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parasitedrag [2018/05/03 15:30]
jgravett Minor changes
parasitedrag [2020/05/27 16:15] (current)
ramcdona
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 $ D \equiv \mbox{Diameter} $\\ $ D \equiv \mbox{Diameter} $\\
 $ X_{area} \equiv \mbox{Max cross sectional area} $\\ $ X_{area} \equiv \mbox{Max cross sectional area} $\\
- ​\equiv \mbox{Fineness ratio} $\\+FR  ​\equiv \mbox{Fineness ratio} $\\
 $ Re \equiv \mbox{Reynolds number} $\\ $ Re \equiv \mbox{Reynolds number} $\\
 $ V_{inf} \equiv \mbox{freestream velocity} $\\ $ V_{inf} \equiv \mbox{freestream velocity} $\\
Line 108: Line 108:
  
 ===== Coefficient of Friction Equations ===== ===== Coefficient of Friction Equations =====
 +
 +Unfortunately,​ there is substantial opportunity for confusion around the equations for the skin friction coefficient. ​ The primary source of confusion is inconsistent nomenclature in the literature around two related quantities -- the local skin friction coefficient vs. the flat plate average skin coefficient. ​ Different references use inconsistent nomenclature to differentiate these quantities. ​ A reader must be pedantic to verify that they understand the notation for any given publication. ​ Often, the nomenclature is defined far away from where the equations are presented.
 +
 +^  Local    ^  Average ​ ^  Reference ​            ^
 +|  $C_f$    |  $C_D$    | White (3)              |
 +|  $C_{\tau}$ ​ |  $C_f$    | Hoerner (1)            |
 +|  ${c_f}'​$ ​  ​| ​ $c_f$    | Schlicting (6)         |
 +|  $c_f$    |  $C_f$    | Anderson (5)           |
 +|  $C_f$    |  $C_D$    | White & Christoph (2)  |
 +
 +This analysis is only concerned with the flat plate average skin friction coefficient,​ which will be denoted $C_f$ herein.  ​
 +
 +
 $ x \equiv \mbox{distance along chord} $\\  $ x \equiv \mbox{distance along chord} $\\ 
 $ k \equiv \mbox{roughness height} $\\  $ k \equiv \mbox{roughness height} $\\ 
Line 126: Line 139:
  
 == Blasius (25) == == Blasius (25) ==
-$$ C_f = \frac{1.32824}{Re^{1/2}}\ $$+$$ C_f = \frac{1.32824}{\sqrt{Re}}\ $$
  
 ==== Turbulent ====  ==== Turbulent ==== 
  
 == Explicit Fit of Spalding (3)== == Explicit Fit of Spalding (3)==
-$$ C_f = \frac{0.455}{\ln^2(0.06\ Re)}\ $$+$$ C_f = \frac{0.523}{\ln^2(0.06\ Re)}\ $$
  
 == Explicit Fit of Spalding and Chi (2) ==  == Explicit Fit of Spalding and Chi (2) == 
-$$ C_f = \frac{0.225}{\log(Re)^{2.32}}\ $$+$$ C_f = \frac{0.430}{\log(Re)^{2.32}}\ $$
  
 == Explicit Fit of Schoenherr (1) == == Explicit Fit of Schoenherr (1) ==
Line 141: Line 154:
 == Implicit Schoenherr (1) ==  == Implicit Schoenherr (1) == 
 $$ \log\left(Re\ C_f\right) = \frac{0.242}{\sqrt{C_f}}\ $$ $$ \log\left(Re\ C_f\right) = \frac{0.242}{\sqrt{C_f}}\ $$
- 
-== Implicit Karman (2) ==  
-$$ \frac{1}{\sqrt{C_f}}\ = 4.15\log\left(Re\ C_f\right) + 1.70 $$ 
  
 == Implicit Karman-Schoenherr (4) ==  ​ == Implicit Karman-Schoenherr (4) ==  ​
Line 149: Line 159:
  
 == Power Law Blasius (2) ==  == Power Law Blasius (2) == 
-$$ C_f = \frac{0.0592}{Re^{1/​5}}\ $$+$$ C_f = \frac{0.072}{Re^{1/​5}}\ $$
  
 == Power Law Prandtl Low Re (5) ==  == Power Law Prandtl Low Re (5) == 
Line 155: Line 165:
  
 == Power Law Prandtl Medium Re (3) ==  == Power Law Prandtl Medium Re (3) == 
-$$ C_f = \frac{0.027}{Re^{1/​7}}\ $$+$$ C_f = \frac{0.0315}{Re^{1/​7}}\ $$
  
 == Power Law Prandtl High Re (3) ==  == Power Law Prandtl High Re (3) == 
-$$ C_f = \frac{0.058}{Re^{1/​5}}\ $$+$$ C_f = \frac{0.0725}{Re^{1/​5}}\ $$
  
 == Schlichting Compressible (6) ==  == Schlichting Compressible (6) == 
 $$ C_f = \frac{0.455}{\log\left(Re\right)^{2.58}}\ $$ $$ C_f = \frac{0.455}{\log\left(Re\right)^{2.58}}\ $$
- 
-== Schlichting Incompressible (7) ==  ​ 
-$$ C_f = \frac{0.472}{\log\left(Re\right)^{2.5}}\ $$ 
- 
-== Schlichting-Prandtl (2) ==  
-$$ C_f = \frac{1}{\left(2\ \log\left(Re\right) - 0.65\right)^{2.3}}\ $$ 
- 
-== Schultz-Grunow High Re (2) ==  
-$$ C_f = \frac{0.370}{\log\left(Re\right)^{2.584}}\ $$ 
  
 == Schlutz-Grunow Estimate of Schoenherr (1) ==  == Schlutz-Grunow Estimate of Schoenherr (1) == 
 $$ C_f = \frac{0.427}{\left(\log\left(Re\right) - 0.407\right)^{2.64}}\ $$ $$ C_f = \frac{0.427}{\left(\log\left(Re\right) - 0.407\right)^{2.64}}\ $$
    
-== White-Christoph Compressible (2) ==  +{{ :parasitefrictioncoefficientsfixed.png?850 |}}
-$$ C_f = \frac{0.42}{\ln^2\left(0.056\ Re\right)}\ $$ +
- +
-{{ :parasitefrictioncoefficients.png?850 |}}+
  
 === Roughness === === Roughness ===
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 == Schlichting Avg (6) == == Schlichting Avg (6) ==
 $$ C_f = \left(1.89 + 1.62\ \log\left(\frac{l}{k}\right)\right)^{-2.5} $$ $$ C_f = \left(1.89 + 1.62\ \log\left(\frac{l}{k}\right)\right)^{-2.5} $$
- 
-== Schlichting Local (6) ==  ​ 
-$$ C_f = \left(2.87 + 1.58\ \log\left(\frac{x}{k}\right)\right)^{-2.5} $$ 
- 
-== White (3) ==  
-$$ C_f = \left(1.4 + 3.7\ \log\left(\frac{x}{k}\right)\right)^{-2} $$ 
  
 == Schlichting Avg Compressible (7) ==  == Schlichting Avg Compressible (7) == 
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 === Body Equations === === Body Equations ===
 +
 +Slender body form factor equations are typically given in terms of the fineness ratio (FR), which is the length to diameter ratio for the body.
 +
 +$$ FR = \frac{l}{d}$$
 +
 +For bodies of arbitrary cross section, an equivalent diameter is calculated based on the cross sectional area.
 +
 +$$ d = 2 * \sqrt{\frac{A_{xsec}}{\pi}}$$
  
 == Schemensky Fuselage (9) == == Schemensky Fuselage (9) ==
Line 264: Line 264:
  
 == Hoerner Streamlined Body (1) == == Hoerner Streamlined Body (1) ==
-$$ FF = 1 + \frac{1.5}{\left(\frac{l}{d}\right)^{1.5}} + \frac{7}{\left(\frac{l}{d}\right)^3} $$+$$ FF = 1 + \frac{1.5}{\left(FR\right)^{1.5}} + \frac{7}{\left(FR\right)^3} $$
  
 == Torenbeek (14) == == Torenbeek (14) ==
-$$ FF = 1 + \frac{2.2}{\left(\frac{l}{d}\right)^{1.5}} + \frac{3.8}{\left(\frac{l}{d}\right)^3} $$+$$ FF = 1 + \frac{2.2}{\left(FR\right)^{1.5}} + \frac{3.8}{\left(FR\right)^3} $$
  
 == Shevell (15) == == Shevell (15) ==
-$$ FF = 1 + \frac{2.8}{\left(\frac{l}{d}\right)^{1.5}} + \frac{3.8}{\left(\frac{l}{d}\right)^3} $$+$$ FF = 1 + \frac{2.8}{\left(FR\right)^{1.5}} + \frac{3.8}{\left(FR\right)^3} $$
  
 == Covert (4) == == Covert (4) ==
-$$ FF = 1.02\left(1.0 + \frac{1.5}{\left(\frac{l}{d}\right)^{1.5}} + +$$ FF = 1.02\left(1.0 + \frac{1.5}{\left(FR\right)^{1.5}} + 
-\frac{7.0}{\left(\frac{l}{d}\right)^3 \left(1.0 - M^3\right)^{0.6}}\right) $$+\frac{7.0}{\left(FR\right)^3 \left(1.0 - M^3\right)^{0.6}}\right) $$
  
 == Jenkinson Fuselage (8) == == Jenkinson Fuselage (8) ==
parasitedrag.1525386606.txt.gz · Last modified: 2018/05/03 15:30 by jgravett