13.1.3: Proposed solution
- Page ID
- 78423
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
Figure 13.2: XLFR5 analysis
The solution to this exercise is left open. The software is very user-friendly. Find here a video tutorial. Extension of this exercise could be the 3D analysis (wing analysis) and the Aircraft analysis. In the sequel of this Section, a very brief and schematic overview of NACA airfoils is given. Also, some XLFR5 plots are presented in Figure 13.2.
On NACA Airfoils
The NACA airfoils are a series of airfoils created by NACA (Nacional Advisory Committee for Aeronautics), including the following series: Four-digit-series; Five-digit-series; Modifications in four and five-digit series; 1-series; 6-series; 7-series; 8-series.
4-digit series:
- First digit: describes the maximum camber as a percentage of the chord (% c).
- Second digit: describes the maximum camber’s distance measured from the leading edge in 1/10 of the percentage of the chord (% c).
- Third and fourth digit: describing the maximum thickness as a percentage of the chord (% c). [By default the maximum thickness is 30% of the chord]
Some examples include:
- NACA 2412
- Maximum camber is 2% of c. (0.02c)
- Maximum camber located at 40% (0.4c) of the leading edge.
- Maximum thickness of 12% of the chord (0.12c) - NACA 0015
- Symmetric airfoil (00)
- Maximum thickness of 15% of the chord (0.15c)
5-digit series:
- First digit: describes the \(C_l\) multiplying the digit by 0.15.
- Second and third digits: dividing them by 2, describes the maximum camber’s distance measured from the leading edge as percentage of the chord (% c).
- Fourth and fifth digits: describing the maximum camber as a percentage of the chord (% c).
- By default the maximum thickness is 30% of the chord.
The following example illustrates it:
- NACA 12345
- \(C_l = 0.15.\)
- Maximum camber located at 11.5% (0.115c) of the leading edge. This implies \(x_{mc} = 0.15\).
- Maximum camber of 45% of the chord (0.45c)
Notice that camber line is defined as follows [\(y\) and \(x\) normalized with the chord]:
\[y = \begin{cases} \tfrac{k_1}{6} \{ x^3 - 3mx^2 + m^2 (3 - m)x \} & 0 \le x \le m, \\ \tfrac{k_1 m^3}{6} (1 - x) & m \le x \le 1; \end{cases}\nonumber \]
with \(m\) is chosen so that the maximum camber takes place in \(x = c_{mc}\).
Modifications in four and five-digit series: The fourth and fifth digit series can be modifies by adding two digits with a dash.
- First digit after the dash: describes how rounded is the shape, being 0 very sharp and 6 exactly as the original airfoil, and 9 more rounded that the original.
- Second digit after the dash: describing the maximum thickness distance measured from the leading edge in 1/10 as a percentage of the chord (% c)
The following example illustrates it:
- NACA 1234-05
- NACA 1234 with sharp leading edge shape.
- Maximum thickness located at 50% c (0.5c) measured from the leading edge.
1 series:
- First digit: describes the series.
- Second digit: describes the minimum pressure’s distance measured from the leading edge in 1/10 as a percentage of the chord (% c).
- Third digit [after a dash line]: describes \(C_l\) in 1/10.
- Fourth and fifth digits [after a dash line]: describe the maximum thickness in 1/10 as a percentage of the chord.
Consider the following example as illustration:
- NACA 16-123
- Minimum pressure located at 60% of the chord.
- \(C_l = 0.1.\)
- \(E_{\max} = 0.23c\) measured from the leading edge.
6 series: It is essentially an improvement of 1-series to maximize the laminar flow:
- First digit: describes the series.
- Second digit: describes the minimum pressure’s distance measured from the leading edge in 1/10 as a percentage of the chord (% c).
- Third digit [typically as a subindex]: describes the fact that drag remains low a number of tenths below of \(C_l\).
- Fourth digit [after a dash line]: describes \(C_l\) in 1/10.
- Fifth and sixth digits [after a dash line]: describe the maximum thickness in 1/10 as a percentage of the chord.
- "a=. . . " [followed by a decimal number]: describes the fraction of chord in which the laminar flow remains. By default a=1.
Please find below an example:
- NACA 61 - 345 a = 0.5
- Minimum pressure located at 10% of the chord.
- \(C_l = 0.3\). What this means is that the airfoil was designed for maximum efficiency at a lift coefficient of approximately 0.3
- \(E_{\max} = 0.45c\) measured from the leading edge.
- The laminar flow is maintained over 50% of the chord.
7 and 8 series: Correspond to additional improvements to maximize the laminar flow both in extrados and intrados.
- First digit: describes the series.
- Second digit: describes the minimum pressure’s distance in the extrados measured from the leading edge in 1/10 as a percentage of the chord (% c).
- Third digit: describes the minimum pressure’s distance in the intrados measured from the leading edge in 1/10 as a percentage of the chord (% c).
- Letter Letter referring to an standard airfoil of previous NACA series
- Fourth digit [after a dash line]: describes Cl in 1/10.
- Fifth and sixth digits [after a dash line]: describe the maximum thickness in 1/10 as a percentage of the chord.
The following example illustrates it:
- NACA 712A345
- Minimum pressure located at 10% of the chord in the extrados.
- Minimum pressure located at 20% of the chord in the intrados.
- \(C_l = 0.3.\)
- \(E_{\max} = 0.45c\) measured from the leading edge.