Euler Equation & Boundary Layers Flashcards
What is ∇.v for an incompressible fluid?
∇.v = 0
What is the equation for the Reynolds number and what do the variables mean?
Re = ρv(av)D/μ, where μ is the dynamic viscosity, D is the scale of interest (like flowing over a step or around a ball of diameter D, and v(av) is the typical flow speed.
In the navier states equation, which are the intertial and viscous terms?
ρ(v.∇)v = inertial term, μ∇^2 *v = viscous term
What can we approximate the inertial and viscous terms to?
|ρ(v.∇)v| ~ ρv(av)^2/D, |μ∇^2 *v| ~ μv(av)/D^2
What can we get from the approximate inertial and viscous terms?
Find that the ratio of them is the Reynolds number, so the Reynolds number measure the relative importance of the inertial to viscous terms.
What is Euler’s Equation?
ρdv/dt + ρ(v.∇)v = -∇P - ρg k
What is the Euler solution to flow over a plate?
Constant velocity past plate, slip at boundary and v=/0
What is the velocity like for the full viscous solution for the flow over a plate?
Euler (constant) up until a point a distance 𝛿 away from the plate, where is is viscous flow, and decreases closer to the plate.
How can we estimate the boundary layer for the plate problem?
Set the approximate inertial and viscous terms equal to each other and rearrange for the ratio of 𝛿/d.
What do we find is the ratio of 𝛿/d?
𝛿/d = 1/Re^1/2, so for large Re, boundary layers are thin.
What are 4 ways to visualise flow?
Streamlines, streaklines, pathlines and streamtubes
What is a streamline?
A line which is everywhere tangential to the instantaneous velocity vector v(r,t)
For 2D flow, what is the equation for the streamline curve?
y = y(x), whose slope dy/dx must be equal to vy/vx
For 3D flow, what is the equation for the streamline curve?
r = (x(λ), y(λ), z(λ)) and dr/dλ is parallel to v
What is a stagnation point?
Stagnation streamline where v=0
