Fluids – Lecture 3 Notes 1. 2-D Aerodynamic Forces and Moments 2. Center of Pressure 3. Nondimensional Coefficients Reading: Anderson 1.5 – 1.6 Aerodynamics Forces and Moments Surface force distribution The fluid flowing about a body exerts a local force/area (or stress) f~on each point of the body. Its normal and tangential components are the pressure p and the shear stress τ . ( magnitude greatly exaggerated) V f f τ RM RL D N A α α p local pressure and shear stress components force/area distribution on airfoil τ r dsr resultant force, and moment about ref. point alternative components of resultant force The figure above greatly exaggerates the magnitude of the τ stress component just to make it visible. In typical aerodynamic situations, the pressure p (or even the relative pressure p − p∞) is typically greater than τ by at least two orders of magnitude, and so f~is very nearly perpendicular to the surface. But the small τ often significantly contributes to drag, so it cannot be neglected entirely. The stress distribution f~integrated over the surface produces a resultant fo r ce ~R, and also a moment M about some chosen moment-reference point. In 2-D cases, the sign convention for M is positive nose up, a s shown in the figure. Force components The resultant force ~R has perpendicular components along any chosen axes. These axes are arbitrary, but two particular choices are most useful in practice. 1� � Freestream Axes: The ~R components are the drag D and the lift L, parallel and perpendic-ular to ~V∞. Body Axes: The ~R components are the axial force A and normal force N, parallel and perpendicular to the airfoil chord line. If one set of components is computed, the other set can then be obtained by a simple axis transformation using the angle of attack α. Specifically, L and D are o bta ined f r om N and A as follows. L = N cos α − A sin α D = N sin α + A cos α Force and moment calculation A cylindrical wing section of chord c and span b has force components A and N, and mo-ment M. In 2-D it’s more convenient to work with the unit-span quantities, with t he span dimension divided out. ′ ′ ′ A ≡ A/b N ≡ N/b M ≡ M/b V α u up (s ) p (s )l l θ θ u uτ (s ) x τl (s )l y θ us uds sl lds uτ up uds b c On the upper surface, the unit-span force components acting on a n elemental area of width dsu are ′ dN = (−pu cos θ − τu sin θ) dsuu ′ dA = (−pu sin θ + τu cos θ) dsuu And on the lower surface they are ′ dN ℓ = (pℓ cos θ − τℓ sin θ) dsℓ ′ dA ℓ = (pℓ sin θ + τℓ cos θ) dsℓ Integration from the leading edge to the trailing edge points produces the total unit-span forces. TE � TE ′ ′ ′ N = dN + dN ℓu LE LE TE � TE ′ ′′ A = dA + dA ℓu LE LE 2� � � � � � � � � The moment about the origin (leading edge in this case) is the integral of these forces, weighted by their moment arms x and y, with appropriate signs. TE � TE TE � TE ′ ′ ′′ ′ M = −x dN + −x dN ℓ + y dA + y dA LE u uℓ LE LE LE LE From the geometry, we have dy ds cos θ = dx ds sin θ = −dy = − dx dx which allows all t he above integrals to be performed in x, using the upper and lower shapes of the airfoil yu(x) and yℓ(x). Anderson 1.5 has t he complete expressions. Simplifications In practice, the shear stress τ has negligible contributions to the lift and moment, giving the following simplified forms. ′ c � c dyℓ dyuL = cos α (pℓ − pu) dx + sin α pℓ − pu dx 00 dx dx � � � � �� ′ c dyu dyℓM = pu x + yu − pℓ x + yℓ dx LE 0 dx dx A somewhat less accurate but still common simplification is to neglect the sin α term in L′ , and the dy/dx terms in M′ . c ′ L ≃ (pℓ − pu) dx 0 c ′ M ≃ −(pℓ − pu) x dx LE 0 The shear stress τ cannot be neglected when computing the drag D′ on streamline bodies such as airf oils. This is because for such bodies the integrated contributions of p toward D′ tend to mostly cancel, leaving the small contribution of τ quite significant. Center of Pressure Definition The value of the moment M′ depends on the choice of reference point. Using the simplified form of the MLE integral, the moment Mref for an arbitrary reference point xref is c ′ = −(pℓ − pu) (x − xref ) dx Mref 0 ′ ′ = MLE + L xref This can be positive, zero, or negative, depending on where xref is chosen, as illustrated in the figure. At one particular reference location xcp, called t he center of pressure, the moment is defined to be zero. ′ ′ ′ Mcp = M + L xcp ≡ 0LE ′ ′ xcp = −MLE/L 3� or or xcp lp −puL L LM < 0 M = 0 M < 0 The center of pressure asymptotes to +∞ or −∞ as the lift tends to zero. This awkward situation can easily occur in practice, so the center of pressure is rarely used in aerodynamics work. For reasons which will become apparent when airfoil theory is studied, it is advantageous to define the “standard” location f or the moment reference point of an airfoil to be at its quarter-chord location, or xref = c/4. The corresponding standard moment is usually written without any subscripts. c ′ ′ ≡ M = −(pℓ − pu) (x − c/4) dx Mc/4 0 Aerodynamic Conventions As implied above, the aerodynamicist has the option of picking any reference point for the moment. The lift and the moment then represent the integrated pl−pu distribution. Consider two po ssible representations: 1. A resultant lift L′ acts at the center of pressure x = xcp. The moment about this ′point is zero by definition: Mcp = 0. The xcp location moves with angle of attack in a complicated manner. 2. A resultant lift L′ acts at the fixed
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