Generalised Bernoulli Equation

Question 1

What does dv/dt equal for irrotational flow?

Answer 1

dv/dt = -∇(P/ρ + gz + v^2/2)

Question 2

What is the generalised Bernoulli equation?

Answer 2

dФ/dt + v^2/2 + P/ρ + gz = const

Question 3

In cylindrical polars, what does r(hat) equal?

Answer 3

r(hat) = cosθ i + sinθ j

Question 4

In cylindrical polars, what does θ(hat) equal?

Answer 4

θ(hat) = -sinθ i + cosθ j

Question 5

In cylindrical polars, what does z(hat) equal?

Answer 5

z(hat) = k

Question 6

What is the potential vortex?

Answer 6

A flow with circular paths around an axis such that ∇v = 0

Question 7

What does ∇v = 0 equal in cylindrical coordinates?

Answer 7

1/r d/dr(rv(θ)) - 1/r *d/dθ(v(r)) = 0

Question 8

For the potential vortex, what does the cylindrical coordinates version of ∇v = 0 equal and why?

Answer 8

Potential vortex only has v(θ)(r), so ∇v = 0 -> d/dr(r*v(θ)) = 0, so v(θ) = const/r = K/2πr

Question 9

What is the equation for v?

Answer 9

v = v(θ) θ(hat) = v(θ)*(-sinθ i + cosθ j)

Question 10

What is the equation for l, the vector to the path?

Answer 10

l = rcosθ i + rsinθ j

Question 11

What is the closed integral of v.dl equal to?

Answer 11

integral from 0 to 2π of v.dl/dθ dθ, then sub in the differentiated equation for l and the equation for v, and simplifies to equal K

Question 12

What does the closed integral of v.dl show us?

Answer 12

That circulation around loop enclosing axis is not zero, and circulation around loop not enclosing axis is zero as ∇v = 0.

Question 13

What is the equation for P far from the centre?

Answer 13

P = P0 - 1/2ρ(k^2/4π^2r^2)

Question 14

What is Ф equal to for the potential vortex v = k/2πr θ(hat)?

Answer 14

v = ∇Ф, so Ф = K/2π θ (because ∇Ф = dФ/dr + 1/r *dФ/dθ + dФ/dx in cylindrical)

Question 15

What is Ф equal to for uniform flow?

Answer 15

v = v i(hat), so Ф = v(x)

Question 16

What is Ф equal to for a point source/sink?

Answer 16

v = 2/2πr r(hat), so Ф = 2/2π *ln(r)

Question 17

For potential flow over a cylinder, what is the first stage of the 3 stage process to build the flow?

Answer 17

Have a source-sink pair (flowing out of one point on left and into other point on right)

Question 18

For potential flow over a cylinder, what is the second stage of the 3 stage process to build the flow?

Answer 18

Add a uniform flow from left to right to the diagram. Will be a stagnation point at each end where the flows flow into eachother. Closed streamline joins stagnation points and acts like a solid object.

Question 19

For potential flow over a cylinder, what is the third stage of the 3 stage process to build the flow?

Answer 19

As source/sink brought together, the closed streamline approximates a circle.

Question 20

What does the diagram of the source sink pair look like after the third stage?

Answer 20

Triangle split in 2 with θ1 on left, θ in middle and θ2 on right (outside of triangle). Same with r1, r and r2, and 2a across top. Point P joining r1 and r2.

Question 21

What is the equation for the added potentials of the source and sink at point P?

Answer 21

Ф = q/2π *ln(r1) - q/2π *ln(r2) = q/2π *ln(r1/r2) = q/4π *ln(r1^2/r2^2)

Question 22

What is the equation for r1, the distance from the source to the point P?

Answer 22

r1^2 = r^2*sin^2(θ) + (rcos(θ)+a)^2

Question 23

What can we approximate the equations for r1 and r2 to?

Answer 23

With r&raquo_space; a, can approximate r1^2 = r^2 +2racosθ, r2^2 = r^2 - 2racosθ

Question 24

What is the equation for the potential Ф after simplifying r1 and r2?

Answer 24

Ф = v(x) + qa/π *cosθ/r, as we add the potential for the uniform flow.

Question 25

What is the final equation for the potential Ф?

Answer 25

Ф = v*r*(1+R^2/r^2)*cosθ, since x = rcosθ, with R^2 = qa/πv

Question 26

How do we find v(r) and v(θ)?

Answer 26

v(r) = dФ/dr, v(θ) = 1/r *dФ/dθ ,so v(r) = v*(1-R^2/r^2)*cosθ and v(θ) = -v*(1+R^2/r^2)*sinθ

Question 27

How can we approximate v(r) and v(θ) for a cylinder surface?

Answer 27

r = R, so v(r) = 0 and v(θ) = -2v*sinθ

Question 28

What is the equation for P(s), the pressure on the surface of a cylinder?

Answer 28

P(s) = P0 + 1/2*ρ*v^2*(1-4*sin^2(θ))

Question 29

What is the equation for the net force on the cylinder?

Answer 29

F = -double integral of P(s) ds, so F(x) = double integral from 0 to L and 0 to 2π of P(s)*(cosθ*R)dθ dz = 0