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CURVATURE LIFT - An Improved Explanation for the Origin of Airfoil Lift

Brief Summary of the Bernoulli Equation

The Bernoulli Equation relates the differences in velocity and pressure between two points in a flow by applying Newtons Laws along a STREAMLINE.

The Bernoulli Equation usually references the velocities and pressures of the streamline points to the far-field static pressure, $$P_{\infty}$$ and the free-stream velocity, $$V_{\infty}$$.

Simply, the Bernoulli Equation states for any general, Point 'A', in the flow field:

$$       P_{A}  =  P_{\infty } +  \left ( \frac{{\rho V_\infty}^2}{2} \right )     - \left ( \frac{\rho {V_A}^2}{2} \right )                     $$

Curvature Lift Theory

Similarly, the Curvature Lift Theory accounts for the difference in velocity and pressure between two points in the flow by applying Newtons Laws along an EQUIPOTENTIAL LINE - a line of constant velocity potential in Potential Flow. Equipotential lines are geometric lines which cross streamlines at right-angles to form a flownet.

Curvature Lift Theory proposes that pressure gradients generated by the radial acceleration of particles moving along curved pathlines in the flow, result in pressure differences in the fluid which cause lifting forces on the airfoil.

Curvature Lift Theory applies Newtons Laws to the radial acceleration of fluid particles, as opposed to the Bernoulli Theory which applies Newtons Laws to the linear acceleration of fluid particles along streamlines. Interestingly, the Curvature Lift Theory produces the same pressure/velocity equation as the Bernoulli Theory.

Curvature Lift Equation and its proof

The Curvature Lift Equation states that for any Point 'A' on the airfoil surface, or anywhere in the flow, that:

$$       P_{A}  =  P_{\infty } +  \left ( \frac{{\rho V_\infty}^2}{2} \right )     - \left ( \frac{\rho {V_A}^2}{2} \right )                     $$

Glossary of Terms Used in Proof

$$ P_{A} $$                ... the static pressure at Point 'A'

$$ P_{\infty} $$           ... the static pressure of the free-stream

$$ \Delta P_{static} $$    ... the static gauge pressure between Point 'A' and the free-stream static pressure

$$ V_{A} $$                ... the velocity at Point 'A'

$$ V_{\infty} $$           ... the free-stream velocity

$$ \rho $$                 ... the density of the fluid (assumed constant density)

$$ dl $$                   ... the length element along the equipotential line integral

$$ a_{radial}$$            ... the radial acceleration of the fluid particle along the streamline

$$ k $$                ... the local curvature of the streamline

$$ R $$                    ... the local radius of curvature of the streamline

Schematic Diagram 1 - General arrangement for proving the Curvature Lift Equation



Proof

The static pressure at any point 'A' in the flow is the static pressure of the free-stream plus the gauge pressure difference to point 'A'

$$P_{A} = P_{\infty} +  \Delta P_{static}   $$                        ......equation [1]

Now, obtaining the static gauge pressure at point 'A' by integrating the pressure gradient along an equipotential line from point 'A' to the far-field

$$\Delta P_{static} = - \int\limits_{A}^{\infty }\frac{dP}{dl}\, dl$$                      ......equation [2]

Newtons Law of motion for forces acting tranversely to the streamlines (or equivalently, forces acting along equipotential lines or the radius of streamline curvature)

Note: aradial is positive if pointing in the direction of line integration

Note: aradial, k and R all have the same sign convention

$$\frac{dP}{dl}\ = -  \rho a_{radial} $$                              ......equation [3]

Combining equation [2] & [3]

$$\Delta P_{static} = \int\limits_{A}^{\infty }\rho a_{radial} \, dl$$

$$=\int\limits_{A}^{\infty }\rho V^2 k \, dl$$  or alternatively   $$=\int\limits_{A}^{\infty }\ \rho  \frac{V^2}{R}  \, dl$$

$$=\rho \int\limits_{A}^{\infty }\ V^2 k \, dl$$  or alternatively   $$= \rho \int\limits_{A}^{\infty }\ \frac{V^2}{R}  \, dl$$

$$=\rho \int\limits_{A}^{\infty }\ V^2 \left ( \frac{1}{V} \frac{dV}{dl} \right ) \, dl$$

$$=\rho \int\limits_{A}^{\infty }\ V \frac{dV}{dl} \, dl$$

$$=\rho \int\limits_{V_A}^{V\infty }\ V \, dV$$

$$=\rho \left [ \frac{V^2}{2} \right ] ^{V\infty}_{V_A} $$

$$\Delta P_{static} = \left ( \frac{{\rho V_\infty}^2}{2} \right )    - \left ( \frac{\rho {V_A}^2}{2} \right )                                  $$                 ......equation [4]

Combining equation [1] & [4] to solve for the actual static pressure at any Point 'A'

$$       P_{A}  =  P_{\infty } +  \left ( \frac{{\rho V_\infty}^2}{2} \right )     - \left ( \frac{\rho {V_A}^2}{2} \right )                     $$             ......the Curvature Lift Equation is PROVEN

CONCLUSION

Many interesting results and insights arise from examining the Curvature Lift Theory. The key conclusions follow:

[1] The Bernoulli Theory is not a unique means for calculating pressures in a known velocity flow field.

[2] Airfoil lift is determined by the integration of the pressure distribution around the airfoil. The Curvature Lift Theory allows the pressure on the airfoil to be calculated for the determination of lift, independently of the Bernoulli Theory.

[3] The Curvature Lift Theory provides a direct means of relating lift forces on an airfoil with the mass and acceleration of fluid responsible for creating these lift forces. No other theory of lift has acheived this in principle.

[4] It can be shown by inspection, the fluid mass directly responsible for causing lift on an airfoil is wholly located between the equipotential lines of the two stagnation points of the airfoil.

[5] It can be shown by inspection, the contribution to lift from the flow decreases with increasing distance along the equipotential lines, away from the airfoil.

AUTHOR: Mark Vincze, Professional Engineer, Brisbane Australia

KEYWORDS: Theory of Lift, Curvature and Lift , Wing , Lift Force , Bernoulli , Aerodynamics.

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