A fractional step method to compute a class of compressible gas–liquid flows
Abstract: We present in this paper some algorithms dedicated to the computation of numerical approximations of a class of two-fluid two-phase flow models. Governing equations for the statistical void fraction, partial mass, momentum, energy are presented first, and meaningful closure laws are given. Then we may give the main properties of the class of two-fluid models. The whole algorithm that relies on the fractional step method and complies with the entropy inequality is presented afterwards. Emphasis is given on the computation of pressure–velocity–temperature relaxation source terms. Conditions pertaining to the existence and uniqueness of discrete solutions of the relaxation step are given. While focusing on some one-dimensional test cases, the true rates of convergence may be obtained within the evolution step and the relaxation step. Eventually, some two-dimensional numerical simulations of a heated wall are shown and are briefly discussed. Some advantages and weaknesses of algorithms are also discussed.
Summary (3 min read)
- Two distinct types of models are used in order to compute liquid-gas or watervapour two-phase flows in industrial codes: the homogeneous approach and the two-fluid approach.
- The two-fluid approach is assumed to be more general, and it is also expected to predict more accurately flows for which the phasic desequilibrium plays a crucial role (see [9, 25, 26]).
- One must be aware that at least two difficulties are hidden in these sets of PDE.
- The second section gives emphasis on the approximation of velocity-pressure-temperature relaxation effects.
2.1 The two-fluid model
- Throughout the paper, indexes l, g refer to the liquid and gas phases; the statistical void fractions of gas and liquid are noted classically αg and αl, which should agree with: αl + αg = 1.
- The so-called conservative variable W will be defined as: W = (αl, αlρl, αlρlUl, αlEl, αgρg, αgρgUg, αgEg) Moreover, PI(W ) and VI(W ) respectively denote in this paper the interfacial pressure and velocity, and will precised afterwards.
- The term S1,l will also be introduced later on.
- External sources might be included but are not considered herein.
- Standard viscous contributions may be included, which of course comply with the entropy inequality that will be detailed in the next subsection.
2.2 Closure laws for interfacial transfer terms
- The same holds when tackling three-phase flows, as emphasized in .
- Now, the second requirement (H1) implies that the field associated with λ = VI should be linearly degenerate.
- As shown in [8, 11], few expressions guarantee this behaviour.
2.4 Main properties of the two-fluid model
- The authors may now recall in brief the main properties of system (2) using the previous closure laws.
- Apart from the field associated with the eigenvalue λ =.
- Otherwise the computation of shock solutions would be meaningless, since multiple shock solutions may be obtained using various -stable- solvers (see for examples ) .
- Obviously, an alternative formulation of jump conditions in genuinely non linear fields that is probably more convenient may be: σ = [ρφUφ]RL/[ρφ].
3.2 Computing the evolution step
- Many solvers have been proposed in the literature for such a purpose.
- Approximate solutions in the evolution step may also be obtained using either the non-conservative form of Rusanov scheme, or the non-conservative form of the approximate Godunov scheme VFRoe-ncv ; the authors refer to [11, 10] for such a description.
- Obviously, the ultimate scheme has not been proposed yet.
- The proof is classical but is briefly recalled.
- The authors detail afterwards the relaxation step, with special focus on the pressure relaxation step which is rather tricky.
3.3 Computing the velocity relaxation step
- It may be easily checked that internal energies remain positive through this step.
- The proof is obvious considering formula (16) and is left to the reader.
3.4 Computing the temperature relaxation step
- The proof provides a practical way to compute solutions of system (18), that is actually used in the code.
- The authors emphasize that other -simpler- algorithms may be exhibited ; however, practical computations seem to show that the non-linear temperature relaxation scheme described here provides better results (see  and section IV).
- The authors turn now to the most difficult part which corresponds to the pressure relaxation step.
3.5 Computing the pressure relaxation step
- The first one is a semi-implicit scheme, that is such that the existence and uniqueness of the discrete solution is ensured, whatever the equations of state would be.
- The authors focus here on the second one which is totally implicit with respect to the unknown (Pl, Pg, αl).
- The main lines of the proof are given below, also known as Proof.
- The implicit scheme (20) is exactly the same as the one introduced for dense granular gas-particle flows in , also known as Remark 2.
4 Numerical results
- In the first subsection, the authors focus on the computation of the evolution step involving all convective terms.
- The following unities are used : m for distances, kg/m3 for densities, m/s for velocities, Pa for pressures, K for temperatures, s for times and J/s/m2 for heating fluxes.
4.1 Verification of the evolution step
- The authors use for this test case perfect gas EOS within each phase, setting γg = 1.4 and γl = 1.1.
- The second test case is another Riemann problem taken from , where initial conditions are given in table 2.
- This Riemann problem also contains a contact wave associated with Ug, and a right-going gas shock wave.
4.2.1 Velocity relaxation substep
- Two different series of verification test cases have been considered in  for the velocity relaxation step.
- The first one refers to a constant time scale τ3.
- In that case, the scheme (16) is perfect, since it computes the exact value at time tn+1.
- In that case, analytic solutions allow computing the true error occuring in the velocity relaxation step.
- Several examples can be found in , which confirm that a first-order rate of convergence is achieved.
4.2.2 Temperature relaxation substep
- The authors present below some tests corresponding to constant time scales τ4, when computing approximate solutions of (17) with the scheme (18).
- Φ, unlike the non-linear scheme (18); actually, round-off errors are found for the linear scheme in that case.
4.2.3 Pressure relaxation substep
- Eventually, the authors provide an example of measured convergence rates in the pressure relaxation step.
- This one is crucial, and should be handled with great care.
- Otherwise, both the present model (where the relaxation time scale τ2 is non-zero) and standard two-fluid models (corresponding to τ2 = 0) may be confused, if inadequate “rough” schemes are used to provide approximate solutions of (19).
- The initial conditions of the test case are given in table 5. Figure 6 and Figure 7 show the behaviour of the scheme (20) and the comparison with another half-implicit scheme introduced in  and recalled in , focusing on void fractions and pressures within each phase at time t = 10−5.
- This enables to retrieve the fact that the initial-value problem associated with τ2 = 0 is ill-posed: spurious oscillations arise when the mesh size is sufficiently small, and this is particularly spectacular for void fraction profiles, since the algorithm guarantees bounded variations owing to properties 4 and 5 (see  and  also for a similar study).
4.3 Two-dimensional numerical results
- The authors consider now the two-dimensional unsteady computation of a heated wall in an almost square domain, where the wall contains a small cavity in the middle of the lower part .
- On the contrary the two cells at the exit corners of the cavity do not receive any heat flux.
- It also seems worth mentionning that almost similar results have been obtained while changing the pressure relaxation time step within the range τ2 ∈ [10−9, 10−6], keeping other relaxation time steps unchanged and using the same mesh.
- Nonetheless, one must be aware that the 3D counterpart of the present 2D mesh would contain more than 150 millions of cells, which is of course far beyond what one can afford in an industrial situation.
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Cites background from "A fractional step method to compute..."
...Actually, comparing with Lax-Friedrichs type schemes is quite significant since for such stiff configurations as vanishing phase cases, these schemes are commonly used in the industrial context because of their known robustness ....
...This is actually the case when, in addition to the convective system (1), zero-th order source terms are added to the model in order to account for relaxation phenomena that tend to bring the two phases towards thermodynamical (T1 = T2 and u1 = u2) and mechanical (p1 = p2) equilibria (see [12, 21] for the models and  for adapted numerical methods)....
...However, this work is mainly concerned with the convective effects and these relaxation source terms are not considered here (see  for some modeling choices of these terms and  for their numerical treatment)....
"A fractional step method to compute..." refers background in this paper
...2 Closure laws for interfacial transfer terms The interfacial transfer contributions have been studied in [4, 5, 17, 10, 28, 29], and a summary can be found in appendix A....
...Among other studies, one should at least point out the contributions [4, 28, 5, 27, 32] which are concerned with the modeling aspects....
"A fractional step method to compute..." refers background in this paper
...with “zero time scales”) one can refer to  (which deals with instantaneous velocity and pressure relaxation) or  (which only deals with instantaneous pressurerelaxation) among others....
...Fewer papers tackle the problem of water-gas or water-vapour flows, among which we must quote [33, 34] and more recently the article  , that examines a medium of small oscillating bubbles in a liquid medium, and also provides a general formalism in order to derive meaningful governing equations....
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