Embed Size (px)
Citation preview
Advances in Applied Mathematics and MechanicsAdv. Appl. Math. Mech., Vol. 4, No. 1, pp. 72-92
DOI: 10.4208/aamm.11-m1160February 2012
A Consistent Characteristic Boundary Condition forGeneral Fluid Mixture and Its Implementation in aPreconditioning Scheme
Hua-Guang Li1,3, Nan Zong2, Xi-Yun Lu1 and Vigor Yang3,∗
1 Department of Modern Mechanics, The University of Science and Technologyof China, Hefei, Anhui 230026, China2 Department of Mechanical and Nuclear Engineering, The Pennsylvania StateUniversity, University Park, PA 16802, USA3 School of Aerospace Engineering, The Georgia Institute of Technology, Atlanta,GA 30332, USA
Received 23 March 2011; Accepted (in revised version) 21 June 2011
Available online 13 December 2011
Abstract. Characteristic boundary conditions that are capable of handling generalfluid mixtures flow at all flow speeds are developed. The formulation is based onfundamental thermodynamics theories incorporated into an efficient precondition-ing scheme in a unified manner. Local one-dimensional inviscid (LODI) relationscompatible to the preconditioning system are proposed to obtain information car-ried by incoming characteristic waves at boundaries accurately. The approach hasbeen validated against a variety of sample problems at a broad range of fluid statesand flow speeds. Both acoustic waves and hydrodynamic flow features can passthrough the boundaries of computational domain transparently without any un-physical reflection or spurious distortion. The approach can be reliably applied tofluid flows at extensive thermodynamic states and flow speeds in numerical simu-lations. Moreover, the use of the boundary condition shows to improve the compu-tational efficiency.
AMS subject classifications: 65M10, 78A48Key words: Real-fluid, preconditioning method, method of characteristics, LODI relations.
1 Introduction
Proper implementation of boundary conditions is of great importance in obtaining re-liable and accurate numerical solutions of compressible flows. Much effort has been
∗Corresponding author.URL: http://soliton.ae.gatech.edu/people/vigor.yang/Email: [email protected] (H. G. Li), [email protected] (N. Zong), [email protected] (X. Y. Lu),[email protected] (V. Yang)
http://www.global-sci.org/aamm 72 c⃝2012 Global Science Press
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 73
applied to develop robust numerical boundary conditions based on the method ofcharacteristics (MOC) [1]. The technology was originated from the classical character-istic solution of differential equations of a hyperbolic system [2]. The MOC methodwas later extended to address multidimensional inviscid system with non-reflectiveacoustic boundary conditions [3]. Poinsot and Lele [4] further developed an approachcapable of treating viscous compressible flows. They derived local one-dimensionalinviscid (LODI) relations to calculate quantities associated with incoming character-istic waves and improved the accuracy of the method. The viscous terms are addedseparately and can be relaxed smoothly to inviscid flows as viscous effects diminish.Except for problems with gently varying flow and weak acoustic wave near bound-aries, the boundary conditions based on characteristic method can usually get moreaccurate and physical results than simple extrapolation method or simplified Riemanninvariants boundary conditions. In addition, the method gives rise to a more robustsolution procedure. Baum et al. [5] extended the characteristic boundary conditionsfrom equations which only can consider perfect gas with constant homogeneous ther-modynamic properties to accommodate multi-component reactive flows with variablethermodynamic properties. Although the characteristic boundary conditions havebeen widely used, most of those studies, however, focused on treating the fluid flowsin the ideal gas regime. Very limited work has been conducted to develop a consistentboundary treatment for fluid flows under high-pressure and low-temperature condi-tions, for which the method based on ideal gas does not work and real fluids effectsmust be considered [6].
Many fluid flows involve thermodynamic states in the trans-critical and super-critical regimes, where the real-fluid effect is notably strong. Fluid properties mayexperience severe variations when approaching the liquid-gas critical point [7] (seeFig. 3). This is important in heat transfer and fluid dynamics and raises a challenge tothe boundary condition treatment. For example, a very small thermal disturbance cangenerate enormous unsteadiness in the form of shock and expansion waves in a near-critical fluid due to the abnormally high fluid compressibility, a phenomenon knownas the piston-effect in near-critical fluids [7]. The boundary conditions must be treatedproperly to avoid unphysical oscillations and reflections that will then propagate intothe computational domain and ruin the solution. Okong’o and Bellan [8] extendedMOC based boundary conditions from ideal gas to multi-component real-fluid mix-tures. The formulations have been validated against the convection of a single vortexin supercritical heptane/nitrogen homogeneous mixture with supersonic flow speeds.Although it has been demonstrated that the extended characteristic boundary condi-tion can successfully treat fluid flows with moderate real-fluid effects in high-speedflow regimes, the method cannot be directly implemented to handle transcritical orsupercritical fluid flows at very low flow speeds because there is no low Mach num-ber flow treatment technique such as the preconditioning method [9–11].
In this paper, we aim at developing consistent boundary conditions for generalfluid mixtures flow at all speeds. The acoustic wave propagation related formula-tions, such as system eigenvalues and eigenvectors, are derived based on the unified
74 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
treatment of general fluid thermodynamics [6, 14] and incorporated into an efficientpreconditioning scheme [11] to deal with low Mach number flows. The LODI rela-tions were proposed for the governing system with preconditioning techniques in aconsistent manner.
The feasibility and robustness of the method are examined through testing caseswith a broad range of fluid state and flow speed. The acoustic wave propagation caseshows the method with real-fluid effects treatment works well at the ideal gas regime.The single vortex convection case demonstrates the capability of handling real-fluidflow with Mach number ranging from 0.1 to 1.6. The near-critical fluid piston effectcase shows the current method works well even when fluid properties exhibit anoma-lies. It is also shown that the implementation of this consistent boundary conditionresults in a fast convergence in the pseudo-time space and thus enhances the compu-tational efficiency.
This paper is organized as follows. In Section 2, theoretical framework is de-scribed, including the improved preconditioning schemes [11] and formulation deriva-tion for the characteristic boundary conditions. Section 3 introduces numerical imple-mentation and carries out numerical tests. Finally a brief summary concludes thepresent work.
2 Theoretical formulation
2.1 Euler equations and preconditioning scheme
To facilitate discussion, only the three-dimensional preconditioned Euler equationswith species equations are considered. The viscous terms can be approached sepa-rately as introduced by Poinsot and Lele [4]. The preconditioned conservation equa-tions used here exhibit the following form,
Γ∂
⌢
Q∂τ
+∂Q∂t
+∂E∂x
+∂F∂y
+∂G∂z
= 0, (2.1)
where Γ represents the preconditioning matrix and τ the pseudo-time. The conserva-tive variable vector, Q is defined as
Q =(
ρ, ρu, ρv, ρw, ρet, ρY1, ρY2, · · · , ρYN−1
)T. (2.2)
Pressure-temperature type of primitive variables is adopted as the independent vari-
ables. The pseudo-time variable vector,⌢
Q is defined as,
⌢
Q =(
p′, u, v, w, T, Y1, Y2, · · · , YN−1
)T, (2.3)
where ρ denotes the density, u, v and w the velocity components. T and et denotethe temperature and the specific total energy, respectively. Yi and N denote the mass
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 75
fraction of species i and the number of species, respectively. Explicit expressions ofthe convective flux vectors E, F and G are given by Hsieh and Yang [10], Meng andYang [6] and Oefelein [15]. The pressure p is decomposed into a constant referencepressure p0 and gauge pressure p′ to circumvent the pressure singularity problem atlow Mach numbers [9],
p = p0 + p′. (2.4)
The details of this method can be found in [11].
2.2 Method of characteristics
The method of characteristics is introduced by transforming the governing equationsto characteristic form. Define Jacobian matrices as
T =∂Q
∂⌢
Q, A =
∂E
∂⌢
Q, B =
∂F
∂⌢
Q, C =
∂G
∂⌢
Q, (2.5)
then Eq. (2.1) can be written as
Γ∂
⌢
Q∂τ
+ T∂
⌢
Q∂t
+ A∂
⌢
Q∂x
+ B∂
⌢
Q∂y
+ C∂
⌢
Q∂z
= 0. (2.6)
The eigen-properties of the preconditioning system are determined by the matrix Γ−1A,Γ−1B and Γ−1C instead of A, B and C for Navier-Stokes equations. For brevity, onlyoperation in the x-direction is discussed. Characteristic form of the preconditioningsystem (2.1) can be obtained through diagonalizing Γ−1 A
Γ−1 A = MΛM−1, (2.7)
where the columns of M consist of eigenvectors of Γ−1A, and diagonal components ofΛ consist of the eigenvalues of Γ−1A. The matrix Γ and A are given in the appendix.Due to involving both preconditioning process and real-fluid effects, the derivation ismore complex than that for ideal gas. With a series of matrix operation, M and Λ canbe obtained. Here we give the matrix in a way convenient for developing code,
M =
M11 M21 0 0 0 0 · · · 0M12 M22 0 0 0 0 · · · 0
0 0 1 0 0 0 · · · 00 0 0 1 0 0 · · · 01 1 0 0 1 0 · · · 00 0 0 0 0 1 · · · 0...
......
......
.... . .
...0 0 0 0 0 0 · · · 1
. (2.8)
76 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
The components of matrix M are given by
M11 =ATγp + BTρ2a2
γρR0, M12 = M11,
M21 =R1(R2 + R3)
ρR0(γ1/2R4 − γR3u/Θ + γR3ua2), M22 =
R1(R2 − R3)
ρR0(γ1/2R4 + γR3u/Θ − γR3ua2),
where a, γ and Θ represent the sound speed, the specific heat ratio and the scalingfactor, respectively,
Θ =ε−2 + (γ − 1)
a2 ,
and ε is a preconditioning factor that can optimally control the convection and diffu-sion process [9, 12, 13]. Optimal values are specified locally by the relation
ε = min{
1, max{ε inv, ε∆t, εvis}}
.
The subscripts inv, ∆t, and vis refer to the inviscid, unsteady, and viscous precondi-tioning factors, respectively
ε inv =
ε2
0, M ≤ ε0,2M2, ε < M < 1,1, M ≥ 1.
The term ε0 is a small number and is included to avoid singularities in stagnationregions where M = 0
ε∆t =1a2
[( lx
π∆t
)+
( ly
π∆t
)+
( lz
π∆t
)]+ M2.
Here, lx, ly and lz represent the characteristic dimensions of the computational domainand M is the Mach number
εvis =[ u2δx(δx − 1)
u2δ2x+ a2 ,
v2δy(δy − 1)v2δ2
y+ a2 ,
w2δz(δz − 1)w2δ2
z+ a2
],
and
δx =ν
uCFL
VNN, δy =
ν
vCFL
VNN, δz =
ν
wCFL
VNN,
where ν is viscous coefficient. The parameters
AT =( ∂p
∂T
)ρi
, BT =( ∂e
∂T
)p,Yi
,
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 77
can be derived from basic thermodynamics relations [11]. The parameters Ri are givenby
R0 =N
∑i=1
Yi⌢e i − e − p
ρ, R1 = BTρ2a2 + ATγ
(pa2 +
γ
ΘρR0
),
R2 = 2γ12 u(1 + ε)R1, R3 = 2R1((1 − ε)2γu2 + 4γεa2)
12 ,
R4 = 2BTρ2a6 +γ3
Θ2 BTρ2a2u2 − 2γ
ΘBTρ2a4u2 + BTγρ2a6u2
+ ATγ2(
pa4u2 + 2γ
ΘBT pa2(a2 − u2) +
γ2
Θ2 (2ρa2R0 + γpu2))
,
where ⌢e i is the partial-density internal energy [6],
⌢e i =(∂ρe
∂ρi
)T,ρj =i
. (2.9)
The inverse of matrix M is given as
M−1 =
M′11 M′
12 0 0 0 0 · · · 0M′
21 M′22 0 0 0 0 · · · 0
0 0 1 0 0 0 · · · 00 0 0 1 0 0 · · · 0
M′51 0 0 0 1 0 · · · 0
0 0 0 0 0 1 · · · 0...
......
......
.... . .
...0 0 0 0 0 0 · · · 1
, (2.10)
where
M′11 =
M22
M11M22 − M12M21, M′
12 =M12
M11M22 − M12M21, M′
21 =M21
M11M22 − M12M21,
M′22 =
M11
M11M22 − M12M21, M′
51 =M21 − M22
M11M22 − M12M21.
The diagonal eigenvalue matrix Λ is
Λ = diag( λ1 λ2 · · · λN+3 λN+4 ), (2.11)
where λ1 and λ2 represent the rescaled acoustic wave speed propagating upstreamand downstream, respectively,
λ1 =12
[(1 + ε)u −
√(1 − ε)2u2 + 4εa2
], (2.12a)
λ2 =12
[(1 + ε)u +
√(1 − ε)2u2 + 4εa2
], (2.12b)
λ3∼N+4 = u. (2.12c)
78 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
All of the eigenvalues in the pseudo-time space are real and have signs consistentwith the directions of characteristic wave propagation. The present scheme not onlypreserves the hyperbolicity of the system, but also gives rise to individual eigenvaluesthat behave in a manner representative of the physical reality involved [10].
Eq. (2.1) is placed in characteristic form through multiplying by M−1Γ−1,
M−1Γ−1(
Γ∂
⌢
Q∂τ
+ T∂
⌢
Q∂t
+ A∂
⌢
Q∂x
+ B∂
⌢
Q∂y
+ C∂
⌢
Q∂z
)= 0. (2.13)
To simplify Eq. (2.13), define variables D and V as
D ≡ Γ−1(
T∂
⌢
Q∂t
+ B∂
⌢
Q∂y
+ C∂
⌢
Q∂z
), (2.14a)
dV ≡ M−1d⌢
Q + M−1Ddτ. (2.14b)
Eq. (2.13) then can be written into characteristic form in terms of V,
∂V∂τ
+ Λ∂V∂x
= 0, (2.15)
or∂V∂τ
+ L = 0, (2.16)
whereL =
(L1, L2, · · · , LN+3, LN+4
)T= Λ
∂V∂x
. (2.17)
The propagating direction of characteristic waves is determined by the sign of thecharacteristic values (λi). The variables V and L corresponding to outgoing wavesmust be specified from solution of interior computational domain, and those corre-sponding to incoming waves must be determined from information of exterior com-putational domain.
2.3 Local one-dimensional inviscid relations
LODI relations are introduced to avoid arbitrariness in deciding information from ex-terior computational domain. Since conservation form of the governing equations isusually preferred in a compressible flow simulation, Eq. (2.15) is written into a moreapplicable form,
Γ∂
⌢
Q∂τ
+ T∂
⌢
Q∂t
= −(
B∂
⌢
Q∂y
+ C∂
⌢
Q∂z
)− ΓML. (2.18)
By neglecting the transverse convective term and viscous term, LODI relations can bederived for the preconditioning system,
Γ∂
⌢
Q∂τ
+ T∂
⌢
Q∂t
+ ΓML = 0. (2.19)
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 79
Considering that the pseudo-time term will converge to zero, Eq. (2.19) becomes
∂Q∂t
+ ΓML = 0. (2.20)
The application of the LODI relations is in the same way as that introduced by Poinsotand Lele [4]. First solve Li corresponding to outgoing characteristic waves from insidecomputational domain solution, then solve the rest Li through LODI relations, andfinally, substitute Li into Eq. (2.18) and solve it to obtain all independent quantities atboundaries.
3 Treatments of various boundary conditions
3.1 Subsonic outflow boundary conditions
For simplicity and not losing generality, we take u > 0, λ1 < 0 and λ2 > 0 for sub-sonic outflow boundary condition. The sign of the eigenvalues indicates the propa-gating direction of characteristic waves. There are N+3 outgoing and one incomingcharacteristic waves at the boundary. For the outgoing waves associated with λ2∼N+4,L2∼N+4 can be obtained from solution of inside computational domain. For the incom-ing wave associated with λ1, we can solve L1 through LODI relations to get partialreflecting boundary condition, or set L1 = 0 to get perfectly non-reflecting boundarycondition. However, simply setting L1 = 0 will raise pressure drifting which couldbecome a serious problem in the calculation. Rudy and Strikerda [16], Poisot andLele [4], and Baum et al. [5] have proposed that for the incoming wave on subsonicoutflow boundary,
L1 = k(p − p∞), (3.1)
should be used to feedback information of infinite field pressure p∞ into the compu-tational domain, where k is a factor to determine the speed with which the averagepressure in the computational domain relaxes towards the imposed pressure at infin-ity. Rudy and Strikerda [16] proposed that optimal value of k is given by
k =2σεa2(1 − M2)
lc√
u(1 − ε)2 + 4εa2, (3.2)
where M represents the maximum Mach number in the computational domain, lc isthe characteristic length of the domain and σ is a scaling factor for optimization. Sub-stitute Li into Eq. (2.18) and solve it combining with equation of state, and then we canobtain all independent physical quantities at the outflow boundary.
3.2 Subsonic inflow boundary conditions
For subsonic inflow boundary conditions we still take u > 0, λ1 < 0 and λ2 > 0. Ac-cordingly, there are N+3 incoming and one outgoing characteristic waves on the sub-sonic inflow boundary. Many physical conditions can be applied to subsonic inflow
80 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
boundary. Here we choose to give u, v, w, T and Yi for the inlet boundary condition. L1can be obtained from solution of inside computational domain, and then L2∼N+4 canbe solved through LODI relations. With the given boundary conditions and solved Li,we can solve the first equation in Eq. (2.18) combining with equation of state to obtaingauge pressure p′ and finally get all independent quantities on the inlet boundary.
3.3 Supersonic outflow boundary conditions
For supersonic outflow boundary conditions with u > 0, λ1 > 0 and λ2 > 0, allcharacteristic waves on the boundary are outgoing. Therefore, all Li can be obtainedfrom inside computational domain solution and no additional boundary condition isrequired.
3.4 Supersonic inflow boundary conditions
For supersonic inflow boundary conditions with u > 0, λ1 > 0 and λ2 > 0, all charac-teristic waves are incoming, so all independent quantities p, u, v, w, T and Yi shouldbe given on this boundary.
4 Numerical method
A dual-time-stepping integration technique is implemented to obtain time-accurateresults [6, 10, 14, 17]. The solution converged in pseudo-time corresponds to a time-accurate solution in physical time. One major advantage of this technique lies inthe fact that the convergence of the iterative process is dictated by the well-behavedeigenvalues in the pseudo-time space, instead of the original eigenvalues that maybecome disparate in certain flow regimes (e.g., low Mach number flows [17]). Astandard fourth-order Runge-Kutta scheme is employed to perform the inner-looppseudo-time integration because of its relatively higher temporal accuracy and greaterstability margin compared to many commonly used explicit schemes [18]. A localpseudo-time step is used to accelerate the convergence. To obtain an accurate result,a convergence criterion requires that relative residual error decreases three orders atleast and is smaller than 10−5. The temporal discretization of the real-time derivativeterm is obtained using a second-order backward difference. Spatial discretization isachieved by means of a fourth-order flux-differencing scheme [19]. Further improve-ment is acquired by adding both the second- and fourth-order artificial dissipationwith a total-variation-diminish (TVD) switch [20,21] to ensure numerical stability andconvergence. The second-order artificial dissipation term is enforced only in regionswith strong gradients, whereas the fourth-order term is applied in smooth regimesto achieve numerical stability. For boundary conditions, the same time integrationscheme as interior field is used and one-sided third-order difference scheme is usedfor special discretization.
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 81
5 Test cases
Three problems have been tackled to demonstrate the validity and capability of thepresent method, including one-dimensional acoustic wave propagation in the mix-ture of nitrogen and oxygen at standard state, one-dimensional piston effect in ni-trogen with pressure at the critical point p0 = pcr and temperature slightly abovethe critical point T0 = Tcr + 0.1K, and two-dimensional single vortex convection inhigh-pressure cryogenic nitrogen/oxygen mixture with a compressibility factor of 0.4.The widely used modified Soave-Redlich-Kwong (SRK) equation of state is adoptedfor the first and third samples to realize the treatment of real fluids [22]. Benedict-Webb-Rubin (BWR) equation of state is adopted for the near-critical nitrogen pistoneffect case because it is more accurate than the SRK equation of state to handle theanomalies exhibited by fluid properties in the close vicinity of the liquid-gas criticalpoint [23]. However, the computational cost of the BWR equation is larger than that ofthe SRK equation, and the latter is as accurate as the former in the regimes except fornear-critical and liquid regimes. To exhibit the flexibility of the present method, thetwo equations of state are used for different cases.
5.1 One-dimensional acoustic wave propagation in ideal gas mixture
One purpose of the non-reflecting MOC boundary condition is to eliminate non-physical reflection of acoustic waves penetrating an outflow boundary. Accordingly,the case of one-dimensional acoustic waves propagating towards an outflow bound-ary is often used to test boundary condition methods. Similar to Baum’s work [5], theone-dimensional acoustic wave is given as follows. The velocity distribution is
u = u0 + b exp[−
(c
x − 0.5ll
)2]. (5.1)
The pressure distribution is
p = p0 + p′ = p0 + ρ0a0(u − u0). (5.2)
The density distribution is
ρ = ρ0 +ρ0(u − u0)
a0. (5.3)
The mass fractions areYO2 = 0.33, YN2 = 0.67. (5.4)
The domain length l measures 1m. The reference velocity u0 is 1m/s. The referencepressure p0 is 1atm. The reference density ρ0 is 1.18kg/m3 . The parameter b and c,taking value of 10 and 5, determine the strength and stiffness of the acoustic wave,respectively.
Fig. 1 shows the process of the acoustic wave propagating towards the outflowboundary. The acoustic wave passed through the outflow boundary smoothly without
82 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
-1.0
0.0
1.0
2.0
3.0
-1.0
0.0
1.0
2.0
3.0-1.0
0.0
1.0
2.0
3.0-1.0
0.0
1.0
2.0
3.0
-1.0
0.0
1.0
2.0
3.0
x, m0.0 0.2 0.4 0.6 0.8 1.0
-1.0
0.0
1.0
2.0
3.0
0.0
2.0
4.0
6.0
8.0
0.0
2.0
4.0
6.0
8.0
0.0
2.0
4.0
6.0
8.0
0.0
2.0
4.0
6.0
8.0
0.0
2.0
4.0
6.0
8.0
x, m0.0 0.2 0.4 0.6 0.8 1.0
0.0
2.0
4.0
6.0
8.0
x (m) x (m)
u (m/s)p’ (kPa)
Figure 1: Propagation of acoustic wave towards outflow boundary at x = 1m in ideal-gas mixture with aninterval 0.3ms: gauge pressure p′ (left column) and velocity u (right column).
noticeable disturbance. More accurately, two probes are set at two positions, x/l = 0.8and the outflow boundary, to record the local pressure and velocity variation throughtime. As shown in Fig. 2, both the strength and stiffness of the acoustic wave arekept well without noticeable change during the entire propagation process. This alsodemonstrates that the current method can relax to ideal gas regime smoothly.
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 83
t, ms
p',kPa
0.0 0.5 1.0 1.5 2.0 2.5-0.5
0.0
0.5
1.0
1.5
2.0
2.5
= 0.8 1.0x/l
t, ms
u,m/s
0.0 0.5 1.0 1.5 2.0 2.50.0
2.0
4.0
6.0
8.0
= 0.8 1.0x/l
t (ms)t (ms) t (ms)t (ms)
u (m/s)
p’(kPa)
Figure 2: Propagation of the acoustic wave recorded by probes at x = 0.8m and outflow boundary x = 1m:gauge pressure (left) and velocity (right).
5.2 One-dimensional piston effect in near-critical nitrogen
To examine the performance of the current boundary treatment in the fluid regimewith strong real-fluid effects, the piston effect induced by thermal heating at theboundary of a near-critical fluid is modeled. The thermodynamic and transportanomalies occur when temperature and pressure approach the liquid-gas critical point(shown in Fig. 3). Under such a condition, the coefficient of volume expansion andisothermal compressibility approach infinite. And consequently, a very small changeof temperature/pressure of a near-critical fluid can lead to a significant volume di-latation. For instance, a more than 50 percent density reduction of nitrogen fluid oc-curs when temperature increases about one Kelvin from the critical value, whereasa one Kelvin temperature increasing can only induce less than 0.5% of density vari-ation at the standard state. In addition, the heat capacity under constant pressurediverges and thermal diffusivity tends to be zero in the regime close to the criticalpoint. For instance, the thermal diffusion of near-critical nitrogen at (Tcr + 1K, pcr)is more than three orders of magnitude lower than that under standard state condi-tions, which makes the conduction process extremely slow in a near-critical fluid. Thehighly variable properties of near-critical fluid have substantial impact in aspects offluid dynamic and heat transfer.
The piston effect is the response of a compressible fluid to a thermal disturbance.For example, in compressible fluids, a hot thermal boundary generated near a heatedwall expands in acoustic time scale and acts like a piston. If the thermal disturbance isstrong and rapid enough, shock waves could be generated. For the highly compress-ible and low thermal diffusive near-critical fluid, this effect becomes prominent, andanomalous heat transport by the piston effect happens [7]. Accurate and consistentboundary conditions are needed to deal with the strong real-fluid effects and strongacoustic waves. Moreover, the preconditioning method is suitable to handle the largedisparity between acoustic time scale, flow time scale and diffusion time scale in thiscase. BWR equation of state is used to calculate the highly variable thermodynamicproperties accurately.
84 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
temperature, K
density,kg/m
3
120 122 124 126 128 130100
200
300
400
500
600
p = pcr
termperature, K
Z
120 122 124 126 128 1300.1
0.2
0.3
0.4
0.5
0.6
0.7
p = pcr
temperature, K
volumeexpansivity,K
-1
122 124 126 128 13010
-2
10-1
100
101
p = pcr
termperature, K
µ,Pa.s
124 126 128 1301E-05
2E-05
3E-05
4E-05
termperature, K
Cp,J/kg.K
124 126 128 13010
3
104
105
p = pcr
p / pcr
isothermalcompressibility,Pa-1
0.996 0.998 1.000 1.002 1.004
10-5
10-4
10-3
T = Tcr
( )
( )
( )
temperature (K)temperature (K)
temperature (K) p/pcr
temperature (K) temperature (K)
T=Tcrp=pcr
p=pcr
p=pcr
p=pcr
p=pcr
Cp( )
μ( )
Figure 3: Thermodynamic and transport properties of nitrogen in the close vicinity of critical point (Z, µand Cp are compressibility factor, dynamic viscosity and specific heat under constant pressure, respectively).
The physical model studied consists of one infinite vertical plane wall and infinitelarge region filled with near-critical nitrogen on the right side of the wall. The initialpressure is critical pressure of nitrogen (p0 = pcr) and the initial temperature is 0.1Kabove the critical temperature of nitrogen (T0 = Tcr + 0.1K), where pcr = 3.4Mpa,Tcr = 126.2K. The wall temperature Tw starts to increase linearly with time from thebeginning (t = 0),
Tw(t) = T0 + 5t. (5.5)
The x axis is normal to the wall and directs from the wall to the fluid. The computa-
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 85
x, µmT,K
0 25 50126.300
126.302
126.304
126.306
126.308
126.310
x ( )
T (K)
Figure 4: Temperature field evolution of side heated near-critical nitrogen from the beginning with an interval0.015µs (the arrow indicates the direction of time evolution).
x, µm
p',kPa
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8
1.0
1.2
x, µm
u,m/s
0 10 20 30 40 500.00
0.01
0.02
0.03
x ( ) x ( )
p’(kPa)
u (m/s)
Figure 5: Propagation of the compression wave in side heated near-critical nitrogen from the beginning withan interval 0.015µs: gauge pressure (left) and velocity (right) (the arrow indicates the direction of timeevolution).
t, µs
p,kPa
0.0 0.1 0.2 0.3 0.4 0.50.0
0.3
0.6
0.9
x=40µm 50µm
t, µs
u,m/s
0.0 0.1 0.2 0.3 0.4 0.50.00
0.01
0.02
50µmx=40µm
t ( ) t ( )
p’(kPa)
u (m/s)
x=40x=40
Figure 6: Propagation of the compression wave recorded by probes at x = 40µm and outflow boundaryx = 50µm: gauge pressure (left) and velocity (right).
tional domain starts from the wall at x = 0 and ends in the nitrogen field at x = 50µm.Non-reflecting MOC boundary condition is used on the right side for the open bound-ary resultant from physical domain truncation.
The piston effect, another heating mechanism other than thermal diffusion andconvection, is illustrated in Fig. 4. The entire field nitrogen was heated within oneacoustic time period. Fig. 5 disclosures the origination of the fast heating mechanism,the strong compressive wave induced by wall boundary heating. The wave trans-
86 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
mitted right-towards and heated the fluid behind the wave front. Accurate and con-sistent boundary conditions must be implemented to avoid disturbance on the openboundary. Fig. 6 shows the local wave propagation process recorded at two samplinglocations at x = 40µm and the open boundary (x = 50µm). The strong compressivewave passed through the open boundary transparently without any disturbance. Thisdemonstrates that the current boundary condition method works well even for thenear-critical fluid with anomalies shown in thermodynamic and transport properties.
5.3 Two-dimensional single vortex convection in real-fluid mixture
Characteristic boundary conditions are theoretically exact only for one-dimensionalproblems. To test the performance of the present method in multi-dimensional simu-lations, two-dimensional single vortex convection in a high-pressure cryogenic nitro-gen/oxygen mixture with strong real-fluid effects is simulated. Similar to Poinsot andLele’s work [4], the vortex is given as follows. The stream function is
ψ = C exp(− x2 + y2
2R2
). (5.6)
The velocity distribution is
u = u0 +∂ψ
∂y, v = −∂ψ
∂x. (5.7)
The pressure distribution is
p = p∞ + p′ = p∞ + p∞[(u − u0)
2 + v2] 12 . (5.8)
The temperature isT = T0. (5.9)
The mass fractions areYO2 = 0.33, YN2 = 0.67. (5.10)
The Mach number of free stream is
Ma =u0
a0. (5.11)
The parameter C taking value of 5 × 10−3 determines the vortex strength. R equalingto one tenth of computational domain length is the characteristic radius of the vortex.u0 is the mean flow velocity with different values for different Mach numbers. To testthe current method for real fluids, the free stream temperature T0 is set to 120K andthe pressure P∞ is set to 100atm. Under these conditions the compressibility factorZ of the nitrogen/oxygen mixture is 0.4, which is considerably far away from idealgas. Both subsonic and supersonic cases are tested. For subsonic cases, both lowMach number case (Ma = 0.1) and moderate Mach number case (Ma = 0.6) are
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 87
1
1 2
4
5
6 8
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
1
1
4 6
7
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
1
4
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
(a) (b)
(c) (d)x ( ) x ( )
x ( )x ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
Figure 7: Evolution of vorticity field (s−1) of a single vortex passing through the outflow boundary on theright-side, Ma = 0.1: (a) t = 0.0ms, (b) t = 0.75ms, (c) t = 1.35ms, (d) t = 2.5ms.
1
1
2
2
3
4 5
5
6
7
8
9
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3
Level p'
9 2.6E+06
8 2.3E+06
7 2.0E+06
6 1.7E+06
5 1.4E+06
4 1.1E+06
3 8.0E+05
2 5.0E+05
1 2.0E+05
1
1
1
11
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
7
7
7
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3
Level p'
9 2.6E+06
8 2.3E+06
7 2.0E+06
6 1.7E+06
5 1.4E+06
4 1.1E+06
3 8.0E+05
2 5.0E+05
1 2.0E+05
1
1
2
2
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3
Level p'
9 2.6E+06
8 2.3E+06
7 2.0E+06
6 1.7E+06
5 1.4E+06
4 1.1E+06
3 8.0E+05
2 5.0E+05
1 2.0E+05
1
2
2
3
3
x, m
y,m
-0.3 0.0 0.3-0.3
0.0
0.3
Level p'
9 2.6E+06
8 2.3E+06
7 2.0E+06
6 1.7E+06
5 1.4E+06
4 1.1E+06
3 8.0E+05
2 5.0E+05
1 2.0E+05
(a) (b)
(c) (d)x ( ) x ( )
x ( )x ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
Figure 8: Evolution of gauge pressure field (Pa) of a single vortex passing through the outflow boundary onthe right-side, Ma = 0.1: (a) t = 0.0ms, (b) t = 0.75ms, (c) t = 1.35ms, (d) t = 2.5ms.
tested. For the supersonic case Ma = 1.6 was applied. Fig. 7 shows the process ofthe single vortex transmitting through the outflow boundary in a uniform mean flow
88 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
11
1
1
34
45
6
6
8
9
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
1
1
2
3
5
6
78
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
1 1
1
6
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
1
12
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level ω
10 90
9 80
8 70
7 60
6 50
5 40
4 30
3 20
2 10
1 -10
(a) (b)
(c) (d)x ( ) x ( )
x ( )x ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
Figure 9: Evolution of vorticity field (s−1) of a single vortex passing through the outflow boundary on theright-side, Ma = 0.6: (a) t = 0.0ms, (b) t = 0.2ms, (c) t = 0.28ms, (d) t = 0.33ms.
1
1
2
2
2
3
3
4
4
56
6
7
7
7
89
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level p'
10 3.5E+06
9 3.2E+06
8 2.8E+06
7 2.5E+06
6 2.2E+06
5 1.8E+06
4 1.5E+06
3 1.2E+06
2 8.3E+05
1 5.0E+05
1
1
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level p'
10 3.5E+06
9 3.2E+06
8 2.8E+06
7 2.5E+06
6 2.2E+06
5 1.8E+06
4 1.5E+06
3 1.2E+06
2 8.3E+05
1 5.0E+05
1
1
1
1
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level p'
10 3.5E+06
9 3.2E+06
8 2.8E+06
7 2.5E+06
6 2.2E+06
5 1.8E+06
4 1.5E+06
3 1.2E+06
2 8.3E+05
1 5.0E+05
1
1
x, m
y,m
-0.1 0.0 0.1-0.1
0.0
0.1Level p'
10 3.5E+06
9 3.2E+06
8 2.8E+06
7 2.5E+06
6 2.2E+06
5 1.8E+06
4 1.5E+06
3 1.2E+06
2 8.3E+05
1 5.0E+05
(a) (b)
(c) (d)x ( ) x ( )
x ( )x ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
y ( )
Figure 10: Evolution of gauge pressure field (Pa) of a single vortex passing through the outflow boundaryon the right-side, Ma = 0.6: (a) t = 0.0ms, (b) t = 0.2ms, (c) t = 0.28ms, (d) t = 0.33ms.
with Mach number 0.1. Since viscosity is neglected in this case, the vortex keeps itsoriginal shape as at the beginning in the whole process and no diffusion happens. The
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 89
iteration steps
iterationerror
0 200 400 60010
-5
10-4
10-3
10-2
MOC BC
extrapolation BC
iteration steps
iterationerror
1000 1200 1400 160010
-5
10-4
10-3
10-2
MOC BC
extrapolation BC
iteration steps
iterationerror
5270 5280 5290 5300 531010
-5
10-4
10-3
10-2
MOC BC
extrapolation BC
(a) (b)
(c)
1 1 1 1 1
Figure 11: Comparison of convergence history of pseudo-time iteration in vortex convection case withMa = 0.6 between current boundary condition method and simple extrapolation boundary method: (a) theearly stage with uniform flow field, (b) the stage when the vortex passes through the outflow boundary, (c)zooming in of figure (b).
vortex passes through the boundary without distortion or reflection. Fig. 8 shows theprocess of the pressure field evolution with time. Due to the pressure difference inthe radical direction of the vortex, the pressure wave spread from the vortex centertowards outside with almost sound speed. The current method allows the pressurewave pass through the boundary smoothly and transparently without distortion orreflection. Satisfying results are also obtained for the moderate Mach number case(Ma = 0.6) as shown in Figs. 9-10 and for the supersonic case. This indicates that thepresent boundary conditions work well for multidimensional multi-species real-fluidflows at all speeds.
Fig. 11 shows the convergence history of pseudo-time iteration in the case withfree stream Mach number equal to 0.6. Comparison was made between the MOCboundary conditions and simple extrapolation boundary conditions. At the begin-ning, the outflow boundary has only uniform flows passing by, and therefore theresidual of both cases are almost the same as shown in Fig. 11(a). As the vortex getscloser to the boundary, different convergence history from two types of boundary con-ditions shows up. The extrapolation boundary condition case needs more iterationsteps to converge to the same residual level as in MOC boundary condition case (seeFig. 11(b)). The reason is that hydrodynamic and pressure fields can not pass theboundary smoothly, and obvious distortion of the flow field occurs in the near regionof the outflow boundary. The distorted flow field induces larger residual error, lead-ing to more iteration steps and computational time. Fig. 11(c) shows that in the MOC
90 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
boundary condition case, the converging speed is faster than that of the extrapola-tion boundary condition case, which is indicated by larger slop of residual converginghistory curve. This demonstrates that the current boundary method improves thecomputational efficiency.
6 Concluding remarks
Consistent boundary conditions based on the method of characteristics have been de-veloped for general fluid mixtures flow at all speeds. The formulation derivation ac-counts for real-fluid effects, and preconditioning process is introduced to deal withflows with very low Mach number. The derivation is complex but the results are sim-ple and easy to apply. Test cases are carried out for a broad range of fluid state andMach number. The results appear to be satisfactory; both acoustic wave and flow fieldcan pass the boundary smoothly without unphysical reflection, distortion or oscilla-tion near the boundary. The current method has also been demonstrated to improvethe computational efficiency of the preconditioning scheme and work well within theentire range of thermodynamic and hydrodynamic states.
Appendix
The preconditioning matrix Γ is given by
Γ =
Θ 0 0 − ATAρ
Θu ρ 0 − AT uAρ
Θv 0 ρ − AT vAρ
Θht +( N
∑i=1
Yi ei − e − pρ
)( ∂ρ∂p)
T,Yiρu ρv ρBT − AT et
Aρ
ΘY1 0 0 − ATY1Aρ
......
......
ΘYN−1 0 0 − ATYN−1Aρ
− AY1Aρ
· · · − AYN−1Aρ
− AY1 uAρ
· · · − AYN−1 uAρ
− AY1 vAρ
· · · − AYN−1 vAρ
ρBY1 −AY1 et
Aρ· · · ρBYN−1 −
AYN−1 et
Aρ
ρ − AY1 Y1
Aρ· · · − AYN−1 Y1
Aρ
.... . .
...
− AY1 YN−1
Aρ· · · ρ − AYN−1 YN−1
Aρ
.
H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92 91
The Jacobian matrix A is
A =
u( ∂ρ∂p )T,Yi ρ 0 0 −u AT
Aρ
1 + u2(∂ρ∂p )T,Yi 2ρu 0 0 −u2 AT
Aρ
uv( ∂ρ∂p )T,Yi ρv ρu 0 −uv AT
Aρ
uw(∂ρ∂p )T,Yi ρw 0 ρu −uw AT
Aρ
u[ht + (N∑
i=1Yi
⌢e i − e − pρ ) + 1]( ∂ρ
∂p )T,Yi ht + ρu2 ρuv ρuw u(ρBT − AT etAρ
)
uY1(∂ρ∂p )T,Yi ρY1 0 0 −u ATY1
Aρ
......
......
...
uYN−1(∂ρ∂p )T,Yi ρYN−1 0 0 −u ATYN−1
Aρ
−uAY1Aρ
· · · −uAYN−1
Aρ
−u2 AY1Aρ
· · · −u2 AYN−1Aρ
−uvAY1Aρ
· · · −uvAYN−1
Aρ
−uwAY1Aρ
· · · −uwAYN−1
Aρ
u(ρBY1 −AY1 et
Aρ) · · · u(ρBYN−1 −
AYN−1 et
Aρ)
u(ρ − AY1 Y1
Aρ) · · · −u
AYN−1 Y1
Aρ
.... . .
...
−uAY1 YN−1
Aρ· · · u(ρ − AYN−1 YN−1
Aρ)
.
Acknowledgments
The authors would like to thank Professor H. Ding for improving the manuscript. Thiswork was supported in part by the National Natural Science Foundation of China(Nos. 11132010 and 11072236).
References
[1] T. COLONIUS, Modeling artificial boundary conditions for compressible flow, Ann. Rev. Fluid.Mech., 36 (2004), pp. 315–345.
[2] G. W. HEDSTROM, Nonreflecting boundary conditions for nonlinear hyperbolic systems, J.Comput. Phys., 30 (1979), pp. 222–237.
[3] K. W. THOMPSON, Time-dependent boundary conditions for hyperbolic systems, J. Comput.Phys., 89 (1987), pp. 439–461.
[4] T. J. POINSOT AND S. K. LELE, Boundary conditions for direct simulations of compressibleviscous flows, J. Comput. Phys., 101 (1992), pp. 104–129.
[5] M. BAUM, T. POINSOT AND D. THEVENIN, Accurate boundary conditions for multicompo-nent reactive flows, J. Comput. Phys., 116 (1994), pp. 247–261.
92 H. G. Li, N. Zong, X. Y. Lu and V. Yang / Adv. Appl. Math. Mech., 4 (2012), pp. 72-92
[6] H. MENG AND V. YANG, A unified treatment of general fluid thermodynamics and its applica-tion to a preconditioning scheme, J. Comput. Phys., 189 (2001), pp. 277–304.
[7] B. ZAPPOLI AND D. BAILLY, Anomalous heat transport by the piston effect in supercriticalfluids under zero gravity, Phys. Rev. A., 41 (1990), pp. 2264–2267.
[8] N. OKONG’O AND J. BELLAN, Consistent boundary conditions for multicomponent real gasmixtures based on characteristic waves, J. Comput. Phys., 176 (2002), pp. 330–344.
[9] Y. H. CHOI AND C. L. MERKLE, The application of preconditioning in viscous flows, J. Com-put. Phys., 105 (1993), pp. 207–223.
[10] S. Y. HSIEH AND V. YANG, A preconditioned flux-differencing scheme for chemically reactingflows at all Mach numbers, Int. J. Comput. Fluid. Dyn., 8 (1997), pp. 31–49.
[11] N. ZONG AND V. YANG, An efficient preconditioning scheme for real fluid mixtures usingprimitive pressure-temperature variables, Int. J. Comput. Fluid. Dyn., 21 (2007), pp. 217–230.
[12] P. E. O. BUELOW, Convergence Enhancement of Euler and Navier-Stokes Algorithms,Ph.D thesis, The Pennsylvania State University, 1995.
[13] S. VENKATESWARAN AND C. L. MERKEL, Dual time stepping and preconditioning for un-steady computations, AIAA Paper, No. 95-0078, 1995.
[14] J. C. OEFELEIN AND V. YANG, Modeling high-pressure mixing and combustion processes inliquid rocket engines, J. Propul. Power., 14 (1998), pp. 843–857.
[15] J. C. OEFELEIN, Mixing and combustion of cryogenic oxygen-hydrogen shear-coaxial jet flamesat supercritical pressure, Combust. Sci. Tech., 178 (2006), pp. 229–252.
[16] D. H. RUDY AND J. C. STRIKWERDA, A nonreflecting outflow boundary condition for subsonicNavier-Stokes calculations, J. Comput. Phys., 18 (1980), pp. 55–70.
[17] J. S. SHUEN, K. H. CHEN AND Y. H. CHOI, A coupled implicit method for chemical non-equilibrium viscous flows at all speeds, J. Comput. Phys., 106 (1993), pp. 306–318.
[18] A. JAMESON, The evolution of computational methods in aerodynamics, J. Appl. Math., 50(1983), pp. 1052–1070.
[19] M. M. RAI AND S. CHAKRAVARTHY, Conservative high-order accurate finite difference meth-ods for curvilinear grids, AIAA Paper, No. 1993-3380.
[20] P. JORGENSON AND E. TURKEL, Central difference TVD schemes for time dependent andsteady state problems, J. Comput. Phys., 107 (1993), pp. 297–308.
[21] R. C. SWANSON AND E. TURKEL, On central difference and upwind schemes, J. Comput.Phys., 101 (1992), pp. 292–306.
[22] G. SOAVE, Equilibrium constants from a modified Redlich-Kwong equation of state, Chem. Eng.Sci., 27 (1972), pp. 1197–1203.
[23] R. T. JACOBSEN AND R. B. STEWART, Thermodynamic of nitrogen including and vapor phasefrom 63K to 2000K with pressure to 10000 bar, J. Phys. Chem. Ref. Data., 2 (1973), pp. 757–922.
