2. Prediction of Minimum Drag
Drag of fighter could be divided to several components measured or predicted by experiment or simulation. Actually, specific details of them are well summarized by Hoerner’s famous book for drag and lift [1]; massive information for the individual drag component and data is specified. Indeed, repeat of the information is not much meaningful in this article. This article will focus on brief information about minimum drag for jet fighter and its flight regime in first, then its prediction methods, experiment and numerical simulation achievement is reviewed.
2.1 Introduction : Components of Drag and Flight Regime
Part A Components of Drag
Importance of the drag could be emphasized by few sensitivities described in Mason’s brief lecture [2]; a one count of drag (0.0001) is equal to the two additional passengers in airliners; 90lb/count, 48lb/count weight sensitivity at sub or transonic cruise respectively. Indeed, prediction of the drag has well established for each component as shown in Fig. 2.1; the components are consisting of pressure and friction drag via cause of the drag. Sometimes these terms are re-organized by zero-lift, due-to-lift or profile, induced, wave drag in terms of flight situation or engineers’ convenience. In following paragraph, I will summarize terms of drags briefly.
Skin friction drag
Skin friction drag is very intuitive term of drag experienced by non-slip condition of any exposed surface in free-stream. Non-slip condition, attachment of air-molecule at the ‘zero-height’ of the surface, causes loss of momentum of the free-stream, and sum of these momentum loss is interpreted as skin-friction drag. Naturally, it is proportional to exposed area and traveling length of air. When we analyzed the characteristic of flow in partial differential equation called Navier-Stokes equation, ratio of the friction, viscosity of flow, and inertia is represented as Reynolds number; larger value of the number means flow is in inertia-driven flow. Indeed, any non-inviscid flow could be normalized by Reynolds number (and Mach number in compressible flow).
Most of the jet fighters are inherited to fly in higher Reynolds number region compared to other aerodynamic vehicles, and portion of the skin-friction drag is decreased in higher speed. However, it does not mean importance of skin-friction drag is negligible. Still, fighters should spend most their time in sub-transonic region, and skin-friction and transition of flow affect terminal configuration of high AoA, compressible or even in hypersonic flow (Impact of transition on high AoA flow was described in Author’s previous articles). Fig 2.2 shows change of skin-friction drag coefficient and thickness of boundary layer in terms of Reynolds number; most of the jet fighters and airliners easily exceed 106 of Reynolds number where transition from laminar to turbulent is inevitable.
Fig. 2.2. Skin friction coefficient and boundary layer thickness change via Reynolds number change for laminar and turbulent flow [1].
Laminar flow is well organized flow with friction where momentum loss of the flow is not severe. When the loss of momentum goes large, adverse pressure gradient or unstable oscillation (Linear, Tollmien-Schlichting wave, or non-linear phenomenon) caused by disturbance breakdown the organization of the flow. Then, small turbulence, usually evaluated in time-averaged characteristic value, fill boundary layer of the flow. So, it is not surprise that turbulent flow has higher skin-friction drag coefficient and thicker boundary layer as shown in Fig. 2.2. Although turbulent or vertical flow is more resistant to flow separation and welcome in high AoA flow, at least in drag perspective, laminar flow shows much better characteristics. So this is why numerous studies and experimental test [4] tried delay of the flow transition.
Most of the reference case studies for skin-friction drag was done for flat plate, ideal situation for flow characteristic development; transition of the real aircraft is much more complex. Curvature, AoA, cross-flow, imperfection of mechanical processes should also be considered; Famous P-51’s laminar airfoil suffers the imperfection and low speed characteristics problem. Imperfection surface of the laminar airfoil often causes quicker transition than lab-situation, and higher drag value than expectation of engineers. As small leading edge radius was chosen for the airfoil for drag reduction, it caused early flow separation via certain AoA as shown in Fig. 2.3. Few cases of cold war jet for skin friction drag coefficient are shown in Fig. 2.4. In rough picture, skin friction coefficient follows trend of flat plate equation in mean length, however, variance from the trend could be for more than few tens of count which is not negligible. Indeed,
Fig. 2.3. Airfoil section of few examples; P-51’s airfoil providing better drag characteristic brought other disadvantages of low speed characteristics while low drag performance could also be harmed by practical surface imperfection.
Fig. 2.4. Summary of few equations for skin friction drag coefficient and example value of few aircraft for skin-friction drag coefficient [3].
Pressure drag
Except skin-friction drag, other form of the drag could be regarded as pressure drag due to the shape of the vehicle in free-stream. Part of pressure drag could be predicted without considering skin-friction effect called induced drag however, drag like form or base drag could not be explained without friction effect. As shown in Fig. 2.5, when the flow turns around the certain object, flow is fully attached and there is no drag without considering skin friction.
When the flow loses part of or whole of its kinetic energy via skin friction, flow is separated because of adverse pressure gradient. The separated region forms large area of low pressure and makes pressure drag. If the body is well streamlined, separated region could be minimized like shape of airfoil. For the convenience, pressure drag is categorized for base-form, induced, and wave drag. Base-form drag could be regarded most classical form of pressure drag which we could imagine; induced drag occurs via wake of lift proportional to square of the lift; wave drag is caused by Mach number shock wave. More detailed explanation related to these terms will be shown in right next.
Fig. 2.5. Flow pattern of two-dimensional object with or without friction effect and change of Reynolds number condition [1]
Terms of Convenience
Some terms of the drag were divided by convenience of engineers or source of drag, and it caused some overlap of the terms as shown in Fig. 2.1. For example, wave drag can be caused by both lift and volume (form), however, wave drag is usually regarded as independent term because of its generating mechanism. In the below, except skin-friction drag, individual terms of drag will be explained.
@ Induced Drag
When the object in the air generates lift, it induces vorticity caused by pressure difference between upper and lower surfaces usually at wing-tip positions. Original direction of the lift force vector is normal to the body axis differentiated by AoA. It means some part of, sine components, the lift force is regarded as drag in stability axis system, most usual coordinate in aircraft analysis. This is shown in Fig. 2.6. Indeed, heritage of the induced drag means it is proportional to the square of the lift; both size of sine components and lift is proportional to the AoA.
Wider wing, high AR, generates smaller induced drag because induced wake affect less as wing-tip is in distance. Other linear term, represented as ‘k’ or ‘1/e’ in classical aerodynamics, determines how shape of the fuselage and wing impact on the induced drag. It considers three-dimensional effect of lift distribution along span-wise direction. it is well known that elliptic shape is the most efficient shape to minimize wing-tip wake degrading total lift.
Most of the classical theories or panel methods is well established to predict induced drag term via shape of arbitrary fuselage and wings. Due to the efforts of these studies, as described in the high AoA articles, except high AoA and minimum drag regions, prediction of intermediate AoA region is highly reliable and accurate than the two extreme regions. This article basically concentrates on minimum drag and no further discussion on induced drag term is considered.
Fig. 2.6. Concept of induced drag in two-dimensional airfoil; this is why induced drag is proportional to the square of lift, sine components of the lift.
@ Wave Drag
When the speed of aircraft in the air become near the speed of sound, compressibility takes certain amount of portion of the drag. Speed of air is basically delivery of pressure change via density variation depending on root mean square of density and temperature; decrease of density and temperature lead to decrease of speed of sound. As the speed of aircraft near the speed of sound (about M0.6), effect of compressibility become reality. Fig. 2.7 showed effect of wave drag in terms of pressure drag. Equation 3-13 in Fig. 2.7 represents the wave drag is proportional to the dynamic pressure and gradient of area of aircraft cross section.
In order to reduce the wave drag of the aircraft, bodyline of the jet fighter should be smoothened as shown in Fig. 2.8. Sudden change of cross section like start of wing or inlet could generate massive drag source for wave point of view. For each Mach number, drag could be calculated from Mach cone angle shown in Fig. 2.8; projected cross section area is important for each angle. This is called ‘Area-rule’ discovered at the development phase of F-102; then most jet fighters considering supersonic performance has been designed to have smooth increase of cross section. Fig. 2.9 showed impact of Area-rule design; in-evitable drag increase at M~1 is significantly reduced via optimization of cross section.
Fig. 2.7. Concept of wave drag [2]
Fig. 2.8. Concept of wave drag in specific Mach number (left) and cross section change via Area rule for YF-16 (right) [2]
Fig. 2.9. Drag change via Area rule for generic body (Left), and F-102A design (right) [2]
@ Zero-Lift drag
Zero-Lift drag is artificial term for drag component, counter-part of induced drag; zero-lift drag include effect of skin-friction and pressure or wave drag. Drag related to the lift is called induced drag while the drag without any lift is summarized as zero-lift drag. As shown in the next articles, it is hard to predict drag in very precise level less than 5% uncertainty for sub and supersonic condition. Indeed, lot of effort to predict zero-lift drag or minimum drag (sometimes it causes confusion as shown in Fig. 2.10) is attempted using wind-tunnel experiment, CFD, Semi-empirical formula and combination of them.
Hardness of drag value near the minimum or zero-lift condition is coming from sophisticated effect of laminar-turbulent transition, interference between components, wave-propagation, spillage air from engine-inlet, and snow-balled combination of them. Because drag value without lift wake is much smaller than that of drag with certain lift condition, small amount of the error caused by previously-acknowledged reasons can take large proportion. For more than skin-friction drag caused by exposed aircraft surface, zero-lift drag consists of base, inlet-spillage, aero-elastic effect. Base or boat-tail drag is related to the cut-off area at the tail which can cause flow separation or re-circulation behind the body. Inlet-spillage drag is caused by interaction between engine air flow and aircraft body. If amount of the mass flow for engine is just in fit, upstream streamline of the inlet seems straight forward. While required flow is too low for designed inlet (low RPM set up or too high freestream speed), excessive air is turning around the inlet usually causing additional drag (so this is why designers try to make whole data-base for thrust-drag bookkeeping). Aero-elastic effect is more complex than previous ones because structural deformation effect caused by dynamic pressure is iteratively considered.
More than those effects, interferences between components (even for their stores) should be considered while Miscellaneous drag usually made by flow of APU or LRUs also contribute their own proportion.
Fig. 2.10. Drag curve of generic aircraft; it shows there is slight difference between minimum drag and zero-lift drag [2]
PART B Flight Regime and Mechanics
In Part A, I introduced components and basic cause of drag categorized in few terms; contribution ratio of these terms is dramatically changed by flight regime, Mach number and AoA. Fig. 2.11 well summarized typical ratio change for few types of aircraft including sub/supersonic airliners, supersonic bombers, and jet fighter class.
If aircraft were in take-off status requiring high lift, induced drag take large proportion compared to zero-lift drag components; additional drag induced by landing gear also contribute their own. In cruise status. Subsonic airliner has much less induced drag than others because it has long wing span to minimize lift wake. In that configuration skin friction and pressure drag without wave one take significant amount of drag, so wing and body having wide exposed area generate lots of drag.
Supersonic aircraft experiences huge amount of wave drag and it also endure large induced drag because most of supersonic jet has smaller AR wing generating strong lift wake and induced drag. Most of the jet fighter should experience both sub and supersonic cruise status, and cruise performance of both regimes should be well considered. For some cases, in order to reduce the drag in transonic region M0.6~0.8, special airfoil called supercritical or laminar airfoil is used. Most conventional airfoil has its peak thickness at 20~30% of its chord length however, the special airfoil has its peak at more than 50% reducing rapid increase of flow speed of upper surface. Although that kind of airfoil is susceptible at high AoA in low speed, it has much lower drag coefficient in high subsonic cruise speed, called drag-bucket as shown in Fig. 2.12.
In hypersonic region, most of the drag is governed by inviscid pressure drag with compressibility effect however, at that stage, gas molecule starts to react via high temperature of kinematic heat. Indeed, gas composition change via chemical reaction is considered. Moreover, interestingly, snow-ball effect of skin-friction still influence on these reaction or nose part of the aircraft.
Fig. 2.11. Proportion of drag contribution for each flight regime [3]
Fig. 2.12. Drag coefficient change for supercritical type airfoil [1]
Drag prediction of the aircraft in design phase could be divided to two techniques, analytic/numerical estimation, and wind-tunnel/flight testing. Recent rapid progress of simulation technology make engineer depend more on CFD though other techniques still large proportion. Because of the difficulties described in previous paragraph, precision of drag prediction is varied from +-20 to 3%; variation is large for supersonic jet as shown in Fig. 2.13. As you can see in the figure, aircraft with simple configuration having cylindrical body and wing without complex joint part is very precise in their drag prediction however,
Phase of the aircraft design for drag is divided to three phases; preliminary, detailed and final design phase as shown in Fig. 2.14. In the preliminary design phase, semi-empirical formula or estimation from past design is broadly accepted because there is no specific design for the aircraft. CFD calculation or extensive wind-tunnel test could be cost in-effective; whole configuration of the aircraft is rapidly changed. In that phase, size of the jet fighter is determined and estimate whether this aircraft could meet the requirement of the customer or not.
In the detailed design phase, more specific layout of the aircraft is figured such as LERX, nose shape, wing-planform with control surfaces, and fuselage shape. Rough layout of the aircraft was determined in the previous steps, then current step evaluate and optimize layout of the jet. More precise estimation of the drag is attempted; CFD and wind-tunnel testing are extensively used to study. Initial estimation of APU or LRU effect is also developed while the fuselage and main-wing is determined. At the final phase of the design, most of the configuration is already fixed, then, more specific term or effect is determined. At this stage, prototype aircraft is built to be evaluated or partial CFD is attempted for precise evaluation.
Amount of probable error for each design phase is shown in Fig. 2.15; Initial phase has a lot of uncertainty margin for its drag and method error. As designed is fixed, uncertainty from design is significantly reduced while method error is maintained about 5% level.
From now on, I will introduce specific examples for each drag prediction technique.
Fig. 2.13. Precision of drag prediction in many jet aircraft development [3]
Fig. 2.14. Design phase effecting drag prediction technique [3]
Fig. 2.15. Amount of error in each design phase, method error and uncertainty [3]
* Reference
[1] Hoerner, S. F., 1965, Fluid-Dynamic Drag: Theoretical, Experimental, and Statistical Information
[2] Mason, W. H., 2006, ConfigAeroDrag
[3] Jobe, C. E., 1984, Prediction of Aerodynamic Drag, AFWAL-TM-84-203
[4] Marino, A ., et al., 1975, Evaluation of Viscous Drag Reduction Schemes for Subsonic Transport, NASA CR-132718
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