Critical Mach Number

111 Additional Aerodynamic Considerations Compressibility With the exception of Gas Dynamics, almost all of the fluid mechanics taken by a UC undergraduate has been for incompressible flows. In addition, Gas Dynamics is devoted mostly to supersonic flow theory, i.e., shock and expansion waves. While these types of phenomena are important and will be discussed in more detail later, there exists a real gap in knowledge between the incompressible and supersonic flows. A great deal can be done in this regime by utilizing thin airfoil theory, in which the governing equations can be written: ∂ 2φ ∂ 2φ β 2 + = 0 (11.1) ∂x 2 ∂y2 2 where β ≡ 1− M ∞ . Equation (11.1) reduces directly to the Laplace equation for incompressible flows. However, a simple transformation exists that allows us to take the flow governed by Eq. (11.1) and transform it into a flow about an “equivalent” incompressible body. The major details of this approach can be found in the Anderson text. However, a compressibility correction can be established from this approach such that if we know the pressure coefficient on an airfoil in an incompressible flow, C , we can obtain the pressure coefficient p o on the same airfoil in a compressible flow via: 112 C C = PO (11.2) P 2 1− M ∞ This is known as the Prandtl-Glauret Rule. As fortune would have it, since lift and drag are related to the pressure coefficient, the same sort of correction holds for Cl and Cm . C C = lO (11.3) l 2 1− M ∞ C C = mO (11.4) m 2 1− M ∞ Two higher order corrections to the Prandtl-Glauret rule exist: The Karman-Tsien Rule C C = PO P M 2 C 1− M 2 + ∞ PO ∞ 2 1+ 1− M ∞ 2 and Laitone’s Rule C PO CP = 2 γ −1 2 M ∞ 1+ M ∞ 2 2 CP 1− M + O ∞ 2 1− M ∞ 2 113 Critical Mach Number These corrections are appropriate until the start of transonic flow, around Mach 0.85. The critical Mach number, M cr , is the freestream Mach number at which the local Mach number at some point on the airfoil becomes sonic. 114 The corresponding pressure coefficient is known as the critical pressure coefficient, C P cr . An estimate for C P cr can be found in a round about manner. We start with the general equation for the pressure coefficient: 2 p C = A −1 (11.5) P , A 2 γM ∞ p∞ p and use isentropic relations to replace A with an expression in p∞ terms of M A and M ∞ . We can then set M A to sonic and get the relationship between the critical Mach number and pressure coefficient. γ γ −1 γ −1 1+ M 2 2 cr C = 2 −1 (11.6) P cr γM 2 γ −1 cr 1+ 2 However, this relation only matches possible critical Mach numbers and pressure coefficients, since it is only an isentropic relationship for the pressure coefficient. The way it can be used is to introduce a given airfoil and find its most negative pressure coefficient. If this is done at incompressible conditions, the Prandtl-Glauret rule can be used to map the variation of that CP with Mach number. The two curves meet at M cr . 115 Consider the following figure Which demonstrates the goodness of thin airfoils, i.e., smaller magnitude negative CP and hence a higher M cr . 116 Drag Divergence Mach Number The M cr is an important demarcation line after which the drag begins to rise, however, a second point of more rapid drag rise is the drag divergence Mach number, M drag−divergence , as illustrated in the figure below: Thin Airfoils Clearly the M cr and M drag−divergence imply that thin airfoils are useful. The trend for recent high performance aircraft is shown in the figure below: 117 This argument also explains why wing sweep is useful, that is, as sweep angle, Ω, increases from zero, the effective chord seen by a streamline increases, but the airfoil thickness does not, thereby reducing the effective thickness to chord ratio and creating a thinner wing. 118 Area Ruling In the early 1950s Küchemann recognized that wing sweep results in 3D flow about a wing that is very different from that of an infinite wing, since the flow tends toward the wing root. However, the fuselage requires that the flow return to parallel, hence, compression waves form which leads to a serious increase in drag. Whitcomb recognized this problem and related it to ballistic experience that showed smooth cross sectional area is important for reduced drag. His basic idea, Whitcomb’s area rule, was to pinch an aircraft fuselage so that its cross sectional area would vary smoothly as the wing was encountered. The resultant “coke bottle” fuselage shape can have as much 2x reduction in drag over a non-area ruled shape. 119 Supercritical Airfoils The basic approach of sweep and thickness reduction is to increase M cr . However, another approach can be taken: increase M drag−divergence directly. Although this happens with an increase in M cr it can also happen independently if the airfoil is designed such that the velocity does not increase substantially in the supersonic region, resulting in a weaker recompression shock and less loss and overall drag. Airfoils of this type are called supercritical and are characteristically very flat on top as shown in the figure below: 120 Dihedral Clearly many factors enter into the choice of the wing and airfoil section. Some based on the desired flow regime, others because of structural considerations. Yet another complication comes about because of stability and control issues. Examples of this are dihedral and anhedral. A wing with dihedral has a positive angle with respect to the horizon: Dihedral angle In a roll, the aircraft looks like this As it rolls clockwise, the higher vertical force of the right wing tends to induce a counter-clockwise rotation to return the aircraft to straight and level. This enhances the stability of the aircraft but makes it “harder” to turn. The opposite is true for an aircraft with anhedral. Perhaps the most famous example of this type is the gull winged F4U Corsair used by the Marines in World War II. In a clockwise roll the left wing produces the vertical force and this tends to enhance the roll characteristics of the aircraft making it more maneuverable but less stable. 121 NACA Series Nomenclature 4 digit series NACA XXXX Max camber in hundredths of chord Max thickness in Location of max camber hundredths of chord in tenths of chord NACA 4412 Max camber 0.04c 4% camber Max camber location 0.4c 40% chord Max thickness 0.12c 12% thick 5 digit series NACA X XX XX x3/2= design lift coefficient in tenths Max thickness in /2=Location of max camber hundredths of chord from LE in hundreths of NACA 23012 Design Cl in tenths 0.3 Max camber location 0.15c 15% chord Max thickness 0.12c 12% thick 122 6 digit series NACA 6X-X XX Location of minimum pressure in tenths of chord from LE Max thickness in Design lift hundredths of chord NACAcoefficient 65-218 in Design Cl in tenths 0.2 Min pressure location 0.5c 50% chord Max thickness 0.18c 18% thick 123 Estimate of 3D Wing Drag You now have all the tools to estimate the drag on a three- dimensional wing. Induced Drag – Use the finite wing theory to calculate the drag due to lift as well as the induced angle of attack. Skin Friction Drag – Use XFOIL to calculate the given wing section at the effective angle of attack as determined above. This provides a distribution of C f that can then be integrated over the surface. Wave Drag – Use compressible flow theory to compute shocks and expansion waves about your airfoil. Then compute surface pressures and the associate drag force. C 2 C = C + L DA/C D,e πeAR Total Aircraft Drag Induced Drag Parasite Drag Profile + Wave Drag Wave Drag Profile Drag C + C d f d P Friction + Pressure.

Load more