The magnitude of these forces depend on the shape of the object, the speed of the object, the mass of the fluid going by the object and on two other important properties of the fluid; the viscosity , or stickiness, and the compressibility , or springiness, of the fluid. To properly model these effects, aerospace engineers use similarity parameters which are ratios of these effects to other forces present in the problem.
If two experiments have the same values for the similarity parameters, then the relative importance of the forces are being correctly modeled. Aerodynamic forces depend in a complex way on the viscosity of the fluid. As the fluid moves past the object, the molecules right next to the surface stick to the surface. The molecules just above the surface are slowed down in their collisions with the molecules sticking to the surface. These molecules in turn slow down the flow just above them.
The farther one moves away from the surface, the fewer the collisions affected by the object surface. This representation is directly connected with the system of equations of motion in the boundary layer 6.
In the compressible boundary layer, density pulsations can be considered to be the result of temperature pulsations. For example,. Boussinesq in On the basis of the concept of similarity of molecular and turbulent exchange similarity theory Prandtl introduced the mixing length die Mischungsweg hypothesis.
In accordance with this condition, he derived an equation which proved to be fundamental for the boundary layer theory:.
For turbulent region of the near wall flow boundary layer, L. Prandtl considered the length 1 proportional to y. For this region, after integrating Eq. Velocity distribution representation in universal coordinates and mathematical models for turbulent viscosity coefficient are dealt with in greater detail in the section of Turbulent Flow.
One of the current versions of the semi-empirical theory of turbulent boundary layer developed by S. Kutateladze and A. The concept of relative friction law, introduced by S. Leontiev , indicates. In Eq. Static pressure and pressure gradient value do not vary across boundary layer thickness. Thus, the relationship:. Hence, the total skin drag coefficient confirms the qualitative observations we made before that the frictional shear stresses in a turbulent boundary layer are greater than those in a laminar one.
The unfortunate fact for aircraft designers is that turbulent flow is much more common in nature than laminar flow. The tendency for flow to be random rather than layered can be interpreted in a similar way to the second law of thermodynamics.
The fact that entropy in a closed system only increases is to say that, if left to its own devices, the state in the system will tend from order to disorder. And so it is with fluid flow.
However, the shape of a wing can be designed in such a manner as to encourage the formation of laminar flow. The problem back then, and to this day, is that laminar flow is incredibly unstable. As a result, most of the laminar flow wings that have been designed based on idealised conditions and smooth wing surfaces in a wind tunnel have not led to the sweeping improvements originally imagined.
Some of their research suggested the wrapping of a glove around the leading edge of a Boeing just outboard of the engine. The modified shape of this wing promotes laminar flow at the high altitudes and almost sonic flight conditions of a typical jet airliner. To prevent the build up of insect splatter at take-off a sheath of paper was wrapped around the glove which was then torn away at altitude.
The active test panels essentially consisted of titanium covers perforated with millions of microscopic holes, which were attached to the leading edge and the top surface of the wing. The role of these panels was to suck most of the boundary layer off the top surface through perforations using an internal pumping system. By removing air from the boundary layer its thickness decreased and thereby promoted the stability of the laminar boundary layer over the wing.
This Supersonic Laminar Flow SLFC project successfully maintained laminar flow over a large portion of the wing during supersonic flight of up to Mach 1.
F XL with suction panels to promote laminar flow. While these elaborate schemes have not quite found their way into mass production probably due to their cost, maintenance problems and risk , laminar flow wings are a very viable future technology in terms of reducing greenhouse gases as stipulated by environmental legislation.
An important driver in reducing greenhouse gases is maximising the lift-to-drag ratio of the wings, and therefore I would expect research to continue in this field for some time to come. Hello, Thank you very much for this article.
I would like to know more about the approximated law given the boundary layer thickness in case of turbulent flow. Hi Kevin, thanks for your message. Only a thought but if viscosity is defined by Sutherlands law I. I guess the question is — is it worth it? Carrying that extra kit — but leading edges of modern aircraft already have heating elements for deicing — but how long would it take to get the viscosity down. Sure, it would definitely be a matter of a cost-benefit analysis. Intuitively, my first thoughts are that the faster you are going the harder it will be to get heat into the boundary layer to reduce viscosity and skin friction.
By analogy I would presume that conduction, which is also based on molecules boundcing into each other, would be low. This looks involved- it will take me a while to absorb it all. I was thinking of just making a simple spreadsheet with the derivation of viscosity from Sutherlands law over a range of temperatures probably air. Plug the resultant viscosities into the derivation of gamma. Rainer — do you or have you worked in Switzerland- I only ask because I used to work with someone called Rainer over there.
Hi David, yes, absolutely. That would be the simplest and quickest way to get a feel for how the viscosity changes the Mach number for a given pressure ratio and fun of course. But I guess the question remains could you actually get sufficient heat flux into the air with something like a de-icing device to cause an appreciable change in temperature?
I guess the fluid right at the surface of the wing will take the temperature of the heating device, but then it might drop off quite dramatically moving away from the surface. Ha, no I have never worked in Switzerland. Heating elements across the body? Given a certain wing skin temperature, how much can you expect to heat up the surrounding fluid? I just have a quick question do you calculate a drag coefficent based on combined boundary layer flow, so laminar going to turbulent flow?
One usually thinks of a boundary layer as being thin compared to the scale of the body on which it develops. This is true at high Reynolds numbers, but it is not true at low Reynolds numbers. I will show you here, by a fairly simple line of reasoning, that the boundary-layer thickness varies inversely with the Reynolds number. The thickness of the boundary layer is determined by the relative magnitude of two effects:.
The greater the second effect compared with the first, the thinner the boundary layer. Think in terms of the downstream component of fluid momentum at some distance away from the solid boundary and at some distance downstream from the leading edge of the boundary layer. The slowing of fluid by friction is a little trickier to deal with. Think back to Chapter 1 , where I introduced the idea that the viscosity can be thought of as a cross-stream diffusion coefficient for downstream fluid momentum.
In line with that idea, within the boundary layer the downstream fluid momentum is all the time diffusing toward the boundary. Fluid dynamicists like to say that the boundary is a sink for momentum.
The rate of thickening of the boundary layer is crudely represented by the ratio of downstream transport of momentum, on the one hand, to the rate of decrease of momentum at a place on account of the diffusion of momentum toward the boundary, both of these quantities having been derived in the last paragraph:. For the flat plate, this Reynolds number is based on the distance from the leading edge; for the sphere, it is based most naturally on sphere diameter.
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