Superposition. Section 8.5.3

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1 Supeposition Section 8.5.3

2 Simple Potential Flows Most complex potential (invicid, iotational) flows can be modeled using a combination of simple potential flows The simple flows used ae: Unifom flows Line souces and sinks Iotational votices Doublets All of these simple flows satisfy the Laplace equation

3 Simple Potential Flows

4 Supeposition of potential flows If ψ 1 and ψ ae two independent solutions of Laplace eqn then ψ ψ 1 +ψ satisfies Laplace eqn. So steam function of a complex flow ψ can be found by adding steam functions (ψ 1, ψ ) of two simple flows Note: The above conclusion is also valid fo potential functions

5 Souces & Sinks If fluid flows adially outwad it is a souce Fo adially inwad it is sink. Let q be the volume flow ate pe unit length of a line passing though cente As thee is no flow in angula diection ψ 1 φ vθ 0 θ Theefoe the volume flow (pe unit length) of a souce/sink is only in the adial diection: *π *v q [Note q is -vefo Sink] Hee q is called Stength v q π 1 ψ θ φ q ψ θ π m θ q φ ln π m ln

6 PLANE - FLOW SOLUTIONS a) Souce and Sink of equal stength Souce, m at (-d,0) and a sink, m at (d,0) Fo souce ψ 1 mθ 1. Fo sink ψ mθ (whee, q stength of souce, m q/π) Fo combined flow ψ ψ 1 +ψ m(θ 1 θ ). ψ m b) Doublet It is a special case of souce and sink when both appoach the oigin while keeping the poduct of stength and distance constant (q*d/π constant µ.) It can be poved that ψ ( θ 1 θ ) µ sin θ φ φ 1 m(ln 1 ln ) µ cosθ

7 Steam & Potential functions of some standad flows Types of flow Steam, ψ Potentialφ Unifom flow in x U sinθ U y U cosθ U x diection Souce mθ m ln Sink - mθ - m ln Fee votex (anticlockwise) Doublet -K ln K θ µ sinθ µ cosθ Whee K (Γ/π), m q /π; Ciculation, Γ can be +ve o ve.

8 Supeposition of simple flows SPIRAL VORTEX (Line sink, m ) + (Fee votex anticlockwise, Γ) ψ ψ sink + ψ votex mθ K ln Similaly φ m ln + Kθ Quiz:How would you fom an outwad spial? FLOW OVER HALF BODY (Unifom flow) + (Souce ) Ψ U sinθ + m θ Φ U cosθ + m ln

9 Flow past a cylinde This can be simulated by supeimposing a doublet on a unifom flow ψ ψ unifom flow + ψ doublet Ψ U sinθ λ sin θ (U.- λ/) sinθ Steamline on object s suface is zeo. So fo the suface sinθ(u. λ /) 0; o (λ/ U) 0.5. It is the equation of a cicle Stagnant points occus at V θ 0; which occus at 0 & 180. v ψ λ θ ( U + ) sinθ Velocity on the cylinde suface is found on substitution v 1 ψ θ U λ cosθ

10 Dag Dag is found by integating foce in flow diection and this integal is found to be zeo D π P ( ba) dθ.cosθ 0 s It suggests that thee is no dag on a cylinde in coss-flow on a stationey cylinde This absud esult is called D Alembet s Paadox. 0

11 ROTATING CYLINDER Flow ove otating cylinde Unifom flow + Fee Votex (anticlockwise) + Doublet Ψ U sinθ + (-K ln ) - (λsinθ/) Velocity at a point v θ ψ U sin θ + Votex flow does not change V. So it emains unchanged. So, the suface is still defined by λ U a Velocity on its suface is theefoe V θ U sinθ+( Κ/a) Stagnation point occus whee V θ 0, So, no ciculation (K0), stagnation occus at θ s 0 and Fo otation If K au, sinθ 0.5; stagnation occus at θ s 30 and K Γ πκ whee Γ is ciculation λ sin θ sin θ s K au

12 Steam lines aound a otating cylinde

13 ROTATING CYLINDER Applying Benoullis eqn. between the suface (whee velocity is v θ ) and feesteam we get whee sin a K U P V P U P s s θ ρ ρ ρ θ ( ) sin 4 4 sin 1 1 β θ β θ ρ + U P P s Ua K β ( ) ( ) ( ) b U b K U d ba P P L s Γ ρ π ρ θ θ π sin 0 Lift, is found by integating foces nomal to the flow Γ U b L ρ whee a adius of cylinde & b length of cylinde Upwad foce, Lift on a otating cylinde is geneated when ciculation is negative. It is called the Magnus effect

14 Developed by M. Flettne in 197

15 Poblem - 1 a) Find the steam function (in pola coodinate) in an unifom flow in x diection. b) In a two-dimensional incompessible flow the fluid velocity components ae given by ux-4y and v-y-4x. Find the steam function and show that it is an iotational flow

16 Poblem - In the ideal flow aound a half body, the fee steam velocity is 0.5 m/s and the stength of the souce is 1/π m /s. Detemine the fluid velocity and its diection at a point 1 m and θ 10

17 Poblem on Magnus Effect In 197 a man named Flettne had a ship built with two otating cylindes to act as sails. The height of the cylindes is 15 m and diamete.75 m, The wind speed is 30 km/h, and the speed of the ship is 4 km/h. The cylindes otate at a speed of 750 pm by the action of a steam engine below deck. Ai density 1.9 kg/m3. Find the maximum possible populsive thust on the ship fom the cylindes. [Shames p-597]

18 Home wok (White 4.7) A coastal powe plant takes in cooling wate though a vetical pefoated manifold. The total volume flow intake is 110 m 3 /s. Cuents of 0.5 m/s flow past the manifold. Estimate how fa downsteam and how fa nomal to the pape the effects of the intake ae felt in the ambient 8 m deep wates. [8.75 m, 55 m]

π(x, y) = u x + v y = V (x cos + y sin ) κ(x, y) = u y v x = V (y cos x sin ) v u x y

π(x, y) = u x + v y = V (x cos + y sin ) κ(x, y) = u y v x = V (y cos x sin ) v u x y F17 Lectue Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V φ = uî + vθˆ is a constant. In 2-D, this velocit