Flowin St Bility

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Nuclear Engineering and Design 215 (2002) 153170

On the analysis of thermal-fluid-dynamic instabilitiesvia numerical discretization of conservation equations

Juan Carlos Ferreri a,1*, Walter Ambrosini b

a Autoridad Regulatoria Nuclear, A6. del Libertador 8250, 1429 Buenos Aires, Argentinab DIMNP, Facolta di Ingegnera, Uni6ersity of Pisa, Via Diotisal6i 2, 56126 Pisa, Italy

Abstract

The numerical aspects involved in the prediction of fluid-dynamic instabilities are shortly reviewed, mainly

addressing single-phase natural circulation flows. In particular, some open literature, produced since the pioneering

work of Welander [J. Fluid Mech. 29 (1967) 17 30] and related to single-phase thermosyphon loops, is firstly

considered; then the analysis turns to summarize some early calculations and examples of code verification results

dealing with the prediction of single-phase natural circulation. The work previously performed by the Authors in

relation to Welanders problem and other relevant single-phase thermosyphon loop cases is reviewed, with main

emphasis on the adopted methodology, based on numerical discretization of governing equations. These results serve

to illustrate how time domain codes may be used to analyze such problems and also to highlight the effect of

considering different constitutive laws in the computed linear stability map. Reference to the application of this

methodology to boiling channel stability analysis is also made to demonstrate its generality. The results show boththe qualitative and quantitative details of the changes that stability maps undergo as a consequence of numerical

discretization of governing equations. The recent attempt to make easier the application of the methodology by the

use of automatic FORTRAN code differentiation tools is finally reported, showing quantitative results concerning

calculation of the sensitivity of simulation results to physical and numerical parameters. 2002 Published by Elsevier

Science B.V.

www.elsevier.com/locate/nucengdes

1. Introduction

The need for modeling of natural circulation

flows in hydraulic circuits is the result of the

importance of the related phenomena in engineer-ing sciences and applications. References concern-

ing this type of flows in the last 20 years are so

numerous as to provide a trivial proof of this

assertion. As a matter of fact, there are several

relevant motivations to deal with this subject: in

addition to the traditional widespread utilization

in industry, natural circulation is receiving atten-

This paper is an updated and condensed joint version of

the invited lecture: Single-phase Natural Circulation in Simple

CircuitsReflections on its Numerical Simulation by J.C.F.

and the paper Parameter Sensitivity in Single-Phase Natural

Circulation via Automatic Differentiation of FORTRAN Codes

by the authors, both presented at the Eurotherm63 Seminar.

* Corresponding author.

E-mail address: [email protected] (J.C. Ferreri).1 Member of CONICET, Argentina.

0029-5493/02/$ - see front matter 2002 Published by Elsevier Science B.V.

PII: S 0 0 2 9 - 5 4 9 3 ( 0 2 ) 0 0 0 4 8 - 1
mailto:[email protected]:[email protected]


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J.C. Ferreri, W. Ambrosini/Nuclear Engineering and Design 215 (2002) 153170154

tion in the nuclear field for passive heat removal

systems in innovative reactors, which makes it a

subject deserving even greater attention from the

technical community.

The problem of simulating natural circulation

flow is a rather complex one. For instance, it is

difficult to establish which are the appropriate

values of system parameters, such as singular anddistributed pressure losses in loop components,

because the flow in pipes and restrictions, like

fittings and valves, is usually non-developed. The

usual practice in this field, consists in adopting

well established results obtained for steady-state

conditions, such as using Idelchik (1966) hand-

book for estimating singular pressure drops; but

there are well known limitations in the application

of these relationships to non-developed or time

dependent flows (these aspects have been reviewed

in many references, e.g. Zvirin (1981)). The samereasoning applies to distributed friction resistance

laws. The balance between friction and driving

forces determines steady-state conditions in natu-

ral circulation flows, then, depending on the

choice for closure laws, the system may present a

different behavior. Obviously, this represents a

key point in determining the capabilities of mod-

els; it will be noted in the following that in the

case of simplified analyses based on dimensionless

equations, the choice of adequate independent

variables may put constraints in the selection of

constitutive laws that have an impact on results.

Scaling natural circulation flows to get an ap-

propriate experimental apparatus representative

of real life installations is not trivial and has been

the subject of many analyses up to the present

time. Relevant information on this subject may be

found in Zvirin (1981), Ishii and Kataoka (1984),

DAuria et al. (1991), Vijayan and Austregesilo

(1994), Wulff and Rohatgi (1999). This is indeed

an important aspect because it defines if a model

will reasonably represent a given physical appara-

tus. Sometimes, scaling leads to the adoption of

the 1-D approximation; this may, in turn, hide

important aspects of the system physics. A simple

example of this situation consists in keeping the

height of the system unchanged to get the same

buoyancy; then, if the system is scaled accordingly

to the power/volume ratio, the cross section area

of the volume will be reduced; this leads to a

much smaller pipe diameter that makes the 1-D

representation reasonable, at the cost of eliminat-

ing the possibility of fluid internal recirculation. A

workaround for this situation is providing paths

for recirculation, in the form of additional, inter-

connected components; however, this solution

may impose the flow pattern in the system and thebalance between these aspects is a challenge to

any practitioner in natural circulation modeling.

An additional important characteristic of natu-

ral circulation flows is that they are non-linear in

essence. The problem is particularly challenging

since no matter how simple the hydraulic loop can

be, its behavior is usually very complex: depend-

ing on the ratio of driving forces (heat sources/

sinks strength) to the integrated friction along the

loop, the flow may become oscillatory, showing

flow reversals. The modeling of the dynamic be-havior of non-linear systems is presently accom-

plished by different techniques, depending on the

characteristics of the problem under consider-

ation. We leave apart lumped parameter models

despite their usefulness for studying, for example,

the system behavior beyond the stability

threshold. It is our interest to focus on the solu-

tion of natural circulation flows as represented by

non-linear partial differential equations (PDEs),

which are the only feasible choice for adequately

modeling the behavior of distributed parameter

systems. Then, appropriate techniques must be

applied to get a reasonable picture of both stabil-

ity conditions and unstable transient behavior.

Discretization of PDEs by numerical schemes

leading to systems of algebraic equations is the

usual technique in complex cases, in order to

provide a realistic prediction of plant behavior.

The large thermal-hydraulic (TH) system codes

adopted for safety analyses of nuclear power

plants are relevant examples of this choice. How-

ever, it is well known that numerical schemes

usually adopted in the simulation of natural circu-

lation flows introduce damping in the solution;

therefore, unstable flows may be turned stable

because of this added diffusion. Numerical

schemes may affect the expected convergence of

computed results, even in simple situations; these

effects may originate, for instance, in the numeri-


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J.C. Ferreri, W. Ambrosini/Nuclear Engineering and Design 215 (2002) 153170 155

cal coupling of the discretised equations or in the

criteria adopted for variable lumping in finite

volume schemes.

Therefore, when applied to fluid-dynamic insta-

bilities, most of the adopted numerical methods

are conditioned by the above mentioned effects of

discretization, that introduce unwanted numerical

diffusion and other spurious effects which maystrongly change the predicted dynamic behavior

of the system. These effects are often found to

make difficult a reliable stability analysis by nu-

merical means (see e.g. Bau and Torrance, 1981;

Misale and Tagliafico, 1987; Vijayan et al., 1995;

Wu and Almenas, 1994; Ambrosini and Ferreri,

1998). Examples of accurate estimates of the ef-

fect of numerical diffusion on the predicted degree

of stability or instability are not so frequent. Basic

references to the effects of the numerics on com-

puted results are the works of Stuhmiller andFerguson (1979), Barre et al. (1993), Mahaffy

(1993), Yih-Yun Hsu (1994), Ginestar Peiro and

March-Leuba (1998). It is, therefore, important to

quantify the sensitivity of the results obtained

from numerical simulations to physical and nu-

merical discretisation parameters.

This paper focuses on the numerical aspects of

1-D calculations of fluid-dynamic instabilities,

with main reference to single-phase natural circu-

lation flows in the context of nuclear safety evalu-

ations. We, therefore, consider systems amenable

to one-dimensional (1-D) modeling, thus disre-

garding a wide class of problems for which this

assumption would be questionable. The aspects

related to the effect of numerical discretisation of

PDEs on the computed system behavior are given

particular attention, summarizing previously pub-

lished work performed with a systematic method-

ology of analysis and presenting recent

developments related to the evaluation of sensitiv-

ity to physical and numerical parameters. The

effect of the choice of constitutive laws for evalu-

ating friction is also addressed. To complete the

panorama of obtained results the extension of the

methodology of analysis also to boiling channel

stability problems is mentioned.

Though the goal of the work is to discuss the

capabilities of system codes in predicting natural

circulation, other aspects relevant in this frame

are not directly addressed. One of these aspects is

the so-called user effect, i.e. the effect that the

modeling choices of a given code user may have

on the computed results. This problem received

careful attention quite recently in the field of

nuclear safety evaluations (e.g. see the papers in

IAEA, 1998). However, the problems here ad-

dressed are generally simple enough as to mini-mize the effect of user choices on the results of

code application.

2. Numerical simulation of 1-D fluid-dynamic

instabilities

2.1. Early experience on single-phase natural

circulation

Single-phase natural circulation flows may bemodeled via the use of two equations:

the integral momentum balance equation:

dW

d~=CB BTCF f(W) WW (1)

the distributed energy balance equation:

(q

(~+W

(q

(s=Q+F

(2q

(s2(2)

In these equations, written in dimensionless

form, W represents some measure of the flow rate,

CB is a coefficient to weight the buoyancy term

(BT) which is usually the integral of temperature

along the loop length, CF is a coefficient in the

friction law and f(W) gives the form of the fric-

tion law. Moreover, q is some measure of the fluid

energy, usually temperature, Q the energy gain/

loss by wall transfers (it includes boundary condi-

tions), F is a coefficient to weight the influence of

axial diffusion and, finally, ~ and s are the non-di-

mensional time and space co-ordinates,

respectively.

In the papers by Ferreri and Doval (1984, 1985,

1988), Doval and Ferreri (1988), one of the au-

thors started to document his experience on the

numerical modeling of natural circulation flows in

nuclear safety related problems. This subject, to-


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gether with the well-posedness of the governing

equations in thermal-hydraulics, used to be (and

continues to be) a field of active work. The prob-

lem faced at the time was using TH system codes

to compute the transition from forced circulation

to natural circulation flow in a nuclear power

plant. The codes used at the time, as described in

Ferreri and Doval (1984), failed to converge to-

ward a solution in the last stage of the simulation.

This fact and the need to get results lead to the

development of a 1-D, single-phase, single-loop

simulation code, including concentrated and dis-

tributed heat sources and sinks to compute natu-

ral circulation flows (see Ferreri and Doval, 1984,

1985, 1988). The obtained results (quoted in Fer-

reri and Doval, 1985) pointed out the damping of

flow perturbations as a function of the adopted

discretization in a closed loop.In Ferreri and Doval (1984), the developed

code consisted in a 1-D version of the iterative,

semi-implicit SOLA algorithm. It adopted an im-

plicit finite-difference method for the energy equa-

tion in the primary loop and heat exchangers. The

momentum equation was solved in two ways: a

fully discrete finite-difference method using the

forward time, upwind-space (FTUS) approxima-

tion and a formulation obtained by the usual

integration along the loop. The cases considered

were:(a) the results in Zvirin and Greif (1979), making

reference to a previous fundamental paper by

Welander (1967), concerning a very idealized

thermosyphon loop with point heating source

and sink;

(b) the results of Zvirin et al. (1981), concerning

the natural circulation flow in a nuclear

loop, formed by parallel loops and plenty of

fittings;

(c) the computational simulations of toroidal

loops by Mertol (1980).These simulations were aimed at providing

some hints on the convergence of the numerical

results to theoretical and experimental results. In

Zvirin and Greif (1979), no instabilities were ob-

tained, even in cases of theoretically unstable

flows. These authors attributed these results to the

adopted analysis technique.

The tests reported in Ferreri and Doval (1984)

allowed for reproducing Welander (1967) results

and studying the effect of the friction correlation

and of the number of nodes (i.e. of the numerical

diffusion) in the transient behavior of the loop.

Consideration was given to the minimum number

of cells providing enough damping to keep the

results stable (in stable cases) and rapidly conver-gent. The economy on cell number was manda-

tory because of CPU time and memory

restrictions by these years.

Keeping in mind the final application of the

code, the results in Zvirin et al. (1981) were then

considered. Surprisingly, the results obtained by

Ferreri and Doval (1984) provided a better ap-

proximation of the experimental data than the

ones by Zvirin et al. (1981), doubtless due to a

fortuitous choice of better system parameters, the

critical aspect mentioned in the Section 1.Concerning the work by Mertol (1980), the task

was determining the minimum number of cells

necessary to keep the system in neutral oscillatory

state around the theoretical steady-state flow rate.

A rule of thumb was learnt from that work: one

to one ratio up to one to three between cell length

and diameter usually assured catching flow oscil-

lations. The time step was set to keep the cell

Courant number near to 0.8, a basic tenet to get

low diffusion while keeping the calculations stable

under the forward time upwind space (FTUS)

approximation. The analysis of non-linear sys-

tems, as applied to natural circulation flows in

loops, was beginning with a theoretical analysis

by Hart (1984, 1985). The conceptual explanation

of flow instabilities in Welanders paper (We-

lander, 1967) remained valid in these analyses.

Bau and Wang (1992) reviewed the literature re-

lated to this way of analysis, that kept growing up

to date.

The aforementioned process of verification and

partial validation of the code allowed to the sin-

gle-phase natural circulation flows in a simplified

version of a nuclear power plant. It must be

mentioned that in this plant (Atucha-I pressur-

ized, heavy-water reactor) the core consists in

nearly 250 vertical channels immersed and packed

in a moderator tank, therefore, the 1-D approxi-

mation was a reasonable one. The first stage of


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the transient consisted in a pump trip without

depressurization; then, after the reactor scram, the

flow decreased up to the point in which natural

circulation flow was established. At this point, the

system code adopted at the time (RELAP4/

MOD5) ceased to converge. The values of the

system parameters were then fed to the in-house

developed natural circulation flow code and, aftersome minor oscillations (in the order of 10%),

stabilized quite nearly to the system code values.

After this, a transient was simulated, whose re-

sults in the long term, i.e. the steady-state flow

rate for residual power, fairly agreed with the

ones known by design, as to testify the appropri-

ate simulation of the ratio between driving forces

to loop friction. Then, despite the limited code

capabilities, results were quite good as a conse-

quence of the time spent during the stage of

verification and partial validation. This code wasnot applied to the case of the Embalse CANDU

pressurized, heavy-water, nuclear power plants;

however, up to recently, this type of configura-

tions (see Vijayan and Date, 1990, 1992) have

been the subject of active research.

The use of TH system codes is essential for the

safety assessment of nuclear power plants but

their application in the analysis of fluid-dynamic

instabilities raises the question of the validity of

the results, as already pointed out in previous

studies (see for e.g. the works by DAuria et al.,

1991, 1997; DAuria and Galassi, 1998; Wulff et

al., 1992). The effects of numerical discretization

is the subject of main interest here and in what

follows attention will be focused on this impor-

tant aspect of code utilization.

2.2. Systematic analysis of truncation error

effects in single-phase flow

Based on the experience gained in the afore-

mentioned early developments, it was decided to

check the applicability of the new series of TH

systems codes like RELAP5/MOD3.2 (see Carl-

son et al., 1990) in the calculation of unstable flow

problems. The motivation for pursuing the work

was the suspicion that the code, making use of

first order numerical methods, should behave as

the programs adopted in previous experience. Ad-

ditional interest for the subject was found consid-

ering information on the observation of similar

behavior in blind inter-comparison experiments

(e.g. the final report by Ferraz Bastos, 1997) and

simultaneous work (Wu and Almenas, 1994)

This study provided the occasion to cope with

the problem at a very basic level, developing an

activity that included the following phases: analysis of the behavior of RELAP5 in the

application to a particular single-phase natural

circulation problem;

study of the same problem by the use of ana-

lytical (modal expansion) technique and by dif-

ferent numerical methods;

systematic analysis of truncation error effects

by linearization of numerically discretized

equations;

consideration of other single-phase ther-

mosyphon loop geometry by the samemethodology;

extension of the methodology to deal with

boiling channel instabilities.

All this work was summarized in Ferreri et al.

(1995), Ambrosini and Ferreri (1997a,b, 1998),

Ferreri and Ambrosini (1999), Ambrosini et al.

(1999, 2000). What follows is a brief mention of

these results, giving explicit consideration to these

references.

In Ferreri et al. (1995), Welanders problem

(Welander, 1967) was put under analysis using the

RELAP5.x series of codes. The problem deals

with natural circulation flow in a very simple

situation: namely, flow driven by buoyancy in a

loop constructed with two parallel, adiabatic, ver-

tical tubes. The loop has a point heat source at its

bottom and a point heat sink at its top. In fact,

the mode of heat transfer is assumed to be such

that the product of the heat transfer coefficient in

the source by the heated length is kept constant

when the heated length tends to zero. The same

holds for the heat sink so that the physically finite

heated lengths become points. Welander per-

formed his analysis only for laminar flow; how-

ever, his results were generalized to consider

turbulent flow. Chen (1985), among many other

authors, performed such analysis for a more gen-

eral loop considering heated (cooled) horizontal

tubes of finite length. In Ferreri et al. (1995), the


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analysis in Welanders paper was generalized in

order to keep strictly the same hypotheses of the

original derivation.

In its form generalized to turbulent flow, the

problem is expressed by Eqs. (1) and (2) where the

Blasius law is adopted to quantify friction in

momentum equation, thus resulting in a depen-

dence of the friction term on flow rate raised tothe power 1.75. Appropriate boundary conditions

are then imposed to translate the effect of source

and sink on the fluid passing through them.

The application of RELAP5 to an unstable case

of Welanders problem showed that (Fig. 1): depending on the node size in the legs and the

adopted time-step the semi-implicit and the

nearly-implicit numerical methods available in

the code may predict both stability and

instability;

the semi-implicit numerical method is less dif-fusive than the nearly-implicit one;

when sufficient detail is adopted in discretizing

the adiabatic legs, unstable behavior is pre-

dicted showing, as expected, successive flow

reversals.

The results shown in Ambrosini and Ferreri

(1998), that reports most of the material obtained

in the study on Welanders problem, dealt with

several of the relevant aspects mentioned in the

Section 1. In particular, it was shown that conver-

gence to exact steady-state in Welanders problem

could only be achieved by giving consideration to

a special form of heat sources/sink boundary con-

dition. This occurs because the single nodes

adopted to simulate the source and the sink can-

not provide the correct boundary conditions at

the inlet of each adiabatic leg owing to the poor

description of the actual temperature distribution

in the heater and cooler. This result had been

previously documented by Grandi and Ferreri

(1991) showing the inter-dependence of boundary

conditions application and the numerical scheme

used.But the crucial topic in Ambrosini and Ferreri

(1998) was to ascertain the reasons for the influ-

ence of nodalization on stability predictions ob-

served in RELAP5. To help in testing and in

having reference results, some simple codes were

Fig. 1. RELAP5 results for an unstable case of Welanders problem (Dt=0.5 s).


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Fig. 2. The flow rate for the FTUS scheme using 1000 nodes and a modal expansion with 500 modes and the FTUS O(1 /1000)

numerical diffusion coefficient.

developed to solve natural circulation flow in this

simple system. One code version was based on the

modal approximation, to get diffusion-free solu-

tions. Other finite-differences code versions im-

plied: the forward-time, upwind method (FTUS)

being first order accurate in both space and time,

an implicit-time upwind method (ITUS), again

first order accurate, and the second order McCor-

mack method and Warming-Beam explicit

methods.

The effects of nodalization on the computed

neutral stability boundaries were quantified and

put into perspective against diffusion-free results.

It is interesting now to compare the results ob-

tained using the explicit upwind method with the

results of a modal solution with a second order

term simulating numerical diffusion. With this

aim, the diffusion coefficient is defined as:

D=q Ds

2

1

q DtDs

(3)

where q is the loop volumetric flow-rate, Ds is the

space interval and Dt is the time step, all ex-

pressed in dimensionless form. As it can be easily

recognized, the particular form adopted for D

comes from the analysis of truncation error in the

FTUS method. Fig. 2 (taken from Ferreri and

Ambrosini, 1999) shows the predicted loop flow

rate variation with time. The diffusion coefficient

was the one corresponding to 1000 nodes under

the FTUS approximation; the number of modes

considered in the modal expansion was 500. As

may be observed, both approximations behave

similarly. Exact coincidence is precluded by the

non-linearity of the system and small differences

in starting conditions. It may be shown that the

predicted degree of instability is very similar for

the nodal and the modal solution with equivalent

dissipative effects and the agreement is improved

by increasing the number of nodes. The impor-

tance of the second order term alone in determin-

ing the overall truncation error effect on stabilitypredictions is once again emphasized.

To get quantitative information on the effect of

nodalization on the neutral stability boundary of

a given 1-D flow system in Ambrosini and Ferreri

(1998), the linearization of numerically discretized

partial differential equations was adopted. This

methodology shares with the usual techniques

adopted for studying stability of physical systems

(e.g. direct linearization of PDEs or of ODEs

obtained by modal or nodal expansion) the idea

to locally perturb the solution around steady-stateconditions to achieve information on linear sta-

bility, i.e. on stability to infinitesimal perturba-

tions. However, the fact that the numerically

discretized equations are perturbed allows to get

information on stability as predicted by the nu-

merical method, including the effects of trunca-

tion error and discretization in general. This is an


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important information when judging about the

acceptability of a given nodalization for transient

stability analyses, as it clearly shows the effect of

numerical discretization in stability maps.

The methodology of analysis addresses the per-

turbations in the variables defining the state of the

considered system. Once the vectors of perturba-

tions of state variables at the nth and (n+1)thtime levels and at each node are defined, a lin-

earized relationship is found between them in the

form:

(ly6

)n+1=(J66 SSn+1)1J6

6 SSn (ly

6

)n (4)

that involves the Jacobeans of the N components

of the vector function representing the numerical

method and the related boundary conditions with

respect to the system state variables. The Ja-

cobeans are evaluated at the steady-state (fixed)

point as a function of grid parameters (in thepresent case the grid step Ds and the time step Dt)

and physical parameters.

In order to set up stability maps, the concept of

a margin in excess to neutral stability is then

introduced. Two different quantities were used in

this purpose:

Dz=z(A66

)1 zR=ln z(A6

6

)

Dt(5)

which are based on the spectral radius z(A66

) of the

matrix A66

=(J66SSn+1)1J

66SSn and are positive for

unstable conditions and negative for stable ones.

The rank of matrix A66

is in the order of the

product of the number of nodes by the number of

independent variables defined at each node.

In a recent paper, Doster and Kendall (1999)

adopted a similar methodology of analysis for

studying single-phase natural circulation flows.

Their technique is based on the assumption of a

complex exponential trend for the space distribu-

tion of state variable perturbations, leading to the

discussion of stability by consideration of the

eigenvalues of a matrix as a function of a wave

number. On the other hand, the methodology

referred to in the present paper is based on the

linearization of the full set of space and time

discretized equations and deals with the overall

amplification or damping of perturbations at each

node.

In the case of Welanders problem (Welander,

1967), the relevant physical parameters are h, the

coefficient corresponding to the buoyancy term

and m, the coefficient corresponding to the friction

term in the momentum equation. Both coefficients

determine the system steady-state conditions and

can be taken as coordinates of a map ofDz or zR.

In this purpose, Dz or zR are calculated through-

out a selected hm rectangular domain, thus iden-

tifying, with the aid of contour plots, regions with

a different degree of stability. This method is easy

to implement in computer programs and has the

advantage to provide a great deal of information,

at the price of a reasonable computing effort. Fig.

3 shows the results obtained for the FTUS

method with 30, 40, 50 and 100 nodes and Dt=

104. The effect of numerical diffusion on the

marginal stability boundary predicted by the first

order explicit numerical method are clear from

the figure and qualitatively justifies the behavior

observed by RELAP5. In relation to the other

numerical methods, it was shown that, as ex-

pected, the implicit first order method is much

more diffusive than the explicit one (as it is in

RELAP5 for the nearly-implicit method with re-

spect to the semi-implicit numerical method) and

Fig. 3. The effect of the number of nodes on the neutral

stability curve using FTUS.


9/18


that second order methods have a superior perfor-

mance in stability analyses, owing to the lack of

the second order derivative term in the modified

equation.

Other effects on the neutral stability boundary

considered in Ambrosini and Ferreri (1998) were

the ones due to the cell Courant number and to

the time discretization of buoyancy and frictionterms in momentum equation. Low values of

Courant number implied increased damping in

the first order explicit method, what is usual in

these finite-differences approximations. Quantita-

tive results in relation to the time discretization of

buoyancy and friction are not so usual (see Gines-

tar Peiro and March-Leuba, 1998; Doster and

Kendall, 1999, for example). In Ambrosini and

Ferreri (1998), it was shown that explicit coupling

between energy and momentum implied a change

in the way of convergence of the neutral stabilityboundary at practical values of cell Courant num-

bers (i.e. $0.8). Implicit coupling allowed recov-

ering the expected diffusive way of map

convergence.

The effect of closure relations was explicitly

mentioned in the Section 1. Ambrosini and Fer-

reri (1997b) dealt with this problem to quantify

their effect on the neutral stability boundary and

to show how the choice of adequate dependent

variables may facilitate this analysis. The analysis

of natural circulation flows in toroidal loops with

imposed heat flux is quite appropriate for this

task. This problem has been studied, both theoret-

ically and experimentally, by several authors con-

sidering different kinds of boundary conditions

(see e.g. Hart, 1984; Sen et al., 1985). Ambrosini

and Ferreri (1997b) considered the specific

boundary conditions addressed by Sen et al.

(1985), involving imposed heat flux heating and

cooling and selected the Reynolds number as

main flow variable in the dimensionless equations

for the toroidal loop making use of three different

frictions laws, including one accounting for a

realistic transition between laminar and turbulent

flow (Churchill, 1977).

Taking advantage of the fact that for this prob-

lem only three ordinary differential equations

fully represent the system dynamics, stability

maps for the spacetime continuous mathemati-

Fig. 4. Effect of friction law on the neutral stability boundary:

modal method vs. FTUS method and 200 nodes (Churchill

friction law with relative roughness 0.0001, torus length/di-

ameter ratio=100, D~=107).

cal problem have been obtained by plotting the

real part of the eigenvalue of the linearized ODE

system having maximum norm, zR

(to be directly

compared with the quantity defined in Eq. (5) for

numerically discretized equations) as a function of

a Grashof number and of the angle of asymmetry

in heating, k. In Fig. 4 it can be noted that the use

of a simple power law for the Fanning factor,

which is the usual choice in literature on ther-

mosyphon loops, gives a very poor representation

of the situation occurring when a realistic law is

used and the regime transition falls in the domain

of dependence of parameters. The curve corre-

sponding to the FTUS scheme with a considerablenumber of nodes shows reasonably converged

results. It is a direct poof of the dramatic influ-

ence of the closure relations, no matter what the

adopted numerical scheme is. It may be argued

that it is an obvious consequence of the modeling

criteria; however, to the authors knowledge, it is

not so usual to find a quantitative determination

of these effects.

The authors also analyzed with the same

methodology the results of Vijayan and Austrege-

silo (1994) for a rectangular thermosyphon loopsobtaining similar results in relation to the effect of

truncation error on stability prediction.

2.3. Numerical analysis of boiling channel

stability

Given the interesting results obtained by the


10/18


methodology developed for studying stability of

single-phase thermosyphon loops, an effort was

then spent to extend its applicability to other

instability problems. Among the several ones hav-

ing an interest in engineering, boiling channel

instability plays a key role in nuclear reactor

safety and has been the subject of in depth investi-

gations (see e.g. Wulff et al., 1992; DAuria et al.,1997 among many others).

In order to test the capabilities of the methodol-

ogy when applied to boiling channel stability, a

simple homogeneous equilibrium model was se-

lected for a single boiling channel with imposed

pressure drop, overall heat flux and inlet subcool-

ing (Ambrosini et al., 1999, 2000). The adoption

of this simple model also allowed comparison

with plenty of results available from previous

works. The problem is anyway far more complex

than in the case of the single-phase thermosyphonloops, as in each node three independent variables

must be now defined, being the local pressure,

specific fluid enthalpy and volumetric flux.

Both nonlinear and linear stability conditions

were addressed setting up, as in the case of ther-

mosyphon loops, twin computer programs based

on a same semi-implicit numerical method: a tran-

sient program provides information on nonlinear

behavior starting from selected initial conditions;

a linearized program supplies the basic informa-

tion about linear stability of steady-state flow

conditions. The adopted numerical method makes

use of standard choices in nuclear reactor system

codes as staggered meshes and upwind

differencing.

Fig. 5 shows a stability map obtained as a

function of the phase change and the subcooling

numbers as defined by classical relationships (see

e.g. Ambrosini et al., 2000):

Npch=

P

Whfg

6fg

6f Nsub=

hfhin

hfg

6fg

6f (6)

where P is the channel power, W the mass flow

rate, hin the inlet fluid enthalpy and hf, hfg, 6f, 6fgare the saturated liquid and differential vapor-to-

liquid enthalpies and specific volumes. The map

was obtained for a relevant boiling channel case,

highlighting the regions of density-wave and

Fig. 5. Typical quantitative boiling channel stability map (96

nodes, max Courant number=0.9): the neutral stability

boundary obtained with 48 nodes is also reported for purpose

of comparison (from Ambrosini et al., 1999).

Ledinegg instabilities. The spatial discretization of

the channel includes 96 nodes and a maximum

Courant number of 0.9 was adopted. For purpose

of comparison, the neutral stability boundary ob-

tained with a lower number of nodes is also

reported, pointing out the effect of numerical

diffusion at low inlet subcooling. The map is a

contour plot of the previously defined variable zR,

representing the degree of damping or amplifica-

tion of perturbations as predicted by the numeri-

cal method. This quantity is obviously strictly

related to the predicted decay ratio of perturba-

tions, i.e. the ratio between the amplitude of

oscillations at two subsequent cycles.

By the use of the linear analysis program, it waspossible to set up a number of stability maps

providing information on the sensitivity of stabil-

ity to different physical and numerical parameters

(Ambrosini et al., 1999). As an example of such

Fig. 6. Quantitative boiling channel stability map (12 nodes,

max Courant number=0.9): the neutral stability boundary for

48 nodes is also reported for purpose of comparison (from

Ambrosini et al., 1999).


11/18


analyses, Fig. 6 reports the results obtained for

the same case as in Fig. 5 by decreasing the

number of nodes down to 12. As it can be noted,

the use of a coarse nodalization has a dramatic

effect on the shape of the neutral stability

boundary, giving rise to alternating stripes of

stable and unstable conditions. These changes can

be related to the interaction of the boilingboundary with node boundaries, as it can be

understood considering that the stripes of alter-

nated stability and instability approximately fol-

low lines of constant boiling boundary position,

represented by the relationship:

NsubLbb

L

=

Lbb

LNpch (7)

where L and Lbb are the channel length and the

boiling boundary position.

This is a well-known effect occurring whenusing the homogeneous equilibrium model and

has been reported in previous works (see e.g.

Podowski and Rosa, 1997) as requiring a special

treatment to be mitigated. The stability map in

Fig. 6 gives a very clear and understandable pic-

ture of this effect, capable to justify previous

observations of local behavior in parameter space.

3. Role of automatic differentiation tools

In the previous sections, it was shown how a

methodology of analysis based on numerical dis-

cretization of partial differential equations gov-

erning fluid-dynamic problems may be useful to

get information on the capabilities of numerical

methods in accurately predicting stability. A key

point of this methodology is the evaluation of

Jacobean matrices of the algebraic relationships

characterizing the numerical method and the re-

lated boundary conditions. This is a basic but

somehow boring and error-prone process, which

represents a difficult task to be completed when

dealing with complex models. It is in this perspec-

tive that the use of automatic FORTRAN differenti-

ation tools may play a considerable role.

In this section, as a first step towards this goal,

the sensitivity of stability results to physical and

numerical discretization parameters is analyzed

using a tool for the automatic differentiation of

FORTRAN codes. ADIFOR (meaning Automatic

DIfferentiation of FORtran), version 2.0D that

has been developed by C. Bischof from the ANL,

USA and A. Carle from Rice University, USA

(Bischoff et al., 1994) is the adopted tool that

allows for evaluating the derivatives of model

variables with respect to model parameters. Re-lated bibliography may be found at the ADIFOR

WWW site (ANL, 1999). Some introductory ex-

amples may also be found in the WWW (see for

example ANL, 1999; Goenka, 1999; Rightley,

1997). This is not the only tool available to auto-

matically calculate derivatives and a comparison

between two of them may be found in Rightley

(1997). Its use is becoming widespread to calculate

derivatives of dependent variables with respect to

parameters. Why it is important to know the

derivatives of dependent variables is closely asso-ciated with the construction of response surfaces

to determine the uncertainties of computed re-

sults, as has been clearly stated by Isukapalli

(1999).

ADIFOR is a pre-processor code which, given a

FORTRAN 77 code that computes a function, auto-

matically generates another, augmented, FOR-

TRAN 77 code. It must be considered that any

code may be put in the form of a function, simply

by introducing a call to a main routine after

setting parameter values. The latter computes the

function and the derivatives with respect to a list

of variables. The user must specify the list of

dependent and independent variables. After gen-

erating the augmented code that calculates the

specified derivatives via ADIFOR, the user must

provide a new driving FORTRAN 77 code that

takes into account the new set of variables. Even

when the codes must comply with FORTRAN 77

standards, ADIFOR accepts some common lan-

guage extensions.

Some simple hints to help on the very first use

of ADIFOR may be readily found in Rightley

(1997). Many references document the accuracy of

the derivatives calculated in this way. This is not

the only way to apply ADIFOR; the reader is

referred to the aforementioned literature. The use

ofADIFOR is straightforward. However, the origi-

nal code must be sometimes prepared to get the


12/18


Fig. 7. Map ofDz for the first-order, explicit momentum, implicit temperature coupling, cell Courant number 0.8, turbulent flow

and Ds=1/10, see Ambrosini and Ferreri (1998).

appropriate functional dependence between the

independent parameters and the dependent ones.This fact turns essential to take some time to verify

the automatically computed derivatives. One im-

portant feature ofADIFOR is the careful treatment

of derivative exceptions when handling intrinsic

functions. Its reporting capability is also of great

importance.

We are now interested in showing the sensitivity

of the neutral stability boundary to nodalization

and system parameters in natural circulation flows.

It may be argued that the effects of discretization

are known a priori for a given numerical model.However, this knowledge may not be simply

achievable in quantitative terms. In the frame of

such studies, the use of automatic differentiation of

FORTRAN codes may be especially useful. The

results reported here deal again with the pioneering

work of Welander (1967), because its stability

properties have been the subject of the above-men-

tioned analytical and theoretical studies.

The analysis was performed using the dimension-

less volumetric flow rate and temperature as system

variables and is focused on the results obtainedusing the implicit coupling of the momentum and

energy equations and the forward time, upwind-

space finite-difference method (FTUS) for the mo-

mentum equation. It may be argued that the

consequences of using this approximation on the

results are well known, i.e. obtaining a first order

accurate solution in the space and time increments.

However, what is particularly addressed here is the

quantification of these effects (and those of thecoupling) on the stability limits in systems working

in the natural circulation flow regime.

Fig. 7 shows the stability map for Welanders

problem in the plane ofh and m, the system physical

parameters. It has been obtained using 11 nodes

(Ds=1/10), cell Courant number (COU)=0.8 and

a value of 1.75 for the exponent of flow rate in

momentum equation x (defined as 2-b, where b is

the exponent in the friction law, assigned in the

present case to 0.25 as in the Blasius relationship

for turbulent flow in smooth pipes). The selectionof the value 0.8 for COU was not arbitrary: despite

the very low number of nodes, it ensures low

numerical diffusivity while keeping the calculations

stable. The theoretical neutral stability boundary

corresponds to Welanders analysis (Welander,

1967) modified for turbulent flow conditions and is

coincident with the one obtained from modal

analysis (see Ambrosini and Ferreri, 1998).

The interest in the analysis to follow is showing

how the solution depends on the cell Courant

number (COU) and x. The above mentioned set ofparameter values in the dependence domain will be

denoted as ()O. This implies calculating the follow-

ing derivatives:

(Dz

(COU

O

and(Dz

(x

O


13/18


In these cases, the value of the space increment,

Ds, will be fixed; however, Ds was also varied in

turn trying to investigate how the computed

trends confirm the results of the authors previous

analyses on the sensitivity to nodalization. In

order to improve the reliability of the obtained

conclusions, it was decided to check every ex-

ploratory calculation made with the help of ADI-

FOR using finite-difference approximations of the

derivatives; this prudent approach was rewarding,

because it allowed eliminating some reasonable

looking, wrong results.

The influence of x is considered first. Fig. 8

shows the map of (#Dz/#x)O as a function of h

and m considering Ds=1/10. It may be observed

that this derivative is not always negative; this is

due to the shift in the neutral stability boundary

for increasing values of x. Fig. 9 shows the same

map, but considering Ds=1/30. Now the negativearea is more smoothly defined. A reasonably con-

verged solution with respect to nodes number

corresponds to Ds=1/100, which was obtained

also by finite-differences as a further check. In this

case, the zero curve remains unchanged, with

small changes in the values of the derivative.

The change ofDz as a function of COU, (#Dz/

#COU)O, obtained using the same set of parame-

ters, is shown in Fig. 10. Decreasing COU should

lead to further damping in zones of positive

derivatives; this is coherent with the analysis

shown in Ambrosini and Ferreri (1998), where a

set of calculations was made imposing a fixed time

interval thus leading to more damped solutions.

Fig. 11 shows the map obtained assuming Ds=1/

30. Now, a shift in the zero curve towards the

right of the map may be observed. Once again, it

is interesting to resort to converged maps; the

map so obtained showed further bending of the

zero curve, leading to positive and negative zones

in the map.

As already mentioned, the effects of Ds on the

neutral stability boundary were also addressed. In

Ambrosini and Ferreri (1998) work was mainly

devoted to this aspect of the computations and

the effects of different approximations were as-sessed. The derivative of Dz in terms of Ds for

Ds=1/100, COU=0.8 and x=1.75 showed con-

vergence. However, the same derivatives com-

puted using ADIFOR were not coincident. Further

analysis is needed to clearly understand the rea-

sons of this discrepancy, but it is likely that the

discrete nature of the node spacing should impose

a limitation to its computation.

Fig. 8. Sensitivity map ofDz to the friction law exponent (#Dz/#x) for the first-order, explicit momentum and implicit temperature

coupling. [COU=0.8, x=1.75 and Ds=1/10].


14/18


Fig. 9. Sensitivity map ofDz to the friction law exponent (#Dz/#x)O for the first-order, explicit momentum and implicit temperature


4. Conclusions

This paper discussed the general aspects of

natural circulation flows in 1-D systems; and then

focused on some early results by the Authors

concerning the development of calculation tools.

The philosophy adopted in these simulations was

to gain experience from the use of in-house devel-

oped codes, adapted to simple situations, in order

to understand the behavior of large system codes

in predicting fluid-dynamic instabilities.System codes must be robust, in the usual

nomenclature; unfortunately, it sometimes means

that they should produce good results (or, simply,

results) in most cases. Experience shows that

avoiding numerical difficulties usually implies the

use of numerical damping, the basic tenet being

the use of full upwinding. Due to the intrinsic

non-linearity of natural circulation flows, coupled

problems of heat and fluid flow use linearization

and/or averaging that imply changing the time

constant of the computed results, adding damp-

ing. Then, the use of sensitivity analysis tech-

niques or looking for convergence of results in a

brute force way is sometimes rewarding. Time

domain codes may be used to analyze unstable

flows, if an adequate nodalization is used.

The methodology adopted for analyzing the

drawbacks involved in the use of numerical dis-

cretization of partial differential equations while

dealing with stability problems provided a quanti-

tative assessment of the major difficulties occur-

ring in the use of transient codes to this purpose.

Most of the observed qualitative trends were ex-

pected, based on the knowledge about the charac-

teristics of the numerical schemes used. The

quantitative information presented provides a

clear picture of the changes that truncation error

and, in general, discretization may induce in pre-

dictive capabilities about stability. In particular,the results obtained can be summarized as

follows:

first order explicit numerical methods are less

diffusive than first order implicit ones, but

attention must be anyway paid to the choice of

the nodalization and of the time step;

second order methods are better suited for

stability analyses and should be considered for

use in system codes wherever possible;

other details of the discretization, e.g. the cou-

pling in time between momentum and energy

equations, may also have an effect on stability

predictions;

additional discretization effects may appear in

two-phase flow stability problems, as it was

observed for the homogeneous equilibrium

model of a boiling channel; of course, for

two-fluid models with complete non-equi-


15/18


librium the situation is likely to be different

and a specific analysis should be made to clar-

ify the effect of discretization choices on stabil-

ity predictions.

A new development of the research has been

introduced here. It consists in the use of an auto-

matic differentiation tool: ADIFOR, version 2.0D,

which proved to be useful to determine the influ-

ence of parameters on the neutral stability

boundary of natural circulation flows. This is not

the most usual use of this tool, since it involves

tracing the influence of parameters on the spectral

radius of the discrete problem. The results ob-

tained confirmed exactly the numerically simu-

lated derivative maps. In this way, the influence of

the cell Courant number and a variable related to

the exponent in the friction law was assessed.

Experience showed that checking results withfinite-differences might be rewarding to get insight

on the results.

The use of ADIFOR is straightforward when

applied to predictive codes that give a result as a

function of a set of parameters. However, it is

essential to know the details of the code to be

differentiated and to explicitly calculate the

steady-state conditions corresponding to the ad-

dressed fixed point. This is particularly difficult

when dealing with unstable steady-state condi-

tions, which cannot be achieved asymptotically by

transient calculations with constant boundary

conditions.

In this respect, the goal of applying this tech-

nique to large TH systems codes seems a really

challenging one. This is partly due to multiple

sources of derivative discontinuities, coming from

physical correlations and intrinsic functions. Tab-

ular data may pose further difficulty to the analy-

sis, not to speak about numerically induced

oscillations like the ones observed on flow rate.

The task would surely require a huge effort.

At present, the short-term perspective of this

research consists in developing more sophisticated

tools for analyzing, in a realistic way, complex

stability problems making use of the described

numerical methodology. In particular, a code for

analyzing general single-phase single-loop naturalcirculation flows with a variety of boundary con-

ditions is being developed. At the time of writing

(June 2000), the transient program is already

completed and the linearized version will follow

soon, possibly owing to the use of ADIFOR. It

is the goal of the Authors to be capable to pro-

duce in the next future an example of coherent

linear and non-linear stability analysis tools for

natural circulation loops being enough flexible

and realistic to be reliably applied to a variety of

Fig. 10. Sensitivity map ofDz to cell Courant number (#Dz/#COU)O for the first-order, explicit momentum and implicit temperature



16/18


Fig. 11. Sensitivity map ofDz to cell Courant number (#Dz/#COU)O for the first-order, explicit momentum and implicit temperature

coupling. [COU=0.8, x=1.75 andD

s=1/30].

conditions of interest for experiments and

applications.

Acknowledgements

The work summarized here was possible due to

the support of institutions and the collaboration

of many colleagues. However, J.C. Ferreri must

specially mention his early work in collaborationwith Alicia S. Doval at CNEA, Argentina. He is

also deeply grateful by the warm and generous

friendship of the staff of the DIMNP, University

of Pisa, along all these years of informal collabo-

ration. Last but not least, comments and criti-

cisms by referees allowed the authors improving

the referred papers.

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