### Table of Contents

# Parasite Drag Tool

Parasite drag is a combination of form, friction, and interference drag that is evident in any body moving through a fluid. Although VSPAERO includes an estimate of parasite drag in the calculation of the zero lift drag coefficient, the Parasite Drag tool provides much more advanced options and capabilities. The Parasite Drag Tool GUI is accessed by clicking “Parasite Drag…” from the Analysis drop-down on the top menu-bar.

## General Overview

$ c_i \equiv \mbox{Chord length at span station} $

$ b_i \equiv \mbox{Section span} $

$ S_i \equiv \mbox{Section area} $

$ S_{total} \equiv \mbox{Total Area} $

$ \bar{c} \equiv \mbox{Weighted chord sum} $

$ L_{ref} \equiv \mbox{Reference length} $

$ D \equiv \mbox{Diameter} $

$ X_{area} \equiv \mbox{Max cross sectional area} $

$ FR \equiv \mbox{Fineness ratio} $

$ Re \equiv \mbox{Reynolds number} $

$ V_{inf} \equiv \mbox{freestream velocity} $

$ \nu \equiv \mbox{kinematic viscosity} $

$ C_{f} \equiv \mbox{friction coefficient} $

$ f \equiv \mbox{flat plate drag} $

$ S_{wet} \equiv \mbox{wetted area} $

$ S_{ref} \equiv \mbox{reference area} $

$ C_D \equiv \mbox{coefficient of drag} $

Visible on all tabs of the GUI is the parasite drag table, which identifies individual components and their inputs in the parasite drag calculation. The table may be sorted by Component, S_wet, or % Total by selecting the toggles at the top of the table. Clicking on a geometry in the Component column of the table will break up the geometry into its surfaces and sub-surfaces. Below the list of components, excrescences are identifies, for which the Excrescence tab is available to add, remove, and adjust excrescences. Once the table has been setup, the “Calculate CDO” button on the bottom left of the GUI will run the parasite drag calculation. Once finished, the results will update in real time in response to changes in input values, such as the flow condition. However, if a component is added, removed, or modified, the tool must be rerun. The entire Parasite Drag table, excrescence list, and total results can be exported by selecting “Export to *.csv”. Next to this button, the Export Sub-Components toggle is available to include or ignore component breakup in the export file. On the bottom right of the GUI, the total form factor, drag coefficient, and percent contribution to total drag is listed for all components, excrescences, and the combination of both.

#### Wetted Area

Wetted areas for all components are taken from executing Comp Geom for the indicated geometry set and as such as are appropriately trimmed.

### Grouping

Utilizing this feature the user is able to combine the wetted area of any geometry with that of another. Currently, geometries can only be grouped with their ancestors and geometries of the same shape type.

This can be used for example if the gear pod is modeled seperately from the fuselage but the wetted area of the gear pod should be applied with the drag qualities (e.g. length, form factor, etc.) of the fuselage.

### Sub Surface Handling

By default, subsurfaces are incorporated as a part of the geometry as a whole. In other words, the surfaces do not subtract any wetted area from the geometry or have any of their own unique properties. However, the parasite drag tool let's the user choose these as options if they desire.

The three options are:

- Treat as Parent
- Separate Treatment
- Zero Drag

**Treat as Parent**: The default option, incorporates the wetted area of the subsurface as part of a continuous geometry.

**Separate Treatment**: Allows the user, to some extent, control the qualities of the subsurface. However, due to limitations of the methodology used, the geometry based qualities (e.g. L_{ref}, Re, etc.) are derived from the parent geometry. The user is allowed to change the form factor equation type, manually set the laminar percentage, and manually set the interference factor for the subsurface.

**Zero Drag**:The subsurface wetted area is subtracted from the total wetted area of the geometry and no longer contributes to the drag of the component.

#### Reference Length

Reference length is calculated according to it's geometry type internal to VSP; either a “Wing” type or a “Body” type. If the result of these calculations are zero, due to rotation or abnormal geometry shape the other equation is attempted as a fail safe. If neither of these methods yield a result greater than zero, a default value of 1.0 is used as to prevent division by zero when calculating fineness ratio.

##### Wing

Wing reference length is found by taking the area weighted chord.

First find the width of the wing section. $$ \Delta_x = |x_{le(1)} - x_{le(end)}| $$ $$ \Delta_y = |y_{le(1)} - y_{le(end)}| $$ $$ \Delta_z = |z_{le(1)} - z_{le(end)}| $$ $$ b_i = \sqrt{ {\Delta_x}^2 + {\Delta_y}^2 + {\Delta_z}^2} $$ $$ S_i = b_i * \frac{c_i + c_{\left(i+1\right)}}{2} $$ $$ S_{total} = \sum\left(S_i\right) $$ $$ \bar{c} = \sum\left(\frac{c_i + c_{\left(i+1\right)}}{2}\right) $$ $$ L_{ref} = \frac{\bar{c}}{S_{total}} $$

##### Body

Body geometry reference lengths are calculated by taking the distance between to the front and back ends of the degenerated stick. $$ \Delta_x = |x_{le(1)} - x_{le(end)}| $$ $$ \Delta_y = |y_{le(1)} - y_{le(end)}| $$ $$ \Delta_z = |z_{le(1)} - z_{le(end)}| $$ $$ L_{ref} = \sqrt{ {\Delta_x}^2 + {\Delta_y}^2 + {\Delta_z}^2} $$

#### Fineness Ratio/Thickness to Chord

Thickness to chord takes the max thickness to chord from the degenerate stick created from Degen Geom.

Fineness Ratio is calculated by taking the diameter calculated using the max cross sectional area taken from the degenerate stick from Degen Geom.

$$ D = 2 * \sqrt{\frac{X_{area}}{\pi}} $$ $$ F = \frac{D}{L_{ref}} $$

#### Reynolds Number

Depending on the atmosphere input type, the kinematic viscosity is calculated accordingly and used to find the Reynolds number for the geometry.

$$ Re = \frac{V_{inf} * L_{ref}}{\nu} $$

#### Laminar Percent

Laminar Percent takes a 0 to 100 value and at default is set to 0% for a fully turbulent flow.According to Raymer, most aircraft have 10-20% laminar flow over lifting surfaces, and almost no laminar flow over the fuselage. However, an aircraft like the Piaggio GP180 can have up to 50% laminar flow over the wings and tail and 20-35% over the fuselage$^{22}$.

The laminar friction coefficient is calculated using the following equation: $$ C_f = C_{f (100\%Turb)} - \left(C_{f (\% Partial Turb)} * \% Lam\right) + \left(C_{f (\% Partial Lam)} * \% Lam\right) $$

where (using a chosen friction coefficient and it's corresponding inputs) $$ C_{f (\% Partial Turb)} = f\left(Re_{Lam}\right) $$ $$ C_{f (\% Partial Lam)} = f\left(Re_{Lam}\right) $$ where $$ Re_{Lam} = Re * \% Lam $$

#### Interference Factor

$ Q \equiv \mbox{Scale factor applied to drag coefficient} $

#### Flat Plate Drag

$$ f = S_{wet} * Q * C_f * FF $$

#### Drag Coefficient

$$ C_D = \frac{f}{S_{ref}} $$

## 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} $

$ k \equiv \mbox{roughness height} $

$ \gamma \equiv \mbox{specific heat ratio} $

$ r \equiv \mbox{recovery factor} = 0.89 $

$ n \equiv \mbox{viscosity power-law exponent} = 0.67 $

$ l \equiv \mbox{length of component} $

$ d \equiv \mbox{diameter of component} $

$ w \equiv \mbox{width at maximum cross sectional area} $

$ h \equiv \mbox{height at maximum cross sectional area} $

$ FR \equiv \mbox{Covert Fineness Ratio} = \frac{l}{\sqrt{wh}} $

$ l_{r} \equiv \mbox{Length of Fuselage} $

$ A_{x} \equiv \mbox{Cross Sectional Area of Fuselage} $

$ Q \equiv \mbox{Interference Factor} $

### Laminar

##### Blasius (25)

$$ C_f = \frac{1.32824}{\sqrt{Re}}\ $$

### Turbulent

##### Explicit Fit of Spalding (3)

$$ C_f = \frac{0.523}{\ln^2(0.06\ Re)}\ $$

##### Explicit Fit of Spalding and Chi (2)

$$ C_f = \frac{0.430}{\log(Re)^{2.32}}\ $$

##### Explicit Fit of Schoenherr (1)

$$ \frac{1}{\sqrt{C_f}}\ = 3.46\log\left(Re\right) - 5.6 $$

##### Implicit Schoenherr (1)

$$ \log\left(Re\ C_f\right) = \frac{0.242}{\sqrt{C_f}}\ $$

##### Implicit Karman-Schoenherr (4)

$$ \frac{1}{\sqrt{C_f}}\ = 4.13\log\left(Re\ C_f\right) $$

##### Power Law Blasius (2)

$$ C_f = \frac{0.072}{Re^{1/5}}\ $$

##### Power Law Prandtl Low Re (5)

$$ C_f = \frac{0.074}{Re^{1/5}}\ $$

##### Power Law Prandtl Medium Re (3)

$$ C_f = \frac{0.0315}{Re^{1/7}}\ $$

##### Power Law Prandtl High Re (3)

$$ C_f = \frac{0.0725}{Re^{1/5}}\ $$

##### Schlichting Compressible (6)

$$ C_f = \frac{0.455}{\log\left(Re\right)^{2.58}}\ $$

##### Schlutz-Grunow Estimate of Schoenherr (1)

#### Roughness

##### Schlichting Avg (6)

$$ C_f = \left(1.89 + 1.62\ \log\left(\frac{l}{k}\right)\right)^{-2.5} $$

##### Schlichting Avg Compressible (7)

$$ C_f = \frac{\left(1.89 + 1.62\ \log\left(\frac{l}{k}\right)\right)^{-2.5}}{\left(1 + \frac{\gamma - 1}{2}\ M_\infty\right)^{0.467}} $$

#### Heat Transfer

##### White-Christoph (2)

$$ f = \frac{\left(1 + 0.22\ r\ \frac{\gamma - 1}{2}\ {M_e}^2\ \frac{Te}{Tw}\right)}{\left(1 + 0.3\ \left(\frac{Taw}{Tw} - 1\right)\right)} $$ $$ C_f = \frac{0.451\ f^2\ \frac{Te}{Tw}}{\ln^2\left(0.056\ f\ \frac{Te}{Tw}^{1+n}\ Re\right)} $$

## Form Factor Equations

#### Wing Equations

##### EDET Conventional Airfoil (10)

$$ FF = 1 + \frac{t}{c}\ \left(2.94206 + \frac{t}{c}\ \left(7.16974 + \frac{t}{c}\ \left(48.8876 + \frac{t}{c}\ \left(-1403.02 + \frac{t}{c}\ \left(8598.76 + \frac{t}{c}\ \left(-15834.3\right)\right)\right)\right)\right)\right) $$

##### EDET Advanced Airfoil (10)

$$ FF = 1 + 4.275\ \frac{t}{c} $$

##### DATCOM (11)

Recreated Data from DATCOM is shown in the Figure and is used to find the Appropriate Scale Factor for use in the DATCOM Equation through interpolation. $$ FF = \left[1 + L\ \left(\frac{t}{c}\right) + 100\ \left(\frac{t}{c}\right)^4\right] * R_{L.S.} $$

##### Hoerner (1)

$$ FF = 1 + 2\ \left(\frac{t}{c}\right) + 60\ \left(\frac{t}{c}\right)^4 $$

##### Shevell (15)

$$ FF = 1 + Z\ \left(\frac{t}{c}\right) + 100\ \left(\frac{t}{c}\right)^4 $$ $$ Z = \frac{\left(2-M^2\right) \cos\left(\Lambda_{\frac{c}{4}}\right)} {\sqrt{1-M^2\cos^2\left(\Lambda_{\frac{c}{4}}\right)}} $$

##### Kroo (13)

$$ FF = 1 + \frac{2.2 \cos^2\left(\Lambda_{\frac{c}{4}}\right)} {\sqrt{1-M^2\cos^2\left(\Lambda_{\frac{c}{4}}\right)}} \left(\frac{t}{c}\right) + \frac{4.84 \cos^2\left(\Lambda_{\frac{c}{4}}\right) \left(1 + 5\cos^2\left(\Lambda_{\frac{c}{4}}\right)\right)} {2\left(1-M^2\cos^2\left(\Lambda_{\frac{c}{4}}\right)\right)} \left(\frac{t}{c}\right)^2 $$

##### Torenbeek (14)

$$ FF = 1 + 2.7\ \left(\frac{t}{c}\right) + 100\ \left(\frac{t}{c}\right)^4 $$

##### Covert (4)

$$ FF = 1 + 1.8\ \left(\frac{t}{c}\right) + 50\ \left(\frac{t}{c}\right)^4 $$

##### Schemensky 6 Series Airfoil (9)

$$ FF = 1 + 1.44\left(\frac{t}{c}\right) + 2\left(\frac{t}{c}\right)^2 $$

##### Schemensky 4 Series Airfoil (9)

$$ FF = 1 + 1.68\left(\frac{t}{c}\right) + 3\left(\frac{t}{c}\right)^2 $$

##### Jenkinson Wing (8)

$$ F^* = 1 + 3.3\left(\frac{t}{c}\right) - 0.008\left(\frac{t}{c}\right)^2 + 27.0\left(\frac{t}{c}\right)^3 $$ $$ FF = \left(F^* - 1\right)\left(cos^2\left(\Lambda_{\frac{c}{2}}\right)\right) + 1 $$

##### Jenkinson Tail (8)

$$ F^* = 1 + 3.52\left(\frac{t}{c}\right) $$ $$ FF = \left(F^* - 1\right)\left(cos^2\left(\Lambda_{\frac{c}{2}}\right)\right) + 1 $$ $$ Q = 1.2 $$

#### 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)

$$ FF = 1 + \frac{60}{FR^3} + 0.0025\ FR $$

##### Schemensky Nacelle (9)

$$ FF = 1 + \frac{0.35}{FR} $$

##### Hoerner Streamlined Body (1)

$$ FF = 1 + \frac{1.5}{\left(FR\right)^{1.5}} + \frac{7}{\left(FR\right)^3} $$

##### Torenbeek (14)

$$ FF = 1 + \frac{2.2}{\left(FR\right)^{1.5}} + \frac{3.8}{\left(FR\right)^3} $$

##### Shevell (15)

$$ FF = 1 + \frac{2.8}{\left(FR\right)^{1.5}} + \frac{3.8}{\left(FR\right)^3} $$

##### Covert (4)

$$ FF = 1.02\left(1.0 + \frac{1.5}{\left(FR\right)^{1.5}} + \frac{7.0}{\left(FR\right)^3 \left(1.0 - M^3\right)^{0.6}}\right) $$

##### Jenkinson Fuselage (8)

$$ \Lambda = \left(\frac{l_{r}}{\frac{4}{\pi}A_{x}}\right)^{0.5} $$ $$ FF = 1 + \frac{2.2}{\Lambda^{1.5}} - \frac{0.9}{\Lambda^3} $$

##### Jenkinson Wing Nacelle (8)

Jenkinson suggests a constant Form Factor for typical nacelles on wings. $$ FF = 1.25 $$

##### Jenkinson Aft Fuselage Nacelle (8)

Jenkinson suggests a constant Form Factor for typical nacelles on aft fuselages. $$ FF = 1.50 $$

## Flow Condition

On the Overview tab, the Parasite Drag Tool includes several options for atmosphere models as well as the option for the user to have manual control over certain atmospheric qualities. The freestream type is identified by the choice labeled “Atmosphere”, and the sliders below will activate or deactivate depending on this selection. The first two freestream types are the US Standard Atmosphere 1976 and USAF 1966 atmospheric models, for which a comparison is shown below$^{16, 17}$. Note, the upper limit for the US Standard Atmospheric model is 84,852 meters, and the upper limit for the USAF model is 82,021 feet. Both require freestream velocity and altitude to be input, but an additional delta temperature input is available to offset temperature from the atmospheric model. The remaining atmosphere options require a series of manual inputs to calculate the atmospheric condition, but will not calculate and update the altitude slider. In addition, if the atmospheric choice type is “Re/L + Mach Control”, no additional properties of the flow will be calculated (i.e. pressure and density).

#### Dynamic Viscosity (16)

$ \beta \equiv \mbox{}1.458E10^{-6} \frac{kg}{(s*m*K^{1/2})} $

$ \mu \equiv \mbox{Dynamic Viscosity} $

$ S \equiv \mbox{Sutherland's Constant = 100.4 K} $

$$ \mu = \frac{\beta \cdot T^{3/2}}{T + S} $$

## Transonic Drag

**Note that the transonic drag formulations have not yet been implemented in the Parasite Drag Tool. However, the descriptions and formulas below will remain here for easy reference.**

$ \frac{t}{c} \equiv \mbox{thickness to chord ratio of selected geometry} $

$ M \equiv \mbox{freestream Mach number for flight condition} $

$ M_{cr} \equiv \mbox{Critical Mach number, the point at which drag creep begins to occur} $

$ M_{DD} \equiv \mbox{Drag Divergence Mach number, the point at which drag significantly begins to rise} $

$ M_{DD,eff} \equiv \mbox{Effective Drag Divergence Mach number, Drag Divergence Mach number with consideration fro wing sweep} $

$ \Delta_{CD} \equiv \mbox{Additional CD due to Transonic Drag effects} $

$ \phi_{25} \equiv \mbox{Average quarter chord sweep of selected geometry} $

$ \gamma \equiv \mbox{Specific heat ratio; typically 1.4} $

$ C_{L} \equiv \mbox{Design Lift Coefficient} $

Example plots are given showing Drag Divergence Mach number using each equation with contours of quarter chord sweep angle at 0°, 15°, 30°, 45°, 60°, and 75° utilizing the following inputs: $$ C_{L} = 1.0 $$ $$ Airfoil Type = Conventional $$ $$ \gamma = 1.4 $$ $$ A_{F} = 0.8 $$ $$ K_{A} = 0.8 $$

#### Delta CD Due to Transonic Drag

$$ \Delta_{CD} = 20 * \left(M - M_{cr}\right)^4 $$

#### Mach Drag Divergence Equations

##### Torenbeek w/o Lift Coefficient (14)

$ M^{*} \equiv \mbox{1.0, conventional airfoils; maximum t/c at about 0.30c} $

$ M^{*} \equiv \mbox{1.05, high-speed (peaky) airfoils, 1960-1970 technology} $

$ M^{*} \equiv \mbox{1.12 to 1.15, supercritical airfoils [Conservative = 1.12; Optimistic = 1.15]} $

$$ M_{DD,eff} = M_{DD} * \sqrt{\cos{\phi_{25}}} $$
$$ \frac{t}{c} = 0.30\cos{\phi_{25}}\left(\left(1 - \left( \frac{5 + {M_{DD,eff}}^2}{5 + {M^*}^2}\right)^{3.5}\right)\frac{\sqrt{1 - {M_{DD,eff}}^2}}{{M_{DD,eff}}^2}\right)^{\frac{2}{3}} $$

##### Torenbeek w/ Lift Coefficient (14)

$ M^{*} \equiv \mbox{1.0, conventional airfoils; maximum t/c at about 0.30c} $

$ M^{*} \equiv \mbox{1.05, high-speed (peaky) airfoils, 1960-1970 technology} $

$ M^{*} \equiv \mbox{1.12 to 1.15, supercritical airfoils [Conservative = 1.12; Optimistic = 1.15]} $

$$ M_{DD,eff} = M_{DD} * \sqrt{\cos{\phi_{25}}} $$
$$ \frac{t}{c} = 0.30\cos{\phi_{25}}\left(\left(1 - \left( \frac{5 + {M_{DD,eff}}^2}{5 + \left(k_{M} -
0.25c_{l}\right)^2}\right)^{3.5}\right)\frac{\sqrt{1 - {M_{DD,eff}}^2}}{{M_{DD,eff}}^2}\right)^{\frac{2}{3}} $$

##### Shevell (18)

Equation solved for M. $$ A = \frac{M^2\cos^2{\phi_{25}}}{\sqrt{1-M^2\cos^2{\phi_{25}}}}\left(\left(\frac{\gamma + 1}{2}\right)\frac{2.64\frac{t}{c}}{\cos{\phi_{25}}} + \left(\frac{\gamma + 1}{2}\right)\frac{2.64\left(\frac{t}{c}\right)\left(0.34C_{L}\right)}{\cos^3{\phi_{25}}}\right) $$ $$ B = \frac{M^2\cos^2{\phi_{25}}}{1-M^2cos^2{\phi_{25}}}\left(\left(\frac{\gamma + 1}{2}\right)\left(\frac{1.32\frac{t}{c}}{\cos{\phi_{25}}}\right)^2\right) $$ $$ C = M^2cos^2{\phi_{25}}\left(1 + \left(\frac{\gamma + 1}{2}\right)\frac{\left(0.68C_{L}\right)}{\cos^2{\phi_{25}}} + \frac{\gamma + 1}{2}\left(\frac{0.34C_{L}}{\cos^2{\phi_{25}}}\right)^2\right) $$

$$ A + B + C - 1 = 0 $$

If a Peaky Airfoil Type is selected $$ M_{DD} = M + 0.06 $$ otherwise $$ M_{DD} = M $$

##### Kroo (13)

$ M_{cc} \equiv \mbox{Crest Critical Mach number} $

$$ a = 0.2 $$
$$ b = 2.131 $$
$$ x = \frac{\frac{t}{c}}{\cos{\phi_{25}}} $$
$$ y = \frac{C_{L}}{{\left(\cos{\phi_{25}}\right)}^2} $$

$$ M_{cc} = \frac{2.8355x^2 - 1.9072x + 0.949 - a\left(1-bx\right)y}{\cos{\phi_{25}}} $$

If Conventional Airfoil Type: $$ \Delta_{CD} = 0.04 $$ If Peaky Airfoil Type: $$ \Delta_{CD} = 0 $$ If Supercritical Conservative: $$ \Delta_{CD} = -0.04 $$ If Supercritical Optimistic: $$ \Delta_{CD} = -0.06 $$

$$ M_{DD} = M_{cc} * \left(1.025 + 0.08\left(1-\cos{\phi_{25}}\right)\right) - \Delta_{CD} $$

##### Howe (19)

$ A_{F} \equiv \mbox{Airfoil Technology Factor, typically between 0.8 and 0.95} $

$$ M_{DD,eff} = M_{DD} * \sqrt{\cos{\phi_{25}}} $$
$$ M_{DD,eff} = A_{F} - 0.1C_{L} - \frac{t}{c} $$

##### Jenkinson (8)

$$ \frac{t}{c} = 0.7185 + 3.107e^{-5}\phi_{25} - 0.1298C_{L} - 0.7210M_{DD} $$

##### Weisshaar (20)

$ K_{A} \equiv \mbox{Airfoil Technology Factor, typically between 0.8 and 0.9} $

$$ M_{DD} = \frac{K_{A}}{cos{\phi_{25}}} - \frac{\frac{t}{c}}{cos^2{\phi_{25}}} - \frac{C_{L}}{10cos^3{\phi_{25}}} $$

##### Bottger (21)

$$ a = -1.147 $$ $$ b = 0.2 $$ $$ c = 0.838 $$ $$ d = 4.057 $$ $$ M_{DD} = a (C_{L} - b)d + c + \frac{30}{27}\left(\frac{t}{c} - 0.113\right) + 0.00288\left(\phi_{25} - 29.8\right) $$

#### Mach Critical And Mach Drag Divergence Relations

##### Boeing And Airbus (22)

$$ M_{cr} = M_{DD} $$

##### Fokker (23)

$$ M_{cr} = M_{DD} - 0.02 $$

##### Roskam (24)

$$ M_{cr} = M_{DD} - 0.1 $$

##### Weisshaar (20)

$$ M_{cr} = M_{DD} - \left(\frac{1}{80}\right)^{\frac{1}{3}} $$

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