lec36 - Aerodynamics Lecture 36 Prandtl-Glauert...

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Aerodynamics Prandtl-Glauert Compressibility Correction Drag-Divergence Mach Number Swept Wings 36.1 Lecture 36 Aerodynamics Subsonic Compressible Flow AE311 Aerodynamics Manoj T. Nair IIST
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Aerodynamics Prandtl-Glauert Compressibility Correction Drag-Divergence Mach Number Swept Wings 36.2 Agenda 1 Prandtl-Glauert Compressibility Correction 2 Drag-Divergence Mach Number 3 Swept Wings
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Aerodynamics Prandtl-Glauert Compressibility Correction Drag-Divergence Mach Number Swept Wings 36.3 Prandtl-Glauert Compressibility Correction I Prandtl-Glauert Compressibility Correction We studied the aerodynamic theory for incompressible flow over thin airfoils at small Aoa Such theory was adequate for aircrafts of the period 1903-1940 During W-WII, the aircraft speed increased to 700 km/hr Towards the end of W-WII, Germany had the first jet propelled airplane - Me 262 This aircraft could reach 900 km/hr The incompressible theory was no longer applicable
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Aerodynamics Prandtl-Glauert Compressibility Correction Drag-Divergence Mach Number Swept Wings 36.4 Prandtl-Glauert Compressibility Correction II A large amount of data was already collected for low speed aerodynamics Therefore, there was a search for methods for correction of this data to include effect of compressibility Such methods are called compressibility corrections Prandtl-Glauert compressibility correction is one such correction
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Aerodynamics Prandtl-Glauert Compressibility Correction Drag-Divergence Mach Number Swept Wings 36.5 Prandtl-Glauert Compressibility Correction III Consider the subsonic, compressible inviscid flow over an airfoil The shape of the airfoil is given by y = f ( x ) Assume that the airfoil is thin and α is small Define β 2 = 1 - M 2 The linearized equation becomes β 2 2 ˆ φ x 2 + ˆ φ y 2 = 0
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Aerodynamics Prandtl-Glauert Compressibility Correction Drag-Divergence Mach Number Swept Wings 36.6 Prandtl-Glauert Compressibility Correction IV Let us transform the independent variables ( x , y ) ( ξ, η ) ξ = x ; η = β y Consider a new velocity potential ¯ φ in this transformed space, such that ¯ φ ( ξ, η ) = β ˆ φ ( x , y ) By chain rule ˆ φ x = ˆ φ ∂ξ ∂ξ x + ˆ φ ∂η ∂η x ˆ φ y = ˆ φ ∂ξ ∂ξ y + ˆ φ ∂η ∂η y But ∂ξ x = 1 ∂ξ y = 0 ∂η x = 0 ∂η y = β Therefore ˆ φ x = ˆ φ ∂ξ ; ˆ φ y = β ˆ φ ∂η
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Aerodynamics Prandtl-Glauert Compressibility Correction Drag-Divergence Mach Number Swept Wings 36.7 Prandtl-Glauert Compressibility Correction V We had ¯ φ ( ξ, η ) = β ˆ φ ( x , y ) Then ˆ φ x = 1 β ¯ φ ∂ξ ˆ φ y = ¯ φ ∂η 2 ˆ φ x 2 = 1 β 2 ¯ φ ∂ξ 2 2 ˆ φ y 2 = β 2 ¯ φ ∂η 2 Substituting this into the small perturbation equation β 2 1 β 2 ¯ φ ∂ξ 2 + β 2 ¯ φ ∂η 2 = 0 2 ¯ φ ∂ξ 2 + 2 ¯ φ ∂η 2 = 0 This is similar to the governing equation for incompressible flow - Laplace equation
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Aerodynamics Prandtl-Glauert Compressibility Correction
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  • Fall '14
  • Manoj
  • Aerodynamics, Incompressible Flow, Compressibility Correction XVI, Drag-Divergence Mach, Compressibility Correction XXI, prandtl-glauert compressibility correction

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