One of the main problems of classical mechanics is that of turbulent fluid flow. The reason for this is that there is the presence of thin no-slip boundary which requires a lot of meshpoints in order for the solution to be arrived. There is a research which has been arrived and researched upon which shows that with the use of slip boundary condition to be the model for friction to be achieved for the small skin, there is the possibility for predictive simulation for mean quantities. An example of such mean quantities is the drag and lift turbulent flow with millions of mesh points. This paper will focus on how the Kutta-Zhukovsky Condition was developed and the contributions this development has made to fluid mechanics.
Introduction
The development of flying objects whose masses are greater than that of air has been a great discovery for many scholars and researchers. There was research that was undertaken in the 1890s by Otto Lilienthal on the bird’s gliding effect and how this could be implemented in the real world. With these studies, he was able to make 2000 successful heavier-than-air flights that used the gliding effects of the birds. This was the start of success in this field. This was stopped in 1896 when he broke his neck after falling. The aircraft stalled at 15m latitude. The developments were enhanced in 1903 after two brothers, Wilbur and Wright managed to use 12 horse-power engines to make their Flyer aircraft off the ground. This was another major contribution and success of the aircraft. There were then some modifications of the flow of air around the section of a wind. This was modified with the introduction of zero lift/drag. This was a major breakthrough in the development of an aircraft wing. This condition was such that there was the introduction of large scale rotation of air around a wing section which was 2-dimentsioanl in nature. This illustration is shown in the diagram below:
Figure 1: Kutta-Zhukosvky theory
The explanation of Kutta-Zhukovsky shows how a lift is generated with the use of the technology that has been proposed. From the diagram, it is clear that there are combinations of the high (H), low (L) pressure combinations form to get the net lift by way of producing low pressure at the top of the wing. High pressure is at the bottom. This will eventually produce a net lift to the aircraft. The suggestion that was made by Kutta-Zhukosvky was that the circulation of the air around the wing section met a counter-rotating which was so-called starting some kind of vortex from behind the wing. This is shown in the diagram. This gives zero total rotation in the wing section. This made use of Kelvin’s theorem. The theorem of Kutta-Zhukosvky to enable lift that was in tandem with the angle of attack was in agreement with the long wings and the small angles of attack. This was not the case for short wings with small angles of attack while the drag was still reading zero. The drag should be some other figure apart from the zero reading. Although there exist such shortcomings, the Kutta-Zhukosvky is the only theory that attempted to solve the lift process and was successful at it. The only problem that is seen with the Kutta-Zhukosvky is that it is fictional theorem in mathematics and does not describe physics. In real life situation, there is nothing like large scale circulation that is found around the wing. There is also no starting vortex that is found behind the wing.
Literature Review
There has been research about the validity of the theory of Kutta-Zhukosvky. One argument regarding this theory is that there is no physical relationship of the technology and that there is no explanation of the origin. This shows that it is not possible to fly with this theory.
There have been other developments to show flight and it takes place. One popular theory is that of solving d’Alembers Paradox which strives to explain that the zero lift/drag is not possible because of the fact that potential flow of air is not in a stable condition when there is separation. In this theory, the flow of air around a wing may be referred to as potential flow of air which has been modified by a mechanism which is non-potential. This is in the form of 3D slip separation and has point stagnation. This brings about the instability of 2d irrotational potential flow separation. In the process there is the generation of lift and drag by way of avoiding the building up of high pressure on potential flow at potential which can be seen at the upper surface of the wing just before the edge which is trailing. This theory is different from that that has been presented by Kutta-Zhukosvky. This is also different from the explanation of the viscosity effect by Prandtl.
In new developments and explanations of the causes of flights, there is the theory that the real cause of flight is caused by zero drift/drag that is caused by a high pressure that forms at the edge of the wing. This effect cancels the positive lift and the positive drag that is found to form at the leading edge. There is then the separation of air flow at non-sharp trailing edge. The separation is 3d rotation. This process builds up the required high pressure of potential flow and in the process generates both the lift and the drag. This is shown in the diagram below.
Figure 2: New theory of lift and drag. There is the creation of potential flow that gets modified by the main mode of instability at the stage of instability
The separation by 2d potential flow can only take place by way of stagnation and there has to be zero flow velocity because of the fact that a solid boundary is a streamline. Lift is achieved by preservation of potential flow without having any form of separation up to the separation of 3d slip separation. This process results in changing the direction of the flow in downwash. This is achieved through the combination of suction that comes from the upper side and the push that is got from the lower side. The rotational slip generation is achieved through the unstable state of potential flow that is found at separation. This is made up from stream wise vorticity rolls of counter-rotating and of low pressure. This is initiated through meeting flows which move in different directions. This is shown clearly in the figure 3 and 4
Figure 3: Turbulent separation that forms counter-rotating low-pressure
Figure 4: Slip separation through trailing edge low pressure
There is now the focus that is directed at attachment and separation of the flow around the wing. The particular point of interest is 3d stability of the flow that is found near the trailing edge of the wing. The phenomenon of the flow of air around the wing uses the simple principle that the air flow will attach at the leading edge and will separate at the trailing edge. This takes place as particles approach and leaves the wing proximity. The two processes of attachments and separation will require that there is zero flow velocity that is achievable by adverse pressure gradient. This is possible when the flow is opposing at separation and from the surface of the wing when there is attachment taking place.
There has been a concern to have improvement on 3d separation. This is because of the danger of the concepts that were passed from 2d. There has been the need to reconsider the reattachment points, the separated bubble, and the zone of recirculation.
There have been developments that have been done in order to improve on Kutta-Zhukosvky condition. The developments have been based on having a new trailing edge that a wing has. In the initial Kutta-Zhukosvky circulation theorem, it is presented that there will be no lift taking place without having a sharp trailing edge. This is because a lift is supposed to come from circulation that is to eliminate the singularity of potential flow that is to be experienced at the sharp edge of the wing.
There are different angles of attacks that are used in the management of the lifts.
There have been various concerns about the dependence of the lift and drag on the radius of the trailing edge. This is another development that was added to the Kutta-Zhukosvky condition. Rom this new development, it was found out that a wing which had a diameter which was less than 1% of the length of the chord showed the same technology and procedure of lift and drag as that of a sharp end which has the maximum sharpness. At the same time, the moderate increase of drag was recorded to be 2%. The new development makes the assumption that the trailing edge is more or less rounded in a smooth manner. This is open for analysis, both computational and mathematical. This is also true in practice. In reality, real wings are not with knife-sharp trailing edges. There is some sort of smoothness that has been integrated. This follows the fact that in practice or in theory, there is no way that there is the use of sharp trailing edges of a wing. There is a need to have some sort of smoothness. This new development gives the possibility of having accurate computations using 3d rotational slip resolution separation pattern. This is possible without having any excessive mesh points.
It is also worth noting that the Kutta-Zhukosvky theory of using sharp trailing ends is tedious from practical and mathematical aspects. From the practical point of view, this is cumbersome because of the fact that real wings do not have sharp ends. From mathematical point of view, this is not possible because the flow has some features of singularity. This is difficult to be analyzed and understood. The reason why this new development has been hard to achieve is due to the fact that Kutta-Zhukosvky claimed that lift can be achieved by having sharp edge with some form of singularity. There is also the issue of Prandlt who claimed that drag is achieved through a layer which is vanishingly thin. These two assumptions and claims made further research on them harder to be undertaken. This sought physics in mathematical evasive singularities.
There is the also the issue of lift and drag invariance which has been advanced development of Kutta-Zhukosvky. The equations of stationary Navier-Stokes which have slip and viscosity which are vanishingly small are not variant to scale because of the fact that if there is a change in space scale leaves the equation in an invariant state. This therefore translates to having the 3d rotational patter of separation that is found at the trailing edge to be the same. There will be no changes that will be experienced in the trailing edge. This is because the radius, r, of the trailing edge tends to zero. The gradient and vorticity scale is now seen to scale to 1/r locally, whereas the local pressure is 1. The total effect being seen to be scaling to r. this, thus, will bring the scale to be tending to zero with r. with the use of 3d rotational separation, the flow will be able to undertake a separation without undergoing the high pressure that is experienced with 2d potential flow. This is seen to be the new development of flight.
In this new development, as the radius tends to zero, the flow is seen to be forming a type of “vortex sheet”. With this formation, there is a complex pattern of vorticity which is counter rotating streamwise. This corresponds to the pressures which oscillates high and low which is found to be small in mean and cancellation. With the similarity in scale, the lift and drag is expected to remain the same as the radius of the trailing edge of the aircraft tends to zero. This is also seen to correspond to the practical essence where the Reynolds number is required to be large enough.
For trailing edges which are blunt, the geometry will result in a separation where there is low pressure wake of equal size as the diameter (height) of the trailing edge which is blunt. As the diameter of the blunt trailing edge vanishes, there is the buildup of high pressure where the potential flow is avoided.
Future Outlook
The future of the developments will be undertaken with further exploration of theories which have not been given enough analysis and research. One of the theories which have not been given enough research time is that of KZ-solution. This forms the future of lift and drag solutions. This is the potential solution that can be used in large scale circulations. This potential solution has high pressure at the trailing edge of the potential solution. This will result in zero drag. There are many aspects that show that the KZ-solution is unphysical. One of the aspects is that there is the missing mechanism that can be used for generating large scale circulation. There is also the lack of high pressure that is seen at the trailing edge of KZ-solution so that it gives zero drag. This is missing in many experiments. Perhaps this is the potential area where further experiments can be undertaken and exploited. There are also missing aspects of this in computational simulations that are based on equations of Navier-Stokes.
The use of KZ-solution came into existence when it was first introduced as a mathematical trick that would generate a lift. This development has not been seen to represent true physics. The survival of this theory has been sustained because there has been the lack of good explanation and definition of how it really works.
Figure 5: Drag and lift coefficients which are from G2 computations utilizing the Unicorn software
There is the promise of new developments with the use of new theory by G2 computations. This is possible with long NACA 0012 wing. The lift/drag is shown in figure 5. The distributions of the force are illustrated in figure 6 and 7.
Figure 6: G2 computation of pressure where the pressure normalization is done along the lower and upper parts of the wing
Figure 7: Velocity magnitude on the surface with isosurfaces shown on the left and surface pressure shown on the right (with isosurfaces)
Concluding Remarks
The Kutta-Zhukosvky theory was one of the breakthroughs that were experienced in lift/drag experiments. There are developments that were built on that theory which have been the basis for further enhancement and better drag and lift theories. The 3d quasi patterns which are unstable are identified in terms which are analytical. The theory has enabled the approximations of aerodynamic forces of wings and aircrafts through the use of Navier-Stokes equation computations with small skin friction boundary condition. With this, there is no resolution of layers which are thin but turbulent. There were then some modifications of the flow of air around the section of a wind. This was modified with the introduction of zero lift/drag. This was a major breakthrough in the development of an aircraft wing. This condition was such that there was the introduction of large scale rotation of air around a wing section which was 2-dimentsioanl in nature. The theories have shown the places where high pressure and low pressure can be built to have stable and reliable lift/drag theories taking place. The only problem that is seen with the Kutta-Zhukosvky is that it is fictional theorem in mathematics and does not describe physics. In real life situation, there is nothing like large scale circulation that is found around the wing. There is also no starting vortex that is found behind the wing. In new developments and explanations of the causes of flights, there is the theory that the real cause of flight is caused by zero drift/drag that is caused by a high pressure that forms at the edge of the wing.
- References
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[2] K. Chang, Staying Aloft: What keeps them up there? New York Times, Dec 9, 2000.
[3] J. S. Calero, The Genesis of Fluid Mechanics 1640-1780, Studies in History and Philosophical Sciense 22, Springer, 2008.
[4] G. Schewe, Reynold’s-number effects in flow around more or less bluff bodies, 4 Intern. Colloquium Bluff Body Aerodynamics and Applications, also in Journ. Wind Eng. Ind. Aerodyn. 89 (2001).
[5] J. Kim and P. Moin, Tackling Turbulence with Supercomputer, Scientific American.
