Scale effect and Reynolds Number
This appendix is an amplification of a short note given about scale effect in Chapter 2. From the earliest days of the science of flight, even before any aeroplane had actually flown, people experimented with small models. The problem is how do we relate the behaviour of the model and the aerodynamic forces that are exerted on it to a full-size aircraft? We know that if we can measure, say, the lift force on a small model, we can work out its lift coefficient using L = 2pv2SCL, and from this we can calculate the lift on the full-size aircraft at any speed. However, we do not always get quite the correct answer, and sometimes we get an answer that is completely wrong. Is there something else that we should be doing? The clue to this was found by Osborne Reynolds some 150 years ago. Reynolds was not interested in aircraft aerodynamics but in the flow of liquids. In particular, he was interested in the conditions that determined whether the flow of water in a pipe was smooth and layered (laminar), which is normally associated with low-speed flow, or turbulent, which is associated with higher-speed flow. What he discovered was that the speed of flow at which the change or transition occurred depended on the value of the quantity:
which is now called a Reynolds Number.
In this quantity, p is the density of the fluid (water in his case) v is the flow speed l is the diameter of the pipe p is the viscosity of the fluid
For the problem of flow in pipes it was found that the transition from one type of flow to the other occurred at a critical value of around 2300. It was discovered that this critical value held regardless of the size of the pipe and what type of fluid was used; it even works for gases. So what does the flow of water have to do with model testing of aircraft? Well, we find that if we test our model at
the same Reynolds Number as the full scale aircraft, then it will behave in the same way. For example its wing will stall at exactly the same angle of attack, and full-scale forces calculated using the lift and drag coefficients will be correct. Of course for aircraft there is no pipe diameter involved, so for the quantity /, we have to use some other characteristic length. This leads to some confusion, because different characteristic lengths have to be used for different types of model. For a wing section, the wing chord is normally used as the characteristic length, but a missile may not have a wing, so in this case we would probably use the overall missile length. It does not actually matter which length we use, as long as we are consistent between model and full-size. It is also important to say exactly what dimension one is using. All too often Reynolds Number values are quoted without this important piece of information. For the testing of a low-speed aircraft then, apparently, all we have to do is to ensure that the Reynolds Number of the model test is the same as that of the flight conditions that we are going to simulate. For example, consider an aircraft as below, flying at sea level
Flight speed v = 30 m/s
Wing chord = 2 m
If we want to test a l/10th scale model under the same sea level conditions, then the Reynolds Numbers must be the same so the speed v that we must test the model at is found from
pv X 2/10 = p 3 0X2
After cancelling out the density and viscosity terms (which for this special case are the same for both model and full scale) we find that the required model test speed v is 300 m/s. This result is both surprising and unfortunate. The model actually needs to fly ten times faster than the real aircraft in order to correctly simulate the flight conditions. This is normally impractical, especially for faster aircraft, because the model would have to be flying at speeds where compressibility would totally change the flow.
We do not, of course, normally fly our models around the room; we put them in a wind tunnel and let the air flow past them. However, this does not immediately solve the problem that the relative air flow speed past a l/»th scale model would need to be n times as fast as the full-scale aircraft. In the example above, this would mean a tunnel speed of 300 m/s, which would not only require a very strong tunnel and a huge amount of power to drive it, but would also mean that the effects of compressibility would be important. One solution is to make the tunnel very strong indeed and raise the air pressure inside. This has the effect of increasing the density, and, as will be seen from the Reynolds Number expression above, if the density is raised, then the speed can be lowered in the same proportion. Fortunately, modern large aircraft mostly cruise at very high altitude, where the density is low, so this also reduces the speed required for the model. Unfortunately, large compressed air wind tunnels are extremely expensive both to build and to run, and for this reason there are very few of them in the world; indeed the number is if anything reducing now. Making the tunnel smaller does not help, because the smaller the model, the greater the speed required. One way to overcome the problem of tunnel size is to test small critical parts of aircraft such as a wing section at large scale in a relatively small compressed air tunnel.
Compressed air tunnels do not unfortunately solve all the problems of similarity because nowadays all but light and a few specialist aircraft need to fly in the transonic region at speeds approaching the speed of sound, and fast military aircraft have to fly supersonically. Under these conditions, getting the Reynolds Number correct is less important than getting the right condition for similarity of compressibility effects. The latter entails getting the same Mach Number on the model and the full-scale aircraft. Mach Number is given simply by the ratio (speed/speed of sound). The speed of sound depends only on the square root of the absolute temperature of the air. The difference between the sea level absolute temperature on a really hot day and the temperature in the upper atmosphere is only a ratio of about 2:3, so broadly speaking, for compressible flows, matching of Mach Numbers requires us to run the tunnel air at a speed that is quite similar to that of the full-size aircraft. Notice that the size of the model does not come into this.
Trying to match both the Mach Numbers and the Reynolds numbers at the same time is very difficult. The most practical solution has been the adoption of the cryogenic tunnel, which is a variation of the compressed air tunnel, in which the air is cooled by injecting liquid nitrogen. Cooling affects the speed of sound (and hence the Mach Number), the density and the viscosity coefficient (and hence the Reynolds Number). By suitably juggling the pressure, temperature and speed, it is possible to get a simultaneous match for both the Reynolds and Mach numbers. Such tunnels are extremely expensive to construct and to run, and only a few exist in the entire world.
Needless to say, not all wind tunnel testing is carried out in such facilities. For practical purposes, we can still make reasonably accurate predictions by use of compromises. It is fortunate that matching the Reynolds Numbers is most important for very low-speed flight, so we can still make useful measurements in low-speed (about 30 to 100 m/s) wind tunnels.
The mismatch in Reynolds Number is particularly significant when it comes to the behaviour of the boundary layers. The position of transition from laminar to turbulent flow and the point at which the flow separates is strongly related to the value of the Reynolds Number, so we can expect reasonably good results if we are testing in situations where these two factors are not likely to be critical. However, when investigating stall behaviour, for example, we may need to use a large higher speed tunnel or rely more on full-scale testing.
In high flight speeds it is the matching of Mach Numbers that is important. Thus for high-speed flight, we use specially designed transonic or supersonic tunnels where we can match the Mach Numbers, but normally have to ignore the Reynolds Number. The expensive cryogenic tunnels are used only where highly accurate work is required on a major airliner or military aircraft. All of this might seem a little baffling, and indeed it requires a great deal of experience and knowledge to achieve really reliable wind tunnel results.
I am frequently asked what is the speed at which you should test a model to simulate a given full-scale speed. The answer, as you may see from the above, is that it is the speed at which the Reynolds and Mach numbers are both simultaneously the same as they would be if full scale. As we have also seen above, though, this is not normally practical unless you have a very large research budget. In practice, the answer normally is to test as fast as your tunnel will allow. The precise speed of the test is not really important, because we can determine the full-scale lift and drag etc. by using the relationships L= 2 pv2SCL and D = 2pv2SCD The wind-tunnel data allow us to work out the lift and drag coefficients, and these can then be used to determine the values of lift and drag that would be obtained at full-scale size air density and speed. We just have to hope that the effects of Reynolds Numbers, are small.
Apart from the problems of trying to match Reynolds and Mach numbers, wind tunnels have some other drawbacks. These mainly arise from the fact that the air is constrained by the tunnel walls and cannot behave exactly as it would if there were no boundary. For example, the model forms partial blockage in the tunnel and thus it causes the flow to speed up in its proximity, thereby giving misleadingly high loads. Explaining the details of these effects is well beyond the scope of this book. What is normally done, however, is that some theoretically based corrections have to be applied. Whole books and many scientific papers have been written in the subject of wind tunnel corrections, and the science is still developing.
With all the difficulties and expense involved in wind tunnel testing it is not surprising that people have sought to find ways around it. Increasing use is now being made of computer modelling or computational fluid dynamics (CFD). After many years of development, CFD can now provide accurate predictions for many aspects of aircraft aerodynamics, but it is not such a cheap or quick solution as might have been hoped for. Also, CFD is not yet reliable for situations where important flow separations occur; it can be quite poor at predicting the drag forces from areas and components where the flow is not streamlined. At the time of writing, both wind tunnel testing and CFD are used, and there is no indication that wind tunnels are about to disappear. Experience shows that there are situations, such as in the behaviour of boundary layers, where wind tunnels work best, and others where computational methods are more appropriate. Finally, it should be mentioned that with modern telemetry and remote guidance systems it is possible to do some useful testing using flying scale models. Apart from the use of radio controlled models, there is a whole new area of testing which involves making piloted scale models, usually with relatively cheap composite material airframes.