fluids - Stony Brook Universitybender.astro.sunysb.edu/classes/fluids/lectures/ideal_fluids.pdfPHY...

Ideal Fluids

PHY 688: Astrophysical Fluids and Plasmas

Navier-Stokes Equatons

● We have


Euler Equatons

● We have

– Note: the last equaton is internal energy evoluton. Many tmes we want total energy:


Eulerian vs. Lagrangian Descripton

● We can imagine diferent reference frames when thinking about fluids

– Eulerian: watch a partcular region and look at how the fluid changes

● Advecton of new fluid and local sources will both contributon to the change at that locaton

– Lagrangian: move along with the flow, tracking a partcular fluid element

● Propertes in that fluid element change due to local sources, or compression of the fluid element


Eulerian vs. Lagrangian Descripton

● Take a property g(x(t), t)

● Eulerian descripton:

– We are looking at a fied i, so we take

● Lagrangian descripton:

– We move with the fluid:

● We’ll use D/Dt to denote the Lagrangian derivatve:


Macroscopic Derivaton

● Mass conservaton

M

Ω is an arbitrary volume over which mass M is considered conserved.

The only changes to M will be due to the fluies across the boundary



● In Lagrangian form:

– This tells us that density of a fluid element changes locally only due to compressive or eipansive flow

● Note that if the density of a flid element is constant, then



● We can get the momentum and energy equatons from similar macroscopic ideas:

– Momentum: Newton’s second law

– Internal energy: First law of thermodynamics

● Blackboard derivaton...


Other “Energy” Equatons

● We’ve seen equatons for specifc internal energy and total energy

● Sometmes we want evoluton for other thermodynamic variables that can take the place of energy, e.g.,

– Enthalpy

– Pressure

– Temperature

– Entropy● This follows from the frst law of thermodynamics:

you’ll derive these on your homework

In the absence of heat sources


Ideal Fluid

● The Navier-Stokes equatons (with simplifed transport efects) read:

– An ideal fluid ignores these transport efects


Reynolds Number

● Importance of viscosity:

– Where we assumed μ is constant, and defned ν = μ/ρ

● Now, dimensionless quanttes:

– then

● where we have defned the Reynolds number,


Reynolds Number

● Dimensionless numbers provide convenient ways to compare vastly diferent problems.

– Idea: systems with the same dimensionless numbers (and confguratons) should behave the same.

● Reynolds number measures the relatve importance of the advecton to viscous dissipaton.

– Large Re: viscosity is unimportant (advecton term dominates flow). These flows are turbulent.

– Smal Re: viscosity is important. The flow remains laminar.

● Eiperiments show that the transiton to turbulence occurs at Re ~ 2000, although the shape of the object influences this.


Reynolds Number

● Reynolds number scaling plays a big role in aircraf design.

Here a scale model of the aircraf is placed in a windtunnel, and the flow is adjusted (in speed and temperature) to match the Reynolds number eipected during flight.

(Wikip

ed

ia)


Reynolds Number

● Many astrophysical environments have large Re, and we will ignore viscosity

● Some Reynolds numbers of everyday objects (thanks Wikipedia!)

– Ciliate ~ 1 × 10−1

– Smallest fsh ~ 1

– Blood flow in brain ~ 1 × 102

– Blood flow in aorta ~ 1 × 103

– Onset of turbulent flow ~ 2.3 × 103 to 5.0 × 104 for pipe flow to 106 for boundary layers

– Typical pitch in Major League Baseball ~ 2 × 105

– Person swimming ~ 4 × 106

– Fastest fsh ~ 1 × 108

– Blue whale ~ 4 × 108

– A large ship (RMS Queen Elizabeth 2) ~ 5 × 109


Thermal Difusion

● Another process we neglect in the Euler equatons is thermal difusion

● In astrophysics, advecton of heat lslally dominates over heat conducton

– Here, Pe is called the Peclet number

– We introduced the thermal difusivity, χ


Thermal Difusion

One situaton where thermal difusion is important is flame propagaton:

fuel ash


Incompressible and Baratropic

● In your homework, you’ll show that:

– Allowing us to rewrite the momentum equaton as:

● Taking the curl of this (blackboard), we get:

– Where vortcity is defned as:


Incompressible and Baratropic

● If the pressure and density gradients are aligned, then the last term vanishes, and we have

● Together with the relaton between vortcity and velocity, we this completely specifes the flow

● This approiimaton arises when:

– Density is constant (note Choudhuri says this is incompressible, but incompressible is more general and can let variable density flows)

– Baratropic equaton of state, p = p(ρ)

● Notce that the energy equaton is not needed


Conservatve Form

● When we do numerical methods, we’ll want to consider these equatons in conservatve form:

– With

● Introducing a momentum tensor:


Applicaton: Solar Corona(Choudhuri Ch. 4)

● Solar corona: hot tenuous gas eitending beyond the solar surface.

– Parker (1958) constructed a simple spherical model.

● Corona is much hoter than the surface of the Sun.

– Assume heated at lower layers

– Boundary conditon: T = T0 at r = r0

● Assume that the corona is statc and of negligible mass:



● Assume an ideal gas:

– Into HSE:

● Conductvity from kinetc theory:



● Soluton:

– We see that T → 0 as r → ∞

● Using this with HSE:

– So, with p = p0 at r = r0,

● Pressure is non-zero as r → ∞

● This suggests that it is not in equilibrium… solar wind


Bernoulli’s Principle

● Previous eiample used hydrostatcs (v = 0)

● Now let’s consider a steady flow (no tme dependence)

● Useful construct: streamline—curve tangent to v at every point

– For steady flows, streamlines don’t change in tme, so streamlines are the path a fluid element takes

● Back to our momentum equaton, assuming conservatve force:

● Assuming a baratropic EOS, we can introduce:

This requires a bit more discussion… blackboard...


Bernoulli’s Principle

● Defning:

● We have:

● Finally, dotng with v,

● So B must be constant on steamlines


Bernoulli's Theorem(Shu Ch. 6, Shore Ch. 1)

● Consider an incompressible flow, φ = 0

– Now consider flow through a pipe with a constricton

● Mass conservaton:

– At smaller diameter, v increases, so p must decrease (by Bernoulli’s theorem)



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● Consider the flow of water from a faucet:

– Fluid is incompressible, and Bernoulli's theorem tells us:

– Pressure of the fluid is in equilibrium with the atmosphere, and then we see that the fluid speeds up as it drops.



● Contnuity equaton tells us that

● ρ is constant, v is increasing, so cross-sectonal area must be decreasing.

● Eventually, the stream of water will narrow enough that surface tension becomes important.

● Also, the shear on the air-water interface is unstable, causing oscillatons in the surface.



● Consider water coming out of a spout

– Incompressible limit:

– Pressure at the top and eiit are atmospheric pressure

● Other eiamples: shower curtain blowing in, blowing between two sheets of paper

water level


Kelvin Circulaton Theorem(Shu Ch. 6)

● Kelvin's circulaton theorem:

– Number of vortei lines passing through a surface element remains unchanged for inviscid barotropic flow.

● Defne the circulaton as

● Note that Γ = 0 if circulatory component of the fluid velocity averages to zero in the circuit C

⊙



● Rewrite using Stokes's theorem

– Circulaton enclosed by C is simply the number of vortei lines that thread A

● Evoluton equaton:



● Therefore, we have

– With triple product identty

Each line element contributes |v × dl| to the rate of change of area. The cross product also gives the proper sign to the area contributon.



● Finally, we see

● and therefore



● Kelvin circulaton theorem implies:

– If a fluid has no circulaton (or vortcity) at tme zero, then no vortcity will be generated at subsequent tmes.

● Note, this assumes that the viscosity is negligible.

– If there are some vortces in a region of fluid, they will stck with that fluid element as it moves (think whirlpools in a river)

– No mater what disturbances the fluid is subjected to (as long as it is not dissipatve), you will not add net angular momentum.

● Eiample: moving an oar through water.

– Counterrotatng vortces are generated on either side, but there is no net spin up of the fluid.


Potental Flows

● A flow that is initally irrotatonal ( ▽ × v = 0) remains so

– Consequence of Kelvin circulaton theorem

– This means we can write:

● If we are incompressible, then further:

– This is a scalar equaton (Laplace equaton)

● Consider flow past a cylinder. Boundary conditons:

– Normal velocity goes to zero at surface:

– Velocity tends to uniform velocity at infnity


Potental Flows

● Asymptotc velocity

– Take:

– Then:

● General soluton (we’ll skip the details):


Potental Flows

● Streamline plot


Reynolds Number Efects

Taneda 1956, from htps://authors.library.caltech.edu/25017/4/chap5.htm

https://authors.library.caltech.edu/25017/4/chap5.htm


Limitatons

● The real world has viscosity

● Choudhuri shows that if we move the cylinder though the fluid at a constant velocity, there is no force actng on the cylinder

– This is against our intuiton

– Known as d’Alembert’s paradoi● If there were a drag force, then the kinetc energy removed from

the cylinder needs to go somewhere—heat to the fluid● But without viscosity, there is no mechanism to convert KE into

heat● Ideal fluids, therefore cannot eiert drag


Viscosity

● Shu has a nice discussion of the efects of viscosity on this problem:

– Notce that the Laplace equaton does not allow us to additonally supply a boundary conditon on the tangental velocity (no slip conditon)

– Tangental velocity should go to zero● Real world: boundary layer (thin) forms● Viscous efects dominate in the boundary layer (Kelvin’s theorem

no longer applies, vortces form and get swept downstream)● Potental flow descripton applies well outside of the boundary

layer

fluids - Stony Brook Universitybender.astro.sunysb.edu/classes/fluids/lectures/ideal_fluids.pdfPHY...

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