3 Advanced Potential Flow

8/3/2019 3 Advanced Potential Flow

1/21

HCMC University of Technology

22/09/200957:020 Fluid Mechanics 1

Advanced Potential Flow

DWFS Tool CANICE 2D/3D


2/21


22/09/200957:020 Fluid Mechanics


1. Eulers equation of motion

2. Velocity potential

3. Stream function4. Complex potential

5. Example of potential flow

6. Conformal mapping


3/21



Re: large diffusion of vorticity: small fluidcan be treated as an ideal fluid

Applicability of ideal flow is large

Potential flow refers to an ideal fluid

Stream function Velocity potential



4/21



Eulers equation of motion

Steady flow:

Continuity equation + Eulers equation of motion-> u, v, p = f(t,x,y)

(Difficult to obtain analytically except for a particular restricted case!)


5/21



Velocity Potential

= gradV

Potential flow = Irrotational motion

Definition

Laplaces equation

Velocities u and v can be obtained bydifferentiating the velocity potential

-> flow pattern is found


6/21



Velocity Potential

Equipotential line: = const

Direction of fluid velocity is at right angles to

the potential line

0=

dsV


7/21



Stream function

Streamline for 2D flow:

consteid == ..,0 : define a streamline

Velocities u and v can be obtained bydifferentiating the stream function

-> flow pattern is found


8/21



Stream function

Relationship between flow rate and stream function


9/21



Complex potential

w(z) complex potential of a flow

: conjugate complex velocity


10/21



Summary


11/21



Examples of Potential Flow

Parallel flowVelocity Potential Stream Function


12/21




SourceDischarge q per unit thickness: constvrq r == 2


13/21




Free vortexCirculation:

O

1

>0

2

3

4

x

y

321


14/21



Synthesizing of flows

By combining two 2D incompressible potential flows,another potential flow can be obtained

Combining a source and a sink (doublet)


15/21




Flow around a cylinder = Parallel flow + Doublet

StreamlinesPressure Coefficient


16/21




Flow around a cylinder with circulation

= 0 < 4Ruo= 4Ruo > 4Ruo


17/21



StreamlinesPressure Coefficient

Flow around a cylinder with circulation


= 0UL

Kutta-Joukowsky Theorem


18/21



Conformal Mapping

Conformal Mapping:

Equiangularly mapped

Similarity


19/21



Conformal Mapping

Joukowskis mapping function:


20/21



Conformal Mapping

Joukowskis mapping function:


21/21



Conformal Mapping

Mapping of flow around cylinder onto flow around wing

Kutta condition

Documents

3 Advanced Potential Flow