Roe-type schemes for dense gas flow computations - CiteSeerX

Computers & Fluids 35 (2006) 1264–1281 www.elsevier.com/locate/compfluid

Roe-type schemes for dense gas flow computations P. Cinnella

*

Universita` di Lecce, Dipartimento di Ingegneria dellInnovazione, via Monteroni, 73100 Lecce, Italy Received 28 June 2004; received in revised form 14 February 2005; accepted 25 April 2005 Available online 24 August 2005

Abstract Dense gas dynamics studies the dynamic behavior of gases in the thermodynamic region close to the liquid–vapor critical point, where the perfect gas law is no longer valid, and has to be replaced by more complex equations of state. In such a region, some fluids, known as the Bethe–Zeldovich–Thompson fluids, can exhibit non-classical nonlinearities, such as expansion shocks, and mixed shock-fan waves. In the present work, the problem of choosing a suitable numerical scheme for dense gas flow computations is addressed. In particular, some extensions of classical Roes scheme to real gas flows are reviewed and their performances are evaluated for flow problems involving non-classical nonlinearities. A simplification to Roes linearization procedure is proposed, which does not satisfy the U-property exactly, but significantly reduces complexity and computational costs. Such simplification introduces an additional error O(dx2), with dx the mesh size, with respect to the first-order accurate Roes scheme, and O(dx6) with respect to its higher-order MUSCL extensions. Numerical experiments, concerning a one-dimensional dense gas shock tube, supersonic flow of a BZT gas past a forward-facing step, and transonic dense gas flow through a turbine cascade, show a negligible influence of the adopted linearization procedure on the solution accuracy, whereas it significantly affects computational efficiency.  2005 Elsevier Ltd. All rights reserved.

1. Introduction The assumption that the fluid behaves like a perfect gas is the basis of classical gas dynamics, and is used in most compressible flow analyses in the engineering sciences. Dense gas dynamics, on the other hand, studies the dynamic behavior of gases in the dense regime, i.e. at thermodynamic conditions close to the liquid–vapor coexistence curve, where the perfect gas law is invalid. The computation of dense gas flows has received increased attention in the last decade, motivated by the fact that some very common fluids employed in engineering applications, mainly heavy polyatomic fluids, can exhibit unusual gas dynamic behavior in the dense gas regime at transonic and supersonic speeds. The most impressive differences occur for the so-called Bethe–Zeldovich–Thompson (BZT) fluids, for which compression *

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shocks violate the entropy inequality over a certain range of temperatures and pressures, and are therefore inadmissible. The dynamics of dense gases is governed by the key parameter [1]:   v 3 o2 p C :¼ 2 ; ð1Þ 2a ov2 s here presented in its non-dimensional form. In Eq. (1), v is the fluid specific volume, p is the pressure, a the sound speed, and s the entropy. C is commonly referred-to as the fundamental derivative of gas dynamics [1]. Remembering that the square of the sound speed a is given by   2 2 op a ¼ v . ov s C can be interpreted as a measure of the rate of change of the sound speed with density due to isentropic perturbations. The sign of C is entirely  2  determined by the sign of the second derivative oovp2 , i.e. the concavity of the s

0045-7930/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2005.04.007

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

constant-entropy lines (isentropes) in the p–v plane. Now, the relationship between the entropy change and the specific volume change through a weak shock wave can be written as [2]:

0.8

1.3

1.8

2.3

2.8

3.3

1.4

1.4

Isentropes

ð2Þ

where D represents a change in a given fluid property through the shock, and T is the absolute temperature. For perfect gases, C is just equal to cþ1 , where c is the 2 specific heat ratio. As c is necessarily greater than one, for thermodynamic stability reasons, than C > 1 also. As a result, a negative change in the specific volume through the shock, i.e. a compression, is required in order to satisfy the second law of thermodynamics. For dense gases, the perfect gas law no longer holds, and more complicated equations of state have to be considered. In this case, the isotherms are no longer concave-up hyperbolas in the p–v plane, but more complicated curves that exhibit negative concavity in the neighborhood of the liquid–vapor coexistence curve, in order to satisfy the thermodynamic conditions of zero slope and zero curvature at the critical point. It is well known that the isentropes tend to coincide with the isotherms as the specific heats tend to infinity; therefore, any fluid with sufficiently large specific heats will necessarily have concave-down isentropes in the dense-gas region of the p–v plane, which immediately implies C < 0 in the same region. The Bethe–Zel’dovich–Thompson fluids (from the names of the researchers who for the first time postulated their existence) are precisely defined as fluids which exhibit a region of negative C above the saturation curve in the vapor phase. The thermodynamic region where C < 0 is called the inversion zone and the curve C = 0 the transition line. If C is negative, from Eq. (2) we conclude that the specific volume must increase through a shock wave in order to have a corresponding increase of the entropy. This indicates that only expansion shocks will be admissible in these regions, whereas discontinuous compression waves will always spread into fans if inserted within the flow. BZT properties are typically encountered in heavy fluids characterized by large cv/R ratios (e.g. [3]), where cv is the constant volume specific heat and R the gas constant. In Fig. 1, the p–v diagram for a van der Waals gas with c = 1.0125 is shown, representative of a typical heavy fluorocarbon. The disintegration of compression shocks and other non-classical effects typical of BZT fluids could find application in technology. Particularly, attractive seems the possibility of reducing losses due to shock waves and boundary layer separation in turbomachines and nozzles. In the past, several numerical methods have been proposed for the computation of the so-called ‘‘real gas flows’’. Such methods are typically extensions of schemes previously developed for perfect gas problems.

1.2

1.2 Γ=0 curve

p/pc

a2 C ðDvÞ3 þ OðDv4 Þ; Ds ¼  3 v 6T

0.3

1265

1

1 Critical isotherm

Saturation curve

0.8

0.8 Concave-down portion

0.6 0.3

0.6 0.8

1.3

1.8

2.3

2.8

3.3

v/vc Fig. 1. p–v diagram for a van der Waals gas with c = 1.0125. The variables are normalized with their critical values.

Roes method being may be the most widely used in perfect gas CFD codes, it is also the one for which more real-gas extensions have been proposed. In fact, it is well known that the linearization procedure of Roes scheme is not uniquely determined when a real gas equation of state is taken into account. Some examples of real-gas generalizations of Roes scheme are given by Refs. [4–8]. In practice, however, no dramatic evidence of the numerical superiority of one formulation over another has been provided, even for very severe applications such as hypersonic flows, characterized by strong shock waves, chemical reactions, ionization and so on, see for example [4,9]. This has even driven some researchers to adopt, in the current use, approximate averages which do not satisfy the U-property, but work fairly fine in practice. An approximate Roe-type scheme has been proposed, for example, in [10,11]. Dense gas flows, on the other hand, can be characterized by quite ‘‘exotic’’ waves, such as shock/fan combinations, expansion shocks, etc.; however, flow discontinuities are generally very weak for a large range of temperatures and pressures. Thus, it seems reasonable to suppose that, for dense gas problems, the choice of a particular Roe linearization would have a quite small influence on the quality of the numerical results. On the contrary, the complexity of the particular formulation does affect computational costs significantly, especially when complicated equations of state are taken into account. The aim of the present work is to provide a theoretical justification for the limited influence of the chosen Roe linearization on the solution accuracy, and to the fact that even ‘‘approximate’’ averages give substantially correct results. To do that, an ‘‘extreme’’ case is considered: a simplified procedure is introduced, which reduces

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computational costs and implementation complexity. The simplified formulation does not satisfy the U-property exactly when real gas flows are considered: thus, it could not be properly included into the class of Roetype schemes; the scheme can be rather be considered a matrix-dissipation scheme, which reduces to Roes scheme for perfect gas flows. In the following of the paper it will be referred-to, with some abuse of language, as the ‘‘simplified Roe scheme’’. It will be shown that, for flows without strong discontinuities such as the dense gas flows of interest, the simplified scheme introduces negligible additional errors. The paper is organized as follows. In Section 2, the governing equations and the thermodynamic models are presented. Since the main objective is to discuss the accuracy and efficiency of differencing methods for the inviscid terms, the analysis is restricted to the Euler equations. The main properties of gas flows in the dense regime are briefly recalled. In Section 3, Roes flux difference splitting scheme and its generalizations to real gas flows are shortly reviewed. Two particular Roe-type schemes are discussed and compared. Then, an approximated linearization procedure is presented, and the additional numerical error introduced by this simplification is evaluated. Finally, numerical results are shown for a dense gas shock tube involving non-classical nonlinearities, for a supersonic BZT flow past a forward-facing step, and for a transonic BZT flow through a turbine cascade.

to generalize Roes scheme for real gases. The flux of takes the form: Jacobian matrix A ¼ ow 1 0 0 1 0 op op C B op C B  u2 þ 2u C. ð5Þ ow ow ow A¼B 1 2 3 C B    A @ op op op H u H þ u 1þ u ow1 ow2 ow3 Following [9], we now introduce the auxiliary vector   op op op qðwÞ ¼ u; H ; ; ; ow1 ow2 ow3 and we write A ¼ AðqðwÞÞ to indicate the explicit occurrence in the Jacobian matrix of the variables u, H, and of the partial derivatives of the pressure with respect to the conservative variables. The eigenvalues of the Jacobian matrix are k1 ¼ u  a;

k2 ¼ u;

k3 ¼ u þ a;

ð6Þ

where the speed of sound a is given by a2 ¼

op op þ ðH  u2 Þ . ow1 ow3

For a perfect politropic gas:   1 w22 pðwÞ ¼ ðc  1Þ w3  2 w1 and op c1 2 u; ¼ ow1 2

2. Governing equations and thermodynamic models The present work deals with inviscid compressible flows of dense gases, governed by the Euler equations. In one space dimension, they can be written as ow of þ ðwÞ ¼ 0; ot ox

ð3Þ

with w ¼ ðw1 ; w2 ; w3 ÞT ; w1 ¼ q;

w2 ¼ qu;

w3 ¼ qE

the conservative variable vector, and 2

f ðwÞ ¼ ðqu; qu þ p; quH Þ

T

the flux function. In the above expressions, q is the fluid density, u is the flow velocity, E is the total energy per unit mass, and H = E + p/q is the total enthalpy per unit mass. The pressure p is related to the conservative variables by a functional relation of the form: p ¼ pðwÞ.

ð4Þ

The preceding equation is not a thermodynamic relation, the pressure being a function of just two independent variables for gases in equilibrium conditions. Eq. (4) represents, however, a convenient functional form

op ¼ ðc  1Þu; ow2

op ¼ ðc  1Þ; ow3

where c is the specific heat ratio, constant for a polytropic gas. In this case, the Jacobian matrix A takes the well-known form: 1 0 0 1 0 C B c3 2 B u ðc  3Þu c 1C C B 2 A¼B C. C B   A @ c1 2 2 u H u H  ðc  1Þu cu 2 Note that, for perfect polytropic gases, just two independent variables, namely u and H, appear explicitly into the definition of the flux Jacobian. This happens because of the first-order-degree homogeneity of the flux function f with respect to w. Consequently, the auxiliary vector q just reduces to qðwÞ ¼ ðu; H Þ.

ð7Þ

On the other hand, if more complex equations of state are considered, the first-order homogeneity is lost in general, and A will depend on three independent variables, e.g. the velocity u and two thermodynamic variables. In the present work, two gas models are considered. The first one, is the van der Waals equation of state

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

for polytropic gases which, written as a function of density q and of internal energy per unit mass e, reads p ¼ ðc  1Þ

qe þ aq2  aq2 ; 1  bq

ð8Þ

where a and b are two gas-dependent constants, related to the fluid critical values by the conditions: a¼

9pc ; 8q2c Z c

b¼

1 ; 3qc

with Zc = pc/(RqcTc) the critical compressibility factor, equal to 3/8 for a van der Waals gas, R the gas constant and pc, qc and Tc the critical pressure, density and temperature, respectively. Normalizing Eq. (8) with respect to qc and pc, and using the critical value of Zc, one finds that, for a generic van der Waals gas, the normalized van der Waals constants are: a ¼ 3;

1 b¼ . 3

Rewritten in the general form (4) as a function of the conservative variables, Eq. (8) becomes p ¼ ðc  1Þ

w3  12

w22 w1

þ aw21

1  bw1

 aw21 .

ð9Þ

The van der Waals gas model is the simplest one that can take into account BZT effects, and has been often used as a qualitative model to study the behavior of non-classical nonlinearities in dense gas flows (see, e.g. [8,12,13]). In fact, a region of negative C values appears above the liquid–vapor coexistence curve for van der Waals gases having a specific heat ratio c less than about 1.06. Nevertheless, it is well known that the van der Waals model is not very accurate for temperatures and pressures close to the saturation curve (see [14]): in particular, it over-predicts the extent of the inversion zone. Consequently, more complex gas models, such as the Martin–Hou [15] or the Benedict–Webb–Rubin [16] equations, have to be used whenever quantitative results are needed. The realistic Martin–Hou equation of state, involving five virial terms and satisfying ten thermodynamic constraints, is one of the best available gas models to manageably compute dense gas effects [17]. This equation reads p¼

5 X RT fi ðT Þ þ ; v  b i¼2 ðv  bÞi

ð10Þ

where T is the absolute temperature, and the functions fi(T) are of the form: T

fi ðT Þ ¼ Ai þ Bi T þ C i ekT c ; with Tc the critical temperature and k = 5.475. The gasdependent coefficients Ai, Bi, Ci can be expressed in terms of the critical temperature and pressure, the criti-

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cal compressibility factor, the Boyle temperature, and one point on the vapor pressure curve. The gas model is completed by a caloric equation of state of the form:  Z T 5  X Tf 0i ðT Þ  fi ðT Þ cv1 ðT Þ dT þ ; ð11Þ e ¼ er þ ðiÞðv  bÞði  1Þ Tr i¼2  n where cv1 ðT Þ ¼ cv1 ðT c Þ TTc and cv1 ðT c Þ and n are gasdependent parameters. It is not straightforward to recast explicitly Eqs. (10) and (11) in the general form (4): nonetheless, such equation can be still introduced from a formal point of view. Let us now briefly review the main properties of gas flows in the dense regime. For more details, we refer to the review articles of Cramer [18], and Menikoff and Plohr [19]. From a mathematical point of view, the appearance of non-classical nonlinearities is related to the change in convexity of the flux function f(w), which occurs in regions where C < 0. Let us consider the nonlinearity factor, ak, associated to the kth characteristic field of system (3): ak ðwÞ :¼

okk ðwÞ  rk ðwÞ; ow

where rk is the kth right eigenvector of the flux Jacobian A(w). The kth characteristic field is said to be genuinely nonlinear if ak 5 0, and linearly degenerate is ak = 0. Factor ak can be rewritten as a function of the fundamental derivative [19]: ak ðwÞ ¼ k CðwÞaðwÞq; where 1 = 1, 2 = 0, and 3 = 1. If C 5 0, only the second characteristic field is linearly degenerate, and the hyperbolic system of conservation laws (3) is said to be convex. On the contrary, for thermodynamic states close to the transition line, ak vanishes on all characteristic fields and genuine nonlinearity is then lost. In this case, system (3) becomes non-convex and non-classical waves can occur. Thompson [1] was the first to show that, for 1D unsteady flows, C is related to the rate of change of the propagation speed of a simple right-running wave:   q oðu þ aÞ C¼ . a oq s Thus, if C > 0 compression waves tend to steepen, and rarefaction waves spread out; if C < 0 rarefaction waves steepen, whereas compression waves spread out; finally, if C = 0, waves have fixed form (all characteristic fields are linearly degenerate). For a left-running wave, it is easy to prove that   q oðu  aÞ C¼ ; a oq s which leads to similar conclusions.

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If 2D steady flows are considered, it is possible to show [1], that C is related to the slope of the flow Mach lines through: dðl þ hÞ ¼

CM 2 dh; ðM 2  1Þ 1

where l = sin (1/M) is the Mach angle, M is the Mach number, and h is the flow deflection angle. If C > 0, then Mach lines will steepen during compressions past concave walls and will spread out for flows past convex walls. The contrary will be true if C < 0. The fundamental derivative also governs other important properties of dense gases. We recall here two of them. Let us re-write the fundamental derivative as a function of the density q and of the sound speed:   q oa C¼1þ . a oq s In regions where C < 1, the sound speed experiences an uncommon variation in isentropic perturbations, i.e. it decreases with isentropic compression, contrarily to what happens for ‘‘common’’, perfect-gas-like fluids, for which C > 1. The second property concerns thermodynamic conditions where C  0. Cramer and Kluwick [2], showed that C = O(Dv) for small volume changes in the vicinity of the transition line. Therefore, for shock waves having jump conditions in the thermodynamic region near the C = 0 contour, the associated entropy change is expected to be reduced up to an order of magnitude from that predicted by Eq. (2), that is, losses are much lower than normal.

The scheme can be written in the semi-discrete conservation form:  ðdhÞjj ow ¼ 0; ð12Þ þ  ot j dx where dx is the space increment, j is a given mesh point, h is a numerical flux, and d is the difference operator upon one cell, defined as ðdwÞjjþ1 ¼ wjþ1  wj 2

for any mesh function wj. It is possible to represent the numerical flux, h, as the sum of a centered approximation (consistency) and a numerical dissipation (stabilization) term: 1 ð13Þ hjþ12 ¼ ðlf Þjjþ1  jAR jjþ1 ðdwÞjjþ1 ; 2 2 2 2 where l represents the average operator upon one cell, 1 ðlwÞjjþ1 ¼ ðwjþ1 þ wj Þ. 2 2 In the equation above, jARj is the dissipation matrix of Roes numerical scheme. The numerical flux (13) is a first-order approximation to the physical flux function evaluated at the interface between two adjacent cells, fjþ12 :   dx ow dx2 o2 f  þ þ Oðdx3 Þ. hjþ12 ¼ fjþ12  jAR jjþ1 2 2 ox  1 8 ox2 jþ1 jþ 2

2

ð14Þ Different formulations for AR just affect the scheme stabilization term, i.e. the error constant, but not modify the order of accuracy. 3.1. Roe’s scheme for perfect gases

3. Roe-type schemes for dense gases Roes approximate Riemann solver is a Godunovtype scheme for hyperbolic systems of conservation laws, based on local linearization of the governing equations in order to avoid the exact solution of the Riemann problem at cell interfaces. The system of conservation laws is locally approximated by the linearized problem: ow ow þ AR ðwL ; wR Þ ¼ 0; ot ox where wL and wR are the left and right states of the considered Riemann problem, respectively, and AR is an average Jacobian matrix which, for every couple of states wL and wR, satisfies the three conditions, sometimes referred-to as the U-property: (a) AR(wL,wR)(wR  wL) = f(wR)  f(wL); (b) AR(w,w) = A(w); (c) AR(wL,wR) has real eigenvalues and can be diagonalized.

Originally, Roes scheme has been formulated for solving the Euler equations completed by the equation of state for perfect polytropic gases [20]. In this case, Roes average AR may be written: wÞÞ; AR ðwL ; wR Þ ¼ Aðqð~ ~ is an intermediate state, w ~¼w ~ ðwL ; wR Þ such where w that condition (a) is satisfied, and the auxiliary vector q is defined as (7). The required averaged values of u and H are given by pffiffiffiffiffi pffiffiffiffiffiffi qL uL þ qR uR ~u ¼ pffiffiffiffiffi pffiffiffiffiffiffi ; ð15Þ qL þ qR pffiffiffiffiffi pffiffiffiffiffiffi qL H L þ qR H R e ¼ H ; ð16Þ pffiffiffiffiffi pffiffiffiffiffiffi qL þ qR 3.2. Roe’s scheme for dense gases When a real gas equation of state is taken into account, matrix AR is no longer uniquely determined,

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

and several different approaches can be followed. A detailed discussion and classification of the different extensions of Roes scheme to real gas flows has been provided in [9,8]. In such references, the Roe-type linearizations for the real gas are divided into two families. A first family, that includes the greatest part of the methods available in the literature, includes linearizations in quasi-Jacobian form. In such approaches, the linearization matrix is assumed to be of the form: AR ðwR ; wL Þ ¼ Að~ qÞ; with ~q ¼

ð17Þ

! g g g op op op e; ~ u; H ; ; . ow1 ow2 ow3

ð18Þ

AR is said to be in quasi-Jacobian form because the op f op average quantities f , op , and f are no longer the parow1

ow2

ow3

tial derivatives of p(w) computed at some intermediate state, but instead additional independent unknowns of the linearization problem. Roes average matrix has then to be determined by specifying the extended intermediate state ~ q. The required intermediate values are typically determined by assuming the linearization matrix to be of the form (17) and using direct substitution into condition (a). As an alternative (see for example [5]), it is possible to assume Eq. (17) and to substitute into the eigenvector expansion of Df and Dw (where D( Æ ) = ( Æ )R  ( Æ )L). Both approaches lead to definitions of the intermediate velocity and total enthalpy per unit mass identical to expressions (15) and (16), respectively, provided that the remaining average values are chosen to satisfy the additional linear condition: Dp ¼

g g g op op op Dw1 þ Dw2 þ Dw3 . ow1 ow2 ow3

ð19Þ

Eqs. (15), (16) and (19) determine a two-parameter family of solutions of the linearization problem: the linearization procedure has then to be completed by evaluating the averaged pressure derivatives in terms of the relevant average thermodynamic variables. In practice, in order to enforce consistency, the following relation is imposed: g g op op ¼ ~ u ; ow2 ow3 derived by applying the chain rule to Eq. (4), which is satisfied pointwise at the left and right states. This reduces the number of degrees of freedom to one. The remaining value is computed by choosing an equation of state for the pressure, typically of the form: pðwÞ ¼ pðwðq; iÞÞ; where variable i is often chosen as the internal energy, either per unit mass e, or per unit volume  = qe. According to which choice is made, the pressure derivaop op tives ow , op , and ow , the auxiliary vector q, and the flux 1 ow2 3

1269

Jacobian A assume a different form, thus influencing the determination of Roes average state. We refer to [9] for a detailed discussion. A drawback of this class of schemes lies in that the intermediate pressure derivatives involved in the linearization problem are artificial unknowns, not retaining their exact thermodynamic significance. This may lead to inconsistencies whenever these quantities are employed to derive other thermodynamic variables such as, for example, the speed of sound. The methods of the second family, on the other hand, wÞÞ, are constructed by assuming AR ðwL ; wR Þ ¼ Aðqð~ ~¼w ~ ðwR ; wL Þ, and where the auxiliary vector q dewith w pends, in general, on three independent variables. The pressure partial derivatives involved into the Jacobian matrix definition are now considered as dependent unknowns, g op op ¼ ð~ wÞ; ow1 ow1

g op op ¼ ð~ wÞ; ow2 ow2

g op op ¼ ð~ wÞ. ow3 ow3 ð20Þ

The intermediate state is obtained by solving the system of equations given by condition (a). As the first equation of the system reduces to an identity, only two independent equations are left, leading to a one-degree-of-freedom family of solutions of the linearization problem. Different methods can be derived according to the way of specifying the additional optional condition. We now consider two generalizations of Roes scheme, belonging each to one of the above families. The first one, due to Glaister [5], uses a linearization in quasi-Jacobian form; the second one, uses a linearization in strict Jacobian form, and is due to Guardone and Vigevano [8]. 3.2.1. Glaister’s solver (GL) In Glaisters extension of the method [5], an equation of state of the form p ¼ pðq; eÞ

ð21Þ

is considered. Written in terms of the chosen thermodynamic variables, the pressure partial derivatives with respect to the conservative variables are   op 1 2 p u e e; ¼ pq þ ow1 2 q op p ¼ u e ; ow2 q

ð22Þ

op p ¼ e; ow3 q where pq and pe are shorthand notations for the partial derivatives of the pressure with respect to density and internal energy. Taking into account Eqs. (22), the intermediate auxiliary vector (18) becomes

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  e;q ~; ~e; ~ ~ ~; H pe q¼ u pq ; ~

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

ð23Þ

and the linearization matrix is expressed by 1 0 0 1 0 ~p e ~ ~pe C u~ p B a2  ~ ~ u2 Þ u2  e ð H 2~ u e C B ~ ~ ~ ~ C; q q q AR ¼ B C B 2 A @ ~ ~ ~ ~ ~ ~ u p u p u p e e e 2 2 eÞ e e ~ ~ ~ uð~ a H ðH uÞ H uþ ~ ~ ~ q q q ð24Þ where the intermediate speed of sound is given by   ~ p~ pe 2 ~ a ¼ þ ~pq ð25Þ ~2 q and the intermediate pressure is   1 2 e ~ H  ~e  ~ ~ p¼q u . 2

ð26Þ

The extended average state ~ q is determined by enforcing condition (a) and using direct substitution into the eigenvector expansion of Df and Dw. The resulting avere are still given by relations (15) and age quantities ~ u; H (16), with the average density given by pffiffiffiffiffiffiffiffiffiffiffi ~ ¼ qL qR ; q ð27Þ Moreover, the additional relation: pffiffiffiffiffi pffiffiffiffiffiffi qL eL þ qR eR ~e ¼ pffiffiffiffiffi pffiffiffiffiffiffi qL þ qR

ð28Þ

is introduced. With this choice, the averaged partial derivatives ~ pq and ~pe are found to be related by the linear condition: pq Dq. Dp ¼ ~ pe De þ ~

ð29Þ

Vice versa, assuming Eq. (29) to hold, the average internal energy has to be defined as in (28) in order to meet condition (a) (see Ref. [5], Eq. (3.51)). Eq. (29) could be obtained equivalently from the general condition (19) by assuming the average partial derivatives to be of the form (22) and evaluating density, velocity, internal energy, and total enthalpy according to Eqs. (27), (15), (28), and (16), respectively. In order to satisfy (29), Glaister evaluates the average pressure derivatives ~ pe and ~ pq through the finite difference approximations: ( pðq ;eR Þþpðq ;eR Þpðq ;eL Þpðq ;eL Þ R L R L if De 6¼ 0; 2De ~ pe ¼ ð30Þ 1 ½p ðq ; e Þ þ p ðq ; e Þ if De ¼ 0; L R L R e e 2 8 pðqR ;eR ÞþpðqR ;eL ÞpðqL ;eR ÞpðqL ;eL Þ if Dq 6¼ 0; < 2ðdqÞjjþ1 2 ~ pq ¼ ð31Þ  :1

p ðq ; e Þ þ p ðq ; e Þ if Dq ¼ 0; L L R R q q 2 that satisfy condition (29) exactly. The preceding formulas have the advantage of generality and can be easily implemented with any equation of state of the form (21); on the other hand, as remarked elsewhere, they

introduce two artificial values that could lead to a non-physical intermediate state. The use of ad hoc projection procedures allows to overcome this problem (see e.g. [6]). Nonetheless, in the present study we restrict to the original Glaisters formulation: numerical results show that, at least for the dense gas problems of interest here, no practical problems arise. If the equation of state is not explicitly available in the form (21), as it happens for the Martin–Hou equation considered in the present work, the following procedure can be adopted: (1) If De 5 0 or Dq 5 0 then • Compute temperatures T(qR,eR), T(qL,eR), T(qR,eL), and T(qL,eL) from the caloric equation of state. • Compute pressures p(T(qR,eR),qR), p(T(qL,eR), qL), p(T(qR,eL),qR), and p(T(qL,eL),qL) from the thermal equation of state. (2) If De = 0 then • Compute partial derivatives pe(qL,eL), pe(qR, eR). (3) If Dq = 0 then • Compute partial derivatives pq(qL,eL), pq(qR, eR). (4) Apply formulas (30) and (31). The temperature can be efficiently computed from the caloric equation of state using a Newton–Raphson iteration. The pressure partial derivatives pe, pq can be computed from the thermal and caloric equations of state using the following thermodynamic relations:     op 1 op pe ¼ ¼ ; ð32Þ oe q cv oT q         op op 1 op oe pq ¼ ¼  . ð33Þ oq e oq T cv oT q oq T 3.2.2. Guardone and Vigevano’s solver (GV) A Roe linearization in strict Jacobian form has been provided in [8]. First, an equation of state of the form p ¼ pðq; Þ

ð34Þ

is selected. With this choice, the pressure partial derivatives with respect to the conservative variables are: op op 1 2 op þ u ; ¼ ow1 oq 2 o op op ¼ u ; ow2 o op op ¼ . ow1 o

ð35Þ

In order to close the system of two equations in three unknowns stemming from condition (a), the

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

linearization problem is incremented with a supplementary equation, of the form: g r w p  Dw ¼ Dp.

ð36Þ

The preceding relation is similar to condition (19), used in deriving linearization matrices in quasi-Jacobian form. Nevertheless, it differs in that the pressure partial derivatives involved in (36) are now evaluated at the ~ , i.e. according to Eqs. (20). The intermediate state w two independent equations obtained from condition (a) lead to explicit expressions for the intermediate velocity and total enthalpy of the standard form (15) and (16); on the other hand, the supplementary equation, rewritten in terms of variables (q,u,H), leads to ~: an equation for the unknown intermediate density q Dp ¼ p ð~ q; ~ÞD þ pq ð~ q; ~ÞDq;

ð37Þ

e Þ related to ~ e by the thermodynamic with ~ ¼ ð~ q; H u, H relation:   ~ u2 e ~ ¼ q ~ H ~; ~Þ.  p ðq 2 For a polytropic van der Waals gas, the resulting equa~, and the relevant tion is a third-order polynomial in q real root is singled out as the one lying within or closer to the interval [qL,qR]. The approach described above leads to a more consis~ , however, tent evaluation of the intermediate state w ~ from Eq. (37) is not always straightforward computing q when complex equations of state are considered: in par~ would require, in general, the soluticular, evaluating q tion of a complex nonlinear algebraic equation, using a suitable iterative technique. In the present implementation, a modified Newton–Raphson iteration (Steffensens method), of quadratic convergence, has been used, which avoids evaluation of the function derivative. 3.2.3. Simplified Roe-type solver (RS) In spite of the great number of Roe-type schemes proposed in the past for the computation of real gas flows, there is no clear numerical evidence of the superiority of one formulation over another. This has already been pointed out, for instance, in [4,9]. Also, in [8] it is remarked that comparisons of several extensions of Roes scheme ‘‘showed almost no differences’’ for dense gas shock tube problems. For this reason, in the present work, a somewhat ‘‘crude’’ simplification of the linearization procedure for real gases is considered. Specifically, the proposed approach stems from Glaisters method by introducing some simplifying assumptions. Let us choose an equation of state of the form p = p(q,e). Then, the pressure partial derivatives with respect to the conservative variables are given by Eqs. (22) and the auxiliary vector ~ q and the matrix AR take the form (23) and (24), respectively. We now need to evalu-

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ate ~q. To do that, we choose to compute the average density, velocity, and total enthalpy via the usual formulas: pffiffiffiffiffiffiffiffiffiffiffi ~ ¼ qL qR ; ð38Þ q pffiffiffiffiffi pffiffiffiffiffiffi qL uL þ qR uR ~u ¼ pffiffiffiffiffi pffiffiffiffiffiffi ; ð39Þ qL þ qR pffiffiffiffiffi pffiffiffiffiffiffi qL H L þ qR H R e H ¼ . ð40Þ pffiffiffiffiffi pffiffiffiffiffiffi qL þ qR The preceding equations allow to completely determine ~ . If, as in [5], we make the addian intermediate state w tional assumption that the average internal energy is defined by relation (28), we would find that the remaining intermediate quantities, i.e. the pressure partial derivatives, are related through the linear condition (29). However, differently from [5], they are now evaluated at the intermediate state, i.e. they are treated as dependent variables: ~pe ¼ pe ð~ q; ~eÞ;

~pq ¼ pq ð~ q; ~eÞ.

ð41Þ

Unfortunately, nothing assures in general, that the partial derivatives (41) verify condition (29) for any equation of state and, ultimately, that the computed intermediate state satisfies condition (a). To overcome this, a subsequent projection step over the straight line defined by (41), as proposed for example in [6], is needed. We now want to show that, even omitting the projection step, the partial derivatives (41) satisfy condition (29) approximately, to within O(kDwk2), where k Æ k is a vector norm. Assuming that the left and right states wL and wR are sufficiently close to each other, it is possible to expand the equation of state for the pressure (of the form ~ , which gives (21)) about the intermediate state w 2

~; ~eÞDe þ pq ðq ~; ~eÞDq þ OðjjDwjj Þ. Dp ¼ pe ðq

ð42Þ

Eq. (42) equals condition (29) to within O(kDwk2). Substituting this equation into the eigenvector expansion of condition (a) (see Ref. [5], Eq. (3.51)), one obtains 2

~ De þ OðjjDwjj Þ ¼ 0. DðqeÞ  ~e Dq  q But, as the internal energy is defined by (28), the term ~ De DðqeÞ  ~e Dq  q vanishes, and we find that condition (a) is verified to within O(kDwk2). In summary, using the proposed simplified procedure, the evaluation of Glaisters extended intermediate state is e, ~, ~u, ~e, and H not modified concerning the quantities q whereas the quantities ~pe and ~pq are evaluated according to Eqs. (41). Such values do not satisfy condition (29) exactly, and consequently do not satisfy the U-property. Instead, they still verify an expansion of the form: Dp ¼ ~pe De þ ~pq Dq þ OðjjDwjj2 Þ.

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We note again that the simplified scheme so obtained is strictly equivalent to that proposed in [6], without the projection stage for the pressure derivatives (see also [9]). The simplified intermediate state is no longer a solution of the linearization problem, and the matrix AR obtained from this procedure is no longer the linearization matrix, but merely the dissipation matrix of the firstorder matrix viscosity scheme (12) and (13). For this reason, as anticipated in Section 1, the resulting scheme is not truly a Roe-type scheme. For instance, it is not able to capture steady shocks exactly. Nevertheless it reduces naturally to Roes scheme for perfect polytropic gases; moreover, in the real gas case, it differs only for higher-order terms from Roe-type schemes, as shown below. In the previous discussion, it has been shown that the simplified intermediate state satisfies condition (29) to within O(kDwk2). This implies that condition (a) is also satisfied to within the same order of approximation: 2

Df ¼ AR ð~ wRS ÞDw þ OðkDwk Þ;

ð43Þ

RS

~ denotes the approximate intermediate state. In where w practice, the approximation only concerns the dissipation matrix of numerical flux (13), and the modified scheme remains conservative, consistent, and first-order accurate. In fact, setting wL = wj and wR = wj+1, the numerical flux associated to an exact extension of Roes scheme is 1 wEX Þjdwjþ12 ; hjþ12 ¼ lfjþ12  jAR ð~ 2 ~ EX is an intermediate state which satisfies, by where w construction, condition (a) exactly. On the other hand, the simplified numerical flux can be written as 1 hRS wRS Þjdwjþ12 . ð44Þ jþ12 ¼ lfjþ12  jAR ð~ 2 Equating the right-hand sides of condition (a) and of relation (43) gives the estimate: AR ð~ wRS Þ ¼ AR ð~ wEX Þ þ OðjjDwjjÞ.

ð45Þ

Substituting into (44), and taking into account that Dw ¼ wR  wL ¼ dwjþ12 , we obtain   1 2 RS EX hjþ1 ¼ lfjþ12  jAR ð~ w Þjdwjþ12 þ Oðjjdwjj Þ; ð46Þ 2 2

In practice, first-order-accurate Roes scheme is too much dissipative, and higher-order extensions have to be used. A standard way to increase the schemes accuracy is MUSCL extrapolation. We now prove that, for higher-order MUSCL extensions of Roes scheme, the use of the simplified average state involves an additional error only O(dx6), which is negligible with respect to the schemes leading truncation error term. Let us consider the higher-order numerical flux: hHjþ1 ¼ 2

f ðwR Þ þ f ðwL Þ 1  jAR ðwR ; wL Þ j ðwR  wL Þ; 2 2 ð49Þ

where wL and wR are the extrapolated left and right states, 1 wL ¼ wj þ ½ð1  kÞðwj  wj1 Þ þ ð1 þ kÞðwjþ1  wj Þ; 4 1 wR ¼ wjþ1  ½ð1  kÞðwjþ2  wjþ1 Þ þ ð1 þ kÞðwjþ1  wj Þ 4 ð50Þ and k is the MUSCL extrapolation parameter. Flux (49) is a second-order approximation to fjþ12 :  k  1 2 o2 w dx hHjþ1 ¼ fjþ12 þ Ajþ12 2 4 ox2 jþ1 2 3  k  1 3 o w dx jAR jjþ1 þ Oðdx4 Þ. ð51Þ 2 8 ox3 jþ1 2

If Roes average AR is computed using the simplified procedure, the corresponding numerical flux is hHRS jþ1 ¼ 2

f ðwR Þ þ f ðwL Þ 1  j AR ð~ wRS Þ j ðwR  wL Þ. 2 2

ð52Þ

Using estimate (45), one obtains 2 H hHRS jþ1 ¼ hjþ1 þ OðjjDwjj Þ. 2

2

ð53Þ

Now, expanding relations (50) in Taylor series, it is easy to show that wR  wL ¼

k  1 3 o3 w dx þ Oðdx5 Þ 4 ox3

ð54Þ

that is

that is, 2

hRS jþ1 ¼ hjþ12 þ Oðjjdwjj Þ;

ð47Þ

2

where ðdwÞjjþ1 ¼ wjþ1  wj ¼ OðdxÞ. Using Eq. (14), one 2 finally obtains  dx ow RS hjþ1 ¼ fjþ12  jAR jjþ1 þ Oðdx2 Þ ¼ hjþ12 þ Oðdx2 Þ. 2 2 ox  1 2 jþ2

ð48Þ In summary, the proposed simplification introduces an additional error O(dx2) which does not modify the leading error term of the scheme.

jjDwjj ¼ Oðdx3 Þ. Injecting the last relation in (53), the final result is H 6 hHRS jþ1 ¼ hjþ1 þ Oðdx Þ 2

2

ð55Þ

as anticipated. The above reasoning is valid in flow regions where the solution is regular, but is likely to fail close to flow discontinuities, where large jumps at cell interfaces are to be expected. It is then possible that the simplified scheme introduces larger numerical error in the vicinity of shocks and contact discontinuities, thus producing

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

less sharp discontinuity profiles than schemes satisfying the U-property exactly. Such a conjecture is confirmed by numerical results presented in Section 5. The above schemes are extended to two-dimensional flow problems using a cell-centered finite volume formulation. Solutions are marched in time using four-stage Runge–Kutta time-stepping, which is fourth-order accurate in time for linear problems and second-order accurate for nonlinear problems [21]. For steady problems, local time stepping, implicit residual smoothing and multigrid are used to speed up convergence. Fully upwind MUSCL extrapolation (k = 1) is used to raise the schemes accuracy to second order. The van Albada slope limiter is applied to prevent spurious oscillations near flow discontinuities. A crucial point for complex BZT fluid flow computations is the selection of the correct, entropic weak solution of system (3). Roes scheme does not satisfy a discrete entropy condition at sonic and stagnation points, where the numerical dissipation term vanishes on some of the characteristic fields. For this reason, eigenvalues of Roes matrix are corrected as proposed by Harten and Hyman [22], using the spectral radius of the Jacobian matrix as the velocity scale; this ensures a right amount of numerical dissipation on all of the characteristic fields, and the convergence of the scheme towards the physical vanishing viscosity solution.

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mann problem have been considered, described in Table 1. The working fluid is a van der Waals gas with c = 1.0125, representative of a heavy fluorocarbon. The initial conditions of the Riemann problem are normalized with respect to critical density and pressure. The newly proposed case DG0, concerning the propagation of two rarefaction fans, is characterized by a continuous solution, and allows to verify the actual order of accuracy of the considered solvers. Shock-tube conditions DG1–DG3, firstly proposed in [13], involve the formation and transport of non-classical waves. The interested reader may refer to [13] for a detailed discussion of the flow physics of these dense gas problems, including reflection and interaction of different families of waves, and their representation on the p–v plane. Results are computed using the real gas solvers of Guardone and Vigevano (GV), Glaister (GL) and the simplified solver described in Section 3 (RS), and computational grids of increasing resolution, composed by 100, 200, and 400 cells, respectively. 4.1.1. Test case DG0 The present test case, involving continuous solutions, has been used to evaluate the order of convergence of the numerical schemes under investigation using

0

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1

4. Results Guardone-Vigevano Glaister Simplified

Reduced density

In the present section, the numerical schemes described above are evaluated for a set of one-dimensional and two-dimensional flow problems. The first series of numerical experiments concerns a dense-gas shock tube. For this problem, numerical results exist in the literature [13,8], which can be used for code validation. The second test case involves supersonic flow of a BZT fluid past a forward facing step. Present results are compared with those of [23], in order to validate the present implementation for two-dimensional problems. The last test case, concerns transonic dense gas flow through the linear cascade of turbine blades VKI LS-59.

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Numerical results are first presented for a one-dimensional shock tube. Several initial conditions of the Rie-

Fig. 2. Case DG0. Density distribution on a 100-cell grid. Comparison of three schemes. t* = 0.15, Dt = 2 · 103.

Table 1 Riemann problem data for the one-dimensional shock tube Case

ql

ul

pl

Cl

qr

ur

pr

Cr

DG0 DG1 DG2 DG3

0.100 1.818 0.879 0.879

0.5 0 0 0

0.100 3.000 1.090 1.090

0.679 4.118 0.031 0.031

0.100 0.275 0.562 0.275

0.5 0 0 0

0.100 0.575 0.885 0.575

0.679 0.703 4.016 0.703

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Richardsons extrapolation. The flow field is characterized by left-running and right-running continuous rarefaction waves and a middle contact discontinuity. The Fundamental Derivative remains positive everywhere. Given three numerical solutions computed by a numerical scheme on uniform grids of increasing spacing, h, 2h, and 4h, it is possible to estimate the schemes order of convergence as  ð4hÞ  F  F ð2hÞ p ¼ log logð2Þ ð56Þ F ð2hÞ  F ðhÞ where F(h), F(2h), and F(4h) denote the computed values of a solution functional F on the three grids, respectively (see [24]). In the present work, F is taken equal to the average density in the tube at non-dimensional time t* = 0.15. Solutions are computed using 100, 200, and 400 computational cells; the ratio Dt/dx is chosen equal

2

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to 0.2 for all computations. The convergence rate for all three schemes was 1.734 ± 0.001, which confirms the reduced influence of the chosen linearization procedure on the solution accuracy. Fig. 2 shows the computed density distribution at t* = 0.15 on the coarser grid. Solutions given by the three schemes under investigation are superposed. 4.1.2. Test cases DG1–DG3 We now turn to flow problems involving non-classical dense-gas effects. The first test case, in the following designated as DG1, represents a Riemann problem where both the left and the right states lie within the C > 0 region. The solution is characterized by a left-running rarefaction wave, a middle contact discontinuity and a right-running compression wave. The left-running wave starts as a rarefaction fan in the positive C region.

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Fig. 3. Results for the DG1 shock-tube. Comparison of three Roe-type schemes. Coarse grid, 100 cells. t* = 0.15, Dt = 5 · 104.

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However, during the flow evolution the transition line is crossed, and the rarefaction fan steepens into a rarefaction shock for states with C < 0. The right-running compression wave also crosses the transition line. Nevertheless, as the Rankine–Hugoniot conditions and entropy inequality are satisfied for this right-running wave, discontinuous compression is still allowed on both sides of the transition line, and no mixed wave is created. For these more challenging problem, the solution of the GV and GL solvers are still superposed; conversely, the simplified solver RS gives a slightly different solution, especially in the vicinity of the transition from continuous rarefaction to the expansion shock. In general, the RS scheme resolves flow discontinuities less sharply than the GV and GL solvers. However, the maximum difference is about 3% on a 100-cell grid (see Fig. 3) and it tends to disappear with grid refinement.

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On a 400-cell grid the three solutions are practically indistinguishable (Figs. 4 and 5). Test case indicated as DG2 has both left and right states lying within the negative C region. During the evolution, the fundamental derivative remains negative everywhere, and the flow behavior is exactly opposite with respect to ‘‘classical’’ Riemann problems in common fluids. Specifically, the solution presents a left-running rarefaction shock, a middle contact discontinuity, and a right-running compression fan. In spite of the non-classical waves characterizing the flow, the problem is not very challenging for numerical schemes, once the entropy behavior has been properly fixed. In fact, the tube has a quite low pressure ratio and only very weak waves are generated. The three schemes under investigation give, independently from the computational grid used, results that only differ on the fifth or sixth decimal

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Fig. 4. Results for the DG1 shock-tube. Comparison of three Roe-type schemes. Medium grid, 200 cells. t* = 0.15, Dt = 5 · 104.

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Fig. 5. Results for the DG1 shock-tube. Comparison of three Roe-type schemes. Fine grid, 400 cells. t* = 0.15, Dt = 5 · 104.

digit (using double precision arithmetics). Results on the finer grid are reported in Fig. 6 for the GV solver only. The last test case, DG3, provides an example of evolution joining a state within the negative C region and a state with positive C. The Fundamental Derivative changes its sign from left to right and a mixed rarefaction wave forms at the crossing of the transition line. The compression wave lies entirely within the classical zone, and it is of discontinuous type. Comparisons between the three Roe-type schemes show differences of about 0.5% at worst on the coarser grid (in the region close to the compression shock). Fig. 7 shows results obtained with the GV solver, the remaining schemes providing very similar solutions. The enlargement in Fig. 8 shows the transition from the rarefaction shock to the rarefaction fan at the location where the Fundamental Derivative changes its sign, computed on a

1000-cell grid. Let us remark that, in spite of the numerical smearing of the expansion shock, due to numerical dissipation, it is possible to identify quite clearly the transition from discontinuous to continuous rarefaction. The transition is correctly predicted by all the considered schemes, whose solutions are superposed. 4.2. BZT supersonic flow past a forward-facing step We now turn to two-dimensional test cases. We consider the supersonic flow of a BZT polytropic van der Waals with inlet Mach number equal to 1.5 past a forward facing step. The computational domain length is two times the initial channel height. The step height is 1/10 of the initial channel height. The domain is covered by an equally spaced Cartesian grid in both directions, with grid size equal to 1/100 the channel height.

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Fig. 6. Results for the DG2 shock-tube. Fine grid, 400 cells. t* = 0.45, Dt = 1.5 · 103.

Characteristic boundary conditions are imposed at the inlet/outlet boundaries, and a slip condition based on two-dimensional linear extrapolation of the pressure from interior points is used at the channel walls. The use of a cell-centered finite volume formulation avoids difficulties at the step corner. The inlet reduced pressure and density are taken equal to 0.98 and 0.62, respectively. The flow is characterized by a detached bowshock upstream of the step, which terminates into a Mach reflection at the upper boundary of the channel. Such a shock compresses the flow into the C > 0 region. At the step corner, a mixed expansion fan/shock wave forms. The expansion shock completely expands the flow through the C < 0 region. Close to the upper boundary, the reflected oblique shock spreads into a compression fan as it enters the C < 0 region downstream of the expansion shock. Numerical isopycnics

for the GV solver are presented in Fig. 9. Thick white lines represent C = 0 contours. Velocity vectors are also represented. The present solution is in good agreement with results of Ref. [23], where a power law variation of the specific heat was used instead of the present polytropic gas assumption. In spite of the complexity of the computed flow field, solutions provided by the GL and RS solvers are practically indistinguishable from the GV solution, as demonstrated by Fig. 10, showing the density distribution at y = 0.5. 4.3. Dense gas flow through a linear turbine cascade The last test case concerns the flow of a dense gas through the linear turbine cascade VKI LS-59. The blade geometry and experimental data for a perfect diatomic gas (air) flow through this cascade can be found in

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x

Fig. 7. Results for the DG3 shock-tube. Fine grid, 400 cells. t* = 0.2, Dt = 103.

[25]. For the computations, a C-grid composed by 192 · 16 cells has been used, showed in Fig. 11. The blade is characterized by a bluff trailing edge, which may cause problems when performing inviscid computations. For this reason, in the present calculations, the blade has been slightly modified in order to obtain a sharp trailing edge. The cascade data are: inlet angle equal to 30, expansion ratio pin/pout ’ 1.81. This corresponds, for air flow through the turbine, to sonic exit conditions. Characteristic boundary conditions are imposed at the inlet and outlet boundaries: the static pressure, density, and flow angle are prescribed at the (subsonic) inlet, whereas the static pressure is specified at the outlet boundary when the flow is subsonic. If the flow reaches supersonic conditions at the outlet, the flow variables are just extrapolated from interior points. At the wall, the slip condition is applied. Finally,

periodic boundary conditions are imposed at the interpassage boundaries. Firstly, perfect gas results obtained using a secondorder Roe-type scheme are shown for reference in Fig. 12. Specifically, such a figure shows the flow contours of the pressure coefficient (C p ¼ 1  p=p0in , with p0in the inlet stagnation pressure). The flow is characterized by a shock wave located at about 70% of the blade chord, and by a second shock attached to the trailing edge. Then, transonic flow of the heavy fluorocarbon PP10 (chemical formula C13F22) is considered. Such a fluid exhibits a region of negative C values above the upper saturation curve (see for example [3] for more details). The comprehensive Martin–Hou equation is now adopted to model the gas behavior. The overall solution for the GV solver is provided in Fig. 13. Solutions

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

1

0.2

1279

0.3 1

Reduced density

Guardone-Vigevano

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4 0.2

0.3

x

0.4

Fig. 11. VKI LS-59 cascade: computational grid.

Fig. 8. Results for the DG3 shock-tube. Enlargement of the transition from discontinuous to continuous evolution in the mixed rarefaction wave. t* = 0.48, Dt = 103.

1 1.5

0.8

1.3 0.6

1.1

y

0.9

0.4

0.7 0.5

0.2

0.3 0

0

0.5

1

x

1.5

2

Fig. 9. Supersonic forward facing step: numerical isopycnics (Dq = 0.05). M1 = 1.5. GV solver.

1.5

0

0.5

1

1.5

2

1.5

Guardone-Vigevano Glaister Simplified

1.25

ρ

1.25

1

1

0.75

0.5

0.75

0

0.5

1

1.5

2

0.5

x Fig. 10. Supersonic forward facing step: density distribution at y = 0.5.

Fig. 12. VKI LS-59 cascade, perfect gas flow. Pressure coefficient contours, DCp = 0.05.

provided by the GL and RS solvers agree with the above to within the fifth–sixth decimal digit (working with double precision arithmetics). Fig. 14 shows the computed Mach distribution at the wall for air and PP10: in the dense gas flow, C < 1 on the suction side: thus, the sound speed drops during the flow expansion and a lower maximum Mach number is reached. Specifically, the compression shock on the blade suction side is completely suppressed, and the strength of the trailing edge

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P. Cinnella / Computers & Fluids 35 (2006) 1264–1281 Table 2 Relative computational times for three test cases Test case

Glaister

Guardone–Vigevano

Simplified

Dense-gas hock tube Forward-facing step Dense-gas turbine

1.18 1.14 1.15

1.12 1.10 1.19

1 1 1

computed cascade efficiency (based on the static real-toideal enthalpy drop through the cascade) is less than 94% when a perfect diatomic gas is considered, and less than 88% if the working fluid is taken to be water vapor, whereas it becomes more than 95% for the same cascade working, with the same pressure ratio, with dense gas PP10. These efficiency gains are only due to the particular properties of the working fluid: there is hope to improve them by combining the proper choice of the working fluid with optimization of the blade shape. 4.4. Computational efficiency

Fig. 13. VKI LS-59 cascade, dense gas flow. Pressure coefficient contours, DCp = 0.05.

1.4

0

0.25

0.5

0.75

Perfect gas BZT gas

1.2

1.2

1

Mach

1 1.4

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.25

0.5

0.75

0 1

x/c Fig. 14. VKI LS-59 cascade. Mach distribution at the wall.

shock is greatly reduced. Even if not shown, the present solution for the perfect gas is in good agreement with experimental data by Kiock et al. [25]. A detailed study of the efficiency of transonic dense gas turbine cascades has been provided in Ref. [26]. In order for the reader to have an idea of the possible efficiency improvements deriving from the use of dense gases, we recall here that

We conclude this section with some final remarks about computational efficiency. Table 2 reports scaled computational times (with respect to the RS scheme) for the three schemes considered. For steady 2D problems, the results presented in Table 2 refer to computational costs per iteration and per point. Convergence histories were found to be very similar for the three schemes, and are not reported. For the one-dimensional problem, and using the relatively simple van der Waals equation, the simplified schemes requires a CPU time about 15% less than the GL solver and 11% less than the GV solver. This gains are due essentially to the lower number of evaluations of the equation of state required. For two-dimensional flows governed by the same equation of state, such gains are slightly reduced, as the computational time required for the evaluation of thermodynamic properties has a lower weight on the overall computational cost. On the other hand, the computational gains obtained with the simplified scheme for a given problem, grow with the complexity of the chosen equation of state. Also notice that, when using the van der Waals equation of state, the GV solver is cheaper than the GL solver, as the intermediate density can be analytically evaluated using Cardanos formulas. The opposite is true when using the Martin–Hou equation: in this case, the GV solver requires an iterative computation of the intermediate density, and computational cost increases.

5. Conclusion In the present work, the performance of some extensions of Roes upwind scheme to the computation of dense gas flows with non-classical nonlinearities has been evaluated. Such flows exhibit complicated wave struc-

P. Cinnella / Computers & Fluids 35 (2006) 1264–1281

tures, including expansion shocks, mixed shock/fan waves, and splitting shocks. On the other hand, flow discontinuities in the dense regime are typically much weaker than in dilute gas flows. In Section 3, the extension of Roes scheme to flows governed by general equations of state has been briefly reviewed. In particular, one standard extension using a linearization procedure in quasi-Jacobian form and a more recent solver using a more rigorous linearization in strict Jacobian form have been analyzed. Then, an approximated linearization procedure reducing implementation complexity has been proposed. Truncation error analysis shows that, if the simplified method does not meet the U-property requirements exactly, nevertheless it introduces additional errors which, at least in smooth flow regions, are higher order infinitesimals with respect to the schemes leading truncation error term. The three schemes have finally been tested for some one-dimensional and two-dimensional problems, governed by different equations of state. Initially, a one-dimensional shock tube problem involving the propagation of non-classical waves has been considered. For flows characterized by continuous solutions, the schemes exhibit an almost identical convergence order. The solutions provided by the first two solvers, which satisfy the U-property exactly, are always superposed. The simplified solver, on the other hand, gives slightly more diffused discontinuities for Riemann problems involving relatively high pressure ratios. For lower pressure ratios, the solution of the simplified scheme coincides to within the sixth decimal digit with those of the exact solvers. For two-dimensional problems, involving non-classical supersonic flow past a forward-facing step, and dense gas flow through a turbine cascade, the three solvers under analysis provide almost identical solutions. Computational costs have also been evaluated. The simplified scheme allows to reduce CPU requirements for all of the test problems considered here. Such gains are higher when using more predictive, complex equations of state involving several virial expansion terms and non-polytropic gas models. In summary, present results show that the particular strategy chosen for extending Roes scheme to flows governed by general equations of state has almost no influence on the solution accuracy, when dense gas flows are computed. Therefore, the solver choice can be oriented towards numerical solvers which minimize implementation complexity and computational costs.

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