Initially, the governing equations are presented without any source terms, which will be included later. This is done in order to make the resulting equations more physically founded
The first is conservation of mass, given as
∂ρ
∂t + ∂
∂x(ρu) = 0 (2.1)
where ρ is density and uvelocity. This simply states that no mass is created within the medium.
The second equation is the Navier-Stokes equation for conservation of momen-tum in a compressible Newtonian fluid, neglecting viscosity and heat-conduction.
∂
∂t(ρu) + ∂
∂x(ρuu+p) = 0 (2.2)
which is an representation of Newton’s second law applied to the fluid control volume in regard.
The third is the energy equation, which is an application of the first law of thermodynamics to the medium. The time rate of change of total energy is equal to the rate of change in internal energy, caused by heat input to the gas, and the work done on the system. In this inviscid fluid, the only work done is by the pressure (and eventually body forces).
∂
∂t(ρE) + ∂
∂x(ρEu+pu) = 0 (2.3)
where the total energyE, the sum of internal energy and kinetic energy of the gas, is given as
E=e+12u2 . (2.4)
2.2 The Euler Equations 7
The internal energy is a function of temperature alone, again by the ideal gas assumption, and with assumed constant thermal properties (i.e. temperature independent heat capacities), gives the expression
e=cVT . (2.5)
The conservation law for total energy (2.3) can be rewritten in terms of internal energy by use of (2.4) and the momentum equation, giving
∂
∂t(ρe) + ∂
∂x(ρue) +p∂u
∂x =ρeheat (2.6)
The internal energy equation could be the appropriate place to include the effects of additional heat release,ρeheatbeing this source. This would dependent on the physics being considered, and in the subject considered here, the heat release is entered differently.
The three equations – (2.1),(2.2) and (2.3) – needs a fourth relation to close the system which will be the thermodynamic equation of state. By the ideal gas assumption, this is given as
p = (γ−1)ρe (2.7)
= ρRT
whereγ=cP/cV is the ratio of specific heats and has the additional relationships R=cP −cV , cV = R
γ−1 , cP = γR
γ−1 . (2.8)
Finally, the three equations form the system of conservation laws, which is how the Euler equations usually are presented. If this system was to contain additional terms (source terms), it would be called a system of balance laws or general conservation laws instead.
∂U
These nonlinear partial differential allows for a variety of different phenomena, even ones which looses uniqueness and develops shocks. Therefore a solution is often sought as a weak solution to the system.
In addition, later when heat release is introduced, it enters as a Dirac’s delta function, which introduces an discontinuity in the solution.
Interlude on weak solutions. The theory regarding weak solutions is a large mathematical area and quite complicated. So this short section is by far an adequate coverage of the theory, but more some principles regarding weak solutions and introduction of some conditions used throughoutly in this project.
A solution that somehow becomes discontinuous, can not fulfill the equation (2.9). Which is why we will seek an less restrictive formulation, the weak for-mulation. This formulation allows for the real solution (or the classic solution) to (2.9), but also for discontinuous solution.
The reasoning towards it can be expressed in different ways, but in abstract terms it tries to solve for a limit function, a function in the vicinity of the ”real”
function in the respective function space. A weak formulation can be defined in different ways, but one is to use test functions. This one relates to the numerical scheme presented later.
Another is presented here, to introduce some conditions used often in dealing with both the mathematical model and the numerical scheme.
The PDE in a weak formulation is defined as I
C
F dt−U dx= 0 (2.11)
withC a positively oriented, piecewise-smooth, and closed curve in (x, t). IfU andF are continuous and differentiable, the Green’s formula in the plane gives
I so it is a solution to the original equation and vice versa.
Now consider the solution being discontinuous across some line in (x, t)-space, and denoting subscript±as the limit state in front and after this shock.
Using the contourC to evaluate the integral (2.11) around the shock, this can be expressed as which leads to that the integrand should vanish. This is known as the Rankine-Hugoniot (abbreviated RH) conditions,
[F]+−=σ[U]+− (2.14)
which has to hold across a shock, if it is to conserve mass, momentum and energy.
The weak solution is not necessarily unique. In order to ensure a unique solu-tion, the weak formulation is usually supplemented by an appropriate entropy
2.2 The Euler Equations 9
condition. In the dG method used to find a weak solution, this is incorporated in the method.
In addition, the system is hyperbolic, i.e. were (2.9) expressed in a quasi-linear form Ut+A(U)Ux = 0, where A(U) is the jacobian matrix. Then the latter will have real and distinct eigenvalues. These eigenvalues reflects the speed at which information travels in the system. This property of hyperbolicity is quite essential in some aspects of the numerical modelling.
These conserved quantities (mass, momentum and energy) are less ”physical”
or natural in regard of what we hear and can measure, so the original conser-vation laws are formulated in a non-conservative form in terms of the primitive variables; densityρ, velocityuand pressurep. No information is lost from the equations during this change of variables.
The mass equation remains unchanged, the momentum equation in terms of velocity is derived from (2.2) by subtracting the continuity/mass conservation equation (2.1). At last, the equation describing the pressure evolution is de-rived from conserved internal energy equation (2.6) and the equation of state (2.7). So the system of conservation equations (2.10) in a non-conservative form, expressed in primitive quantities, is
∂ρ
∂t +u∂ρ
∂x+ρ∂u
∂x = 0 (2.15a)
∂u
∂t +u∂u
∂x+1 ρ
∂p
∂x = 0 (2.15b)
∂p
∂t +u∂p
∂x+γp∂u
∂x = 0 (2.15c)
This change of variables does change the elements in the coefficient matrix of the system (2.15), but not its structure, since this coefficient matrix and the jacobian matrix mentioned earlier are conjugate.
The eigenvalues of this system are
λ1=u−c , λ2=u , λ3=u+c (2.16)
which reflects two sound waves travelling at their characteristic speeds λ1 and λ3down- and upstream respectively, and one convective-acoustic waveλ2in the system.