donna calhoun

IPDE Summer School

Tuesday, June 21

Donna Calhoun

Linear advection

Tuesday, June 21, 2011

with an advective flux to get the advectionequation

Linear advection

IPDE SS : Linear Advection

Consider the conservation law

qt + f(q)x = 0

We might also consider an equation in advective form, givenby


Linear advection


In one dimension, these two forms are equivalent if or is a constant.

It is easy to verify that

Consider the constant initial value problem

solves the initial value problem.

qt + uqx = 0


Scalar advection


We can also describe the problem in terms of how the solutionbehaves along curves in the x-t plane.

We might look for curves along which the solution is constant or

Then we would get


Characteristic curves


But this is true only if

or

Solution is constant along characteristic curves


or

Characteristic curves


The solution can be traced back along characteristics. That is, can be found by determining the from which thesolution propagated. Solve


Scalar advection


Consider the scalar advection equation :

The solution travels along characteristic rays in the (x,t) planegiven by . For u < 0 :

t = 0→

t = 1→

qt + uqx = 0


Initial boundary value problem


Infinite domain problem :

Boundary value problem




characteristicboundary

For

Solution




Periodic boundary conditions

Solution


Tangent vectors to the curve are then

Variable coefficient case


Now let’s go back to the variable coefficient case.

if

Again, we have




Solution is constant along characteristics




For equations in conservation form, we have

as a result,

and the solution is not constant along characteristics.

or

Characteristics curves are the same, however.


Physical interpretation - constant solution


Equation in advective form - the color equation

Conservative form

Constant solution remains constant

Constant solution may be compressed or expanded by the velocity field.


Examples


Constant velocity field and periodic boundary conditions


Variable coefficient velocity field

IPDE SS : Linear AdvectionTuesday, June 21, 2011

Two space dimensions


A model of transport of q(x,t) in a control volume C :

q(x, t)


Two space dimensions


d

dt

�

Cq(x, t) dA = −

�

C∇ · f(q(x, t)) dA

= −�

∂Cf(q(x, t)) · n dL

A model of transport of q(x,t) in a control volume C :

q(x, t)


Two space dimensions...


Advective form

Velocity field is given by

Conservative form

Two forms are equivalent if . In this case, the flow is said to be incompressible.


Two dimensions - advective form


In analogy with the 1d case, the two dimensional solution follows characteristic curves in x-y-t space.

if

Velocity field traces out characteristic paths in x-y plane


Two dimensions - conservative form


Conservative case :

We don’t get conservation along characteristic curves unlessthe velocity field is divergence-free (i.e. incompressible).


Flow in periodic box


Constant velocity field and periodic boundary conditions.


Incompressible flow field


A convenient way to define an incompressible flow field in two dimensions is to use a streamfunction.

For example :

Then

and we automatically get

Go to this slide to read more about the streamfunction


Two dimensions


Incompressible velocity field

Variable coefficient incompressible flow field

Contours of the streamfunction


Incompressible Navier-Stokes


ωt + u ·∇ω = µ∇2ω

We can convert this to stream-function vorticity formulation by taking the curl of the first equation :

In 2d, this leads to the following advection equation for the vorticity :

where the vorticity is the scalar . ω = vx − uy


Computing the velocity from a streamfunction


�

C∇ · u dA =

�

∂Cu · n dS = 0

ψO(P ) =� P

O

u · n dS

For any simply connected region C in an incompressible two-dimensional flow field, we have

where is a unit vector normal to the boundary of C. This implies that for two points O and P in the flow field, the value of

n

is independent of the path between O and P.


Existence of a streamfunction


ψO(P �) =� P

�

O

u · n dS = 0

Since we have , the value of does not change along streamlines starting at point O. It is straightforward to show that if the streamline does not contain the point O, the value along the streamline is still constant (although no longer equal to zero).

ψO(P �)ψO(O) = 0

ψO(P ) =� P

O

u · n dS

An interesting feature of this function is that it is constant along streamlines of the flow. Suppose we have chosen a path between O and P such that along this path, we always have . Then for any along this path, we have

ψO(P )

u · n = 0 P �




We can then compute and by choosing paths betweenpoints O and P and O and Q along which it is easy to evaluate

ψx ψy

u · n

∆x

∆y

P

Q

O

n

n

ψx(P ) = lim∆x→0

ψO(P )− ψO(O)∆x

= lim∆x→0

1∆x

� P

O

u · n dS

= lim∆x→0

1∆x

� P

O

(−v)dx

= −v

As the path between O and P, we choose the straight line along the coordinate line y = constant.




We can then compute and by choosing paths betweenpoints O and P and O and Q along which it is easy to evaluate

ψx ψy

u · n

∆x

∆y

P

Q

O

n

n

ψy(Q) = lim∆y→0

ψO(Q)− ψO(O)∆y

= lim∆y→0

1∆y

� Q

O

u · n dS

= lim∆y→0

1∆y

� P

O

u dx

= u

As the path between O and Q, we choose the straight line along the coordinate line x = constant.




ψ(x, y)

Because our choice of origin O only changes the value of the streamfunction by a constant, we will drop the dependence on the point O and write streamfunction as


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