3.4
Hydrodynamic conservation laws (Euler equations)
The Euler equations3 are a set of coupled non-linear hyperbolic4 conservation laws, which can be
used as a simplified model to describe the dynamics of compressible fluids. The Euler equations are only a first approximation. They neglect body forces (body forces – e.g. gravity – have to be added as source terms to the solution of the hyperbolic PDEs), viscosity (included in the Navier Stokes equations) or magnetic fields (treated in MHD). The Euler equations support sound waves and an entropy wave.
The derivation of the hydrodynamic conservation laws (Eq.3.3to3.6) is based on the conservation of mass, energy and momentum. Using an integration of those quantities over a cell and Gauß’s theorem, leads to the differential form of the Euler equations via the vanishing integrands. The detailed derivations can be found in any book on hydrodynamics, e.g.Shu(1992, Chapter 2 pages 20 to 23 and Chapter 4 pages 45 to 46).
In addition to these hydrodynamic conservation laws, pressure and energy have to be connected with an equation of state (EOS). This closure relation – i.e. the relation between pressure and energy – is required to solve the system of hydrodynamic conservation laws, since there are more unknowns than equations. A possible choice is the adiabatic5EOS of an ideal gas:
Equation of State (EOS): p= (γ−1)ρein or pV =N kBT . (3.2)
The adiabatic exponent γ = cp
cV =
f+2
f is the ratio of the specific heats (cp, cV). It is a constant
that depends on the degrees of freedom (f) in the chosen type of gas. In a monoatomic ideal gas
the energy per degree of freedom is kBT /2. Thus, for increasing the temperature we can write
cV∆T =f k/2∆T, which has been normalized with the number of particlesN. If the volume has
to be adjusted to keep the pressure constant, we find from the EOS thatV is proportional toT and
thusp∆V = pVT ∆T = kBT
T ∆T, which leads tocp =cv+kand cp
cV =
f+2
f . For monoatomic gases
(e.g. atomic hydrogen, HI) the adiabatic exponent attains the valueγ = 5
3. Twoatomic gases and
linear molecules (e.g. gases like air or H2) have an adiabatic exponent of γ = 75 = 1.4. In the
isothermal case (i.e. constant temperature) the pressure is a function of density onlyP =c2
sρ.
ein=E/ρ−0.5|~v2|is the specific internal energy density. In Equation3.2einhas to be multiplied
with the density, sinceeinis the internal energy per unit mass and not an internal energy volume
densityEtherm =ρein.
The hydrodynamic equations without gravity and viscosity are: mass : ∂ρ ∂t + ∂ρvk ∂xk = 0 (3.3) momentum : ∂ρvi ∂t + ∂ρvivk ∂xk =−(γ−1)∂ρein ∂xi (3.4) energy : ∂Etot
∂t + ∂Etotvk ∂xk =−(γ−1)∂ρeinvk ∂xk (3.5) internal energy : ∂ρein
∂t + ∂ρeinvk ∂xk =−(γ−1)ρein ∂vk ∂xk . (3.6)
3Strictly speaking, only Equation 3.8is the Euler equation, but many authors call the whole system of partial
differential equations (PDEs) Euler equations.
4A PDE for a function u(x,t) the formAu
tt+ 2Butx+Cuxx+... = 0is called hyperbolic, ifAC−B2 <0.
This kind of PDEs behaves like a wave equation and has real Eigenvalues. It describes a phenomenon with finite propagation speed.
In vector notation and with the EOS (Eq.3.2), the system of these three conservation laws can be written as:
∂tρ+∇ ·(ρ~v) = 0 [conservation of mass] (3.7)
∂t(ρ~v) +∇ ·(ρ~v⊗~v) +∇p = 0 [conservation of momentum] (3.8)
∂tEtot+∇ ·[~v(Etot+p)] = 0 [conservation of energy] . (3.9)
In this system of coupled nonlinear partial differential equations Etot denotes the total energy
density,pis the pressure,~vis the velocity vector,ρis the density,∂ab = ∂a∂b are partial derivatives
and⊗is the tensor product.
The technical terms “diffusive” and “convective” terms, often used in context of hydrodynamical simulations, refer to parts of such equations: A transport equation for a general flow quantity Φ
and a diffusion coefficientΓtypically consists of four terms:
∂tρΦ | {z } time +∇ ·(~vρΦ) | {z } convection =∇ ·(Γρ∇Φ) | {z } diffusion + SΦ |{z} source .
The first term on the left side describes the net gain or net loss per unit volume and unit time. The convective term covers thedownstreamtransport with velocity~v. A nonuniform spatial distribution
ofΦleads to a diffusive term on the right hand side. All sources and sinks are collected inSΦ.
Basically, the flow can be described either with the vector of primitive variablesW~ T = (ρ, ~v, P)or
with the vector of conservative variablesU~T = (ρ, ρ~v, E). The latter is favorable for computations
(see Sect.3.2where conservative methods like finite volumes are discussed), as it directly uses the conservation of mass, momentum and total energy.
Generally, a system of conservation laws can be written in a compact form using the flux vectors
~ Fi
~
U, the vector of conservative variablesU~ and Einstein’s summation convention: ~ F1TU~ = ρv1, ρv12+p, ρv1v2, ρv1v3, v1(E+p) ~ F2TU~ = ρv2, ρv1v2, ρv22+p, ρv2v3, v2(E+p) ~ FT 3 ~ U = ρv3, ρv1v3, ρv2v3, ρv23+p, v3(E+p) ∂tU~ +∂xiF~i ~ U= 0 . (3.10)
With the Jacobians of the flux functionsJ~iU~= ∂ ~Fi
∂ ~U it can be rewritten as:
∂tU~ +J~i∂xiU~ = 0 (3.11)
This system is hyperbolic if the matrix J~ has real and distinct Eigenvalues λi. Physically, the
Eigenvalues represent velocities of propagation of information. The same type of system can be written down for the primitive variables.
An important concept for the numerical solution are the characteristic curves. These curves are possible trajectories of a signal in the space-time diagram. Fig. 3.5 shows this diagram for a hyperbolic PDE. The real Eigenvalues of this PDE correspond to wave families with finite speeds. These signal propagation velocities lead to a domain of dependence and a domain of influence.
3.4 Hydrodynamic conservation laws (Euler equations) 45 Time Space xj−atn xj+atn tn P (xj, tn) xj C1 C2 Domain of Domain of dependence influence Riemann invariants are constant along the
characteristic curves(C1, C2).
For an entropy wave:P, u.
For a sound wave: s, u± 2cs
γ−1.
Figure 3.5: Hyperbolic partial differential equations have real Eigenvalues (λ) with the physical
interpretation of finite wave speeds of different wave families. The characteristic curves(C1, C2)–
which are possible wave trajectories (i.e.x(t) =xj+λ(t−tn)) – limit the domain of dependence
and the domain of influence of the pointP(xj, tn)in the space-time diagram. The sketch also lists
the Riemann Invariants for two wave families.
Since Eq. 3.11 will lead to a velocity of the wave λ(U~), which will be rather a function of the
solution than constant, compression and expansion of the wave can be observed. For example, if the velocity increases for largerU~, the characteristics in the space-time plane are steeper in a region
with smallerU~ than in a region with largerU~ (in this diagram the slope is indirectly proportional
to the velocity). This is shown in Fig. 3.6. Diverging characteristics indicate a rarefaction fan – shown in Fig.3.6in the higherU~ region. The solution for such waves is discussed in Sect.4.1.2.
Converging characteristics lead to a shock – shown in Fig.3.6in the lowerU~ region. The shock is
a discontinuity and the integral forms of the conservation equations lead to the Rankine Hugoniot shock jump conditions, as shown in Sect.4.1.3. If characteristics on both sides of an interface are parallel, acontact discontinuitycan develop (see Sect.4.1.1).
Time Space (a) Time Space (b) Time Space (c)
Figure 3.6: The dependence of the Eigenvalues on the vector of conservative variables leads to non-constant slopes of the characteristics. We assume two constant states. This leads to two sets of parallel characteristics shown in blue and black. They can interact in three ways: (a) a shock forms if characteristics of the same wave-type intersect; (b) diverging characteristics produce a rarefaction fan; (c) finally, regions with parallel characteristics can harbor a contact discontinuity.
Time Space xj+1/2 pL pR ρL ρR vL= 0 vR = 0 head RF RFtail CD shock Time Space x−λ3t P (x, t) x−λ2t x−λ1t λ2 λ3 λ1 RF CD Shock
Figure 3.7: The Riemann problem. The top left panel shows the initial conditions of the Sod shock tube (see Sect. 4.2), which is a Riemann problem: a discontinuity separates a left and a right state. Pressures are shown in green, densities in blue and velocities in red. Details on the initial conditions can be found in Fig. 4.3. In the middle left panel the time evolved solution of the Riemann problem is shown: we see the propagation of a shock and a contact discontinuity to the right and a rarefaction wave propagating to the left. The dashed lines indicate the location of the head and the tail of the rarefaction fan (RF), thecontact discontinuity (CD) and the shock. The lowest left panel shows the characteristics for the different waves. The right panel shows the domain of influence (solid lines) for the point P(x,0) and the domain of dependence (dashed lines) for the point P(x,t), which is similar to Fig.3.5.