Onedimensional Hydrodynamics

Joachim Köppen Kiel/Strasbourg/Illkirch Sept. 2006

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The Hydro Applet is the simulation of the movement of gas in a linear tube. It shows the evolution of the density, velocity, and temperature of the gas, starting from an initial situation which the user can specify.

The applet starts with the default model which is a rather simple but fundamental example: In the beginning, the left half is filled with gas of high density (value = 1), while the right half is partial vacuum (gas density at a lower value of 0.1), separated by a thin membrane. In both parts, the gas is at rest, i.e. there is no systematic flow in either direction. At time zero, this membrane is suddenly removed, and the gas rushes from the left side to fill the vacuum. This happens with some interesting features, which can be nicely seen in this simulation. The linear tube is a classical example to show the formation of shocks in hydro- and aerodynamical flows. When you click Clear Screen you get this view of the initial density profile:

The left hand panel allows the user to operate the simulation and specify some parameters and options:

Here is the result of the simple shock tube - in the adiabatic case - which we stopped at 0.08 time units

In the velocity plot, one sees the flow speed of the gas to the right (positive velocity) and to the left (negative velocity). The gas in the left hand half starts to move slowly to the right, at the position where the partition had been, the speed reaches 1.0, it stays constant right up to the shock (at position 0.7) which marks the extreme position up to which gas has penetrated the empty region. The density profile shows the rarefraction wave - where the gas moves with increasingly greater speed - the contact discontinuity - at position 0.56 - and the shock as two jumps in density. In truth, these jumps are discontinuities, but in the numerical solution there are somewhat smeared out, because it is difficult for our simple method to treat accurately any sharp jumps in the solution. The temperature profile shows that the gas between shock and contact discontinuity has been heated up.

The shock tube in the isothermal case - the temperature is set to 1.0 - looks similar: There is the shock and the rarefraction wave, but the contact discontinuity is absent.

Some experiments:

About the units: We use normalized units:

About the method:

The model is described by a system of two (or three) partial differential equations, the conservations of mass and momentum (and energy). This system is formulated in a conservative way, by considering how the mass and momentum (and energy) of each parcel of gas are changed by the interaction with the neighbouring parcels. A simple explicit method is used to compute the time evolution.

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