f07mathdivcurl_fall

f07mathdivcurl_fall - Review Div Curl etc Velocity eld...

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Review: Div, Curl, etc. Velocity field Figure 1 shows a Control Volume or circuit placed in a velocity field vector V ( x, y ). To evaluate the volume flow through the CV, we need to examine the velocities on the CV faces. It is convenient to work directly with the x , y velocity components u , v . Velocity Field Velocities on faces of y Control Volume x Velocity components or Circuit V u v Figure 1: Velocity field and Control Volume. Volume outflow – infinitesimal rectangular CV Let us now assume the CV is an infinitesimal rectangle, with dimensions dx and dy . We wish to compute the net volume outflow (per unit z depth) out of this CV, vector ˆ d V ˙ = V · n ds = ( u n x + v n y ) ds where ds is the CV side arc length, either dx or dy depending on the side in question. Figure 2 shows the normal velocity components which are required. The velocities across the opposing faces 1,2 and 3,4 are related by using the local velocity gradients ∂u/∂x and ∂v/∂y . 9 9 9 Normal velocity components v + dy y u + u dy v dx v 9 Normal vectors y u dx x n 1 2 4 3 x Figure 2: Infinitesimal CV with normal velocity components. The volume outflow is then computed by evaluating the integral as a sum over the four faces. �� �� �� �� d ˙ V = vector V · ˆ n ds + vector V · ˆ n ds + vector V · ˆ n ds + vector V · ˆ n ds 1 2 3 4 1
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vector vector = = u dy + u + ∂u ∂x dx dy + v dx + v + ∂v ∂y dy dx ∂u ∂x + ∂v ∂y dx dy or d ˙ V = ∇ · vector V dx dy (1) where ∇ · V is a convenient shorthand for the velocity divergence in the parentheses. This final result for any 2-D infinitesimal CV can be stated as follows: 2-D : d(volume outflow/depth) = (velocity divergence) × d(area) (2) For a 3-D infinitesimal “box” CV, a slightly more involved analysis gives d V ˙ = ∂u + ∂v + ∂w dx dy dz = ∇ · V dx dy dz (3) ∂x ∂y ∂z 3-D : d(volume outflow) = (velocity divergence) × d(volume) (4) Volume outflow – infinitesimal triangular CV Although statements (2) and (4) were derived for a rectangular infinitesimal CV,
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