Mathematical Models for High Speed Flows

Contents 1 - introduction ....................................................................................................................................... 2

2 - Governing Equations of Gas Dynamics ............................................................................................ 2

2.1 - Introduction ................................................................................................................................... 2 2.1.1 - The mass conservation law ................................................................................................... 4

2.1.2 - Conservation of Momentum ................................................................................................. 5

2.1.3 - Conservation of Energy ......................................................................................................... 7

2.1.4 - The Integral and Differential Form of the Governing Equations .......................................... 9

2.2 - The Euler Equations .................................................................................................................... 10

2.3 - Thermodynamic Relations .......................................................................................................... 13 2.3.1 - Equilibrium Real Gas ........................................................................................................... 13

2.3.2 - The specific heat capacities and the speed of sound ......................................................... 15

2.3.3 - Non-Equilibrium Real Gas ................................................................................................... 17

2.4 - Simplified Euler equations .......................................................................................................... 18 2.4.1 - 1D Compressible Flows with Area Variation ....................................................................... 18

2.4.2 - Cylindrical and Spherical Symmetry .................................................................................... 19

2.4.3 - Axi-Symmetric and Plane (2D) Flows .................................................................................. 20

2.5 - Mathematical properties of the Euler equations.......................................................................... 21 2.5.1 - The 1D Euler equations ....................................................................................................... 21

2.5.2 - Two-Dimensional Euler Equations ...................................................................................... 26

2.5.3 - The 3D Euler Equations ....................................................................................................... 29

2.5.4 - Conclusion ........................................................................................................................... 31

3 - References ....................................................................................................................................... 33

1 - INTRODUCTION

The role of this work is to review systematically the well-known Euler equations of gas dynamics in the frame of actual aerothermodynamics. In the frame of the present work, the supersonic/hypersonic flows under consideration are flows dominated by the occurrence of (strong) shock waves and, for hypersonic flows, significant local high temperature effects.

2 - GOVERNING EQUATIONS OF GAS DYNAMICS

2.1 - Introduction

In this part we recall the standard form of the classical fluid dynamics equations written in eulerian form. The fundamental assumption is that the fluid is a continuum (see the considerations about the low-density case before). For the evolution of a fluid in 1N spatial dimensions, the description involves ( )2N + fields. These are the mass density, the velocity field and the energy. The fundamental equations of fluid dynamics are derived following the next three universal conservation laws:

1. Conservation of Mass. 2. Conservation of Momentum or the Newtons Second Law. 3. Conservation of Energy or the First Law of Thermodynamics.

The resulting equations are the continuity equation, the momentum equations and the energy equation, respectively. The system of governing equations is closed by:

1. Statements characterizing the thermodynamic behavior of the fluid, which are called state equations. The state equations do not depend on the fluid motion and represent relationships between the thermodynamic variable: pressure (p), density ( ), temperature (T), specific internal energy per unit mass (e) and total enthalpy per unit mass (h). The microscopic transport properties like the dynamic viscosity ( ) and the thermal conductivity (k) are also related to the thermodynamic variables (see for instance Sutherlands relations.

2. Constitutive relations, which are postulated relationships between stress and rate of strain and heat flux and temperature gradient. The previously mentioned microscopic

transport properties appear in the constitutive relations as proportionality factors (between stress and rate of strain and heat flux and temperature gradient) and must be determined experimentally.

The flow is considered as known if, at any instant in time, the velocity field and a precise number of thermodynamic variables are known at any point. This number equals two for a real compressible fluid in thermodynamic equilibrium (for example, the pressure and the density). There is some freedom in choosing a set of variables to describe the fluid flow. A possible choice is the set of primitive or physical variables, namely the density, pressure and velocity

components along the three directions of a reference frame, i.e. the set ( ), , , ,p u v w . Another choice is the set of the so-called conservative variables which are the density, the three

momentum components and the total energy per unit mass, i.e. the set ( ), , , ,u v w E . The set of conservative variables results naturally from the application of the above fundamental laws of conservation. From a computational point of view, there are some advantages in expressing the governing equations in terms of the conservative variables. We will call conservative methods the class of numerical methods based on the formulation of the governing equations in conservative variable. The primitive and conservative variables are dependent variables. They are depending on time and space coordinates, i.e. on the

independent variables. For example, the density is ( ), , ,t x z y = . Recall that the number of the spatial coordinates is N. The reader is invited here to notice an important difference between the governing equations and the closure conditions. The governing equations are exact mathematical conditions that must be satisfied by the flow variables. The state equations and the constitutive relations are only practical models and they depend on the degree of knowledge the user has about the fluid and flow under consideration. Thus, they represent sources of uncertainties that are introduced in the mathematical model of the flow.

Let us now consider the new scalar field

( ) ( , , , )V

t x y z t dV = (2-1)

where the volume of integration V is enclosed by a piece-wise smooth boundary surface A V= that moves with the material under consideration. It can be shown, [1, 2] that the

material derivative of is given by:

( )V AD dV dADt t = +

n V (2-2)

where 1 2 3( , , )n n n=n is the outward pointing unit vector normal to the surface A. This equality can be generalized to vectors ( , , , )x y z t as follows:

( )V AD dV dADt t

= +

n V

(2-3)

The surface integrals in the last two equalities may be transformed to a volume integral by virtue of Gausss theorem, [1]. This states that for any differentiable vector field

1 2 3( , , ) = and a volume V with smooth bounding surface A the following identity holds:

( )A V

dA div dV = n . (2-4)

The above identity also applies to differentiable scalar and tensor fields. If we use it, for example, to transform the second term of the right hand side of equation (1-5) into a volume integral, one obtains:

( )VD div dVDt t = + V .

(2-5)

In the above relations the volume V is arbitrary but moving with the fluid particles. Let J be the Jacobian of the transform that links two successive positions of the volume V. There is a theorem due to Euler which states that the time derivative of the Jacobian is, [2]:

D J J divDt

= V . (2-6)

2.1.1 - The mass conservation law

The law conservation of mass can now be stated in integral form by identifying the scalar in (1-2) as the fluid density . By assuming that no mass is generated or annihilated within the material control volume V, we have:

( ) 0V AdV dAt + = n V .

(2-7)

This is the integral form of the law of the conservation of mass. This integral conservation law may be generalised to include sources of mass, which will then appear as additional integral

terms. A useful reinterpretation of the integral form results if now V is a fixed control volume and we rewrite it as:

( )V V A

ddV dV dAt dt = = n V .

(2-8)

Thus, the time-rate of change of the mass enclosed by the volume V, in the absence of sources or skins, is due to the mass flow through the boundary of the control volume V. One obtains from (2-7) that:

( ) 0V

div dVt + = V

(2-9)

As V is arbitrary it follows that the integrand must vanish, that is:

( ) ( ) ( ) 0t x y zu v w + + + = (2-10)

which is the differential form of the continuity equation.

2.1.2 - Conservation of Momentum

As done for the mass equation, we now provide the foundations for the law of conservation of momentum, derive its integral form in quite general terms and show that under appropriate smoothness assumptions the differential form is implied by the integral form. A control volume V with bounding surface A is chosen and the total momentum in V is given by:

( )V

t dV= V (2-11) The law of conservation of momentum results from the direct application of Newtons law: the time rate of change of the momentum in V is equal to the total force acting on the volume V. The total force is divided into surface forces fs and volume fv given by:

, vA VdA dV= = sf S f g (2-12) Here g is the specific volume-force vector and may account for inertial forces, gravitational forces, electromagnetic forces and so on. S is the stress vector, which is given in terms of a stress tensor as = S n . The stress tensor can be split into a spherical symmetric part due to pressure p, and a viscous part :

p= I (2-13)

Application of Newtons Law for the momentum of the fluid in the fixed volume V (independent of time) gives:

( ) ( ) ( ) sV V Ad dV dV dAdt t

= = + + vV V V n V f f

(2-14)

We interpret the above as saying that the time rate of change of momentum within the fixed control volume V is due to the net momentum inflow over momentum outflow, given by the first term in (2-14), plus the effects of surface and volume forces. Substituting the stress tensor into (2-12) and writing all surface terms into a single integral we have:

[ ]( ) ( )V A V

dV p dA dVt = + +

V V n V n n g (2-15)

This is a general or integral statement of the momentum conservation law and it is valid even for the case of discontinuous solutions.

The differential conservation law can now be derived from (2-15) under assumption that the integrand in the surface integral is sufficiently smooth so that Gausss theorem may be invoked. The first term of the integrand of the surface integral can be rewritten as:

( ) , = V n V n V V

where the dyadic (tensor) product VV is a tensor. The three columns of the left hand side are:

2

2

2

( ) , , ,

( ) , , ,

( ) , , .

T

T

T

u u uv uw

v uv v vw

w uw vw w

=

=

=

n V n

n V n

n V n

Using the Gausss theorem for each of the surface terms:

[ ]( ) ( )V V V

div gradp div dV dVt = + +

V V V g (2-16)

As this is valid for any arbitrary volume V the intregrand must vanish, i.e.:

[ ]( ) div pt + + =

V V V I g

(2-17)

This is the differential conservative form of the momentum equation, including a source term due to volume forces. In order to use it for a given fluid flow the components of the viscous stress tensor must be provided through a constitutive relation.

2.1.3 - Conservation of Energy

As done for mass and momentum we now consider the total energy in a control volume V, that is:

( )V

t EdV = (2-18)

where E is the total energy per unit volume:

( )2 2 2 21 12 2E e e u v w = + = + + +

V . (2-19)

Here e is the specific internal energy and the second term represents the specific kinetic energy.

Before we discuss the equations in more detail, let us make a physical comment: the derivation of the energy equation relies upon the assumption that, in a fluid in motion, the fluctuation around thermodynamic equilibrium are sufficiently weak so that the classical thermodynamics results hold at every point and at all times. In particular, the thermodynamic state of the fluid is determined by the same state variables as in classical thermodynamics, and these variables are determined by the same state equations (see the devoted paragraph in this chapter).

The energy conservation law states that the time rate of change of total energy )(t is equal to the work done, per unit time, by all forces acting on the volume plus the influx of energy per unit time into the volume. The surface and volume forces in (2-12) give rise to the following two terms:

( ) ( )surf A AE p dA dA= + V n V n (2-20)

and

( )vol VE dV= V g . (2-21)

In (2-20) surfE is the work done by the surface forces: the first term corresponds to the work

done by the pressure while the second term corresponds to the work done by the viscous stresses. In (2-21) volE is the work done by the volume force g.

To account for the influx of energy into the volume we denote the energy flux vector by

1 2 3( , , )q q q=Q . The flow of energy per unit time across a surface dA is given by the flux ( )dA n Q . This gives the total influx of energy to be included in the equation of balance of

energy:

inf ( )AE dA= n Q (2-22)

The total internal energy variation, i.e. the energy conservation law in integral form for a fluid volume V fixed in space and independent of time, is therefore:

[ ]( ) ( ) ( ( )V A A V

EdV E dA p dA dVt

+ = + + + n V n V Q V n V g

(2-23)

The differential form of the conservation of energy law can now be derived by assuming sufficient smoothness and applying Gausss theorem to all surface integrals of (2-23). Direct application of Gausss theorem gives:

( ) ( )A V

p dA div p dV = n V V

A VdA div dV = n Q Q

The work done by the viscous stresses can be transformed by first observing that ( ) ( ). = V n n V This follows from the symmetry of the viscous stress tensor . Hence:

( ) ( )A V

dA div dV = V n V

Substitution of these volume integrals into the integral form of the law of conservation of total internal energy gives:

( ) ( ) 0V

E div E p dVt

+ + + = V V Q V g

(2-24)

Since the volume V is arbitrary the integrand must vanish identically, that is:

( ) ( ) 0E div E pt

+ + + = V V Q V g

(2-25)

This is the differential conservative form of the law of conservation of energy with a source term accounting for effect of body forces; if these are neglected we obtain the homogeneous energy equation corresponding to the Navier-Stokes equation. The energy flux vector Q must be specified through a constitutive relation. Further, when viscous and heat conduction effects are neglected one obtains the energy equation corresponding to the compressible Euler equations.

2.1.4 - The Integral and Differential Form of the Governing Equations

The previous derivation of the governing equations of compressible flows, such as the compressible Euler and Navier-Stokes equations stated earlier, is based on integral relations on control volume and their boundaries. The differential form of the equations results from further assumptions on the flow variables (the smoothness of the flow field variables is assumed). In the case of the Navier-Stokes equations, the smoothness of the solution is naturally assured by the diffusion due to the viscosity and heat conductivity. On the contrary, in the absence of viscous diffusion and heat conduction one obtains the Euler equations. These admit discontinuous solutions and the smoothness assumption that leads to the differential form no longer holds true. Thus, one must return to the more fundamental integral form involving integrals over control volumes and their boundaries. However, for most of the theoretical developments the differential form of the equations is preferred. From a computational point of view there is another good reason for returning to the integral form of the equations.

We have determined previously the integral form of the governing equations, as well as the conservative differential form of the equations. One notices that the name Euler and/or Navier-Stokes originally given to the momentum equations (or equations of motion) transfers to the entire system of equations. We summarize below the general laws of conservation of mass, momentum and total energy, written both in differential and integral form:

a) the continuity equation:

( ) 0t div + =V or

( ) 0V A

d dV dAdt

+ = n V

b) the momentum equation:

[ ]( ) ,t div p + + =V V V I g or

[ ]( ) ( ) .V V V

div gradp div dV dVt + + =

V V V g c) the energy equation

[ ]( ) ( )tE div E p + + + = V V Q V g or

[ ]( ) ( ( )V A V

d EdV E p dA dVdt

+ + + = n V V Q V n V g where ),,( 321 gggg = is a body force vector.

At this point, all the terms, such as the viscous stress tensor, the heat flux, the pressure, the temperature and the specific internal energy are expressed as functions of the conservative variables.

2.2 - The Euler Equations

In this section we consider the time-dependent Euler equations. The continuity, momentum and total internal energy equations are forming a system that takes the name of the inviscid momentum equations, [1]. This system of non-linear hyperbolic conservation laws governs the dynamics of a compressible fluid, such as gases (or liquids at high pressure). This is possible only within the framework of the hypothesis that the fluid is an ideal one, without viscosity and without heat conduction. This means that both microscopic transport properties of the fluid (i.e., the dynamic viscosity and the heat conduction coefficients) are completely neglected, by the previous assumption.

When body forces are included via a source term vector but viscous and heat conduction effects are neglected the mass, momentum and total energy conservation laws become the so called Euler equations that may be written in conservative form as:

( ) ( ) ( ) ( )t x y z+ + + =U F U G U H U S U (2-26)

where the vector of conservative variables is U and the convective fluxes are F, G, H in the x, y and z directions, respectively. Their explicit forms are:

2

2

2

, , ,

( ) ( ) ( ).

wu vuwu u p uvvwv uv v p

w puw vw w pE u E p v E p w E p

+ = = = =+

+ + + +

U F G H (2-27)

First, it is important to note that the fluxes are nonlinear functions of the conserved variable vector. Any set of partial differential equations written in the form (2-26) is called a system of conservation law (in differential formulation). Recall that the differential form assumes smooth solutions, that is, partial derivatives are assumed to exist. It is also clear now that the integral form of the equations is an alternative way of expressing the conservation laws in which the smoothness assumption is relaxed so that to include discontinuous solutions.

The term ( )=S S U is a source or forcing term. Due to the presence of the source term, the equations are said to be inhomogeneous. There are several physical effects that can be included in the forcing term, like: body forces such as gravity, injection of mass, momentum and/or energy. Usually, ( )S U is a prescribed algebraic function of the flow variables and does not involve derivatives of these, but there are exceptions. If ( ) 0=S U one speaks of homogeneous Euler equations. This is the case of aerodynamic applications, when the body forces due to gravity and other sources are negligible. We also mention here that there are situations in which source terms arise as a consequence of approximating the homogeneous equations to model situations with particular geometric features (axy-symmetric flows, for instance). In this case the source term is of geometric character, but we shall still call it a source term.

Sometimes it is convenient to express the equations in term of the primitive or physical variables , u, v, w and p. By expanding derivatives in the conservation law form and using the mass equation into the momentum equations and in turn using these into the energy equation one can re-write the thee-dimensional Euler equations for ideal gases with a body-force source term as:

1

2

3

( ) 0

1 ,

1 ,

1 ,

( ) 0

t x y z x y z

t x y z x

t x y z y

t x y z z

t x y z x y z

u v w u v w

u uu vu wu p g

v uv vv wv p g

w uw vw ww p g

p up vp wp p u v w

+ + + + + + =

+ + + + =

+ + + + =

+ + + + =

+ + + + + + =

(2-28)

For computational purposes it is the conservation law form (2-26) that is most useful. The formulation in primitive variables (2-28) is more convenient to theoretical developments.

We have presented in this section the time-dependent Euler (and Navier) equations of fluid dynamics. The equations are accompanied by equations of state and by constitutive relations and also by initial and boundary . Both models can be used for homogeneous gases and/or liquids at high pressure. We appreciate that for supersonic/hypersonic flows the Euler equations are a representative and useful model for both theoretical developments, [3], and computational purposes, [1, 2]. From a mathematical point of view, these models are systems of partial differential equations. Since with a few exceptions one cannot find exact solutions of these equations for practical problems, the only way is to solve them is to use numerical techniques. It is generally accepted that the discretization techniques must be based not only on the underlying physics bust also on the mathematical properties of the partial differential equations. It is therefore useful to begin with the analysis of the mathematical nature of the governing equations of the flow before trying to solve them numerically. Further, the equations of compressible fluid flow reduce to hyperbolic conservation laws (i.e. the Euler system) when the effects of viscosity and heat conduction are neglected. The hyperbolic part represents the convection and pressure gradient effects and it can be identified in the equations even the above physical effects are not negligible. This is the reason why we allocate further a special section to study the hyperbolic partial differential equations.

We now conclude this section with a brief discussion of initial and boundary conditions. Usually, we solve the systems of equations for 0t prescribing the values of all the unknowns at 0t = . The question of the boundary condition is much more delicate. Simply said, one impose physical boundary condition at the solid boundaries for some of the variables, and these can be of Dirichlet or Neumann type (depending on the variable type). Since the boundary conditions are replacing the governing equations on the boundaries, the rest of the variables are determined via numerical techniques. These are strongly related to the numerical

solvers and thus we will discuss the imposition of the boundary conditions in the context of numerical solvers.

A short comment on the Navier-Stokes model is also necessary. The equations presented in this chapter can be used for the numerical simulation of laminar flows only. At practical Reynolds numbers, the effects of the turbulence must be taken into account. The Reynolds Averaged Navier-Stokes (RANS) equations and the Large Eddy Simulation (LES) equations have formally the same mathematical aspect, with minor differences. The huge difference comes from the necessity of modeling the turbulence effects, mainly in the vicinity of solid boundaries. As we already mentioned, it is beyond the purposes of this book to discuss about turbulence modeling and turbulence models. More details about the derivation of these and other equations can be found in [1, 2, 4, 5].

2.3 - Thermodynamic Relations

2.3.1 - Equilibrium Real Gas

In thermodynamic equilibrium there are two independent variables, [3, 4, 7]. Thus, two equations are necessary combining the four variables, which are the pressure (p), the density (), the absolute temperature (T) and the internal energy per unit mass (e). These equations are the so-called thermal and the caloric equations of state.

The thermal equation of state (EOS) for a real gas (like the air at usual and high temperatures is) can be written as

( ),p RT Z T=

, (2-29)

where R is the specific gas constant per unit mass and Z is the compressibility factor. For non-dissociating and non-ionizing gases and if the pressure is low-enough so that the van de Waals effects are negligible, the compressibility factor 1Z and thus the thermal ideal gas equation can be applied:

p RT=

. (2-30)

The caloric equation of state is a relation between the internal energy per unit mass e and two dependent variables, i.e. the density and the temperature and can be written in general as:

( ),e e T= . (2-31)

In a non-dissociating and non-ionizing gas the specific internal energy does not depend on the density and thus the caloric equation of state of a real gas is

( )e e T= . (2-32)

It can be shown that (2-32) is, [2, 3, 4]:

( ) ( )0

T

Ve e T c T dT= = , (2-33)

where ( )Vc T is the specific heat at constant volume.

Below 775 K (about 5000C) the specific heat in air is independent of the temperature and the air behaves as a calorically perfect/ideal gas, i.e. the air is a thermally and calorically ideal gas and:

( ) 5,2 1V V

e e T c T c R R= = = =

. (2-34)

Above 775 K molecular vibrations are excited and become important and the specific heats at constant pressure and volume become dependent on the temperature. The thermal and caloric equations of state can be combined to give the pressure in terms of the density and internal energy per unit mass similar to (2-27):

( ) ( ), ,p RT e Z e=

. (2-35)

The implementation of the state equation (2-35) of real gases in equilibrium in a code requires

the knowledge of pressure function ( ),p p e= and of its partial derivatives, e

p

and pe

,

if the speed of sound is needed by the algorithm. For air the real gas description is given by Mollier, [10] or from tables/routines approximating the pressure as function of density and internal energy.

2.3.2 - The specific heat capacities and the speed of sound

The heat capacity at constant pressure cp and the heat capacity at constant volume cv (specific heat capacities) are now introduced. In general, when an addition of heat dQ changes the temperature by dT the ratio dTdQc /= is called the heat capacity of the system. For a thermodynamic process at constant pressure one obtains ( ) ,dQ de d pv dh= + = where the definition of the specific enthalpy is

ph e= +

(2-36)

Assuming ),( pThh = , the heat capacity cp at constant pressure becomes:

pp

hcT =

(2-37)

The heat capacity cv at constant volume may be written, following a similar argument, as:

vv

ecT =

(2-38)

The speed of sound is a variable of fundamental interest. For flows in which particles undergo unconstrained thermodynamic equilibrium one defines a new state variable a, called the equilibrium speed of sound or just speed of sound. Given a caloric equation of state such as ),,( spp = one defines the speed of sound a as:

s

pa

=

(2-39)

where s = the specific entropy. This basic definition can be transformed in various ways using established thermodynamic relations. For instance, given a caloric EOS in the form ),,( phh = we can also write, [2]:

+

+=

+

ds

spdpTdsdhdp

ph

spp

1

Settings ds=0 (isentropic conditions) and using the definition (2-39) we obtain successively:

12

=

ph

h

a p ,

and for a thermally ideal gas (perfect gas) characterized by the EOS of the form h= h(T):

12

=

p

pp

h

h

a

From pp

ch =

(by definition) and for a thermally ideal gas one obtains the widely

used relation for the speed of sound:

( ) ( ) pa R

= = (2-40)

Remark: One notices the dependence of the ratio of the specific heats on the temperature.

For a general material (gas, but not only) the caloric EOS is a functional relationship involving the variables p--e. The derived expression for the speed of sound a, depends on the choice of dependent variables. Two possible choices and their respective expression for a are:

2

2

( , ),

( , ),

e

p p

pp p e a p p

epe e p ae e

= = +

= =

(2-41)

where subscripts denote partial derivates. Various isentropic exponents can be defined, depending on the assumption of the state

of the gas, [4]. For instance, for a thermally ideal gas:

p

v

hc d hT

ec deT

= = =

,

(2-42)

or for a perfect gas:

p

v

c hc e

= = . (2-43)

In the case of perfect gas all the possible definitions are identical, [4]. However, the implementation of the state equation of real gases in equilibrium in an Euler model requires the

knowledge of ( ),p p e= and the derivatives e

p

and pe

, if the speed of sound is

needed by the numerical algorithm. For air , the real gas description can be taken from [10, 9].

2.3.3 - Non-Equilibrium Real Gas

The composition of the dry air atmosphere is given in Table 2.1. The high temperature occurring in hypersonic flows cause the excitation of vibrations and dissociation of N2 and O2 and the formation of N, O and NO. At even higher temperatures it occurs also ionization. The argon an carbon dioxide are considered non-reacting components.

Table 2.1. Composition of atmospheric dry air, [6].

From [4], for instance, on a typical shuttle re-entry trajectory, the real gas effects have to be taken into account for Mach numbers greater than 4. At altitudes below 40 km the air is approximately in local thermal and chemical equilibrium. At altitudes above 40 km, due to the decreasing free stream density the chemical relaxation time increases to the same order of magnitude as the characteristic flight time leading to a non-equilibrium state of the gas. Finite chemical reaction rates lead to chemical non-equilibrium and finite energy exchange rates of vibrational degrees of freedom lead to thermal non-equilibrium, [4]. The bow shock layer consists of a thin shock, a nearly inviscid region and a thin boundary layer, [3, 4]. However, the

influence of thermal non-equilibrium is expected to be rather small since the thickness of the thermal non-equilibrium layer is only a few millimeters, [4].

We consider that the reacting air neglecting ionization can be modeled using five species (i.e. nitric oxide NO, molecular N2 , O2 , and atomic N, O). Each species behaves as a perfect gas and the Dalton law allows the calculation of the mixture pressure equation of state as a sum of species partial pressures. Then, a chemical reactions model that includes a number of reactions must be added, [4, 6]. There are models with 17 equations, [4] or less, see [6]. The computational efficiency is of primary importance for the use of such a model in a coupled manner with an Euler solver.

2.4 - Simplified Euler equations

In this section we wish to consider successively simplified versions, or submodels, of the 3D Euler governing equations and their closure conditions. There are many reasons and many ways to use and derive simplified models. On one hand, it is clear that any simplification leads to the reduction of the generality of the equations. On the other hand, a rational simplification may lead to a significant reduction of the computational effort without penalties on the quality of the solution. There are some possibilities that can be exploited to derive simplified models, and perhaps the most common of them are the reduction of the dimensionality of the problem and the introduction of some new assumptions/hypothesis. One can also augment the basic equations by adding source terms to account for additional physics. This is not exactly a simplification of the basic model; on the contrary, adding some semi-empirical source terms may enlarge the area of practical applications that can be solved with that model.

2.4.1 - 1D Compressible Flows with Area Variation

Compressible inviscid flows with area variation arise naturally in the study of fluid flow phenomena in shock tubes and nozzles. One may start from the three dimensional homogeneous version of Euler equations to produce, under the assumption of smooth area variations, a quasi-one dimensional system with a geometric source term ( )S U , namely:

( ) ( )t x+ =U F U S U , (2-44)

where

2 21, ,( ) ( )

x

x

x

u uAu u p pu A

AE u E p u E p A

= = + = + +

U F S .

(2-45)

Here x denotes distance along the tube, nozzle, etc.; A is the cross-sectional area and in general is a function of only space and time, that is ( )A A x= . The flow variables are averaged quantities on the transversal cross-section area. In this common case one can re-write the equations in the more convenient form:

( ) ( )t x S+ =U F U U , (2-46)

where

2

0, ( ) ,

( ) 0x

A A uA u A u p pAAE Au E p

= = + = +

U F S .

(2-47)

The simplified version of (2-44) is the case without the geometric source term, namely:

) 0t x+ =U F(U (2-48)

These equations are useful for solving shock-tube type problems. Further, under suitable physical assumptions they produce even simpler mathematical models. In all the submodels studied so far we have assumed some thermodynamic closure condition given by equations of state.

2.4.2 - Cylindrical and Spherical Symmetry

Cylindrical and spherical symmetric wave motion arises naturally in the theory of explosion waves in air and other compressible media. In these situations: a) the microscopic transport properties may be neglected due to the very fast physical processes and b) the multidimensional Euler equation may be reduced to essentially one-dimensional equations with a geometrical source term vector S(U) to account for the second and third spatial dimensions. One can write therefore:

( ) )t r+ =U F U S(U (2-49)

where

2 2, ,( ) ( )

u uu u p u

rE u E p u E p

= = + = + +

U F S

(2-50)

Here r is the radial distance from the origin and u is the radial velocity. When 0= one deals with plane one-dimensional flow. When 1= one deals with a simplified cylindrically symmetric flow, which is an approximation to two-dimensional axy-symmetric equations when no axial variations are present )0( =v . For 2= we have spherically symmetric flow, an approximation to three-dimensional flow. Approximations of this kind can easily be solved numerically to a high degree of accuracy by a good one-dimensional numerical method. These accurate one-dimensional solutions can then be very useful in partially validating two and three dimensional numerical solutions of the full models.

2.4.3 - Axi-Symmetric and Plane (2D) Flows

Here we consider domains that are symmetric around a coordinate direction. We choose this coordinate to be the x-axis and is called the axial direction. The second coordinate is r, which measures distance from the axis of symmetry z and is called the radial direction. There are two component of velocity, namely the axial ( , , )u t x r and the radial ( , , )v t x r velocities. Then the three dimensional (homogeneous) conservation laws are approximated by a two dimensional (inhomogeneous) problem with geometric source terms S(U) , namely:

( ) ( ) ( )t r z+ + =U F U G U S U (2-51)

where

22

2, , ,

( )( ) ( )

u v uuvu uu p

v uvruv u pE u E pu E p v E p

+ = = = = + ++ +

U F G S (2-52)

When 0= one deals with plane, two-dimensional flow. When 1= one deals with an axy-symmetric flow.

2.5 - Mathematical properties of the Euler equations

In this section we apply some mathematical tools to find the basic properties of the time-dependent Euler equations. As seen in previous sections, the Euler equations result from neglecting the effects of viscosity, heat conduction and body forces on a compressible medium. Here we show that these equations form a system of hyperbolic conservations laws. In particular, we study those properties that are essential for the numerical solution of the equations: the eigenstructure of the equations (that is, the eigenvalues and eigenvectors), the properties of the characteristic fields and the basic relations across rarefactions, contacts and shock waves and finally the solution of the Riemann problem.

We assume in what follows that the working gas is thermally and calorically perfect. This only simplifies the analysis but do not represent a particular case. The use of more complicated state equations only changes the way the sound speed is determined but does not change the conclusions obtained within the perfect gas assumption [1, 2, 4, 5].

2.5.1 - The 1D Euler equations

The conservative differential form of the 1D Euler equations was derived before and is:

0t x+ =U F(U) , (2-53)

where U and F(U) are the vectors of conserved variables and fluxes, given respectively by:

1 12

2 2

3 3

,( )

u f uu u f u pu E f u E p

= = + +

U F (2-54)

Here is density , p is the pressure, u is particle velocity and E is total energy per unit volume,

)21( 2 euE += , where e is the specific internal energy given by the caloric equation of state

),( pee = . For ideal gases one has:

)1(),(

==

ppee , (2-55)

with vp cc= denoting the radio of specific heats. From (2-55) and using equation (2-40) one

has the speed of sound a as:

peepa p == )(

2 . (2-56)

The 1D conservation laws may also be written in quasi-linear form:

0t x+ =U A(U)U , (2-57)

where the coefficient matrix A(U) is the Jacobian matrix:

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

def f u f u f uF f u f u f uU

f u f u f u

= =

A(U) .

(2-58)

After some manipulations, one obtains using (2-58) the Jacobian A in the form:

( ) ( )

2

2 2

1 1

3 2

2 3 32 2 221 1 1 1 1

0 1 0

1 ( 3) (3 ) 12

31 12

u uu u

u u uu u uu u u u u

= +

A(U) , (2-59)

or using the speed of sound and the fluid velocity:

( )

2

2 23 2

0 1 01 ( 3) (3 ) 12

1 3 222 1 2 1

u u

a u au u u

=

+

A(U) . (2-60)

The Euler equations with the ideal-gas satisfy the homogeneity property:

( ) =F U A(U)U . (2-61)

The homogeneity property can be proof by direct calculations and is useful in deriving the family of numerical schemes called Flux Vector Splitting, [1, 2, 4, 5].

In what is concerning the eigenstructure of the 1D Euler equations, the eigenvalues of the Jacobian matrix A are:

1

2

3

,,

.

u auu a

= == +

. (2-62)

The corresponding right eigenvectors are, using the total enthalpy ( )H E p = + :

(1) (2) (3)

2

1 1 1, ,

12

u a u u aH ua H uau

= = = + +

R R R . (2-63)

The eigenvalues are all real and the eigenvectors (1) (2) (3), ,R R R form a complete set of linearly independent eigenvectors. Following the theory of systems of first order partial differential equations, the time-dependent, one-dimensional Euler equations for ideal gases are strictly hyperbolic, because the eigenvalues are all real and distinct. It is worth to remark that hyperbolicity remains a property of the Euler equations for more general equations of state.

The Euler equations in general and the 1D Euler equations in particular may be written in terms of variables other than the conserved variables used in (2-53) and (2-54). These new formulations are called non-conservative formulations and one can notice that for smooth solutions all formulations are equivalent. For theoretical developments the non-conservative formulations have some advantages over their conservative counterpart. For the one-

dimensional case one possibility is to choose a vector ( ), , Tu p=V of primitive or physical variables, with p given by the equation of state. Expanding derivatives in the mass equation, we obtain:

0t x =V + A(V)V , (2-64)

where

2

0, 0 1

0

uu up a u

= =

V A(V) . (2-65)

The change of variable does not change the (hyperbolic) nature of the 1D Euler equations. Thus the matrix A(V) has three real eigenvalues and three linearly independent eigenvectors, that is:

1

2

3

,,

,

u auu a

= == +

(2-66)

and

(1) (2) (3)

2 2

1 1 1, 0 ,

0a aa a

= = =

K K K . (2-67)

By changing the variables, the eigenvalues remain the same but the eigenvectors have other structure. The left eigenvectors of the matrix A(V) are:

( )( )( )

(1)1 1

(2) 22 2

(3)3 3

0, , ,

, 0, ,

0, , .

a

a

a

= =

=

L

L

L

(2-68)

where 321 ,, are scaling factors or normalisation parameters.

For 1 2 3, 1,2 2a a = = = (such that L(i)K(i) = 1, a normalization criteria) one

obtains:

( )( )( )

(1) 2

(2) 2

(3) 2

0, 2 , 1 2 ,

1, 0, 1 ,

0, 2 , 1 2 .

L a a

L a

L a a

= =

=

(2-69)

The theory of systems of first order partial differential equations uses the concepts of characteristics and characteristics equations, respectively. Thus, the characteristic directions in the space-time domain are defined by:

3,2,1for == idtdx i , (2-70)

and the corresponding characteristic equations are defined by:

( ) 0, along , for 1, 2,3i id dx dt i = = =L V . (2-71)

In the case of the Euler equations, one obtains the following three characteristic equations (in fact, they are differential relations that hold true along characteristic directions):

12

2

3

0 along dx dt ,

0 along dx dt ,0 along dx dt

dp adu u adp a du udp adu u a

= = = = = = + = = = +

. (2-72)

The characteristic directions and equations are useful and necessary in the numerical solution of the Euler and Navier-Stokes equations, being used for the correct implementation of the boundary conditions as well as for the solution of local Riemann problems.

The Riemann problem for the 1D, time dependent Euler equations (2-53) with initial data ( )RL UU , is the Initial Value Problem (IVP) defined by the governing equations and the constant initial conditions:

(0)

0, if 0,

( 0, ) ( ) if 0.

t x

L

R

xx

t x xx

+ = < <

U F(U)U

U UU

,

(2-73)

The physical analogue of the Riemann problem is the shock-tube problem, in which the left and right side velocities are zero. The Riemann problem is a key problem in the numerical solution of compressible flows, [1, 2, 5]. Its exact solution is complicated, but a linearized one can be obtained and solved using acceptable approximations, [1, 2]. This is key problem in the so-called Flux Difference Splitting family of solvers used in numerical solution of compressible flows.

Remark on the entropy. The entropy is defined by:

0 lnVps s c

=

,

(2-74)

where s0 is a constant. It can be shown that along the characteristic line 2dx dt u= = the

entropy is constant, i.e. 0t xs u s+ = so long as the flow remains smooth.

2.5.2 - Two-Dimensional Euler Equations

The two-dimensional version of the Euler equations in differential conservative form is:

( ) ( ) 0t x y+ + =U F U G U , (2-75)

with

( ) ( )

2

2, ,

u vu p uvu

uv v pvu E p v E pE

+ = = = + + +

U F G . (2-76)

The Jacobian matrix ( )A U corresponding to the flux ( )F U is given by (the Jacobian of the other flux may be calculated similarly):

( )( ) ( ) ( )

( ) ( ) ( )

2 2

2 2

0 1 0 01 1 3 1 12

01 1 1 12

u V u vd

uv v ud

u V H H u uv u

+ = =

FA UU

. (2-77)

The eigenvalues of A and the corresponding right eigenvectors are:

1

2 3

4

,,

u au

u a

= = == +

(2-78)

and

( ) ( ) ( ) ( )1 2 3 4

2

11 0 10

, , ,1

12

uu a u avv v

H au v H uaV

+ = = = = +

R R R R (2-79)

Thus, the Jacobians of the fluxes have real eigenvalues and linearly independent eigenvectors. In contradistinction with the 1D case, one of the eigenvalue is multiple.

The Jacobian matrix ( )A U is diagonalizable, because there is a similar transform so that it can be written as:

( ) ( ) ( ) ( )1= A U R U U R U , (2-80)

where ( )R U is the non-singular matrix the columns of which are the right eigenvectors ( ) ( ) ( )1,i R U R U is its inverse and ( ) U is the diagonal matrix with the eigenvalues ( )i U

given by (2-78) as the diagonal entries.

Remark on the Rotation Invariance. The fluxes and the vector of conservative variable given in (2-80) satisfy an important property, called the rotational invariance. Thus, given an

unit vector ( )cos ,sin =n normal to a line, where is the angle formed by x-axis and n, 20 , the normal flux ( )nF U to the line is:

( ) ( ) ( ), cos sinn = = + F F G n F U G U , (2-81)

The rotational invariance means that the 2D Eulers equations satisfy for all angles and vectors U the property:

( ) ( ) ( )cos sinn = + = -1F F U G U T F TU . (2-82)

The matrix ( )T = T is the rotation matrix and ( )-1T is its inverse:

1

1 0 0 0 1 0 0 00 cos sin 0 0 cos sin 0

, .0 sin cos 0 0 sin cos 00 0 0 1 0 0 0 1

= =

T T (2-83)

The rotational invariance property of the two-dimensional time dependent Euler equations is fundamental in numerical calculation but also allows us to proof that the equations are hyperbolic in time.

By definition, the 2D time dependent Euler system (2-75) is hyperbolic in time if for all

admissible states U and angles the Jacobian of the normal flux ( ), cos sinn = + A U A B is diagonalizable ( A and B are respectively the Jacobian matrices of the fluxes ( )F U and

( )G U ), [1, 2, 5]. That is, there exist a diagonal matrix ( ), U and a non-singular matrix ( ),R U such that:

( ) ( ) ( ) ( )1, , , ,n = A U R U U R U . (2-84)

where ( )R U is the non-singular matrix the columns of which are the right eigenvectors ( ) ( ) ( )1,i R U R U is its inverse and ( ) U is the diagonal matrix with the eigenvalues ( )i U

given by (2-78) as the diagonal entries. By taking the derivative with respect to U of the normal flux given by (2-82) one obtains:

( )( ) ( )

cos sinndd

= + = =-1 -1F TU

A A B T T T A TU TTU

. (2-85)

Further, using (2-80) it results that the equality (2-88) may be obtained:

( ) ( ) ( )( ) ( )( ) ( ) ( )( )1 1n = =-1 -1A T R TU TU R TU T T R TU TU R TU T , (2-86)

provided that:

( ) ( )( )( ) ( )( )

, ,

, .

=

=

-1R U T R T U

U T U

(2-87)

Thus, by direct matrix manipulations we have proofed that the condition (2-84) holds and therefore the two dimensional Euler equations are hyperbolic time, [1. 2].

2.5.3 - The 3D Euler Equations

The previous results proved for the one- and two-dimensional Euler equations can be extended to the time-dependent three dimensional Euler equations, (2-26) and (2-27). One may recall them here:

( ) ( ) ( ) ( )t x y z+ + + =U F U G U H U S U ,

where:

2

2

2

, , ,

( ) ( ) ( ).

wu vuwu u p uvvwv uv v p

w puw vw w pE u E p v E p w E p

+ = = = =+

+ + + +

U F G H .

The Jacobian matrix A corresponding to the flux ( )F U is given by:

( )

( )

2 2

2 2

0 1 0 0 0 3

0 00 0

1 32

H u a u v wuv v uuw w u

u H a H u uv uw u

= =

FAU

, (2-88)

where ( ) ( )2

2 2 2 2 21 , , 12 1

aH E p V V u v w

= + = + = + + =

. This matrix has five real

eigenvalues:

1

2 3 4

5

,,

u au

u a

= = = == +

(2-89)

The matrix of corresponding right eigenvectors is

2

1 1 0 0 10 01 00 1

12

u a u u av v vw w w

H ua V v w H ua

+

=

+

R (2-90)

while its inverse is

( )

( )

( )

2

2 21

2

2 2

1

42 2 2 2 2

1 2 20 0 02

2 20 0 0

1

a aH u a u v w

H a u v w

va aa

wa a

a aH u a u v w

+ +

+

= +

R (2-91)

The Jacobian matrix ( )A U given by (2-88) is diagonalizable, because there is a similar transform so that it can be written as:

( ) ( ) ( ) ( )1= A U R U U R U , (2-92)

where ( )R U is given by (2-90), ( )1R U is given by (2-91) and ( ) U is the diagonal matrix with the eigenvalues ( )i U given by (2-89) as the diagonal entries.

The time-dependent three dimensional Euler equations are also rotationally invariant, that is for the normal flux through a surface they satisfy, for all angles ( ) ( )zy , and vectors U, the equality:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )cos cos cos sin siny z y z yn = + + = -1F F U G U H U T F TU . (2-93)

Here the rotation matrix ( ) ( )( ),y z =T T is the product of two simple rotations matrices, [2]:

( ) ( )( ) ( ) ( ),y z y z = =T T T T . (2-94) where

( )

( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

1 0 0 0 0 1 0 0 0 0

0 cos 0 sin 0 0 cos sin 0 0,0 0 1 0 0 0 sin cos 0 0

0 sin 0 cos 0 0 0 0 1 00 0 0 0 1 0 0 0 0 1

y y z z

y z z z

y y

T T

= =

(2-95)

and

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )

=

100000cossinsincossin000cossin00sinsincoscoscos000001

yzyzy

zz

zyzy yT

(2-96)

Similar to the two-dimensional case, the three-dimensional Euler equations are proofed to be hyperbolic in time, [1, 2].

2.5.4 - Conclusion

The Euler equations are hyperbolic in time irrespective the space dimension is. The inviscid fluxes are homogeneous and the variables and multi-dimensional fluxes satisfy the rotational invariance condition. These mathematical properties are useful for the development of the numerical algorithms dedicated to the solution of the Euler equations for supersonic/hypersonic flows.

3 - REFERENCES

1. Danaila, S., Berbente, C., Metode Numerice in Dinamica Fluidelor, Editura Academiei Romane, Bucuresti , 2003.

2. Toro, E.F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin, Heidelberg, 1997.

3. Anderson, J.D., Jr., Hypersonic and High Temperature Gas Dynamics, McGraw-Hill Series in Aeronautical and Aerospace Engineering, 1998.

4. Weiland C. and all, Numerical Methods for Aerodynamic Design I, Space Course 1991, RWTH Aachen, 1991.

5. Feistauer, M., Felcman, C., Straskraba, I., Mathematical and Computational Methods for Compressible Flow, Clarendon Press, Oxford, 2003.

6. Iannelli, J., Characteristics Finite Element Methods in Computational Fluid Dynamics, Computational Fluid and Solid Mechanics Series, Springer, 2006.

7. Masatsuka, K., I do like CFD, Vol. 1, Yorktown, 2009. 8. Park, C., Hypersonic Aerothermodynamics: Past, Present and Future, Intl J. of Aeronautical

and Space Sci. 14(1), pp. 1-10, 2013. 9. Barbante, P.F., Magin, T.E., Fundamentals of hypersonic flight Properties of high

temperature gases, RTO-EN-AVT-116, Paper presented at RTO AVT Lecture Series on Critical Technologies for Hypersonic Vehicle Development, held at the von Krmn Institute, Rhode-St-Gense, Belgium, 10-14 May, 2004.

10. Srinivasan, S., Tannehill, J.C. and Weilmuenster, K.J. Simplified Curve Fits for the Thermodynamic Properties of Equilibrum Air, by S., ISU-ERI-Ames-86401, Iowa State University, 1986.

1 - introduction2 - Governing Equations of Gas Dynamics2.1 - Introduction2.1.1 - The mass conservation law2.1.2 - Conservation of Momentum2.1.3 - Conservation of Energy2.1.4 - The Integral and Differential Form of the Governing Equations

2.2 - The Euler Equations2.3 - Thermodynamic Relations2.3.1 - Equilibrium Real Gas2.3.2 - The specific heat capacities and the speed of sound2.3.3 - Non-Equilibrium Real Gas

2.4 - Simplified Euler equations2.4.1 - 1D Compressible Flows with Area Variation2.4.2 - Cylindrical and Spherical Symmetry2.4.3 - Axi-Symmetric and Plane (2D) Flows

2.5 - Mathematical properties of the Euler equations2.5.1 - The 1D Euler equations2.5.2 - Two-Dimensional Euler Equations2.5.3 - The 3D Euler Equations2.5.4 - Conclusion

3 - References