Fluids Lecture 15 Notes


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1 Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit field is specified eithe b the feesteam velocit components u, v, o b the feesteam speed V and flow angle α. u = u v = v = V cosα = V sin α Note also that V 2 = u2 + v2. The coesponding potential and steam functions ae φ(, ) = u + v = V ( cosα + sin α) ψ(, ) = u v = V ( cos α sin α) V α v u Zeo Divegence A unifom flow is easil shown to have zeo divegence V = u + v = 0 since both u and v ae constants. The equivalent statement is that φ(, ) satisfies Laplace s equation. 2 φ = 2 (u + v ) (u + v ) 2 = 0 Theefoe, the unifom flow satisfies mass consevation. Zeo Cul A unifom flow is also easil shown to be iotational, o to have zeo voticit. V ξ = ( v u ) ˆk = 0 1
2 The equivalent iotationalit condition is that ψ(, ) satisfies Laplace s equation. Souce and Sink 2 ψ = 2 (u v ) (u v ) 2 = 0 Definition A 2D souce is most cleal specified in pola coodinates. The adial and tangential velocit components ae defined to be V = Λ, V θ = 0 whee Λ is a scaling constant called the souce stength. The volume flow ate pe unit span V acoss a cicle of adius is computed as follows. V = 0 V ˆn da = 0 V dθ = 0 Λ dθ = Λ Hence we see that the souce stength Λ specifies the ate of volume flow issuing outwad fom the souce. If Λ is negative, the flow is inwad, and the flow is called a sink. V θ V θ Catesian epesentation The catesian velocit components of the souce o sink ae u(, ) = Λ v(, ) = Λ and the coesponding potential and steam functions ae as follows. φ(, ) = Λ ln = Λ ln ψ(, ) = Λ actan(/) = Λ θ 2
3 It is easil veified that apat fom the oigin location (, ) = (0, 0), these functions satisf 2 φ = 0 and 2 ψ = 0, and hence epesent phsicallpossible incompessible, iotational flows. Singulaities The oigin location (0, 0) is called a singula point of the souce flow. As we appoach this point, the magnitude of the adial velocit tends to infinit as V 1 Hence the flow at the singula point is not phsical, although this does not pevent us fom using the souce to epesent actual flows. We will simpl need to ensue that the singula point is located outside the flow egion of inteest. Unifom Flow with Souce Two o moe incompessible, iotational flows can be combined b supeposition, simpl b adding thei velocit fields o thei potential o steam function fields. Supeposition of a unifom flow in the diection and a souce at the oigin theefoe has u(, ) = Λ v(, ) = Λ V o φ(, ) = Λ ln V = Λ ln + V cos θ o ψ(, ) = Λ actan(/) + V = Λ θ + V sin θ The figue shows the steamlines of the two basic flows, and also the combined flow. The bulletshaped heav line on the combined flow coesponds to the dividing steamline, which sepaates the fluid coming fom the feesteam and the fluid coming fom the souce. If we eplace the dividing steamline b a solid semiinfinite bod of the same shape, the flow about this bod will be the same as the flow outside the dividing steamline in the supeimposed flow. 3
4 Unifom Flow with Souce and Sink We now supeimpose a unifom flow in the diection, with a souce located at ( l/2, 0), and a sink of equal and opposite stength located at (+l/2, 0), plus a feesteam. ψ = Λ (θ 1 θ 2 ) + V sin θ 1 2 ψ θ1 l θ 2 The figue on the ight shows the steamlines of the combined flow. The heav line again indicates the dividing steamline, which taces out a Rankine oval. All the steamlines inside the oval oiginate at the souce on the left, and flow into the sink on the ight. The net volume outflow fom the oval is zeo. Again, the dividing steamline could be eplaced b a solid oval bod of the same shape. The flow outside the oval then coesponds to the flow about this bod. Doublet Conside a soucesink pai with stengths ±Λ, located at ( l/2, 0). Now let the sepaation distance l appoach zeo, while simultaneousl inceasing the souce and sink stengths such that the poduct κ lλ emains constant. The esulting flow is a doublet with stength κ. ψ = lim κ l 0 l θ = κ κ=const. sin θ ψ θ l 4
5 A simila limiting pocess can be used to poduce the doublet s potential function. φ = κ cosθ The steamline shapes of the doublet ae obtained b setting whee ψ = κ = d sin θ d = κ c sin θ = c = constant In pola coodinates this is the equation fo cicles of diamete d, centeed on, = (0, ±d/2). Nonlifting Flow ove Cicula Clinde Flowfield definition We now supeimpose a unifom flow with a doublet. o ψ = V sin θ ψ = V sin θ whee R 2 κ/(v ) ( κ sin θ ) 1 R2 2 ( = V sin θ 1 This coesponds to the flow about a cicula clinde of adius R. ) κ V 2 The adial and tangential velocities can be obtained b diffeentiating the steam function as follows. V = 1 ( ) ψ = V cosθ 1 R2 θ 2 V θ = ψ ( ) = V sin θ 1 + R2 2 5
6 Suface velocities and pessues On the suface of the clinde whee = R, we have V = 0 V θ = 2V sin θ The maimum suface speed of 2V occus at θ = ±90. The suface pessue is then obtained using the Benoulli equation p(θ) = p o 1 2 ρ ( V 2 + V ) θ 2 Substituting V = 0 and V θ (θ), and using the feesteam value fo the total pessue, p o = p ρv 2 gives the following suface pessue distibution. ( 1 4 sin 2 θ ) p(θ) = p ρv 2 The coesponding pessue coefficient is also eadil obtained. C p (θ) p(θ) p 1 2 ρv 2 = 1 4 sin 2 θ 6
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