Tetra
Tetra is an unsteady three dimensional
Navier-Stokes
solver, written by Dr. Frank Muldoon at Louisiana State University. Its purpose is the efficient (see article, dissertation) solution of the unsteady Navier-Stokes equations for Direct
Numerical Simulation (DNS). It is also used as a test bed for numerical
algorithm research and development by the author. Tetra is written in Fortran
95 and
is parallelized using the Message Passing Interface (MPI). Tetra
solves the incompressible Navier-Stokes equations in non conservative
form
along with an arbitrary number of scalar equations also in non
conservative
form. It allows the selection of numerous finite difference
schemes
which can be of high order. The highest
order terms are 6th order accurate in space on an even grid.
In addition, Total Variation
Diminishing
(TVD) schemes can be applied to the convective terms. Numerous different
implicit and explicit time integration schemes can be used. The
immersed
boundary method is used to handle complicated and/or moving
geometries. For analyzing turbulence models using DNS (see article),
it can compute the numerous terms in the equations for turbulent
kinetic energy and dissipation by advancing only once in time, without
the need to first determine a mean flow field state. Tetra
incorporates the ability to track and indentify massless passive
particles which are convected with the flow. These particles are
extremely useful for visualizing flow patterns and understanding the
creation,
evolution and destruction of dynamic flow structures in an unsteady
flow. Recent additions have included the ability to solve the adjoint
equations so as to determine the gradient of an objective function with
respect to control variables and various optimization algorithms which use this gradient; this is intended for the solution of
unsteady optimization/control problems. Tetra has been
run on a large number of platforms at High
Performance
Computing Centers including the Aeronautical Systems Center (ASC), Naval
Oceanographic Office (NAVO), Arctic Region Supercomputing Center
(ARSC), Maui High Performance Computing Center (MHPCC), Louisiana Optical Network Initiative (LONI) and the Center
for Computation and Technology (CCT) at Louisiana State University.
Animations of optimization and flow control using Tetra
Recent usage of Tetra has concentrated on the control (or optimization) of unsteady flows. The
following are some movies of two-dimensional flow control problems.
Mathematically the problems shown in the movies are posed as follows; functions of space
and time which describe wall-normal velocity boundary conditions for
the Navier-Stokes equations are sought which minimize a specified objective function. More details here.
Center
for Computation and Technology
Grid dimensions are 350x88 for the simulation of flow around the words "Center
for Computation and Technology". The temporal dimension of the optimal control is 1600 time steps per event horizon (there are 17 event horizons), while the spatial dimension of the control is every discrete wall-normal velocity at both walls; the total number of control variables is 2x350x1600x17 = 19x10^6. Taking
into consideration the energy input of the control, turning on the
optimal control results in a 7.3% increase in the extractable power of
the system. The optimal control decreases the drag of the system and increases the flow rate (by 4.9%).
More details here.
Here the contours are of the component of velocity in the flow direction. The vectors which appear at the top and bottom walls are the control variables, magnified by a factor of 15. Click the picture to see the movie
Navier-Stokes and temperature equations
Grid dimensions are 350x127 for the simulation of flow around the Navier-Stokes and temperature equations. The temporal dimension of the optimal control is 1600 time steps per event horizon (there are 19 event horizons), while the spatial dimension of the control is every discrete wall-normal velocity at both walls; the total number of control variables is (2x350x1600x19) = 21.3x10^6. More details here.
Here the contours are of the component of velocity in the flow direction. The vectors which appear at the top and bottom walls are the control variables, magnified by a factor of 30. Click the picture to see the movie
Note: It
is commonly believed that the diffusive (2nd derivative) terms
damp unsteadiness and vortex shedding, but here it can be clearly seen
that significant vortex shedding occurs off of the "2" in the diffusive
terms!?!
Here the contours are of the temperature. The vectors which appear at the top and bottom walls are the control variables, magnified by a factor of 30. Click the picture to see the movie
Here the contours are of the magnitude of the gradient of temperature. The vectors which appear at the top and bottom walls are the control variables, magnified by a factor of 30. Click the picture to see the movie
Analysis of the heat transfer rate shows that turning on the optimal control results in a 28.8% decrease in the rate of heat
transfer across the channel. Taking into consideration the energy
input of the control, this significant decrease in the rate of heat transfer is accomplished with only a 1.7% decrease in the extractable power of
the system.
Conclusion
Optimization/control of the unsteady Navier-Stokes and temperature equations has been demonstrated on two complicated geometries. This has been accomplised using a reasonable amount of computing resources, indicating that control
of the unsteady Navier-Stokes and temperature equations in a numerical
context is a practical tool for investigating physical systems which
can be accurately modeled with using these equations.
Acknowledgments
All simulations were run using a single processor on the AIX/IBM computer "Zeke" of the Louisiana Optical Network Initiative (LONI). The movies were created using the post-processing software Tecplot 360 (link). The help of Michael Saunders at Tecplot concerning improvements in the capabilities of Tecplot 360 regarding unsteady data sets and in providing suggestions on the use of Tecplot 360, is greatly appreciated.
