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Mathematics and Computers in Simulation 80 (2010) 1624–1635

Numerical modeling of two-phase transonic flow

Jan Halama a,b,∗, Fayssal Benkhaldoun c,d, Jaroslav Fort a,b

a Department of Technical Mathematics, FME, CTU Prague, Czech Republicb Institute of Thermomechanics, Academy of Sciences, Prague, Czech Republic

c LAGA, Universite Paris 13, Franced CMLA, ENS Cachan, France

Received 30 October 2008; received in revised form 6 February 2009; accepted 11 February 2009Available online 21 February 2009

Abstract

Our work is aimed at the development of numerical method for the modeling of transonic flow of wet steam including conden-sation/evaporation phase change. We solve a system of PDE’s consisting of Euler or Navier-Stokes equations for the mixture ofvapor and liquid droplets and transport equations for the integral parameters describing the droplet size spectra. Numerical methodis based on a fractional step technique due to the stiff character of source terms, i.e. we solve separately the set of homogenousPDE’s by the finite volume method and the remaining set of ODE’s either by explicit Runge-Kutta or implicit Euler method. Thefinite volume method is based on the Lax-Wendroff scheme with conservative artificial dissipation terms for structured grid. Wealso note result achieved by recently developed finite volume method with VFFC scheme. We discuss numerical results of steadyand unsteady two-phase transonic flow in 2D nozzle, 2D and 3D turbine cascade and 2D turbine stage with moving rotor cascade.© 2009 IMACS. Published by Elsevier B.V. All rights reserved.

PACS: 02.60.Cb; 64.60.Q; 64.70.F

Keywords: Two-phase flow; Condensation; Transonic flow; Fractional step

1. Introduction

Transonic flow is typical in many practical applications like an internal flow in channels or turbine cascades. Theflowing media in such cases often contains vapor component, which may condense due to the strong pressure andtemperature variations. The structure of flow field is influenced by condensation through the released latent heat. Thislatent heat release can also induce some oscillations. The condensation is a non-negligible phenomenon, especiallyin the case of flow in last stages of steam turbines, where even small drop in efficiency due to condensation brings asignificant loss in output energy.

Here we deal with the compressible flow of wet steam, which is the mixture of vapor and condensed droplets.The flow of this mixture is approximated by an inviscid or laminar flow models (Euler or Navier-Stokes equationsrespectively). The laminar flow model is considered as an intermediate step for future extension into turbulent flow.We consider following simplifications: condensation is homogenous; the droplets are convected by the vapor; the mass

∗ Corresponding author at: Department of Technical Mathematics, FME, CTU Prague, Czech Republic.E-mail address: [email protected] (J. Halama).

0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2009.02.004


J. Halama et al. / Mathematics and Computers in Simulation 80 (2010) 1624–1635 1625

fraction of liquid phase (droplets) is small, i.e. the volume of droplets can be neglected; the pressure of the mixture isapproximated by the pressure of vapor; droplet size spectra is described by the Hills approximation [7].

2. Flow model

We distinguish two mechanisms of phase change: the creation of new droplets (nucleation) and the growth orevaporation of existing droplet. Due to high velocities and relatively small size of physical domain (very fast temperaturevariations) we deal with non-equilibrium case. Therefore new droplets with the size of critical radius rc start to appear,when the vapor temperature drops sufficiently below the saturation temperature (a typical sub-cooling for steam isbetween 30 and 40 K). The number of new droplets per unit volume and per second is given by the nucleation rate Jof [2]

J =√

2σ

πm3v

· ρ2v

ρl

· exp

(−β · 4πr2

cσ

3kBTv

), rc = 2σ

ρlRvTv ln(pv/ps), (1)

where the symbol mv denotes mass of water molecule, ρl water density, kB Boltzmann constant, Rv vapor gas constant,ρv vapor density computed as ρv = (1 − w)ρ, where w gives the mass fraction of liquid in mixture (so called wetness)and ρ is the mixture density. The vapor pressure pv is considered equal to the pressure of the mixture from Eq. (6)and the vapor temperature Tv is calculated using a perfect gas law Tv = pv/(ρvRv). The value of surface tension σ iscorrected by the coefficient β = 1.328p0.3

cor, see [11], where pcor (bar) denotes the pressure at the intersection of theisentropic expansion from reservoir conditions and the steam saturation line, see Fig. 1.

The growth r of existing droplet depends on the temperature of vapor surrounding the droplet and on droplet radius,see [19].

r = λv(Ts − Tv)

Lρl(1 + 3.18 · Kn)· r − rc

r2 , Kn = νv · √2πRvTv

4rpv

, (2)

where λv is vapor thermal conductivity, Ts is saturation temperature, L latent heat, r is average radius, νv vapor viscosity.The whole size spectra of droplets is described by three moments Q0, Q1 and Q2 proposed in [7](this approximationyields a relatively simple model, which can be applied for multidimensional flow problems):

Q0 = N, Q1 =N∑

i=1

ri, Q2 =N∑

i=1

2ri, r =

⟨0, w ≤ 10−6

√Q2/Q0, w > 10−6

, (3)

where N denotes the total number of droplets per unit mass of mixture, ri is the radius of i-th droplet and r is the averageradius. The limit value 10−6 for wetness was chosen to avoid a small numbers in denominator. The relation (2) takes

Fig. 1. Definition of pcor.


1626 J. Halama et al. / Mathematics and Computers in Simulation 80 (2010) 1624–1635

into account also the evaporation of droplet (in such case we set rc = 0). The evaporating droplet should disappearwhen it reaches its starting size to conserve the energy. Due to the used averaging it is not possible to track each droplet(and of course if it would be possible, it would be computationally expensive), so we let the liquid phase to disappear,when the average radius reaches artificially chosen minimum rmin = 10−12m. According to [13] we set r = 0, if theaverage radius is below the critical radius and the temperature of the vapor is below saturation temperature to avoidevaporation of droplets with ri > rc.

If we omit convection, then the time derivatives of Q0, Q1, Q2 and w are driven by the nucleation and dropletgrowth terms

Q0 = J

ρ

Q1 = rcJ

ρ+ Q0r

Q2 = (rc)2 J

ρ+ 2Q1r

w = ρl

4

3π(rc)3 J

ρ+ ρl4πQ2r.

(4)

The Navier-Stokes equations for the 2D flow (equations for 3D case can be written analogically) of mixture andthe transport equations for variables Q0, Q1, Q2 and w can be written together as a one system of partial differentialequations in form

∂tW + ∂xF + ∂yG = Q, where F = Fc − Fv and G = Gc − Gv, (5)

W = [ρ, ρux, ρuy, e, ρw, ρQ2, ρQ1, ρQ0

],

Q =[

0, 0, 0, 0,4

3πr3

cρlJ + 4πρQ2rρl, r2cJ + 2ρQ1r, rcJ + ρQ0r, J

],

Fc = [ρux, ρu2

x + p, ρuxuy, (e + p)ux, ρwux, ρQ2ux, ρQ1ux, ρQ0ux

],

Fv = [0, τxx, τxy, uxτxx + uyτxy − qx, 0, 0, 0, 0

],

Gc =[ρux, ρuyux, ρu2

y + p, (e + p)uy, ρwuy, ρQ2uy, ρQ1uy, ρQ0uy

],

Gv = [0, τxy, τyy, uxτxy + uyτyy − qy, 0, 0, 0, 0, 0

],

τxx=2η

3

(2∂xux−∂yuy

), τxy = τyx = η

(∂yux + ∂xuy

), τyy = 2η

3

(2∂yuy − ∂xux

), qx = −λ∂xT qy = −λ∂yT.

The equations of the system (5) are coupled by the equation for the pressure according to [15] or [12]:

p = (γ − 1)(1 − w)

1 + w(γ − 1)

[e − 1

2ρ(u2

x + u2y) + ρwL

]. (6)

The specific heat ratio is taken as a function of temperature: γ = (cp(T )/cp(T ) − Rv). The domain of solutionD ⊂ R2 (or R3) has boundary typically with four different parts: an inflow boundary (or inlet), an outflow boundary(or outlet), a non-permeable wall boundary (or wall) and periodical boundary. The solution of considered problem hasto satisfy the integral form (7) of Eq. (5) for any V ⊂ D

∂t

∫ ∫V

WdV = −∮∂V

(F(W), G(W)) �n∂V dS −∫∫

V

Q(W)dV. (7)

and proper boundary conditions along boundary ∂D.



Fig. 2. Computational domain D —one blade channel.

We use following boundary conditions for the inviscid flow model: Since the velocity component, which is perpendic-ular to the inlet boundary Γi, is subsonic for all considered flow cases, ‘the number of unknowns minus one’ parametershave to be set according to the theory of characteristics. We prescribe a constant values of reservoir conditions T0 andp0, the flow direction, w = 0, Q0 = 0, Q1 = 0 and Q2 = 0. The non-permeability condition (ux, uy)�n = 0, where �ndenotes the unit vector normal to boundary, is considered along walls Γw. The velocity component, which is perpen-dicular to the outlet boundary Γo, is subsonic in all cases of turbine flows (according to the theory of characteristics oneparameter has to be specified—we prescribe an outlet static pressure) and supersonic in all cases of nozzle flows (noparameter can be set). The computational domain for turbine flow calculations consists of one blade passage, i.e. weconsider a spatial periodicity of solution W(A) = W(B) along Γp The boundary conditions for viscous flow are similar,the system of the above mentioned conditions is completed by proper Neumann’s conditions. The non-permeabilitycondition along walls is of course replaced by the no-slip condition ux = 0, uy = 0 and the adiabatic wall condition∂T/∂�n = 0 (Fig. 2).

3. Numerical method

Phase change time scale, which can be estimated, e.g. by the relaxation time τ [20] is typically one or two orderssmaller compared to the scale of convection

τ = (1 − w)cpr2ρl(1 + 3.18Kn)

3λvw. (8)

Therefore we use symmetrical (or Strang) splitting method, see [16] in form

∂tW∗ = Q(W∗) (9)

∂tW∗∗ = −∂xFc(W∗∗) + ∂xFv(W∗∗) − ∂yGc(W∗∗) + ∂yGv(W∗∗) (10)

∂tW∗∗∗ = Q(W∗∗∗), (11)

i.e. we solve successively three initial value problems (9)–(11) with following initial conditions: W∗(t) = W(t),W∗∗(t) = W∗(t + t/2), W∗∗∗(t) = W∗∗(t + t) and the solution W∗∗∗(t + t/2) corresponds to the solutionW(t + t) of the original unsplit system (5) plus some splitting error. A finite volume cell-vertex method, describedlater, is used to solve the convection–diffusion part (10) and the explicit two-stage Runge-Kutta method is applied ineach grid point to compute condensation parts (9) and (11). Let us denote one step of finite volume method for Eq. (17)by CV(Wn

i,j, t), where the first argument corresponds to ‘initial data’ and the second one to used time step. Further,lets denote one step of the Runge-Kutta method by RK(Wn

i,j, t). Proposed numerical method then reads



W(0)i,j = Wn

i,j

W(k+1)i,j = RK

(W(k)

i,j , t

2N)

, k = 0, . . . ,N− 1

W(N+1)i,j = CV(W(N)

i,j , t)

W(k+1)i,j = RK

(W(k)

i,j , t

2N)

, k = N+ 1, . . . , 2N

Wn+1i,j = W(2N+1)

i,j ,

(12)

where the number of sub-steps N fits the relaxation time τ, N = tτ and t comes from the stability condition ofcell-vertex method. Due to the stability problems of this algorithm in three-dimensional case, the explicit method forthe parts (9) and (11) was replaced by the implicit Euler method, where the nonlinear equation has been solved byiterative method.

As an initial conditions W0i,j we usually prescribe the reservoir conditions inside the whole domain. Analysis of

simple scalar case with Strang splitting shows, that the truncation error introduced by the splitting is of the secondorder for steady and the first order for the unsteady case, see [9].

4. Cell-vertex method for convection–diffusion part

The presented finite volume method has the origin in the scheme of Ni [10]. The nodes of a structured quadrilateralgrid are indexed by subscripts i and j, see Fig. 3. The method is based on the Taylor expansion for W, where the timelevel is denoted by the superscript n.

Wn+1i,j = Wn

i,j + t∂tW∣∣∣ni,j

+ t2

2∂ttW

∣∣∣∣ni,j

(13)

The first order time derivative is obtained by a standard finite volume approach for non-moving cell-vertex finitevolumes

∂tW|i,j = − 1

μ(Vi,j)

∫ ∫Vi,j

(∂xF + ∂yG)dV. (14)

Fig. 3. Finite volumes Vi,j and V ∗i,j for the grid point Pi,j .



Using the mean value theorem for the second derivative of W we get

Wn+1i,j = Wn

i,j − t

μ(Vi,j)

∫ ∫Vi,j

(∂xF + ∂yG)ndV + t2

2μ(V ∗i,j)

∫ ∫V ∗

i,j

∂ttW|ndV ∗, (15)

where the second derivative of W can be expressed in the following way

∂t(∂tW) = −∂t(∂xF + ∂yG) = −∂x(∂tF) − ∂y(∂tG), (16)

where the time derivatives of F and G can be calculated numerically, so finally

Wn+1i,j = Wn

i,j − t

μ(Vi,j)

∮∂Vi,j

(F(Wn), G(Wn))�n∂V d

Wn+1i,j = W

n+1i,j − t2

2μ(V ∗i,j)

∮∂

∗Vi,j

(f, g)�n∂V ∗dS∗ +n

DISi,j

,(17)

with

f = [F(Wn+1

) − F(Wn)]

t, g = [G(W

n+1) − G(Wn)]

t(18)

and the term DISni,j is an artificial dissipation term. The first order term is approximated in the following way∮

∂Vi,j

(F, G) · (nx, ny)dl =∮∂Vi,j

(F, G) · (dy, −dx) ≈8∑

k=1

(Fk yk − Gk xk), (19)

where Fk and Gk are averages of F and G on cell edges coinciding with the boundary of volume Vi,j , e.g. F1 =(Fi,j−1 + Fi−1,j−1)/2. The symbols xk and yk denote x- and y-displacements of mentioned cell edges, e.g. x1 =(xi,j−1 − xi−1,j−1)/2. The numerical approximation of the second order term is done analogously by summation offluxes through eight edges of ∂V ∗

i,j in order to preserve conservativity. To stabilize this central type scheme, we use aconservative form [17] of artificial dissipation proposed by [8].

DISi,j =4∑

β=1

[(2)εβ

(AVGβ(W) − Wi,j

) − (4)εβ

(AVGβ(δ(2)W) − (2)

δi,j

W)]

,

where δ(2)i,j W =

4∑β=1

(AVGβ(W) − Wi,j

),

(20)

where AVGβ(W) are the averages of W in the centers of cells surrounding the point indexed i, j. The first part of theDIS term in Eq. (20) approximates the Laplacian of W and the second one approximates the sum of fourth orderspatial derivatives. The coefficient ε

(2)β is controlled by the second derivative of pressure and stabilizes the solution on

steep gradients and goes to zero for smooth parts of solution. The coefficient ε(4)β acts in opposite way (it stabilizes the

smooth parts).The space derivatives of all velocity components and of temperature T, which are necessary for the evaluation of

shear stresses τ and heat fluxes q are calculated as follows

∂x

⎡⎢⎢⎢⎣ux

uy

T

⎤⎥⎥⎥⎦∣∣∣∣∣∣∣∣∣i,j

= 1

μ(Vi,j)

∫∫Vi,j

∂x

⎡⎢⎢⎢⎣ux

uy

T

⎤⎥⎥⎥⎦ dV = 1

μ(Vi,j)

∮∂Vi,j

⎡⎢⎢⎢⎣ux

uy

T

⎤⎥⎥⎥⎦ nx∂V dS, (21)



Fig. 4. Pressure distribution along the nozzle axis (left) and wetness isolines (right) for T01 = 380.50 K.

where nx∂V denotes the x-component of �n∂V . The y-derivatives are calculated analogically. Numerical method has been

validated for several cases of inviscid transonic flow, see, e.g. [5].

5. Numerical results for flow in a convergent–divergent nozzle

We consider the model of two-phase inviscid flow in convergent–divergent nozzle with the geometry publishedby Barschdorff [1]. Numerical results along nozzle axis are compared with the measurement of [1]. Three differentregimes have the same inlet total pressure p01 = 78, 390 Pa and axial inlet flow direction. They differ in the value oftotal inlet temperature. The heat released by the nucleation slows down the supersonic flow resulting in a pressure jumpcalled the condensation shock. The results in Figs. 4 (left) and 5 (left) show a well captured position of this pressurejump.

The decrease of the total inlet temperature shifts the nucleation zone upstream and leads to the higher values ofwetness. The flow in the cases with T01 = 380.50K and T01 = 373.15K is steady see also [3]. Further decrease ofinlet total temperature to T01 = 367.00K yields oscillations of nucleation shock. This behavior has been observed insimilarly arranged experiments. Computed pressure distribution of these oscillations along the nozzle axis during onetime period T = 1.4 × 10−3 s is in Fig. 5 on the right. Interesting is also an effect of numerical viscosity, whereless dissipative VFFC method (for details about this method see, e.g. [21]) in Fig. 5(on the left) yields more sharpcondensation shock, for details see [6].

Fig. 5. Pressure distribution along the nozzle axis for steady flow with T01 = 373.15.50K (left) and five distributions during the period of unsteadyflow with T01 = 367.00K (right).



Fig. 6. Numerical results achieved by the model of inviscid one-phase (a) and two-phase flow model (b) and (c). Isolines of Mach number (a) and(b) with M = 1 denoted by bold line and M = 0.05, isolines of wetness (c).

6. Transonic flow in the axial turbine cascade SE1050

The structure of two-phase flow even in a such simple geometry like previous case of 2D flow in the nozzle dependsstrongly on the quality of numerical method. The boundary between steady and unsteady solution depends, e.g. onthe amount of numerical dissipation introduced by used numerical method. The case of turbine flow is much morecomplex, we can observe in addition evaporation of droplets. Figs. 6 and 7 show numerical results of two-phaseflow in turbine cascade SE1050 (for geometry see [14]) using an inviscid and a laminar (Re = 1.5 × 106) 2D flowmodels. We consider following boundary conditions: the inlet angle 19.3◦ from axial direction, the inlet total pressure

Fig. 7. Numerical results achieved by the model of laminar one-phase (a) and two-phase flow model (b) and (c). Isolines of Mach number (a) and(b) with M = 1 denoted by bold line and M = 0.05, isolines of wetness (c).



Fig. 8. Stator cascade, wetness isolines ( w = 0.005). (a) Explicit method for condensation and (b) implicit method for condensation.

p01 = 36, 730 Pa and temperature T01 = 340 K and the mean outlet pressure poutlet/p01 = 0.423 for all presentedcases. The effect of molecular viscosity is more pronounced in the case of two-phase flow model compared to theone-phase flow model. This viscosity enlarges the nucleation zone. Note also a big difference in the shape of sonicline and of the flow structure around trailing edge caused by the latent heat effect (compare corresponding one andtwo-phase results).

7. Numerical results of flow in 3D axial turbine cascade

Fig. 8 shows results achieved by 3D inviscid two-phase flow model in turbine stator cascade by the presentednumerical method with explicit and implicit solver for the condensation part. The explicit method yields numericalsolution with oscillations, see the wetness isolines in Fig. 8 left. The use of an implicit method (backward Euler) in3D fixed this problem. Note that all 2D results were computed by fully explicit method without this problem.

8. Numerical solution of unsteady flow in 2D axial turbine stage

We consider domain with two parts: stator part Ωs (bounded by inlet Γi and interface Γc) and the rotor one Ωr

(bounded by Γc and outlet Γo), see Fig. 9 (a).

Fig. 9. Description of computational domain: (a) boundaries and (b) periodicity for stage.



Fig. 10. The interface between the stator and the rotor grid for an unsteady stage calculations.

The governing equations are solved in coordinate frames attached to the blade cascade. Boundary conditions alongstage inlet boundary Γi and stage outlet boundary Γo are the same like in the case of single cascade. We furtherconsider the continuity of W (in common coordinate frame) along Γc. Concerning periodicity conditions we havechosen a simple approach, where the stator and the rotor parts Ωs and Ωr have the same dimension in tangentialdirection, see Fig. 9(b), so simply W(A) = W(B) and W(C) = W(D). Domain for presented case has 4 stator and 7rotor blade channels (approximation of real stage with 32 stator and 56 rotor blades). The stator and rotor grids havethe same number of equally distributed points along Γc, they are therefore directly connected. The relative movementbetween them is done by deformation of one row of cells along Γc. To keep the deformation bounded, the grids arereconnected in a proper moment, see Fig. 10. This algorithm was originally proposed by Giles [4].

Due to moving grid in y direction, the integral form of governing Eq. (7) has to be changed into

∂t

∫∫V

WdV = −∫∫

V

(∂xF + ∂yG − Q

)dV +

∮∂V

W�u∂V �n∂V ds, (22)

where �u∂V is the velocity of ∂V and �n∂V is the outer normal vector of ∂V . The stator and rotor grids are overlapped overthree cells, see Fig. 11, where ‘the last point in the stator’ is Bs (the finite volume for Bs is denoted by grey color) and‘the first point for the rotor’ is Cr. The solution from points Cr and Dr is transferred after each time step into ‘ghost’points Cs and Ds and similarly the information from As and Bs is transferred into ‘ghost points’ Ar and Br see also [18].

The presented case is a 2D cylindrical cut of steam turbine stage at radius 0.74 m. The rotor has RPM = 3000.The inlet total pressure is p01 = 78, 390 Pa and the mean outlet pressure is poutlet = 51, 737 Pa. The inlet angle is 0◦from axial direction. Numerical results of flow with neglected phase change are denoted as ‘without condensation’and the case with phase change as ‘with condensation’. Comparison of these two cases helps to understand mainlythe effect of heat released by condensation. Numerical results in form of pressure and wetness isolines are shown inFigs. 12 and 13. The flow is subsonic in the whole domain, i.e. the released heat accelerates locally the flow. Moreoverthe flow fluctuations can travel up to the stage inlet (the stator cascade is not choked).

Computed pressure distributions along stator and rotor blade profiles are plotted in Figs. 14 and 15. The pressurefluctuations are stronger on the rotor blade profile. A slight flow acceleration downstream the nucleation zone (i.e. alsohigher expansion) causes lower expansion along the suction side of stator blade for the case with condensation, seeFig. 14.

Fig. 11. Overlapping of stator and rotor grids at the interface.



Fig. 12. The instantaneous structure of unsteady flow field in turbine stage: (a) without condensation, isolines p/p0 (b) with condensation, isolinesp/p0 (c) with condensation, wetness isolines.

Fig. 13. Continuation of Fig. 12 after rotor shift of half rotor pitch: (a) without condensation, isolines p/p0 (b) with condensation, isolines p/p0

(c) with condensation, wetness isolines.

Fig. 14. Pressure distribution along stator blade profile in five consecutive steps during the rotor shift of one rotor pitch (basic period of pressurefluctuation on stator blade), without condensation dashed line, with condensation continuous line.

Fig. 15. Pressure distribution along rotor blade profile in five consecutive steps during the rotor shift of one stator pitch (basic period of pressurefluctuation on rotor blade), without condensation dashed line, with condensation continuous line.



9. Conclusions

The algorithm of presented numerical method solves ODE’s for condensation in each grid point independently,i.e. the time step for the condensation can be refined locally and the extension into multi-D case is straightforward.The agreement between numerical results and experimental data in the distribution of static pressure along axis of theBarschdorff nozzle is good. The numerical results of 2D two-phase flow in SE1050 turbine cascade are physicallyexpectable and show that the present method is able to model also an evaporation. The problems with numericalstability numerical method in 3D case were solved by replacing the explicit Runge-Kutta method for ODE’s part bythe implicit Euler method. The numerical results of two-phase unsteady flow in stator–rotor interaction were obtainedby the fully explicit method. The realization of connection between stator and rotor parts by direct connection of gridsand one row of deforming cells seems sufficiently robust.

Acknowledgments

This work has been supported by the Research plan of MSM CR No. 6840770003 and by the grant No. 201/08/0012of GACR.

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Author's personal copy Numerical modeling of two-phase transonic flow

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