By Sears W.M.
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Extra resources for Introduction to theoretical aerodynamics and hydrodynamics
1) must be augmented by the addition of four new terms, in general, to constitute a differential equation for q ′( x ′, y ′, z ′, t ). However, space derivatives such as grad, div, and curl are unaffected in form by the transformation of axes. The only changes in such terms therefore arise from the process of carrying out these operations on quantities, such as q, which have additional terms. There are no such terms in the scalar quantity r; hence, Eq. 1) is altered only by the addition of the four new left-hand terms mentioned.
1), multiplied by dt and integrated over a time interval: τ τ τ τ ∂q 1 ∫0 ∂t dt + ∫0 q ⋅∇q dt = − ∫0 ρ grad p dt + ∫0 F dt Assume, for simplicity, that ρ = constant. The first term can be integrated and the pressure term simplified: τ q (τ ) − q (0 ) + ∫ q ⋅ ∇q dt = − 0 τ τ 1 grad ∫ p dt + ∫ F dt ρ 0 0 Now suppose that τ → 0 while both p and F are increased indefinitely so as to result in impulsive pressures ϖ and force J, respectively. 15) But for any irrotational flow, q = grad φ . Thus, f(x, y, z, t) can be interpreted as [minus (1/ ρ ) times] that distribution of impulsive pressures that would be required to generate impulsively the flow existing at time t.
2 Volume V at times t1 and t1+ Dt. and finally, making use of the general equation of continuity, Eq. 5), D Dq D ρq ρ q dτ = ∫ + ρ q div q dτ = ∫ ρ dτ ∫ Dt V Dt Dt V V Now equating the two expressions for D m / D t and transforming the pressure term to a volume integral, we have − ∫ grad p dτ + ∫ ρ F dτ = ∫ ρ V V V Dq dτ Dt Because V is arbitrary, the integrands must be equal, and results in Eq. 1). 4 Other Forms of the Equations of Motion The term q ⋅ ∇q that occurs in the dynamical equations, Eq.