By Raymond L. Bisplinghoff, Holt Ashley, Robert L.Halfman
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First, we define the stream functions for two-dimensional flows. Examining the continuity equation for an incompressible, two-dimensional flow in the Cartesian frame ∇. �⃗ = �� �� + = 0. 26) It is obvious that the equation is satisfied identically by a stream function �, such that �= �� �� and �=− �� . 27) For polar coordinate system, we can obtain the radial and azimuthal components of velocity in terms of the stream function as �� = 1 �� � �� and �� = − �� .
7 Circulation and Rotationality in Flows This is defined as the closed contour integral of the velocity field given by, Γ = ∮ �⃗ ⋅ �⃗�, which is considered as positive, when the contour is traced in a counter-clockwise direction. 6. 6 Defining circulation (ΔΓ) around an infinitesimal fluid element 36 Theoretical and Computational Aerodynamics the sides and proceeding counter-clockwise, we obtain the circulation for this infinitesimal element as ) ( �� Δ� Δ� ΔΓ = � Δ� + � + �� ( ) �� − � + Δ� Δ� − � Δ� �� ( = ) �� �� − Δ� Δ�.
4) Basic Equations of Motion 23 As one defines the Mach number by, � = lent to � � , the condition of incompressibility is equiva- 1 2 � ≪ 1. 05 or 5%. 3. 2 Conservation Principles For low-speed flights, the energy equation may not be necessary to solve unless some explicit heat transfer problems are addressed. Additionally, conservation of linear momentum equations can be simplified by successively making inviscid and irrotational flow assumptions. These lead to significant ease of analysis and have led to development of simple, yet very elegant aerodynamics theories.