Embed Size (px)
Citation preview
Shock Waveshttps://doi.org/10.1007/s00193-019-00934-y
ORIG INAL ART ICLE
Heat-induced planar shock waves in supercritical fluids
M. T. Migliorino1,2 · C. Scalo3
Received: 2 February 2019 / Revised: 31 October 2019 / Accepted: 6 December 2019© Springer-Verlag GmbH Germany, part of Springer Nature 2019
AbstractWe have investigated one-dimensional compression waves, produced by heat addition in quiescent and uniform initial condi-tions, in six different supercritical fluids, each taken in four states ranging from compressible pseudo-liquid fluid to ideal gas.Navier–Stokes simulations of a canonical semi-infinite domain flow problem, spanning five orders of magnitude of heatingrate, are also carried out to support the theoretical analysis. Depending on the intensity of the Gaussian-shaped energy source,linear waves or shock waves due to nonlinear wave steepening are observed. A new reference heating rate parameter allowsto collapse in the linear regime the whole dataset, together with the existing experimental data, thanks to its absorption ofreal-fluid effects. Moreover, the scaling strategy illustrates a clear separation between linear and nonlinear regimes for allfluids and conditions, offering motivation for the derivation of a unified fully predictive model for shock intensity. The latteris performed by extending the validity of previously obtained theoretical results in the nonlinear regime to supercritical fluids.Finally, thermal to mechanical power conversion efficiencies are shown to be proportional, in the linear regime, to the fluid’sGrüneisen parameter, which is the highest for compressible pseudo-liquid fluids, and maximum in the nonlinear regime forideal gases.
Keywords Scaling · Blast waves · Supercritical fluids · Heat addition · Peng–Robinson EoS · Navier–Stokes equations
1 Introduction
Whencompressiblefluids are thermally perturbed, amechan-ical response is generated in the form of waves [1,2], whichcan range in intensity from acoustic compressions [3] toshock waves [4]. Heat-induced waves appear in a wide vari-ety of natural and artificial phenomena, such as the soundproduced from asteroids’ impact on Earth’s atmosphere[5,6], blast waves from nuclear explosions [7], thermo-
Communicated by M. Brouillette.
B M. T. [email protected]
1 School of Mechanical Engineering, Purdue University, WestLafayette, IN 47907, USA
2 Present Address: Department of Mechanical and AerospaceEngineering, Sapienza University of Rome, via Eudossiana18, 00184 Rome, Italy
3 School of Mechanical Engineering, and Aeronautical andAstronautical Engineering (by courtesy), Purdue University,West Lafayette, IN 47907, USA
phones [8], shock waves induced by spark discharge [9],thunder [10], and spacemanufacturing processes [11].More-over, heat-induced fluctuations, referred to by some authorsas thermoacoustic waves [12,13] or thermoacoustic sound[8], are the governing mechanism for thermal relaxationin near-critical fluids in enclosed cavities [14–20]. Thisphenomenon, referred to as the piston effect [16] or ther-moacoustic convection [21], has been widely investigatedthrough numerical simulations [13,22–24] and experiments[3,21,25,26].
In spite of such efforts, fully predictive models for heat-induced wave intensity in supercritical fluids have beenlimited to low-amplitude acoustic waves [3], with less atten-tion devoted to shock waves. Moreover, the previouslyreported scaling relationships for heat-induced waves [27]fail in the case of supercritical fluids, because of the lackof consideration for real-fluid effects in the selection of thereference heating rate parameter. Furthermore, an open fun-damental question is whether supercritical fluids guaranteehigher heat to mechanical power conversion efficiency withrespect to ideal gases. Given the wide interest of the scien-tific community in non-ideal compressible fluid dynamics,
123
Author's personal copy
M. T. Migliorino, C. Scalo
Fig. 1 aPhase diagram of CO2showing flooded contours ofreduced density ρ/ρcr;b reduced density of CO2 versusreduced temperature T /Tcr forp = pcr (dashed line) andp = 1.1pcr (solid line). Bothplots are generated with thePeng–Robinson equation ofstate. PL, PB, and PG indicatepseudo-liquid, pseudo-boiling,and pseudo-gaseous conditions,respectively
0.8 0.9 1.0 1.1 1.2 1.3T/Tcr
0.0
0.5
1.0
1.5
2.0
2.5
ρ/ρ
cr
PG
PL
PB
0
200
400
600
800
1000
ρ(k
g/m
3 )
(a) (b)
especially in shock waves [28–32], it is desirable to addressthe aforementioned literature gaps.
The present manuscript is structured as follows. The prob-lem under consideration is formulated in Sect. 2, with theselection of thermodynamic states and fluids of interest inSect. 2.1 and the description of the governing equations andof the computational setup in Sect. 2.2. The results section(Sect. 3) covers: (1) the linear regime of heat-induced waves(Sect. 3.1), where the new reference heating rate parameteris proposed; (2) the nonlinear regime (Sect. 3.2), where wavesteepening is followed by the formation of shock waves, theamplitude ofwhich is accurately predictedwith a newmodel;and (3) the analysis of thermal to mechanical power con-version efficiency (Sect. 3.3). Finally, the manuscript’s mainfindings are summarized in Sect. 4.
2 Problem formulation
2.1 Fluidmodel
Hereafter, we refer to supercritical fluids as fluids pressur-ized above their critical point, p > pcr (Fig. 1a), indicatedby the subscript cr. Such fluids exhibit variations in den-sity, speed of sound, and thermal capacity, ranging fromliquid-like to gas-like values depending on their temperature,making them an attractive choice for the scope of the presentinvestigation. In fact, for a given supercritical pressure (cho-sen as p0 = 1.1pcr in this study), starting from cold andheavy pseudo-liquid conditions (PL, for T < Tcr), the den-sity rapidly drops for increasing temperatures (Fig. 1b) via apseudo-boiling (PB), or pseudo-phase transitioning, process[33–35], after which the fluid reaches a pseudo-gaseous (PG)state, and then eventually a near-ideal-gas (IG) state for suf-ficiently high temperatures. This behavior can be capturedby adopting analytically defined equations of state for realfluids, which enable a thermodynamically consistent closureof the governing flow equations.
Table 1 Marker legend for the selected six fluids, each consideredin four different supercritical thermodynamic states represented bygrayscale levels
Fluid PL PB PG IG Tcr(K) pcr(MPa)
CO2 304.13 7.3773
O2 154.58 5.043
N2 126.20 3.398
CH3OH 512.64 8.097
R-134a 374.26 4.059
R-218 345.10 2.68
T0/Tcr 0.89 1.02 1.11 2.20
p0/pcr 1.1 1.1 1.1 1.1
Black: pseudo-liquid fluid (PL); dark gray: pseudo-boiling (PB) fluid;light gray: pseudo-gaseous (PG) fluid; white: fluid near to ideal-gas (IG)conditions . Such reference conditions are indicated with the subscript0 and are used as initial conditions. All cases are considered at p0 =1.1pcr . Values of fluid-specific critical properties (Tcr , pcr) [36,37] arealso reported
The fluids considered in this study (Table 1) are: car-bon dioxide (CO2), oxygen (O2), nitrogen (N2), methanol(CH3OH), 1,1,1,2-tetrafluoroethane (CH2FCF3 or R-134a),and octafluoropropane (R-218). The Peng–Robinson equa-tion of state [38], hereinafter called PR EoS, is chosen forthe present investigation because of its simplicity and accept-able accuracy for the range of flow conditions considered.A Newton–Raphson-based iterative method is employedto obtain temperature from density and internal energy.Real-fluid dynamic viscosity and thermal conductivity areestimated via Chung et al.’s method [39], whose accuracyonly affects the simulation of thermoviscous processes (seeappendix of [40]). However, the jumps of the thermo-fluid-dynamic quantities across the shocks as well as the profilesof the compression waves in the isentropic wave propa-gation regime—both comprising the focus of the presentmanuscript—are independent from the values of viscosityand conduction coefficient. The prediction and analysis ofthe details of the shock profile structure is out of the scopeof this work.
123
Author's personal copy
Heat-induced planar shock waves in supercritical fluids
Fig. 2 Isobaric thermal expansion coefficient (1) versus reduced tem-perature obtained through the PR EoS. The symbols correspond to thefluids and conditions in Table 1
The isobaric thermal expansion coefficient (Fig. 2),
αp = − 1
ρ
(∂ρ
∂T
)p, (1)
is the thermodynamic variable of choice to guide the selectionof the different base conditions shown in Table 1: state in thepseudo-boiling (PB) region, in the immediate vicinity of themaxima of αp (T0 = 1.02Tcr), state in the pseudo-liquidregion (PL, T0 = 0.89Tcr), and state in the pseudo-gaseousregion (PG, T0 = 1.11Tcr). For the ideal-gas (IG) region,we choose the state at T0 = 2.2Tcr, for which αp0T0 ≈ 1.The IG state is thus modeled with the perfect ideal-gas EoS,p = ρRT , and with a constant ratio of specific heats γ ,taken equal to the ratio of cp and cv given by the PR EoS atT = 2.2Tcr and p = 1.1pcr.
The selection in Table 1 results in a total of 24 distinctthermodynamic initial states, spanning an overall pressure,temperature, and density range, respectively, of approxi-mately p0,max− p0,min = 6MPa, T0,max−T0,min = 1016K,and ρ0,max − ρ0,min = 1340 kg/m3.
2.2 Governing equations and computational setup
The governing equations for a fully compressible one-dimensional viscous flow read
∂C∂t
+ ∂F∂x
= S, (2)
where C = (ρ, ρu, ρE)T is the vector of conservative vari-ables, F = (ρu, ρu2 + p − τ, ρuE + pu − uτ + q)T isthe vector of fluxes, S = (0, 0, Q̇)T is the vector of sources.Time is indicated with t , x is the spatial coordinate, u is thevelocity, ρ is the density, p is the thermodynamic pressure,and E = e + u2/2 is the specific, i.e., per unit mass, totalenergy (sum of specific internal energy and specific kinetic
energy). TheNewtonian viscous stresses, expressed in accor-dance with Stokes’s hypothesis, and the heat flux, modeledwith Fourier heat conduction, are
τ = 4
3μ
∂u
∂x, q = −k
∂T
∂x, (3)
where μ is the dynamic viscosity, k is the thermal conduc-tivity, and T is the absolute or thermodynamic temperature.Bulk viscosity effects were neglected due to the lack of avail-able data in the literature for the fluids and flow conditionsinvestigated in this work. The spatiotemporal distribution ofthe imposed volumetric heating rate Q̇ is given by
Q̇(x, t) = ω̇g(x), g(x) = 1
�√2π
e− 12 (x/�)2 , (4)
respectively, where ω̇ (W/m2) is the planar heating rateand g(x) (m−1) is a Gaussian function with unitary (non-dimensional) integral on the real axis (
∫ ∞−∞ g(x)dx = 1)
with characteristic width � = 0.75µm, inspired by the thinfoil heater used in previous experiments [3]. Initial conditionsare quiescent and uniform. For each initial condition, simu-lations are performed at several values of planar heating ratespanning five orders of magnitude, with upper bound limitedby the occurrence of near-complete rarefaction of the fluid(ρ → 0) at the location of heat injection. The values of ω̇ con-sidered are 105, 107, 109, 1010, 3 ·1010, and 6 ·1010 W/m2
for all initial conditions in Table 1 and 1011, 3 · 1011, and6·1011W/m2 only for pseudo-liquid (PL) conditions. In total,162 computations are analyzed in this study.
The computational setup (Fig. 3), which considers onlyx ≥ 0 with symmetry conditions imposed at x = 0, issufficiently long to allow waves to form and track themnumerically to determine their propagation Mach number,M . Simulations are halted before perturbations reach theright boundary of the computational domain (located atx = 20µm). The analysis below focuses solely on the peakwave speed, measured upon coalescence of compressionwaves when and if shock formation occurs, but before theonset of wave amplitude decay due to thermoviscous losses.As such, the reported pressure jump values are independentfrom viscosity and conductivity.
Fig. 3 Computational setup for heat-induced wave generation. Thepost-compression state is denoted with the subscript 1, while pre-compression states are denoted with the subscript 0
123
Author's personal copy
M. T. Migliorino, C. Scalo
(a)
(b)
Fig. 4 Grid convergence analysis for Navier–Stokes simulationsof heat-induced shock waves in R-134a at PL conditions forω̇ = 6 · 1010W/m2. The grid selected for this case is the one withΔx = 0.5nm, and the reference solution employed for the convergencestudy is with Δx = 0.33nm
Fully compressible Navier–Stokes simulations are per-formed with the solver Hybrid [41], which has extensivelybeen used in the past for computations of shock waves.A fully conservative spatial discretization is employed topredict the correct shock speed [42], and a fourth-orderRunge–Kutta scheme is employed for timeadvancement. Thecompression wave fronts are resolved with a minimum of 14grid points (for the strongest shock) and with a time stepof at most Δt = 0.01 ns. A uniform grid was employedfor all cases, ranging in grid spacing from Δx = 8 nm(coarsest grid, used for weak shocks) to Δx = 0.5 nm(finest grid, used for strong shocks). While the thus obtainedshock profile, especially for the higher-Mach-number cases,is not physically accurate [43], this strategy avoids theadoption of shock-capturing schemes and still accurately pre-dicts the jump of the thermo-fluid-dynamic quantities acrossthe shock. Moreover, because of the numerical resolutionemployed, the results were not hindered by spurious pres-sure oscillations typically associated with transcritical flows[44,45], which are commonly contained by adopting dissi-pative and/or non-conservative schemes [46–48].
To assess the suitability of the grid resolution employed inthis study, a grid convergence analysis has been performedon themost computationally demanding case (Fig. 4). Under-
resolved computations, yielding spurious oscillations aroundthe shock location, are avoided by resolving with a sufficientamount of grid points all the computed shocks considered inthis work (Fig. 4a). Away from the shock, the flow is wellresolved even on the coarsest grids considered; furthermore,the numerical method is third-order accurate for this flowconfiguration (Fig. 4b). This analysis confirms that the flow-fields presented in this study are grid independent.
3 Results
3.1 Linear regime
Combining the linearized mass and momentum equations,with the assumption of isentropic flow, yields
∂2
∂t2δ p = a20
∂2
∂x2δ p, (5)
where δ p indicates a small pressure perturbation from thebase state. The initial conditions for (5) are
(δ p)t=0 = 0, (6)
and
(∂
∂tδ p
)t=0
= αp0a02
cp0ω̇g(x), (7)
where the latter is derived from the pressure evolutionequation [49] evaluated at the initial time. With the initialconditions imposed by (6) and (7), d’Alembert’s analyticalsolution of (5) is given by
δ p
ρ0a20= Ω
2
[(√2
2
x + a0t
�
)− erf
(√2
2
x − a0t
�
)]. (8)
The dimensionless linear pressure perturbation [left handside of (8)] depends on the dimensionless space and timecoordinates x/� and a0t/�, respectively, and on the dimen-sionless heating rate Ω; this normalization removes thedependence from the base state, and hence the specific fluid,achieving full collapse of the data as discussed below. Thedimensionless heating rate is obtained by normalizing ω̇withthe reference parameter plotted in Fig. 5, yielding
Ω = ω̇
2ρ0a0cp0/αp0= G0ω̇
2p0a0γ̃0, (9)
123
Author's personal copy
Heat-induced planar shock waves in supercritical fluids
Fig. 5 Reference scaling parameter for heating rate versus reducedtemperature obtained through the PR EoS. The symbols correspond tothe fluids and conditions in Table 1
where cp is the isobaric specific thermal capacity, theGrüneisen parameter [50] is
G = αpa2
cp= γ − 1
αpT, (10)
which reverts to γ − 1 for ideal gases, where γ = cp0/cv0
is the ratio of specific isobaric and isochoric thermal capac-ities, the relation a2Tαp
2/cp = γ − 1 has been used, andγ̃0 = ρ0a20/p0 is the isentropic exponent [51]. We restrictthe applicability of the present scaling strategy, and hence ofthe reference parameter used in (9), only to thermodynamicstates with positive Grüneisen parameter G0. The scalingdoes not apply, for example, to (near) isothermal acousticwaves (i.e., γ0 tending to one) in dense gases or to fluidsexhibiting non-standard Riemann problem solutions [52].
The dimensionless pressure jump across the compressionwave,
Π = p1 − p0ρ0a20
, (11)
where the subscript 1 indicates the post-compression state(Fig. 3), is the classic definition of shock strength [53]. Com-paring (8) and (11) yields
Π = Ω. (12)
This result is equivalent to the one obtained by Miura etal. [3], who predicted the amplitude of heat-induced linearacoustic waves via
p1 − p0 = ρ0a0T0
(∂T
∂ p
)s,0
ω̇
2, (13)
which, using the thermodynamic relation (∂T /∂ p)s =αpT /(ρcp), becomes
p1 − p0 = a0αp0
cp0
ω̇
2, (14)
which, made dimensionless with (9) and (11), reverts to (12).Chu [27] used 2p0a0 as the reference heating rate param-
eter for ideal gases. His scaling strategy, applied to thespatial pressure profiles considered in this study (Fig. 6a, b),does not collapse the data for all conditions considered inTable 1, not even among the IG cases. This is because Chu’sscaling parameter does account for variations in γ , thereforecollapsing only perfect ideal gases with the same value ofγ . The IG cases considered in this work possess differentvalues of γ , and thus, they can be properly scaled only ifthe reference heating rate is 2ρ0cp0a0T0 (Fig. 6c), equal top0a0γ /(γ − 1) for perfect ideal gases. Still, this upgradedscaling parameter, which collapses well IG data, does nottake into account real-fluid effects. This issue is resolved byreplacing T0 with 1/αp0 , yielding full collapse of the numer-ical data across all fluids and conditions (Fig. 6d). For suchlow heating rates (Ω ≤ 3.634·10−5), the compressionwavessatisfy the hyperbolic problem described by the linearizedwave equation (5) and their dimensionless profiles all col-lapse into the functional law of (8). Notice that other scalingstrategies not taking into account compressibility effects can-not achieve collapse of the data analyzed in this study (seeAppendix 1).
The scaling parameters used in Fig. 6, whose data arelimited to Ω ≤ 3.634 · 10−5, are also applied to the com-plete dataset (Figs. 7 and 8). Experimental data from Miuraet al. [3] have also been included, using the provided dimen-sionless value of the density jump, δρ/ρ0 = 1.1 · 10−7, andthe dimensional planar heating rate ω̇ = 1830W/m2. Thedimensionless density jump is converted to a dimensionlesspressure jump assuming isentropicity of the transformation(δρ/ρ0 = δ p/(ρ0a02)), and the provided heating rate ismadedimensionless via (9). Both numerical and experimental dataare well collapsed (Fig. 7d) by the present scaling strategy[(9) and (11)] up to Π = 5.09. This result is not achievedby the other scaling strategies (Fig. 7b, c). Shock speedsextracted from the Navier–Stokes simulations (Fig. 9a) arealso partially collapsed by following the proposed normal-ization of the heating rate (Fig. 9b). Moreover, even at thehighest wave Mach number obtained, the flow inside theshock-wave thickness has been found to be in the continuumlimit (see Appendix 2).
The scaling also allows to clearly observe(Figs. 8 and 9b) the departure from the linear prediction of(12), and to define linear (Ω ≤ 10−1) and nonlinear (Ω ≥10−1) regimes (Fig. 8). The dimensionless collapse obtainedin the nonlinear regime suggests that an approximate pre-diction of the shock strength Π from the knowledge of thedimensionless heating rate Ω only is possible. However, inthe nonlinear regime, data follow a fluid- and state-specificdeparture from the linear prediction (see inset in Fig. 8), cre-ating the need for a unified (linear and nonlinear) predictivemodeling framework.
123
Author's personal copy
M. T. Migliorino, C. Scalo
−3 −2 −1 0 1 2 3x − a0t (μm)
0
20
40
60
80
100
120
140δp
(Pa)
(a)
−4 −3 −2 −1 0 1 2 3 4(x − a0t)
0.0
0.1
0.2
0.3
[δp/
(ρ0a
2 0)]/
[ω̇/(
2p0a
0)]
(b)
−4 −3 −2 −1 0 1 2 3 4(x − a0t)
0
1
2
3
4
[δp/
(ρ0a
2 0)] /
[ ω̇/(
2 ρ0c
p 0a
0T0)
]
(c)
−4 −3 −2 −1 0 1 2 3 4(x − a0t)
0.0
0.2
0.4
0.6
0.8
1.0
[δp/
(ρ0a
2 0)]/
[ω̇G
0/(2
p 0a
0γ̃0)
]
(d)
Fig. 6 Scaling of linear pressure waveforms for heating rate ω̇ =105 W/m2. aDimensional pressure profiles; b scaling proposed byChu [27]; c scaling only collapsing IG data; d the proposed scaling(9, Fig. 5). The solution of the wave equation (8) is plotted as a thickdotted line ind. The maximum wave Mach number of the data shown
herein is M = 1.00003, and the dimensionless heating rate Ω rangesfrom 1.033 · 10−7 to 3.634 · 10−5, well within the linear regime (seeFig. 8). The symbols correspond to the fluids and conditions in Table 1and are used here to distinguish the curves from one another
3.2 Nonlinear regime
We initially focus on the heat-induced flowfield in R-134a,shown in Fig. 10, as a representative case to guide our anal-ysis and modeling of the nonlinear regime. The shocks,propagating to the right with speed us, are identifiable assharp gradients in all thermo-fluid-dynamic quantities. Byinspecting the dimensionless density variation profiles, whatis identified as a smeared contact discontinuity (c.d.) is foundmoving approximately at the shock-induced Eulerian veloc-ity u1. Based on inviscid theory (Fig. 11a), across the c.d.pressure and velocity are continuous, but density and totalenergy jump from a steady post-shock state value to a zoneat increasingly lower density, where heat is continuously sup-plied, referred to as the heated zone.
For a perfect ideal gas (IG, the first column of Fig. 10), thepressure and total energy in the post-shock and heated-zoneregions are approximately uniform, and their values are set bythe shock jump. In fact, IG pressure and volumetric internalenergy are directly proportional via ρe = p/(γ − 1). Thisis one of the key assumptions made by Chu [27] in deriving
his predictive model for heat-induced waves in ideal gases,which is discussed later in (29).
For a supercritical fluid, instead, the relationship betweenpressure and volumetric internal energy is nonlinear andalso dependent on another thermodynamic quantity such astemperature or density; hence, pressure and total energy per-turbations exhibit real-fluid effects. Moreover, their spatialprofiles between the shock wave and the heated-zone edgeare not uniform. Finally, in the heated zone, total energyvaries significantly with time and space, especially in PLconditions (the last column of Fig. 10), and pressure is notconstant. All of the above reasonings set the stage for themodeling procedure outlined below.
The following analysis considers the mirrored extensionof the semi-infinite problem in Fig. 3, from −ε to ε, where ε
is an arbitrarily large number such that ε > ust . Inside thisdomain, the integral of the vector of conservative variables[see (2)] is
123
Author's personal copy
Heat-induced planar shock waves in supercritical fluids
103 104 105 106 107 108 109 1010 1011 1012
ω̇ (W/m2)
100
101
102
103
104
105
106
107
108
109
p 1−
p 0(P
a)
(a)
10−7 10−5 10−3 10−1 101 103
ω̇/(2p0a0)
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Π
(b)
10−11 10−9 10−7 10−5 10−3 10−1 101
ω̇/(2ρ0cp0a0T0)
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Π
(c)
10−6 10−4 10−2 100 102
Ω
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Π
(d)
Fig. 7 Scaling of pressure jumps obtained from theNavier–Stokes sim-ulations versus heating rate. aDimensional pressure jump Π versusdimensional heating rate ω̇;b scalingproposedbyChu [27]; calternativescaling collapsing only IG data; d the proposed scaling (9, Fig. 5).The prediction of (12) is shown with the dashed line ind. The star
at Ω = 1.27 · 10−7 and Π = 1.1 · 10−7 represents the T0 − Tcr = 150mK case by Miura et al. [3]. The other symbols correspond to pressurejumps for the fluids and conditions in Table 1. All data are included insupplementary material
Fig. 8 Shock strength Π versus dimensionless heating rates Ω . Star:T0−Tcr = 150mKcase byMiura et al. [3]with reportedΠ = 1.1·10−7
and estimated Ω = 1.27 · 10−7; all other symbols: Navier–Stokes datafor all combinations of conditions in Table 1. The linear predictionof (12) is shown with a dashed line. The highest shock strength isΠ = 5.09, and the highest wave Mach number is M = 2.95
∫ ε
−ε
C(x, t)dx =∫ −us t
−ε
C(x, t)dx
+∫ us t
−us tC(x, t)dx +
∫ ε
us tC(x, t)dx . (15)
It should be noted that in the following derivation ε is not afunction of time. In fact, a sufficiently large, fixed value ofε can always be found for a given time window of analysis,[0, t f ], where t f is picked sufficiently large to capture shockformation in all cases. The time variation of the left hand sideof (15) is, after using Leibniz’s rule for each term on its righthand side,
d
dt
∫ ε
−ε
C(x, t)dx =∫ ε
−ε
∂C(x, t)
∂tdx, (16)
which becomes, after exploiting theNavier–Stokes equations(2),
d
dt
∫ ε
−ε
C(x, t)dx = −∫ ε
−ε
∂F∂x
dx +∫ ε
−ε
Sdx . (17)
Using the definition of Q̇ in (4), and considering the flow
123
Author's personal copy
M. T. Migliorino, C. Scalo
104 105 106 107 108 109 1010 1011 1012
ω̇ (W/m2)
102
103
us(m
/s)
(a)
10−6 10−4 10−2 100 102
Ω
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
M2−
1
(b)Mmax = 2.95
Fig. 9 Scaling of shock speeds obtained from the Navier–Stokes simu-lations versus heating rates for all combinations of conditions reportedin Table 1. a shock speed versus dimensional heating rate ω̇; bMachnumber squared minus unity versus dimensionless heating rate Ω
uniform and at rest for −ε ≤ x < −ust and ust < x ≤ ε,(17) becomes
d
dt
∫ ε
−ε
C(x, t)dx = (0, 0, ω̇)T, (18)
where ε is large enough so that∫ ε
−εg(x)dx � 1. Even if the
heat input introduces a net integral of positive momentum inthe x > 0-portion of the domain considered by the Navier–Stokes simulations, this analysis refers to the −ε < x < ε
domain, for which the net change in the integral value of themomentum is zero, as stated in (18), due to symmetry of theproblem about the x = 0 location. Integrating (18) in timeyields
∫ ε
−ε
C(x, t)dx −∫ ε
−ε
C(x, 0)dx = t(0, 0, ω̇)T, (19)
which can be further manipulated to finally obtain
∫ ust
−ustC(x, t)dx = t(2ρ0us, 0, 2ρ0e0us + ω̇)T. (20)
If the profiles of ρ and ρE are assumed to be symmetric withrespect to the plane x = 0, (20) becomes
1
t
∫ ust
0ρdx = ρ0us,
1
t
∫ ust
0ρEdx = ρ0e0us + ω̇
2,
(21)
and only x ≥ 0 is analyzed. For any given time, (21) can bewritten as
∫ M
0ρdξ = ρ0M,
∫ M
0ρEdξ = ρ0e0M + ω̇
2a0, (22)
where M = us/a0 is the wave Mach number and
ξ = x
a0t. (23)
By inspecting Fig. 10, the idealized scenario illustrated inFig. 11b is assumed, allowing to further develop (22), yield-ing
ρ3U +∫ M
Uρ2dξ = ρ0M, (24)
and
ρ3E3U +∫ M
Uρ2E2dξ = ρ0e0M + ω̇
2a0, (25)
whereU = u1/a0. Only a suitable choice ofρ2(ξ) in (24) and(25) is left for the prediction of heat-induced wave strength.We make the simplifying assumption of a linear functionalform for ρ2(ξ),
ρ2(ξ) = ρ3 + ξ −U
M − U (ρ1 − ρ3), (26)
which inserted in (24) yields the value of the heated zone’sdensity (Fig. 11b),
ρ3 = ρ1M − UM + U . (27)
A different choice for the functional form of ρ2(ξ) in (26)would change (27), but not the overall mass balance imposedby (24).
123
Author's personal copy
Heat-induced planar shock waves in supercritical fluids
(p−
p 0)/
(ρ0a
2 0)IG PG PB PL
ρE
/(ρ
0E0)
−1
u/a
0
0 2 4 6 8 10 12
ρ/ρ
0−
1
0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12
Fig. 10 Rows: dimensionless perturbations of pressure, total energy,velocity, and density in R-134a, extracted from the Navier–Stokes sim-ulations, plotted versus the dimensionless space coordinate x/� andshifted upward by an arbitrary value for every 1 ns of temporal evolu-tion. Values of heat deposition rate for IG, PG, and PB: ω̇ = 1010W/m2;
for PL: ω̇ = 6 · 1010W/m2. Similar behavior is found for the other flu-ids and conditions (not shown). Solid oblique lines: maximum shockvelocity (u = us) obtained with a tracking algorithm; dashed obliquelines: heated-zone edge velocity (c.d., u = u1) computed through thenumerical solution of the RH equations with us as input
123
Author's personal copy
M. T. Migliorino, C. Scalo
Fig. 11 Schematics ofheat-induced right-travelingshock wave and contactdiscontinuity (c.d.). aShock andc.d. trajectory in the space–timeplane; bassumed flow structurefor heat-induced field in asupercritical fluid. The densityprofile ρ2(ξ) is assumed to belinear (26), and the variable ξ isdefined in (23)
(a) (b)
In fact, the assumption ρ2(ξ) = ρ1 made by Chu [27]allows, together with the first RH jump condition (ρ1(us −u1) = ρ0us), to simplify (24) directly to
ρ3 = 0. (28)
Furthermore, when perfect ideal gases are considered,ρ2E2 = p1/(γ − 1) + ρ1u21/2 and the other RH equationssimplify (25) to
p1u1 = γ − 1
2γω̇, (29)
which coincideswith Eq. (34) of Chu’s article [27]. Given theuniform pressure of state 2, ρ2E2 is equal to its correspond-ing post-shock value, ρ1E1, consistently with the flowfieldobserved in the IG column of Fig. 10. Equation (29) can beconverted to the dimensionless form, yielding
Ω =√2Π(γΠ + 1)√Π(γ + 1) + 2
, (30)
which matches the parametrization derived in Eq. (39) ofChu’s paper [27], but recast following the normalization andsymbology used in this paper [see (9) and (11)]. For Π 1,(30) reverts to (12), removing the dependency from the ratioof specific heats γ . Equation (30) is only valid for perfectideal gases, for which ρ3e3 is only a function of p1.
Notice that ρ3 does not appear in (29); hence, ρ3 = 0(28) is not relevant for the modeling of perfect ideal gases.However, ρ3 = 0 is non-physical for a real fluid and preventsthe evaluation of the first term on the left hand side of (25),hence our different modeling assumption on the profile ofρ2 (26), which yields our assumed profile of ρ3 (27), whichis not null. This modeling strategy is valid for supercriticalfluids and can be used for perfect ideal gases as well, hencegeneralizing the model of Chu [27].
Equations (24) and (25) are solved with the follow-ing procedure, assuming all pre-shock values are known:first, for a given Mach number M , treated as input, the
Rankine–Hugoniot (RH) equations are solved numericallyvia a root finding algorithm, yielding all the post-shock quan-tities (ρ1, p1, u1, T1); then, ρ3 is computed from (27), ande3 = e(ρ3, p1) is obtained through the equation of state;finally, with the knowledge of ρ2(ξ) (26), p1, and u1, theintegral on the left hand side in (25) is computed, provid-ing a value for ω̇. In summary, the result is the numericaldetermination of the following relation:
ω̇ = ω̇(M, ρ0, p0, u0), (31)
where, for the particular setup considered here, u0 = 0.During the iterative procedure carried out on the RH equa-
tions, the admissibility problem of the non-unique solutionsneeds to be tackled. This is done by ensuring that Smith’sstrong condition for uniqueness of the Riemann problemsolution [52,54], γ̃ > G, holds for all thermodynamic statesconsidered. This corresponds to the medium condition ofMenikoff et al. [52]. Furthermore, we also confirm that thefundamental derivative of gasdynamics [53] is positive (i.e.,all isentropes are convex) for all cases considered in thiswork. Therefore, shock waves are of the compression typeand yield a positive entropy jump.
The proposed model for heat-induced waves accuratelypredicts the fluid- and state-specific trends exhibited by dataextracted from the Navier–Stokes simulations (Fig. 12), forall initial conditions in Table 1. In the nonlinear regime, theratio of the dimensionless pressure jump to the dimensionlessheating rate, Π/Ω , is maximum for perfect ideal gases andminimum for fluids in pseudo-liquid conditions (Fig. 12) andis, in all cases, always upper-bounded by the linear predictionΠ/Ω = 1.
The information gathered from the plots of the ratioΠ/Ω
is not sufficient to establish a quantitative measure for theefficiency of thermal to mechanical power conversion, whichis instead given in the next section.
123
Author's personal copy
Heat-induced planar shock waves in supercritical fluids
Fig. 12 Fluid-by-fluid shock strength versus dimensionless heating rateshowing deviation from the prediction of (12) (dashed line): Navier–Stokes simulations corresponding to the fluids and conditions in Table1 (symbols) and results from the modeling strategy ((24) and (25), solidlines). IG data (white-filled symbols) lay on the curve defined by 30.The linear scale in this figure hides data from the low-amplitude linearcompression wave cases (compare with Fig. 8)
3.3 Efficiency of thermal tomechanical powerconversion
Aquantitativemetric for the efficiency of thermal tomechan-ical power conversion can be obtained by dividing themechanical power carried by the shocks, 2(p1 − p0)u1, bythe total heat power input, ω̇,
η = (p1 − p0)u1ω̇/2
. (32)
(32) can be recast in the dimensionless form, after using theRankine–Hugoniot equations, yielding
η = G0Π2
ΩM, (33)
where G0 is the base Grüneisen parameter (10), which isused, in the context of thermoacoustic energy conversion[55], as the ratio between the work parameter γ − 1 andthe heat parameter αpT .
The efficiency computed from the Navier–Stokes data isaccurately predicted by the model outlined in Sect. 3.2, asshown in Fig. 13. For linear waves, the efficiency in (33) isapproximately given by
η = G0Π (34)
and is the highest for fluids in PL conditions, followed byPB, PG, and then IG conditions. On the other hand, as thedimensionless heating rate increases and the waves become
Fig. 13 Fluid-by-fluid shock thermal to mechanical energy conversionefficiency (32) versus dimensionless heating rate Ω: Navier–Stokessimulations corresponding to the fluids and conditions in Table 1 (sym-bols) and results from the modeling strategy (24 and 25, solid lines)
nonlinear, the efficiency of perfect ideal gases grows, until itbecomes the highest among all fluid conditions considered.The highest efficiencies are achieved inN2 andO2, the lowestin R134a and R218, with CO2 and methanol in between,indicating that heavier fluids convert heat into mechanicalwork more poorly.
In the limit of Ω → ∞, both the perfect ideal gas EoSand the PR EoS cease to be physically representative of theexisting fluids. Nonetheless, it is of theoretical interest toinvestigate the asymptotic behavior of heat-induced wavesin supercritical fluids, as done by Chu [27] for ideal gases.
For Ω 1 and, thus, shock strength Π 1, (30), validfor a perfect ideal gas, reverts to the two-thirds law proposedin Eq. (40) by Chu [27], which can be recast as
Π =(
γ + 1
2
)1/3 (Ω
γ
)2/3
. (35)
Thus, fluids in IG conditions tend to an asymptotic thermalto mechanical power conversion efficiency,
η = 1 − 1
γ, (36)
where we used the RH equations, (35), and (33).For supercritical fluids, instead, the ratio Π/Ω tends to
values lower than the ones obtained with fluids in IG condi-tions (Fig. 14a), and the efficiency η eventually decreasesto values even inferior to the ones of the linear regime(Fig. 14b). The asymptotic behavior shown in Fig. 14 con-cerns the jumps across the shocks, which rapidly abandonthe heated region due to their supersonic propagation speed.
123
Author's personal copy
M. T. Migliorino, C. Scalo
10−1 100 101 102 103 104 105 106 107 108
Ω
10−1
100
101
102
Π
(a)
∝ Ω2/3Π = Ω
PL
PB
PGIG
10−7 10−5 10−3 10−1 101 103 105 107
Ω
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
η
(b)
η = Ω
PL
PB
PG
IG
Fig. 14 Results from the predictivemodel (24 and 25) for R-134a in dif-ferent thermodynamic states. aShock strength Π versus dimensionlessheating rate Ω; befficiency of thermal to mechanical power conversionη (32) versus Ω . Equations (12) and (35) are plotted with a dashed lineina, and (36) is plotted with a dashed-dotted line inb
Hence, there should be no practical or theoretical expecta-tion of Ω → ∞ representing an ideal gas case for the shockdynamics per se. This can be further explained, keeping (25)in mind, by observing the profiles in Fig. 10, imagined inthe limit of Ω 1. For IG conditions, the heated zone’stotal energy does not increase significantly in time, regard-less of the imposed heating rate, implying that all of theinjected power directly sustains shock-wave propagation. Onthe other hand, for real fluids, the nonlinear relationshipsbetween thermodynamic variables entail a significant changeof volumetric internal energy, even with the slight variationsof pressure present in the heated zone. For PL conditions,in particular, most of the injected power is retained by theheated zone, allowing less power to flow toward the shockwave.
4 Conclusions
In summary, we have investigated heat-induced planar com-pression waves, generated by a Gaussian-shaped sourceterm in the total energy equation, in six different fluids at
supercritical pressure and in pseudo-liquid, pseudo-boiling,pseudo-gaseous, and ideal-gas conditions.
The reference heating rate parameter, expressed appro-priately in terms of the base isobaric thermal expansioncoefficient, collapses data from one-dimensional Navier–Stokes numerical simulations and from experimental resultsof Miura et al. [3]. The scaling strategy also outlines thedivision between linear and nonlinear regimes.
The real-fluid effects on the structure of thenonlinearflow-field have been revealed, and a fully predictive model, basedon global mass and energy conservation, has been proposed,generalizing the results of Chu [27] to supercritical fluids.
Finally, the thermal to mechanical power conversion effi-ciency in the linear regime has been shown to be proportionalto the base Grüneisen parameter, which is maximum for flu-ids in pseudo-liquid conditions. However, the efficiency inthe nonlinear regime is eventually the highest in ideal gases.In supercritical fluids, in fact, the nonlinear energy pathwaysinside the heated zone lead to a thermodynamic bottleneckfor energy conversion.
Future challenges for this work lie in addressing two-dimensional and three-dimensional effects, including and/orobtaining bulk viscosity data for each fluid at various shockamplitudes, and derive appropriate thermodynamic closureswhen and if flow conditions entail loss of equilibrium at amolecular scale.
Acknowledgements MTM acknowledges the support of the FrederickN. Andrews and Rolls-Royce Doctoral Fellowships from Purdue Uni-versity. The authors thank Pat Sweeney (Rolls-Royce) and Stephen D.Heister (Purdue) for the fruitful discussions that have inspired the needfor scaling across different fluids. The computing resources were pro-vided by the Rosen Center for Advanced Computing (RCAC) at PurdueUniversity and Information Technology at Purdue (ITaP). The authorsfinally thank the two anonymous reviewers and the editor for their veryuseful comments during the review process.
Appendix 1: Application of the Widom-linesimilarity law
The post-shock states, indicated with a subscript 1, obtainedin this work span the ranges p1/pcr = 1.1−69 and T1/Tcr =0.89− 4.34 (Fig. 15a), covering a large portion of the super-critical state space. As expected, fluids in PL conditionstend to exhibit the largest pressure response to small heat-induced temperature changes, while fluids in IG conditionsexhibit the weakest response. The dataset, if scaled usingthe recently proposed similarity law for Widom lines [56](Fig. 15b), does not show collapse onto a fluid-independentcurve for each initial condition (see Table 1). The Widom-line similarity proposed in [56], which was derived focusingon the p/pcr < 3 region, is able to only partially scale someof the data for PG fluids (see inset of Fig. 15b); overall, it
123
Author's personal copy
Heat-induced planar shock waves in supercritical fluids
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5T1/Tcr
100
101
102p 1
/pcr
(a)
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5T1/Tcr
100
101
102
(p1/
p cr)
A0/
As
(b)
0.9 1.0 1.1 1.2 1.31.01.11.21.31.41.5
0.9 1.0 1.1 1.2 1.31.01.11.21.31.41.5
Fig. 15 Post-shock thermodynamic states plotted in the reduced pres-sure versus reduced temperature space (a) and in the scaled reducedpressure [56] versus reduced temperature space (b). A0 = 5.52 forall fluids, while As = 6.57 for CO2, As = 5.63 for O2, As = 5.7for N2, As = 7.02 for R-218, As = 7.03 for R-134a, and As = 8.06for CH3OH. Solid lines and symbols (see Table 1) correspond to thenumerical solution of the Rankine–Hugoniot equations
lacks the ability to successfully scale data from simulationsof acoustic and shock waves.
Appendix 2: Knudsen number analysis
Supercritical fluids undergoing pseudo-phase change arecharacterized by a molecular reorganization process [34,57]with an associated finite timescale. This raises concernsregarding excessively thin shocks, yielding short fluid par-ticle residence times, challenging the validity of the ther-modynamic equilibrium assumption implicit in the presentcalculations.
In this study, the predicted particle residence times in theshock range from 4.7 ·10−3 ns to 0.8 ns, with fluids in PLconditions yielding the shortest residence times. Due to thelack of data on molecular relaxation times for all of the fluidsanalyzed here, we revert to an analysis based on the Knud-sen number to establish the likelihood of loss of equilibriumin the shocks considered. An excessively high Kn number,approaching the rarefied regime, would imply a low num-ber of collisions for a given shock thickness, hence longerrelaxation times. The relevant Knudsen number in this caseis
10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102
Ω
10−5
10−4
10−3
10−2
10−1
100
Kn
Fig. 16 Knudsen numbers for the heat-induced shocks computedthrough Navier–Stokes simulations for the fluids and conditions inTable 1
Kn = λ
δs, (37)
where δs is the shock thickness, asmeasured from theNavier–Stokes simulations, and an order-of-magnitude approximateestimate for the mean free path [58] λ is
λ = Mm√2πd2ρ0NA
, (38)
where NA = 6.02214076 · 1023 1/mol is the Avogadro num-ber, Mm is the molar mass, and d is the molecular kineticdiameter, which is d = 0.33nm for CO2; d = 0.364nm forN2; d = 0.346nm for O2 [59]; d = 0.41 nm for methanol(CH3OH) [60]; and d = 0.5nm for R-134a and R-218 (esti-mated from data of [61] on similar molecules).
The shock thicknesses extracted from the simulations δsrange from 4.6nm to 5.4µm, with the important caveat thatfully resolved shock profiles fromNavier–Stokes simulationsare only physically relevant for Mach numbers sufficientlyclose to unity [43] and, moreover, that the values of δs stemfrom a thermodynamic equilibrium assumption. The maxi-mum wave Mach number observed in this study is 2.95 (seeFig. 9b). As such, the values of δs used here serve as esti-mates, which are expected to be sufficiently accurate for thisanalysis as the details of the inner shock profile depend onthe choice of the EoS and the transport quantities which doaccount for real-fluid effects.
As apparent from the data shown in Fig. 16, the vastmajority of the shocks are well within the continuum regime(Kn << 1), primarily due to the high base pressures anddensities. We hence conclude that a fluid parcel undergo-ing compression across the shock is experiencing enough
123
Author's personal copy
M. T. Migliorino, C. Scalo
collisions at a molecular level to stay in thermodynamicequilibrium. The highest values of Kn, of around 0.03, areachieved by fluids in PL conditions for dimensionless heatdeposition rates of Ω > 0.1 and pose the most significantchallenge to the assumption of thermodynamic equilibrium.
References
1. Kassoy, D.R.: The response of a confined gas to a thermal distur-bance. I: Slow transients. SIAM J. Appl. Math. 36(3), 624–634(1979). https://doi.org/10.1137/0136044
2. Sutrisno, I., Kassoy, D.R.: Weak shocks initiated by power depo-sition on a spherical source boundary. SIAM J. Appl. Math. 51(3),658–672 (1991). https://doi.org/10.1137/0151033
3. Miura, Y., Yoshihara, S., Ohnishi, M., Honda, K., Matsumoto,M., Kawai, J., Ishikawa, M., Kobayashi, H., Onuki, A.: High-speed observation of the piston effect near the gas–liquid criticalpoint. Phys. Rev. E 74, 010101 (2006). https://doi.org/10.1103/PhysRevE.74.010101
4. Clarke, J.F.,Kassoy,D.R., Riley,N.: Shocks generated in a confinedgas due to rapid heat addition at the boundary. II. Strong shockwaves. Proc. R. Soc. Lond. A 393(1805), 331–351 (1984). https://doi.org/10.1098/rspa.1984.0061
5. Boslough, M., Crawford, D.: Low-altitude airbursts and the impactthreat. Int. J. Impact Eng. 35(12), 1441–1448 (2008). https://doi.org/10.1016/j.ijimpeng.2008.07.053
6. National Research Council: Defending Planet Earth: Near-Earth-Object Surveys and Hazard Mitigation Strategies. The NationalAcademies Press, Washington, D.C. (2010). https://doi.org/10.17226/12842
7. Taylor, G.: The formation of a blast wave by a very intense explo-sion. I. Theoretical discussion. Proce. R. Soc. Lond. A: Math.Phys. Eng. Sci. 201(1065), 159–174 (1950). https://doi.org/10.1098/rspa.1950.0049
8. Suk, J.W., Kirk, K., Hao, Y., Hall, N.A., Ruoff, R.S.: Thermoacous-tic sound generation from monolayer graphene for transparent andflexible sound sources. Adv. Mater. 24(47), 6342–6347 (2012).https://doi.org/10.1002/adma.201201782
9. Liu, Q., Zhang, Y.: Shock wave generated by high-energy electricspark discharge. J. Appl. Phys. 116(15), 153302 (2014). https://doi.org/10.1063/1.4898141
10. Temkin, S.: A model for thunder based on heat addition. J.Sound Vib. 52(3), 401–414 (1977). https://doi.org/10.1016/0022-460X(77)90567-3
11. Krane, R.J., Parang, M.: Scaling analysis of thermoacoustic con-vection in a zero-gravity environment. J. Spacecr. Rockets 20(3),316–317 (1983). https://doi.org/10.2514/3.25598
12. Carlès, P.: Thermoacoustic waves near the liquid–vapor criticalpoint. Phys. Fluids 18(12), 126102 (2006). https://doi.org/10.1063/1.2397577
13. Shen, B., Zhang, P.: Thermoacoustic waves along the critical iso-chore. Phys. Rev. E 83, 011115 (2011). https://doi.org/10.1103/PhysRevE.83.011115
14. Onuki, A., Hao, H., Ferrell, R.A.: Fast adiabatic equilibration in asingle-component fluid near the liquid–vapor critical point. Phys.Rev. A 41, 2256–2259 (1990). https://doi.org/10.1103/PhysRevA.41.2256
15. Boukari, H., Briggs, M.E., Shaumeyer, J.N., Gammon, R.W.: Crit-ical speeding up observed. Phys. Rev. Lett. 65, 2654–2657 (1990).https://doi.org/10.1103/PhysRevLett.65.2654
16. Zappoli, B., Bailly, D., Garrabos, Y., Le Neindre, B., Guenoun,P., Beysens, D.: Anomalous heat transport by the piston effect in
supercritical fluids under zero gravity. Phys. Rev. A 41, 2264–2267(1990). https://doi.org/10.1103/PhysRevA.41.2264
17. Amiroudine, S., Zappoli, B.: Piston-effect-induced thermal oscil-lations at the Rayleigh-Bénard threshold in supercritical 3He.Phys. Rev. Lett. 90, 105303 (2003). https://doi.org/10.1103/PhysRevLett.90.105303
18. Zappoli, B.: Near-critical fluid hydrodynamics. Comptes RendusMécanique 331(10), 713–726 (2003). https://doi.org/10.1016/j.crme.2003.05.001
19. Onuki, A.: Thermoacoustic effects in supercritical fluids near thecritical point: resonance, piston effect, and acoustic emission andreflection. Phys. Rev. E 76, 061126 (2007). https://doi.org/10.1103/PhysRevE.76.061126
20. Carlès, P.: A brief review of the thermophysical properties of super-critical fluids. J. Supercrit. Fluids 53(1–32), 2–11 (2010). https://doi.org/10.1016/j.supflu.2010.02.017
21. Brown, M.A., Churchill, S.W.: Experimental measurements ofpressure waves generated by impulsive heating of a surface. AIChEJ. 41(2), 205–213 (1995). https://doi.org/10.1002/aic.690410202
22. Zappoli, B., Durand-Daubin, A.: Direct numerical modelling ofheat and mass transport in a near supercritical fluid. Acta Astro-naut. 29(10/11), 847–859 (1993). https://doi.org/10.1016/0094-5765(93)90167-U
23. Nakano, A.: Studies on piston and soret effects in a binary mixturesupercritical fluid. Int. J. Heat Mass Transf. 50(23–24), 4678–4687(2007). https://doi.org/10.1016/j.ijheatmasstransfer.2007.03.014
24. Hasan, N., Farouk, B.: Thermoacoustic transport in supercriticalfluids at near-critical and near-pseudo-critical states. J. Supercrit.Fluids 68, 13–24 (2012). https://doi.org/10.1016/j.supflu.2012.04.007
25. Hwang, I.J., Kim, Y.J.: Measurement of thermo-acoustic wavesinduced by rapid heating of nickel sheet in open and confinedspaces. Int. J. Heat Mass Transf. 49(3–4), 575–581 (2006). https://doi.org/10.1016/j.ijheatmasstransfer.2005.08.025
26. Hasan, N., Farouk, B.: Fast heating induced thermoacoustic wavesin supercritical fluids: experimental and numerical studies. J. HeatTransf. 135(8), 081701 (2013). https://doi.org/10.1115/1.4024066
27. Chu, B.T.: Pressure waves generated by addition of heat ina gaseous medium. NASA Technical Report, NACA-TN 3411(1955). https://ntrs.nasa.gov/search.jsp?R=19930084155
28. Argrow, B.M.: Computational analysis of dense gas shock tubeflow. Shock Waves 6(4), 241–248 (1996). https://doi.org/10.1007/BF02511381
29. Schlamp, S., Rösgen, T.: Flow in near-critical fluids induced byshock and expansion waves. Shock Waves 14(1), 93–101 (2005).https://doi.org/10.1007/s00193-004-0241-6
30. Mortimer, B., Skews, B., Felthun, L.: The use of a slow soundspeed fluorocarbon liquid for shock wave research. Shock Waves8(2), 63–69 (1998). https://doi.org/10.1007/s001930050099
31. Kim, H., Choe, Y., Kim, H., Min, D., Kim, C.: Methods forcompressible multiphase flows and their applications. ShockWaves 29(1), 235–261 (2019). https://doi.org/10.1007/s00193-018-0829-x
32. Re, B., Guardone, A.: An adaptive ALE scheme for non-ideal com-pressible fluid dynamics over dynamic unstructuredmeshes. ShockWaves 29(1), 73–99 (2019). https://doi.org/10.1007/s00193-018-0840-2
33. Fisher,M.E.,Widom, B.: Decay of correlations in linear systems. J.Chem. Phys. 50(20), 3756–3772 (1969). https://doi.org/10.1063/1.1671624
34. Tucker, S.C.: Solvent density inhomogeneities in supercritical flu-ids. Chem. Rev. 99(2), 391–418 (1999). https://doi.org/10.1021/cr9700437
35. Banuti, D.T.: Crossing the widom-line—supercritical pseudo-boiling. J. Supercrit. Fluids 98, 12–16 (2015). https://doi.org/10.1016/j.supflu.2014.12.019
123
Author's personal copy
Heat-induced planar shock waves in supercritical fluids
36. Lemmon, E.W., McLinden, M.O., Friend, D.G.: Thermophysi-cal Properties of Fluid Systems in NIST Chemistry WebBook,NIST Standard Reference Database Number 69. National Instituteof Standards and Technology, Gaithersburg MD, 20899 (2019).https://doi.org/10.18434/T4D303
37. Poling, B.E., Prausnitz, J.M., O’Connell, J.P.: The Properties ofGases and Liquids, 5th edn. McGraw-Hill Education, New York(2001)
38. Peng, D.Y., Robinson, D.B.: A new two-constant equation of state.Ind. Eng. Chem. Fundam. 15(1), 59–64 (1976). https://doi.org/10.1021/i160057a011
39. Chung, T.H., Ajlan, M., Lee, L.L., Starling, K.E.: Generalizedmultiparameter correlation for nonpolar and polar fluid transportproperties. Ind. Eng. Chem. Res. 27(4), 671–679 (1988). https://doi.org/10.1021/ie00076a024
40. Kim, K., Hickey, J.P., Scalo, C.: Pseudophase change effects inturbulent channel flowunder transcritical temperature conditions. J.Fluid Mech. 871, 52–91 (2019). https://doi.org/10.1017/jfm.2019.292
41. Larsson, J., Lele, S.K.: Direct numerical simulation of canonicalshock/turbulence interaction. Phys. Fluids 21(12), 126101 (2009).https://doi.org/10.1063/1.3275856
42. Toro, E.F.:Anomalies of conservativemethods: analysis, numericalevidence and possible cures. Comput. Fluids Dyn. J. 11(2), 128–143 (2002)
43. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation ofGas Flows. Oxford University Press, Oxford (1994)
44. Karni, S.: Multicomponent flow calculations by a consistent primi-tive algorithm. J. Comput. Phys. 112(1), 31–43 (1994). https://doi.org/10.1006/jcph.1994.1080
45. Abgrall, R.: How to prevent pressure oscillations in multicompo-nent flow calculations: a quasi conservative approach. J. Comput.Phys. 125(1), 150–160 (1996). https://doi.org/10.1006/jcph.1996.0085
46. Kawai, S., Terashima, H., Negishi, H.: A robust and accuratenumerical method for transcritical turbulent flows at supercriti-cal pressure with an arbitrary equation of state. J. Comput. Phys.300(Supplement C), 116–135 (2015). https://doi.org/10.1016/j.jcp.2015.07.047
47. Ma, P.C., Lv, Y., Ihme, M.: An entropy-stable hybrid schemefor simulations of transcritical real-fluid flows. J. Comput. Phys.340(Supplement C), 330–357 (2017). https://doi.org/10.1016/j.jcp.2017.03.022
48. Migliorino, M.T., Chapelier, J.B., Scalo, C., Lodato, G.: Assess-ment of spurious numerical oscillations in high-order spectraldifference solvers for supercritical flows. 2018 Fluid Dynam-ics Conference, AIAA Aviation Forum, AIAA Paper 2018-4273(2018). https://doi.org/10.2514/6.2018-4273
49. Migliorino,M.T., Scalo, C.: Dimensionless scaling of heat-release-induced planar shock waves in near-critical CO2. 55th AerospaceSciences Meeting, AIAA SciTech Forum, AIAA Paper 2017-0086(2017). https://doi.org/10.2514/6.2017-0086
50. Grüneisen, E.: Theorie des festen Zustandes einatomiger Ele-mente. Ann. Phys. 344(12), 257–306 (1912). https://doi.org/10.1002/andp.19123441202
51. Iberall, A.S.: The effective “Gamma” for isentropic expansions ofreal gases. J. Appl. Phys. 19(11), 997–999 (1948). https://doi.org/10.1063/1.1698089
52. Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow ofreal materials. Rev. Mod. Phys. 61, 75–130 (1989). https://doi.org/10.1103/RevModPhys.61.75
53. Thompson, P.A.: A fundamental derivative in gasdynamics.Phys. Fluids 14(9), 1843–1849 (1971). https://doi.org/10.1063/1.1693693
54. Smith, R.G.: The riemann problem in gas dynamics. Trans. Am.Math. Soc. 249(1), 1–50 (1979). https://doi.org/10.1090/S0002-9947-1979-0526309-2
55. Swift, G.W., Migliori, A., Hofler, T., Wheatley, J.: Theory andcalculations for an intrinsically irreversible acoustic prime moverusing liquid sodium as primary working fluid. J. Acoust. Soc. Am.78(2), 767–781 (1985). https://doi.org/10.1121/1.392447
56. Banuti, D.T., Raju, M., Ihme, M.: Similarity law for Widom linesand coexistence lines. Phys. Rev. E 95, 052120 (2017). https://doi.org/10.1103/PhysRevE.95.052120
57. Raju,M., Banuti, D.T., Ma, P.C., Ihme,M.:WidomLines in BinaryMixtures of Supercritical Fluids. Scientific Reports 7(1), 3027(2017). https://doi.org/10.1038/s41598-017-03334-3
58. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge University Press, Cambridge (1970)
59. Ismail, A.F., Khulbe, K., Matsuura, T.: Gas SeparationMembranes—Polymeric and Inorganic. Springer, Cham (2015).https://doi.org/10.1007/978-3-319-01095-3
60. Van der Bruggen, B., Schaep, J., Wilms, D., Vandecasteele, C.:Influence of molecular size, polarity and charge on the retention oforganic molecules by nanofiltration. J. Membr. Sci. 156(1), 29–41(1999). https://doi.org/10.1016/S0376-7388(98)00326-3
61. van Leeuwen, M.E.: Derivation of Stockmayer potential parame-ters for polar fluids. Fluid Phase Equilib. 99, 1–18 (1994). https://doi.org/10.1016/0378-3812(94)80018-9
Publisher’s Note Springer Nature remains neutral with regard to juris-dictional claims in published maps and institutional affiliations.
123
Author's personal copy
