Hi i'm Benjamin Herrmann and I will be showing how you can perform resolvent analysis using
time resolved flow field measurements exclusively and without recourse to the governing equations.
This is work presented in a recent paper titled data-driven resolved analysis
which I wrote in collaboration with Peter Baddoo from Imperial college; Richard Semaan from TU
Branschweig; Steve Brunton from University of Washington; and Beverley McKeon from Caltech.
The paper is available on archive and you can find a link to it in the description below.
So say you have a linear stable system that's being periodically forced,
then the long term response in the frequency domain is governed by this expression, where
this is the resolvent operator. This operator encodes how any disturbances to the system are
filtered by the linear dynamics to produce a response in the context of non-normal systems.
We know that the resultant behaves as a very selective amplifier of disturbances and because
of this it is critical to understand the dynamics of such systems. This was first pointed out in the
work of Trefethen encountered in the context of shear flows. Since then there has been a growing
interest in using this method mainly due to its relevance to the design of flow control strategies
by using the input output framework developed by Jovanovich and Bamieh and also to do its relevance
to identifying structures and scale interactions in turbulent flows by using the interpretation of
non-linearities as forcing proposed by McKeon and Sharma. And even though we know it's important and
there is this great interest from the community in applying resolvent analysis to flows with two
or three inhomogeneous directions, this is still not commonplace and one of the main reasons for
this is that resolvent analysis requires under hood access to high fidelity numerical solvers
of the linearized governing equations. In this work we develop a purely data-driven method
that can work with black box numerical solvers and doesn't require access to adjoint solutions,
so it opens up the possibility of eventually doing resolvent fully from experimental data.
So now we'll be showing you how does the method work and for that i will be using data from
direct numerical simulations of plane channel flow and all the details of the numerical setup
are in our paper the first step is to collect a sequence of state snapshots
from recording of a transient simulation starting from a perturbation uh to the stable system
so in this case the state corresponds to the three velocity components in this
three-dimensional flow field which here i'm visualizing as velocity iso surfaces
then we can enrich our data set by considering
additional recordings of transients starting from different initial conditions
the second step is to assemble our data matrix matrices which is just stretching the state into
a very long vector and horizontally concatenating these states according to your temporal sequence
followed by the temporal sequence corresponding to the next initial condition
and the next initial condition and so on and actually the ordering in fact it's not really
important what's important is that all the columns in the y matrix correspond to the columns in the x
matrix but everything shifted one time step into the future and the f matrix that you see there
is just a waiting to adjust the inner product so that the l2 norm measures
a physically meaningful quantity which in this case is the perturbation kinetic energy
now that we have these matrices we use dynamic mode decomposition to approximate the leading
eigenvectors and eigenvalues of the underlying linear operator that involves our measurements
and we use this learned eigen decomposition
to build our resultant operator but projected onto eigen coordinates
and we perform resultant analysis so the singular value decomposition of the resultant on these
coordinates this projection of the resolvent was first used by radiant henningsen a while back
in 1993 where they used it to reduce the computational cost of computing resolvent gains
but in their case they used the actual eigen decomposition of the system from a known operator
whereas here we're using learned eigenvectors and eigenvalues that we learned from data using dmd
so the last step is to synthesize our resultant modes back into
physical coordinates so if we had enough data we we will get
a good approximation of the actual resultant modes of the system and here i'm showing the
optimal forcing where i'm showing a world normal velocity iso surfaces
and the optimal response for channel flow where i'm showing stream wise velocity iso surfaces
so these are the familiar stream waste voltages causing stream west tricks
and here are some more results so these are the operator-based resultant modes just for
ground truth comparison and these are the ones obtained from data using a data set that was
generated starting from these initial conditions where this middle plot shows the learned dmd
spectrum overlaid on top of the true spectrum and these are the data-driven resulting modes
for this data set we needed to retain 200 dmd modes to get this approximation now what happens
if we use the optimal forcing as an initial condition to generate our data set well in that
case we only need 20 dmd modes to generate an even better approximation of the true resolvent modes
and what's more interesting is that we are actually learning a very specific region of
the spectrum which is something that we noticed so the last result that i want to show you
is what happens when you use a localized disturbance as the initial condition
to generate the data set here we're using this doughnut-shaped perturbation which is
something that resembles what you would get in an experiment with a well normal jet so in this case
we get these modes which are quite different from the true resultant modes
so this the question that arises is are these results useful how do they
relate to the operator the actual operator of the system and how can we interpret these modes
it turns out that if we do an operator-based resolvent analysis but where we restrict the
input and output spaces to be in the span of the data snapshots then we get the same modes
so this means this is very encouraging because it means that these results are still very valuable
we just need to be aware that what we are getting are these forcing and response modes
that are constrained to live in the span of our data
which completely makes sense okay so these are some points that i want to leave you with
first this data-driven resolvent analysis inherits its capabilities and limitations from dmd
so computational cost noise robustness data efficiency all depend on what dmd variant
you're using also we want to point out that we found that it's very important that we develop
more effective ways to explore the state space of dynamical systems
i also want to point out that this is the first step towards a fully experimental
resolvation analysis and we believe that this work contributes to lowering
the barrier of entry to resultant research and applications and finally a final remark is that
we recognize that there is still a critical step missing before we can make the leap to
data-driven resultant of turbulent flows which is we need to disseminate the linear
from the non-linear dynamics and that's actually something that we're working on at the moment
okay so thanks for watching again there are a lot more details
in the paper and you can find a link to it in the description below thank you