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Equation Sheet - LOW-SPEED AERODYNAMICS EQUATIONS OVERVIEW...

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LOW-SPEED AERODYNAMICS EQUATIONS OVERVIEW Continuity Equation t ρ dV CV ∫∫∫ + ρ ( r V r n ) dA CS ∫∫ = 0 Cartesian coordinates: ∂ρ t + ( ρ u ) x + ( ρ v ) y + ( ρ w ) z = 0 cylindrical coordinates: ∂ρ t + 1 r ∂ ρ rV r ( ) r + 1 r ∂ ρ V θ ( ) ∂θ + ∂ ρ V z ( ) z = 0 Momentum Equation ρ r f dV + r F visc CV ∫∫∫ - P r n dA CS ∫∫ + r R = t ρ r V dV CV ∫∫∫ + ρ r V ( r V r n ) dA CS ∫∫ Cartesian coordinates: ( ) ( ) ( ) ( ) ( ) ( ) 0 ~ ) ( 0 ~ ) ( 0 ~ ) ( = - - + + - = - - + + - = - - + + - z visc z y visc y x visc x F f z P V w t w direction z F f y P V v t v direction y F f x P V u t u direction x ρ ρ ρ ρ ρ ρ ρ ρ ρ r r r r r r Wake-Rake: - = 2 2 1 2 2 ) ( dy u u u D ρ Substantial Derivative D Dt = t + u x + v y + w z Streamlines d r s × r V = 0 Cartesian coordinates: w dy - vdz = 0 u dz - w dx = 0 v dx - udy = 0 for 2-D flow: dy dx = v u Vorticity r ξ 2 r ω = r ∇ × r V = curl r V = w y - v z ˆ i + u z - w x ˆ j + v x - u y ˆ k
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