donna calhoun
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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
Tuesday, June 21, 2011
Linear advection
IPDE SS : 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
Tuesday, June 21, 2011
Scalar advection
IPDE SS : Linear 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
Tuesday, June 21, 2011
Characteristic curves
IPDE SS : Linear Advection
But this is true only if
or
Solution is constant along characteristic curves
Tuesday, June 21, 2011
or
Characteristic curves
IPDE SS : Linear Advection
The solution can be traced back along characteristics. That is, can be found by determining the from which thesolution propagated. Solve
Tuesday, June 21, 2011
Scalar advection
IPDE SS : Linear 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
Tuesday, June 21, 2011
Scalar advection
IPDE SS : Linear 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
Tuesday, June 21, 2011
Scalar advection
IPDE SS : Linear 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
Tuesday, June 21, 2011
Initial boundary value problem
IPDE SS : Linear Advection
Infinite domain problem :
Boundary value problem
Tuesday, June 21, 2011
Initial boundary value problem
IPDE SS : Linear Advection
characteristicboundary
For
Solution
Tuesday, June 21, 2011
Initial boundary value problem
IPDE SS : Linear Advection
Periodic boundary conditions
Solution
Tuesday, June 21, 2011
Tangent vectors to the curve are then
Variable coefficient case
IPDE SS : Linear Advection
Now let’s go back to the variable coefficient case.
if
Again, we have
Tuesday, June 21, 2011
Variable coefficient case
IPDE SS : Linear Advection
Solution is constant along characteristics
Tuesday, June 21, 2011
Variable coefficient case
IPDE SS : Linear Advection
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.
Tuesday, June 21, 2011
Physical interpretation - constant solution
IPDE SS : Linear Advection
Equation in advective form - the color equation
Conservative form
Constant solution remains constant
Constant solution may be compressed or expanded by the velocity field.
Tuesday, June 21, 2011
Examples
IPDE SS : Linear Advection
Constant velocity field and periodic boundary conditions
Tuesday, June 21, 2011
Variable coefficient velocity field
IPDE SS : Linear AdvectionTuesday, June 21, 2011
Two space dimensions
IPDE SS : Linear Advection
A model of transport of q(x,t) in a control volume C :
q(x, t)
Tuesday, June 21, 2011
Two space dimensions
IPDE SS : Linear Advection
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)
Tuesday, June 21, 2011
Two space dimensions...
IPDE SS : Linear Advection
Advective form
Velocity field is given by
Conservative form
Two forms are equivalent if . In this case, the flow is said to be incompressible.
Tuesday, June 21, 2011
Two dimensions - advective form
IPDE SS : Linear Advection
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
Tuesday, June 21, 2011
Two dimensions - conservative form
IPDE SS : Linear Advection
Conservative case :
We don’t get conservation along characteristic curves unlessthe velocity field is divergence-free (i.e. incompressible).
Tuesday, June 21, 2011
Flow in periodic box
IPDE SS : Linear Advection
Constant velocity field and periodic boundary conditions.
Tuesday, June 21, 2011
Flow in periodic box
IPDE SS : Linear Advection
Constant velocity field and periodic boundary conditions.
Tuesday, June 21, 2011
Incompressible flow field
IPDE SS : Linear Advection
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
Tuesday, June 21, 2011
Two dimensions
IPDE SS : Linear Advection
Incompressible velocity field
Variable coefficient incompressible flow field
Contours of the streamfunction
Tuesday, June 21, 2011
Two dimensions
IPDE SS : Linear Advection
Incompressible velocity field
Variable coefficient incompressible flow field
Contours of the streamfunction
Tuesday, June 21, 2011
Incompressible Navier-Stokes
IPDE SS : Linear Advection
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
Tuesday, June 21, 2011
Computing the velocity from a streamfunction
IPDE SS : Linear Advection
�
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.
Tuesday, June 21, 2011
Existence of a streamfunction
IPDE SS : Linear Advection
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 �
Tuesday, June 21, 2011
Existence of a streamfunction
IPDE SS : Linear Advection
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.
Tuesday, June 21, 2011
Existence of a streamfunction
IPDE SS : Linear Advection
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.
Tuesday, June 21, 2011
Existence of a streamfunction
IPDE SS : Linear Advection
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
Tuesday, June 21, 2011
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