Abstract Here, the fundamental problem of Rayleigh–Taylor instability (RTI) is stud-ied by direct numerical simulation (DNS), where the two air masses at differenttemperatures, kept apart initially by a non-conducting horizontal interface in a 2Dbox, are allowed to mix. Upon removal of the partition, mixing is controlled by RTI,apart from mutual mass, momentum, and energy transfer. To accentuate the instability,the top chamber is filled with the heavier (lower temperature) air, which rests atop thechamber containing lighter air. The partition is positioned initially at mid-height ofthe box. As the fluid dynamical system considered is completely isolated from out-side, the DNS results obtained without using Boussinesq approximation will enableone to study non-equilibrium thermodynamics of a finite reservoir undergoing strong
This article is part of the 13th Joint European Thermodynamics Conference Special Issue.
irreversible processes. The barrier is removed impulsively, triggering baroclinic insta-bility by non-alignment of density, and pressure gradient by ambient disturbances viathe sharp discontinuity at the interface. Adopted DNS method has dispersion rela-tion preservation properties with neutral stability and does not require any externalinitial perturbations. The complete inhomogeneous problem with non-periodic, no-slip boundary conditions is studied by solving compressible Navier–Stokes equation,without the Boussinesq approximation. This is important as the temperature differ-ence between the two air masses considered is high enough (�T = 70 K) to invalidateBoussinesq approximation. We discuss non-equilibrium thermodynamical aspects ofRTI with the help of numerical results for density, vorticity, entropy, energy, andenstrophy.
Rayleigh–Taylor instability [9,25,38] is a prototypical event, occurring when quies-cent equilibrium condition of heavier fluid resting on top of a lighter fluid is studiedwith a large initial temperature difference. Such discontinuous temperature differencewill not allow one to apply Boussinesq approximation [9,21], which has been used forover a century now for solving mixed convection problem. Mikaelian [21] notes thatuse of this approximation leads to about 40 % error for incompressible flows. Thusin the present investigation, we use a compressible formulation which obviates thenecessity for adopting Boussinesq approximation. Although the fluid ensemble startsfrom a static condition, convection sets in by creation of vortices by a baroclinic sourceterm (−∇ p × ∇ρ/ρ2) triggered by background disturbances which are omnipresent,otherwise the pressure and density gradients terms are perfectly aligned for initialequilibrium state. This only points to the fact that the equilibrium arrangement is notstable and the instability is triggered by unsettling moment on any control volumeat the interface caused by buoyancy, once the partition separating the air masses isremoved. This temporal instability has been originally studied by an inviscid mech-anism [14,25,38,40] and the corresponding viscous instability has been reported in[8–10], among many other references.
Fluids of different densities and/or different temperatures are destabilized at theinterface, as the system seeks a least combined potential energy configuration. Thisonset is very clearly visualized experimentally in [1,18,26] for various combinationof fluids. The onset of instability at the interface is at very high wavenumbers orig-inating at the junction, where the fluids meet the side-wall. An arrangement of suchan experiment is shown in the schematic of Fig. 1. Although the onset of instabilityoccurs at very high wavenumbers, it subsequently manifests in lower wavenumbers—an attribute of transfer of energy from high to low wavenumbers, which is termed asthe inverse cascade for turbulence [12]. This observation indicates one aspect of vari-ous computations reported in the literature. If one uses a dissipative numerical method
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Fig. 1 Schematic of computational domain to study RTI in a box. The mixed region (hm ), the length ofthe box (L1), the height of the box (H ), boundary conditions, and depth of penetration of both fluids (hm )have been depicted
or a method which cannot resolve higher wavenumbers, then one is forced to applyexcitation to trigger instability, as in [8,10,11,24,27,39,40]. This is also accentuatedby the fact that all the authors have used periodic boundary conditions in the horizontalplane for these 3D simulations. In 3D simulations, one notices a forward cascade ofenergy due to vortex stretching term, which is absent in 2D Navier–Stokes equation(NSE). A forward cascade of enstrophy is possible for both 2D and 3D flows, as hasbeen explained in [35]. In the backdrop of this scenario, we have used a non-periodicformulation here with very high resolution for 2D simulation, as was reported in [6].
RTI can be viewed as a special case of Kelvin–Helmholtz instability (KHI), whichis also an interfacial event noted between two layers of fluids convecting with differ-ent speed (as in a mixing layer) and/or different densities. In the context of historicaldevelopment using inviscid mechanism, it can be easily shown [14,30] that 2D dis-turbances are more destabilizing, as compared to 3D disturbances. Apart from thisreason, it is also possible to perform 2D analysis with higher resolution, in compar-ison to poorly resolved 3D computations. It is symptomatic of this problem, whichprompted researchers [8] to state that the availability of even more powerful computershas led to a somewhat ironic state of affairs, in that agreement between simulationsand experiments is worse today than it was several decades ago. However, in thepresent work, we will establish that this is not true due to the following reasons: (i)The authors used methods which had poor numerical dispersion and dissipation prop-erties; (ii) The authors used periodic boundary conditions in horizontal planes, whichprecluded physical onset of the instability phenomenon, as shown here in Fig. 2; and(iii) The authors also used incompressible flow formulation with Boussinesq approx-imation. For 3D simulations performed with highly dissipative numerical methods,
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Fig. 2 (i) Present numerical simulations for comparison with Read’s experiment [26], using (3000×1800)grid points shown for non-dimensional density at non-dimensional times: (a) t = 3; (b) t = 5.5; (c) t = 8; (d)t = 12; (e) t = 21, and (f) t = 34 (ii) Validation of numerical simulation in Fig. 2(i) with Read’s experiment[26] shown here at times: t = 0 ms; t = 16.6 ms; t = 22.1 ms; t = 27.6 ms; t = 44.1 ms; and t = 60.8 ms forqualitative comparison
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Fig. 2 continued
various researchers have used long-wavelength initial perturbations with the expecta-tion that higher wavenumbers will be created by nonlinear cascade mechanism. Presentcomputations and experimental visualizations [1,18,26] indicate the opposite direc-tion of migration of energy from small to large scale—noted above also as inversecascade. However, it is worthwhile to remember that present simulations are for 2Dflows and their match with quasi-2D experiments [26] has to be understood in itsproper perspective.
If we define ρ as the fluid density and use subscripts u and l to indicate upper andlower fluids, then in the literature [18], a non-dimensional Atwood number is definedas: A = (ρu − ρl)/(ρu + ρl) which is less than one for the arrangement shown inFig. 1. It is also interpreted that A effectively lowers the action of gravity. In thepresent study, we consider a case of A = 0.105583, for the steep-jump of temperaturecorresponding to 70 K across the interface initially and we have noted already that
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Boussinesq approximation will incur large unacceptable error. It has been noted in[39] that there are higher order effects which are obscured by Boussinesq approxima-tion for low A flows. To avoid the sources of errors in [8], here a compressible flowformulation is used, which does not require one to make Boussinesq approximation.Also the present computations use a neutrally stable, near-spectral accurate compactdispersion relation preservation (DRP) scheme without making periodic flow assump-tion in the horizontal direction [31]. In justifying the present research, we note thatMikaelian [21] specifically noted that the issue of compressibility is highly challengingand direct numerical simulations are the best way to assess the validity of Boussinesqapproximation.
The partition separating the air in two compartments at different temperatures ini-tially is removed impulsively. A schematic of the problem is shown in Fig. 1, displayingfeatures of the flow instability, following the removal of the partition at t = 0 (whichis shown by a dotted line). Such impulsive movement of the partition causes excitationof wide band of wavenumbers due to density and temperature discontinuities. In thefigure, the evolving mixing region is shown by hatched lines, characterized by thewidth of the mixing region (H ). Different boundary conditions for no-slip and zerotransfer of energy at the wall of the box are also indicated in the figure. The box isshallow, with height equal to half the width of the box. Exact dimensions and physicalvariables are listed in the subsection on boundary and initial conditions.
The large temperature differential between the two air masses initially, triggers athermodynamic process which can be classified as occurring in far-from equilibriumstates. We also note that at t = 0, the isolated thermodynamic system has zero kineticenergy and the onset of instability gives rise to non-zero kinetic energy. However, mostof the energy is manifested in rotationality of the system characterized by enstrophy,which is the square of vorticity of the flow [2,13,35]. We also note that the initial stateof the system is characterized by perfect order (which is tantamount to zero entropy tobegin with), while the subsequent mixing is symptomatic of disorderly state. However,RTI also creates spikes and bubbles (as explained later) after the primary instability,triggered by creation of coherent vortices. In fluid dynamics, coherent structures inturbulent regime is commonplace, despite the apparent chaotic dynamics of the system.It is interesting to note that Prigogine [22] has noted that non-equilibrium state canbecome a source of order. He also noted that if the system is perturbed, the entropyproduction will increase, but the system reacts by coming back to the minimum value ofentropy production. Far-non-equilibrium thermodynamics has been studied in recenttimes [3] using steepest entropy ascent model, which is a unified treatment of theso-called maximum entropy production principle, as reviewed in [20]. In evaluatingthe concept of negentropy introduced by Schroedinger [29] for evolution in biologicalsystem, Mahulikar and Herwig [19] proposed the negentropy principle for dynamicorder existence and principle of maximum negentropy production (PMNEP), whichare the counterparts of the existing entropy principle and the law of maximum entropyproduction, respectively.
The paper is formatted in the following manner. In the next section, governing equa-tions and auxiliary conditions are introduced. In Sect. 3, numerical methods are statedbriefly [31,32]. Various physical features of RTI involving different elements are pro-vided in Sect. 4. The energy consideration of this far-non-equilibrium thermodynamic
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process is discussed in Sect. 5, relating important fluid dynamical considerations ofenstrophy transport equation and Crocco’s theorem relating vorticity with entropy.This is followed by summary and conclusion.
2 Formulation of the Problem
The flow in the rectangular shallow box is shown in Fig. 1, with adiabatic walls insulat-ing the thermodynamical system completely from the surroundings. Thus, the presentsystem falls in the category of thermodynamic systems without heat and thermal reser-voir, as described in [4] from the perspective of non-equilibrium thermodynamics. Thethought experiment performed here involves filling the two chambers of a box withair at different constant temperatures, separated initially by an insulating partition. Atthe onset of the numerical experiment (t = 0), this partition is removed impulsively,so that the flow system with density and temperature discontinuities experiences per-turbations in space, given by a Heaviside function for these variables, while the timevariation also mimics a jump discontinuity. In an ideal condition, such an initial per-turbation is capable of exciting all possible spatial and temporal scales, as governed bythe initial destabilizing potential energy. However, RTI has its own dispersion relationwhich relates wavenumbers with circular frequency for the instability.
One of the aims of the present investigation is to study a truly isolated system by veryaccurate formulation and numerical method. The accuracy of the method is ensuredby monitoring conservation laws for the dynamical system and also by providing anestimate of entropy generation that will be associated with vorticity generation at no-slip wall and mixing of the shear layer via RTI. The governing equations of motion arethe 2D unsteady NSE which conserve mass, momentum, and energy. This approachwould account for the density gradients existing within the flow. A set of four nonlinearpartial differential equations for the flow have been expressed in divergence form. Thesystem of equations can be expressed in terms of fluxes as follows [16]:
∂ Q
∂t∗+ ∂ E
∂x∗ + ∂ F
∂y∗ = ∂ Ev
∂x∗ + ∂ Fv
∂y∗ , (1)
where the conserved variables are given by
Q =
⎡⎢⎢⎣
ρ∗ρ∗u∗ρ∗v∗ρ∗e∗
t
⎤⎥⎥⎦ . (2)
The convective flux variables E and F are given as
E =
⎡⎢⎢⎣
ρ∗u∗ρ∗u∗2 + p∗ + (ρ∗ − ρs)gx x
ρ∗u∗v∗(ρ∗e∗
t + p∗)u∗
⎤⎥⎥⎦ , (3)
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F =
⎡⎢⎢⎣
ρ∗v∗ρ∗u∗v∗
ρ∗v∗2 + p∗ + (ρ∗ − ρs)gy y(ρ∗e∗
t + p∗)v∗
⎤⎥⎥⎦ , (4)
and the viscous fluxes Ev and Fv are
Ev =
⎡⎢⎢⎣
0τ ∗
xxτ ∗
xyu∗τ ∗
xx + v∗τ ∗xy − q∗
x
⎤⎥⎥⎦ , (5)
Fv =
⎡⎢⎢⎣
0τ ∗
yxτ ∗
yyu∗τ ∗
yx + v∗τ ∗yy − q∗
y
⎤⎥⎥⎦ . (6)
In Eq. 2, the variables ρ∗, p∗, u∗, v∗, T ∗, and e∗t represent the density, the fluid
pressure, Cartesian components of fluid velocity, the absolute temperature, and thespecific internal energy of the fluid, respectively, and τ ∗
xx , τ ∗xy , τ ∗
yx , and τ ∗yy are the
components of the symmetric viscous stress tensor and are related to the gradients ofvelocity as
τ ∗xx =
(2μ
∂u∗
∂x∗ + λ∇∗.V∗)
, (7)
τ ∗yy =
(2μ
∂v∗
∂ y∗ + λ∇∗.V∗)
, (8)
τ ∗xy = τ ∗
yx = μ
(∂u∗
∂ y∗ + ∂v∗
∂x∗
). (9)
Fluid property of thermal conductivity (κ) is kept constant in this formulation.Specifically, we use the Stokes’ hypothesis, which relates μ with λ via the relation:λ = −2μ/3. This is routinely used for both incompressible and compressible flowcomputations, yet its generic proof is still missing. See for example, the discussiongiven in [7,15,23] about the applicability of Stokes’ hypothesis. It is understood that thehypothesis is valid for dilute monoatomic gases, its generalization would require bettertheoretical understanding of non-equilibrium aspects of fluid flow and/ or experimentaldata which can provide alternative(s) to this hypothesis. In the absence of any suchdata, here Stokes’ hypothesis has been used for the presented simulations. The heatconduction terms, q∗
x and q∗y are given by
q∗x = −κ
∂T ∗
∂x∗ , (10)
q∗y = −κ
∂T ∗
∂y∗ . (11)
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Closure of system of equations is provided by the ideal gas equation given by
p∗ = ρ∗ R∗T ∗, (12)
and it is used to define specific energy, e∗t as
e∗t = CvT ∗ + (u∗2 + v∗2)
2. (13)
The presented governing equations and numerical methods have been successfullyimplemented for DNS of transonic flow past aerofoils in [36]. The governing equationsare nondimensionalized so that the solution is not restricted to a particular geometryor a set of flow conditions. The independent and dependent variables are nondimen-sionalized with appropriate scales as:
t = t∗Us
L, x = x∗
L, y = y∗
L, u = u∗
Us, v = v∗
Us, (14)
ρ = ρ∗
ρs, θ = T ∗ − Ts
�T, p = p∗ − ps
ρsU 2s
, et = e∗t − ets
U 2s
, (15)
where L , Us , �T , and ρs are the characteristic length, velocity, temperature difference,and density, respectively. Here, the standard conditions are chosen as that of air at nor-mal temperature (Ts = 298 K) and pressure (ps = 101 kN·m−2), ρs = 1.225 kg·m−3.For ensuring complete similarity of 2D compressible flows, five characteristic num-bers have to be identical, which are: Reynolds number (Re), Prandtl number (Pr ),Froude number (Fr ), Eckert number (Ec), and Gay–Lussac number (Ga) given by,
Re = ρsUs L
μs, Pr = μsC p
κ, Fr = U 2
s
gL, Ec = U 2
s
C p�T, Ga = �T
Ts,
(16)
where Froude and Eckert numbers represent the ratio of inertial to gravitational forceand enthalpy difference, respectively. The Gay–Lussac number or thermal expansionnumber relates the temperature difference from the reference temperature taken as,298 K. The reference velocity is taken as 26.52357 m · s−1 and the length scale ischosen as 5.5262 m, so that the Reynolds number used is 107. Other reference valuesused in the present simulations are given by, μs , Cp and κ , which have been calculatedfrom the fixed value of Ec = 0.01, Re and Pr = 0.7. Similarly, reference pressureand Mach number are calculated from ρs and �T .
The nondimensionalized governing equations are thus obtained as,
∂ Q
∂t+ ∂ E
∂x+ ∂ F
∂y= ∂ Ev
∂x+ ∂ Fv
∂y. (17)
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The viscous shear stress components with Stokes hypothesis applied are given by
τxx = 1
Re
[4
3
∂u
∂x− 2
3
∂v
∂y
], (18)
τyy = 1
Re
[4
3
∂v
∂y− 2
3
∂u
∂x
], (19)
τxy = τyx = 1
Re
[∂u
∂y+ ∂v
∂x
]. (20)
The heat conduction terms are given by
qx = − 1
RePr Ec
∂θ
∂xand qy = − 1
RePr Ec
∂θ
∂y. (21)
The equation of state translates to p = ρRθ − RGa (1 − ρ), where R = R∗�T/U 2
s .The present problem is solved in the physical Cartesian grid, circumventing the prob-lem of aliasing [31].
2.1 Physical Domain and Auxiliary Conditions
The flow is computed in the domain: 0 ≤ x ≤ 60 and 0 ≤ y ≤ 30, discretizedwith uniformly spaced grid with 3000 points in the x-direction and 1800 points in they-direction. Energy cascade follows the creation of small scales at the interface withthe side-walls as a billowing motion, and this is followed by creation of larger scalesby bowing motion, as noted in experiments [1,26]. We use compact scheme alongwith compact filters and for time integration a two-time level method has been used.
The horizontal interface separating the fluid of different densities is exactly at halfthe height of the box (AB), as shown in Fig. 1. The boundary conditions are appliedon the variables u, v, p, θ , which are deduced from the conserved variables Q,at every time-step. Along the walls (AB, BC, CD, AD) for x ≥ 0 and y ≥ 0, no-slip adiabatic conditions are imposed at the wall, which makes the flow in the boxcompletely isolated from the surrounding. The present compressible formulation willyield solution to study RTI in all essential details from linear to nonlinear stages,as well as, estimating rotationality and entropy generation for the non-equilibriumthermodynamic system. The non-dimensional wall boundary conditions are givenby:
u = 0; v = 0; ∂θ
∂n= 0 and
∂p
∂n= 0.
In simulating the flow, the heavier air on top chamber at t = 0 is taken as T = Ts
(298 K), so that θ = 0 for the colder air and θ = 1 for the warmer air at the bottom.For such a discontinuous θ distribution, the interface non-dimensional temperature att = 0 is given as 0.5. The reference density is taken as ρs and the temperature on thebottom of the interface at the onset (t = 0) is Tl = 368 K. The densities of air above
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and below the interface are obtained at t = 0, in terms of reference density ρs as,ρu = ρs and ρl = ρs
TuTl
. The Neumann boundary condition on temperature indicatesno heat transfer with the surrounding, while the pressure boundary condition implyapplication of boundary layer assumption at the wall.
3 Numerical Methods and Validation
The numerical methods used to solve the governing equations are described in [6].Briefly, high-resolution optimized upwind compact scheme (OUC S3) [31] is used forconvective flux derivative calculations, second-order central difference (C D2) schemeis utilized for viscous flux calculations, and time integration has been performed usingOC RK3 scheme, an optimized version of three-stage Runge–Kutta method, whichensures better properties as compared to classical four-stage fourth-order Runge–Kuttamethod [34].
3.1 Validation of RTI Results with Experimental Observation
In Fig. 2(i), density contours are shown for early times from t = 3–8 to observe theonset of RTI characterized by billowing along the interface starting from the side-walland progressing inward. Also later stages of RTI have been captured with densitycontours at t = 12, 21, and 34. Additionally, frames from [26] have been shown inFig. 2(ii), to facilitate qualitative comparison of present simulation with an experiment.In the first frame of Fig. 2(i) for t = 3 and the frame from [26] in Fig. 2(ii), at t = 0ms, one notes the perturbations to originate at the side-walls, which are of very smalllength scales. However, all along the interface one can also note much smaller lengthscale disturbances, which are triggered by discontinuous temperature and density jumpacross the interface in subsequent frame at t = 8 in Fig. 2(i). In the experimental resultof Fig. 2(ii), this is seen to occur at t = 22.1 ms. Similar correspondence of subsequentframes show qualitative match between sequence of events in computation and the 2Dexperiment in [26]. The computational time scale is noted to be approximately about200 ms, which is slower compared to the experimental value. This good qualitativematch suggests that solving a non-periodic problem here is more relevant than solvingperiodic problem [8].
In later frames of Fig. 2(i, ii), one notices contamination from the side-walls andat the same time disturbances in the interior increase in size, along with spike andbubble formation. The physical evidence of spikes and bubbles are shown in Fig. 3,where a small section of the interface is zoomed in at t = 6.379. The appearanceof spikes and bubbles are the harbingers of formation of the mixing region. Thespikes refer to packets of heavier fluid parcel sinking in the lower half of the box asidentified in the center of Fig. 3. The bubbles refer to packets of lighter fluid penetratingthrough the heavier fluid kept above the interface. One of the striking features of thespikes and bubbles at onset is their perfect geometrical symmetry. Also, Lawrie [17]has noted that the bubble and spike structures eventually overturn due to their ownvorticity and acquire a mushroom-like appearance, which is clearly noted in Fig. 3.
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Fig. 3 Illustration of spikes and bubbles, during primary stage of RTI, in the zoomed non-dimensionaldensity contour at the non-dimensional time, t = 6.379
By t = 8, the nonlinear stage is already reached, as noted in Fig. 2(i). By the later timeevolution shown (from t = 12 to 34) in Fig. 2(i), and the corresponding experimentalframes at t = 27.6, 44.1, and 60.8 ms from [26] in Fig. 2(ii), the mixing is stronglyevident covering the full domain. Also in Fig. 3, one notices a front above the bubbles,with fluid density that is more than what was present in the top half of the box att = 0, indicating a compressed wave-front which precedes the sequence of bubbles.In instability of different flows, presence of such spatio-temporal fronts is known tocause transition, as shown in [5] for evolution of disturbance field for zero pressuregradient boundary layer.
4 Primary Stage of RTI
In Fig. 3, we have noted the presence of three most important elements of primaryinstability: spikes, bubbles and spatio-temporal wave-fronts, by looking at a narrowregion of the interface. We have also identified in the introduction section the term
−∇ p×∇ρ
ρ2 as the trigger for instability onset. We emphasize that in the equilibrium stateat t = 0, the terms in the numerator are perfectly aligned in the same plane for 2Dflow geometry, and this baroclinic term by itself will be identically zero at the equi-librium state. It is only the omnipresent background disturbances, will misalign ∇ pwith ∇ρ, which will work as the seed of instability. One of the special feature of the
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adopted numerical method is its ability to resolve very high numbers without numer-ically attenuating these disturbances—unlike other low accuracy methods or higherorder methods which include attenuation through filtering, sub-grid scale models inLES.
4.1 Creation of Density Fronts During the Primary Instability
In Fig. 4, we show the zoomed interface over the length 4.4 ≤ x ≤ 5.2 to furtherexplain the onset of primary instability. The frame at t = 5.019 shows the beginningof the instability, taking the appearance of viscous fingering, and the misalignment isevident from the slant of these structures. The slant is noted in either direction aboutthe middle of the interface at x = 4.8 shown in Fig. 4, and this is more prominentlyvisible at the next frame at t = 5.499. The appearance of symmetric structures oneither side is very striking, and the center-point has the first spike that provides theleading point of spikes on either side. One also notices bubbles forming on either sideof the center-point in a sequence and in Fig. 4; one observes the two most prominentbubbles at x = 4.57 and at 5.12. We would like to emphasize another aspect of theonset of primary instability, which is the formation of two density fronts with distinctlydifferent density from its ambience. For example, in the top left frame, one notices adensity front (at y = 2.76) which has a density higher than the maximum density ofthe two layers at t = 0. However, below the spatio-temporal density wave-front andjust above the viscous fingers, one notices even higher density. This implies that theinterface is the source of density wave-fronts.
This shock-wave like feature explained above is matched by a rarefaction frontbelow the interface visible at y = 2.6. One notices that these rarefaction fronts alsohave density which is slightly more than the minimum density at t = 0. In this earlystage of development, the flow can be viewed as developing in a quasi-steady mannerand a true rarefaction front would lead to drop in entropy and is not feasible, whenone is very near the equilibrium state. However, this will not be precluded, when thesystem departs significantly far away from the equilibrium stage. Also, this rarefactionfront is not a continuous one during the primary stage; it is punctuated by connectinglinks with the viscous fingers in the center, as seen at t = 5.019.
As time progresses, the spikes and bubbles grow and the spatio-temporal frontabove the mean interface conforms to the growing shapes of bubbles and spikes, whilemaintaining left-right symmetry in the flow. However, due to finite size of the box onall sides, growing primary structures are affected by the no-slip walls on all four sides,which eventually leads to breakdown of symmetries, as shown in later time frames ofFig. 4.
4.2 Generation of Vorticity During Primary Instability
The onset of RTI is due to instability of the quiescent fluids by omnipresent distur-bances which causes destabilizing moment by gradient of density and pressure. Thisinstability results in the distortion of the interface by the commencement of the baro-
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Fig. 4 Physical events occuring during RTI, from early viscous fingering to later development of bulbous,mushroom-like bubbles and spikes and eventual breakdown of symmetry, observed in the density contoursat times (a) t = 5.019, (b) t = 5.499, (c) t = 6.019, (d) t = 6.499, (e) t = 7.019, (f) t = 7.499,(g) t = 8.019, and (h) t = 8.499. All quantities are non-dimensional
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clinic generation of vorticity as a consequence of pressure and density gradients notremaining aligned.
The onset of baroclinic vorticity generation during the instability is concentratedat the junction of side-walls with the interface of the fluid. This eventually advectsleading to interface becoming increasingly stratified in space in a slanted manner. InFig. 5, the vorticity contours are plotted during early times only in a region spanned by3.05 ≤ x ≤ 3.85 and 2.4 ≤ y ≤ 2.9. In the first frame at t = 4.7, one notices alternaterectangular slanted roll, in alternate directions. Within each roll, one notices elongatedstrips of vortices of opposite signs stacking one after the other. This striking symmetrycontinues at early times, seen up to t = 5.019 in Fig. 5. One also notices that the slantedvortex elements which are initially linear and narrow, with time, these elements sufferan oblique instability. The linear elements develop waviness or undulations, indicatinga secondary stage of instability, which progresses along the vortical elements. In thefollowing, we describe the sequence in which vortical blobs are formed from theslanted vortical elements, without showing any figures.
One notices that each of the vortex elements stretch in their respective directions ofalignment, which eventually leads to a neck formation at the center with low vorticityvalue there. At the same time, the two ends of the linear element grows in size andstrength. This is true for both signs of vortical elements. Eventually these two endsdisconnect from each other, forming rounded vortex blobs. Further growth of waveamplitude of these elements leads to roll-up of these into distinct blobs of vortices;some of which can be noted in the frame at t = 5.499 in Fig. 5, which shows anintermediate stage.
5 Non-equilibrium Stage of RTI and Evolution of Translational andRotational Energy
One of the defining feature of the flow field in a confined domain is the role ofrotational motion of the fluid particles, as compared to their rectilinear motion. Here,the dynamics is initiated by the destabilizing potential energy associated with heavierfluid resting on top of lighter fluid. This can be readily evaluated as 439 005.84 Joules.The translational motion is quantified in terms of specific kinetic energy, as shown inthe top frame of Fig. 6, which starts from a zero value, as would be expected. However,it is not easy to quantify angular momentum and rotational energy for fluid flow, asidentity of any distinct fluid blob is not been tracked for various reasons. Like the massdetermining the kinetic energy of rigid body, the same cannot be used for rotationalfluid flow, as the rotational energy depends upon moment of inertia tensor, which fora moving fluid, changes with time, as a fluid blob changes size, shape, and orientationwith time.
To avoid this problem of quantification, one can use enstrophy as the metric of spe-cific rotational energy, as one uses specific kinetic energy to characterize translationalmotion. Enstrophy is plotted as a function of time in the middle frame of Fig. 6. Itis readily apparent that the rotational motion consumes more energy than rectilinearmotion, as one can notice from the scales of the ordinate. The middle and bottom framelook indistinguishable, as their difference is given by the top frame, which is of the
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Fig. 5 Vorticity contours at times (a) t = 4.7, (b) t = 4.85, (c) t = 5.019, (d) t = 5.499, (e) t = 6.019,
(f) t = 6.499, (g) t = 7.019, and (h) t = 7.499 to emphasize the beginning of primary instability, neckingof angular strands of vortices into discrete vortex blobs as the secondary instability and development ofspikes and bubbles. All quantities are nondimensionalized
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timeΔEqu
ivalen
tSpe
cific
Ene
rgy
20 40 60-2.0x10+06
-1.0x10+06
0.0x10+00
1.0x10+06
2.0x10+06
3.0x10+06
time
ΔKE
20 40 600
200
400
600
time
ΔEns
troph
y
20 40 60-2.0x10+06
-1.0x10+06
0.0x10+00
1.0x10+06
2.0x10+06
3.0x10+06
Fig. 6 Change in specific properties for kinetic energy in J·kg−1 (top), non-dimensional enstrophy (middle),and equivalent specific energy in J/kg (bottom) plotted as function of non-dimensional time to characterizeand quantify energy dynamics during the instability and emphasize dominance of rotational energy overtranslational energy
order of hundreds, while the rotational energy is of the order of millions. This aspecthas been also highlighted in the lid-driven cavity problem in [33], where one noticesorbital and spinning motion of fluids in the cavity with polygonal vortices formingat the center. Further properties of enstrophy are described in [35] where a transportequation for enstrophy is developed to explain the cascading process in fluid flow.Thus, energetics of fluid flow can be represented by adding specific kinetic energywith the enstrophy, and the resultant is shown in the bottom frame of Fig. 6, and weterm it as total specific energy associated with motion. In Figs. 6, 7, 8, and 9, the post-processed data for energy and entropy are specific quantities expressed in dimensionalunits of J·kg−1 and J·kg−1·K−1, respectively, while the computed quantities shownare all in non-dimensional units.
Fig. 7 Specific entropy (in J·kg−1·K−1) evaluated at i = 1501, j = 901 (top), i = 1501, j = 900(middle) and i = 1501, j = 899 (bottom) and its non-dimensional time evolution. The interface instabilityis quantified by entropy by comparing events occurring above and below the interface
6 Entropy and Vorticity Creation During RTI
One of the consequences of RTI is the mixing of two fluids with dissimilar densityand temperature, segregated from each other initially. While the flow in the initialequilibrium state represents perfect order, subsequent mixing through RTI makes theisolated system disorderly and in the process, the entropy of the system increases. Forthe air, segregated initially at different temperatures by a barrier, removal of which
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Fig. 8 Crocco’s Theorem [37] determines the relationship between entropy (left, in J/(kg·K)) and non-dimensional vorticity (right) at non-dimensional times (a) t = 5.019, (b) t = 10.019, and (c) t = 24.999
will cause mixing and heat transfer. During a time-step of �t for a perfect gas, withtemperature-dependent specific heat at constant pressure (cp), one defines �smn =s(xm, yn, t + dt) − s(xm, yn, t) as the change in instantaneous entropy at that cell as
�smn(t) = (a − R)
[Ln
T (xm, yn, t + �t)
T (xm, yn, t)
]
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time
Entropy
( ΔS)
0 1 2 3 4 5 6 7
-2
0
2
4
6
8
10
12
DOMAIN SIZE : 60 x 30DOMAIN SIZE : 60 x 60
Fig. 9 Total entropy in J·K−1 for rectangular and square domain plotted as function of non-dimensionaltime. The interface instability causes spikes and bubble formation in the presence of mixing. The punctuatedfluctuations are caused due to predominance of pressure waves, while the background mixing due to diffusionshows up as continuously increasing entropy
+ b
[T (xm, yn, t + �t) − T (xm, yn, t)
]
+ c
2
[T 2(xm, yn, t + �t) − T 2(xm, yn, t)
]
+ d
3
[T 3(xm, yn, t + �t) − T 3(xm, yn, t)
]
− R Lnρ(xm, yn, t + �t)
ρ(xm, yn, t). (22)
The time variation of cp as a cubic polynomial is given in [28]. Using R as thespecific gas constant for dry air (287 J/kg·K) to be constant, one can evaluate cv alsoa function of time. In [28], the coefficients of the cubic polynomial for dry air is givenas, a = 970.649 171 3 J/(kg·K), b = 0.067 921 27 J/(kg·K), c = 1.658 149 17 ×10−4 J/(kg·K), and d = −6.788 674 0 × 10−8 J/(kg·K), in Appendix A10 of thereference. In Fig. 7, entropy is evaluated using Eq. 22 at a location just above the initialinterface (i = 1501, j = 901) in the top frame, at the interface (i = 1501, j = 900)in the middle frame and at a point just below the interface (i = 1501, j = 899) in thebottom frame and its time evolution is tracked. In the top frame, a positive fluctuating
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value of entropy is obtained. A marked dip in the entropy is visualized at t = 6.2associated with the secondary instability and subsequent development of spikes andbubbles. A similar trend is noticed in the middle and bottom frames of Fig. 7. Inthe middle frame (at the interface), one notices both positive and negative values ofentropy, while in the bottom frame (below the interface), one notices negative values ofentropy. This brings more understanding to events occurring during RTI, particularlythe reduction in entropy during flow instability. One notices that, in the domain, whileat certain locations one may observe an increase in entropy, this is associated witha reduction in entropy at another location. Although the entropy at a location in thedomain can increase, it can do so only by decreasing the entropy at another locationby the same or greater amount. In closing this discussion, we also note that the entropyincrement reported in Fig. 6 does not change much, if cp and cv are taken as constants.In the following, we will take cp and cv as constants, in relating entropy and vorticitycreation during RTI.
6.1 Relationship Between Entropy and Vorticity
One can estimate the specific entropy change in a cell by smn identified by its co-ordinate as (xm, yn), with respect to the datum state. For compressible flow, the entropygeneration is concomitant with vorticity generation, as dictated by Crocco’s theorem[37]
T ∇S = ∇h0 + ω × V − 1
ρ∇
[λ(∇ · V)
]. (23)
In Fig. 8, entropy and vorticity contours of the system are shown on left and rightside, respectively. Here, the entropy of the system at any point (xm, yn) is given by
smn(t) = cp LnT (xm, yn, t)
Ts− R Ln
p(xm, yn, t)
ps, (24)
where Ts and ps have been defined before as the standard temperature and pressure.Vorticity generation via spikes and bubbles formation also leads to entropy generationand reduction. The reduction in entropy for an isolated system is due to formationof coherent vortices, which leads to enhanced order in the dynamics of the system.This is the reason for fluctuating entropy variation (and not a monotonic increase inentropy, which one would expect to see in an irreversible process with infinite massand no instability, according to the second law of thermodynamics). This is typical ofnon-equilibrium thermodynamics associated with isolated system affected by physicalinstability. As noted before also, Prigogine [22] stated that if the system is perturbed,the entropy production will increase, but the system reacts by coming back to theminimum value of entropy production. He also acknowledged that non-equilibriummay be a source of order. These observations can be further verified by calculatingthe total entropy of the system as a function of time, as shown in Fig. 9, explained inthe next paragraph.
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Fig. 10 Non-dimensional density contours for the present simulations for (a) Rectangular box (60 × 30)and (b) Square box (60×60) at non-dimensional time t = 004.500. Frames (c) and (d) show zoomed viewof the same for instability and mixing at the interface of the rectangular and the square domain. Twistingand stretching at the interface can be visualized in frames (b) and (d) which are absent in frames (a) and(b)
The present simulations are also performed for a square box of dimension (60×60),where L1 ≡ H as compared to the rectangular domain considered earlier where L1 �H . In Fig. 9, we have compared the evolution of total entropy of the isolated systemfor two sizes of the box. All the figures (Figs. 2, 3, 4, 5, 6, 7, 8) are for the rectangularshallow box (60×30) except Fig. 9, where the total entropy of the system is comparedwith a square box (60 × 60). The solid line in Fig. 9 with diamond-shaped symbolsrepresents the total entropy for the rectangular box (60 × 30) and the dashed line withsquare symbols for the square box (60×60). As the box is completely insulated, there
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is no energy exchange with the surrounding and in the primary instability stage thetotal entropy shows continual growth, in the mean. It is interesting to see the effect ofH on the atypical pattern observed for the total entropy of the system. Fluctuations inthe form of letter ‘M’are observed for total entropy for the rectangular and the squarebox. The total entropy of the square box has distorted M-shape fluctuations. The meanrate of increase of total entropy is also greater and approximately double as comparedto the rectangular case. In Fig. 10, density contour plot for the complete domain forthe rectangular and the square box is shown in frames (1a) and (1b) at the same timet = 004.500. The zoomed view of the interface for the rectangular box is shown inframe (2a) and for the square box in frame (2b). The mean rate of increase of totalentropy for the square box (60 × 60) is greater due to the fact that mixing occurs earlyand at very small scales as compared to the rectangular box seen in the zoomed viewof the interface in Fig. 10 depicting the evolution of instability. The pressure pulses,propagating right from the onset of the numerical experiment to the top and bottom ofthe box, are responsible for formation of the atypical M-shape fluctuations in Fig. 9.The increase in amplitude of fluctuation in the square box in Fig. 9 is due to large H ofthe square box. The pressure pulse takes a longer time to reach the top of the box andreflect from the wall and to move back in an opposite direction causing rarefaction ofwaves. In frame (2b) of Fig. 10, diffusion at small scales along the interface occursmore rapidly to the center from both the end points. Twisting and stretching of theinterface is also evident.
7 Summary and Conclusions
Here RTI has been studied in a 2D box, using high-accuracy compact scheme to solvecompressible flow formulation of NSE, allowing us to work without the restrictiveBoussinesq assumption. Thus in Fig. 1, we can afford to solve a problem with a tem-perature differential of 70 K, which otherwise cannot be solved with incompressibleflow formulation with Boussinesq approximation. However, solving NSE using com-pressible flow formulation for almost zero Mach number is known to cause seriousnumerical problems.
The main advantage of the present compressible flow formulation is the satisfac-tion of all conservation laws, without the need to make Boussinesq approximation toaccount for heat transfer effects. Also, we have not made any periodicity assumptionfor numerical expediency, as we have attempted to replicate the experimental con-ditions in [1,26]. At the same time, to avoid pitfalls of numerical error in trackinginterface of different fluids, we consider air separated by a horizontal partition ini-tially at significantly different temperatures. The flow configuration shown in Fig. 1displays heavier (colder) air resting atop lighter (hotter) air separated initially by anon-heat conducting plane barrier. On removal of this barrier impulsively, vorticitystarts forming at the triple points where the interface meets the side-walls.
Present inhomogeneous simulation results shown in Fig. 2(i) match qualitativelyvery well with experimental flow visualization [26] shown in Fig. 2(ii). The non-periodic domain does not require initial perturbations, as the detailed sequence ofphysical processes during RTI is tracked here starting from inclined viscous fingers
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forming in very regular patterns initially. Its subsequent stretching in lengthwise direc-tion leads to necking of the element into two distinct vortices from each finger—asshown in Fig. 5. Even though these events occur earlier, but in the literature, attentionis focused on spikes’ and bubbles’ formation, and these are shown here in Figs. 3 and 4.Thus, the events leading up to Fig. 5 are related to primary and secondary instabilities.
Thereafter nonlinearity takes over and leads to non-equilibrium thermodynamicprocesses, for which energetics of the flow field is shown in Fig. 6, emphasizingthe role of rotational energy over kinetic energy. The rotationality of the flow field isdescribed in terms of enstrophy, whose creation and time evolution are shown in Fig. 7.The relationship of entropy with vorticity is explained by Crocco’s theorem, and itsmanifestation in the contour plots of these two dependent variables is shown in Fig. 8.The present study of RTI in a truly inhomogeneous system has yielded valuable insighton this fundamental fluid dynamical problem and its relationship to non-equilibriumthermodynamics via high-accuracy computations without Boussinesq approximation.In Figs. 9 and 10, effects of geometry of the box is studied, with respect to time evo-lution of the total entropy and density contours, respectively. Larger entropy is notedfor the deeper box. This is a novel approach not reported before, where the entropyof the total system completely isolated from the surrounding can be demonstrated ina non-equilibrium thermodynamic framework.
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