Euler equations (fluid dynamics)

Euler's equations or Euler's equations ( by Leonhard Euler ) is a mathematical model to describe the flow of inviscid fluids. Is a partial differential equation system first order, which is obtained as a special case of the Navier -Stokes equations, if the internal friction (viscosity) and the thermal conductivity of the fluid is negligible.

Derivation

The Euler equations can be derived in several ways: A common approach applies the transport theorem of Reynolds to the second Newtonian axiom. The transport theorem describes the temporal change of a physical quantity in a moving control volume.

Another approach is based on the Boltzmann equation: The collision operator is multiplied there with three possible terms, the so-called Kollisionsinvarianten. After integration over the particle arise continuity equation, momentum equation and energy balance. Finally, a scaling for large time and space dimensions is performed ( Hydrodynamic Limites ), and the result is the extended Euler equations.

Formulation

Momentum equation

The essential part of the Euler equations is the pulse rate, which reads, neglecting external forces in differential form:

Wherein the speed vector, the density, the pressure, the position vector of the time and the nabla operator. In Cartesian coordinates, this equation is the two-dimensional case for and written out in full:

The left side describes this in both cases, the substantial acceleration, consisting of local and convective acceleration.

Flow formulation

Using the continuity equation is obtained from the momentum equation, the balance equation of the pulse density of the Pulse Density:

Or alternative spelling

This is the unit tensor also means stress tensor. The tensor is the convective transport of momentum density, its divergence

Is the convective momentum flux.

By integrating over a fixed volume V, and applies the Gaussian integral theorem, we obtain:

Where V is the volume with the surface S, and n is the unit normal vector to the surface element dS. This formulation of the equation demonstrates the conservation of momentum with the introduction of the static pressure p. The pressure is a surface force and takes effect on the pulse by exchange with the environment, the pressure is not an independent source of momentum.

Full system of equations

Also a common formulation of the Euler equations in addition includes the scalar equations for mass conservation ( continuity equation) and energy conservation. In the two-dimensional case, the four coupled differential equations arise as

Where the vector of conservation variables and the river are given with the enthalpy H by the following expressions:

Together with an equation of state is obtained a closed system of equations to calculate the five unknown quantities speed and then, pressure, density and temperature.

Boundary conditions

At solid walls is set as a condition that the velocity in the normal direction is zero. Then no additional condition can be set to the tangential component of velocity. This is in contrast to the Navier -Stokes equations, in which the no- slip condition and the second-order terms create a boundary layer. Together with the absence of turbulence makes this from the significant difference between the Euler and the Navier -Stokes equations.

Mathematical properties

The Euler equations belong to the category of non-linear hyperbolic conservation laws. This usually occur after a finite time even with smooth initial data discontinuities, such as shocks ( shock waves ). Under strong conditions in the relevant case exist global smooth solutions, such as when the solution moves away in a kind of rarefaction wave. In the stationary case, the equation is depending on the Mach number elliptic or hyperbolic. In a transonic flow then occurs in both subsonic and supersonic regions, and the equation has mixed character.

The eigenvalues ​​of the equations, the velocity in the normal direction ( with multiplicity the dimension) and this plus or minus the speed of sound. So that the Euler equations as a function of the one-dimensional even strictly hyperbolic, using the ideal gas equation, so it is useful existence and uniqueness results there. In multidimensional they are no longer strictly hyperbolic because of the multiple eigenvalue and the mathematical solution is extremely difficult. It revolves mainly around determining physically meaningful weak solutions, ie those which can be interpreted as solutions of the Navier -Stokes equations with vanishing viscosity.

In addition, the Euler equations are rotationally invariant, so it holds for arbitrary unit vectors and rotations of velocities in the direction of normal vectors:

In addition, the flow functions are homogeneous, it is so.

Numerical Solution

Since the Euler equations representing conservation equations, they are usually solved using finite volume method. Conversely, the effort in the field of aerodynamics, since the 1950s was the Euler 's equations to simulate numerically driving forces in the development of the finite volume method. Since, in contrast to the Navier -Stokes equations without the boundary layer must be taken into account, this can happen at a relatively coarse computational grids. The central difficulty is the treatment of the Euler flow, which is usually treated with the help of approximate Riemann solvers. These provide an approximation to the solution of Riemann problems along the cell edges. The Riemann problem of Euler equations is solved exactly even, but the calculation of this solution is extremely complex. Since the 1980s, numerous approximate solvers were therefore developed, starting with the Roe solver up to the AUSM - family in the 1990s.

At the time of integration, the CFL condition is observed. Just in the range of Mach numbers close to zero or one, the equations are due to the different eigenvalue scales very stiff, which makes the use of implicit time integration methods necessary: the CFL condition based on the largest eigenvalue ( ), while relevant for the simulation parts of the flow with moving. An explicit method needed in most cases unacceptably so many time steps.

The resulting solution occurring systems of nonlinear equations is then performed either by means of preconditioned Newton - Krylov method or with special nonlinear multigrid method.

Derived relations

From the Euler equations can be extended - in particular through appropriate scaling - a series gasdynamic basic equations are derived:

• The Bernoulli equation,
• The acoustic wave equation,
• The incompressible Euler equations.