3.4.3: Increase in CLmax
- Page ID
- 78114
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Table 3.1 shows the typical values for the increase of coefficient of lift in airfoils.
| High-lift devices | \(\Delta c_{L_{\max}}\) |
| Trailing edge devices | |
| Plain flap and intrados flap Slotted flap Fowler flap Doble slotted Fowler flap Tripple slotted Fowler flap |
0.9 1.3 \(1.3 c'/c^*\) \(1.6 c'/c\) \(1.9 c'/c\) |
| Leading edge devices | |
| Slot Krueger and drop flap Slat |
0.2 0.3 \(0.4 c'/c\) |
| * \(c'\) is the extended chord and \(c\) to the nominal chord. | |
Table 3.1: Increase in \(C_{l_{\max}}\) of airfoils with high lift devices. Data retrieved from \(F_{\text{RANCHINI}}\) et al. [4].
The increase in the maximum coefficient of lift of the wing (\(\Delta C_{L_{\max}}\)) can be related with the increase of the maximum coefficient of lift of an airfoil (\(\Delta c_{L_{\max}}\)). For slotted and Fowler flaps, the expression is:
\[\Delta C_{L_{\max}} = 0.92 \Delta c_{l_{\max}} \dfrac{S_{fw}}{S_w} \cos \wedge_{1/4}, \nonumber \]
where \(\wedge_{1/4}\) refers to the swept measured from the locus of the \(c/4\) of all airfoils and \(S_{fw}\) refers to the surface of the wing between the two extremes of the flap. If the flap is a plain flap, the expression is:
\[\Delta C_{L_{\max}} = 0.92 \Delta c_{L_{\max}} \dfrac{S_{fw}}{S_w} \cos^3 \wedge_{1/4}. \nonumber \]
In the Table 3.2 the typical values of \(C_{L_{\max}}\) and flap deflections in different configurations are given.
| High-lift device | \(\delta_f\ TO^*\) | \(\delta_f\ LD\) | \(\tfrac{C_{L_{\max}}}{\cos \wedge_{1/4}}\ TO\) | \(\tfrac{C_{L_{\max}}}{\cos \wedge_{1/4}}\ LD\) |
| Plain flap Slotted flap Fowler flap Doble slotted** flap Tripple slotted flap and slat |
\(20^{\circ}\) \(20^{\circ}\) \(15^{\circ}\) \(20^{\circ}\) \(20^{\circ}\) |
\(60^{\circ}\) \(40^{\circ}\) \(40^{\circ}\) \(50^{\circ}\) \(40^{\circ}\) |
1.4-1.6 1.5-1.7 2-2.2 1.7-1.95 2.4-2.7 |
1.7-2 1.8-2.2 2.5-2.9 2.3-2.7 3.2-3.5 |
| * \(TO\) and \(LD\) refers to take off and landing, respectively. ** Double and triple slotted flaps have always Fowler effects increasing the chord. |
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Table 3.2: Typical values for \(C_{L_{\max}}\) in wings with high-lift devices. Data retrieved from \(F_{\text{RANCHINI}}\) et al. [4].