoulli in Turbulent Flows

• dtms1
In summary, the Bernoulli principle is only applicable in laminar flows and airplanes typically fly in turbulent conditions. More complicated mathematics is required for analysis in turbulent flow.f

#### dtms1

I have a question but it's not so much a specific HW problem as it is with trying to conceptualize the physics behind it.

So when dealing with aerodynamics, let's take for example a wing / airfoil.

I understand due to a difference in pressure lift is generated. Higher pressure wants to naturally go to lower pressure thus creating a net force of an upwards (lifting) motion. I also know that the higher pressure is on the underside of the wing because of lower local velocity the air travels as oppose to the higher velocity of air on the top side of the wing.
Which follows Bernoulli's principle.

However, here is where I start to get confused. From my understanding Bernoulli's equation is only applicable in laminar flows where you don't have significant shear losses caused by turbulent flow.
If that is the case, airplanes typically fly at high speeds which can safely be assumed to be flying in turbulent conditions (Reynold's number proportional to speed thus with high velocity high Reynold's number which would mean turbulent conditions). IF that is the case, how is Bernoulli still applicable when analyzing airfoils/wings?

Even in a car, you already have turbulent flow and the streamline is no longer constant/smooth. You have wakes and circulation. Can we safe to say that Bernoulli is no longer valid in these situations?

In turbulent flow, Bernoulli's equation is not applicable to properly analyzing airfoils and wings. Much more complicated mathematics is required. For certain idealized flow conditions, computational fluid dynamics can now be used for this analysis, where, prior to high-speed computers, testing of proposed airfoils in wind tunnels was the only method available to determine the lift and drag characteristics for the foils. In many respects today, analysis of turbulent flow numerically is still not completely solved, and quite a bit of model testing still goes on.

Ludwig Prandtl was one of the first engineers to propose a theory of how airfoils generate lift:

http://en.wikipedia.org/wiki/Ludwig_Prandtl

Prandtl originally worked in the field of solid mechanics and took up studying airfoils when aircraft first began to be built in the early twentieth century.

Hmm, interesting.

Question when you say
In turbulent flow, Bernoulli's equation is not applicable to properly analyzing airfoils and wings.

Can the same be said for turbulent flow in internal flow (such as pipes)? If Bernoulli is applicable in internal turbulent flow, why is that okay but not in analyzing airfoils/wings?

In analyzing the internal flow of ducts and pipes using the Bernoulli equation, the flow must be incompressible and the losses due to friction must be negligible. When the flow is turbulent, other equations, like Darcy-Weisbach, can be used to account for the frictional losses and incorporated into a Bernoulli-type analysis.

The flow around airfoils is much more complex than Bernoulli can account for. The energy of the fluid flowing over the wing is not constant, and that essentially is one point which invalidates the Bernoulli principle. More details can be found here:

http://en.wikipedia.org/wiki/Lift_(force)

A discussion of Prandtl's theory of lift is given here:

http://en.wikipedia.org/wiki/Lifting-line_theory

This theory also contains significant limitations, as mentioned at the bottom of the article. There are more complex theories which have been developed to overcome these limitations.

A full accounting of all the physics in aerodynamics can be exhausting e.g. reading the wikipedia articles suggested. It helps that many of the effects are secondary, and one can make some assumptions and be generally correct. So Bernoulli's equation applies to laminar flow, but it does give the right trends in pressure and velocity in turbulent flow. In analyzing lift on a wing, for example, we simplify by saying that air is almost inviscid except for the little bit in the boundary layer and where there is flow separation. Then we can use potential flow theory and get results that are off by a bit but still display the right patterns of flow over most of the flow field. Since you are trying to understand the physics, it helps to use such simplifications first, and then build up the understanding by adding more effects e.g. compressibility effects at high Mach numbers.

By the way, it is more correct to say that lift is caused by suction over the top surface of a wing. There is a large drop in pressure as air flows over the top of a wing - see the attached diagram (adapted from NACA report 563, also attached). The long vectors over the top are pulling on the top surface, a suction. The shorter vectors pushing on the bottom are indeed the higher pressure from slower fluid flow. See how much more suction there is compared with the push from below. The same diagram also compares experimental results with theoretical calculations (each contour line has different simplifications and assumptions, while the vectors are measured results). As you can see, the overall shape is correct, but better assumptions agree better with measured results. See the report for more details.

#### Attachments

• Figure 3.1.1.png
23.7 KB · Views: 436
• NACA report 563 4412 press dist.pdf
1.2 MB · Views: 970
Bernoulli's equation is derived from F=ma, applied along a streamline.
A streamline is a curve at AN INSTANT defined that the velocity field is always tangent to it.

When the field is stationary, streamlines also represents particle trajectories. For non-stationary flows, such as during turbulunce, streamlines lose this nice additional representation, and are fairly useless mathematical abstractions (who cares how field lines look at a particular instant, when these are something totally different the next moment?)

Although Bernoulli's equation, as a first integral, is only derivable when there are no dissipative forces present, as long as the field is stationary, I've seen engineering formulae in which one adds a dissipative term which effectively is to be interpreted how much energy a particle going from one location on the streamline to the next is likely to lose.

While such engineering formulae are hard to justify theoretically, some of them might be applicable for analysis of a restricted class of problems.
------------------------
I have not seen this applied to problems of air flow though, but believe it was related to flow of fluids through channels.