integrated simulations for multi-component analysis of gas...

Integrated Simulations for Multi-Component Analysis

of Gas Turbines : RANS Boundary Conditions

Sangho Kim∗ Jorg Schluter† Xiaohua Wu‡ Juan J. Alonso§ and Heinz Pitsch¶

Stanford University, Stanford, CA 94305, U.S.A

The aero-thermal computation of the flow path of an entire gas turbine engine canbe performed using multiple flow solvers, each specialized to a component of the engine.Here, we present an approach to integrate a Large Eddy Simulation (LES) solver and aReynolds Averaged Navier-Stokes (RANS) solver. Challenges arise, when the LES solveris based on a low-Mach number approximation and can not deliver all variables needed fora compressible RANS solver. This study investigates the choice of boundary conditionsapplied to the RANS interface. We propose the use of inlet/exit boundary conditions andinvestigate the effect on simple pipe geometries.

INTRODUCTION

Computational Fluid Dynamics (CFD) has been used in the analysis of single components of the gasturbine engines as an aid in the design process. However, the simulation of the entire flow path including

the compressor, the combustor, and the turbine etc. has not been feasible due to the enormous computationalcosts and the wide variety of the flow phenomena that have to be modeled. Recently the challenge tosimulate an entire system of physical and/or geometrical complexity is motivated by the availability of highperformance computing platforms and the development of new and efficient analysis algorithms.

The goal of the Advanced Simulation and Computing Initiative (ASC) of the Department of Energy(DoE) at Stanford is to develop high-performance flow solvers which are able to use highly parallel super-computers for the simulation of an entire engine. Multi-Component Integrated Simulation is one of themain tasks in the ASC effort of the DoE at Stanford. The prediction of multi-component phenomena,such as compressor/combustor instability, combustor/turbine hot-streak migration, and main/secondary flowingestion/interaction, can be improved by a simultaneous simulation of all components (Fig. 1). For realizinga simultaneous simulation, new routines coupling different codes and handing the interface communicationshas been developed by using Message Passing Interface (MPI) standard.1 Several flow solvers has successfullybeen integrated using these routines.2, 3, 4 The routines were written in a very generic fashion such that theyestablish a communication contact between any arbitrary flow solvers ensuring the necessary information onthe interface boundaries.

The recent effort has focused on coupling the compressor, gas-turbine combustor, and turbine main flowpath. For the turbomachinery parts, the Reynolds-Averaged Navier-Stokes (RANS) approach was usedwith an appropriate turbulence model since the flow in this domain is characterized by both high Reynoldsnumbers and high Mach numbers. On the other hand the flow in the combustor is characterized by detachedflows, chemical reactions and heat release. The prediction of detached flows and free turbulence is greatlyimproved using flow solvers based on Large-Eddy Simulations (LES).

The integration of the structured LES flow solver with the unsteady RANS flow solver has recently beenimplemented.4 The actual computation using this integration of RANS and LES codes has been performedfor the coupled NASA stage 35 compressor/prediffuser configurations.5, 6 More recently the coupling methodhas also been applied to a Pratt & Whitney gas turbine.7 The RANS flow solver used for these works is the

∗Postdoctoral Fellow, Department of Aeronautics and Astronautics†Research Associate, Center for Turbulence Research‡Research Associate, Center for Turbulence Research§Assistant Professor, Department of Aeronautics and Astronautics¶Assistant Professor, Center for Turbulence Research

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American Institute of Aeronautics and Astronautics

TFLO code developed in the Aerospace Computing Lab (ACL) at Stanford. The LES flow solver chosen isthe CDP code developed at the Center for Turbulence Research (CTR) at Stanford. Both TFLO and CDPhave been proven in the past to their efficiency and accuracy in modeling of physical effects.8, 9

Figure 1. Computation of the flow path of an entire gasturbine: Domain decomposition for Pratt & Whitneygas turbine

Assuming that all the necessary flow infor-mation is exchangeable, the communication be-tween two separate flow solvers can be very sim-ilar to the information exchange between proces-sors of a parallel computation in the multi-blockapproach10 or/and the Overlapped (Chimera) gridapproach.11, 12 Meanwhile, in the coupling of LESand RANS flow solvers whose mathematical formu-lations are different, the formulation of LES andRANS interface boundary conditions becomes a newand very challenging issue to be resolved. The fol-lowing problems, then, can be addressed: first, un-steady LES boundary conditions have to be gener-ated which produce the statistical properties of thetime-averaged solution delivered by the RANS flowsolver. Even if an unsteady RANS computation isassumed, the time-step of the unsteady RANS com-putation can be larger than the LES time-step byseveral orders of magnitude. The LES boundary condition then has to correspond to the ensemble-averagesdelivered by the RANS computation. LES inflow conditions from a RANS interface and LES outflow condi-tion from a RANS zone has also recently investigated and presented.13, 14, 15 Second, all necessary flow datamay not always be available for the coupling of such different codes. In the current study, for example, thecompressible formulation of the RANS flow solver and the low-Mach-number formulation of the LES codeposed one of the challenges since the density fields of two codes are not compatible. While the RANS codeallows for acoustic waves to propagate within the limits of its domain, the density field of the LES solutionis uniform since it is entirely defined by mixing and the combustion process, not by acoustics.

To compensate any shortages of the necessary flow information at the RANS interface due to the issuejust mentioned, we suggest that the typical inlet/exit RANS boundary analysis is to be employed. We will,for a convenience, call the RANS inlet/exit boundary conditions as the “interface inflow/outflow” boundaryconditions when they are used at the interface. Research in this paper focuses on the RANS interfaceinflow/outflow boundary conditions for simple pipe flows. This paper presents a numerical experiment thatjustifies the treatment of the interface communication as a boundary condition problem. Though the presentwork is to eventually provide the interface boundary conditions for general cases of RANS-LES coupling, aRANS-RANS coupling with the same interface communication routines was used for representing the RANS-LES coupling in order to explicitly exclude sources of error that may stem from the different mathematicalformulation in RANS and LES. Further simplification was made by considering only the steady state inorder to help recognize sources of problem. The specification of RANS interface boundary conditions isalso presented. The following sections will briefly describe the RANS and LES flow solvers, the interfaceframework, and the LES boundary conditions. Then intensive discussions and results for RANS interfaceboundary conditions are presented.

FLOW SOLVERS

RANS Flow Solver

RANS flow solvers are solving the classical Reynolds-Averaged Navier-Stokes equations for turbulent flows.Here, the flow variables are split into a mean and a fluctuating part vi = vi + v′i and the Navier-Stokesequations are time-averaged. This delivers a set of equations for the mean velocities, but leaves an unclosedterm v′iv

′

j , which has to be modeled with a turbulence model. Turbulence models are commonly based onthe eddy viscosity approach, where the eddy viscosity can be modeled in varying levels of complexity. Themost applied models for RANS flow solvers are two-equation models, such as the k − ε or k − ω models,where two additional transport equations are solved in order to determine the eddy viscosity. These modelsare accepted as a good compromise for turbo-machinery applications between efficiency and accuracy.

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The flow solver computes the unsteady Reynolds Averaged Navier-Stokes equations using a cell-centereddiscretization on arbitrary multi-block meshes. The solution procedure is based on efficient explicit mod-ified Runge-Kutta methods with several convergence acceleration techniques such as multi-grid, residualaveraging, and local time-stepping. These techniques, multi-grid in particular, provide excellent numericalconvergence and fast solution turnaround. Turbulent viscosity is computed from a k − ω two-equation tur-bulence model. The dual-time stepping technique16, 17, 18 is used for time-accurate simulations that accountfor the relative motion of moving parts as well as other sources of flow unsteadiness.

LES Flow Solver

LES flow solvers solve for the filtered Navier-Stokes equations. The filter ensures that the large scaleturbulence is resolved in time and space resulting in a decomposition of the variables in a resolved and asubgrid part vi = vi + vi”. For practical purposes, usually the mesh filter is applied, which means thatthe cell size defines the filter at each location. Applying the filter to the Navier-Stokes equation leaves an

unclosed term ˜vi”vj”, which defines the subgrid turbulence. As opposed to the similar unclosed term v′

iv′

j

from the RANS flow solver, which includes the turbulent motions of all scales, the LES term describes onlythe subgrid turbulence. With a sufficiently high mesh resolution, the LES solution is rather robust againstthe chosen subgrid model. Most models use an eddy viscosity approach to model the subgrid stresses. Here,the eddy viscosity can be determined by algebraic models such as the standard Smagorinsky model,19 or,as used in this study, by a dynamic procedure, where the solution of the high frequent resolved flow field isused to determine the subgrid stresses20 . The filtered momentum equations are solved on a cell-centeredunstructured mesh and are second-order accurate. An implicit time-advancement is applied. The subgridstresses are modeled with a dynamic procedure.

INTERFACE

Part of the efforts to integrate these flow solvers is the definition of the interface. The optimization ofthe communication and the processing of the exchanged data to meaningful boundary conditions are someof the encountered challenges. In previous work interface routines have been established and validated withsimple geometries 2, 4, 5 .

The interface used for establishing a connection between the flow solvers consists of routines following anidentical algorithm in all flow solvers. The message passing interface MPI is used to create communicators,which are used to communicate data directly between the individual processors of the different flow solvers.This means that each processor of one flow solver can communicate directly with all of the processors of theother flow solvers. This requires the interface routines to be part of the source code of all flow solvers. Adetailed description of the common algorithms can be found in Schluter et al.21

In a handshake routine, the processors on the interface establish a direct communication with the peerprocessors. This allows an efficient communication during the actual flow computation.

The approach of embedding the interface into the source code of each flow solver has been chosen forits efficiency in the communication process. Alternative solutions would be to use a third code, whichorganizes the communication between the flow solvers, or to limit the peer-to-peer communication to theroot processes of each flow solver. While the latter two solutions are usually easier to implement, they causemore communication processes and slow down the computation.

INTERFACE BOUNDARY CONDITIONS

LES Boundary Conditions

The definition of the boundary conditions requires special attention especially on the LES side due to thedifferent mathematical approaches. Since on the LES side part of the turbulent spectrum is resolved, thechallenge is to regenerate and preserve the turbulence at the boundaries. At the LES outflow, a body forcemethod has been developed to impose RANS solutions at the outflow of the LES domain .15

At the LES inflow boundary, the challenge is to prescribe transient turbulent velocity profiles fromensemble-averaged RANS data. Simply adding random fluctuations to the RANS profiles miss the temporaland spatial correlations of real turbulence and are dissipated very quickly. Instead, a data-base of turbulentfluctuations is created by an auxiliary LES computation of a periodic turbulent pipe flow. The LES inflow

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boundary condition can then be described as :

ui,LES(t) = ui,RANS︸︷︷︸I

(t) + (ui,DB(t) − ui,DB)︸︷︷︸II

·

√u′2

(i)RANS

(t)√

u′2(i)

DB︸︷︷︸III

(1)

with the sub-script RANS denoting the solution obtained from the RANS computation and quantities withsub-script DB are from the database. Here, t is the time, ui(u1 = u, u2 = v and u3 = w) stands for thevelocity components, and ui is the ensemble average of the velocity component ui.

Term II of Eq. (1) is the velocity fluctuation of the database. This turbulent fluctuation is scaled to thedesired value by multiplication with term III , which ensures that the correct level of velocity fluctuation isrecovered.

RANS Boundary Conditions

Except that the unsteady LES flow data is time-averaged over the time-step applied by the RANS flow solverand is employed directly as a boundary condition, the specification of RANS interface boundary conditionsfrom the LES data is essentially the same as the specification of the inlet and exit boundary conditions of asingle RANS simulation. Therefore, in this section, the treatment of typical inflow and outflow ( inlet andexit ) boundary conditions will be briefly described in order to help readers understand the discussion onthe RANS interface boundary conditions in the following result section.

Although the concept of “characteristic lines” may be questionable for the Navier-Stokes equations, westill deal with the hyperbolic propagation-dominated systems and specify boundary conditions for hyperbolicsystem using the relations based on the characteristic lines, i.e., on the analysis of the different waves crossingthe boundary. This method has been extensively studied for the Euler equations and the mathematicalbackground of boundary conditions based on characteristic wave analysis can be found in the references.22, 23

First of All, we must distinguish two classes of boundary conditions: “physical” boundary condition and“numerical” boundary condition. The physical boundary condition specifies the known physical behaviorof the dependent variables at the boundaries. In order to solve a problem numerically the total number ofboundary condition should be equal to the number of primitive variables. When the number of physicalboundary conditions(say Np) is less than the number needed for the numerical methods(say Nt), Nn =Nt − Np numerical boundary conditions should be introduced. Now the following two essential issues haveto be addressed:

(1) How many physical boundary conditions are to be imposed at a given boundary?The number of physical boundary conditions to be imposed will be equal to the number of waves propa-

gating with each characteristic velocity, λi into the interior domain. These characteristic velocities are

λ1 = un − c, λ2 = λ3 = λ4 = un, and λ5 = un + c,

where un is normal flow velocity and c is speed of sound.The number of necessary and sufficient physical boundary conditions are summarized in Table 1. The

detailed theoretical analysis is found in the reference.24

Table 1. Number of Physical Boundary Conditions

Boundary Type Euler Navier-Stokes

Supersonic inflow 5 5

Subsonic inflow 4 5

Supersonic outflow 0 4

Subsonic outflow 1 4

(2) How are the remaining numerical boundaryconditions to be defined at the boundaries?

An arbitrary and the simplest way is to use ex-trapolation for variables which are not imposed bythe physical boundary conditions. A method withmore theoretical base is the characteristic bound-ary method, which uses the characteristic variables(equivalently called Riemann invariants), or thecompatibility equations. These two methods aremost general but many other methods has been sug-

gested. In addition, there are many more issues regarding to how the physical and numerical boundaryconditions are to be formulated and discretized in order to be compatible with the order of accuracy andthe stability conditions of the numerical scheme. These issues and some good references are summarized byHirsch.25

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RESULTS

��

��

x

r

x=0D x=3D

diameter D

��

��

x=2D x=5D

D=1cm=0.01m

Ubulk= 34.4 m/s

Ma = 0.1

Re = 22630

Figure 2. Geometry for RANS/RANS coupled compu-tation

For the numerical experiments, we consider apipe with a constant diameter D and a length of5D. We then split the pipe into two domains. Thefirst domain is a pipe from x = 0D to x = 3D, thesecond domain a pipe from x = 2D to x = 5D.Hence, the two domains have an overlap of 1D be-tween x = 2D and x = 3D (Fig. 2).

A subsonic case is considered and Mach numberis set as 0.1. The Reynolds number is 22630. Basedon this Reynolds number the mesh for each pipe wasgenerated. The meshes of two pipes are mismatchedin axial direction over the overlapped region of eachpipe.

Here, boundary conditions developed for Eulerequations have been applied to Navier-Stokes com-putations.

Four cases of subsonic inflow boundary condi-tions and one case of subsonic outflow boundary

condition listed in Table 2 have been numerically tested for single and coupled pipe calculations.

Table 2. Boundary Conditions for Pipe Flow.

Boundary case Physical boundary condition Numerical boundary condition

Inflow case 1 u, v, w and T imposed ρ extrapolated

Inflow case 2 u, v, w and ρ imposed p(or T ) extrapolated

Inflow case 3 pt, Tt, and inflow angles imposed Characteristic Method using Riemann invariants

Inflow case 4 ρu, ρv, ρw and T imposed ρ extrapolated

Outflow pexit imposed Extrapolation

where u, v, w–flow velocity in x, y, z directions, T–temperature, ρ–density,p–pressure, pt–total pressure,Tt–total temperature, and pexit–exit pressure.

Reference Test Cases: Single RANS and coupled RANS-RANS

Before RANS interface inflow and outflow boundary conditions were investigated, the first three inflowcases and the outflow case in Table 2 were tested for a single pipe with the same diameter, D and thetotal length, 5D. These boundary conditions for a single pipe calculation are nothing else than typicalinlet and exit boundary conditions for a RANS flow simulation. These computations allow an assessmentof the accuracy of the coupled simulations. Numerical results show that the solutions of the single pipecalculations are converged well with all the boundary cases tested. For example, with the inflow case 3 andoutflow boundary conditions the solution converges to 10−8 level in 5000 multigrid cycles, the pipe flow isfully developed as shown in pressure contour and mass flow rate is effectively constant while ρu varies locally( Fig. 3).

A RANS-RANS computation has also been carried out by coupling two TFLO codes. For this case, inflowboundary case 3 was selected for the inlet of the pipe 1 ( the pipe from x = 0D to x = 3D) and the onlyoutflow was used for the exit of the pipe 2 ( the pipe from x = 2D to x = 5D). This inlet and exit boundarycondition setting will be the same for all the coupled computations presented in this paper for consistencybetween single and coupled computations. All the necessary data was exchanged at the interfaces in thistest. While the exchange of all data is not possible for coupled RANS-LES computations, since the low-Machnumber LES code can not provide all necessary data, this test case allows to demonstrate the accuracy ofthe coupling approach itself. Here, the coupled computation is similar to a classic domain decomposition.The convergence level of this coupled case is 3 orders higher than of single pipe test case and this higher

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convergence level results in the offset of mass flow rate from the single pipe case as shown in Figure 4. Theamount of perturbation in mass flow rate of each pipe for this coupled case is less than of the single pipecase but there is a difference over the overlapped region between the pipe 1 and the pipe 2. This differenceis due to the numerical error resulting from any combination of the difference in convergence between thetwo pipes, the linear interpolation used for data communication. However, the total value is very small at0.0273% of the average value of the mass flow rate.

The results from these single and coupled pipe calculations will be used as the base references for theremainder of the current study. A feasibility of using the RANS boundary conditions for the interface bound-aries of coupled simulations has been investigated and the results will be presented in the next subsections.

RANS Outflow Interface Boundary Condition

In order to test the outflow interface boundary condition, a RANS-RANS (TFLO-TFLO) coupled computa-tion was carried out such that the pipe 1 received pressure from the pipe 2 and sent all data that the pipe 2needed. This allows to have a well-posed problem in pipe 2 similar to the previous test-case, and all effectscan be now associated to the variation of the exit boundary condition in pipe 1.

For a well-posed problem in the pipe 1, the data for the rest of primitive variables, ρ, u, v, and w, atthe interface boundary of the pipe 1 were updated by zero order extrapolation from the first interior cellsof the pipe 1. This interface outflow boundary treatment is essentially a typical exit boundary treatment ofa RANS calculation. The only difference is that the pressure at the interface of the pipe 1 was imposed bythe value at the same location in the pipe 2 instead of imposing the exit pressure, pexit, which is normallyspecified by a known experimental value. As shown in Figure 5, the solution converged well but around oneorder higher than the previous coupled case in which all variables required at the interface boundaries werefully exchanged. Though the mass flow rate deviation becomes bigger (0.0754%), the error is still within thesame order of magnitude and satisfies an engineering accuracy.

RANS Inflow Interface Boundary Conditions

Interface inflow boundary condition, case 1 in Table 2, was tested similarly(Fig. 6). This time T, u, v, and w

for the pipe 2 interface were imposed by the values from the pipe 1 and density, ρ, was extrapolated for thepipe 2. At the exit of pipe 1, the entire data set was imposed. This test case allows a well-posed problem inpipe 1, and the solution is now only influenced by the choice of inflow boundary conditions at the pipe 2.

The coupled solution converged as well as the case exchanging all data. Interestingly the difference inmass flow between two pipes are even smaller(0.0121%).

The inflow boundary case 2 in Table 2 was tested and the similar results were obtained as shown inFigure 7. This interface inflow boundary condition guarantees a mass conservation across the interface ifthere is no interpolation error. However, this boundary condition may not be useful, since in Low-Machnumber LES flow solvers, the density is not a function of the flow and, hence, does not represent a fullycompressible solution.

Currently in the applications of LES-RANS integration, the mass-flux vector at every point of inlet isbeing specified corresponding to the value delivered by the LES computation. This means ρu, ρv, ρw and T

are imposed and ρ is extrapolated at the boundary. This allows the density ρ to fluctuate to account for thepassing of acoustic waves. The velocity components u, v and w are adjusted accordingly in order to conservethe mass-flux. This current implementation, case 4, was also tested and the smallest difference in mass flowrate was observed as shown in Figure 8.

Justification of RANS Interface Boundary Conditions

In the final set of simulations we now apply the tested boundary conditions on both sides of the domainsimultaneously. We apply inflow boundary case 1 for the pipe 2 and outflow boundary condition was appliedat the exit of the pipe 1 (Fig. 9). As expected, the convergence level is a little higher. However, the error inmass flow rate is still low (0.0143%) and within engineering accuracy. The fact that the error is even lowerthan in the reference test-case demonstrates that the here achieved levels of mass conservation are withinthe machine accuracy.

In order to demonstrate the necessity of the proposed boundary treatment, we exchanged the samedata set between the two codes, but instead of applying RANS interface boundary conditions, the missing

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variables are updated by the numerical scheme. As can be seen in Fig. 10, the error in mass flux is increasedby two orders of magnitude and the flow solution in the pipe differs considerably from all previous solutions.

Similarly the inflow condition, case 2, and the outflow condition were tested and the results are shownin Figure 11. Interestingly we observed that the solution of this testing case was dependent on the choice ofvariables( p or T ) to be extrapolated for the pipe 2. With the choice of pressure as the variable extrapolatedthe solution became unstable. A theoretical source has not been found but the result is shown in Figure 12.

CONCLUSIONS

In this study we investigated compressible RANS boundary conditions intended for application in coupledRANS-LES computations. The different mathematical modeling of compressibility (fully compressible onthe RANS side, low-Mach number approximation on the LES side) creates challenges in the prescription ofRANS boundary conditions.

We presented a number of inlet and exit boundary conditions and tested them on a simple geometry. Inthe formulation of the compressible boundary conditions we essentially apply Euler characteristic boundaryconditions. Numerical experiments using RANS-RANS (TFLO-TFLO) coupled simulations were set up inorder to simulate the incomplete interface data communication for a coupled LES-RANS and/or RANS-LESsimulation.

The results show that the usage of RANS inlet/exit boundary conditions are able to model the interfaceconditions. We propose the use of these boundary conditions in order to minimize the error introduced bythe decomposition of a given domain into RANS and LES domains. The determination of these boundaryconditions allows the application of the coupled RANS-LES approach to gas turbine engines.

ACKNOWLEGMENT

The support by the US Department of Energy within the ASC program is gratefully acknowledged.

References

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0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l

0 0.01 0.02 0.03 0.04 0.05 0.063.35

3.3501

3.3502

3.3503

3.3504

3.3505

3.3506x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0146%

Figure 3. Single TFLO computation : convergencehistory, pressure contour, x-momentum (ρu) contourand mass flow rate

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l

Single PipeCoupled Pipe 1Coupled Pipe 2

0 0.01 0.02 0.03 0.04 0.05 0.063.334

3.336

3.338

3.34

3.342

3.344

3.346

3.348

3.35

3.352x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0273%


Figure 4. TFLO-TFLO coupled computation exchang-ing all variables : convergence history, mass flow rateand x-momentum (ρu) contour

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0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.344

3.345

3.346

3.347

3.348

3.349

3.35

3.351

3.352x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0754%


Figure 5. TFLO-TFLO coupled computation for test-ing outflow boundary condition : convergence history,mass flow rate and x-momentum (ρu) contour

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.334

3.336

3.338

3.34

3.342

3.344

3.346

3.348

3.35

3.352x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0121%


Figure 6. TFLO-TFLO coupled computation for test-ing interface inflow boundary condtion, case 1 : con-vergence history, mass flow rate and x-momentum (ρu)contour

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0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.334

3.336

3.338

3.34

3.342

3.344

3.346

3.348

3.35

3.352x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0112%


Figure 7. TFLO-TFLO coupled computation for test-ing interface inflow boundary condition, case 2 : con-vergence history, mass flow rate and x-momentum (ρu)contour

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.34

3.342

3.344

3.346

3.348

3.35

3.352x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0069%


Figure 8. TFLO-TFLO coupled computation for test-ing interface inflow boundary condition, case 4 : con-vergence history, mass flow rate and x-momentum (ρu)contour

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0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.35

3.3502

3.3504

3.3506

3.3508

3.351

3.3512x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0143%Single PipeCoupled Pipe 1Coupled Pipe 2

Figure 9. TFLO-TFLO coupled computation for test-ing inflow (case 1)/outflow boundary condition test :convergence history, mass flow rate and x-momentum(ρu) contour

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.32

3.33

3.34

3.35

3.36

3.37

3.38

3.39

3.4x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 1.9691%

∆ (ρ u ⋅ A) = 2.4510%


Figure 10. TFLO-TFLO coupled computation ex-change T, u, v and w for p without boundary treatment :convergence history, mass flow rate and x-momentum(ρu) contour

12 of 13


0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.346

3.3465

3.347

3.3475

3.348

3.3485

3.349

3.3495

3.35

3.3505

3.351x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)

∆ (ρ u ⋅ A) = 0.0510%


Figure 11. TFLO-TFLO coupled computation fortesting interface inflow (case 2 with T extrapola-tion)/outflow boundary conditions : convergence his-tory, mass flow rate and x-momentum (ρu) contour

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Multigrid Cycles

Ave

rage

Den

sity

Res

idua

l


0 0.01 0.02 0.03 0.04 0.05 0.063.25

3.3

3.35

3.4

3.45

3.5

3.55x 10

−3

x location

Mas

s F

low

Rat

e (ρ

u ⋅

A)


Figure 12. TFLO-TFLO coupled computation fortesting interface inflow(case 2 with p extrapola-tion)/outflow boundary conditions : convergence his-tory, mass flow rate and x-momentum (ρu) contour

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