Chapter 6 Differential Analysis of Fluid Flow

Transcription

1 1 Chapte 6 Diffeential Analysis of Fluid Flow Inviscid flow: Eule s equations of otion Flow fields in which the sheaing stesses ae zeo ae said to be inviscid, nonviscous, o fictionless. fo fluids in which thee ae no sheaing stesses the noal stess at a point is independent of diection: p = σ xx = σ yy = σ zz Fo an inviscid flow in which all the sheaing stesses ae zeo, and the noal stesses ae eplaced by p, the Navie-Stokes Equations educe to Eule s equations ρg p ρ V = + ( V ) V t In Catesian coodinates: p u u u u ρgx = ρ + u + v + w x t x y z p v v v v ρgy = ρ + u + v + w y t x y z p w w w w ρgz = ρ + u + v + w z t x y z The Benoulli equation deived fo Eule s equations The Benoulli equation can also be deived, stating fo Eule s equations. Fo inviscid, incopessible fluids, we end up with the sae equation p V gz const ρ + + = 1

2 It is often convenient to wite the Benoulli equation between two points (1) and () along a stealine and to expess the equation in the head fo by dividing each te by g so that p1 V1 p V z1 z γ + g + = γ + g + The Benoulli equation is esticted to the following: inviscid flow steady flow incopessible flow flow along a stealine The Iotational Flow and coesponding Benoulli equation If we ake one additional assuption that the flow is iotational V = 0 the analysis of inviscid flow pobles is futhe siplified. The Benoulli equation has exactly the sae fo at that fo inviscid flows: p1 V1 p V + + z1 = + + z γ g γ g but it can now be applied between any two points in the flow field, not liited to applications along a stealine.

3 3 Vaious egions of flow: (a) aound bodies; (b) though channels The Velocity Potential Fo an iotational flow: w v u w v u V = i+ j+ k = 0 y z z x x y So we have w v u w v u =, =, = y z z x x y It follows that in this case the velocity coponents can be expessed in tes of a scala function φ (x, y, z, t), called velocity potential, as φ φ φ u =, v =, w= x y z In vecto fo: V = φ 3

4 4 The velocity potential is a consequence of the iotationality of the flow field, wheeas the stea function is a consequence of consevation of ass. It is to be noted, howeve, that the velocity potential can be defined fo a geneal thee-diensional flow, wheeas the stea function is esticted to two-diensional flows. Fo an incopessible flow we know fo the consevation of ass: V =0 and theefoe fo incopessible, iotational flow, it follows that φ = 0 The velocity potential satisfies the Laplace equation. In Catesian coodinates: φ φ φ + + = 0 x y z In cylindical coodinates: 1 φ 1 φ φ = θ z Soe Basic, Plane Potential Flows Fo potential flow, basic solutions can be siply added to obtain oe coplicated solutions because of the ajo advantage of Laplace equation that it is a linea PDE. Fo siplicity, only plane (two-diensional) flows will be consideed. Since we can define a stea function fo plane flow, ψ ψ u =, v = y x 4

5 5 If we now ipose the condition of iotationality, it follows u v = y x and in tes of the stea function ψ ψ = y y x x ψ ψ + = 0 x y Thus, fo a plane iotational flow we can use eithe the velocity potential o the stea function both ust satisfy Laplace's equation in two diensions. It is appaent fo these esults that the velocity potential and the stea function ae soehow elated. It can be shown that lines of constant (called equipotential lines) ae othogonal to lines of constant ψ (stealines) at all points whee they intesect. Recall that two lines ae othogonal if the poduct of thei slopes is 1, as illustated by this figue Along stealines ψ=const: dy dx along ψ = const v = u Along equipotential lines = const φ φ dφ = dx + dy = udx + vdy = x y 0 5

6 6 dy u = dx along φ = const v Unifo flow at angle α with the x axis Velocity potential: φ = U( xcosα + ysinα) Stea function: ψ = U( ycosα xsinα) Velocity coponents: u= Ucos α, v= Usinα Souce o sink ( > 0 souce; < 0 sink) Velocity potential: φ = ln Stea function: ψ = θ Velocity coponents: v =, vθ = 0 6

7 7 Fee votex (Γ > 0 counteclockwise; Γ < 0 clockwise) Γ Velocity potential: φ = θ Γ Stea function: ψ = ln Velocity coponents: v = 0, vθ = Γ Doublet (with stength k=a/π) K cosθ Velocity potential: φ = K sinθ Stea function: ψ = Kcosθ Ksinθ Velocity coponents: v =, v θ = 7

8 8 Supeposition of Basic, Plane Potential Flows Souce in a Unifo Stea Half-Body Flow aound a half-body is obtained by the addition of a souce to a unifo flow. The flow aound a half-body: (a) supeposition of a souce and a unifo flow; (b) eplaceent of stealine ψ = πbu with solid bounday to fo half-body. Velocity potential: φ = U cosθ + ln Stea function: ψ = U sinθ + θ Velocity coponents: v =, vθ = Usinθ Rankine Ovals Rankine ovals ae foed by cobining a souce and sink with a unifo flow. 8

9 9 The flow aound a Rankine oval: (a) supeposition of souce sink pai and a unifo flow; (b) eplaceent of stealine ψ = 0 with solid bounday to fo Rankine oval. Velocity potential: cos φ = U θ ( ln ln 1 ) 1 asinθ Stea function: ψ = U sinθ tan a 1 a Body half length: l = + a πu Body half width: h a π h = tan Uh a Flow aound a Cicula Cylinde A doublet cobined with a unifo flow can be used to epesent flow aound a cicula cylinde. The flow aound a cicula cylinde K cosθ Velocity potential: φ = U cosθ + 9

10 10 K sinθ Stea function: ψ = U sinθ Velocity coponents: a a v = U 1 cos θ, v U 1 sin θ = + θ 10