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Chapter%205%20Inviscid%20Flow--2

Chapter%205%20Inviscid%20Flow--2 - V General Features of...

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V. General Features of Inviscid Flows There are many situations in which it is more convenient to study the fluid motion in terms of vorticity ϖ rather than in terms of velocity u . Flow over a solid wall at high Reuynolds number can be characterized by a near wall region of large vorticity and a region of zero vorticity. To study the dynamics associated with the vorticity ϖ , we start with kinematics of the rotational flow. 5.1 Vortex Field and Its Properties Definition of the vorticity vector: ϖ = x u . Consequently, one has the following mathematical identity, ∇ ⋅ ϖ = 0. (5.1) Hence, the vorticity field is always divergence free or ϖ is solenoidal. Vortex line : A vector line in the vortex field whose tangent coincides with the direction of the vortex vector ϖ . Based on the definition of vortex line and the physical meaning of ϖ , it is observed that the vortex line is also the instantaneous axis of rotation of the fluid element. Similarly to the equation for the streamlines given by (3.16), the equation for the vortex line is = = . (5.2) A vortex line is an instantaneous curve. ϖ Fig. 5.1 A vortex line. vortex line
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Vortex tube : selecting a small closed contour, c 1 , in the vortex field and passing through each point of this contour c 1 a vortex line, as shown in Fig. 5.2, we obtain a tubular surface called vortex tube . Fig. 5.2 A vortex tube. ϖ c 1 C irculation Γ : along any segment of a curve define the line integral as Γ = u d l = u i dx i . (5.3) The circulation Γ is closely related to the vortex field ϖ . Consider a closed curve C; the circulation around it is Γ . Using the Stokes theorem, it follows that Γ = C Ñ u d l = C Ñ u i dx i = ∫∫ A x u d S = ∫∫ A ϖ d S . (5.4) where A is any surface bounded by the curve C, and d S is the element of area of the surface shown below. C S Fig. 5.3 Surface S bounded by the closed curve C. ϖ
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Intensity of the vortex tube = vortex intensity =   ∫∫ A ϖ d S S = open surface bounded by a closed curve lying entirely in the fluid C. circulation Γ around the closed curve C = vortex intensity. Does the vortex intensity vary along the vortex tube? ϖ n 3 c 2 A 1 A 2 c 1 d S 1 d S 2 n 2 A 3 n 1 Fig. 5.4 Constancy of vortex intensity along a vortex tube. dS 3 Referring to Fig. 5.4, we define A 1 and A 2 = areas of the two cross sections cutting the vortex tube, A 3 = area of the vortex tube wall, A = the union of A 1 , A 2 and A 3 , n 1 & n 2 = unit vectors of A 1 & A 2 in the same sense relative to the vortex tube, n 3 = unit vector of A 3 . C A n dS u d l C or l Γ
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We note that ∫∫∫ ∇⋅ ϖ = 0 because of (5.1). Applying the divergence theorem to the volume of the flow field the inside the entire tube A, we obtain ∫∫∫ v ∇⋅ ϖ d v = A ∫∫ Ò ϖ d S = 0, (5.5) i.e. ∫∫ A 1 ϖ d S + ∫∫ A 2 ϖ d S + ∫∫ A 3 ϖ d S = 0.
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