gasdynamics riemmangf

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-1

Dicembre 21, 2011

A-1

Appendix Q

Riemann problem of gasdynamics Introduction This appendix is devoted to the study of the Riemann problem of gasdynamics. This is an initial value problem for the Euler equations governing compressible flows in one dimension under a very particular initial condition that consists in a jump in the variables between two states, with a uniform distribution on the left of the discontinuity and another still uniform but possibly different distribution on the right, to the infinity. The solution to the Riemann problem depends on the variables of the left and right states and is always similar, namely the values of the all variables of the solution are constant on any ray issuing from the initial position of the jump. Moreover, the solution consists in general in three distinct waves emerging from the disintegration of the initial discontinuity and which propagate with definite speeds. The study of the solutions to Riemann problems is therefore very important for understanding the propagation of nonlinear waves in compressible flows. There exists however another very important reason for studying the Riemann problem and for developing methods suitable for its solution, for arbitrary initial values: the modern numerical methods for solving the the Euler equations in transonic, supersonic and hypersonic flows are based on the solution of Riemann problems. In fact, after the computational domain has been subdivided into nonoverlapping elementary cells, called finite volumes, the equations are integrated spatially over each cell. One obtains a discrete form of the equations that is ready for the time integration by solving at each time level all the Riemann problems for all interfaces between pairs of adjacent cells. The techniques for the numerical simulation of compressible flows in the aforementioned regimes employ therefore algorithms relying upon the exact or approximate solution of a very large number of different Riemann problems. In these pages the Riemann problem is tackled considering initially a gas endowed with arbitrary thermodynamic properties. The solution to this problem in a general setting is represented by a system of two nonlinear equations in two unknowns, as originally featured by the first author of this text and Alberto Guardone. This original formulation of the Riemann problem is then particularized to the simple situation of the ideal gas, first polytropic, which is very relevant for the applications, and second nonpolytropic. In the last section the formulation is extended to the van der Waals gas model, both polytropic and nonpolytropic.

Q.1 Eigenvalues and eigenvectors of Euler equations Derivation of the equation governing the specific volume v ∂ρ ∂t

=

∂ ∂t

¡1¢ v

¡1¢

= − v12

∂v ∂t

∇ρ = ∇ v = − 12 ∇v v

− v12

∂ ∂t

¡1¢ v

−

1 u · ∇v v2

+ v1 ∇· u = 0

∂v + u · ∇v − v ∇· u = 0 ∂t

To formulate the Riemann problem of gasdynamics it is convenient to write the Euler equations in quasilinear form choosing as variables the specific volume v (instead of the density ρ), the velocity u of the fluid and the specific entropy s (instead of its internal energy e). Since v = 1/ρ, it is immediate to transform the mass conservation equation ∂ρ/∂t + ∇· (ρu) = 0 into an equation for variable v: ∂v + u ·∇v = v ∇· u. ∂t Moreover, let us recall that, in flows without shock and discontinuity, the balance equation for the internal energy, whenever the viscosity and heat conductivity of the fluid can be considered negligible, is equivalent to the transport equation for entropy s: ∂s + u ·∇s = 0. ∂t

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-2

A-2

APPENDIX Q:

Dicembre 21, 2011

RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

As a consequence the system of Euler equations for one-dimensional flows can be written as ∂v ∂v ∂u +u −v = 0, ∂t ∂x ∂x ¶ ¶ µ µ ∂s ∂u ∂ P ∂v ∂P ∂u +u +v +v = 0, ∂t ∂x ∂v s ∂ x ∂s v ∂ x ∂s ∂s +u = 0, ∂t ∂x where P = P(s, v) represents an equation of state of the fluid. The choice of the thermodynamic variables v and s in place of ρ and e is not mandatory, but allows it to obtain some results more immediately. By introducing the vector of the unknowns and the matrix     u −v 0 v µ ¶  µ∂ P ¶   ∂P   v   u v and A(w) =  w = u   ∂v s ∂s v   s 0 0 u the considered nonlinear hyperbolic system can be written compactly ∂w ∂w + A(w) = 0. ∂t ∂x

Eigenvalue problem and characteristic speeds As seen in section 9.10, the eigenvalues of matrix A(w) of the hyperbolic system provide the propagation speeds of the informations within the fluid. The eigenvalues λ of a matrix A are obtained by solving the characteristic equation |A − λI| = 0. In the present fluid dynamic case, the matrix of the hyperbolic system depends on the vector w and thus also its eigenvalues and eigenvectors will be function of w. This dependence is fundamental for ascertain the different physical characteristics of the waves that can propagate in the considered fluid. The characteristic equation of our eigenvalue problems will be written in the following way ¯ ¯ ¯ ¯ u−λ −v 0 ¯ µ µ ¶ ¶ ¯¯ ¯ ∂P ¯ ∂P ¯ u−λ v |A(w) − λ(w) I| = ¯ v ¯ = 0. ∂v s ∂s v ¯ ¯ ¯ ¯ ¯ 0 0 u−λ ¯ By developing the determinant along the elements of the last row we obtain ¯ u−λ ¯ ¶ ¯ µ (u − λ)¯ ∂P ¯v ∂v s

−v ¯¯ ¯ ¯ = 0, u − λ¯

and by calculating the 2 × 2 determinant we have µ · ¶¸ ∂P (u − λ) (u − λ)2 + v 2 = 0. ∂v s By introducing now the sound speed sµ s µ ¶ ¶ ∂P ∂P c≡ =v − , ∂ρ s ∂v s

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-3

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.2: Genuine nonlinearity and linear degeneracy

A-3

the characteristic equation assumes the following form £ ¤ (u − λ) (u − λ)2 − c2 = 0,

and the eigenvalues are the solutions of the two equations u−λ=0

and

(u − λ)2 − c2 = 0.

The three eigenvalues, ordered increasingly, are therefore λ1 = u − c,

λ2 = u,

λ3 = u + c.

When the dependence on the fluid state w is indicated explicitly, we have the more precise writing λ1 (w) = u − c(s, v),

λ2 (w) = u,

λ3 (w) = u + c(s, v).

The comparison with the results obtained in section 9.10 shows that the eigenvalues of the hyperbolic system do not depend on the particular choice of the variables, as for instanc v, u and s, in the present calculation. To clarify how the nonlinear character of the hyperbolic system is implemented in the different eigenvalues it is necessary to determine the eigenvectors associated to the eigenvalues. To do that it is sufficient to replace the eigenvalue λi (w) into the matrix |A(w) − λ(w) I| and then solve the resulting linear system. Let us consider the first eigenvalue λ1 = u − c(s, v); the substitution provides the following (singular) homogeneous linear system    V1 c(s, v) −v 0 ¡ ¢    ¡∂P ¢  v ∂v s c(s, v) v ∂∂sP v   U1  = 0,

0 0 c(s, v) S1 in the three unknowns V1 , U1 and S1 , where c(s, v) is the sound speed. Thanks to the third equation S1 = 0 and the system reduces to ¶µ µ ¶ V1 c(s, v) −v = 0. −c2 (s, v)/v c(s, v) U1 The two equations are one and the same equation, c(s, v) V1 − vU1 = 0, so that the eigenvector is defined up to a constant factor and, without any loss of generality, the first eigenvector can be chosen as (V1 , U1 , S1 ) = (v, c(s, v), 0). In similar way the eigenvector associated to the third eigenvalue λ3 = u + c(s, v) can be taken as (V3 , U3 , S3 ) = (v, −c(s, v), 0). The eigenvector associated to the intermediate eigenvalue λ2 = u is obtained by means of the system    V2 0 −v 0 ¡ ¢    ¡∂P ¢  v ∂v s 0 v ∂∂sP v   U2  = 0. 0 0 0 S2 The first equation gives U2 = 0 and the system reduces to à !à ! 0 0 V2 = 0. ¡∂P ¢ ¡∂P ¢ v ∂v s v ∂s v S2

The first equation is satisfied trivially while the second requires that ¡∂P ¢ ¡∂P ¢ ∂v s V2 + ∂s v S2 = 0.

w3

w ri (w) w1 Figure Q.1

w2

Local eigenvector ri (w) at w

¡ ¢¢ ¡ ¡ ¢ Therefore, the second eigenvector can be taken in the form (V2 , U2 , S2 ) = − ∂∂sP v , 0, ∂∂vP s . Written all together, the three eigenvectors are  ¡∂P ¢      − ∂s v v v       r1 (w) =  c(s, v)  , r2 (w) =  0  , r3 (w) =  −c(s, v)  . ¡∂P ¢ 0 0 ∂v s

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-4

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APPENDIX Q:

Dicembre 21, 2011

RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

Q.2 Genuine nonlinearity and linear degeneracy w2

∇w λ(w)

λ = costante

w1

Contour curves of the eigenvalue λ and vector field of its gradient ∇w λ

Figure Q.2

Each of the eigenvectors ri (w) represents a vector field in the space of state vectors w = (v, u, s). Each of the eigenvalues λi (w) is instead a scalar field in the same threedimensional space. The gradient of any eigenvalue is another vector field. ∇w λ(w), always in the same space, with ∇w denoting the gradient operator in the space of vectors w. In the case of the Euler equations of gasdynamics the vector fields of the three eigenvalues are  ¡ ∂c ¢     ¡ ∂c ¢  − ∂v s 0 ∂v s       ∇w λ1 (w) =  1  , ∇w λ2 (w) =  1  , ∇w λ3 (w) =  1  . ¡ ¢ ¡ ∂c ¢ − ∂c 0 ∂s v ∂s v

The nonlinear or linear nature of the eigenvalues depends on a well defined geometrical relation between the vector field of a considered eigenvector and the vector field of the gradient of the corresponding eigenvalue. Consider now the following scalar product in the vector space of w beween the vector filed of an eigenvector and the gradient field of the corresponding eigenvalue: r(w) · ∇w λ(w).

The expression linearly degenerate is confusing because the attribute “degenerate” is employed normally to denote a multiple eigenvalue, namley, when two or more eigenvalues are coincident. Actually, “linearly degenerate” means that the eigenvalue is simply linear. This nomenclature comes from a very important theorem due to Boillat that states that every degenerate eigenvalue, i.e. multiple, of a hyperbolic system is also necessarily linear—degeneray implies linearity.

The vanishing of the scalar product in a point w means the orthogonality of the two vector fields in this point. When this scalar product never vanishes for any w, then the eigenvalue is said to be genuinely nonlinear. In this case the sign of the scalar product does not change for any state w allowed. On the contrary, when the scalar product is zero for all w, the eigenvalue is said linearly degenerate. For the Euler equations of gasdynamics, we see immediately that the first and third eigenvalues are such that ¡ ∂c ¢ ¡ ∂c ¢ r1 (w) · ∇w λ1 (w) = −v ∂v + c, r3 (w) · ∇w λ3 (w) = v ∂v − c. s s

Therefore, these eigenvalues will¡ be ¢genuinely nonlinear or will be not depending on ∂c . It is immediate to obt the value of the expression c − v ∂v s c(s, v) − v

∂c(s, v) ∂c(s, ρ) ∂[ρ c(s, ρ)] = c(s, ρ) + ρ = = Γ c, ∂v ∂ρ ∂ρ

where in the last passage one has introduced the dimensionless quantity Γ ≡

1 ∂[ρ c(s, ρ)] , c ∂ρ

called fundamental derivative of gasdynamics. The scalar products involving the first and third eigenmodes can be written as r1 (w) · ∇w λ1 (w) = Γ c

w3

w

∇w λi (w) ri (w)

w1

w2

Genuine nonlinearity: ri (w) · ∇w λi (w) 6 = 0, ∀w

Figure Q.3

and

r3 (w) · ∇w λ3 (w) = −Γ c.

As the eigenvalues, also the properties of genuine nonlinearity or of linear degeneracy are intrinsic properties of the hyperbolic system, which are invariant with respect to any change of variables used to represent the equation system. In particular, the genuine nonlinearity depends only on the vanishing of the scalar products above, and the latter is controlled by the function Γ , which depends only on the thermodynamic properties of the considered fluid. In the following it will be always assumed that Γ 6= 0 in the whole domain of permitted values of the thermodynamical variables. This condition is always satisfied by the polytropic ideal gas for which it is easily found Γ = 12 (γ + 1). Also for a van der Waals gas the condition is satisfied, except for molecular gases with more than 7 atoms and only in a very small region of thermodynamical conditions near the critical point. The loss of genuine nonlinearity will be anyhow left outside the present analisys.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-5

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.3: Generality of the Riemann problem

It can be noted that the vanishing of the expression c − v

by an examination of the second derivative

∂ 2 P(s,v) . ∂v 2

¡ ∂c ¢

∂v s

A-5

can be established also

In fact, a direct calculation shows

· ¸ ∂ 2 P(s, v) 2 ∂ P(s, ρ) 2 ∂ −ρ = −ρ ∂ρ ∂ρ ∂v 2 ¸ · 2 ∂ P(s, ρ) 2 ∂ P(s, ρ) 2 +ρ = ρ 2ρ ∂ρ ∂ρ 2 · ¸ ∂ 2 P(s, ρ) ∂ P(s, ρ) 3 +ρ =ρ 2 ∂ρ ∂ρ 2 · µ ¶¸ · µ 2¶ ¸ ∂c ∂c 3 2 3 = 2ρ c c + ρ = ρ 2c + ρ ∂ρ s ∂ρ s ¸ · 2c ∂c(s, v) = 3 c−v , ∂v v where one can notice the slight abuse of mathematical notation of denoting the two functions P(s, v) and P(s, ρ) by the same symbol, and similalrly for c(s, ρ) and 2 is nothing but the third c(s, v). It is worth observing that the second derivative ∂ P(s,v) ∂v 2 derivative of the fundamental relation ∂ 2 P(s, v) ∂2 = ∂v 2 ∂v 2 w3

∇w λ2 (w) w

¶ µ ∂e(s, v) = −evvv (s, v), − ∂v

so that the loss of genuine nonlinearity is strictly related to the thermodynamical properties of the gas under examination, but it cannot be assured by its thermodynamic 2 (s, v) > 0, and stability, which requires ess (s, v) > 0 and ess (s, v) evv (s, v) − esv therefore depends only on the second order derivatives. Let us consider finally the intermediate eigenvalue. It is easy to show that

r2 (w) w2

w1

Linear degeneracy: r2 (w) · ∇w λ2 (w) = 0, ∀w

Figure Q.4

¡ ¢ ¡ ¢ r2 (w) · ∇w λ2 (w) = − ∂∂sP v × 0 + 0 × 1 + ∂∂vP s × 0 = 0,

so that the second eigenvalue is always linearly degenerate irrespective of the thermodynamic properties of the fluid. This result shows that the Euler equations are a hyperbolic system with a hybrid character, simultaneously linear and nonlinear. This a structural property of the equations of gasdynamics in one dimension which has an even more fundamental counterpart for the multidimensional equations. In the one-dimensional case, the presence of the linearly degenerate intermediate eigenvalue with its own mathematical characteristics allows one to formulate the equations defining the Riemann problem and to solve them in a quite natural way, as it will be shown below.

Q.3 Generality of the Riemann problem As anticipated in the introduction, the Riemann problem of gasdynamics is an initial value problem for the system of the one-dimensional Euler equations supplemented by a quite particular initial datum: the variables of the system have a jump at one point, normally chosen as x = 0, and are for the rest uniform on the left and on the right of the discontinuity. Since no spatial scale exists in mathematical statement of the problem, the solution must be similar, namely all variables will have a constant value along any ray of the space-time half-plane, going out from the initial discontinuity. The solution to the Riemann problem will depend on the values of the variables in the left and right states.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-6

A-6

APPENDIX Q:

RIEMANN PROBLEM OF GASDYNAMICS

Dicembre 21, 2011

ISBN XX-abc-defg-h

At a first sight, one could think that the fluid motion resulting from the initial discontinuity should its mere propagation with some definite speed. In fact, from the theory of the contact discontinuity given in section 11.2, we know that, for the particular jump in which fluid velocity and pressure do not vary across the jump while the density is discontinuous, the solution to the evolutionary Euler equations is a contact discontinuity translating with the same uniform velocity of the gas. However, we are interested to the Riemann problem with arbitrary initial values for the three variables on both sides of the jump, while in the particular case just mentioned the initial values of two variables were fixed by other initial data. A similar situation would be met also if the data on the left and right of the discontinuity would be chosen so as to satisfy the Rankine–Hugoniot jump conditions for the normal shock discussed in section 11.3, or more precisely their version written in a frame in which the shock wave is moving, to avoid the restriction of a stationary shock. In the arbitrary frame, the solution to the Riemann problem would be a single shock wave propagating with a fixed speed. Also in this case all the initial values would not be arbitrarily independent, but constrained by two relationships. Finally, a third situation with the initial values of the variables across the jump are subjected still to two conditions is that of the instationary rarefaction waves studied in section 11.5. In this case the specific entropy of the fluid is one and the same on the left and on the right and a further constraint must be respected in order that the solution of the problem can be actually a single rarefaction wave. In all these special cases examined, the existence of two constraints on the initial data of the Riemann problem implies that the solution is simply a single wave—contact discontinuity, shock wave or rarefaction wave. When a problem with generic data is examined, with all data specified in a completely arbitrary and independent way, then the solution will be more complicated and will include in general more than only a single wave. When only one of the two constraints is removed, the solution will comprise two waves and finally when both constraints are removed the solution will consist generally of three waves. In other words, the solution of the Riemann problem of gasdynamics consists normally of three distinct waves, althuogh each of them can be of zero intensity and therefore may be absent in the solution in very special cases. By recalling now the shock tube problem and its analytical solution discussed in section 11.6, we have found that the initial discontinuity disintegrates in three waves. The intermediate wave is a contact discontinuity propagating with the local velocity u of the fluid, which coincides with the intermediate eigenvalue (linearly degenerate) of the hyperbolic system. The other two waves propagate with speeds which are in relation with the characteristic speeds u − c and u + c, representing the two extremal eigenvalues: in the shock tube examined, the waves are a rarefaction wave propagating into the gas on the left and a shock wave propagating toward the right. Of course, by inverting the values of the initial data, namely with an overpressure in the gas on the right region with respect to the gas on the left region, the shock wave would propagate in the left direction and the rarefaction in the right direction. This kind of solution, consisting in a rarefaction wave together and a shock wave, is frequent but is not the only possibility when three waves are present. In fact, taking for granted that the intermediate wave is always a contact discontinuity, the solution of the Riemann problem can contain also two shock waves or two rarefaction waves. These solutions may occur provided the values of the two initial velocities of the gas are such to imply respectively the collision (when u ℓ > 0 and u r < 0) or the detachment (if u ℓ < 0 and u r > 0) of the fluid portions on the two sides of the jump. In particular, in the last case of formation of two rarefaction waves it is possible to reach also the limit situation of an extreme rarefaction with the formation of a vacuum zone between the tails of the waves. In conclusion, the solution of the Riemann problem will be composed of three waves, with always a contact discontinuity as the middle one while the other two are indifferently a rarefaction or shock wave. Each of the wave might be of zero intensity. Finally, when both external waves are rarefactions, it might be occur the formation of a vacuum region between the two parts of the gas receding from each other.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-7

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.4:

Contact discontinuity

A-7

Geometrical interpretation As it will be shown later, the initial values of the Riemann problem allow one to discover the type of waves of the solution without having to solve the problem explicitly. In the other side, the complete solution of the Riemann problem can be dtermined without knowing in davance whether the first and third waves are a rarefaction fan or a shock, but their kind is discovered at the end of the solution procedure. We will follow the second approach and adopt a geometrical interpreation of the Riemann problem that exploits the structural properties of the intermediate wave which, as already said, is in any case a contact discontinuity. Let us consider the three-dimensional space of the variables (v, P, u) and express the initial data of the problem in this space, so that the two points (vℓ , Pℓ , u ℓ ) and (vr , Pr , u r ) represent the initial data. The contact discontinuity separates two states that can be indicated by (vℓ⋆ , P ⋆ , u ⋆ ) and (vr⋆ , P ⋆ , u ⋆ ), since velocity and pressure are equal on the two sides and only the specific volume is discontinuous. The solution of the Riemann problem consist therefore in determing the values vℓ⋆ , vr⋆ P ⋆ and u ⋆ and in the same time finding the type of the waves—shock or rarefaction—generated at the left and right of the contact discontinuity. We can formulate the problem as the search in the space of the variables (v, P, u) of the two points (vℓ⋆ , P ⋆ , u ⋆ ) and (vr⋆ , P ⋆ , u ⋆ ) corresponding to the two intermediate states of the gas beween the initial states (vℓ , Pℓ , u ℓ ) and (vr , Pr , u r ). The first intermediate state (vℓ⋆ , P ⋆ , u ⋆ ), to the left of the contact discontinuity, will be connected to the left initial state (vℓ , Pℓ , u ℓ ) by a shock wave or a rarefaction wave; similarly, the second intermediate state (vr⋆ , P ⋆ , u ⋆ ), to the right of the contact discontinuity, will be connected to the right initial state (vr , Pr , u r ) by a shock wave or a rarefaction wave. The type of the left wave and of the right wave depends on the unknown values vℓ⋆ , vr⋆ , P ⋆ and u ⋆ , and therefore they will emerge at the end of the solution procedure. On the other hand, the choice of the wave type that occurs on the left and on the right will depend on the fact that the wave can be a shock only when the entropy increases, while in the opposite case the wave will be a rarefaction fan. For gases satisfying the condition of genuine nonlinearity and such that Γ > 0, as the polytropic ideal gas, the shock wave is necessarily compressive with the gas density increasing, while the rarefaction fan is always rarefactive, with the density decreasing. Thus, it is possible to write the functions incorporating this selection by comparing the value of the specific volume variable v with the known values of vℓ and of vr .

Q.4 Contact discontinuity Let us now find the characteristics of the contact discontinuity. Consider a wave propagating with the speed corresponding to the intermediate eigenvalue os the hyperbolic system of Euler equations and determine how the variables change along a wave of this type. This process consists in determining the vetor function w = w(q) that satisfies the following firts-order system of differential equations dw = α(q) r2 (w), dq where q is a parameter representing the independent variable while α(q) is an arbitrary function whose choice determines the parametrization of the solution. Due to the form of the second eigenvector r2 (w) of Euler equations, the equations of the system are written as  dv ∂ P(s, v)   = −α(q) ,   dq ∂s     du = 0,  dq     ds ∂ P(s, v)    = α(q) . dq ∂v

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-8

A-8

APPENDIX Q:

Dicembre 21, 2011

RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

Therefore the velocity u is constant inside the considered wave. Moreover, taking the ration of the first equation to the third, one one obtains dv ∂ P(s, v) =− ds ∂s

Á

∂ P(s, v) . ∂v

After introducing the definition of the function of three variables Φ(s, v, P) ≡ P(s, v) − P, the equation Φ(s, v, P) = 0 represents the implicit definition of a function of two variables v = v(s, P). The partial derivative ∂v(s, P)/∂s of this function is evaluated by means of the rule of partial differentiation for implicit functions: µ

∂v ∂s

¶

P

∂Φ(s, v, P) ∂ P(s, v) ∂s ∂s =− =− . ∂Φ(s, v, P) ∂ P(s, v) ∂v ∂v

Since the expression of the implicit partial derivative is coincident with that of the (ordinary) derivative dv/ds along the considered wave, the pressure P is constant in the wave, as already seen in section 11.2. Thus in the contact discontinuity u(q) = u ⋆ and P(q) = P ⋆ , where u ⋆ and P ⋆ denote the constant values of velocity and pressure in the contact discontinuity. Of course, the third variable v may be discontinuous and we will denote by vℓ⋆ and vr⋆ the specific volume of the gas respectively on the left and on the right of the contact discontinuity.

Q.5 Equations of the Riemann problem At this point the structural properties of the intermediate wave can be exploited to write the equations that define the Riemann problem. Let P(v; ℓ) and u 1 (v; ℓ) denote respectively the pressure and velocity of the fluid in the wave that connects the left state ℓ = (vℓ , Pℓ , u ℓ ) with a state having some specific volume v. The two function P = P(v; ℓ) and u = u 1 (v; ℓ) will be mathematically different depending on whether the wave is a shock rather that a rarefaction wave and this will depend on the fact that v will assume a value smaller (shock) or larger (rarefaction) than vℓ . Analogously, let P(v; r) and u 3 (v; r) be, respectively, the pressure and velocity in the wave that connects the right state r = (vr , Pr , u r ) with a state of specific volume v. Also the form of the two possible functions for v < vr and v > vr will be different according to whether the wave is a shock or a rarefaction. Precisely, the functions representing the pressure in the fluid will have the following form P(v; ℓ) ≡

(

P rar (v; ℓ˜) if v > vℓ P RH (v; ℓ˜) if v < vℓ

and

P(v; r) ≡

(

P rar (v; r˜ ) if v > vr

P RH (v; r˜ ) if v < vr

where the superscripts rar and RH denote, respectively, the solution of the rarefaction wave and the solution of the shock wave provided by the Rankine–Hugoniot conditions, which will be calculated below. These functions depend actually only on the thermodynamic variables of the states ℓ and r and not on the velocity values u ℓ and u r , so that they are purely thermodynamic relations. To make this point clear the vectors ℓ˜ = (vℓ , Pℓ ) and r˜ = (vr , Pr ), that depend only on (vℓ , Pℓ ) and (vr , Pr ) have been introduced.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-9

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.6: Rarefaction waves

A-9

The functions for the fluid velocity in the wave are ( rar ( rar u 1 (v; ℓ) if v > vℓ u 3 (v; r) if v > vr u 1 (v; ℓ) ≡ and u 3 (v; r) ≡ RH u 1 (v; ℓ) if v < vℓ u RH 3 (v; r) if v < vr The left and right functions for velocity are different since they depend also on the eigenvalue: for instance, the function u 1 (v; ℓ) refers to the first eigenvalue λ1 and is therefore different from the function u 3 (v; r) that refers to the third eigenvalue λ3 . Solving the Riemann problem means to find the values vℓ⋆ , vr⋆ , P ⋆ and u ⋆ that characterize the fluid conditions across the contact discontinuity. The equality of the pressure values and of the velocity values on the two sides of the contact discontinuity requires that vℓ⋆ and vr⋆ be solution of the following system of two equations (

P(vℓ⋆ ; ℓ˜) = P(vr⋆ ; r˜ ),

u 1 (vℓ⋆ ; ℓ) = u 3 (vr⋆ ; r),

which can be written in the standard form: (

⋆ ⋆ Φ(ℓ,˜ ˜ r ) (vℓ , vr ) = 0,

Ψ(ℓ,r ) (vℓ⋆ , vr⋆ ) = 0,

after introducing the two functions Φ(ℓ,˜ ˆ ≡ P(v; ℓ˜) − P(v; ˆ r˜ ), ˜ r ) (v, v) Ψ(ℓ,r ) (v, v) ˆ ≡ u 1 (v; ℓ) − u 3 (v; ˆ r). To solve this system of two nonlinear equations by means of the Newton iterative method it is necessary to evaluate the Jacobian matrix: ¡ ¢ µ ′ ¶ P (v; ℓ˜) −P ′ (v; ˆ r˜ ) ∂ Φ(ℓ,˜ ˜ r ) , Ψ(ℓ,r ) , = ∂(v, v) ˆ u ′1 (v; ℓ) −u ′3 (v; ˆ r) with the prime denoting the derivative with respct to the first variable of the functions. After the solution (vℓ⋆ , vr⋆ ) of the nonlinear system has been found, the final elements of the solution of the Riemann problem will be provided by the values P ⋆ = P(vℓ⋆ ; ℓ˜) = P(vr⋆ ; r˜ ) and u ⋆ = u 1 (vℓ⋆ ; ℓ) = u 3 (vr⋆ ; r). The existence and uniqueness of the solution to the Riemann problem of gasdynamics can be proved under the condition ∂e(P, v)/∂v > 0. In this case the Newton method will converge to the solution provided the initial guess is sufficiently near to it. In particular, it is possible and turns out to be also effective to take as initial guess the average value of the specific volume of the left and right states, namely, to take ⋆ ⋆ vℓ,init = vr,init = 12 (vℓ + vr ).

Q.6 Rarefaction waves Let us now determine the behavior of the variables in the rarefaction wave. Consider a wave that connects a given state (vi , u i , si ) of the fluid with the states belonging to an integral curve, namely a curve that is tangent in any point to the direction of the considered eigenvector, assuming that the eigenvalue is genuinely nonlinear. The integral curves w = w(q) associated with the first and third eigenvalues, denoted by λ1|3 (w), are solution of the system of ordinary differential equations dw = α(q) r1|3 (w), dq

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-10

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Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

where α(q) is an arbitrary function that determines a particular parametrization of the curve, supplemented by the initial condition w(0) = wi = (vi , u i , si ).

Since the two considered eigenvalues are genuinely nonlinear, it is r1|3 (w) · ∇λ1|3 (w) 6= 0, ∀w. In this case the parameter of the curve can be chosen in such a way that it can be coincident with the eigenvalue itself along the entire curve and will indicated by ξ , that is we let ξ = λ1|3 (w(ξ )).

Therefore, the adopted parametrization corresponds to the similarity variable met in section 11.5, in the study of the propagation of the rarefaction waves, where it has been found to coincide with the eigenvalue. The derivative of this relation with respect to ξ provides 1 = ∇λ1|3 (w) · dw/dξ = ∇λ1|3 (w) · α(ξ ) r1|3 (w) = ±α(ξ ) Γ (s, v) c(s, v) with subsequent simple implication that α(ξ ) = ±1/[Γ (s, v) c(s, v)], with Γ (s, v) denoting the fundamental derivative of gasdynamics Γ (s, v), introduced in section Q.2. Thus, the system that defines the integral curve can be written in canonical form ±r1|3 (w) dw = , dξ Γ (s, v) c(s, v) where the superior sign refers to the first eigenvalue λ1 while the inferior to the third eigenvalue λ3 . The system must be solved with the initial condition w(ξi ) = wi , where ξi = λ1|3 (wi ), so that the initial value of the independent variable is fixed by the initial datum of the system. For the special hyperbolic system of the Euler equations, thanks to the characteristics of the eigenvalues and of the vector fields of the eigenvectors of the two considered modes, the system assumes the form  dv ±v   =  dv   dξ Γ (s, v) c(s, v) ±v    =    du  dξ Γ (si , v) c(si , v) 1 = H⇒   dξ Γ (s, v) du 1     =   dξ Γ (si , v)  ds   = 0. dξ

since the third equation has the immediate solution s = constant = si , and thus the rarefaction wave is isentropic. The system reduces therefore to only two equations supplemented by the initial conditions v(ξi ) = vi and u(ξi ) = u i , with ξi = λ1|3 (wi ). Instead of calculating the solution v = v(ξ ), u = u(ξ ), we can determine initially the direct relationship between the two variables v and u by solving the differential equation that is obtained by dividing the second equation of the system by the first one, to give ±c(si , v) du = . dv v

This equation for the unknown u = u(v) is separable and is integrated immediately: u rar 1|3 (v; i)

= ui ±

Z

v

vi

c(si , v ′ ) ′ dv , v′

provided that v > vi . The pressure along the rarefaction wave, which is isentropic, is obtained immediately from the equation of state P = P(s, v): P rar (v; ˜˙i) = P(si , v),

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-11

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.6:

Rarefaction waves

A-11

where ˜˙i donotes only the thermodynamic part of the initial state. The link between the specific volume v and the originary independent variable ξ is obtained by integrating the first equation, which is still separable, by means of a simple quadrature and imposing the initial condition ξi = λ1|3 (wi ) = u i ∓ c(si , vi ) Z v Γ (si , v ′ ) c(si , v ′ ) ′ ξ = ξ(v) = u i ∓ c(si , vi ) ± dv . v′ vi By inverting the function ξ = ξ(v), the solution v = v(ξ ) is finally obtained. This function can be substituted into the second equation of the reduced system obtaining another first-order equation, which is still separable, for the unknown u = u(ξ ), and which can be solved by means of another quadrature: Z ξ dξ ′ . u = u(ξ ) = u i + ′ ξi Γ (si , v(ξ ))

Vacuum formation The expression just found of the solution of the rarefaction wave allows one to define the condition for the formation of a region of vacuum between the two rarefactions waves. This situation occurs when the two velocities u ℓ and u r of the data of the Riemann problem are such that its solution consists of two rarefaction fans, as shown in figure Q.5, and when moreover the density at the contact discontinuity goes to zero. Then, the tails of the two rarefaction waves, which are moving apart from each other, leave behind them a region without gas, namely the vacuum, as shown in figure Q.6. The condition for the formation of vacuum, expressed in terms of the specific volume, rar is equivalent to the relation u rar 1 (∞; ℓ) = u 3 (∞; r), namely Z ∞ Z ∞ c(sr , v) c(sℓ , v) dv = u r − dv. uℓ + v v vr vℓ For greater values of the initial relative velocity νr ℓ ≡ u r − u ℓ , namely when νr ℓ ≥

Z

∞ vℓ

c(sℓ , v) dv + v

Z

∞

vr

c(sr , v) dv ≡ νvacuum , v

the contact discontinuity degenerates into a vacuum region and the solution of the Riemann problem consists of two rarefaction waves with the extremes moving in opposite directions with the velocities Z ∞ Z ∞ c(sℓ , v) c(sr , v) u vacuum (ℓ ℓ) = u ℓ + dv and u vacuum (r) = u r − dv. v v vℓ vr At these extremities, the density, the temperature and the pressure of the gas are zero.

P

P Pr

Pr

Pℓ Pℓ

P⋆

uℓ

ur

uℓ x

Figure Q.5

Two rarefaction waves

ur u vacuum (ℓ ℓ)

Figure Q.6

Vacuum formation

u vacuum (r)

x

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-12

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Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

Q.7 Shock waves In this section we derive the shock wave solution. In general, the shock will move with a speed σ 6= 0 with respect to the reference frame in which the Riemann problem is formulated. The jump conditions of Rankine–Hugoniot between an initial state wi and a generic state w consist in the following ¡ ¢ ¡vector relation f(w)−f(wi ) =¢σ [w−wi ], where f(w) = ρu, ρu 2 + P, (E t + P)u = m, m 2 /ρ + P, (E t + P)m/ρ is the flux vector of the hyperbolic system expressed in terms of the vector state w = (ρ, m = ρu, E t ) of the conservative variables. For a given wi , we have a system of three equations in four unknowns ρ, m, E t and σ , so that the Rankine–Hugoniot relations admit a one-parameter family of solutions. Let us rewrite the jump conditions in the reference frame of the shock, where the fluid velocity is indicated by U = u − σ . By ¡the relativity principle, in this system ¢ the relations are F(W) = F(Wi ), where W = ρ, M = ρU, E t = ρe + 21 ρU 2 and ¡ ¢ ¡ ¢ F(W) = ρU, ρU 2 + P, (E t + P)U = M, M 2 /ρ + P, (E t + P)M/ρ . Employing the variable v instead of ρ, the Rankine–Hugoniot jump conditions of the stationary shock assume the following form

U 1 2 2U

± 2

U/v = Ui /vi , ± v + P = Ui2 vi + Pi ,

+ e + Pv = 21 Ui2 + ei + Pi vi ,

where e = e(P, v) is the equation of state of the considered fluid. In the energy equation, it has been assumed that the discontinuity is not a contact discontinuity, so that the common factor U/v = Ui /vi 6= 0 has been already simplified. Let J = U/v = Ui /vi 6= 0 denote the mass flux across the shock surface. Consider the first equation and express the fluid velocity U in terms of that, u, in the laboratory frame u−σ ui − σ =J= . v vi By solving the second part of the relation for σ yields σ = u i − J vi . By eliminating σ in the first part gives J = (u − u i + J vi )/v from which, by solving for J , we obtain J=

u − ui . v − vi

Let us now rewrite the second equation of the jump conditions preceding as Ui2 U2 v + P = vi + Pi . v2 vi2 Since Ui /vi = U/v = J , this equation is equivalent to J 2 v + P = J 2 vi + Pi

H⇒

J2 = −

P − Pi . v − vi

Finally, the third equation, namely 1 2 2U

+ e + Pv = 21 Ui2 + ei + Pi vi ,

thanks Ui /vi = U/v = J , can be written equivalently 1 2 2 2J v

+ e + Pv = 12 J 2 vi2 + ei + Pi vi .

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-13

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.7:

Shock waves

A-13

By eliminating J 2 by means of the relation J 2 = −(P − Pi )/(v − vi ), we obtain e − ei + 12 (Pi + P)(v − vi ) = 0,

where ei = e(Pi , vi ). This relation is purely thermodynamical and is called Hugoniot equation or also Rankine–Hugoniot equation. By exploiting the thermodynamic equation of state e = e(P, v), the Hugoniot equation defines implicitly the value of the pressure P after the shock as a function of v, for any volume–pressure pair (vi , Pi ), e(P, v) − ei + 12 (Pi + P)(v − vi ) = 0 and therefore defines a function which will depend on the form of the state equation e = e(P, v). In particular, the solution guaranteing an increase of the fluid entropy across the shock will be indicated by P = P RH (v; ˜˙i), for v < vi , where ˜˙i = (vi , Pi ). By recollecting the three equations above all together, we have  1    e(P, v) − ei + 2 (Pi + P)(v − vi ) = 0,  u − ui  J = , v − vi

J2 = −

P − Pi . v − vi

Since J depends on σ , the solution of this system represents a one-parameter family of states that satisfy the Rankine–Hugoniot jump conditions. The second and third equations allow one to express the fluid velocity u after the shock in the originary reference frame as p u = u i + sign(J ) −(P − Pi )(v − vi ) .

This equation is only an implicit definition of the velocity u since the sign of the mass flux J across the shock depends on u itself. The ambiguity of the sign can be resolved by exploiting the knowledge of which wave corresponds to the considered shock. Recalling that when Γ > 0 the shock is compressive, it can be shown that J < 0 for the wave associated to the first eigenvalue and J > 0 for the wave associated to the third eigenvalue. Thus, the velocity u of the gas after the shock is u RH 1|3 (v; i) = u i ∓

q £ ¤ − P RH (v; ˜˙i) − Pi (v − vi ) ,

v ≤ vi ,

where the subscripts 1 and 3 refer to the first and third eigenvalue, respectively. The speed σ of the shock in the originary reference frame will be finally σ =

ρu(ρ) − ρi u i vi u(v) − vu i = . vi − v ρ − ρi

Exploration 1

Properties of the linearly degenerate waves We have seen that the solutions associated with a genuinely nonlinear eigenvalue may be either a rarefaction wave or a shock wave. It is therefore legitimate to ask which are the properties of a solution to the hyperbolic system associated with a linearly degenerate eigenvalue. The answer is rather surprising, in the sense that the linear eigenvalue are found to be characterized by waves sharing the properties of both rarefaction waves and shock waves. This result is a consequence of the fact that along the integral curves of a linearly degenerate mode also the Rankine–Hugoniot jump conditions are satisfied. This important property can be demonstrated formally as follows. Let us suppose that the k-th eigenvalue λk (w) of a nonlinear hyperbolic system is linearly degenerate, with w indicating the state vector of the system, for instance that of the conservative variables. Let rk (w) denote the vector field of the eigenvector associated to the eigenvalue λk (w). The linear degeneracy of this eigenvalue means that rk (w) · ∇w λk (w) = 0, for any w in the domain of the admissible states of the physical system.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-14

A-14

Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

Let us now consider the integral curve of the eigenvector rk (w) passing through a ˘ This integral curve will be described by the vector function w = wk (q), given point w. where q is a parameter chosen to follow the curve and that is chosen to assume the value ˘ at point w. ˘ The integral curve is the solution to the ordinary differential q˘ = λk (w) initial-value problem dwk = α(q) rk (wk (q)), dq

˘ wk (q) ˘ = w,

where α(q) is an arbitrary function, whose form determines the parametrization of the curve. The solution w = wk (q) of this probelm can be probe to satisfy also the Rankine–Hugoniot jump condition, namely, £ ¤ ˘ = σ (q) wk (q) − w ˘ , f(wk (q)) − f(w)

where f(w) is the flux vector of the hyperbolic system and σ (q) is the propagation speed of the wave along the integral curve. Let us define the scalar function £ ¤ ˘ − λk (wk (q)) wk (q) − w ˘ G(q) ≡ f(wk (q)) − f(w)

and prove that G(q) ≡ 0. By the chain rule, the derivative of the function is ¤ dG(q) ∂f(wk (q)) dwk (q) dwk (q) £ ˘ wk (q) − w = − ∇w λk (wk (q)) · dq ∂w dq dq − λk (wk (q))

dwk (q) . dq

But ∂f(w)/∂w = A(w), so that, by writing dwk /dq as wk′ and omitting the dependence on variable q to have a simpler mathematical expression, we have ¤£ ¤ £ dG ˘ − λk (wk ) wk′ = A(wk ) wk′ − ∇w λk (wk ) · wk′ wk − w dq £ ¤ £ ¤£ ¤ ˘ , = A(wk ) − λk (wk ) I wk′ − ∇w λk (wk ) · wk′ wk − w

where I is the identity matrix of the same order of the hyperbolic system. The vector wk′ is parallel to the eigenvector rk (wk ), since wk (q) is an integral curve, so that the first term on the right-hand side is zero. Furthermore, the eigenvalue is linearly degenerate so that also the second term vanishes. It follows that dG/dq = 0, namely ˘ − f(w) ˘ − λk (w)[ ˘ w ˘ − w] ˘ = 0, which G = constant. On the other hand, G(q) ˘ = f(w) implies G ≡ 0. ˘ the function λk (wk (q)) of the Finally, if wk (q) is an integral curve through w, ˘ linearly degenerate eigenvalue is constant along it and λk (wk (q)) = λk (w). As a consequence £ ¤ ˘ . ˘ = λk (w) ˘ wk (q) − w f(wk (q)) − f(w)

Therefore, along the integral curve the Rankine–Hugoniot jump condition is satisfied ˘ = constant, which does not change along the with a propagation speed σ = λk (w) curve. It follows that the waves associated with a linearly degenerate eigenvalue can have discontinuous components but the propagation speed does not vary along them, so that all variables translate rigidly without any deformation. In other words, these waves can contain discontinuities but they propagate undeformed exactly as solutions to the simple linear advection equation with a constant advection speed.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-15

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.9: Limit values of the relative velocity

A-15

Q.8 Nonlinear functions of Riemann problem At this point we can write the functions describing the two waves propagating on the two sides of the contact discontinuity. For the left wave, the pressure and velocity functions are defined by the relations

P(v; ℓ˜) ≡

(

P(sℓ , v)

if v > vℓ

P RH (v; ℓ˜) if v < vℓ Z v  c(sℓ , v ′ ) ′   dv  uℓ + v′ vℓ u 1 (v; ℓ) ≡ q £   ¤  u ℓ − − P RH (v; ℓ˜) − Pℓ (v − vℓ )

if v > vℓ if v < vℓ

while for the right wave the functions are

P(v; r˜ ) ≡

(

P(sr , v)

if v > vr

P RH (v; r˜ ) if v < vr  Z v c(sr , v ′ ) ′   dv  ur − v′ vr u 3 (v; r) ≡ q    u + −£ P RH (v; r˜ ) − P ¤(v − v ) r r r

if v > vr if v < vr

Thus, these four functions provide a complete definition of the Riemann problem of gasdynamics provided that the condition Γ > 0 is satisfied.

Q.9 Limit values of the relative velocity The functions just introduced, with their inner choice between shock wave or rarefaction wave, generate, at convergence of the Newton iterative method, the proper combination of waves propagating on the left and the right of the contact discontinuity. However, the initial data of the Riemann problem allow an a priori realization of the kind of waves emerging from the disintegration of the initial discontinuity. As described in Landau and Lifshitz, the physical quantity whose value allows one to determine which kind of external waves occurs in the solution of Riemann problem is the relative velocity νr ℓ ≡ u r − u ℓ existing between the two regions of fluid, initially. This quantity is an invariant with respect to the transformation between inertial frames, known as Galilei transformations. In other words, the velocity difference u r − u ℓ is independent of the inertial frame in which the fluid velocity is measured, and has one and the same (absolute) value in any of them. At the end of section Q.6 it has been already seen that the condition for the vacuum formation assumes the form νr ℓ > νvacuum , where νvacuum represents a value that depends only on the initial thermodynamic states of the problem. Analogously, two other limit values of the relative valocity exist, ν2r and ν2s (with ν2s < ν2r ), that define intervals of ν in which the solution will contain two shock waves, or one shock together with one rarefaction, or two rarefaction waves, as shown by the following argument. Let us suppose, for convenience, to consider the specific case in which Pℓ < Pr and let us determine the first treshold ν2s such that the solution contains two shock waves, as shown in figure Q.7. Thanks to the Galilean invariance, the limit values of ν are independent from the frame of reference, and the disintegration of the initial discontinuity is described most simply in the frame of the contact discontinuity.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-16

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Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

P

ISBN XX-abc-defg-h

P Pr P⋆ Pr

P⋆

Pℓ uℓ

Pℓ ur

uℓ

ur

x Figure Q.7

Two shock waves

x Figure Q.8

One shock and one rarefaction

The fluid velocity u ⋆ℓ after the shock moving away from the left side of the contact discontinuity is given by q u ⋆ℓ = u ℓ − −(P ⋆ − Pℓ )(vℓ⋆ − vℓ ),

where P ⋆ and vℓ⋆ refer to the fluid after the shock, at the left of the contact surface. But in its reference frame of the contact u ⋆ℓ = u ⋆ = 0, and one has q u ℓ = −(P ⋆ − Pℓ )(vℓ⋆ − vℓ ).

When two shock waves emerge, the pressure P ⋆ on both sides of the contact discontinuity will overcome the greater pressure Pr , see figure Q.7. The minimum value that P ⋆ can assume, for the given values Pℓ and Pr , without contradicting the assumption Pℓ < Pr < P ⋆ , is P ⋆ = Pr . When P ⋆ = Pr , the value of vℓ⋆ is on the adiabatic curve connecting the state ℓ, and precisely for P = P ⋆ = Pr . By supposing that the function P = P RH (v; ℓ) of the Hugoniot adiabat is invertible, its inverse function can be indicated by v = v RH (P; ℓ). Moreover, if P ⋆ = Pr , the shock wave propagating on the right has a zero intensity and thus u r = u ⋆ = 0. Thus, the limit value of the relative velocity for the occurrence of two shock waves is νr ℓ = u r − u ℓ = −u ℓ , where u ℓ is obtained by substituting P ⋆ = Pr and vℓ⋆ = v RH (Pr ; ℓ) in the previous relation, to yield q £ ¤ ν2s ≡ − −(Pr − Pℓ ) v RH (Pr ; ℓ) − vℓ .

Therefore, when the relative velocity νr ℓ = u r − u ℓ of the Riemann problem is such that νr ℓ < ν2s , the solution will consist of two shock waves, beyond the contact.

When the solution contains a rarefaction wave, the assumption Pℓ < Pr implies that the isentropic wave will occur on the right, as shown in figure Q.8. Thus, as before, the left velocity (always in the reference frame of the contact) is such that q u ℓ = −(P ⋆ − Pℓ )(vℓ⋆ − vℓ ), while the total variation of the fluid velocity inside the rarefaction wave will be Z vr c(sr , v) dv. ur = − v vr⋆

As a consequence, the relative velocity νr ℓ ≡ u r − u ℓ is given by Z vr q c(sr , v) ⋆ ⋆ νr ℓ = − −(P − Pℓ )(vℓ − vℓ ) − dv. ⋆ v vr

For Pℓ and Pr specified, the value of P ⋆ is between Pℓ and Pr . By substituting P ⋆ = Pℓ in the first contribution to the difference νr ℓ , we obtain the inequality assuring the existence of a rarefaction wave together with a shock wave: Z vr Z vr⋆ c(sr , v) c(sr , v) νr ℓ < − dv = dv. ⋆ v v vr vr

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-17

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.10:

Riemann problem for the polytropic ideal gas

A-17

But the value vr⋆ is unknown a priori and since the limit value is wanted in terms of the Riemann problem data, it is possible to change the integration variable by choosing the pressure which is constant through the contact discontinuity and P ⋆ = Pℓ . Therefore, the limit value of the relative velocity in order that the solution containing a rarefaction and a shock wave becomes a solution with two rarefaction waves is defined by µ ¶ Z Pr Z Pℓ v(sr , P) c(sr , P) ∂v dP = ν2r ≡ d P. v(s , P) ∂ P r Pℓ c(sr , P) Pr sr The expressions of the relative velocities for Pℓ and Pr arbitrary can be obtained by introducing the explicit definition of the smaller and larger values of the initial pressures, as follows: Pm ≡ min(Pℓ , Pr )

and

PM ≡ max(Pℓ , Pr ),

and by labelling also the other initial data vℓ , sℓ and vr , sr in conformity with the order of the pressure values. If we now remind at this point also the limit value of νr ℓ for vacuum formation found at the end of section Q.6, we can write the three limit values of the relative velocity, in increasing order, q £ ¤ ν2s ≡ − −(PM − Pm ) v RH (PM , m) − vm , Z PM v(s M , P) dP , ν2r ≡ Pm c(s M , P) Z ∞ Z ∞ c(sℓ , v) c(sr , v) νvacuum ≡ dv + dv . v v vℓ vr In terms of these values, the kind of waves present in the solution of a given Riemann problem are immediately recognized and the solution will be one of the following four different possibilities νr ℓ < ν2s ν2s < νr ℓ < ν2r

−→ two shock waves −→ one rarefaction and one shock

ν2r < νr ℓ < νvacuum −→ two rarefaction waves without vacuum νr ℓ > νvacuum −→ two rarefaction waves with vacuum

Q.10 Riemann problem for the polytropic ideal gas Let us now develop the equations of the Riemann problem for the particular case of the polytropic ideal gas. The fundamental relation at the basis of this thermodynamic gas model is, in the energy representation, ³ v ´γ −1 0 , e = e(s, v) = K γ (s) v where the shorthand notation

K γ (s) ≡ e0 exp[(γ − 1)(s − s0 )/R] has been introduced and where γ is the constant of the specific heat ratio, defined by γ = c P /cv . The quantities e0 and s0 are suitable constants. The thermodynamic equations of state of the polytropic ideal gas have been provided in the first section of appendix E. For convenience we recall here the relations that are necessary to formulate the Riemann problem for this kind of gas.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-18

A-18

Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

A direct differentiation yields ev (s, v) = −(γ − 1) evv (s, v) = γ (γ − 1)

K γ (s) ³ v0 ´γ , v0 v K (s) ³ v ´γ +1 γ

0

v02

evvv (s, v) = −γ (γ 2 − 1)

v

,

K γ (s) ³ v0 ´γ +2 . v v03

Thus evvv (s, v) 6= 0 and the waves in a polytropic ideal gas associated with the extreme eigenvalues are genuinely nonlinear. To determine the rarefaction wave, the equations of state and the expression c = c(s, v) of the sound speed are required. The former are provided by the first-order partial derivatives of the fundamental relation for the energy, namely: ³ v ´γ −1 γ −1 γ −1 ∂e(s, v) 0 = = K γ (s) e, ∂s R v R K γ (s) ³ v0 ´γ ∂e(s, v) e P=− = (γ − 1) . = +(γ − 1) ∂v v0 v v

T =

By combining the two relations we obtain also the equation of state e = e(P, v) = Pv/(γ − 1). For the sound speed, c = (s, v), a direct calculation gives: c2 (s, v) =

µ

∂P ∂ρ

¶

s

= γ (γ − 1)K γ (s)

³ v ´−(γ −1) . v0

Thus, the integrals involved by the solution of the rarefaction wave can be evaluated analytically, and one obtains, assuming v > vi , Z

v vi

s

³ v ′ ´−(γ −1) dv ′ v0 v′ vi Z v q γ −1 γ −1 − −1 v′ 2 dv ′ = γ (γ − 1) K γ (si ) v0 2

c(si , v ′ ) ′ dv = v′

Z

v

γ (γ − 1) K γ (si )

vi

·

³v ´ 2 p i = γ Pi vi 1 − γ −1 v

γ −1 2

¸ .

Thus, the fluid velocity inside a rarefaction wave from the state (vi , si ) in a polytropic ideal gas depends on the specific volume v as follows u rar 1|3 (v; i)

· ³ v ´ γ −1 ¸ 2ci 2 i , = ui ± 1− γ −1 v

√ where ci = γ Pi vi . The second integral found in section Q.6, which gives the similarity variable ξ inside the rarefaction in terms of v, can be evaluated much in the same manner to give: ξ(v) = u i ±

γ + 1 ³ vi ´ 2ci ∓ ci γ −1 γ −1 v

γ −1 2

,

having used Γ = 12 (γ + 1). This function cabe inverted by solving it for v: ¶ ¸ −2 · µ γ − 1 ξi − ξ γ −1 , v(ξ ) = vi 1 ± γ +1 ci

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-19

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.10:

Riemann problem for the polytropic ideal gas

A-19

where ξi = u i ∓ ci . Finally, the third integral giving the fluid velocity inside the rarefaction is immediate and provides the function u(ξ ) = u i +

2 (ξ − ξi ). γ +1

As far as the condition for vacuum formation is concerned, the limit relative velocity is evaluated easily νvacuum =

2(cℓ + cr ) γ −1

and the fluid velocity at the boundaries of the vacuum region is found to be u vacuum (ℓ ℓ) = u ℓ +

2cℓ γ −1

and

u vacuum (r) = u r −

2cr . γ −1

Coming to the Hugoniot relation, for the polytropic ideal gas e = Pv/(γ − 1), so that the adiabat is Pv Pi vi 1 − + (Pi + P)(v − vi ) = 0, γ −1 γ −1 2 and holds only for v < vi . This relation is a linear equation in P and its solution reads (γ + 1) vi − (γ − 1) v P = P RH (v; ˜˙i) = Pi , (γ + 1) v − (γ − 1) vi

v ≤ vi .

We note that there is a lower bound for the variable v implied by P > 0 and this bound is easily found to be (v/vi ) > (γ − 1)/(γ + 1).

The inverse function of the Hugoniot adiabat is obtained by solving the previous equation for v, to give v = v RH (P; ˜˙i) = vi

(γ + 1)Pi + (γ − 1)P , (γ + 1)P + (γ − 1)Pi

and from this result the limit value of the relative velocity for the existence of two shocks is easily determined: ν2s = −(PM − Pm )

s

2vm . (γ − 1)Pm + (γ + 1)PM

When the data are such that the vacuum is not formed, the solution of the Riemann problem for a polytropic ideal gas is obtained by solving the system of two nonlinear equations established in section Q.5, employing the following functions defined by  ³ v ´γ i  if v > vi   Pi v ˜ P(v; ˙i) ≡ γ −1 (γ + 1) vi − (γ − 1) v    Pi if vi < v < vi (γ + 1) v − (γ − 1) vi γ +1  √ · ³ v ´ γ −1 ¸ 2 γ Pi vi 2  i  u ± 1− if v > vi    i γ −1 v s u 1|3 (v; i) ≡   2γ Pi γ −1   if vi < v < vi  u i ∓ (vi − v) (γ + 1) v − (γ − 1) vi γ +1

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-20

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Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

The limit values of the relative velocity to determine a priori the kind of waves occurring in the solution to the Riemann problem for a polytropic ideal gas can be ¡ ¢−1/γ calculated easily, reminding the equation of state v(s, P) = v0 [K γ (s)]1/γ PP0 √ and the relation c(s, P) = γ v(s, P)P. We obtain: s 2vm , ν2s = −(PM − Pm ) (γ − 1)Pm + (γ + 1)PM   ¶γ −1 µ 2γ 2c M  Pm , ν2r = 1− γ −1 PM νvacuum =

2(cℓ + cr ) . γ −1

Parametrization of the specific enthalpy The independent variable to be used in the integrals requested by the solution of the Riemann problem can be chosen freely. When the specific enthalpy h is chosen together with the entropy s as second independent variable, the√sound speed of the polytropic ideal gas is expressed by the relation c = c(s, h) = (γ − 1) h = c(h), so that it depends on h alone. This makes the calculation of the involved integrals easier. In fact, thanks to the change of variables v → h one has µ ¶ Z Z Z c(s, v) c(h) ∂v c(h) dv = dh = ¡ ¢ dh. v v ∂h s v ∂h ∂v s Moreover, the differentiation rule of composed functions implies µ µ ¶ µ ¶ ¶ µ ¶ d ³1´ [c(h)]2 ∂h ∂P ∂P ∂h = =v =− , ∂v s ∂ P s ∂v s ∂ρ s dv v v

where one has used the convolutory character of the Legendre transformation and the definition of the sound speed. The indefinite integral becomes √ Z Z Z dh c(s, v) dh 1 2 h dv = − = −√ + C. √ = −√ v c(h) γ −1 γ −1 h The limit values of the relative velocity are therefore found to be q ¡ ¢√ 2 PM hm γ Pm − 1 , ν2s ≡ − q γ +1 PM γ −1 Pm + 1 νvacuum

¡√ √ ¢ 2 h ℓ + hr , ≡ √ γ −1

ν2r ≡ 2

s



hM  1− γ −1

µ

Pm PM

¶γ −1 2γ



.

The nonlinear functions that define the Riemann problem for the polytropic ideal gas can be written also using h as the independent variable:  µ ¶ γ  h γ −1   if h < h i Pi hi P(h; ˜˙i) ≡    RH ˜ if h > h i P (h; ˙i) √ ¢ ¡√  2 h − h i   ui ± if h < h i γ −1 u 1|3 (h; i) ≡   if h > h i u i ∓ ∆u RH (h; ˜˙i)

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-21

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.11: Riemann problem for the nonpolytropic diatomic gas

A-21

The use of h as parameter makes the solution of the rarefaction wave particularly simple, while the relations for the shock wave are made slightly more difficult. In fact, the pressure that satisfies the Hugoniot relation starting from the state i is the positive root of the following quadratic equation µ ¶2 µ ¶ h P P γ +1 h 1− + − =0 Pi γ −1 h i Pi hi and the corresponding velocity is given by s · µ ¶ µ ¶¸ P Pi γ −1 hi −1 +h −1 . u= γ Pi P

Q.11 Riemann problem for the nonpolytropic diatomic gas When the specific heat depends on temperature, the density or specific volume is no more the variable most convenient for describing the rarefaction waves and detecting the kind of waves issuing from the initial discontinuity. In fact, considering for instance the case of a diatomic ideal gas with molecular oscillations contributing to the internal energy, the specific energy and entropy of the gas will be e(T ) = e0 +

RTv RT + , γ − 1 exp(Tv /T ) − 1

µ ¶ 1 µ ¶ Tv /T T γ −1 v s(T, v) , = s0 + ln − ln[1 − exp(−Tv /T )] + ln + R Tv exp(Tv /T ) − 1 v0 where γ = 7/5 and Tv is the vibrational temperature of the molecule. Clearly, it is not possible to derive the function T = T (s, v) as an explicit analytical formula describing the isentropic rarefactions. As a consequence, the Riemann problem for nonpolytropic ideal gases requires to use the temperature as the independent variable for parametrizing the states of the rarefaction waves. Thus, the left and right states of the Riemann problem will be denoted by (vℓ , Tℓ , u ℓ ) and (vr , Tr , u r ) and the two values of temperature Tℓ⋆ and Tr⋆ will be chosen as its unknown variables. For any other aspect, the procdure for solving the Riemann problem of the nonpolytropic ideal gas is resembles closely that for the polytropic case. First one introduces the functions P(T ; ℓ˜) and u 1 (T ; ℓ) providing, respectively, the pressure and velocity of the one-parameter family of states that can be connected to the left state ℓ = (vℓ , Tℓ , u ℓ ) of the Riemann problem by a rarefaction wave or a shock wave, depending on the value of T with respect to Tℓ . In an analogous way, P(T ; r˜ ) and u 3 (T ; r) denote the functions of the one-parameter family of states that can be connected with the right state r = (vr , Tr , u r ). These two pairs of functions are defined by ( rar ( rar P (T ; r˜ ) if T < Tr P (T ; ℓ˜) if T < Tℓ ˜ and P(T ; r ) ≡ P(T ; ℓ˜) ≡ RH P RH (T ; r˜ ) if T > Tr (T ; ℓ˜) if T > Tℓ and u 1 (T ; ℓ) ≡

(

u rar 1 (T ; ℓ) if T < Tℓ

u RH 1 (T ; ℓ) if T > Tℓ

and

u 3 (T ; r) ≡

(

u rar 3 (T ; r) if T < Tr

u RH 3 (T ; r) if T > Tr

The equality of the pressure values and of the velocity values on the two sides of the contact discontinuity means that Tℓ⋆ and Tr⋆ are solution to the system of two equations ( ⋆ ⋆ Φ(ℓ,˜ ˜ r ) (Tℓ , Tr ) = 0, Ψ(ℓ,r ) (Tℓ⋆ , Tr⋆ ) = 0,

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-22

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Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

in which the functions of the two variables T and Tˆ are defined as follows ˜ ˆ ˆ ˜ ), Φ(ℓ,˜ ˜ r ) (T, T ) ≡ P(T ; ℓ) − P(T ; r Ψ(ℓ,r ) (T, Tˆ ) ≡ u 1 (T ; ℓ) − u 3 (Tˆ ; r). The solution of this nonlinear system can be tackled by Newton iterative method, conveniently written in incremental form. and tthe following Jacobian matrix ¡ ¢ µ ′ ¶ P (T ; ℓ˜) −P ′ (Tˆ ; r˜ ) ∂ Φ(ℓ,˜ ˜ r ) , Ψ(ℓ,r ) , ≡ ∂(T, Tˆ ) u ′1 (T ; ℓ) −u ′3 (Tˆ ; r) must be evaluated at each iteration. This calculation is more complicated than in the polytropic case since the derivatives involve composed functions: in fact the pressure the equation of function P = P(T ; ˜˙i) is obtained from function v = v(T ; ˜˙i) through ¢ ¡ state P = P(T, v) to yield the composed function P(T ; ˜˙i) = P T, v(T, ˜˙i) , as it will be described below. To simplify the description of the procedure for the particular case of the diatomic gas with vibrational energy dependent on the temperature it is convenient to employ the dimensionless temperature t = T /Tv and introduce the dimensionless counterpart also of the specific energy and entropy defined by ǫ = e/(RTv ) and σ = s/R, respectively. In terms of the dimensionless variables the equations of states will assume the form: ǫ(t) = ǫ0 +

t 1 + , γ − 1 exp(1/t) − 1

µ ¶ 1 v ln t . + − ln[1 − exp(−1/t)] + ln σ (t, v) = σ0 + γ − 1 t[exp(1/t) − 1] v0 The dimensionless specific heat will be cv (t) =

Einstein function E(x) =

x 2 ex x 2 e−x = (ex − 1)2 (1 − e−x )2

dǫ(t) 1 = + E(1/t), dt γ −1

where E(x) = x 2 ex /(ex − 1)2 is the Einstein function. The rarefaction√wave is determined starting from the relation for the dimensionless velocity w ≡ u/ RTv w = wi ±

Z

v

vi

c(si , v ′ ) ′ dv . v′

The most convenienet independent variable along a isentropic transformation for a nonpolytropic ideal gas is temperature t. To perform the change of variables v → t, it is necessary to find the function v = v rar (t, ˜˙i) along the isentropic transfornation passing through the thermodynamic state ˜˙i = (ti , vi ), and the function is found to be µ ¶ 1 exp ¡ 1/t ¢ − exp ¡ −1/t ¢ ti γ −1 1 − e1/t 1 − e−1/t v (t, ˜˙i) = vi ¡ 1/ti ¢ ¡ −1/ti ¢ . t exp − exp 1/ti −1/ti rar

1−e

1−e

The derivative of this function can be evaluated directly " 1 dv rar (t; ˜˙i) rar ˜ = v (t; ˙i) − + dt (γ − 1)t

d dt

¢# ¡ 1/t ¢ d − dt exp 1 −−1/t 1 − e1/t e−1/t . ¡ ¢ ¡ ¢ exp 1 −1/te1/t − exp 1 −−1/t e−1/t

exp

¡

By differentiating the exponential functions and simplifying we obtain " dv rar (t; ˜˙i) 1 rar = v (t; ˜˙i) − + dt (γ − 1)t

¡ 1/t ¢ ¡ 1/t ¢ # d −1/t d dt 1 − e1/t + e dt 1 − e−1/t , 1 − e−1/t

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-23

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.11: Riemann problem for the nonpolytropic diatomic gas

A-23

and by expressing the derivative of the first ratio so as to eliminate the next term · ¸ dv rar (t; ˜˙i) 1 cv (t) v rar (t; ˙˜i) =− + E(1/t) = −v rar (t; ˜˙i) , dt t γ −1 t namely cv (t) dv rar (t; ˜˙i) =− . dt t v rar (t; ˙˜i) 1

The fluid velocity along the rarefaction wave is therefore Z

w = wi ±

t

ti

Z t c(t ′ ) cv (t ′ ) ′ dv rar (t ′ ; ˜˙i) ′ dt = w ∓ dt , i ′ dt t′ v rar (t ′ ; ˙˜i) ti c(t ′ )

where also the sound speed c(t) is dimensionless and is defined by

c(t) =

s·

1+

¸ 1 t. cv (t)

By substituting this function and by taking into account the form of the specific heat cv (t), the velocity along the rarefaction wave assumes the following form Z tp

w rar (t, i) = wi ∓

ti

[1 + cv (t ′ )] cv (t ′ )/t ′ dt ′

Z tq £

= wi ∓

ti

γ (γ −1)2

¤ + E(1/t ′ ) + E2 (1/t ′ ) /t ′ dt ′ .

As far as the shock wave is concerned, the Rankine–Hugoniot equation can be written in dimensionless form in terms of a new scaled pressure variable p = P/(RTv ), with dimension equal to the inverse of the specific volume, to give 1 ǫ(t) − ǫ(ti ) + ( pi + p) 2

µ

ti t − p pi

¶

= 0.

This is simply the quadratic equation µ

p pi

¶2

+ 2Bi (t)

p t − = 0, pi ti

where µ ¸ ¶ · γ +1 t 1 1 1 Bi (t) ≡ 1− , + − 2(γ − 1) ti ti exp(1/ti ) − 1 exp(1/t) − 1 with γ = 7/5. The solution of the quadratic equation provides the pressure along the Hugoniot adiabatic transformation: p RH (t; ˜˙i) = pi

hq

i Bi2 (t) + t/ti − Bi (t) .

All these results allow one to write the functions that define the Riemann problem for the nonpolytropic diatomic gas. The √ functions for the scaled pressure p = P/(RTv ) and dimensionless velocity w ≡ u/ RTv are

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-24

A-24

Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

 µ ¶ γ exp ¡ 1/ti ¢ − exp ¡ −1/ti ¢   t γ −1  1 − e1/ti 1 − e−1/ti  ¡ −1/t ¢ ¡ 1/t ¢  pi t i exp 1 − e1/t − exp 1 − e−1/t p(t; ˜˙i) ≡  q  h i    pi Bi2 (t) + t/ti − Bi (t) w1|3 (t; i) ≡

if t ≤ ti if t > ti

 Z tq ¤ £ γ   + E(1/t ′ ) + E2 (1/t ′ ) /t ′ dt ′  wi ∓ (γ −1)2 ti

  

q £ ¤ ¤£ wi ∓ − p RH (t; ˜˙i) − pi v RH (t; ˜˙i) − vi

if t ≤ ti if t > ti

where v RH (t; ˜˙i) = t/ p RH (t; ˜˙i).

To complete the solution of the Riemann problem, let us determine the dependence of the similarity variable ξ on the specific volume along the rarefaction wave. This variable coincides √ with the eigenvalue and for the left wave its dimensionless counterpart ξa = ξ/ RTv is expressed by the following relation rar

ξa (v) = w (v; ℓ) − c(sℓ , v) = wℓ +

Z

v

vℓ

c(sℓ , v ′ ) ′ dv − c(sℓ , v). v′

On the other hand, the natural variable for following the rarefaction has been seen to be the temperature t, so that it is convenient to perform the change of variables v → t, to give

ξa (t) = wℓ +

Z tq £ tℓ

γ (γ −1)2

¤ + E(1/t ′ ) + E2 (1/t ′ ) /t ′ dt ′ −

s·

¸ 1 t, 1+ cv (t)

since the sound speed depends only on the fluid temperature. Finally, the relationship between the similaritty variable ξ and the dimensionl velocity of the fluid is expressed by rar

u (ξ ; i) = u i +

Z

ξ ξi

dξ ′ , Γ (si , v(ξ ′ ))

where Γ represents the fundamental derivative of gasdynamics of the nonpolytropic ideal gas. It is therefore necessary to evaluate Γ along the rarefaction wave. Also in this case it is useful the change of variables v → t. A direct calculation provides the function Γ (t) =

3 +£ 2

E′ (1/t)/t γ (γ −1)2

+ E(1/t) + E2 (1/t)

¤£

1 γ −1

+ E(1/t)

¤,

which is actually independent from the entropy, as the sound speed, and therefore the sought for solution of the velocity is u rar (ξ ; i) = u i +

Z

ξ ξi

dξ ′ . Γ (t (ξ ′ ))

The Riemann problem solver for the nonpolytropic ideal gas described so far contains all the elements required to address the Riemann problem for a mix of chemically reacting gases.

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-25

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.12: Riemann problem for the van der Waals gas

A-25

A final aspect concerning the solution of the Riemann problem for the nonpolytropic diatomic ideal gas is the determination of the three relative velocities that delimit the range in which the external waves of the solution do not change their type. A direct calculation allows one to find the following values of the relative velocity variable in √ dimensionless form ν = (u r − u ℓ )/ RTv = wr − wℓ s µ ¶p 2 pM Bm (t M ) + t M /tm − Bm (t M ) − t M /tm p , ν2u ≡ − tm −1 pm Bm2 (t M ) + t M /tm − Bm (t M ) Z

tM

ν2r ≡

tℓ

νvacuum ≡

Z

+

Z

Remark

tm

0 tr

0

q£

q£

q£

γ (γ −1)2

γ (γ −1)2 γ (γ −1)2

¤ + E(1/t) + E2 (1/t) /t dt,

¤ + E(1/t) + E2 (1/t) /t dt ¤ + E(1/t) + E2 (1/t) /t dt.

The solution that has been found assumes that the gas after the shock is in a condition of complete thermodynamic equilibrium. This assumption is not satisfied in the real phenomenon because the relaxation time needed for molecular vibrations to reach equilibrium with the translational and rotational excitations is finite. Anyhow, the length scale of the nonequilibrium region behind the shoch front is usually small enough that the proposed method can provide at least a first order approximation to the physical process within the a diatomic gas when the molecular oscillations contribute significantly to the internal energy variations.

Q.12 Riemann problem for the van der Waals gas Let us now consider the Riemann problem for a gas described by the van der Waals equation of state, called van der Waals gas, according to the definition provided in section E.3 of the appendix on the thermodynamical properties of the fluids. In the present section the equations of state of this fluid are briefly recalled assuming first that its specific heat at constant volume is constant, which leads to the model denoted as polytropic van der Waals gas. Then we compute the solutions representing rarefaction waves and shock waves and write the nonlinear functions that define the Riemann problem for this gas model. At the end, the Riemann problem is reformulated in the more general case when the specific heat depends on temperature. In this nonpolytropic situation, also for the van der Waals gas the temperature is the most convenient independent variable for a mathematical formulation of the problem.

Thermodynamic relations The fundamental relation of the van der Waals gas in the energy representation is ³ a ´ ³ v0 − b ´δ a e = e(s, v) = e0 + exp[δ (s − s0 )/R] − , v0 v−b v

where R = R/m is the constant associated to the considered gas of molecular weight m, with R = 8.314 J/(mol K) denoting the universal gas constant. In the previous expression a and b are the dimensional constants of the considered van der Waals gas, while δ = R/cv is the dimensionless parameter with cv being the specific heat at constant volume, assumed to be independent from temperature (for a = b = 0, δ = γ − 1). The other quantities e0 , v0 and s0 denote the values of energy, volume and entropy per unit mass in a fixed reference state. To shorten mathematical expressions, the following function ³ a´ (v0 − b)δ exp[δ (s − s0 )/R] K δ (s) ≡ e0 + v0

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-26

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Dicembre 21, 2011

APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

is introduced, so that the fundamental relation above reduces to: e(s, v) =

a K δ (s) − . δ (v − b) v

The equations of state of the polytropic van der Waals gas are derived from the fundamental relation T = es (s, v) =

δ K δ (s) R (v − b)δ

P = −ev (s, v) = δ

and

K δ (s) a − 2. (v − b)1+δ v

The elimination of variable s in favour of e provides an alternative form of the equations of state T =

a´ δ ³ e+ R v

P =δ

and

e + av a − 2. v−b v

To solve the Riemann problem the expression of the sound speed is required: c(s, v) ≡

r

−v 2

¸1 · K δ (s) 2a 2 ∂ P(s, v) . = δ(1 + δ) − ∂v v (v − b)2+δ

By eliminating the variable s in favour of P with the aid of one of the equations of state provides Pv 2 + a 2a c(P, v) = (1 + δ) − v−b v ·

¸1

2

.

Nonlinear functions for the Riemann problem The fundamental derivative of gasdynamics for the polytropic van der Waal gas is obtained easily by a direct calculation

Γ (P, v) ≡ Γ (s(P, v), v) =

6a P + a/v 2 − 4 (v − b)2 v . 2 P + a/v 4a 2(1 + δ) − 4 v(v − b) v

(1 + δ)(2 + δ)

It is possibe to prove that for δ > δ˘ = 1/16.66 = 0.06 it is Γ > 0 for any thermodynamic state of the fluid in the region outside the coexistence curve of the liquid and vapour phases. The limit value δ˘ = 1/16.66 = 0.06 is overcome when the gas molecules have at least 7 atoms. In this case the fundamental derivative becomes negative in a very small region near to the coexistence curve and the critical point. Therefore, excluding this particular situation, Γ > 0 and the eigenvalues λ1 and λ3 are genuinely nonlinear: the solution of any Riemann problem for the polytropic van der Waals gas has the same characteristics of the ideal gas. By taking into account the link between P and v in the isentropic transformations of the van der Waals gas, the pressure in the rarefaction wave is given by P (v; ˜˙i) = rar

µ

a Pi + 2 vi

¶µ

vi − b v−b

¶1+δ

−

a , v2

and, thanks to the expression found of the sound speed c = c(P, v), the fluid velocity of the van der Waals model within the rarefaction wave is given by the integral u rar 1|3 (v; i)

= ui ±

Z

v

vi

·

¸1 ¶ 2a 2 ′ a (vi − b)1+δ − 3 dv . (1 + δ) Pi + 2 vi (v ′ − b)2+δ v′ µ

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-27

Dicembre 21, 2011

ISBN XX-abc-defg-h

PARAGRAFO Q.12: Riemann problem for the van der Waals gas

A-27

The Hugoniot adiabat of the polytropic van der Waals gas is still a linear equation for the pressure and its solution is

P

RH

(v; ˜˙i) =

ei −

¢ ¡ Pi 1 1 2 (v − vi ) + a 1 − δ v ¡ vi ¢ ¡1 1¢ b 2 + δ v− 2 + δ

+

ab δ v2

.

From this solution, the velocity of the states that can be connected to the state i by a shock wave has the following general expression: u RH 1|3 (v; i) = u i ∓

q £ ¤ − P RH (v; ˜˙i) − Pi (v − vi ) .

By writing the solutions of the rarefaction wave and the shock wave together, one obtains the nonlinear functions that define the Riemann problem for the polytropic van der Waals gas µ ¶ ¶µ a a vi − b 1+δ   Pi + 2 − 2    v − b v vi P(v; ˜˙i) ≡ ¢ ¡ P   ei − 2i (v − vi ) + a 1 − 1δ v1 + δab  v2  if ¡ vi ¢ ¡1 1¢  b 2 + δ v− 2 + δ  ¶ µ Z v·  a (vi − b)1+δ    ui ± − (1 + δ) Pi + 2  vi (v ′ − b)2+δ vi u 1|3 (v; i) ≡ q £   ¤   if  u i ∓ − P RH (v; ˜˙i) − Pi (v − vi )

if v > vi b + δvi /2 < v < vi 1 + δ/2 2a v′3

¸1 2

dv ′ if v > vi

b + δvi /2 < v < vi 1 + δ/2

These functions allow one to solve the Riemann problem for the considered gas model provided the condition for genuine nonlinearity is satisfied, namely when δ > δ˘ = 0.06, which implies that the molecules of the gas contain less that 7 atoms. On the contrary, if δ < δ˘ = 0.06, namely for a gas whose molecules contain 7 or more atoms, the acoustic eigenvalues loose the genuine nonlinearity As anticipated, the violation is however limited only to a very samll region of the thermodynamic plane (P, v) near the coexistence curve and the critical point. As a consequence, whenever the solution of the Riemann problem involves states belonging or near to this region the mathematical analysis of the Riemann problem must be extended. In fact, aside the usual waves represented by the rarefaction fan and the compression shock, it is necessary to consider waves of different and more complicated kind. In particular, waves consisting of compression fan and rarefaction shock can be present in the solution of the Riemann problem. Moreover, also composite waves exist which consist of two or three components which way be either continuous or discontinuous. The reader interested in this development is referred to a recent work by M. Fossati, P. Di Lizia and L. Quartapelle.

Nonpolytropic case Let us now consider a van der Waals gas whos specific heat at constant volume depends on temperature and formulate the Riemann problem for this gas model. In this model the internal energy of the gas includes a contribution due to the molecular vibrations, yielding the so called nonpolytropic van der Waals gas. The greater complexity with respect to the polytropic case is due to the fact that, as shown in section paragrafo E.3, it is not possible to write the fundamental thermodynamic relation as an explicit analytical relation, and one must resort to a parametric representation in terms of both functions for energy and entropy with T and ρ as independent variables. The contribution of molecular vibrations to the internal energy is described starting from the specific heat at constant volume, whose mathematical expression is found to be, by

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-28

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APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

quantum mechanical reasons,   L vib X ∂e(T, v) cv (T ) ≡ = R ǫ + E(Tn /T ) , ∂T n=1

where ǫ = 52 for a (recti)linear molecule while ǫ = 3 for a nonlinear molecule. Moreover E(x) denotes the Einstein function introduced previously. Since cv (T ) depends only on the temperature, the internal energy and entropy per unit mass of the nonpolytropic van der Waals gas are given by (see the end of section E.3) e(T, v) = ǫ RT − +

L vib X n=1

a RTn − , exp(Tn /T ) − 1 v

¾ µ ¶ǫ X L vib ½ ¡ ¢ s(T, v) Tn /T v−b T −Tn /T , + = ln + ln − ln 1 − e R v0 − b T0 eTn /T − 1 n=1

where we have introduced the value T0 ≡ min Tn n

to make the argument of the second logarithm dimensionless, by changing the constant s0 . As shown at the end of section E.3, the sound velocity of the nonpolytropic van der Waals gas is given by the relation c(T, v) =

½· 1+

¸ ¾1 RT v 2 R 2a 2 , − cv (T ) (v − b)2 v

which, similarly to that of the polytropic gas, depends both on temperature and specific volume, but with a slightly more complicated dependence on T due to the nonconstant character of the specific heat cv = cv (T ).

As in the nonpolytropic ideal gas, the rarefaction wave in the nonpolytropic van der Waals gas is described conveniently using temperature as the integraton variable. Much in the same manner, the whole Riemann problem is formulated more easily using the temperatures Tℓ⋆ and Tr⋆ on the two side of the contact discontinuity as the unknown variables. The rarefaction and shock waves of the considered gas model are now determined to obtain the analytical functions needed for the Riemann problem formulation. The rarefaction wave of the nonpolytropic van der Waals gas is obtained by calculating the integral defining the velocity function along the integral curve, after the change of variables v → T has been adopted and by exploiting the constancy of the entropy in the rarefaction wave. One sets s = constant = si in the relation s = s(T, v) and expresses si in terms of the values (Ti , vi ). From this one obtains the relation that represents the specific valume v of the one-parameter family of states of the rarefaction wave that can be connected with the initial thermodynamic state ˜˙i = (vi , Ti ): ln

¾ µ ¶ǫ X L vib ½ v−b 1 − e−Tn /T Tn /Ti Tn /T Ti ln + . = ln + − vi − b T 1 − e−Tn /Ti 1 − eTn /Ti 1 − eTn /T n=1

This relation can be solved with respect to v and yields the following explicit function of v(T ; ˜˙i): ¢ ¡ −Tn /T ¢ ¡ µ ¶ǫ Y L vib exp 1 −Tne/T Ti Tn /T − exp 1 − e−Tn /T ˜ v(T ; ˙i) = b + (vi − b) ¡ Tn /Ti ¢ ¡ −Tn /Ti ¢ . T n=1 exp 1 − eTn /Ti − exp 1 − e−Tn /Ti

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PARAGRAFO Q.12: Riemann problem for the van der Waals gas

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This function is used, together with is derivative, inside the integral that expresses the fluid velocity in the rarefaction wave, where there is also the sound speed given the previous relation. The derivative of v(T ′ ; i) appearing under the integral is calculated in the same way as for the nonpolytropic diatomic gas considered in the previous section, and is found to be · ¸ L vib X 1 dv(T ; ˜˙i) cv (T ) ˜ = − [v(T ; ˙i) − b] ǫ + . E(Tn /T ) = −[v(T ; ˜˙i) − b] dT T RT n=1

where the Einstein function E(x) is used and cv (T ) is the function provided explicitly at the beginning of this section. Using this result in the relation that defines the velocity along the rarefaction wave we obtain

u rar 1|3 (T ; i)

= ui ∓

Z

T Ti

)1 ¸ 2a[v(T ′ ; ˜˙i) − b]2 2 cv (T ′ ) dT ′ R ′ RT − 1+ . cv (T ′ ) RT ′ [v(T ′ ; ˜˙i)]3

(·

The pressure in the rarefaction wave is determined in terms of T only by means of the function v(T ; ˜˙i), through the equation of state of pressure, namely, P rar (T ; ˜˙i) =

a RT − , ˜ v(T ; ˙i) − b [v(T ; ˜˙i)]2

T < Ti .

The nonpolytropic van der Waals gas presents a further difficulty with respect to its polytropic conterpart, due to the fact that the Rankine–Hugoniot relation cannot be solved in closed form. It is therefore necessary to resort to an inner iteration, inside that used to solve the system of two nonlinear equations of the Riemann problem. To find the solution of the Rankine–Hugoniot jump conditions in the case of the nonpolytropic van der Waals gas, the usual form of the Hugoniot equation with the variables P and v is not convenient since it is not possible express the equation of state e = e(P, v) in an explict analytic form. It is instead avalilable the explicit expression of the equation of state e = e(T, v), so that the Hugoniot equation can be formulated in the following way e(T, v) − ei +

1 2

£

¤ Pi + P(T, v) (v − vi ) = 0,

where ei = e(Ti , vi ) and Pi = P(Ti , vi ). Using the equation of state P = P(T, v) in this relation we obtain ǫ RT +

L vib X n=1

¶¸ · µ a RTn RT 1 a (v − vi ) − − ei = 0. Pi + + − eTn /T − 1 2 v − b v2 v

This relation is of the form H(T, v; ˜˙i) = 0 with H(T, v; ˜˙i) ≡ ǫ RT +

L vib X n=1

· µ ¶¸ RT 1 a a RTn Pi + + − 2 (v − vi ) − − ei , T /T n e −1 2 v−b v v

and represents therefore the implicit definition of the function v = v RH (T ; ˜˙i), for T > Ti . For any T > Ti the solution v can be determined by means of Newton iterative method, taking as the initial guess the solution of the equation for the particular case with a = b = 0. In fact, when a = b = 0 one obtains the following algebraic equation for v = vin ¶ ¶ µ L vib µ X RTn RTi RTn 1 RT (v − vi ) + ǫ R(T − Ti ) + + − = 0, 2 v vi eTn /T − 1 eTn /Ti − 1 n=1

Quartapelle e Auteri: FLUIDODINAMICA. Appendice Q – pagina A-30

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APPENDIX Q: RIEMANN PROBLEM OF GASDYNAMICS

ISBN XX-abc-defg-h

which is a quadratic equation for the dimensionless unknown ν ≡ v/vi : ν 2 + 2β(τ )ν − τ = 0,

where τ ≡ T /Ti and β(τ ) ≡

¡1 2

L vib X ¢ Tn + ǫ (τ − 1) + Ti n=1

µ

1 e(Tn /Ti )/τ − 1

−

1 eTn /Ti − 1

¶ .

Therefore, the starting value of v is taken as the physically admissible solution of the equation, namely, h i p vin = vi −β(τ ) + β(τ )2 + τ .

From the solution of the nonlinear equation H(T, v; ˜˙i) = 0 for the unknown v, the post-shock pressure is easily found to be a RT − , P RH (T ; ˜˙i) = RH RH ˜ v (T ; ˙i) − b [v (T ; ˜˙i)]2 and the corresponding post-shock velocity is calculated by means of the relation q £ ¤ ¤£ u RH (T ; i) = u i ∓ − P RH (T ; ˜˙i) − Pi v RH (T ; ˜˙i) − vi , always assuming T > Ti .

By writing the solutions of the rarefaction wave and the shock wave together, one obtains the nonlinear functions that define the Riemann problem of the nonpolytropic van der Waals gas

P(T ; ˜˙i) ≡

u 1|3 (T ; i) ≡

        

RT a − ˜ v(T ; ˙i) − b [v(T ; ˜˙i)]2 RT

v RH (T ; ˜˙i) − b

 Z      ui ∓

T Ti

(·

−

a

[v RH (T ; ˜˙i)]2

if T < Ti if T > Ti

)1 ¸ ′ ; ˜˙i) − b]2 2 c (T ′ ) 2a[v(T R v RT ′ − dT ′ 1+ ′ 3 ′ ˜ cv (T ) RT ′ [v(T ; ˙i)]

  q    u ∓ −£ P RH (v; ˜˙i) − P ¤£v RH (T ; ˜˙i) − v ¤ i i i

where the conditions T < Ti and T > Ti for the velocity function have been understood, for the lack of space. To determine the elements of the Jacobian matrix it is necessary to compute the derivative with respect to T of the pressure along the considered wave. By exploiting the equation of state P = P(T, v) and the differentiation rule for a composed function we obtain " #¯ ∂ P(T, v) ∂ P(T, v) dv(T ; ˜˙i) ¯¯ d P(T ; ˜˙i) = + ¯ ˜ dT ∂T ∂v dT v(T ;˙i) ¾ ½ dv(T ; ˙˜i) 2a −RT R . + + = 2 3 dT v(T ; ˜˙i) − b [v(T ; ˜˙i) − b] [v(T ; ˜˙i)]

The derivative of function v(T ; ˜˙i) with respect to T for the rarefaction wave is simply the previously written formula, while for the shoch wave is calculated by the rule of differention of an implicit function: ¯ ¯ Á ∂H(T, v; ˙˜i) ¯¯ dv(T ; ˙˜i) ∂H(T, v; ˙˜i) ¯¯ =− ¯ RH ˜ ¯ RH ˜ . dT ∂T ∂v v

(T ;˙i)

v

(T ;˙i)

In conclusion, the nonpolytropic character of the gas implies that the Riemann solver for the van der Waals model is based on using the temperature as the independent variable instead of the specific volume, but it does not modify the general structure of the algorithm.