Unsteady Lift due to the Interaction of Incidence Turbulence with an Airfoil Sparsh Sharma

Unsteady Lift due to the Interaction of Incidence Turbulence with an Airfoil Sparsh Sharma

¹

, Ennes Sarradj

²

, Heiko Schmidt

¹

1 Fachgebiet Technische Akustik, BTU Cottbus-Senftenberg, 03046 Cottbus, E-Mail: sparsh.sharma@b-tu.de

2 Fachgebiet Technische Akustik, TU Berlin, 10587 Berlin, E-Mail: ennes.sarradj@tu-berlin.de

Introduction

The acoustic signature from an airfoil downstream an incoming turbulent flow is a complex mathematical phenomenon which requires solving the nonlinear governing equations with higher resolutions. The problem is illustrated in Fig. 1. With the advent of high computing resources, it's possible, but at a cost of higher calculation time. For the preliminary analysis in the design process, a method is needed to calculate unsteady-lift due to the airfoil-turbulence interaction with the following characteristics:

- fast and accurate

- need minimum computing resources

- compares more characteristics trade-off or parametric study

Figure 1: Incidence turbulence interacting with the leading edge of an airfoil

Unsteady Effects

By the definition, an unsteady flow is one where the flow field variables at any point are changing with time which means all the aerodynamic parameters fluctuate with time too, and are certainly the inhibitors of all the disturbances which causes the generation of noise from an airfoil. It is instructive to recall the major historical development in the unsteady aerodynamics. Standard work by Karman, Sears, Amiet etc.

shows that the presence of unsteady lift or pressure fluctuation over the lifting surface is the main cause of the noise generation. In 1922, Prandtl¹ suggested to neglect the influence of viscosity while treating the problem of incompressible flow past an oscillating airfoil, and thus to take Laplace equation as governing equation. It was pointed out that every change in the lift must be accompanied by the detachment of a vortex from the airfoil's trailing edge.

As it is shown in Fig-1 that the incoming turbulent eddies upstream the airfoil strikes with the leading edge of the airfoil and this is heard as noise; whereas because of the boundary layer growth chordwise, the vortex shedding takes place which is another source of noise. An important parameter, reduced frequency, k, was introduced by Birnbaum²,

𝑘 = 𝜔𝑐/𝑢

where k is reduced frequency, c is the chord length and u the flow speed. This parameter is a measure of unsteadiness.

When an oscillating airfoil sheds a vortical wake with a certain wavelength, the reduced frequency compares this wavelength with the airfoil chord because during one oscillation a vortex shed from the trailing edge travels the distance u/w. This means, the higher the reduced frequency the smaller the wavelength.

- Steady state aerodynamics, k = 0 - Quasi-steady aerodynamics, 0 ≤ k ≤ 0.05

- Unsteady aerodynamics, k > 0.05 [k > 0.2 is considered highly unsteady

Methodology

To mathematically derive the problem, Potential theory for unsteady flow³ and Panel method⁴ is used. The nobility of the work is to have the least assumptions and account for:

- finite thickness of the airfoil - high subsonic speeds - deformation of the airfoil - cross wind effects

Fig. 2 shows the steps taken to model the incidence turbulence. Sources and vortices are used as the singularities.

The wake is modelled using the Helmholtz circulation theorem⁵. This leads to the calculation of unsteady pressure fluctuation on the surface of the airfoil, which can further, be used to calculate the noise parameters using the acoustic analogies.

Figure 2: Flowchart of the steps

Incoming Turbulence

Airfoil Noise

Aircraft

Wind Turbines

Submarines Vortex Shedding in Wake:

Tonal Noise Interaction of Incoming Turbulence

with Leading Edge:

Broadband Noise Leading Edge

This section shows the steady-state results. The entire code is written in Python, and it is planned to write the steady code in the first phase and then develop it for the unsteady-flow conditions. Following figures show the airfoil discretization, streamlines around the airfoil, and pressure field around the airfoil. The computations will be validated using the experimental data from the wind tunnel tests at the university’s acoustic facility

Aerodynamics Acoustics

ATIN

(Airfoil Turbulence Interaction Noise)

Unsteady Lift due to the Interaction of Incidence Turbulence with an Airfoil

Sparsh Sharma

¹

; Ennes Sarradj

²

; Heiko Schmidt

¹

1

Fachgebiet Technische Akustik, Brandenburgische Technische Universität Cottbus-Senftenberg

2

Fachgebiet Technische Akustik, Technische Universität Berlin

INTRODUCTION METHODOLOGY

STEADY-STATE RESULTS

Contact:

Sparsh Sharma

Fachgebiet Technische Akustik, Brandenburgische Technische Universität Cottbus-Senftenberg Email: sparsh.sharma@b-tu.de Phone: +49 (0) 355 69 4098 Website: www.aeroakustik.de ABSTRACT

Dieser Beitrag zeigt den ersten Schritt der an der Brandenburgischen Technischen Universität Cottbus - Senftenberg durchgeführten Forschungsarbeit "Stochastische Modellierung von

Vorderkantenschall".

Die für die Schallentstehung an der Vorderkante verantwortlichen Schwankungen der Auftriebskraft (Druckschwankungen) werden dabei mit der Theorie eines Tragflügels in instationärer Strömung berechnet.

Dazu wird die klassische Hess- und Smith-Panel-Methode für instationäre Strömungsverhältnisse erweitert, wobei der Tragflügel in Spannweitenrichtung in Scheiben (Panels) mit aerodynamischen Quellen und Wirbelstärken als Singularitäten diskretisiert wird. Die vorliegende Arbeit legt damit den Grundstein für den nächsten geplanten Schritt, bei dem die akustischen Analogien integriert werden sollen, um die akustische Signatur eines Tragflügels modellieren zu können.

Keywords: leading edge noise, turbu- lence, unsteady lift, broadband noise

References:

1. TH. Von Karman. ”Airfoil Theory for Non-Uniform Motion”, Journal of the Aeronautical Sciences, Vol.

5, No. 10 (1938), pp. 379-390

2. Smith, A.M.O., The Panel Method: Its Original Development. In Applied Computational Aerody- namics, Vol. 125

3. Katz, Plotkin: Low-Speed Aerodynamics, 2^nd edition, Cambridge University Press

To mathematically derive the problem, Potential theory for unsteady flow¹and Panel method²is used. The nobility of the work is to have the least assumptions and account for:

1. finite thickness of the airfoil 2. high subsonic speeds 3. deformation of the airfoil 4. cross wind effects

Fig. 3 and Fig. 4 show the steps taken to model the incidence turbulence.

Sources and vortices are used as the singularities. The wake is modelled using the Helmholtz circulation theorem³. This leads to the calculation of unsteady pressure fluctuation on the surface of the airfoil, which can further, be used to calculate the noise parameters using the acoustic analogies, as shown in Fig. 2

The acoustic signature from an airfoil downstream an incoming turbulent flow is a complex mathematical phenomenon which requires solving the nonlinear governing equations with higher resolutions. The problem is illustrated in Fig. 1. With the advent of high computing resources, it's possible, but at a cost of higher calculation time. For the preliminary analysis in the design process, a method is needed to calculate unsteady-lift due to the airfoil-turbulence interaction with the following characteristics:

1. fast and accurate

2. need minimum computing resources 3. compares more characteristics trade-off

Figure 1. Incidence turbulence interacting with the leading edge of an airfoil

Apply the Flow-Tangency (Neumann Condition !"# $% = 0)_"

Distribute sources & vortices on the panels

Calculate the Influence coefficients, source strengths (q1, q2,...,qN), vorticity strength γ

Use the last step to calculate Cp followed by Cl Figure 3.Flowchart of the steps Airfoil interaction

with nonuniform flow, unsteady Lift

Vortex-source method for unsteady flow

Lighthill’s, FWH, LEE Octave band &

Power Spectral Density (PSD) analogies

Figure: 2Roadmap for calculating the noise from an airfoil

!( α(t)

y

x j+ j

1

)

*+$,- / 01234, 56783692861$ (;<)>

!1386468? 56783692861$ @> Γ>BC− Γ>BE

Γ>BF− Γ>BC Γ>BG− Γ>BF

Incoming Turbulence

Airfoil Noise

Aircraft

Wind Turbines

Submarines Vortex Shedding in Wake:

Tonal Noise Interaction of Incoming Turbulence

with Leading Edge:

Broadband Noise Leading Edge

Boundary Layer Growth

Figure 5.(top to bottom) Airfoil discretization into panels, acoustic wind tunnel, streamlines around the airfoil, pressure contour around the airfoil Enforce the Kutta-Condition

!_E# 8% = !_{E H}# 8I_H

Figure 4.Extension of the panel method for unsteady-flow

Stagnation line at the leading edge

Stagnation line at the trailing edge

DAGA 2017 Kiel

1489

Figure 3: Representation of smooth airfoil with nodes and panels

Figure 3 illustrates the representation of a smooth surface by a series of line segments. The numbering system starts at the lower surface trailing edge and proceeds forward, around the leading edge and aft to the upper surface trailing edge. N+1 points define N panels.

𝜙 = 𝑉₎ 𝑥𝑐𝑜𝑠𝛼 + 𝑦𝑠𝑖𝑛𝛼 + 𝑞 𝑠

2𝜋 ln 𝑟 − 𝛾 2𝜋𝜃 𝑑𝑆

=>?@A C D

CEF

The approach (depicted in Fig. 4) is to

1. break up the surface into straight line segments, 2. assume the source strength is constant over each line

segment (panel) but has a different value for each panel

3. the vortex strength is constant and equal over each panel

Roughly, think of the constant vortices as adding up to the circulation to satisfy the Kutta condition. The sources are required to satisfy flow tangency on the surface (thickness).

Figure 4: Extension of the panel method for unsteady-flow Flow tangency boundary condition

A constant vortex strength 𝜸 will be added to each panel (all panels have the same, constant vortex-sheet strength). The flow tangency boundary condition is applied at every panel center:

0 = 𝑉 ∙ 𝑛_J= 𝜕

𝜕𝑛_L 𝜙 𝑥_ML, 𝑦_ML Enforcing the Kutta Condition

In case of steady flow, the Kutta condition is expressed as the zero velocity at the trailing edge or no vortex sheet or no pressure difference at the wake region.

In case of unsteady flow, however, there is a nonzero velocity at the trailing edge and non-zero vortex sheet at the near wake.

Therefore, the unsteady Kutta condition is expressed as the zero-pressure difference at the wake:

𝑑𝛤_>

𝑑𝑡 + 𝜕

𝜕𝑡 𝛾_Q 𝜉, 𝑡 𝑑𝜉

)

S

+ 𝑈𝛾_Q 𝑥, 𝑡 = 0

Steady-State Results

This section shows the steady-state results. The entire code is written in Python, and it is planned to write the steady code in the first phase and then develop it for the unsteady-flow conditions. Following figures show the airfoil discretization, streamlines around the airfoil, and pressure field around the airfoil. The computations will be validated using the experimental data from the wind tunnel tests at the university’s acoustic facility.

Figure 5: Streamlines around the airfoil (top), pressure contour around the airfoil (bottom)

Reference:

[1] Prandtl, L.: “Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben, ” Hydro und Aerodynamik, Berlin, Julius Springer Verlag, pp. 18-33, 1924

[2] Birnbaum, W.: “Das eben Problem des schlagenden Flüels, Zeitschrift fur angewandte Mathematik und Mechanik (ZAMM), Vol. 4, pp. 277-292, 1924

[3] TH. Von Karman. “Airfoil Theory for Non-Uniform Motion”, Journal of the Aeronautical Sciences, Vol. 5, No. 10 (1938), pp. 379-390

[4] Smith, A.M.O., The Panel Method: Its Original Development. In Applied Computational Aerodynamics, Vol. 125

[5] Katz, Plotkin: Low-Speed Aerodynamics, 2nd edition, Cambridge University Press

This section shows the steady-state results. The entire code is written in Python, and it is planned to write the steady code in the first phase and then develop it for the unsteady-flow conditions. Following figures show the airfoil discretization, streamlines around the airfoil, and pressure field around the airfoil. The computations will be validated using the experimental data from the wind tunnel tests at the university’s acoustic facility

Aerodynamics Acoustics

ATIN

(Airfoil Turbulence Interaction Noise)

Unsteady Lift due to the Interaction of Incidence Turbulence with an Airfoil

Sparsh Sharma

¹

; Ennes Sarradj

²

; Heiko Schmidt

¹

1

Fachgebiet Technische Akustik, Brandenburgische Technische Universität Cottbus-Senftenberg

2

Fachgebiet Technische Akustik, Technische Universität Berlin

INTRODUCTION METHODOLOGY

STEADY-STATE RESULTS

Contact:

Sparsh Sharma

Fachgebiet Technische Akustik, Brandenburgische Technische Universität Cottbus-Senftenberg Email: sparsh.sharma@b-tu.de Phone: +49 (0) 355 69 4098 Website: www.aeroakustik.de ABSTRACT

Dieser Beitrag zeigt den ersten Schritt der an der Brandenburgischen Technischen Universität Cottbus - Senftenberg durchgeführten Forschungsarbeit "Stochastische Modellierung von

Vorderkantenschall".

Die für die Schallentstehung an der Vorderkante verantwortlichen Schwankungen der Auftriebskraft (Druckschwankungen) werden dabei mit der Theorie eines Tragflügels in instationärer Strömung berechnet.

Dazu wird die klassische Hess- und Smith-Panel-Methode für instationäre Strömungsverhältnisse erweitert, wobei der Tragflügel in Spannweitenrichtung in Scheiben (Panels) mit aerodynamischen Quellen und Wirbelstärken als Singularitäten diskretisiert wird. Die vorliegende Arbeit legt damit den Grundstein für den nächsten geplanten Schritt, bei dem die akustischen Analogien integriert werden sollen, um die akustische Signatur eines Tragflügels modellieren zu können.

Keywords: leading edge noise, turbu- lence, unsteady lift, broadband noise

References:

1. TH. Von Karman. ”Airfoil Theory for Non-Uniform Motion”, Journal of the Aeronautical Sciences, Vol.

5, No. 10 (1938), pp. 379-390

2. Smith, A.M.O., The Panel Method: Its Original Development. In Applied Computational Aerody- namics, Vol. 125

3. Katz, Plotkin: Low-Speed Aerodynamics, 2^nd edition, Cambridge University Press

To mathematically derive the problem, Potential theory for unsteady flow¹and Panel method²is used. The nobility of the work is to have the least assumptions and account for:

1. finite thickness of the airfoil 2. high subsonic speeds 3. deformation of the airfoil 4. cross wind effects

Fig. 3 and Fig. 4 show the steps taken to model the incidence turbulence.

Sources and vortices are used as the singularities. The wake is modelled using the Helmholtz circulation theorem³. This leads to the calculation of unsteady pressure fluctuation on the surface of the airfoil, which can further, be used to calculate the noise parameters using the acoustic analogies, as shown in Fig. 2

The acoustic signature from an airfoil downstream an incoming turbulent flow is a complex mathematical phenomenon which requires solving the nonlinear governing equations with higher resolutions. The problem is illustrated in Fig. 1. With the advent of high computing resources, it's possible, but at a cost of higher calculation time. For the preliminary analysis in the design process, a method is needed to calculate unsteady-lift due to the airfoil-turbulence interaction with the following characteristics:

1. fast and accurate

2. need minimum computing resources 3. compares more characteristics trade-off

Figure 1. Incidence turbulence interacting with the leading edge of an airfoil

Apply the Flow-Tangency (Neumann Condition !"# $% = 0)_"

Distribute sources & vortices on the panels

Calculate the Influence coefficients, source strengths (q1, q2,...,qN), vorticity strength γ

Use the last step to calculate Cp followed by Cl Figure 3.Flowchart of the steps Airfoil interaction

with nonuniform flow, unsteady Lift

Vortex-source method for unsteady flow

Lighthill’s, FWH, LEE Octave band &

Power Spectral Density (PSD) analogies

Figure: 2Roadmap for calculating the noise from an airfoil

!( α(t)

y

x j+ j

1 )

*+$,- / 01234, 56783692861$ (;<)>

!1386468? 56783692861$ @>

Γ>BC− Γ>BE

Γ>BF− Γ>BC

Γ>BG− Γ>BF

Incoming Turbulence

Airfoil Noise

Aircraft

Wind Turbines

Submarines Vortex Shedding in Wake:

Tonal Noise Interaction of Incoming Turbulence

with Leading Edge:

Broadband Noise Leading Edge

Boundary Layer Growth

Figure 5.(top to bottom) Airfoil discretization into panels, acoustic wind tunnel, streamlines around the airfoil, pressure contour around the airfoil Enforce the Kutta-Condition

!_E# 8% = !_{E H}# 8I_H

Figure 4.Extension of the panel method for unsteady-flow

Stagnation line at the leading edge

Stagnation line at the trailing edge

This section shows the steady-state results. The entire code is written in Python, and it is planned to write the steady code in the first phase and then develop it for the unsteady-flow conditions. Following figures show the airfoil discretization, streamlines around the airfoil, and pressure field around the airfoil. The computations will be validated using the

experimental data from the wind tunnel tests at the university’s acoustic facility

Aerodynamics Acoustics

ATIN

(Airfoil Turbulence Interaction Noise)

Unsteady Lift due to the Interaction of Incidence Turbulence with an Airfoil

Sparsh Sharma

¹

; Ennes Sarradj

²

; Heiko Schmidt

¹

1

Fachgebiet Technische Akustik, Brandenburgische Technische Universität Cottbus-Senftenberg

2

Fachgebiet Technische Akustik, Technische Universität Berlin

INTRODUCTION METHODOLOGY

STEADY-STATE RESULTS

Contact:

Sparsh Sharma

Fachgebiet Technische Akustik, Brandenburgische Technische Universität Cottbus-Senftenberg Email: sparsh.sharma@b-tu.de Phone: +49 (0) 355 69 4098 Website: www.aeroakustik.de

ABSTRACT

Dieser Beitrag zeigt den ersten Schritt der an der Brandenburgischen Technischen Universität Cottbus - Senftenberg durchgeführten Forschungsarbeit "Stochastische Modellierung von

Vorderkantenschall".

Die für die Schallentstehung an der Vorderkante verantwortlichen Schwankungen der Auftriebskraft (Druckschwankungen) werden dabei mit der Theorie eines Tragflügels in instationärer Strömung berechnet.

Dazu wird die klassische Hess- und Smith-Panel-Methode für instationäre Strömungsverhältnisse erweitert, wobei der Tragflügel in Spannweitenrichtung in Scheiben (Panels) mit aerodynamischen Quellen und Wirbelstärken als Singularitäten diskretisiert wird. Die vorliegende Arbeit legt damit den Grundstein für den nächsten geplanten Schritt, bei dem die akustischen Analogien integriert werden sollen, um die akustische Signatur eines Tragflügels modellieren zu können.

Keywords: leading edge noise, turbu- lence, unsteady lift, broadband noise

References:

1. TH. Von Karman. ”Airfoil Theory for Non-Uniform Motion”, Journal of the Aeronautical Sciences, Vol.

5, No. 10 (1938), pp. 379-390

2. Smith, A.M.O., The Panel Method: Its Original Development. In Applied Computational Aerody- namics, Vol. 125

3. Katz, Plotkin: Low-Speed Aerodynamics, 2^nd edition, Cambridge University Press

To mathematically derive the problem, Potential theory for unsteady flow¹and Panel method²is used. The nobility of the work is to have the least assumptions and account for:

1. finite thickness of the airfoil 2. high subsonic speeds 3. deformation of the airfoil 4. cross wind effects

Fig. 3 and Fig. 4 show the steps taken to model the incidence turbulence.

Sources and vortices are used as the singularities. The wake is modelled using the Helmholtz circulation theorem³. This leads to the calculation of unsteady pressure fluctuation on the surface of the airfoil, which can further, be used to calculate the noise parameters using the acoustic analogies, as shown in Fig. 2

The acoustic signature from an airfoil downstream an incoming turbulent flow is a complex mathematical phenomenon which requires solving the nonlinear governing equations with higher resolutions. The problem is illustrated in Fig. 1. With the advent of high computing resources, it's possible, but at a cost of higher calculation time. For the preliminary analysis in the design process, a method is needed to calculate unsteady-lift due to the airfoil-turbulence interaction with the following characteristics:

1. fast and accurate

2. need minimum computing resources 3. compares more characteristics trade-off

Figure 1. Incidence turbulence interacting with the leading edge of an airfoil

Apply the Flow-Tangency (Neumann Condition !_"# $% = 0)_"

Distribute sources & vortices on the panels

Calculate the Influence coefficients, source strengths (q1, q2,...,qN), vorticity strength γ

Use the last step to calculate Cp followed by Cl Figure 3.Flowchart of the steps Airfoil interaction

with nonuniform flow, unsteady Lift

Vortex-source method for unsteady flow

Lighthill’s, FWH, LEE Octave band &

Power Spectral Density (PSD) analogies

Figure: 2Roadmap for calculating the noise from an airfoil

!₍

α(t)

y

x j+ j

1

)

*+$,- / 01234, 56783692861$ (;<)>

!1386468? 56783692861$ @_>

Γ>BC− Γ>BE

Γ_>BF− Γ_>BC Γ>BG− Γ>BF

Incoming Turbulence

Airfoil Noise

Aircraft

Wind Turbines

Submarines Vortex Shedding

in Wake:

Tonal Noise Interaction of Incoming Turbulence

with Leading Edge:

Broadband Noise Leading Edge

Boundary Layer Growth

Figure 5.(top to bottom) Airfoil discretization into panels, acoustic wind tunnel, streamlines around the airfoil, pressure contour around the airfoil Enforce the Kutta-Condition

!_E# 8% = !_{E H}# 8I_H

Figure 4.Extension of the panel method for unsteady-flow

Stagnation line at the leading edge

Stagnation line at the trailing edge

DAGA 2017 Kiel

1490