Comparison of Natural Convection Flows Under VHTR Type Conditions Modeled

7/25/2019 Comparison of Natural Convection Flows Under VHTR Type Conditions Modeled

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Nuclear Engineering and Design 240 (2010) 13711385

Contents lists available atScienceDirect

Nuclear Engineering and Design

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

Comparison of natural convection flows under VHTR type conditions modeled byboth the conservation and incompressible forms of the NavierStokes equations

Richard C. Martineau, Ray A. Berry, Aurlia Esteve, Kurt D. Hamman, Dana A. Knoll,HyeongKae Park, William Taitano

Fuels Modeling and Simulation, Nuclear Fuels and Materials Division, Idaho National Laboratory, P.O. Box 1625, Idaho Falls, ID 83415-3860, United States

a r t i c l e i n f o

Article history:Received 28 October 2009

Received in revised form

27 December 2009

Accepted 5 January 2010

a b s t r a c t

This paper illustrates a comparative study to analyzethe physical differences between numericalsimula-tions obtained with both the conservation and incompressible forms of the NavierStokes equations for

natural convection flows in simple geometries. The purpose of this study is to quantify how the incom-

pressible flow assumption (which is based upon constant density advection, divergence-free flow, and

the Boussinesq gravitational body force approximation) differs from the conservation form (which only

assumesthatthe fluid isa continuum)whensolvingflowsdriven bygravityactingupondensityvariations

resulting from local temperature gradients. Driving this study is the common use of the incompressible

flow assumption in fluid flow simulations for nuclear power applications in natural convection flows

subjected to a high heat flux (large temperature differences). A series of simulations were conducted on

two-dimensional, differentially heated rectangular geometries and modeled with both hydrodynamic

formulations. From these simulations, the selected characterization parameters of maximum Nusselt

number, average Nusselt number, and normalized pressure reduction were calculated. Comparisons of

these parameters were made with available benchmark solutions for air with the ideal gas assumption

at both low and high heat fluxes. Additionally, we generated specific force quantities and velocity and

temperature distributions to provide a basis for further analysis. The simulations and analysis were then

extended to include helium at the Very High Temperature gas-cooled Reactor (VHTR) normal operating

conditions. Our results show that the consequences of incorporating the incompressible flow assump-tion in high heat flux situations may lead to unrepresentative results. The results question the use of the

incompressible flow assumption for simulating fluid flow in an operating nuclear reactor, where large

temperature variations are present.

2010 Elsevier B.V. All rights reserved.

1. Introduction

Natural convection is a highly reliedupon cooling mechanismin

nuclear reactor safety. Natural convection heat transfer is defined

as mass and energy transport driven by buoyancy forces due to

density variations acted upon by gravitation. The density varia-

tions result from local temperature gradients generated by heat

conduction and internal energy advection. Typical natural convec-tion problems result in flow speeds which are relatively slow, i.e.

lowMach number.This often encourages theuse of theincompress-

ible flow assumption to reduce computational effort. However, the

single fluid incompressible flow model assumes constant density

advection and, thus, one is forced to use an additional approxima-

Corresponding author at: MaterialsDepartment,Idaho National Laboratory, P.O.

Box 1625, M.S. 3860, Idaho Falls, ID 83415, United States. Tel.: +1 208 526 2938;

fax: +1 208 526 2930.

E-mail address:[email protected](R.C. Martineau).

tion for the gravitational bodyforce to simulate natural convection.

Typically, the Boussinesq approximation is employed to model the

gravitationalbody force for incompressible natural convection sim-

ulations. The Boussinesq approximation uses a first-order Taylor

series to approximate the density variations based upon the differ-

ence between local temperature and a reference temperature.

This first-order approximation brings into question the validity

of the incompressible flow assumption incorporating the Boussi-nesq approximation for simulating natural convection flows in

certain nuclear power applications where temperature distribu-

tions can be highly nonlinear. This question of validity has not

gone unnoticed in the nuclear engineering community, given two

recent workshops sponsored by CEA and INRIA on the subject

(CEA/Nuclear Reactor Division, Numerical Workshop, 2000; INRIA

andMAB, NumericalWorkshop, 2004). Thegoalof theseworkshops

was to generate benchmark reference solutions on non-Boussinesq

natural convection flows by extending the well-known de Vahl

Davis(de Vahl Davisand Jones,1983; de Vahl Davis,1983) differen-

tiallyheated squarecavityproblem to thecaseof large temperature

0029-5493/$ see front matter 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.nucengdes.2010.01.022


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1372 R.C. Martineau et al. / Nuclear Engineering and Design240 (2010) 13711385

differences (high heat flux) for which the Boussinesq approxi-

mation is no longer valid. The simulation domain consisted of a

two-dimensional square cavity containing air differentially heated

along the vertical walls with specified temperatures. The various

test cases were defined by Rayleigh number, constant or variable

transport coefficients, and non-dimensional temperature differ-

ence. The chosen benchmark solution parameters were integrated

Nusselt numbers alongboththe hot and coldwalls andthe ratio

of maximum steady-state pressure to initial pressure. The results of

these workshops are tabulatedin Le Quere et al.(2005) and Paillere

et al. (2005).

The contributors to the CEA and INRIA workshops had already

concluded that the incompressible Boussinesq approximation was

invalid for high heat flux boundary conditions, which can cause

significant variations in fluid density. Here, we desire to quantify

the differences by comparing incompressible and compressible

solutionsfor bothsmall and large temperature difference boundary

conditions. The motivation for conducting thesecomparisonsis due

to the common misuse of the incompressible flow assumption in

fluid flow simulations for nuclear power applications,including the

gas-cooled Next Generation Nuclear Plant (NGNP) reactor designs

(MacDonald, 2003). In this study, we extended the goals of the CEA

and INRIA workshops to include specific force distributions and

centerline velocity and temperature distributions.Our approach is to perform simulations with two computa-

tional fluid dynamic (CFD) computer codes with one solving the

conservation form of the governing hydrodynamic equations and

the other solving the incompressible form. We first reproduced the

results of the de Vahl Davis benchmark with a small temperature

difference to demonstrate the equivalency of both formulations

in the incompressible limit. Next, we duplicated two of the high

heat flux test cases for air from the CEA and INRIA workshops. The

two test cases chosen varied only in that one incorporated con-

stant transport coefficients and the other employed temperature

dependent transport coefficients. The reason for choosing these

two test cases was twofold: First, they provided an opportunity

to validate our conservation CFD code against an accepted numeri-

cal benchmark experiment for variable density, low-Mach numberflow. And second, thetwo test cases provided an avenue to quantify

the differing results obtained with an incompressible formulation

when compared to the high heat flux benchmark solutions. Our

simulations culminated by applying our analysis criteria to helium

natural convection at the global operating conditions of the Very

High Temperature gas-cooled Reactor (VHTR) (MacDonald, 2003),

i.e. maximum coolant temperature difference and average reactor

pressure.

2. Governing equations and definitions

This section details the governing hydrodynamic models

employed in this study. We define the conservation and incom-

pressible forms along with a strict mathematical definition of

the incompressible flow assumption. Definitions of various flow

parameters pertinent to this study are also included.

2.1. The conservative NavierStokes equations

The conservation form of the governing hydrodynamic equa-

tions are defined in terms of density, momentum, and total energy

per unit volume. Expressed in two-dimensional planar spacetime

coordinates (x, y, t), these equations are the conservation of massequation:

t +

u

x +

v

y = 0, (1)

the balance of momentum in thex-direction:

u

t +

(u2 + P)

x +

uv

y =

xxx

+yx

y + gx, (2)

the balance of momentum in they-direction:

v

t +

uv

x +

(v2 + P)

y =

xyx

+yy

y + gy, (3)

and the conservation of total energy equation:et

t +

uhtx

+vht

y = (ugx + vgy) +

x(uxx + vxy qx)

+

y(uxy + vyy qy). (4)

In Eqs. (1)(4), thevariable is density, u and vare the componentsof the velocity vector uin thex andy directions, respectively, Pis

the thermodynamic pressure,gxandgy are the gravitational vectorcomponents, and et is the specific total energy. The specific totalenthalpy htis defined by

ht=et+ P

. (5)

Fora Newtonianfluidwiththe Stokeshypothesis, theviscousstresstensor components are defined as

xx = 2

3

2

u

x

v

x

, xy = yx =

u

y+

v

x

,

and yy = 2

3

2

v

y

u

x

, (6)

where is the viscous transport coefficient, dynamic viscosity. Thecomponents of the heat flux vector qare defined as

qx = kT

x and qy = k

T

y, (7)

wherek is the thermal transport coefficient, or thermal conduc-tivity, andTis the absolute temperature. The ideal gas equation of

state is employed in this the study:

P= ( 1)e = RcT, (8)

where e is the specific internal energy, is the ratio of specific

heats, andRcis the specific gas constant per unit mass. With Eqs.(1)(4), noassumptionsabout theflow aremadeoutside ofthe con-

tinuum assumption. These equations govern all fluids, even those

traditionally considered incompressible.

2.2. The strict definition of incompressible flow

Here, we will strictly define the incompressible flow assump-

tion in order to help quantify the physical differences with theconservative form, Eqs.(1)(4). FollowingPanton (1984), the term

incompressible flow is applied to any situation where change in

the density of a fluid particle along a pathline is negligible. A math-

ematical definition can be derived from the conservation of mass

equation. Recasting Eq.(1)in vector form:

t + u = 0, (9)

and applying the chain rule to the spatial derivative, yields:

t + u = u. (10)

The left-hand side of Eq.(10)represents density advection and is

commonly known as the material derivative, the time derivative of


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R.C. Martineau et al. / Nuclear Engineering and Design240 (2010) 13711385 1373

density following a material particle along a pathline in space and

time. The material derivative is denoted by

D

Dt =

t + u , (11)

and, thus, Eq.(10)becomes:

D

Dt = u. (12)

The strict mathematical definition of incompressible flow is

1

D

Dt = 0. (13)

Notice that Eq. (13) does not declare that density must be constant.

In fact, multi-component flows withdiffering densities, suchas two

immiscible fluids (like oil and water) are commonly modeled with

the incompressible flow assumption. The only requirement is that

the density of each fluid particle remain unchanged along a path-

line. Therefore, the local pressure and energy have no influence

upon the density of the fluid particle. Thus, density in an incom-

pressible flow is not described by an equation of state but only by

the initial conditions. Satisfying Eq. (13) requires that, fromEq. (12),

the flow field become divergence free, or

u = 0. (14)

With Eq.(14), the mass conservation Eq.(9) reduces to a mathe-

matical constraint imposed upon the flow field for incompressible

flow.

In reality, a divergence free flowfield is non-physical, especially

for non-isothermal flow fields. As a material particle changes posi-

tion in a flow field, the volume of the fluid particle may change

due to mechanical or thermodynamic influences. If the particle vol-

ume is changing(expandingor contracting), it is doing work on the

surrounding fluid. In Section 4.2, it will be shown that u is asignificant quantity for variable density flows, even for low Mach

numbers.We candescribe thisquantity in physicalterms by return-

ing to the mass conservation Eq. (9). First, define specific volume

as the volume of a material particle occupied by a unit of mass, = 1/. Substituting the definition for into Eq.(9), yields:

u = 1

D

Dt, (15)

which gives a volumetric rate interpretation for u and is knownas the rate of expansion of a fluid particle.

Forthis study, it is also usefulto analyze another consequence of

the incompressible flow assumption. The gravitational body force

terms of the conservation form of the NavierStokes Eqs. (2)and

(3) are reliant uponvariations in density due to thermo-mechanical

effects. Ignoring the mechanical (pressure) effects, the density

dependence on internal energy (temperature) must be approxi-

mated for incompressible flow. Referring to Burmeister (1983), the

definition of the coefficient of thermal expansion is

= 1

T

P

. (16)

We canapproximate the partial differential in Eq. (16) with a Taylor

series expansion about the reference density:

= o +

T(T To) +

2

T2(T To)

2 + , (17)

where o and To are the reference density and temperature,respectively. Ignoring second-order and higher terms, a first-order

approximation for the density difference is obtained by incorporat-

ing Eq.(16):

o o(T To). (18)

The approximate gravitational body force is then

g og og(T To). (19)

Eq. (19) is commonly referred to as the Boussinesq buoyancy model

for natural convection.Obviously, must haveunitsof inverse tem-perature.We can easilysee this by operating on Eq.(8) withEq. (16)

to obtain:

= 1

o P

T2Rc=

oT. (20)

For a single fluid incompressible flow, = o, so that

= 1

To. (21)

Note thatthe coefficient of thermal expansion must be a constant

for incompressible flow. is based upon the assumption that thefluid particles density must remain a constant in space and time.

Therefore, we cannot assume a temperature dependency for asthis implies variable particle density. Also, a higher-order approxi-

mation in terms of the temperature difference wont fully alleviate

the impact of a constant.

2.3. The incompressible NavierStokes equations with thermal

energy

The incompressible form of the governing hydrodynamic equa-

tions may be derived from Eqs.(1)(4)by imposing the definition

of incompressibleflow, Eqs. (13) and (14). As shownin theprevious

section, the conservation of mass equation becomes a mathemati-

cal constraint on velocity. In two dimensions, this constraint is

u

x+

v

y= 0. (22)

The two-dimensional incompressible NavierStokes equations

with the Boussinesq buoyancy model are the balance of x-

momentum:

ou

t+ou

u

x+ ov

u

y=

xxx

+yx

y

p

x+ ogx[1 (T To)],

(23)

and the balance ofy-momentum:

ov

t+ ou

v

x+ ov

v

y=

xy

x +

yy

y +

p

y+ ogy[1 (T To)],

(24)

wherep (lower case)is the incompressible pressure (notthermody-

namic pressure) and is a variance from the reference pressure Po.The components of the incompressible viscous stress tensor, xx ,

xy,yx , andyy, are defined as

xx = 2

u

x

, xy = yx =

u

y+

v

x

,

and yy = 2

v

y

. (25)

Neglecting viscous heating, the thermal energy transport equation

in an incompressible gas can be written in the form:

ocp

T

t + u

T

x+ v

T

y

=

qxx

qy

y, (26)

wherecpis specific heat at constant pressure.


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Fig. 1. Differentially heated square cavity domain with generic specification of

boundary conditions and gravitational vector.

3. Computational domain and test cases

This comparative study will be composed of five simulation

test cases performed on a common computational domain. The

domain is comprised of a two-dimensional square cavity with the

four walls being zero-mass flux boundaries. The vertical walls are

each maintained isothermal at separately specified temperatures.

The horizontal walls are defined as adiabatic (zero heat flux). Ini-

tial and boundary conditions for the first three test cases are taken

from established benchmark reference solutions for natural con-

vection of air. These three test cases provide a solution baseline

for the comparative analysis between the conservative and incom-

pressible flow models. The final two test cases are designed to

analyze helium natural convection flow at the global normal oper-

ating conditions of the VHTR (MacDonald, 2003), i.e., maximum

coolant temperature difference and average reactor pressure. Thefirst helium test case is a steady problem with variable transport

properties. Thesecond heliumtest case is transientwitha temporal

cosine wave defining the hot wall temperature.

3.1. Domain description and physical parameters

The computational domain for the comparison study presented

here is based upon the de Vahl Davis differentially heated square

cavity problem (de Vahl Davis andJones, 1983; de Vahl Davis, 1983)

as shown inFig. 1. The square domain is L width and height. The

natural convection problem is described by two vertically heated

walls with prescribed temperatures,ThandTc, which are the hotand cold wall temperatures, respectively. Adiabatic heat trans-

fer boundary conditions (qy = 0.0) are applied along the horizontalwalls. All four walls have no-slip, zero-mass flux boundary condi-

tions, u = 0. The gravitational vector, g, is applied anti-parallel tothey-coordinate. Initial conditions are composed of constant dis-

tributions of reference pressure and temperature, Po andTo, andstationary flow, u = 0.

The flow field for this natural convection problem can be

characterized by two non-dimensional parameters. The first is a

non-dimensional temperature difference, given by

=Th TcTh + Tc

=Th Tc

2To, (27)

whereTo is the reference temperature taken to be the average ofThand Tc. The second parameter is Rayleigh number and is defined

as the product of the Grashof number, which is the ratio of the

buoyancyand viscous forces within a fluid,and the Prandtl number,

which is the ratio of the momentum and thermal diffusivities. The

Rayleigh number for a square cavity is expressed as

Ra = GrPr= Prg2o (Th Tc)L

3

To2o

, (28)

whereo is the reference density determined from the equationof state at the reference pressure and temperature, Po and To,

respectively, o is the reference dynamic viscosity determinedfrom thermal dependence at the reference temperature, and gis

the magnitude of the gravitational vector.

Allthree workshopbenchmark solutions are in theform of non-

dimensional parameters. Non-dimensional wall heat flux, or local

Nusselt numberNu, is defined as

Nu(y) = 1

(Th Tc)

k(T)

k(To)

T

n

w

, (29)

where nis the outward normal vector to the vertical walls. FromEq.(29)we can compute the integrated (average) Nusselt number

along the wall:

Nu = 1

L y=L

y=0Nu(y) dy. (30)

TheCEA andINRIA workshops required thecalculation of a thermo-

dynamic pressure ratio. The thermodynamicpressure of this closed

system is defined as

Pth = mo

1

RTd

1, (31)

where is the volume of the system and mois defined as the mass

of the system:

mo =

od = 1

R

PoTo

d. (32)

The thermodynamic pressure ratio is then defined asPth/Po.This study will include one air benchmark from the CEA and

INRIA workshops with variable transport properties given by

Sutherlands law (White, 1991):

(T)

Suth=

T

TSuth

1.5 TSuth + ST+ S

, k(T) =(T)cp

Pr . (33)

For this case, Sutherland coefficients for air are S = 110.5 K,TSuth = 273.15K, and Suth = 1.68 10

5 kg/(ms) (INRIA and MAB,

Numerical Workshop, 2004).

3.2. Comparative study approach and test cases

As stated before, the goal of this comparative study is to quan-

tify the differences between incompressible and compressible

solutions for high-heat flux natural convection simulations. Our

comparative study approach is to solve both hydrodynamic for-

mulations to

1. Reproduce the results of the de Vahl Davis benchmark de Vahl

Davis and Jones (1983)andde Vahl Davis (1983)with a small

temperature difference to demonstrate the equivalency of both

model formulations in the incompressible limit.

2. Duplicate two of the high heat flux benchmark cases for

air from the CEA and INRIA workshops (CEA/Nuclear Reactor

Division, NumericalWorkshop, 2000; INRIA andMAB, Numerical

Workshop, 2004). This will provide baseline comparisons


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Table 1

Air benchmark test cases (Ra = 106,To = 600K, andPo = 101.325 kPa).

Test case # Th(K) Tc(K) Properties

TC-1 0.01 606.0 594.0 Constant

TC-2 0.6 960.0 240.0 Constant

TC-3 0.6 960.0 240.0 Eq.(33)

Table 2

Helium test cases (Ra = 106,To = 1148.15 K, andPo = 7.0 MPa).

Test case # Th(K) Tc(K) Properties

TC-4 0.3267 1523.15 773.15 Eq.(33)

TC-5 0.00.3267 Eq.(34) 773.15 Eq.(33)

between the conservative and incompressible solutions against

accepted solutions.

3. Generate additional analysis criteria to include such terms as

specific force distributions and centerline velocity and temper-

ature distributions in the immediate vicinity of the vertical hot

wall boundary layer for the steady-state cases.

4. Finally, apply the comparative analysis criteria to helium natural

convection test cases at the global normal operating conditions

of the VHTR (maximum coolant temperature difference andaverage reactor pressure).

Thefirstfourtestcases definedin thisstudyarethe constant wall

temperature, differentially heated square cavity geometry of de

Vahl Davis at a chosen Rayleigh number ofRa = 106. We chose thisvalue as it ensures a laminar flow field and avoids the problem of

turbulence modeling. A fifth test is definedwitha transienthot wall

boundary condition (seeTable 2). For this transient test case the

Rayleigh number varies between Ra = 0 106. Inall test cases, theinitial conditions are composed of constant distributions of refer-

ence pressure and temperature, Poand To (given in Tables 1 and 2),and stationary flow, u = 0.

The test cases for the comparisons with air benchmark solu-

tions are tabulated inTable 1and are defined by non-dimensionaltemperature difference , specified hot and cold wall temperaturesTh and Tc, respectively, and whether the transport properties areconstant and variable. Test case TC-1, with its small temperature

difference, is used to approximate the de Vahl Davis reference ( de

Vahl Davis and Jones, 1983; de Vahl Davis, 1983) solution.

Similarly, the helium test cases are tabulated in Table 2. The

transienttest caseis TC-5 withthe transienttemperature boundary

condition for the hot wall given by

Th(t) = 1148.15+ 375.0 cos(0.4t) K, (34)

wherethe periodis 5 s. Variable transportproperties forthe helium

cases aredefined by Eq.(33). The Sutherland coefficients for helium

(White, 1991) are S = 97.4 K, TSuth = 273.15K, and Suth = 1.864104 kg/(m s).

4. Simulation results and analysis

In thissection, thesimulation andanalysis results of thecompar-

ative study to quantify the physical differences between numerical

simulations obtained with both the conservation and incompress-

ible forms of the NavierStokes equations for natural convection

flows is presented.

4.1. Solution methods and computational meshes

The conservativesolver employedin thisstudy is the PCICE-FEM

scheme (Martineau and Berry, 2004; Berry and Martineau, 2008).

The PCICE-FEM scheme is a finite element method (FEM) spatial

discretization of the Pressure-Corrected Implicit Continuous-fluid

Eulerian (PCICE) algorithm. The PCICE algorithm defines the tem-

poral discretization and hydrodynamic coupling procedure for

the PCICE-FEM scheme. It is a semi-implicit, mass-momentum

coupled pressure-based scheme. The governing hydrodynamic

equations for this scheme are the conservative form of the bal-

ance of momentum equations (NavierStokes), mass conservation

equation, and the total energy equation. An operator splitting

process is performed along explicit and implicit operators of the

temporal discretization to render the PCICE-FEM scheme in the

class of predictorcorrector schemes. The complete set of semi-

implicit governing equations in the PCICE-FEM scheme are cast in

this form, an explicit predictor step and a semi-implicit pressure-

correction step with an elliptic pressure Poisson solution coupling

the predictorcorrector steps. The result of thispredictorcorrector

formulation is that the pressure Poisson equation in thePCICE-FEM

scheme is provided with sufficient internal energy information toavoid an iterative scheme.

The incompressible studies were conducted using the commer-

cial CFD package STAR-CCM+ (CD-adapco, 2008). The incompress-

ible governing equations are discretized using the finite volume

method (FVM). The default interpolation approach used for the

convective terms is a second-order upwind scheme while the dif-

fusive terms are interpolated by a linear approach. The discretized

Fig. 2. Computational meshes for square cavity domain (shown at one-half resolution). (a) Triangular finite element mesh. (b) Quadrilateral finite volume mesh.


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Fig. 3. Density comparison for test case TC-2. (a) Density for conservative solution. (b) Density for incompressible solution.

equations are solved with the segregated solver using algebraic

multigrid. The segregated solver solves the governing equations in

an uncoupled manner in which the linkage between the momen-

tum and continuity equations occurs using a predictorcorrector

approach. The complete process is developed using collocated

variables and a RhieChow (Rhie and Chow, 1983) type pressure-velocity coupling combined with the SIMPLE algorithm Patankar

(1980).

Fig. 2illustrates the two computational meshes utilized in this

study. Both meshesare clusterednear thesolidwallsto more effec-

tively capture boundarylayer effects. The meshshown in Fig.2(a)is

essentially a structured 120 120 FEM mesh consisting of 28,800

orthogonal triangular elements defined by 14,641 nodal points. It

is upon this mesh that the conservative hydrodynamic formula-

tion is solved for the various test cases. The conservative solution

variables computed on this FEM mesh are located at the nodes of

the triangular element vertices. The mesh shown inFig. 2(b) is of

a structured 120 120 quadrilateral FVM mesh. The incompress-

ible formulation is solved by STAR-CCM+ on this mesh where the

incompressible variables are computed at the cell centers. Thesetwo meshes are essentially the same mesh as the cell vertices for

both meshes are at identical coordinate locations.

4.2. General solution of the differentially heated cavity problem

In this section, we will provide an example of a natural convec-

tion solution in a differentially heated square cavity. The solution

profiles for all the steady-state test cases are similar in form (but

not in magnitude) as the solution to the square cavity problem

is in large part defined by the Rayleigh number, irrespective of

non-dimensional temperature difference or working fluid (air or

helium). We chose test case TC-2 to provide a graphical illustra-

tion of a Rayleigh numberRa = 106 natural convection solution ina square cavity for both conservative and incompressible formula-

tions.This test case, with its large temperature difference ( = 0.6),clearly demonstrates some of the physical differences in solutions

between the formulations. Even with the large temperature dif-ference, this flow field is relatively slow compared to that of a

natural convection scenarioin a gas-cooled reactor.The peakveloc-

ity magnitude for test case TC-2 corresponds to an approximate

Mach number ofM= 2.0 103.The density solution comparison, shown in Fig. 3, highlights

the most striking physical difference between the two model for-

mulations. Fig. 3(a) reflects the density solution obtained with

the conservative formulation. With = 0.6 the air density variesapproximately by a factor of four. For helium natural convection

in the VHTR reactor ( 0.33), density varies by a factor of two. Incontrast,Fig.3(a)illustrates theincompressible renditionof density

which is a specified constant for a single-phase, single-component

fluid.

In Fig. 4, we see a side-by-side comparison of the absolute pres-sure solutions. Fig. 4(a) shows the pressure distribution for the

conservative solution andFig. 4(b) represents the pressure distri-

bution for the incompressible solution. While at first glance, the

solutions appear to be similar because both pressure solutions

appear to be nearly hydrostatic. Both simulations were started

at a prescribed pressure of one atmosphere (101,325 Pa). How-

ever, the steady-state solutions have significantly different domain

integrated (average) pressures. The average pressure for the con-

servative solution is P 87,012 Pa while the average pressure

for the incompressible solution is p 101,328Pa. The incom-pressible average pressure is essentially the sum of the initial

Fig. 4. Pressure comparison for test case TC-2. (a) Pressure for conservative solution. (b) Pressure for incompressible solution.


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Fig. 5. Divergence of velocity comparison for test case TC-2. (a) Conservative divergence of velocity. (b) Incompressible mass imbalance.

pressure and half of the hydrostatic pressure difference (linear

distribution).

The reason for these differences in integrated pressure is purely

physical. The conservative pressure is thermodynamic (physical)

and requires an equation of state to define its functional depen-

dency upon density and energy. In order to conserve mass in thisclosed system, the thermodynamic pressure of the conservative

solution has to adjust to the total integrated energy in the domain

and, in this case, causes an overall reduction in absolute pressure.

In contrast to thermodynamic pressures functional dependency,

the incompressible absolute pressure (non-physical) is dependent

upon an arbitrary reference value. Considering the incompressible

flow Eqs. (22)(26), only the momentum Eqs. (23) and (24) rely

upon pressure in the form of a differential pressure gradient.

Because the density and thermal energy Tdo not depend uponpressure (basis of the incompressible flow assumption, see Section

2.2), the average initial (or reference) pressure is purely arbitrary.

Typically, incompressible flow simulations do not employ a refer-

encepressureintermsofabsolutepressure.Azerogaugepressure

is quite common. Thus, the pressure determined at the computa-tional points are actually departures from this gauge pressure.

The last quantity that we will be comparing is local divergence

of velocity (orrate of expansion). Fig.5(a) illustrates the divergence

of velocity distribution for test case TC-2 conservative solution. The

divergence of velocity represents a physical quantity that we can

see demonstrated by the expansion of the gas along the hot wall

(positive values) as well as the contraction of the fluid along the

cold wall (negative values). The magnitude of the expansion and

contraction of the gas is on the order of 2.02.5 s1.

While divergence of velocity is a physical quantity for the con-

servative solution, the numerical divergence of velocity profile for

the incompressiblesolution, shown in Fig.5(b), is based upon noth-

ing more than the spatial discretization error and the convergence

error in the iterative solution method used to drive the divergence

of velocity towards zero. Ideally, the mathematical divergence ofvelocity is zero across the domain but is never perfectly achieved

in a numerical approximation. The plot shown inFig. 5(b) is actu-

ally the mass imbalance field, which represents the total mass flux

in each cell. Computing the divergence of velocity is not an option

available in STAR-CCM+. The mass imbalance field is representa-

tive of the divergence of velocity at steady-state.

Fig. 6is a side-by-side comparison of the steady-state stream-

line solutions. The obvious difference between the two solutions

is the asymmetry of the conservative streamline solution shown

inFig. 6(a) versus the symmetric incompressible streamline solu-

tion shown in Fig.6(b). The asymmetry of the conservative solution

is a direct result of the expansion of the gas (decrease in density)

along the hot wall and the contraction (increase in density) of the

gas along the cold wall. With the material derivative forced to bezero in the incompressible formulation resulting in a divergence-

free flow field (no expansion or contraction), the incompressible

solution illustrated inFig. 6(b) is then symmetric.

4.3. Comparison with benchmark results

In the followingthree sections,direct comparisons fortest cases

TC-1 through TC-3 are made between the solutions obtained with

the conservative and incompressible solvers and the established

Fig. 6. Streamline comparison for test case TC-2. (a) Streamlines for conservative solution. (b) Streamlines for incompressible solution.


8/15


Fig. 7. Test case TC-1 hot and cold wall Nusselt distributions.

benchmark reference solutions. The de Vahl Davis benchmark

comparison results are described in Section4.3.1, where the ref-erence values are all in terms of various non-dimensional wall heat

(Nusselt numberNu) parameters. For graphical purposes, the cold

wall Nusselt numbers are presented as absolute values. With Eq.

(29), the cold wall Nusselt numbers would have been negative,

whichdenotes heat flowingout of the boundary. Sections 4.3.2 and

4.3.3contain the comparisons with the CEA and INRIA benchmark

results. There, the reference solutions are in terms of integrated

wallNuand non-dimensional thermodynamic pressure reduction.

4.3.1. Test case TC-1, de Vahl Davis air benchmark

Fig. 7illustrates the Nusselt number Nu comparison results for

the de Vahl Davis problem,testcaseTC-1. The line plots are of local

Nusselt number Nu computed along the vertical hot and cold walls

according to Eq.(29). For this low-temperature difference (nearlyisothermal) case, it is difficult to delineate between the conserva-

tive and incompressible solutions. This is clear evidence that, for

nearly isothermal density-driven flow, the formulations are nearly

equivalent. Note that the solutions are nearly symmetric about

y = 0.5 m, a result of constant density definition for the incom-

pressible formulation and nearly constant density distribution for

the conservative formulation.

Table3 gives the computedbenchmark values of various param-

eters for both the conservative and compressible formulations and

theircorresponding relative errorcompared to theacceptedbench-

mark reference values. InTable 3,Nuh

andNuc

are the integrated

(average) Nusselt numbers along the hot and cold walls, respec-

tively, determined by Eq. (30). For natural convection problems,

ifNuh = Nuc, then we have conserved energy due to the adiabatictop and bottom horizontal walls. Nuhmax andNu

cmax are the max-

Table 3

Test case 1 comparison (Ra = 106, = 0.01, and constant transport coefficients).

Parameter Conservative Incompressible Reference %REC %REI

Nuh

8.79721 8.83511 8.8 0.032 0.399

Nuc

8.79721 8.83534 8.8 0.032 0.407

Nuhmax 17.5370 17.5891 17.925 2.165 1.874

Nuhy=0.5

8.31960 8.37688 8.799 5.448 4.797

Nuhmin

0.95816 0.97981 0.989 3.118 0.929

Nucmax 17.5370 17.5913 17.925 2.165 1.862

Nucy=0.5

8.31961 8.37693 8.799 5.448 4.797

Nucmin

0.95816 0.98088 0.989 3.118 0.821


imum Nusselt values along the hot and cold walls, respectively,

with corresponding minimum Nusselt values Nuhmin

and Nucmin

.

Nuhy=0.5andNu

cy=0.5are the Nusselt values aty = 0.5 m for the hot

and cold walls, respectively. %REC and %REI are the percent rela-

tive errors compared to the reference values for the conservative

and incompressible formulations, respectively. With two different

source codes containing different governing equations, algorithms,

and spatial discretization methods, the two solutions we obtained

for the de Vahl Davis problem are nearly identical, which implies

thatthe twoformulationsachieve comparable results in the incom-

pressible limit.

There is one note to make about the given number of signifi-

cant digits in the reference column ofTable 3. The de Vahl Davis

(de Vahl Davis and Jones, 1983; de Vahl Davis, 1983) required grid

converged results on uniform meshes of 11

11, 21

21, 41

41,and 81 81 grid points for this Ra = 106 problem. Given com-monly available computer hardware and the algorithms of 1980,

an 81 81 grid required significant computational effort for this

non-trivial problem. However, a uniform grid of this size is notade-

quate to accurately capture the five vortices of this flow field. Nor

does this coarse grid have the required resolution for accurate heat

transfer calculationsat the boundary. There wasa wide variationin

contributor solution results, thus thelack of significant digitsin the

given reference solutions forNuh

andNuc

. The attention should be

more focused upon the comparison between our solutions, which

were computed on a clustered 121 121 grid. With two different

source codes containing different governing equations, algorithms,

and spatial discretization methods, the two solutions we obtained

for the de Vahl Davis problem are nearly identical, which impliesthatthe twoformulationsachieve comparable results in the incom-

pressible limit.

4.3.2. Test case TC-2, CEA and INRIA air benchmark (constant

transport properties)

Fig. 8illustrates the Nusselt number Nu comparison results for

the hightemperature difference, constanttransport property prob-

lem of test case TC-2. The line plots are of local Nusselt numberNu

computed along the vertical hot and cold walls. Here we begin to

see significant variations in the solutions between the conservative

andincompressible formulations for high-heatflux situations. Note

that the incompressible Nusselt distribution is symmetric and the

conservative Nusselt distribution is not. In this constant transport

property case, the variance in the solutions can be traced back to


9/15


Table 4

Test case 2 comparison (Ra = 106, = 0.6, and constant transport coefficients).


Nuh

8.85439 8.82412 8.85978 0.061 0.402

Nuc

8.85449 8.82418 8.85978 0.060 0.402

Pth/Po 0.85871 NA 0.85634 0.277 NA

Nuhmax 19.5409 17.5929 19.5964 0.283 10.223

Nuhy=0.5

7.80950 8.37693 7.81938 0.127 7.130

Nuhmin

1.07294 0.97917 1.07345 0.048 8.783

Nucmax 16.3407 17.5929 16.3623 0.132 7.521Nuc

y=0.5 8.81809 8.37664 8.79636 0.247 4.772

Nucmin

0.85819 0.97916 0.85512 0.359 14.506

Table 5

Test case 3 comparison (Ra = 106, = 0.6, and variable transport coefficients).


Nuh

8.6810 8.5888 8.6866 0.064 1.126

Nuc

8.6815 8.5888 8.6866 0.059 1.126

Pth/Po 0.923853 NA 0.924487 0.069 NA

Nuhmax 20.2725 17.6753 20.2704 0.010 12.80

Nuhy=0.05

7.4455 8.0161 7.4593 0.185 7.464

Nuhmin

1.0765 0.9815 1.0667 0.919 7.987

Nucmax 15.4255 16.3898 15.5194 0.605 5.608

Nucy=0.05 8.6195 8.2566 8.6372 0.205 4.407Nuc

min 0.7577 0.8539 0.7575 0.026 12.726

the strong variation in density for the conservativeformulation (see

Fig. 3). The variation in density increases the buoyancy of the gas

in this region and creates a significant rate of expansion (non-zero

divergence of velocity) of the gas (seeFig. 5), accelerating the gas

upward.

Thebenchmarkreference solutions forthe CEAand INRIA work-

shops are in terms of integrated wall Nusselt numbers Nuh

and

Nuc

and thermodynamic pressure ratio Pth/Po, determined by Eq.

(31). These reference values are provided inTables 4and 5. The

thermodynamic pressure ratio Pth/Po is not a parameter that can

be applied for the incompressible solution and is denoted by NA.Additional variables that are consistent with the de Vahl Davis

benchmark reference solution were also provided by various work-

shop contributors but were not defined as reference values. One of

the contributors to the workshops, Vierendeels, produced results

on a 2048 2048 grid. Vierendeels solution (Paillere et al., 2005)

agrees exactly with benchmark values (for the given number of

significant digits). Additional non-reference variables provided by

Vierendeels are then assumed to be correct for this study. They are

denoted by an * inTables 4and5and used as reference values for

our comparison study.

Overall,the conservativesolutionobtained on a relativelycoarse

mesh agrees well with the benchmark reference values given in

Table 4, well within 1%. This is important as we extend the simula-

tion analysis to heliumVHTR conditions of test cases TC-4 andTC-5.

We canthus expect equally accurateresultsfor theVHTR conditions

where no reference solutions exist. As expected, the constant den-

sity profile and first-order body force approximation (Boussinesq)

of the incompressible formulation produced less representative

results with peak errors on the order of 10%.

4.3.3. Test case TC-3, CEA and INRIA air benchmark (temperature

dependent transport properties)

Fig.9 illustrates the wall Nusselt number Nu comparison results

for thehigh temperature difference, temperature-dependenttrans-

port property problem of test case TC-3. The line plots are of local

Nusselt number Nu computed along theverticalhot and cold walls.

Now note that, for variable transport coefficients, neither solution

is symmetric.


The reference values, including the Vierendeels values, for test

case TC-3 are provided in Table 5. Note that the peak hot wallNusselt numberNuhmaxare higher than test case TC-2 and the cen-terline hot wall Nusselt number Nuh

min are lower than test case

TC-2. Also, The average temperature of the domain for the con-

servative solution is higher than the average temperature for the

constant transport property test case TC-2. This has resulted in

a higher thermodynamic pressure ratio, Pth/Po = 0.923853, than

pressure reduction ratio of test case TC-2 (see Table 4), Pth/Po =0.85871. Thisthermodynamic phenomena willbe explainedin Sec-tion4.4.3.

As with test case TC-2, the conservative solution agrees well

with the air benchmark reference values for the variable transport

property test case TC-3. Again, the conservative values are within

1% error. The peak errors on the incompressible solution Nusselt

numbers have now risen to approximately 12%.

4.4. Extended analysis

Here, we further extend the air benchmark reference solution

analysis of test cases TC-1 through TC-3 to include the examination

of individual termsof the governing equationsand to compare their

effects upon the viscous and thermal boundary layers. Specifically,

we compare the individual termsof they-component of thebalance

of momentum equations at steady-state in the vicinity of the hot

wall boundary layer. The steady-state form of Eqs.(3)and(24)can

be recast in terms of the summation of specific forces:

0 = uv

x

(v2)

y

P

y+

xyx

+yy

y + gy, (35)

and

0 = ouv

x ov

v

y

p

y+

xyx

+yy

y + ogy[1 (T To)].

(36)

Eqs.(35)and (36)are the y-components of the steady-state bal-

ance of momentum for the conservative and incompressible forms,

respectively.In thefollowing analysis,the sumof thefirst twoterms

on the right-hand-side of Eqs.(35)and(36)are referred to as the

inertia of the conservative and incompressible forms, respectively.

They-component of the pressure gradient terms of Eqs.(35)and

(36)appear identical. However, it must be kept in mind that only

the pressure gradient of the conservative form is thermodynamic.

The sum of the two stress components, fourth and fifth terms,


10/15


Fig. 10. Test case TC-1 specific force distribution in the hot wall boundary layer

taken along the centerline at y = 0.5 m. The solid lines refer to the conservative

profiles of (a) inertia, (b) thermodynamic pressure gradient, (c) stress component,

and (d) gravitational body force. The dashed lines refer to the incompressible pro-

files of (e) inertia, (f) pressure gradient, (g) stress component, and (h) Boussinesq

gravitational body force approximation.

are referred to as the stress component of the conservative and

incompressible forms, respectively. The final momentum terms to

compare are they-component of the body force.

4.4.1. Test case TC-1, de Vahl Davis air benchmark

Fig. 10 illustrates the distributions of the specific force terms

of Eqs.(35)and(36)along the centerline (y = 0.5 m) in the vicin-ity of the hot wall boundary layer for test case TC-1. For this

nearly isothermal, constant propertytest case,the conservativeand

incompressible solutions nearly overlap. The Boussinesq gravita-tional bodyforce (line h) approximates the conservative body force

(line d) very well, whichimplies that this approximation is validfor

small temperature differences. The equivalency of the stress and

inertia terms for nearly isothermal flows results in nearly equiva-

lent velocity profiles in the boundary layer as shown in Fig. 11(a).

With the convective velocities nearly equivalent in the boundary

layer, the temperature profiles in the thermal boundary layer are

nearly indistinguishable, as shown in Fig. 11(b). The end result is

the nearly identical Nusselt distributions ofFig. 7.

Fig. 12. Test case TC-2 specific force distribution in the hot wall boundary layer

taken along the centerline at y = 0.5 m. The solid lines refer to the conservative

profiles of (a) inertia, (b) thermodynamic pressure gradient, (c) stress component,

and (d) gravitational body force. The dashed lines refer to the incompressible pro-

files of (e) inertia, (f) pressure gradient, (g) stress component, and (h) Boussinesq

gravitational body force approximation.

4.4.2. Test case TC-2, CEA and INRIA air benchmark

The specific force distribution near the hot wall for test case

TC-2 is shown inFig. 12. For this high temperature difference test

case, we see a relatively wide (compared to test case TC-1) vari-

ation in the specific force distributions between the conservative

and incompressible solutions. With identical transport properties

utilized for both simulations, these specific force variations are

primarilydue to the variabledensity profileof theconservative for-

mulation, which varies by a factor of four across the domain (see

Fig. 3). InFig. 12, we see that the inertia profiles are still relatively

insignificant to the balance of specific forces. Even though peakboundary layer velocity for this test case is an order of magnitude

higherthan the boundarylayervelocityof test case TC-1, the veloc-

ity magnitudes and variations in velocity for these square cavity

problems are too low for the inertia terms to contribute signifi-

cantly to the viscous andthermalboundary layerprofiles.However,

in the near-wall region (0< x


11/15


Fig.13. Viscous andthermal boundarylayer profile comparisonfor testcase TC-2.(a) Hotwall viscous boundary layercomparison takenalongy = 0.5 m.(b) Hotwallthermal

boundary layer comparison taken alongy = 0.5.

order of 10%, is evident acrossthe domain. Globalintegrationof the

body force terms indicates that thedomain is subjectedto different

total body forces determined by the two formulations.

Fig. 13(a)plots the velocity distribution in thevicinity of the hot

wall boundary layer for test case TC-2. Here, the peak conserva-

tive velocity is 11%higher than the peak incompressible velocity. Itis interesting that in the near-wall region (0.0< x


12/15



boundary layer comparison taken along y = 0.5 m.

Thespecificforce distributionnearthe hot wall fortestcaseTC-4

isshownin Fig. 17. For thisheliumhigh-temperature difference test

case, we see a relatively narrow variation in the specific force dis-

tributions between the conservative and incompressible solutions.

The widest disparity between the solutions occurs in the near-wallregion (0.0< x


13/15



boundary layer comparison taken alongy = 0.5 m.

Fig.19. TransientNusseltcomparison fortestcase TC-5. (a)Hot wall Nusselt at t= 2.5s. (b) Hot wallNusselt at t= 5.0s. (c) Hot wallNusselt at t= 7.5 s. (d)Hot wallNusselt

att= 10.0 s. (e) Hot wall Nusselt att= 12.5 s. (f) Hot wall Nusselt att= 15.0 s.


14/15


As volume (and mass) is constant in this closed system, the

internal energy of the system is proportional to the average pres-

sure of the system at any point in time during the transient period.

The system pressure remains in near thermodynamic equilibrium

with the system internal energy and the constant mass of the sys-

tem. Dynamic pressure effects are almost non-existent because

the time scale of the pressure propagation speeds are four orders

of magnitude smaller than the period of the thermal transient

being investigated here. Thus, the pressure distribution maintains

a hydrostatic profile even as the system, or average, pressure rises

and falls. The lowest system internal energy occurs at an approxi-

mate time oft= 13.425 s. At this point in time, thethermodynamicpressure ratio is Pth/Po = 0.82220for a system pressure of 5.75467MPa versus an initial thermodynamic pressure ratio of Pth/Po =0.97506 with a system pressure of 6.82542 MPa at the beginningof the simulation. At the end timet= 15.0 s, the thermodynamicpressure ratio is nowPth/Po = 0.86057 with the system pressurerising to 6.02401 MPa. There is no reflection of this transient ther-

modynamic response in the incompressible flow results.

Thetransient thermodynamic responseeffects local phenomena

as well, such as in the near wall regions of the hot and cold walls.

Near these walls, the temperature of the gas closely follows the

transient hot wall temperature and remains close to the constant

cold wall temperature. As the system thermodynamic pressurerises and falls through the transient cycle, the local density in these

near wall regions must react accordingly. This can be explained in

terms of the equation of state by differentiating Eq. (8)over some

differential change in time:

P=

P

e

+

P

e

e, (37)

or,

P= RcT + RcT, (38)

whereP,, andTare temporal increments of pressure, den-sity and temperature, respectively. Solving for ,

= P

RcT

T T, (39)

we can see that the change in density near the wall is a nonlinear

combination of the changes in pressure and temperature, and we

know from above that the change in pressure over time is not zero.

As the temperature of the hot wall boundary condition increases,

the local density near the wall must decrease even as the systems

thermodynamic pressure must increase. In order to conserve mass

of the system, the density must increase elsewhere in the system.

This is accomplished by the increasing system pressure.

The transient nature of the near wall density must also sat-

isfy the conservation of mass Eq.(1). Locally, the divergence of the

mass fluxumust be equal to the time rate of change in density.

Therefore, the velocity field must compensate for the expansionand contraction of the fluid. Conversely, the Boussinesq approxi-

mation assumes that the local change in density is approximated

only by a function of the difference between the local tempera-

ture and a reference temperature. Thus, the velocity profile in the

incompressible solution only responds to the effects of the tran-

sient temperature boundary condition along the hot wall on the

system, a thermal response.

The results of the transient thermodynamic response of the con-

servative solution versus the transient thermal response for the

incompressible solution are clearly displayed inFig. 19.Fig. 19(a),

(c), and (e) are the Nusselt number distribution along the hot wall

at timest= 2.5, 7.5, and 12.5 s, respectively. Note that the Nusseltnumber distributions at these solution times are negative along the

wall indicating thatheatis flowingoutof thehotwallboundary (the

interior is at a highertemperature than the specified wall tempera-

ture). While the Nusselt number profiles for both the conservative

and incompressible solutions appear similar, the magnitude of the

Nusselt numbers are decreasing in time. The differences in peak

Nusselt numbers between the solutions are Nu = 2.5, 1.8, and1.5 at timest= 2.5, 7.5, and 12.5s, respectively. At these solution

times, the difference in peak Nusselt numbers translate into 14.8%,

13.2%, and 13.2%, respectively. Fig. 19(b), (d), and (f)are the Nusselt

numberdistribution along the hotwallat times t= 5.0 s, t= 10.0 s,and t= 15.0 s, respectively. At these simulation times, the specified

temperature along the hot wall is equal to the constant specified

hotwall temperature oftest cases TC-4 andTC-5, Th = 1523.15K.Atthese solution times, the Nusselt number profiles are similar with

the peak Nusselt numbers for both solutions slightly increasing in

time (indicating that the interior near wall temperature is slightly

decreasing withtime). Also,the peak difference in Nusselt numbers

remain fairly constant in time, approximately 13.2%.

5. Summary and conclusions

In summary, we presented a comparative study of the physi-

cal differences between numerical simulations obtained with both

the conservation and incompressible forms of the NavierStokesequations for natural convection flows in simple geometries with

the goal of investigating the possible consequences of assum-

ing incompressible flow in Next Generation Nuclear Plant (NGNP)

simulations, specifically for helium-cooled reactor concepts. The

purpose of this study was to quantify how the incompressible

flow assumption (which is based upon constant density advec-

tion, divergence-free flow, and the Boussinesq gravitational body

force approximation) differs from the conservation form (which

only assumes that the fluid is a continuum) when solving flows

driven by gravityactingupon densityvariations resultingfrom local

temperature gradients.

The solutions obtained with both the conservative and incom-

pressible formulations for the de Vahl Davis air benchmark test

case TC-1 were nearlyidentical andcompared well with thebench-marksolutions. Therewas onlya 1.0%variationin local densityfrom

the incompressible reference density o for this low-temperaturedifference ( = 0.01), constant transport property test case. Thesmall density variation seems to have negligibly effected the rel-

ative accuracy when compared to the incompressible benchmark

solution (see Table 3). Our conservative and incompressible results,

obtained on a grid with much finer boundary layer resolution (see

Fig. 2(a) and (b)), agree more closely with each other than the

benchmarksolutionfor some of the specific values of Nusselt num-

bersNu. The extended analysis for this nearly isothermal test case

shows that the near wall specific force distributions and viscous

and thermal boundary layer profiles are nearly identical for both

hydrodynamic formulations (see Figs. 10 and 11). Overall, the solu-

tionobtainedwith the conservative method showed goodbehaviorand accuracy in the nearly incompressible flow regime.

As expected, the comparison results diverged significantly

for the high-temperature difference ( = 0.6) CEA and INRIA airbenchmark test cases TC-2 (constant transport properties) and

TC-3 (temperature dependent transport properties). The solutions

obtained with the conservative method compared very well with

the benchmark Reference values (seeTables 4and5) with the rel-

ative errors in both test cases less than 1%. It is possible in these

two test cases for an ideal gas fluid particle following a pathline in

space and time to undergo a factor of four change in fluid density.

As discussedin Section 2.2, this magnitudeof change in density of a

fluidparticleclearlyresults in a violationof theincompressible flow

assumption, Eq.(13). The inappropriateness of the incompressible

flow model for these two test cases is manifest in our incompress-


15/15


ible results with the relative errors in the magnitude hot and cold

wall Nusselt numbersNu approaching 12%. The extended analysis

for these two air test cases also supports this conclusion.

With confidence in our conservative code gained by generating

representative comparison results for the high-temperature differ-

ence air benchmark test cases TC-2 and TC-3, we applied both the

conservative and incompressible codes to the high-temperature

difference helium test case TC-4 whose initial and boundary con-

ditions are loosely based upon the VHTR operating conditions. We

computed the relative error of the incompressible solution based

upon the conservative solution. The various Nusselt numbers Nu

obtained with theincompressible code were on the order of 5% dif-

ferent from the results obtained with the conservative code (see

Table 6). These results and the extended results are much closer

than those obtained for the air test cases TC-2 andTC-3. The reason

for this is two-fold: First, while the hot and cold wall temperature

differences between the air and helium test cases were roughly

the same (720.0K for the air test cases and 750.0 K for helium test

cases), the non-dimensional temperature difference is approxi-mately half for the helium test cases ( = 0.6 for the air test casesand 0.33 for the helium test cases). The result is that the varia-tion in density acrossthe heliumflow field is approximately a factor

of two versus a factor of four for the high-temperature difference

air test cases. And second, helium can be thought of as a thermo-dynamically stiff gas. If we consider Eq. (39), the firstterm onthe

right-hand side defines the change in densitys dependency upon

the change in thermodynamic pressure. The coefficient 1/(RcT)represents the sensitivity of this dependency. Heliums specific gas

constant is Rc= 2077J/(kg K) and airs is Rc= 287J/(kg K).RcT isapproximately equal to the local sound speed squared. For helium,

this term translates into RcT 3.16 106 m2/s2 atT= 1523.15 Kandair is anorderof magnitudeless with RcT 2.75 10

5 m2/s2 at

T= 960 K, which implies that fora given change in density, the cor-responding change in pressure for helium is an order of magnitude

greater than that forair.The influenceof this sensitivity is exhibited

in the thermodynamic pressure reduction ratiodifference between

the air and helium test cases. For the air test case TC-2, the pres-

sure reduction ratio is Pth/Po = 0.85634, or 14.4% deviation fromthe reference pressure. For the helium test case TC-4, the pres-

sure reductionratiois Pth/Po = 0.97506,or 2.5% deviationfrom thereference pressure.

From the results of the transient helium test case TC-5, it is

apparent that the thermodynamic effects upon the velocity field

may be more important for time-dependent flows than for steady-

state flows. These effects, that are physically neglected with the

incompressible flow assumption, basicallyresulted in two different

temporal solutions. Fig. 19(a)(f) clearly shows that the incom-

pressible solution consistently underpredicts the magnitude of hot

wallheat flux during this transientsimulation. The thermodynamic

response forthe conservativesolutionis significantlydifferent from

the thermal response of the incompressiblesolution. The conserva-

tive velocity distribution differs from the incompressible velocitydistribution due to expansion and contraction of the gas and the

different representations of the gravitational body force. While the

steady-state temperature distributions for both the conservative

and incompressible solutions for test case TC-4 were relatively

close, the transient temperature distributions for both solutions

of test case TC-5 are significantly different in time. Therefore, the

temperature dependent transport properties differ accordingly.

In general, the incompressible flow assumption is valid in the

case of isothermal, or nearly isothermal, single phase flows under

relatively small pressure gradients. In reality, truly incompressible

flows are nonexistent. The assumption is non-physical and noth-

ing more than a mathematical approximation when wechooseto

ignore variations in density that are always present. At times in

the decision process, it is somewhat arbitrary to determine if the

incompressible flow assumption has been violated. However, it

seems unreasonable to neglect the large variation in density of the

helium that will be present in NGNP reactor simulations. There-

fore, the heat transfer results obtained with the incompressible

flow assumption are also called into question for the helium test

cases. If we were to apply typical reactor heat fluxes at the bound-

ary instead of specified wall temperatures, wall temperatures may

vary significantly between the conservative and incompressible

solutions.

Acknowledgments

The submitted paper has been authored by a contractor of

the U.S. Government under Contract No. DE-AC07-05ID14517

(INL/CON-08-15003). Accordingly, the U.S. Government retains a

non-exclusive,royalty-freelicenseto publish or reproducethe pub-

lished form of this contribution, or allow others to do so, for U.S.Government purposes.

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Comparison of Natural Convection Flows Under VHTR Type Conditions Modeled

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