6. Bernoulli's Equation

Question 1

The Bernoulli Function

Answer 1

Η(|x,t) = p/ρ + 1/2||u||² - |g.|x

Question 2

The Bernoulli Equation

Answer 2

(|u . ∇) Η = 0

Question 3

Bernoulli Function for an Ideal Fluid in Steady Flow

Answer 3

-for an ideal steady flow, the Bernoulli function:
Η(|x,t) = p/ρ + 1/2||u||² - |g.|x
-is constant along a streamline

Question 4

Streamfunction and Bernoulli Function

Answer 4

-if a streamfunction ψ(|x) can be defined, Η is a function of ψ:
Η(|x) ≡ Η(ψ)

Question 5

Bernoulli’s Theorem for Irrotational Flows

Answer 5

-for irrotational flows,
∂φ/∂t + Η = ∂φ/∂t + p/ρ + 1/2||∇φ||² - |g.|x
-is a function of time f(t), independent of position |x

Question 6

Bernoulli’s Theorem for Steady, Irrotational Flows

Answer 6

-if the flow is steady as well as irrotational,
Η = p/ρ + 1/2||∇φ||² - |g.|x
-is constant, i.e. Η has the same value on all streamlines

Question 7

Write down the velocity potential of a uniform stream U ^ez past a stationary sphere of radius a

Answer 7

-in cylindrical polar coordinates:
φ(r,z) = Uz(1 + a³/[2(r²+z²)^(3/2)]

-in spherical polar coordinates:
φ(r,z) = Ucosθ(r + a³/2r²)

Question 8

D’Alembert’s Paradox

Answer 8

• it can be demonstrated that the drag force on any 3D solid body moving at uniform speed in a potential flow is zero
• in reality this is not true in reality since flows past 3D solid bodies are not potential

Question 9

What is the pressure distribution on the surface of a solid sphere placed in a uniform stream?

Answer 9

p(θ) = p∞ + 1/2 ρU²(1 - 9/4 sin²θ)

Question 10

Separation and Pressure

Answer 10

• an adverse pressure gradient, (|u.∇)p > 0
  (i. e. pressure increasing in the direction of the flow along the surface) is bad news, it causes the flow to separate leaving a turbulent wake

Question 11

How to reduce drag?

Answer 11

• the way to reduce drag is to reduce separation
• this can be done in two ways:
  1) streamlining
  2) surface roughness

Question 12

Streamlining

Answer 12

• separation occurs because of adverse pressure gradients on the surface of solid bodies
• these can be reduced by using more ‘streamlined’ shapes that avoid diverging streamlines

Question 13

Surface Roughness

Answer 13

-paradoxically, a rough surface can reduce drag by reducing separation

Question 14

Bubble Oscillations

Answer 14

• the sound of a ‘babbling brook’ is due to the oscillation of air bubbles entrained into the stream
• the pitch depends on the size of the bubbles

Question 15

What is the velocity potential for a sphere of radius a moving with velocity U in still water?

Answer 15

φ = - Ua³/2r² * cosθ