Essential Formulae

Note: All symbols have their usual meanings as used in the text. These essential formulas may be made available for use on quizzes and exams with the permission of the instructor.

  • Equation of state:

    \[ p = \varrho R T \]

  • Hydrostatic pressure in a stagnant fluid:

    \[ \nabla p = \varrho \vec{f_b} -\varrho g\, \vec{k} \]

  • Hydrostatic pressure in scalar form:

    \begin{eqnarray*} \frac{\partial p}{\partial x} & = & \varrho f_x \nonumber \\[18pt] \frac{\partial p}{\partial y} & = & \varrho f_y \nonumber \\[18pt] \frac{\partial p}{\partial z} & = & \varrho f_z -\varrho g \nonumber \end{eqnarray*}

  • Hydrostatic equation:

    \[ \frac{dp}{dz} = -\varrho g \]

  • Unique solutions to the hydrostatic equation:
    • For a constant-density fluid:

    \[ p + \varrho g z = \mbox{constant} \]

    • Pressure change with a linear thermal gradient with temperature lapse B between two heights z_1 and z_2:

    \[ \frac{p_2}{p_1} = \left|\frac{T_0 - \alpha z_2}{T_0 - \alpha z_1}\right|^{g /R \alpha} \]

    • Pressure change with an isothermal temperature T_0 between two heights z_1 and z_2:

    \[ \frac{p_2}{p_1} = \exp\bigg( \frac{ -g (z_2 - z_1)}{R T_0}\bigg) \]

  • Specific gravity SG of a liquid = \varrho/\varrho_{\rm H_2 0}

    \[\]

  • Specific weight \gamma of a liquid = \varrho g

    \[\]

  • Temperature conversions:
    • Convert temperature T to Kelvin K from Centigrade or Celsius ^{\circ}\mbox{C}:

    \[ \mbox{K} = \, ^{\circ}\mbox{C} + 273.15 \]

    • Convert temperature T to Rankine R from Fahrenheit ^{\circ}\mbox{F}:

    \[ \mbox{R} = \, ^{\circ}\mbox{F} + 459.67 \]

  • Speed of sound:

    \[ a = \sqrt{ \gamma R T} \]

  • Mach number:

    \[ M = \frac{V}{a} \]

  • Reynolds number based on length l:

    \[ Re = \frac{\varrho V l}{\mu} \]

  • Kinematic viscosity = \mu/\varrho
  • Shear stress in a Newtonian fluid:

    \[ \tau=\mu \left( \frac{du}{dy} \right) \]

  • Equations of a streamline:

    \begin{gather*} d\vec{l} \times \vec{V} = \vec{0} \\[12pt] \frac{dy}{dz}=\frac{v}{w},~~\frac{dz}{dx}=\frac{w}{u},~~\frac{dy}{dx}=\frac{v}{u} \end{gather*}

  • Continuity equation for a fluid:

    \[ \frac{\partial}{\partial t}\oiiint_{\cal{V}} \varrho \, d {\cal{V}} + \oiint_S \varrho \, \vec{V} \bigcdot d\vec{S} = 0 \]

  • Momentum equation for a fluid:

    \[ \vec{F} = \oiiint_{\cal{V}} \varrho \vec{f} _b d{\cal{V}} -\oiint_S p dS + \vec{F}_{\mu} = \frac{\partial}{\partial t}\oiiint_{\cal{V}} \varrho \, \vec{V} d {\cal{V}} + \oiint_S (\varrho \, \vec{V} \bigcdot d\vec{S}) \vec{V} \]

  • The energy equation for a fluid:

    \begin{eqnarray*} \oiiint_{\cal{V}} \overbigdot{q} \varrho \, d {\cal{V}} + Q_{\mu} -\oiint_{S} (p d\vec{S}) \bigcdot \vec{V} + \oiiint_{\cal{V}} ( \varrho \vec{f}_b d{\cal{V}}) \bigcdot \vec{V} + \overbigdot{W}_{\mu} + \overbigdot{W}_{\rm mech} = && \nonumber \\ \frac{\partial}{\partial t} \oiiint_{\cal{V}} \varrho \left( e + \frac{V^2}{2} + g z\right) d{\cal{V}} + \oiint_{S} (\varrho \, \vec{V} \bigcdot d\vec{S}) \left( e + \frac{V^2}{2} + g z\right) & \nonumber \end{eqnarray*}

  • The energy equation for the single-stream system:

    \[ w_{\rm mech} = \frac{V_2^2 - V_1^2}{2} + g ( z_2 - z_1 ) + \left( \frac{p_2}{\varrho_2} - \frac{p_1}{\varrho_1} \right) \]

  • Bernoulli equation:

    \[ p + \frac{1}{2} \varrho V^2 + \varrho g z = \mbox{constant} \]

  • The hydraulic diameter of a pipe:

    \begin{equation*} D_h = \frac{4 \times \mbox{Area~of~cross-section}}{\mbox{Perimeter}} = \frac{4 A_c}{p} \end{equation*}

  • Pressure drop \Delta p_L for a pipe flow along a uniform diameter pipe of length L:

    \begin{equation*} \Delta p_L = \frac{1}{2} \varrho V_{\rm avg}^2 \, f \left( \frac{L}{D} \right) \end{equation*}

  • Head loss for a pipe flow:

    \[ h_L = \frac{\Delta p}{\varrho \, g} \]

  • Pumping power for a pipe flow:

    \[ P = Q \, \Delta p \]

  • Area of a wing:

    \begin{equation*} S = 2 \int_{0}^{s} c \, dy = 2 \int_{0}^{b/2} c \, dy \end{equation*}

  • The aspect ratio of a wing:

    \begin{equation*} A\!R = \frac{b^2}{S} = \frac{4s^2}{S} = \frac{4s^2}{2 \displaystyle{\int_{0}^{s} c \, dy}} \end{equation*}

  • Mean aerodynamic chord (MAC) of a wing:

    \begin{equation*} {\rm MAC} = \overline{\overline{c}} = \frac{2 \displaystyle{\int_{0}^{s} c^2 dy}}{S} \end{equation*}

  • Free stream dynamic pressure:

    \[ q_{\infty} = \frac{1}{2} \varrho \, V_{\infty} ^2 \]

Aerodynamic coefficients for an airfoil section:

    • Lift coefficient, C_{l} = \displaystyle{\frac{L/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{L'}{q_{\infty} c}}
    • Drag coefficient, C_{d} = \displaystyle{\frac{D/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{D'}{q_{\infty} c}}
    • Moment coefficient at some point a, C_{m_{a}} = \displaystyle{\frac{M_a/ \mbox{unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_a}{q_{\infty} c^2}}

Aerodynamic coefficients for a finite wing:

    • Lift coefficient, C_{L} = \displaystyle{\frac{L}{q_{\infty} S}}
    • Drag coefficient, C_{D} = \displaystyle{\frac{D}{q_{\infty} S}}
    • Moment coefficient about some point a, C_{M_{a}} = \displaystyle{\frac{M_a}{q_{\infty} S \overline{\overline{c}}}}
  • Induced drag coefficient of a finite wing:

    \begin{equation*} C_{D_{i}} = \frac{(1 + \delta) {C_L}^2}{\pi A\!R} \end{equation*}

  • The drag coefficient for an airplane:

    \begin{equation*} C_D = C_{D_{0}} + \frac{ {C_L}^2}{\pi \, A\!R \, e} = C_{D_{0}} + k {C_L}^2 \end{equation*}

  • General equations of motion of an airplane:

    \begin{eqnarray*} \mbox{$\parallel$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{d V_{\infty}}{dt} = T \cos \epsilon - D - W \sin \theta \\[18pt] \mbox{$\perp$ to flight path:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{V_{\infty}^2}{r_1} = L \cos \phi + T \sin \epsilon \cos \phi - W \cos \theta \\[18pt] \mbox{Horizontal plane:} \quad && \hspace*{-7mm} \left(\frac{W}{g}\right) \frac{(V_{\infty} \cos \theta)^2}{r_2} = L \sin \phi + T \sin \epsilon \sin \phi \end{eqnarray*}

  • Breguet endurance and range equations for a propeller aircraft:

    \[ E = \frac{ \eta_p}{{\rm BSFC}} \left(\frac{C_L^{3/2}}{C_D}\right) \sqrt{2 \varrho S} \left( \frac{1}{\sqrt{W_0 - W_f}} - \frac{1}{\sqrt{W_0} } \right) \]

    \[ R = \frac{\eta_p}{{\rm BSFC}} \left( \frac{C_L}{C_D}\right) \ln \left( \frac{W_0}{W_0-W_f}\right) \]

  • Breguet endurance and range equations for a jet aircraft:

    \[ E = \frac{1}{{\rm TSFC}} \left( \frac{C_L}{C_D} \right) \ln \left(\frac{W_0}{W_0 - W_f} \right) \]

    \[ R = \frac{2}{{\rm TSFC}} \sqrt{ \frac{2}{\varrho S}} \left(\frac{C_L^{1/2}}{C_D} \right) \left( W_0^{1/2} - W_1^{1/2} \right) \]

  • The equivalent exit velocity of a rocket in terms of specific impulse:

    \[ V_{\rm eq} = I_{\rm sp} \, g_0 \]

  • Thrust equation for rocket:

    \[ T = \overbigdot{m} \, V_{\rm eq} \]

  • Rocket equation (with gravity term):

    \[ \Delta V = V_{\rm eq} \ln \left( \frac{M_0}{M_b} \right) - g_0 t_b \]

  • Conversion factors:
    • 1 knot = 1.68781 ft/s
    • 1 knot = 0.51444 m/s
    • 1 mph = 0.44704 m/s
    • 1 mph = 1.46667 ft/s
    • 1 ft = 0.3048 m
    • 1 statute mile = 5,280 ft
    • 1 nautical mile = 6,076 ft
    • 1 hp = 550 ft-lb s^{-1}
  • Table of ISA Properties at MSL:
Property Symbol SI units USC units
Pressure p_0 1.01325\times10^{5} N m^{-2} 2116.4 lb ft^{-2}
Density \varrho_0 1.225 kg m^{-3} 0.002378 slugs ft^{-3}
Temperature T_0 15.0^{\circ}C = 288.15 K 59.0^{\circ}F  = 518.67 R
Dynamic viscosity \mu_0  1.789\times10^{-5} kg m^{-1} s^{-1} 3.737\times10^{-7}slug ft^{-1} s^{-1}
Speed of sound a_0 340.3 m s^{-1} 1116.47 ft s^{-1}
Gas constant R 287.057 J kg^{-1} K^{-1} 1716.49 ft-lb slug^{-1}R^{-1}
  • Moody chart for the estimation of the Darcy-Weisbach friction factor:
Click on the graph for a larger size.

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Introduction to Aerospace Flight Vehicles Copyright © 2022, 2023, 2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.