Generalised Bernoulli Equation Flashcards
What does dv/dt equal for irrotational flow?
dv/dt = -∇(P/ρ + gz + v^2/2)
What is the generalised Bernoulli equation?
dФ/dt + v^2/2 + P/ρ + gz = const
In cylindrical polars, what does r(hat) equal?
r(hat) = cosθ i + sinθ j
In cylindrical polars, what does θ(hat) equal?
θ(hat) = -sinθ i + cosθ j
In cylindrical polars, what does z(hat) equal?
z(hat) = k
What is the potential vortex?
A flow with circular paths around an axis such that ∇v = 0
What does ∇v = 0 equal in cylindrical coordinates?
1/r d/dr(rv(θ)) - 1/r *d/dθ(v(r)) = 0
For the potential vortex, what does the cylindrical coordinates version of ∇v = 0 equal and why?
Potential vortex only has v(θ)(r), so ∇v = 0 -> d/dr(r*v(θ)) = 0, so v(θ) = const/r = K/2πr
What is the equation for v?
v = v(θ) θ(hat) = v(θ)*(-sinθ i + cosθ j)
What is the equation for l, the vector to the path?
l = rcosθ i + rsinθ j
What is the closed integral of v.dl equal to?
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
What does the closed integral of v.dl show us?
That circulation around loop enclosing axis is not zero, and circulation around loop not enclosing axis is zero as ∇v = 0.
What is the equation for P far from the centre?
P = P0 - 1/2ρ(k^2/4π^2r^2)
What is Ф equal to for the potential vortex v = k/2πr θ(hat)?
v = ∇Ф, so Ф = K/2π θ (because ∇Ф = dФ/dr + 1/r *dФ/dθ + dФ/dx in cylindrical)
What is Ф equal to for uniform flow?
v = v i(hat), so Ф = v(x)
What is Ф equal to for a point source/sink?
v = 2/2πr r(hat), so Ф = 2/2π *ln(r)
For potential flow over a cylinder, what is the first stage of the 3 stage process to build the flow?
Have a source-sink pair (flowing out of one point on left and into other point on right)
For potential flow over a cylinder, what is the second stage of the 3 stage process to build the flow?
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.
For potential flow over a cylinder, what is the third stage of the 3 stage process to build the flow?
As source/sink brought together, the closed streamline approximates a circle.
What does the diagram of the source sink pair look like after the third stage?
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.
What is the equation for the added potentials of the source and sink at point P?
Ф = q/2π *ln(r1) - q/2π *ln(r2) = q/2π *ln(r1/r2) = q/4π *ln(r1^2/r2^2)
What is the equation for r1, the distance from the source to the point P?
r1^2 = r^2*sin^2(θ) + (rcos(θ)+a)^2
What can we approximate the equations for r1 and r2 to?
With r»_space; a, can approximate r1^2 = r^2 +2racosθ, r2^2 = r^2 - 2racosθ
What is the equation for the potential Ф after simplifying r1 and r2?
Ф = v(x) + qa/π *cosθ/r, as we add the potential for the uniform flow.
What is the final equation for the potential Ф?
Ф = vr(1+R^2/r^2)*cosθ, since x = rcosθ, with R^2 = qa/πv
How do we find v(r) and v(θ)?
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
How can we approximate v(r) and v(θ) for a cylinder surface?
r = R, so v(r) = 0 and v(θ) = -2v*sinθ
What is the equation for P(s), the pressure on the surface of a cylinder?
P(s) = P0 + 1/2ρv^2(1-4sin^2(θ))
What is the equation for the net force on the cylinder?
F = -double integral of P(s) ds, so F(x) = double integral from 0 to L and 0 to 2π of P(s)(cosθR)dθ dz = 0
