9.1 Used symbols Appendix

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Appendix

9.1 Used symbols

𝑝, static pressure [Pa] 𝑝_𝑡, total pressure [Pa] 𝑇, static temperature [°K] 𝑇_𝑡, total temperature [°K]

𝑇_𝑟𝑒𝑓 = 273.11, refence temperature [°K] 𝑆 = 110.56, constant in Sutherland law [°K] 𝛾, specific heats ratio

𝑅, gas constant [J/(Kg*°K)] 𝜌, density [Kg/m3_]

𝑎, sound speed [m/s] 𝑡, time [s]

𝑈 or 𝑈_∞, velocity of undisturbed flow [m/s] 𝑀, Mach number

𝑅𝑒𝑐, Reynolds number base on chord length

𝑐, chord length [m] 𝑥/𝑐, dimensionless chord 𝑅𝐻, roughness height [m] 𝜇, dynamic viscosity [Pa*s]

𝜇𝑟𝑒𝑓= 1.716 ∗ 10−5, reference dynamic viscosity [Pa*s]

𝑊𝑑, wall distance [m]

𝑦+_{, dimensionless distance [m]}

𝑈𝑓𝑟𝑖𝑐𝑡, friction velocity [m/s]

𝐶_𝑓, wall friction coefficient

𝜏_{𝑤𝑎𝑙𝑙}, friction force per unit surface [N/m2] 𝑑𝑖𝑟 (𝑋 𝑜𝑟 𝑌), direction cosines

𝑞𝑢𝑎𝑟𝑡𝑒𝑟 𝑐ℎ𝑜𝑟𝑑, x-coordinate of moment centre 𝐾_𝐴𝑙, Aluminium heat transfer coefficient [W/(m*°K)] 𝐾𝑎𝑖𝑟, air heat transfer coefficient [W/(m*°K)]

𝑐𝑙, lift coefficient

𝑐_𝑑, drag coefficient 𝑐_𝑚, moment coefficient 𝑓, frequency [Hz]

𝑓̅ = 2 ∗ 𝜋 ∗ 𝑓 ∗ 𝑐/𝑈_∞, reduced frequency 𝐹𝐹𝑇, Fast Fourier Transform

213 𝑃𝑆𝐷, Power Spectral Density [1/Hz]

𝑅𝑀𝑆, Root Mean Square 𝐿𝐶𝑂, Limit Cycle Oscillation

𝑐_𝑝, specific heat at constant pressure [J/(Kg*°K)] 𝑐𝑝, pressure coefficient

No confusion should be made between the last two, but context well clarify their meaning. Other symbol used in this thesis are explained in the text.

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9.2 Used formulas

• Wall distance computation:

𝑅𝑒_𝑐 = 𝜌 ∗ 𝑈∞∗ 𝑐 𝜇 𝐶_𝑓 = 0.026 𝑅𝑒_𝑐 1 7 𝜏_{𝑤𝑎𝑙𝑙} =1 2∗ 𝜌 ∗ 𝑈∞ 2 _{∗ 𝐶} 𝑓 𝑈_{𝑓𝑟𝑖𝑐𝑡} = √𝜏𝑤𝑎𝑙𝑙 𝜌 𝑊𝑑 = 𝑦 +_{∗ 𝜇} 𝑈_{𝑓𝑟𝑖𝑐𝑡}∗ 𝜌

• Flow parameters computation:

𝑀 =𝑈∞ 𝑎 𝑅𝑒𝑐 = 𝜌 ∗ 𝑈_∞∗ 𝑐 𝜇 𝑎 = √𝛾 ∗ 𝑅 ∗ 𝑇 𝑝𝑡 𝑝 = ( 𝛾 − 1 2 ∗ 𝑀 2₎𝛾−1𝛾

215 𝑇_𝑡 𝑇 = ( 𝛾 − 1 2 ∗ 𝑀 2₎ 𝑝 𝜌= 𝑅 ∗ 𝑇 𝜇 = 𝜇_𝑟𝑒𝑓∗ ( 𝑇 𝑇_𝑟𝑒𝑓) 3 2 ∗𝑇𝑟𝑒𝑓+ 𝑆 𝑇 + 𝑆

Seven equations in seven variables (𝑝, 𝑇, 𝑇_𝑡, 𝜇, 𝑎, 𝜌, 𝑈). Given total pressure and Mach number is possible to compute static pressure using fourth equation. Combining second and sixth relations it can be write:

√𝑇 ∗ 𝜇 =𝑝 ∗ 𝑀 ∗ 𝑐√𝛾 𝑅𝑒_𝑐 ∗ √𝑅

Combining the last with Sutherland law it can be obtained:

𝜇 =𝑝 ∗ 𝑀 ∗ 𝑐 ∗ √𝛾 𝑅𝑒_𝑐√𝑅 ∗ 1 √𝑇= (𝜇𝑟𝑒𝑓∗ 𝑇 𝑇_𝑟𝑒𝑓) 3 2 ∗𝑇𝑟𝑒𝑓+ 𝑆 𝑇 + 𝑆 After some manipulations:

(𝑝 ∗ 𝑀 ∗ 𝑐 ∗ √𝛾 𝑅𝑒_𝑐∗ √𝑅 ∗ 𝑇_𝑟𝑒𝑓 3 2 (𝑇_𝑟𝑒𝑓+ 𝑆) ∗ 𝜇_𝑟𝑒𝑓) = 𝑇2 (𝑇 + 𝑆)

Substituting K instead of left hand side:

𝐾 = (𝑝 ∗ 𝑀 ∗ 𝑐 ∗ √𝛾 𝑅𝑒_𝑐 ∗ √𝑅 ∗ 𝑇_𝑟𝑒𝑓 3 2 (𝑇_𝑟𝑒𝑓+ 𝑆) ∗ 𝜇_𝑟𝑒𝑓)

It can be obtained a relation to compute static temperature: 𝑇2− 𝐾 ∗ 𝑇 − 𝐾 ∗ 𝑆 = 0

216 whose solutions are:

𝑇 = 𝐾 ± √𝐾

2_{+ 4 ∗ 𝐾 ∗ 𝑆}

2

Obviously only temperature obtained using plus sign has physical meaning.

Obtained static temperature can be computed dynamic viscosity, total temperature, density, sound speed and flow velocity using seventh, fifth, sixth, third and first equation respectively.