Relativistic irrotational fluids: 3D simulations in flat space

The following paragraphs will describe the project I am currently working on, whose final aim is to study the behaviour of relativistic irrotational fluids in a curved spacetime, such as Schwarzschild or Kerr. In order to do this, I am developing a code within the Cactus environment, to study numerically the problem. I present the recent results obtained in flat space, which show that the code is valid and ready to be extended to general metrics.

General equations

In order to define an irrotational fluid, we have to introduce the vorticity tensor:

where is the projection tensor, h is the enthalpy of the fluid and is the four velocity. The vorticity tensor is identically null for an irrotational fluid. From the Euler's equations for a perfect fluid:

we can derive a simple expression for the vorticity tensor which we write below :

Equation tells us that, if , then we can write the quantity as the gradient of a scalar field, that is . We then obtain the following expression for the conservation of particle density, in terms of :

This equation must be matched with the one coming from the normalization condition for the four velocity :

Equations and have been studied analytically in literature to study fluids in curved metrics such as Schwarzschild and Kerr. A first attempt to solve these equations can be found in [2] where equation is linearized by setting n=h. This further approximation corresponds to the speed of sound being equal to the speed of light. A perturbative approach can instead be found in [1]. Our aim is to solve numerically equation , starting from the simple case of a flat space, and then moving to curved space-times, thus dropping the further approximation .

Thermodynamic considerations

We define the pressure to be given by the following polytropic expression:

Some straightforward thermodynamic calculations lead to the following equations which relate the particle density n to the enthalpy h.

The speed of sound can also be calculated and it turns out to be:

The conservation law system

In order to study numerically equation , we have to reduce it in first order conservation form (for further details, see [3] or [4]). We can do it for a flat space metric as follows: we write equation in explicit form:

We introduce the variables:

We thus obtain the following conservation law system:

Equation can be rewritten in term of the new variables in the following manner:

The conservation law system is evolved numerically using a second order numerical method (Lax-Wendroff, Mac-Cormack). At the end of each timestep, the new value of h is computed using a Raphson-Newton iteration on equation . The value of n is then computed by using equation .

Numerical Results

In figures and some 1D results are shown. The evolution of an enthalpy perturbation with some initial velocity has been studied. The code reproduces quite well our expectations for 1D evolution. We in fact expect the initial data to evolve following the characteristics given by the following expression:

Following equation , we see that in figure , the initial data are subsonic, so that the initial perturbation is divided into two waves propagating in both directions. Figure represents supersonic initial data, so the two characteristics are both in the same direction.

    
Figure: Fluid simulation with initial data: ,
Figure: Fluid simulation with initial data: ,