__REYNOLDS NUMBER__

During the 19th Century a physicist named Osborne Reynolds was involved in experiments with the flow of fluids in pipes and he made the important discovery that the flow changed from streamlined to turbulent when the velocity reached a value which was inversely proportional to the diameter of the pipe. The larger the diameter of the pipe, the lower the velocity at which the flow became turbulent, e.g. if the critical velocity in a pipe of 2.5 cm diameter was 6 m/s then 3 m/s would be the critical velocity in a pipe of 5 cm diameter. He also discovered that the rule applied to the flow past any body placed within the stream. For example, if two spheres of different sizes were placed within a flow, then the transition to turbulence would occur when the velocity reached a value which was inversely proportional to the diameter of each sphere. Turbulence would therefore occur at a lower speed of flow over the larger sphere than over the smaller, and furthermore, the transition point would be at the point of maximum thickness of the body, relative to the flow. He observed that the density, viscosity and the velocity of fluid and the diameter of the tube, all played a part in determining whether the flow would be laminar or turbulent. He realized that he could combine the influence of all these factors in one non-dimentional parameter viz. __ρ__ __V l__ / µ which subsequently became known as the Reynolds Number, denoted by RN. It is dimensionless

RN = ρ __V l__ / µ Where

ρ = density in Kg/ M³

v = velocity of the test in metres per second

l = a dimension of the body (for aerofoil the chord length is used),

μ = viscosity of the fluid.

Considering the units involved it is not surprising to see test results quoted at : RN = 4 X 10^{6}, or even 12 X 10 ^{6} (12,000,000).

If RN were relatively small (less than ½ Million) the flow was always laminar, if RN were relatively large (more than 10 million) the flow was always turbulent. If RN had some intermediate value (between one and five million), the flow might be either laminar or turbulent, according to the other conditions of the experiment.

__Effect of RN on Boundary Layer__

At low RN the flow is normally laminar and at high RN the flow is turbulent. In general flow problems RN is defined as ρVL / µ where L is any representative length of the body. In the flow past an aerofoil there are two such concepts to take account of:

(a) Overall RN defined as:

RN = ρVC / μ Where C is the Chord length and is used in place of L.

(b) Local RN (Rx) at a point, a distance x behind the leading edge denoted as:

Rx = ρ V x/ μ

The relationship between RN and the nature of the boundary layer explains why in the flow past an aerofoil, the boundary layer is usually laminar to begin with but turbulent further downstream. Near the leading edge Rx is relatively small which implies laminar flow. Further downstream x increases until eventually Rx is so large that the flow must become turbulent. At RN less than half a million the boundary layer will be entirely laminar unless there is extreme surface roughness or turbulence induced in the air stream. RN between one and five million produce boundary layer flow which is partly laminar and partly turbulent. At RN above ten million the boundary layer characteristics are predominantly turbulent.

From the known variation of boundary layer characteristics with RN certain general effects may be anticipated. In the first graph of Fig , the effect of RN on the C_{L} curve shows that at a high value of RN the stall is delayed to a higher angle of attack and that the C_{L max} is increased. This merely means that if the pilot flies his aircraft to the high speed stall the wings will produce more lift (and, incidentally, less drag) because, if the aircraft is flown at a given height the density and temperature (which affects viscosity) are constant, and the only variable is speed. The simple reason is that the turbulent boundary layer has greater inertial force, or greater kinetic energy, the effect of which is to delay boundary layer separation. This results in a higher stalling angle, higher C_{L max}, and to complete the picture, a lower value of section drag coefficient (Fig ).

__Critical Reynolds Number__

In general, turbulent boundary layers result in much more drag than laminar ones, and it may seem to be an advantage to ensure laminar flow if at all possible. However, there is one respect in which the reverse may well be the case. If the flow separates, the resulting increase in drag is many times greater than the increase resulting from transition, and there are additional adverse effects. Also, a laminar boundary layer tends to separate much more readily than a turbulent one. Thus, if one has a flow with a laminar boundary layer which separates early to give high drag, transition to turbulent flow with the resultant delay in separation may well result in a substantial reduction in drag.

This is well illustrated by the example of a circular cylinder (or sphere) in a real fluid. The boundary layer is laminar, and early separation takes place, giving a large dead air region (known as a Von Karman vortex street) and a high drag coefficient. At higher Reynolds numbers, transition takes place at a point upstream of the separation point. As a result, the turbulent boundary layer remains attached until a point much further round is reached. The size of the dead air region is much reduced, as shown in the lower diagram of Fig and so is the drag coefficient. This phenomenon occurs quite suddenly as the Reynolds number is increased through a certain value which is known as the critical Reynolds number or transition RN. In the same way, transition from laminar to turbulent flow in the flow past an aerofoil may often delay the onset of stall. The affect of large RN can be generated by artificially inducing turbulence, like what is caused by the dimples on a golf ball, where these cause early transition resulting in a delayed separation and thus lesser drag and a longer shot.

The swing of a cricket ball is another example of this kind of effect. Consider the ball with its seam at an angle to the flow direction. The boundary layer at the front on the side away from the seam is laminar, early separation occurs on that side of the ball. On the other side, the roughness of the seam causes early transition so that separation is delayed. The result of this asymmetry in the flow is a side force which produces sideways movement of the ball. The speed, and the angle of the seam, is crucial. Also, the ball must be fairly new because, if the surface is too rough, there will be turbulent boundary layers on both sides, without the aid of the seam

** Variation of Airflow over an Aerofoil with Changing RN. **The characteristics depicted by Fig below are for a typical ‘conventional’ aerofoil section. The lift curve shows a steady increase in C

_{L max}with increasing RN. However, note that a smaller change in C

_{L max}occurs between Reynolds numbers of 6.0 and 9.0 million than occurs between 2.6 and 6 million. In other words, greater changes in C

_{L max}occur in the range of Reynolds number where the laminar (low energy) boundary layer predominates. The drag curve for the section show essentially the same feature i.e. the greatest variations occur at very low Reynolds numbers. Typical full scale Reynolds numbers for aircraft in flight may be 3 to 500 million where the boundary layer is predominantly turbulent. Scale model tests may involve Reynolds number of 0.1 to 5 million where the boundary layer will be predominantly laminar. Hence, the ‘scale’ corrections are very necessary to correlate the principal aerodynamic characteristics.

The most direct use of Reynolds number is the indexing or correlating the skin friction drag of a surface. Fig 4-8 illustrates the variation of the surface friction drag of a smooth flat plate with Reynolds’s number, which is based on the length or chord of the plate. The graph shows separate lines of drag coefficients if the flow should be entirely laminar or entirely turbulent. The two curves for laminar and turbulent friction drag illustrate the relative magnitude of friction drag coefficient if either type of boundary layer could exist. The drag coefficients for either laminar or the turbulent flow decrease with increasing Reynolds’s number since the velocity gradient decreases as the boundary layer thickens.

In order to obtain low drag sections, the transition from laminar to turbulent must be delayed so that a greater portion of the surface will be influenced by the laminar boundary layer. The conventional, low speed aerofoil shapes are characterized by minimum pressure points very close to the leading edge. Since high local velocities enhance early transition, very little surface is covered by the laminar boundary layer. A comparison of two 9 percent thick symmetrical aerofoils and their drag characteristics are presented in Fig below. One section is the conventional NACA 0009 section which has a minimum pressure point at approximately 10 percent chord at zero lift. The other section is the NACA 66-009 which has a minimum pressure point at approximately 60 percent chord at zero lift. The lower local velocities at the leading edge and the favorable pressure gradient of the NACA 66-009 delay the transition to some point further aft on the chord. The subsequent reduction in friction drag at low angles of attack is called “Drag Bucket”.

Skin Friction Drag

__Scale Effect__

Variation of aerodynamic characteristics with RN is called Scale Effect and is a very important consideration in wind tunnel testing. The two most important section characteristics affected by scale effects are drag and maximum lift – the effect on pitching moments usually being negligible. All experiments are carried out with models. Models have three drawbacks. These are:

- The smaller the model, the more difficult it is to make it accurate.

- Larger models would be constrained by wind tunnel dimensions- larger wind tunnels would be required.

- Consider a 1/10
^{th}Scale model. All linear dimensions are 1/10^{th}, but the areas are 1/100^{th}and the mass (assuming same material for construction) is 1/1000^{th}. So the model is to scale in some respects but not in others. This is one of the difficulties in trying to learn from flying models of aircraft, and unless the adjustment of weights is very carefully handled, the results of tests in manoeuvre and spinning may be completely false. If we consider two bodies of identical shape but different sizes placed at the same attitude in a uniform stream (possibly of different fluids, under different conditions and at different speeds), then we say that the two flows are geometrically similar. If however the flows, though incompressible, are not inviscid, the presence of a boundary layer must be taken into account. Even if the flows are geometrically similar, the boundary layer may be laminar in one case and largely turbulent in the other, so that the drag coefficients will be different. The way in which RN influences aerodynamic characteristics is, as mentioned before, referred to as scale effect, since scale largely determines the value of RN. If however, transition takes place at corresponding points on the two bodies (say one-third of the way back from the leading edge) then the two flows will be dynamically similar. Assuming that the transition RN is the same in both the cases, this implies that the overall RN was also the same. Thus the condition for dynamic similarity is geometric similarity plus identity of RN. In this case the aerodynamic force coefficients are the same for the two bodies and there is no scale effect.__Geometric and Dynamic Similarity.__

If the flow patterns over a full scale aircraft are not the same as being produced by the model, then fundamental laws of aerodynamics (L= C_{L} ½ ρ V ^{2} S and D = C_{D} ½ ρ V ^{2} S) will not hold good. In order to make these laws hold good, certain conditions need to be applied. These conditions are based on Reynolds’s experiments, i.e. product of velocity (V) and linear dimension (L) must remain constant in order to produce a similarity of flow. This is also referred to as the VL law. If it were necessary to carry out a test on a 1/10^{th} scale model to determine what would happen on the full size aircraft at 400 kmph, the wind-tunnel speed would have to be 4,000 kmph. Furthermore, the wind-tunnel would have to be very large, to prevent interference between the tunnel walls and the flow over the model, especially at such high Mach numbers. In addition, the area of the model would be 1/100^{th} that of the aircraft and the wing of the model would have to support forces equal to those on a full size aircraft.

Fortunately, Reynold also discovered that if different fluids were used, the type of flow was affected by the density and viscosity of the fluid. In fact, he established that similarity of flow pattern

would be achieved if the value of density X velocity X size/ viscosity was constant.