The next assumptions we make are that the flows being treated are steady and

The next assumptions we make are that the flows being

This preview shows page 116 - 118 out of 164 pages.

The next assumptions we make are that the flows being treated are steady and irrotational. Recall that the second of these implies that ∇ × U = 0, which in 2D collapses to u y = v x . With these simplifications the above equations can be expressed as uu x + vv x = 1 ρ p x , uu y + vv y = 1 ρ p y g , and application (in reverse) of the product rule for differentiation yields ρ 2 ( u 2 + v 2 ) x = p x , (4.10a) ρ 2 ( u 2 + v 2 ) y = p y ρg . (4.10b) At this point we introduce some fairly common notation; namely, we write U 2 = u 2 + v 2 , which is shorthand for U · U , the square of the flow speed. Also, on the right-hand side of Eq. (4.10b) because ρ is constant by incompressibility, we can write p y ρg = ∂y ( p + ρgy ) = ∂y ( p + γy )
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4.2. BERNOULLI’S EQUATION 111 But we can employ the same construction in Eq. (4.10a) to obtain p x = ∂x ( p + ρgy ) = ∂x ( p + γy ) because the partial derivative of γy with respect to x is zero. We now use these notations and rearrangements to write Eqs. (4.10) as ∂x bracketleftBig ρ 2 U 2 + p + γy bracketrightBig = 0 , (4.11a) ∂y bracketleftBig ρ 2 U 2 + p + γy bracketrightBig = 0 , (4.11b) where we have again invoked incompressibility to move ρ/ 2 inside the partial derivatives. It should be clear at this point that if we had considered the 3-D case, still keeping the gravity vector aligned with the negative y direction, we would have obtained an analogous result containing yet a third equation of exactly the same form as those given above, with a z -direction partial derivative. (But note that there is more to be done in this case at the time the irrotational assumption is used.) Equations (4.11) are two very simple partial differential equations that, in principle, can be solved by merely integrating them. But a better way to interpret their consequences is to note that the first equation implies that ρ 2 U 2 + p + γy is independent of x while the second equation implies the same for y . But this analysis is 2D; x and y are the only independent variables. Hence, it follows that this quantity must be a constant; i.e. , ρ 2 U 2 + p + γy = C const. (4.12) throughout the flow field. This result can be written for every point in the flow field, but the constant C must always be the same. Thus, if we consider arbitrary points 1 and 2 somewhere in the flow field we have ρ 2 U 2 1 + p 1 + γy 1 = C , and ρ 2 U 2 2 + p 2 + γy 2 = C . But because C must be the same in both equations, the left-hand sides are equal, and we have p 1 + ρ 2 U 2 1 + γy 1 = p 2 + ρ 2 U 2 2 + γy 2 , (4.13) the well-known Bernoulli’s equation. There are a number of items that should be noted regarding this equation. First, it is worthwhile to summarize the assumptions that we made to produce it. These are: steady, incompressible, in- viscid and irrotational flow. Associated with the last of these, we note that an alternative derivation can be (and usually is ) used in which the irrotational assumption is dropped and application of Eq.
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