# On the Incipient Indicial Lift of Thin Wings in Subsonic Flow: Acoustic Wave Theory with Unsteady Three-Dimensional Effects

## Abstract

**:**

## 1. Introduction

## 2. Governing Physics: From Nonlinear Compressible to Linear Incompressible Flow

## 3. Incompressible Potential Flow: Lifting-Line Theory

#### 3.1. Thin Aerofoils: Two-Dimensional Wake Inflow and Chordwise Gust Penetration

#### 3.2. Thin Wings: Three-Dimensional Downwash Angle and Spanwise Gust Penetration

#### 3.2.1. Elliptical Wings: Curved Taper

#### 3.2.2. Trapezoidal Wings: Swept Taper

## 4. Compressible Potential Flow: Acoustic Wave Theory

#### 4.1. Thin Aerofoils: Two-Dimensional Acoustic Waves Propagation and Chordwise Gust Penetration

#### 4.2. Thin Wings: Three-Dimensional Downwash Angle, Spanwise Gust Penetration, and Mean Wing Section

#### 4.2.1. Elliptical Wings: Curved Taper

#### 4.2.2. Trapezoidal Wings: Swept Taper

## 5. Results and Discussion

## 6. Conclusions

## Funding

## Conflicts of Interest

## Acronyms

AOA | Angle of Attack |

CFD | Computational Fluid Dynamics |

FSI | Fluid–Structure Interaction |

MDO | Multidisciplinary Design Optimisation |

PDE | Partial Differential Equation |

ROM | Reduced-Order Model |

SEG | Sharp-Edged Gust |

## Symbols

a | sound speed |

A | wing area |

${B}_{n}^{1}$ | Bessel’s functions of first type and n-th order |

c | wing section chord |

${c}_{p}$ | isobaric heat capacity |

${c}_{v}$ | isochoric heat capacity |

${C}_{\gamma}$ | wing section circulation coefficient |

${C}_{\gamma /\alpha}$ | wing section circulation derivative |

${C}_{\mathsf{\Gamma}}$ | wing circulation coefficient |

${C}_{\mathsf{\Gamma}/\alpha}$ | wing circulation derivative |

${C}_{l}$ | wing section lift coefficient |

${C}_{l/\alpha}$ | wing section lift derivative |

${C}_{L}$ | wing lift coefficient |

${C}_{L/\alpha}$ | wing lift derivative |

${C}_{p}$ | pressure coefficient |

${C}_{E}$ | Sears’s circulation-deficiency function |

${C}_{S}$ | Sears’s lift-deficiency function |

${C}_{T}$ | Theodorsen’s lift-deficiency function |

e | edge-velocity factor |

E | airflow internal energy |

f | wing area impingement factor |

g | downwash gradient factor |

H | airflow enthalpy |

${H}_{n}^{2}$ | Hankel’s functions of second type and n-th order |

I | complete elliptic integral of the second kind |

k | reduced frequency |

l | wing semi-span |

M | Mach number |

$\mathit{n}$ | normal vector |

N | expansion terms |

p | airflow pressure |

r | mean chord ratio |

R | gas constant |

s | non-dimensional elliptic integral argument |

$\mathit{s}$ | space vector |

S | airflow entropy |

t | time |

T | airflow temperature |

U | horizontal free-stream |

x | chordwise coordinate |

y | spanwise coordinate |

z | vertical coordinate |

Greek | |

$\alpha $ | angle of attack |

$\beta $ | airflow compressibility factor |

$\varphi $ | airflow disturbance potential |

$\phi $ | airflow velocity potential |

$\eta $ | wing aspect ratio |

$\kappa $ | shape factor (downwash tuning) |

$\gamma $ | airflow heat capacity ratio |

$\mathsf{\Gamma}$ | wing section circulation |

$\iota $ | wing span impingement fraction |

$\lambda $ | wing taper ratio |

$\mathsf{\Lambda}$ | sweep angle |

$\mathit{\nu}$ | airflow velocity vector |

$\psi $ | spanwise angle |

$\rho $ | airflow density |

$\varsigma $ | sound-travelled distance |

$\tau $ | reduced time |

$\upsilon $ | non-dimensional wake development |

$\omega $ | angular frequency |

$\mathit{\omega}$ | airflow vorticity vector |

Superscripts | |

${\chi}^{\perp}$ | angle of attack step |

${\chi}^{\Vert}$ | gust front parallel to leading edge |

${\chi}^{\u22a3}$ | gust front normal to reference airspeed |

$\stackrel{\xb4}{\chi}$ | airflow normal to aerodynamic axis |

$\stackrel{\u02c7}{\chi}$ | circulatory |

$\widehat{\chi}$ | non-circulatory |

$\overline{\chi}$ | steady |

Subscripts | |

${\chi}_{a}$ | average |

${\chi}_{FC}$ | foremost cone |

${\chi}_{G}$ | gust |

${\chi}_{i}$ | induced |

${\chi}_{LE}$ | leading edge |

${\chi}_{r}$ | root |

${\chi}_{RC}$ | rearmost cone |

${\chi}_{s}$ | section |

${\chi}_{t}$ | tip |

${\chi}_{TE}$ | trailing edge |

${\chi}_{w}$ | wing |

${\chi}_{0}$ | initial |

${\chi}_{\infty}$ | unperturbed |

## Appendix A

## Appendix B

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**Figure 1.**The indicial circulatory lift for elliptical thin wing in incompressible flow: AOA (

**left**) and SEG (

**right**).

**Figure 2.**The indicial airload derivatives (

**left**) and approximate incipient behaviours (

**right**) for thin aerofoil in incompressible flow.

**Figure 3.**The approximate downwash gradient (

**left**) and incipient angle (

**right**) from a unit AOA step for thin elliptical wings in incompressible flow.

**Figure 4.**The incipient indicial lift for elliptical thin wings in incompressible flow: AOA (

**left**) and SEG (

**right**).

**Figure 5.**The gust penetration delay for thin elliptical and trapezoidal wings: taper (

**left**) and sweep (

**right**) effects along the span.

**Figure 6.**The Mach effect on the incipient indicial lift for thin aerofoil in compressible flow: AOA (

**left**) and SEG (

**right**), with $\mathsf{\Lambda}={0}^{\circ}$.

**Figure 7.**The sweep effect on the incipient indicial lift for thin aerofoil in compressible flow: AOA (

**left**) and SEG (

**right**), with ${M}_{\infty}=0.5$.

**Figure 8.**The incipient indicial lift for elliptical wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =6$.

**Figure 9.**The incipient indicial lift for elliptical wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =8$.

**Figure 10.**The incipient indicial lift for elliptical wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =12$.

**Figure 11.**The incipient indicial lift for elliptical wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =20$.

**Figure 12.**The incipient indicial lift for rectangular straight wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =8$ and $\mathsf{\Lambda}={0}^{\circ}$.

**Figure 13.**The incipient indicial lift for rectangular swept wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =8$ and $\mathsf{\Lambda}={30}^{\circ}$.

**Figure 14.**The incipient indicial lift for rectangular straight wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =20$ and $\mathsf{\Lambda}={0}^{\circ}$.

**Figure 15.**The incipient indicial lift for rectangular swept wing in compressible flow: AOA (

**left**) and SEG (

**right**), with $\eta =20$ and $\mathsf{\Lambda}={30}^{\circ}$.

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**MDPI and ACS Style**

Berci, M. On the Incipient Indicial Lift of Thin Wings in Subsonic Flow: Acoustic Wave Theory with Unsteady Three-Dimensional Effects. *Acoustics* **2022**, *4*, 26-52.
https://doi.org/10.3390/acoustics4010003

**AMA Style**

Berci M. On the Incipient Indicial Lift of Thin Wings in Subsonic Flow: Acoustic Wave Theory with Unsteady Three-Dimensional Effects. *Acoustics*. 2022; 4(1):26-52.
https://doi.org/10.3390/acoustics4010003

**Chicago/Turabian Style**

Berci, Marco. 2022. "On the Incipient Indicial Lift of Thin Wings in Subsonic Flow: Acoustic Wave Theory with Unsteady Three-Dimensional Effects" *Acoustics* 4, no. 1: 26-52.
https://doi.org/10.3390/acoustics4010003