Prepared by:Wubishet Degife.
E-mail: wubesvictory@gmail.com
Wollo University
Kombolcha Institute of Technology
School of Mechanical & Chemical Engineering
Chapter 2 part -2:
2-D Potential Flow Theory
- continued
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Potential Flow Theory: Solving Velocity Potential and Stream Function in 2D Flows, Schemes and Mind Maps of Aerodynamics
The 2D Potential Flow Theory, a mathematical method used to solve flow problems approximated as potential flows. It covers the concepts of velocity potential function, Laplace equation, stream function, and superposition of elementary potential flow models. Examples include Uniform Flow and Source Flow, Uniform Flow and Doublet, and Uniform Flow, Doublet, and Free Vortex. The document concludes with the Kutta-Joukowski Theorem.
Typology: Schemes and Mind Maps
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Prepared by: Wubishet Degife. E-mail: [email protected] Wollo University Kombolcha Institute of Technology School of Mechanical & Chemical Engineering
Chapter 2 part - 2:
2 - D Potential Flow Theory
- continued
2 - D Potential Flow Theory
- continued
Reference :
1. Frank M. White, Chapter 4, Sections 4.6 & 4.7; 2. Frank M. White, Chapter 8, Sections 8.1 to 8.
2 - D Potential Flow Theory
- Potential flow is
- an inviscid, incompressible, irrotational, steady flow
- Potential Flow Theory is
- a mathematical method developed to solve flow problems that can be closely approximated as a potential flow
2 - D Potential Flow Theory Question What do we mean by ‘ solve’?
Airfoil :
- Known:
- The forces exerted on bodies moving through a fluid are caused only by two reasons : 1. Pressure distribution ( p ) over the body surface,
2. Shear-stress distribution ( ) over the body
surface. i.e., pressure and Shear-stress are the only two mechanisms nature has for ‘communication’ of force & moment between the body and the moving fluid.
2 - D Potential Flow Theory … The starting point for solving a potential flow problem is the set of eqns governing the flow.
2 - D Potential Flow Theory …
- Recall the set of Differential Eqns for a ( general ) fluid flow :
- The flow is steady. Therefore, the non-steady term of the continuity eqn can be left out. (Friction does not affect continuity eqn).
- The flow is inviscid & incompressible. Therefore, the momentum eqn reduces to :
- We know, the energy eqn is not required for the study /analysis of incompressible flow. So, forget the 3 rd eqn.
2 - D Potential Flow Theory …
Velocity Potential Function (symbol, )
- Known – Potential flow is irrotational flow ➔ Curl U = 0 ;
- Known: if Curl U = 0, then, there exists a scalar function (x, y) such that: U = , where:
- Comparing & we note that the following relationship exists b/n the velocity components of the flow and the function : and j y i x
= j y i x U
= x u = y v = U = ui + vj
2 - D Potential Flow Theory … (Repeated), and where = (x,y) represents the scalar function , named ‘ velocity potential function’.
➢ Now, these velocity components computed from
must of course satisfy the continuity eqn:
U = 0 ,
i.e., , x u = y v = = 0
y v x u ( ) ( )= 0
x x y y 0 2 2 2 2 =
x y
2 - D Potential Flow Theory …
- and therefore,
- the problem of solving the potential flow is now reduced to that of solving simultaneously Laplace and Bernoulli eqns.
2 - D Potential Flow Theory …
- Another Laplace eqn can be obtained in a similar way, using stream function as follows:
- Stream Function (symbol, )
- is defined based on a 2D continuity eqn
and a simple (but clever) mathematical
manipulation.
2 - D Potential Flow Theory …
- Again,
- the problem of solving the potential flow is reduced to that of solving simultaneously another Laplace and Bernoulli eqns.
2 - D Potential Flow Theory …
Geometric interpretation of velocity potential
function and stream function :
- Const- lines represent stream lines of the flow; no flow across the stream lines; flow is along these lines.
- const- lines represent equipotential lines; no flow along equipotential lines; flow is across these lines.
- Const- and const- lines are perpendicular to each other.