Low Speed Flow Past a Symmetric Airfoil

Sujal Machhale - 22B0001

Laboratory Guide: Ashutosh
Instructor: Prof. Vineeth Nair
September 2024

Department of Aerospace Engineering

Indian Institute of Technology, Bombay

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Contents
1 INTRODUCTION TO AIRFOILS 4
1.1 Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Experimental Calculations . . . . . . . . . . . . . . . . . . 4
1.1.2 Theoretical Predictions . . . . . . . . . . . . . . . . . . . 4
1.2 Apparatus Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Airfoil Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Ambient Data and Associated Errors . . . . . . . . . . . . . . . . 4

2 THEORETICAL ASSUMPTIONS AND THEIR SIGNIFICANCE 4

3 PROCEDURE 4
3.1 Measurement of Airfoil Surface Pressure Distribution . . . . . . . 4
3.2 Measurement of Airfoil Wake Velocity Profile . . . . . . . . . . . 4

4 EXPERIMENTAL CALCULATIONS 4
4.1 Sample Calculation Approach . . . . . . . . . . . . . . . . . . . . 4
4.2 Tabulated Calculations . . . . . . . . . . . . . . . . . . . . . . . . 4

5 RESULTS 4
5.1 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5.1.1 Representative Cp Plots . . . . . . . . . . . . . . . . . . . 4
5.1.2 Representative Wake Velocity Profile . . . . . . . . . . . . 4
5.2 Comparison with Literature and Theoretical Data . . . . . . . . 4
5.2.1 Lift Coefficient (Cl ) vs α . . . . . . . . . . . . . . . . . . 4
5.2.2 Drag Coefficients (Cd ) vs α . . . . . . . . . . . . . . . . . 4
5.2.3 Moment Coefficients (Cm ) vs α . . . . . . . . . . . . . . . 4
5.2.4 Center of Pressure (xcp ) vs α . . . . . . . . . . . . . . . . 4
5.3 Estimation of Stall Angle . . . . . . . . . . . . . . . . . . . . . . 4
5.4 Estimating Location of Aerodynamic Center . . . . . . . . . . . . 4
5.5 Error Analysis (Correct Significant Digits) . . . . . . . . . . . . . 4
5.6 Tabulated Dependence on Angle of Attack and RMS Errors . . . 4

6 CONCLUSIONS & DISCUSSION 4

6.1 Comparison Between Plots . . . . . . . . . . . . . . . . . . . . . 4

7 OBJECTIVES 4

8 INTRODUCTION TO AIRFOILS 5
8.1 Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
8.1.1 NACA 4-Series Airfoil . . . . . . . . . . . . . . . . . . . . 5
8.2 Theoretical Predictions . . . . . . . . . . . . . . . . . . . . . . . . 6

9 EXPERIMENTAL CALCULATIONS 6

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10 RESULTS 7
10.1 Representative Cp Plots . . . . . . . . . . . . . . . . . . . . . . . 7
10.2 Comparison with Theoretical and Numerical Data . . . . . . . . 7

11 CONCLUSIONS 8

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1 INTRODUCTION TO AIRFOILS
1.1 Airfoils
1.1.1 Experimental Calculations
1.1.2 Theoretical Predictions

1.2 Apparatus Used

1.3 Experiment Setup
1.4 Wind Tunnel
1.5 Airfoil Geometry
1.6 Ambient Data and Associated Errors

2 THEORETICAL ASSUMPTIONS AND THEIR

SIGNIFICANCE
3 PROCEDURE
3.1 Measurement of Airfoil Surface Pressure Distribution
3.2 Measurement of Airfoil Wake Velocity Profile

4 EXPERIMENTAL CALCULATIONS
4.1 Sample Calculation Approach
4.2 Tabulated Calculations

5 RESULTS
5.1 Plots
5.1.1 Representative Cp Plots
5.1.2 Representative Wake Velocity Profile

5.2 Comparison with Literature and Theoretical Data

5.2.1 Lift Coefficient (Cl ) vs α
5.2.2 Drag Coefficients (Cd ) vs α
5.2.3 Moment Coefficients (Cm ) vs α
5.2.4 Center of Pressure (xcp ) vs α

5.3 Estimation of Stall Angle

5.4 Estimating Location of 4Aerodynamic Center
5.5 Error Analysis (Correct Significant Digits)
5.6 Tabulated Dependence on Angle of Attack and RMS
Errors

6 CONCLUSIONS & DISCUSSION

 2. To capture and analyze an incompressible flow past a symmetric airfoil in
the high Reynolds’ number regime by measuring the pressure distribution
along the surface of the airfoil.
3. To predict the sectional coefficients of lift (Cl ), pitching moments about
the leading edge (Cm,LE ) and quarter chord point (Cm,c/4 ).

4. To estimate the location of the aerodynamic center for thin symmetrical

airfoils using the computed values of sectional coefficients of lift (Cl ) and
pitching moments about the quarter chord point (Cm,c/4 ).

8 INTRODUCTION TO AIRFOILS
8.1 Airfoils
An airfoil is a two-dimensional shape that interacts with aerodynamic flows to
generate forces both tangential and normal to the flow direction. The geomet-
rical characteristics of a typical airfoil shape are shown in Figure 1.

airfoil_nomenclature.png

Figure 1: Airfoil nomenclature

The forces on an airfoil can be divided into lift (L) and drag (D) components:

L′ = N ′ cos(α) − A′ sin(α)

D′ = N ′ sin(α) + A′ cos(α)

8.1.1 NACA 4-Series Airfoil

The NACA 4-digit series airfoil shape is characterized by parameters like max-
imum camber (m), its location (p), and maximum thickness (t) as a percentage

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of the chord length. The upper and lower surface coordinates can be computed
by:
zu = zc + zt , zl = zc − zt

8.2 Theoretical Predictions

Using thin airfoil theory, the sectional coefficient of lift (Cl,theory ) is given by:

Cl,theory = 2πα

The moment coefficients can also be derived:

Cl
Cm,LE = − , Cm,c/4 = 0
4

9 EXPERIMENTAL CALCULATIONS
Using experimental data, the aerodynamic characteristics are computed with
the following formulae:

1 c
Z
Cn = (Cp,l − Cp,u ) dx
c 0

1 c
Z  
dzu dzl
Ca = Cp,u − Cp,l dx
c 0 dx dx
Z c 
1
Cm,LE = 2 (Cp,u − Cp,l )x dx
c 0

The sectional lift and drag coefficients are computed as:

Cl = Cn cos(α) − Ca sin(α), Cd,p = Cn sin(α) + Ca cos(α)

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10 RESULTS
10.1 Representative Cp Plots

cp_vs_xc.png

Figure 2: Cp vs x/c for α = 0◦

10.2 Comparison with Theoretical and Numerical Data

The lift coefficient can be compared to the theoretical value:

Cl,theo = 2πα

The experimental values deviate slightly from theory at higher angles of attack
due to flow separation effects.

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 cl_vs_alpha.png

Figure 3: Cl vs α comparison with theoretical and numerical data

11 CONCLUSIONS
The experiment successfully demonstrated the principles of low-speed flow over
a symmetric airfoil. The measured coefficients of lift, drag, and moment were
found to be in reasonable agreement with theoretical predictions, though some
deviations were observed due to flow separation and other real-world effects not
accounted for in thin airfoil theory.