5. Irrotational Flow - Complex Potential and Aerofoils Flashcards
velocity potential, planar potential flows, the complex derivative, the complex potential, imposing boundary conditions, forces on streamlined bodies, origins of circulation around a wing, three-dimensional aerofoils
High Reynolds Number Flows Past a Streamlined Object
Summary
- vorticity is only generated at the object’s surface
- it is therefore confined to a thin boundary layer around the surface of the object and to a narrow wake behind it
- consequentially, outside these boundary layers the flow has no vorticity and is described as being irrotational
Velocity Potential
Definition
-if ∇xu_=0, we can write: u = ∂φ/∂x v = ∂φ/∂y w =∂φ/∂z -i.e. u_= ∇φ -note that we can add any arbitrary function of t to φ
Incompressible and Irrotational Flows
- if a flow is also incompressible then: ∇.u_ = 0 => ∇.(∇φ)=0 => ∇²φ = 0 -i.e. φ satisfies the Laplace equation
Neumann Boundary Conditions
-since we are now considering the edge of the boundary layer as the boundary. we can no longer impose the u_=U_ condition for a rigid boundary
-however since the volume of the boundary layers is small we can assume mass is conserved:
u_ . n_ = U_ . n_
-thus, the boundary condition on the velocity potential is:
n_ . ∇φ = φ/n_ = U_.n_
-this a Neumann boundary condition
The Euler Equation
-since we neglect the effect of viscosity, the Navier-Stokes equation reduces to the Euler equation:
ρ Du_/Dt = ρg_ - ∇P
The Bernoulli Equation Derivation
-starting with the Euler equation
-use the vector identity:
u_x(∇xu_) = 1/2∇(u_.u_) - (u_.∇)u_
-for a potential flow, u_=∇φ and ω_=0
Dynamic Pressure
Definition
p = P - ρg_.x_
The Bernoulli Equation
∇(∂φ/∂t+P/ρ + 1/2 u_.u_ - g_.x_) = F(t)
=>
∂φ/∂t+P/ρ + 1/2 u_.u_ - g_.x_ = F(t)
OR
∂φ/∂t+p/ρ + 1/2 u_.u_=F(t)
-i.e. for a steady irrotational flow, p+1/2u_.u_ is a constant
-the pressure is lower in regions of higher flow speed
What is the relationship between pressure and flow speed?
-the pressure is lower in regions of higher flow speed
Planar Potential Flows
-for planar flow: u_ = (u(x,y),v(x,y),0) -if the flow is incompressible: u=∂ψ/∂y , v= - ∂ψ/∂x -if the flow is also irrotational: ω = 0 = ∂v/∂x - ∂u/∂y => 0 = -∂²ψ/∂x² - ∂²ψ/∂y² -i.e. the streamfunction, ψ, satisfies the Laplace equation
Cauchy-Riemann Equations
∂φ/∂x = ∂ψ/∂y ∂φ/∂y = - ∂ψ/∂x
Laplace Equation from the Cauchy-Riemann Equations
- sum the x derivative of the first Cauchy-Riemann equation with the y derivative of the second to show that φ satisfies the Laplace equation
- sum the y derivative of the first and x derivative of the second to show ψ satisfies the Laplace equation
The Complex Derivative
Definition
-let f be a complex function of z where z=x+iy
-the derivative of f with respect to z is defined:
df/dz = lim f(z+δz)-f(z) / δz
-where the limit is taken as δz->0
-note that this definition requires that the limit is the same for all increments δz which could be in any direction in the complex plane
-thus differentiability of a complex function imposes restrictions on its real and imaginary parts
The Complex Derivative
Restrictions
-if f(z) = g(x,y) + ih(x,y) is complex differentiable, where g and h are both real functions, then we must get the same result for δz=δx as δz=iδy
=>
∂g/∂x = ∂h/∂y
∂g/∂y = - ∂h/∂x
-the Cauchy-Riemann equations, therefore both g and h must satisfy the two-dimensional Laplace equation
-i.e. the real and imaginary parts of any complex differentiable function f(z) are solutions of the Laplace equation
The Complex Potential
Definition
w(z) = φ(x,y) + iψ(x,y)
The Complex Potential
Fluid Velocity
dw/dz = ∂φ/∂x + i ∂ψ/∂x
= u - iv
-OR
u + iv = (dw/dz)*
-where * indicates complex conjugation
|u_| = √[u²+v²] = √[dw/dz {dw/dz)*] = |dw/dz|
-and velocity direction:
α = arg((dw/dz)*) = -arg(dw/dz)Examples of Complex Potentials
Uniform Flow
-if u=Uo cosα, v=Uo sinα
-then:
dw/dz = Uo cosα - iUo sinα = Uo exp(-iα)
-and so the corresponding complex potential is:
w = Uo exp(-iα) z
Examples of Complex Potentials
Saddle Point Flow
w = Az²
Examples of Complex Potentials
Source/Sink
w = Alog(z) = Alog(r) + iθ
Examples of Complex Potentials
Vortex
-complex potential for a vortex of circulation Γ at z=zo:
w(z) = - iΓ/2π log(z-zo)
Examples of Complex Potentials
Dipole
w = m/z
Imposing Boundary Conditions
- we can only impose the continuity of the normal component of velocity for potential flows
- in order to describe a fixed boundary, it is sufficient to impose that the streamfunction is constant along the boundary
- we therefore need to ensure that the imaginary part of the complex potential is constant on the boundary
Method of Images
-e.g. for a vortex, place an image vortex at the mirror image position on the otherside of the boundary with opposire strength
-more generally for any complex potential f(z):
w(z) = f(z) + f(z)
Milne-Thompson Circle Theorem
-for the case where the boundary is a circle, |z|=a:
w(z) = f(z) + f(a²/z)*
-e.g. on the circle z=e^iθ:
w(ae^iθ) = f(ae^iθ) + f(ae^iθ)*
Conformal Mapping
-using a transformation that maps the boundary onto that of a different problem for which we already know the solution Z = g(z) -so Z is the transformed problem -the corresponding complex potential: W(Z) = w(Z) => W(Z) = w(g^(-1)(Z))
The Joukowski Transformation
Z = z + c²/z
-maps a circle, |z|=a, onto an ellipse with semi-axes (a+c²\a) and (a-c²\a)
Flow Past a Finite Flat Plate
dw/dz = dW/dZ dZ/dz
-the velocity components u and v are given by:
U - iV = dW/dZ = dw/dz (dZ/dz)^(-1)
The Joukowski Aerofoil
-we can generate a symmetric aerofoil of this kind by moving the circle centre along the x-axis to z=-b before applying the Joukowski transformation
-the result is an aerofoil whose shape is given by:
Z = -b + (a+b) exp(iθ) + a²/[-b+(a+b)exp(iθ)]
Forces on Streamlined Bodies
-to calculate the force exerted by the flow on the aerofoil, we only need to consider fluid pressure since we are ignoring viscous effects
-the force from the fluid pressure is given by:
F = -∫ pn_ ds
-where n_ is normal to the surface
