|The simulator produces a series of images in sequential timesteps
which are then assembled into an animated movie file to show the
evolution of the wave over the length of time under evaluation.|
Each image file shows eight plots. Each plot represents a different property of the wave or its medium and shows how its value changes over a one-dimensional distance for that timestep. The first frame (like the one shown) will show the wave under its initial equilibrium conditions. The simulator will then introduce a perturbation and the following frames show how that perturbation affects the wave.
The eight properties are, in order from left to right, top to bottom, density, potential, pressure, x velocity, y velocity, z velocity, y magnetic field and z magnetic field. Which properties exhibit variations depends on the type of wave being propagated; for instance, this image illustrates the first frame of a sound wave mpeg in a medium with no magnetic fields or background potentials (such as gravity). In this case only the density, pressure and x velocity plots show any variance.
OK, this is supposed to be a one-dimensional simulation, so why are there plots for velocities in the y and z directions? Good question! For this simulator, "one-dimensional" refers to the wave vector k of the wave being propagated. However, the presence of magnetic fields will cause transverse motion in a charged medium, so plots are included to show the velocities of the transverse waves in the y and z directions caused by these fields. Obviously, the y and z velocity plots will only exhibit motion when a non-zero magnetic field value has been included in the simulation.
To create a linear wave, the equations of motion for a fluid wave have been "linearized;" ie, in solving the equations variables higher than first order have been dropped. Several different linear waves, both hydrodynamic and magnetohydrodynamic (ie, influenced or produced by magnetic fields), are available in the simulator:
Alfven Wave: Alfven waves result from oscillations caused by the transverse motion of magnetic induction lines. Magnetic tension provides a restoring force to the induction lines to bring them back to a straight line, producing the transverse Alfven waves. The simulator introduces a perturbation into the magnetic field and plots the resulting disturbance.
Entropy Wave: Entropy waves, which can be either hydrodynamic or magnetohydrodynamic, are compressional waves (ie, fluctuations in pressure and density) in which the internal energy and velocity remains constant (unless a gravitational potential is introduced).
Sounds waves, like entropy waves, are hydrodynamic compressional waves
that propagate a fluctuation in pressure and density. Unlike entropy
waves, the internal energy of sound waves varies while entropy remains constant.
A shock tube is less a wave than it is a moving wall of force. The shock itself is a discontinuity, having an instantaneous jump in fluid pressure, density and velocity on either side of the shock. The simulator therefore requests values for these parameters both on the left and right sides of the shock as part of the initial conditions. The default conditions used by the simulator match the Sod hydrodynamic shock tube problem described by Brio & Wu (see Journal of Computational Physics, 1988, volume 75, page 400).
Two basic types of parameters are used in the One-Dimensional
Hydrodynamics simulator: physical and computational. The parameters
that the simulator will request from the user varies based on the type of
wave selected and the level of knowledge the user chooses. In some
cases a parameter's value may be hidden or unnecessary for the
particular simulation. Default values for all parameters are provided.
Computational parameters are used by the simulator to perform calculations and also to set up conditions and ranges on the simulation.
Boundary Conditions: The boundary conditions tell the simulator what to do with the wave at the edges of the plot. Three options are available: periodic, reflecting and standard. Periodic boundary conditions cause the wave to appear on the opposite side of the graph after it moves past a boundary. With reflecting conditions the wave will "bounce back" from the edges (careful - this condition may cause the wave to "interfere" with itself). Under standard conditions the wave will simply move into a "ghost zone" after passing a boundary.
Courant Number: The Courant number helps determine the size of dt, or the magnitude of the timestep between frames. A larger value of the Courant number causes a larger value in dt, and therefore less frames in the final simulation.
Dumping Frequency: The dumping frequency parameter tells the simulator how often to create an image file. Smaller values result in larger numbers of images since the dumping frequency is calculated as a multiple of the wave frequency. A value of 0.1 will result in 10 images for every wave oscillation, and will therefore require additional computation time and produce a larger mpeg file.
Integration Time: The integration time parameter determines the length of the time scale run by the simulator, and, therefore, the number of frames in the resulting movie file. A larger value for the integration time will require a longer wait as the simulator performs computations and will also result in a larger mpeg. Users with slow internet connections may want to opt for smaller integration times.
The numerical resolution determines the "smoothness" of the
wave by setting the number of points plotted. The distance scale is
divided by the numerical resolution to calculate dx, or the amount of change in
"x", for each point plotted. The image above was plotted
with a numerical resolution of 128, which is a good default value.
Larger values for this parameter greatly increase the length of time
required to perform calculations.
Physical parameters determine the properties of the wave and medium being evaluated.
Adiabatic Index: In an adiabatic process the entropy of each fluid particle remains constant and therefore no heat exchange occurs between different fluid elements. The adiabatic index, denoted γ, describes the specific heat ratios of fluid elements in the Poisson adiabatic formula: pVγ=constant. The typical value for this parameter is 5/3.
Amplitude: The amplitude parameter determines the maximum and minimum height of the wave. The simulator associates the amplitude value with one of the physical properties of the wave used in its eigenvector and may be different depending on the type of wave selected. For instance, the amplitude of a sound wave determines the maximum and minimum values of the wave velocity.
Artificial Viscosity: Only an ideal fluid or gas will move with no energy lost to the irreversible transfer of momentum from internal friction between fluid elements with differing velocities. The artificial viscosity determines the extent to which this energy is lost during fluid flow.
Density: The density defines the number of fluid particles in a set volume (in this case a volume containing a unit mass of fluid) or the "thickness" of the fluid element.
Gravitational Potential: On a small scale, gravity has a negligible effect in the fluid equations of motion. However, on large scales such as those found in astrophysical applications gravitational potential can have a dramatic effect. The gravitational potential is given by the Laplacian equation: ∇2Φ = 4πGρ.
Pressure: Pressure is the amount of force exerted on the fluid by the wave.
Magnetic Fields: Magnetic fields can have a dramatic effect on a wave composed of charged particles, and vice versa. A non-zero value for the magnetic field will affect the wave's velocity according to the "right-hand rule," causing transverse motion.
Wave Vector: The wave vector defines the quantum energy state of the wave, and must be an integer value of one or higher. Any decimal inputs for the wave vector parameter will automatically be adjusted to the next lowest integer energy state by the simulator.
The velocity tells the direction and speed of the wave. Since this
simulator is one-dimensional, all wave velocities will be in the
"x" direction unless magnetic field values are included that
cause transverse motion in the wave. Any components of the wave
velocity in the "y" or "z" directions will be
indicated by the y and z velocity plots.
Students working on the simulators available in the Digital Demo Room
participate in the "Research Experience for Undergraduates",
or REU, program in the Physics Department of the University of
Illinois. The students give presentations on their projects at the
midpoint and end of the summer research program and also write a
paper. The presentations and paper for the One-Dimensional
Hydrodynamic simulator, which was built by Conley "Lee" Ditsworth during the 2002 REU program,
are available here for anyone interested in seeing the development of this pedagogical project:
Return to the simulator!