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Potential Flow 1.0 POTENTIAL FLOW One of the most important applications of potential flow theory is to aerodynamics and marine hydrodynamics. Key assumption. 1. Incompressibility – The density and specific weight are to be taken as constant. 2. Irrotationality – This implies a nonviscous fluid where particles are initially moving without rotation. 3. Steady flow – All properties and flow parameters are independent of time. (a) (b) Fig. 1.1 Examples of complicated immersed flows: (a) flow near a solid boundary; (b) flow around an automobile. In this section we will be concerned with the mathematical description of the motion of fluid elements moving in a flow field. A small fluid element in the shape of a cube which is initially in one position will move to another position during a short time interval as illustrated in Fig.1.1. Fig. 1.2 1.1 Continuity Equation v u = velocity component x direction y v y y y v = velocity component y direction u u x u x x x v x Continuity Equation Flow inwards = Flow outwards u v uy vx u xy v yx x y u v 0 - 2D x y u v w 0 - 3D x y z 1.2 Stream Function, (psi) y B Stream Line B A A u x -v The stream is continuity d vdx udy if (x, y) d dx dy x y u and v x y Integrated the equations dx dy C x y vdx udy C Continuity equation in 2 2 y x 0 or x y xy yx if 0 not continuity Vorticity equation, (rotational flow) 2 1 v u ; angular velocity (rad/s) 2 x y v u x y or substitute with 2 2 x 2 y 2 Irrotational flow, 0 Rotational flow. 0 2 2 2 0 x 2 y 2 1.3 Cylindrical Polar Coordinate r y 2 2 y V ur u u u B P(r, ) d r u A r x x r The stream is continuity d ur (r) u (r) ----------------------(1) 2-D equation if (r, ) d d dr ------------------------(2) r compare equation (1) and ( 2) 1 u and u r r r 1.3.1 Continuity Equation u u y ur r 2 ur r 2 r r r 2 P(r, ) r r 2 u r u r ur u r 2 2 r x r Flow inwards = Flow outwards ur r r u u r r r u ur . r u r ur . r u . r r 2 2 2 r 2 2 2 u u u r r 0 r r r 1.4 Velocity Potential, (phi) Occur when some vorticity flow are irrotational, 0 Velocity component; u , v and w x y z 1.4.1 Potential Flow The flow that consist potential flow Occur when some flow are irrotational and consist potential flow. The relationship between velocity potential and stream function are known as Cauchy- Rieman Equation. , y x y x 1.4 Summary 1.4.1 Uniform Flow in 3-D y u = U (constant) v = 0 w = 0 x 1.4.2 The relationship between potential flow and stream function 1 2 3 4 1 4 x U, 0 x y U x C U, 0 y x U y C x U (xkos y sin ) U (ykos xsin ) 1.4.3 Continuity Equation 1.4.3.1 Cartesian Coordinate. u v 0 x y 1 1.4.3.2 Polar Coordinate 2 3 u u 1 u r r 0 r r r 4 1 4 1.4.4 Stream Function 1.4.4.1 Cartesian Coordinate u and v y x Resultant velocity, q u 2 v2 If we know velocity component So vx uy C 1.4.4.2 Polar Coordinate 1 u and u r r r 2 2 Resultant velocity, q u r u If we know velocity component u r ru r 1.4.5 Circulation, 2r 2 1.4.6 Vorticity, 2r 2 2 A r 2 1.4.7 The relationship between vorticity with velocity component and stream function. 1.4.7.1 Cartesian Coordinate. v u ABCD xy x y we know that, u and v y y 2 2 x 2 y 2 1.4.7.2 Polar Coordinate A rr u u 1 u r r r r if we know that, 1 u and u r r r 2 2 1 1 2 2 2 2 r r r r 1.4.8 Potential Flow, 1.4.8.1 Cartesian Coordinate u and v x y 1.4.8.2 Polar Coordinate 1 u and u r r r 1.4.9 The relationship between Stream Function and Potential Flow 1.4.9.1 Cartesian Coordinate u and v y x x y 1.4.9.2 Polar Coordinate 1 1 u and u r r r r r Example 1 Given a flow field k(x 2 y 2 ). Shows that a flow is irrotational. Solution Irrotational is 0 v u 0------------------- irrotational x y u and v y x k(x2 y 2 ) kx2 ky2 u u 2ky so 2k y y v v 2kx so 2k x x v u 2k (2k) 0 = irrotational x y Examples 2 For a certain two-dimensional flow field the velocity is given by the equation V 4xyi 2(x2 y 2 ) j is this flow irrotational? Solution u = 4xy and v 2(x2 y 2 ) 2x2 2y 2 v v 4x 4x y x v u 0 4x 4x 0 = irrotational x y 1.1 Source and Sink Consider a fluid flowing radially outward and inward from a line through the origin perpendicular to the x–y plane as is shown in Fig. 1.3. y Source y Sink = constant = constant ur x x r Fig. 1.3 The streamline pattern for a source and a sink 1.1.1 Stream Function, Let Q be the volume rate of flow emanating from the line (per unit length), and therefore to satisfy conservation of mass Q (2r)ur or Q u r 2r we know that 1 Q u and u 0 r r 2r (r, ) Q r (0r) (r ) r 2r Q r C 2r Q C 2 If 0 at 0 C=0 Q FOR SOURCE 2 Q FOR SINK 2 1.1.2 Velocity potential, Q 1 u , U 0 r r 2r r r r Q r (0) 2r Q 1 . dr (0)d C 2r r Q Q r log r C atau ln r C y 2 e 2 Q log (x 2 y 2 ) 4 e x 2 2 2 r =x +y Q Q 2 2 source log r log (x y ) 2 e 4 e Q Q 2 2 sink log r log (x y ) 2 e 4 e E XAMPLE 1.1 A nonviscous, incompressible fluid flows between wedge-shaped walls into a small opening as shown in Fig. E1.1. The velocity potential (in ft2/in),which approximately describes this flow is 2ln r Determine the volume rate of flow (per unit length) into the opening. Solution Fig.E1.1 The components of velocity are 2 1 u , U 0 r r r r which indicates we have a purely radial flow. The flow rate per unit width, q, crossing the arc of length R / 6 can thus be obtained by integrating the expression / 6 / 6 2 3 Q u R R 2 r -1.05 ft /s 0 0 R Note that the radius R is arbitrary since the flow rate crossing any curve between the two walls must be the same. The negative sign indicates that the flow is toward the opening, that is, in the negative radial direction. 1.2 Free Vortex and Forced Vortex The irrotational vortex is usually called a free vortex. The swirling motion of the water as it drains from a bathtub is similar to that of a free vortex, whereas the motion of a liquid contained in a tank that is rotated about its axis with angular velocity corresponds to a forced vortex. A combined vortex is one with a forced vortex as a central core and a velocity distribution corresponding to that of a free vortex outside the core. (r dr)d u du dr r rd u d d Fig. 1.4 A vortex represents a flow in which the streamlines are concentric circles. d (u du )(r dr)d 0 u (rd) 0 du and d is a small components so it can be neglect. A mathematical concept commonly associated with vortex motion is that of circulation The circulation, is defined as the line integral of the tangential component of the velocity taken around a closed curve in the flow field. In equation form, can be expressed as u .dl C 2r.u u r 2r dr C ln r C 2r 2 so for free vortex is shows below ln r (ccw) f .vortex 2 ln r (cw) f .vortex 2 . (ccw) f .vortex 2 . (cw) f .vortex 2 and for force vortex is shows below r 2 forced vortex = 2 E XAMPLE 1.2 A liquid drains from a large tank through a small opening as illustrated in Fig.E1.2. A vortex forms whose velocity distribution away from the tank opening can be approximated as that of a free vortex having a velocity potential . 2 Determine an expression relating the surface shape to the strength of the vortex as specified by the circulation . Fig E1.2 Solution Since the free vortex represents an irrotational flow field, the Bernoulli equation p V 2 p V 2 1 1 Z 2 2 Z 2g 1 2g 2 can be written between any two points.
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