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13.1.3: Proposed solution

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    78423
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    截屏2022-03-15 下午7.42.59.png
    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.

    13.1.3: Proposed solution is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.