� � � � Review: Div, Curl, etc. Velocity field Figure 1 shows a Control Volume or circuit placed in a velocity field ~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 componentsor Circuit V u v Figure 1: Velocity field and Cont r ol Vo lume. 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, ′ ~ˆdV˙= 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 t he normal velocity components which are required. The velocities a cross the opposing faces 1,2 and 3,4 ar e related by using the local velocity gradients ∂u/∂x and ∂v/∂y. 99 9Normal velocity components v + dyy u +udy v dx v9 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 ′ = ~V · ˆn ds + ~V · ˆn ds + ~V · ˆn ds + ~V · ˆn ds 1 2 3 4 1~� � � � ~� � � � = = � −u � dy + � u + ∂u ∂x dx � dy + � −v � dx + � v + ∂v ∂y dy � dx � ∂u ∂x + ∂v ∂y � dx dy or d ˙V ′ = � ∇ · ~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 fo llows: 2-D : d(volume outflow/depth) = (velocity divergence) × d(area) (2) For a 3-D infinitesimal “box” CV, a slightly more involved analysis gives dV˙= ∂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, they are in fact correct for any infinitesimal shape. For illustration, consider the volume outflow from a right-triangular CV, pictured in F ig ure 3. One complication is that the unit normal vector ˆn on the hypotenuse is not parallel to one of the axes. However, its n x,n y components are easily obtained from the geometry as shown in t he figure. y Velocity components Normal vectors v9 99 9 dx dsny=dy dsdyv + y 2 n dxdy u u dx dyu + nx= x 2 ds 13 v 2 x dx Figure 3: Infinitesimal triangular Control Volume. The volume outflow per unit depth is now �� � � �� � � �� � � ′ ~ˆ~ˆ~ˆdV˙= V · n ds + V · n ds + V · n ds 12 3 �� � � � � ∂u dx dy ∂v dy dx = −u dy + −v dx + u ++ v + ds ∂x 2 ds ∂y 2 ds 2� � � � � ~~� � ∂u ∂v dx dy = + ∂x ∂y 2 ′ Vor dV˙= ∇ · ~� dx dy 2 The area of the triangular CV is dx dy/2, so statement (2) is still correct for this case. Circulation – infinitesimal rectangular cir cuit We now wish to compute the clockwise circulation dΓ on the perimeter of the infinitesimal rectangular CV, called a circuit by convention. −dΓ = V · d~s = V · sˆ ds This now requires knowing the tangential velocity components, shown in Figure 4. Tangential velocity components Tangential vectors 9 9 y dy 9 9 x dx dy v u + u u v + v y 4 s 1 2 3 dx x Figure 4: Infinitesimal CV with tangential velocity components. The circulation is then computed by evaluating the integral as a sum over the four faces. −dΓ = = = �� ~V · ˆs � ds � 1 + �� ~V · ˆs � ds � 2 + �� ~V · ˆs � ds � 3 + �� ~V · ˆs � ds � 4 � −v � dy + � v + ∂v ∂x dx � dy + � u � dx + � −u − ∂u ∂y dy � dx � ∂v ∂x − ∂u ∂y � dx dy or � − dΓ � = � ∇ ×~V � ·ˆk dx dy (5) ˆwhere ∇ ×~V · k is a convenient (?) shorthand f or the z-component of the velo city curl in the parentheses. For a t r ia ngular circuit, equation (5) would have dx dy/2 as expected. The general result for any infinitesimal circuit can be stated as follows: -d(circu lation) = (normal component of velocity curl) × d(area) (6) 3�� �� � � �� � � � � � ��� �� �� �� Volume outflow – finite CV We now wish to integrate the velocity divergence over a fin i te CV. � � ∂u ∂v ∇ · ~V dx dy =+ dx dy ∂x ∂y As shown in Figure 5, this is equivalent t o summing the volume outflows fr om all the in-finitesimal CVs in the finite CV’s interior, either rectangular or triangular in shape, as needed to conform to the bo undary. We then note that the contributions of all the interior faces ΔV dx dy)( )(V n dsinterior face integrations cancel= Sum )(V n ds Figure 5: Summation of divergence over interior of a finite CV cancel, since these have directly opposing ˆn normal vectors, leaving only the boundary faces in the overall summation which the give the net outflow out of the CV. V dx dy = V · ˆ∇ · ~ ~n ds This general result is known as Gauss’s Theorem, and it applies to any vector field ~υ, not just fluid velocity fields. The general Gauss’s Theorem in 3-D is υ dx dy dz = ~ n dA ∇· ~ υ · ˆn υ dA dx dy dz υ where the volume integral is over the interior of the CV, and the area integral is over the surface of the CV. Circulation – finite circuit We now wish to integrate the normal component of t he curl over the interior of a finite circuit. � � � � ∂v ∂u ˆ∇ ×~V · k dx dy = − dx dy ∂x ∂y As shown in Figure 6, this is equivalent
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