Aerodynamic equation of state

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Notes to reviewer: My user name is Aerophiliac. This is my first Wikipedia article. The subject matter of this article, of which I am the sole author was published in an SAE paper in April 2023.

Aerodynamic Equation of State

For the use of this in thermodynamics, see Equation of state.

In physics, an equation of state relates selected intensive state variables that describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. The term intensive means that the state variable is a physical property that does not change with size, volume, or mass of the system. The thermodynamic equation of state relates three intensive state variables; a pressure $p$, a specific volume $\nu$, and a temperature $T$ , the average kinetic energy of the fluid's molecules.

This article covers the aerodynamic equation of state, an equation analogous to the thermodynamic equation of state in that it relates two intensive state variables of the surrounding fluid, the pressure $p$ and its specific volume $\nu$ (the inverse of density $\rho$) with the specific kinetic energy $e_k$ of a lifting surface, an intensive, macroscopic state variable. The lifting surface can be a fixed wing, any number of rotary blades, any pair of flapping wings, a cylinder in Magnus effect, a boomerang, or a Frisbee.

The aerodynamic equation of state for lift is:

$$p\nu =n_L{e}_k$$

where

• $p$, anisotropic pressure on the wetted surface of a translating, rotating, or flapping surface by the surrounding fluid
• $\nu$, specific volume of surrounding fluid (= 1/density)
• $n_L$, normalized lift (dimensionless, typically < 1)
• $e_k$, specific kinetic energy due to translation, rotation, roto-translation, or flapping

An isotropic pressure $p$ on a fluid molecule is the surrounding hydrostatic pressure exerted on it as it pushes on it equally in all directions. This same pressure is applied to the wetted surface of a static lifting surface (i.e., $e_k$ = 0) as the same atmospheric pressure pushes on the lifting surface equally in all directions, resulting in zero lift by the lifting surface. As the lifting surface acquires specific kinetic energy (i.e., $e_k$ > 0), the wetted surface of the lifting surface exerts work on the surrounding flow field, resulting in a transition from an isotropic to an anisotropic pressure $p$ by the surrounding fluid onto the lifting surface, resulting in a resultant force, or lift L.

As the thermodynamic equation of state is a function of temperature T, an intensive microscopic state variable; the aerodynamic equation of state is a function of an intensive macroscopic state variable, the specific kinetic energy (per unit mass) $e_k$ of the lifting surface.:

$$e_k=\frac{1}{2}{v}^2+\frac{1}{2}\frac{I}{m}{\omega }^2$$

where

• $e_k$, specific kinetic energy of the lifting surface
• $v$, velocity during translation ($v$ = 0 during hover)
• $\frac{I}{m}$, specific moment of inertia ($=1/3R^2$ for a propeller and rotor blade, and a flapping wing)
• $\omega$, angular velocity of the propeller or rotor rotating blades (=$2f\varphi$ for flapping wings, where f is the flapping frequency, and $\varphi$ the stroke angle of the upstroke or the downstroke of flapping wings)

The first term on the right-hand side of the equation is the kinetic energy due to translation. The second term is the kinetic energy due to rotation, flapping, (or undulating of a caudal fin when calculating thrust $T$). The simultaneous use of these two terms describes the specific kinetic energy available in, say, a roto-translating propeller or helicopter rotor, or in the flapping wings of a forward-flying bird, a boomerang, a Frisbee, or a Samara seed, etc. If the velocity due to translation is zero, the equation is being applied to the hovering flight of a helicopter rotor, or a hummingbird

The aerodynamic equation of state for lift is valid for viscous flow fields. The equation (with its one or two-term kinetic energy $e_k$) is also valid in the subsonic realm. With the appropriate addition of a third term $e_{kMN>1}$, the aerodynamic equation of state for lift may, hypothetically, be applied to transonic, supersonic, and hypersonic flight as this added term would, again, hypothetically, account for the presence of higher surrounding temperatures above ambient due to the presence of shock waves.

Overview

The aerodynamic equation of state relates two intensive state variables, $p$ and $\nu$, to a third, macroscopic, intensive state variable, the specific kinetic energy $e_k$ of a lifting surface. The left-hand product of pressure $p$ and specific volume $\nu$ in the aerodynamic equation of state equals the lift work $w_L$(addressed sometimes as flow work ), the work exerted onto the surrounding flow field by the presence of a lifting surface possessing a specific kinetic energy $e_k$> 0, resulting in lift $L$:

$$w_L=p\nu$$

where

• $w_L$, lift work
• p, anisotropic pressure on the wetted surface of a translating, rotating, or flapping surface by the surrounding fluid (= Lift / wetted surface)
• $\nu$, specific volume

This lift work $w_L$ exerted by a moving lifting surface induces the transition from an isotropic pressure distribution (when static) to an anisotropic pressure distribution when both of its simultaneous values of $e_k$, and the value of the angle of attack $\alpha$ are both higher than zero, resulting in the generation of lift $L$.

The exertion of lift work w_L onto the surrounding flow field by the moving lifting surface has an energy cost referred to as the energy spent ${\psi }_L$, and quantified by the product $p$ $v$.

$${\psi }_L=n_L{e}_k$$

where

• ${\psi }_L$, spent kinetic energy
• $n_L$, normalized lift
• $e_k$, kinetic energy available at the lifting surface

The aerodynamic equation of state is next expressed in its implicit form, which is the work-energy theorem.

$$w_L=\psi$$

The normalized lift $n_L$ is a dimensionless number and is expected to be below 1. The normalized lift $n_L$ is dependent on the geometric planform of the lifting surface, its sweep, airfoil, and angle of attack, $\alpha$ .

The normalized lift $n_L$not only quantifies the allocation of the amount of kinetic towards the exertion of lift work $w_L$, but is also a lift efficiency, a dimensionless figure of merit calculated as the ratio of the aerodynamic work $w_L$ exerted by, and the available kinetic energy $e_k$ possessed by the lifting surface

$$n_L=\frac{w_L}{e_k}=\frac{pv}{e_k}=\frac{work}{energy}$$

As the reference (wetted) surface is double the planform area of a wing, then, as $S$/$S^{\prime }$ ≈ 2, then $C_L$/$n_L$ ≈ 2. The typical maximum lift coefficient $c_{lmax}$ of a two-dimensional airfoil is 1.6, which translates into a two-dimensional normalized lift $n_{lmax}$ of 0.8, which is interpreted as the lift work $w_L$ exerted during the generation of lift being 80% of the available kinetic energy at the airfoil, $e_k$.

Whereas the normalized lift $n_L$ is expected to be lower than 1, higher values are expected (i) for an airfoil with leading and/or trailing edge high lift devices deflected, and/or (ii) airfoils that possess an inbuilt flap (e.g., NACA SC(2)-0714, which reaches a 2-d lift coefficient $c_{lmax}$= 2.2, or a 2-d normalized lift $n_{lmax}$ = 1.1 @ 9.0x10⁶)

The maximum 2-d normalized lift $n_l$ a cylinder in Magnus effect is approximately 0.14, whereas its 2-d lift coefficient $c_l$ is higher than 9.There are two reasons for the significantly higher value: (i) this value is calculated without considering its rotational kinetic energy, and (ii) its reference is the planform area $S^{\prime }$ of the cylinder, which is smaller than its wetted area $S$.

The wetted surface S as a geometric reference

File:Molecule and Wing.jpg

Particle and wing submerged in a fluid, affected by pressure and specific volume

The presence of a geometric reference is indispensable when using both the legacy lift equation and the aerodynamic equation of state for lift. Whereas the projected planform area $S^{\prime }$ of a lifting surface is widely accepted as the geometric reference for the lift equation, its wetted surface best describes the physics underlying the generation of lift as it is this surface that is subjected to an all-around anisotropic pressure, with lift $L$ being its resultant.

In this section attention is drawn to the difference between two terms; the legacy, two-dimensional area, an the three-dimensional surface.

The equilibrium equation of a static elementary cube of fluid found neutrally buoyant within the fluid is subjected to an isotropic pressure all around it, that is, a pressure of the same magnitude over its six sides. The numerical treatment of this static fluid parcel is treated by the fluid-dynamic equation of state . This equilibrium scenario and its numerical treatment would not be possible when using the planform area of this cube, namely, using one of its six sides.

A static, infinitesimally thin flat plate, submerged in a fluid, and with no kinetic energy, $e_k$= 0 (regardless of its angle of attack $\alpha$), does not generate lift as its wetted surface is subjected to an isotropic pressure. The presence of this isotropic pressure distribution is evidence that the flat plate does not exert any spent kinetic energy $\psi$ on its surrounding flow field. If the flat plate acquires an kinetic energy, $e_k$ > 0, and its angle of attack, $\alpha$ is still zero, there is still an isotropic pressure onto it, and there is still no spent kinetic energy $\psi$. The isotropic pressure field still persists.

If the flat plate has $e_k$ > 0, and its angle of attack $\alpha >0$, there is spent kinetic energy $\psi$ by it onto the surrounding flow field by its wetted area, that results in an anisotropic pressure distribution, and a non-zero force resultant, an essential condition for lift.

An aerodynamic cycle

File:P vs v version 2.jpg

a p vs. v history of an ascending & descending airplane

A sequence of thermodynamic processes in an Otto cycle is plotted on a $p$ vs. $\nu$ diagram, and so can the aerodynamic processes undergone by, say, a wing as it rotates, ascends to a certain altitude, descends, and touches down. The variations of the pressure $p$ on the lifting surface (as the weight or lift decreases as fuel is consumed) can be plotted against specific volume $\nu$, as shown in the figure. The plot shows a wing as it rotates at sea level, process A-B, and ascends to an altitude, point C. A descent follows to sea level, process C-D. Process D-A shows the transfer of the wing lift to the landing gear and the ground.

If the ascending flight occurs at a constant pressure $p$ ($L$/$S$ = const.), the wing undergoes an isobaric process, represented by an horizontal curve B-C'.

The area under the curve B-C represents positive aerodynamic work, +$w_L$, whereas the area under the curve C-D is negative aerodynamic work, -$w_L$.

An electric analogy

File:Electric Analog.jpg

This is an electric analog model representing the aerodynamic equation of state for a hovering hummingbird

The aerodynamic equation of state of, say, a hovering hummingbird, is calculated by extracting the characteristic equation of its electric circuit analog model or ECAM, where the voltage in the "circuit" is replaced by the spent energy $\psi$, the "resistor" by the pressure $p$ (the lift or weight $L$ of the bird divided by the wing's reference wet surface $S$ $,L/S$), and the "current" by the specific volume $\nu$ of the surrounding air. The characteristic equation is obtained by traversing the circuit counterclockwise (or clockwise), adding the "voltage rise" (- to +) and subtracting the "voltage drops" (+ to -) to a null total.

$$-{\psi }_L+p_{L}v=0:characteristicequation;p_{L}v={\psi }_L:aerodynamicequationofstate$$

The aerodynamic equation of state of the hovering hummingbird (right-hand side equation) is obtained by rearranging the characteristic equation (left-hand side equation) by placing the spent energy $\psi$ to the right of the equal equal sign, and the product $p_L$ $\nu$, the lift work $w_L$, to the left:.

Legacy Origins

The aerodynamic equation of state for lift is deconstructed to reveal its ubiquitous ancestor, the lift equation. The anisotropic pressure $p$ exerted by the surrounding fluid on a moving, three-dimensional reference wetted surface $S$ is converted to a lift loading, the lift applied to a two-dimensional planform area $S^{\prime }$. This lift loading is dismembered into the lift L, placed on the left-hand side of the equal sign, and the planform area $S^{\prime }$ on the right-hand side. The specific kinetic energy due to translation $1/2v^2$ is multiplied by the inverse of the specific volume, the density $\rho$, to obtain the dynamic pressure $q$ (=$1/2\rho v^2$), and placed on the left-hand side of the equal sign.

The last step reverts the normalized lift $n_L$ to the legacy lift coefficient $C_L$. The resultant lift equation is

$$L=1/2\rho v^2{S}^{\prime }{C}_L$$

where

• $L$, lift
• $\rho$, air density
• $v$, true airspeed
• $S^{\prime }$, projected planform area
• $C_L$, lift coefficient

The application of the lift equation is limited to translating lifting surfaces, and does not account for the 3-d pressure $p$ exerted by the surrounding fluid onto the submerged surface, nor lift work, kinetic energy, and the lift efficiency. Its maximum value $C_{Lmax}$ may reach values higher than 9. In contrast, the normalized lift $n_L$ is the ratio of lift work and kinetic energy, a lift efficiency, with a typical value less than 1.

The lift coefficient $C_L$ is the ratio of the wing loading $L$/$S^{\prime }$ and the dynamic pressure $q$ (=$1/2\rho v^2$),

$$C_L=\frac{\frac{L}{S^{\prime }}}{1/2\rho v^2}=\frac{wingloading}{dynamicpressure}$$

As the wing loading, in the numerator, is a dimensional figure of merit, the lift coefficient $C_L$ is a dimensionless figure of merit of a dimensional figure of merit. Its usefulness is limited when applied to a single type of lifting device but misleading when comparing dissimilar devices. The maximum lift coefficient $C_{Lmax}$ of a cylinder in Magnus effect may reach values of ≈ 9 , whereas an airfoil (no flaps) is typically ≈1.6. This does not mean that a cylinder is capable of attaining a maximum lift that is 5.6 times higher than that of an airfoil. Quite the contrary, according to their corresponding maximum normalized lift $n_{Lmax}$, which for a cylinder in Magnus effect reaches ≈ 0.14 (typ.), and that of an airfoil is 0.8 (= $C_{Lmax}$ / 2 = 1.6 / 2). According to the normalized lift, an airfoil generates a maximum lift that is ≈ 5.7 higher that that of the cylinder in Magnus effect.

There are two factors affecting adversely the lift coefficient $C_L$ of a cylinder in Magnus effect, standing in the way of it being a physics-based number, and contribute to its overestimation: (i) its formulation does not include the rotational term of its kinetic energy, and (ii) its geometric reference area is its planform area, not its wetted surface.

In contrast to dimensionless figures of merit in other fields of science and engineering, the lift coefficient $C_L$ cannot be used to compare the lift of aircraft of differing configurations, as the lift coefficient of, say, an aircraft of canard configuration cannot be compared to any other aircraft configuration as the planform area of its canard surface is not accounted for in the reference area $S^{\prime }$ .

Range of applications

The aerodynamic equation of state evaluates a wide range of engineered and biomechanical flyers, and may include the evaluation of thrust $T$ of propellers (with rotating blades during static thrust testing, or roto-translating blades during flight) and fish (with its undulating caudal fin). These thrust-generation scenarios are calculated using the same equation of state format, using the proper subscript $T$, to avoid confusion. The resulting normalized lift $n_L$ of an aircraft and the normalized thrust $n_T$ of its propeller(s) can be compared meaningfully as both share the same definition, namely, the ratio of work and energy.

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