Veri cation and Validation of the ISIS CFD Code for Fire Simulation
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Verification and Validation of the ISIS CFD Code
for Fire Simulation
S. Suard, L. Audouin, F. Babik, L. Rigollet, J.C. Latché
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)
Direction de la Prévention des Accidents Majeurs (DPAM)
BP 3 – 13115 St. Paul-lez-Durance, France
{sylvain.suard, laurent.audouin, fabrice.babik,
laurence.rigollet, jean-claude.latche}@irsn.fr
Abstract
Fire is a major concern for the nuclear safety due to potential severe
consequences of an uncontrolled fire on the surroundings of a nuclear
plant. Since more a twenty years, a research program addressing this
topic is in progress at the french ”Institut de Radioprotection et de
Sûreté Nucléaire” (IRSN). Within this framework, a computational code,
named ISIS, dedicated to the simulation of buoyant fire in a compartment
mechanically ventilated, is developed.
Physical models of this code are based on the Reynolds-Averaged
Navier-Stokes equations, supplemented by a two-equation closure for
turbulent flows and the eddy viscosity model. The turbulent production
term is adapted to cope with buoyancy effects. Combustion modeling
relies on classical eddy dissipation approaches and the flux-method
is employed to treat radiation exchanges. Both incompressible and
low Mach number flows are dealt with. For the numerical solution, a
fractional step algorithm has been developed. To ensure stability and
positivity of the discrete operators, the spatial discretization combines
mixed finite element for the Navier-Stokes equations and finite volumes
scheme for transport (advection-diffusion-reaction) equations.
For the verification of the code, a wide range of techniques is employed:
comparison to analytical solution for model problems, use of manu-
factured solution and comparison to benchmark result. We detail in
this paper particular application of each kind; in all cases, convergence
properties of the scheme are assessed.
Validation is now underway, and is based on the so-called building-block
approach. We shortly describe some of the obtained results, first for an
unit problem and then for a large-scale realistic experiment.
Key words : verification, validation, computational fluid dynamics, fire
simulationVerification and Validation of the ISIS CFD Code
1 Introduction
The use of computational engineering, and in particular computational fluid dynam-
ics, nowadays increasingly widespreads, a fundamental reason for this matter of fact
being that the capacities of the computers currently allow to simulate large scale
and complex systems evolving in time. In particular, safety studies addressing fire
propagation in a nuclear plant, a tunnel or a shopping center use more and more
simulation tools. This in turn motivates considerable research efforts in complex
phenomena modelling, to evenly increase the field of investigation of computational
applications.
In view of this central role played by computer predictions, a major questions rises,
namely the assessment of the reliability of the simulations. More precisely, the
problem posed is how to assess the degree of accuracy and validity of results given
by a computer code; this is the aim of the verification and validation process (V&V).
The Defense Modeling and Simulation Office (DMSO) of Department of Defense
(DoD), [5, 6, 7] were the leaders in the development of concepts and terminology
used in V&V. Nowadays, the most commonly referenced and agreed literature on
this topic is probably the work of Roache [20], of Oberkampf et al. [16] and the AIAA
guide [17]. Other works on verification and validation can be found in [2, 21, 22, 23].
Among the concepts clarification brought by these authors, the most important may
be the following basic definition for verification and validation:
- Verification: the process of determining that a model implementation accu-
rately represents the developer’s conceptual model and the solution of the
model.
- Validation: the process of determining the degree to which a model is an
accurate representation of the real world from the perspective of intended uses
of the model.
The verification and the validation of the ISIS CFD code are presented in this pa-
per, closely following the terminology introduced above. In the first two sections,
we briefly describe the numerics and physics of this simulation tool, then we give in
the two last sections examples of the verification and validation work up to now per-
formed. For the verification step, three tests are presented, which roughly cover the
range of techniques developed to this purpose: comparison to analytical solutions,
use of manufactured solutions and comparison to benchmark results. Concerning
validation, two cases are described: a simple unit problem, then the simulation of an
experiment addressing the fire behavior in a large-scale compartment of a nuclear
power plant.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 2Verification and Validation of the ISIS CFD Code
2 Numerical Methods
We address in this section the numerical solution of the following class of problems:
∂ρ
+ ∇ · ρv = 0 (1a)
∂t
∂ρv
+ ∇ · (ρv ⊗ v) = −∇p + ∇ · τ + (ρ − ρ0 )g (1b)
∂t
∂ρφ µe
+ ∇ · ρvφ = ∇ · ∇φ + Sφ (1c)
∂t σφ
ρ = G(φ, · · · ) (1d)
where t stands for the time, v for the fluid velocity, p for the dynamic pressure, and
ρ for the fluid density. The tensor τ represents either the viscous stress tensor or in
turbulent regime, the Reynolds stress tensor. The density is supposed to be given
explicitly as a function of the unknown field φ, and the volumic diffusion coefficient
µe /σφ is strictly positive. The parameter σ φ stands, here, for a Prandtl or a Schmidt
number. The problem is supposed to be posed over Ω, an open bounded connected
subset of Rd with d = 2 or d = 3. It must be supplemented by initial and boundary
conditions for v and φ.
Several physical problems enter this abstract framework. For instance, taking for φ
the temperature and for G(·) the equation of state of perfect gases evaluated at a
fixed pressure yield the asymptotic governing equations of natural convection flows
in the low Mach number limit. With φ equal to a concentration and a simple mixing
rule to evaluate the density, we recover the system of equation modelling solutal
convection of liquids. Last but not least, a model for the computation of diffusion
flames is obtained when considering φ as a Zeldovitch’s variable. The expression of
G(·) can then be derived by calculating the temperature and the concentrations of
the components as a function of φ and using the mixture equation of state. This
computation yields the density value provided that the pressure used in the equation
of state can be considered as constant, i.e. provided that we address once again low
Mach number flows. In fact, this system can be viewed as central for the ISIS code,
in the sense that the more complex models considered in practical applications can
be obtained by complementing the system (1) by transport equations, the numerical
solution of which closely follows the ideas developed for equation (1c) described
hereafter.
To design a numerical scheme for the solution of system (1), one is faced to at
least two difficulties. First, the unknown φ can be expected, from both physical and
mathematical reasons, to meet L∞ and L2 stability properties. This suggests to build
a finite volume numerical scheme which reproduces these features at the discrete
level. Second, as the density of the fluid can be a function of the variable φ, the
model at hand shares the mathematical properties of incompressible flow problems:
no independent evolution equation can be stated for the fluid density; instead, the
mass balance equation rather seems to act as a constraint on the velocity field which
determines the pressure. This justifies the use of fractional step schemes issued from
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 3Verification and Validation of the ISIS CFD Code
the incompressible flow numerics, namely (extension of) projection methods, based
on inf-sup stable pairs of velocity and pressure approximation spaces. Among these
last, nonconforming velocity approximations with degrees of freedom located at the
center of the faces seem to be well suited to a coupling with a finite volume method
for the advection-diffusion-reaction of φ. A fractional step schemes relying on these
basic ingredients, namely finite volumes for the computation of φ and finite element
projection scheme for the solution of the momentum and mass balance equation, is
presented below.
From a physical point of view, it seems natural for the field φ to satisfy a maximum
principle (for instance, both a concentration and a Zeldovitch’s variable must remain
in the [0, 1] interval). However, as the velocity field v is not divergence free, this
is not a direct consequence of equation (1c), but of the system (1a)-(1c). More
precisely, for any given regular velocity field v, a solution (ρ, φ) of (1a)-(1c), with a
positive initial condition for ρ and an initial condition for φ taking values in [0, 1],
takes its values in R?+ × [0, 1]. The purpose of the developments presented hereafter
is to design a numerical scheme which enjoys the same properties. The central role
played by the mass balance relation in both cases suggests the following three-step
implicit Euler algorithm:
ρ̃ − ρn
+ ∇U · ρ̃v = 0 (2a)
∆t
ρ̃φn+1 − ρφn
+ ∇U · ρ̃φn+1 v − ∆C φn+1 = Sφn+1 (2b)
∆t
ρn+1 = G(φn+1 , · · · ) (2c)
where the preceding relations must be understood in the fully discrete sense, with the
standard finite volume discretization for the time derivative term, ∇ U · (·) stands
for the upwind (with respect to the velocity) finite volume discretization of the
divergence operator and ∆C (·) is the usual centered discretization of the diffusion
terms (see [8] for a precise definition).
The first step of this algorithm is a prediction of the density, Eq. (2a). One has to
note, in particular, that in view of the global algorithm, this evaluation of ρ̃ differs
from a second order Richardson’s extrapolation only by the implicitation of ρ and
the upwinding of the divergence term.
A numerical scheme for the solution of the full system of equations is simply ob-
tained by complementing equations (2a)-(2c) by an incremental projection method.
Writing this algorithm in a time-semidiscrete setting, this yields the following five
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 4Verification and Validation of the ISIS CFD Code
step scheme:
ρ̃ − ρn
solve for ρ̃ + ∇U · ρ̃vn = 0 (3a)
∆t
ρ̃φn+1 − ρn φn
solve for φn+1 + ∇U · ρ̃φn+1 vn − ∆C φn+1 = Sφn+1 (3b)
∆t
ρn+1 given by ρn+1 = G(φn+1 , · · · ) (3c)
ρ̃ṽ − ρn vn
solve for ṽ + ∇ · (ρ̃vn ⊗ ṽ) − ∇ · τ (ṽ) + ∇pn = 0 (3d)
∆t
ρn+1 vn+1 − ρ̃ṽ
+ ∇(pn+1 − pn ) = 0
solve for p n+1
, v n+1 ∆t (3e)
ρn+1 − ρn
∇ · ρn+1 vn+1 = −
∆t
The momentum and mass balance equations are discretized by a mixed finite ele-
ment method based on a structured grid, using the so-called rotated bilinear element
for each velocity component and a piecewise constant pressure. The stability of this
element is proven in [18]. The implementation of projection methods for incom-
pressible flows, in particular with this element, is discussed in [24]. An extension of
the projection method to dilatable flows is presented in [12, 4]. It is to note that the
generalization of the scheme presented here to unstructured simplicial triangulations
appears to be straightforward, as the used finite volume method handle this type of
meshings and the rotated bilinear element can be replaced by the Crouzeix-Raviart
one [9, p.132].
For the discretization of the time derivative in the velocity prediction and projection
step, we use the standard trick which consists in lumping the mass matrix, i.e.
adding the contribution of the extra-diagonal terms of the diagonal and setting
them to zero. As a consequence, the discrete counterpart of the first equation of
(3e) yields directly un+1 and substituting the obtained expression in the second
relation yields a Poisson discrete problem for the pressure.
An upwind finite element discretization for the advective terms in equation (3d) is
implemented using the flux correction tools presented in [13].
As the approximate pressure and density are constant over each element, the dis-
cretization of the second relation of (3e) yields the same algebraic relation as a
centered finite volume discretization, which can be written:
ρn+1 − ρn
+ ∇C · ρn+1 vn+1 = 0 (4)
∆t
Since the density is updated after the computation of z n+1 , this scheme does not
conserve the quantity ρz. The conservativity property can be recovered, at the cost
of loosing the property ρn+1 = G(z n+1 ), by solving once again the balance equation
for z at the end of the time step; these schemes are described in [1].
3 Physical Modelling
The basic model of the ISIS code relies on a low Mach number formulation of the
Navier-Stokes equations, which is justified by the fact that buoyant fires are unani-
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 5Verification and Validation of the ISIS CFD Code mously described as low-speed flows. This model is obtained by formally developping the compressible Navier-Stokes equations as a function of the Mach number M and making M tends to zero. This process splits the pressure in a quantity constant with respect to the space variables, termed the thermodynamic pressure, and of order M −2 , and a variable part, named the hydrodynamic pressure, of order M 0 ; the fluid density then depends only on the thermodynamic pressure, which removes the compressibility effects on the flow. The turbulence modelling, for variable density flow, is based on the Favre average or density-weighted average technique. Then the Boussinesq’s eddy viscosity concept is employed to model the turbulent transport, which allows to express the turbulent shear stress as a function of the mean velocity gradients and a turbulent viscosity, for which an additional modelling is required. The scalars fluxes are modelled by the gradient diffusion assumption with turbulent Prandtl or Schmidt number. In fine, a k − ε model is employed to predict the turbulent viscosity, in which the turbulence production term is modified to take into account buoyancy effects. The turbulent combustion model, based on the conserved scalar approach, assumes a fast chemistry and relies on the eddy dissipation concept (EDC), which is an exten- sion of the eddy break up model (EBU), devoted to premixed turbulent combustion. Due to the wide range of physical problems addressed by the ISIS code, it is dif- ficult to go further in the description of the physical modelling in this section; in counterpart, we will provide a self-consistent presentation of each verification and validation case, including the set of solved conservation equations. 4 Verification tests We present in this section three numerical verification tests; following the termi- nology of reference documents [11, 17], they assess consistency and convergence properties of the fractional step schemes and spatial approximation schemes above described. These verification tests are chosen to cover the range of available tech- niques developed to this purpose: first a comparison with an analytical solution, the so-called Taylor-Green vortices, then a manufactured solution of a natural con- vection problem under the Boussinesq hypothesis and, finally, a natural convection benchmark problem in the low-Mach number regime. For each case, we first give the set of governing equations, then we precise the numerical method and, finally, we present the observed convergence results. 4.1 A comparison to an analytical solution Testing against analytical solutions, also called method of exact solutions, is per- haps the simplest mean to perform code verification. The major limitation of this technique is that governing equations admitting an analytical solution are usually obtained for physical problems much simpler than the real situation of interest, often leaving the physical phenomena uncoupled. ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 6
Verification and Validation of the ISIS CFD Code
A - Problem description
This test case addresses on the two dimensional Taylor-Green vortices, which is a
widely used benchmark problem for Navier-Stokes solvers verification. The com-
putational domain is a unit square [0,1] x [0,1] and the set of partial differential
equation considered here is:
∇·v =0 (5a)
∂v
+ v · ∇v = −∇p + ν∆v (5b)
∂t
where ν is the kinematic viscosity, defined as the ratio of the viscous force to the
inertial force. This parameter is constant in this test: ν = 0.01 m 2 /s.
This system admits the following analytical solution:
− cos(πx) sin(πy) exp(−2π 2 νt)
v(x, y, t) =
sin(2πx) cos(2πy) exp(−2π 2 νt)
p(x, y, t) = −0.25 [cos(2πx) + cos(2πy)] exp(−4π 2 νt)
provided that boundary and initial conditions are consistent with these expressions.
We choose here to prescribe the velocity on the whole boundary of the computational
domain.
B - Numerics
Uniform meshes are considered for this test case with resolutions of 10 × 10, 20 × 20,
40 × 40, 80 × 80 and 320 × 320. Numerical parameters and main features of the
numerical scheme are gathered in the following table.
Initial and final time 0, 1
Time-step ∆t = 0.1/n with n = 1, 2, 4, 8, 16, 32
semi-implicit fractional step scheme, obtained from
Solution algorithm
equations (3d-3e) by setting ρ = 1
Time discretization first (backward Euler) and second order (BDF 2) scheme
Spatial discretization no discrete upwinding
C - Results
Spatial and temporal convergence for velocity and pressure are presented in Fig. 1
measured by the L2 error norm, defined as:
sZ
L2 (φ) = (S(x) − φh (x))2 (6)
Ω
where S(x) refers to the analytical solution and φ h to the computational solution.
Regarding, first, the spatial convergence, (Fig. 1a), the L 2 norm of the error evolu-
tion as a function of the mesh step size for velocity and pressure are in agreement
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 7Verification and Validation of the ISIS CFD Code
with the theoretical results known for the considered spatial approximation scheme:
second order convergence for the velocity, fist order one for the pressure. Temporal
convergence is more difficult to check for the coarsest meshing because the spatial
error is more important for this verification case than the temporal discretization
error. With a 320 × 320 grid and the BDF2 time discretization, an order two time
convergence is recovered, Fig. 1b.
0.1 0.001
velocity
pressure
Velocity L error norm
1e-04
L2 error norm
0.01
2
1e-05
0.001 1e-06
0.01 0.1 0.01 0.1
Grid spacing Time-step
(a) (b)
Figure 1: (a): Velocity and pressure L2 error norm vs. grid spacing; (b): Velocity L2 error norm
vs. the time-step.
4.2 An example of use of the manufactured solutions technique
The method of manufactured solutions is a more general approach for code verifi-
cation. Owing to the fact that the principal disadvantage of the methods of exact
solutions is to find an analytical solution to a given complex system of partial dif-
ferential equations, an alternative consists in building an analytical source term by
applying the considered partial derivation operators to a given solution. The man-
ufactured solution is then the exact solution of the partial differential equations
plus the analytical source terms. This method was first formalized by Roache and
Steinberg [19] and a recent review is performed in [21]. A broad overview of code
verification procedures based on the manufactured solution technique can also be
found in [22].
A - Problem description
We address in this section a natural convection problem, within the Boussinesq
assumption framework. In this test, the computational domain is the unit square
and Navier-Stokes equation are supplemented by a balance energy equation:
∇·v=0 (7a)
∂v 1 2 Ra
+ (v · ∇)v = −∇p + ∇ v+ θk + Sv (7b)
∂t Re P rRe2
∂θ 1
+ v · ∇θ = ∇2 θ + S θ (7c)
∂t P rRe
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 8Verification and Validation of the ISIS CFD Code
The unknown fields v, p and θ stand respectively for the dimensionless velocity
vector, pressure and temperature and the parameters Re, Ra and P r are respectively
the Reynolds, Rayleigh and Prandtl numbers, which take the following constant
values: Re = 100, Ra = 5000 and P r = 0.7. The vector k is the unit vector in the
vertical direction.
A two-dimensional manufactured solution for the steady problem is derived from
the 3D solution proposed in [3], by choosing the velocity, pressure and temperature
as follows:
[1 − cos(2πx)] sin(2πz)
v0 =
sin(2πx)[cos(2πz) − 1]
p0 = sin(πx + π/2) sin(πz + π/2)
θ0 = 1 − z + x(1 − x)z(1 − z)
and building the source terms Sv and Sθ from the expression of (v0 , p0 , θ0 ):
∂v0 1 2 Ra
Sv = + (v0 · ∇)v0 − ∇p0 − ∇ v0 − θ0 k
∂t Re P rRe2
∂θ0 1
Sθ = + v0 · ∇θ0 − ∇2 θ0
∂t P rRe
Boundary conditions (Dirichlet) are provided by the analytical manufactured solu-
tion.
We suppose that the solution of the transient problem with these constant-in-time
source terms and boundary condition tends toward the steady solution (v 0 , p0 , θ0 ),
irrespectively of the initial condition, and we then arbitrarily set the initial velocity
and temperature at zero: v(x, y) = θ(x, y) = 0.
B - Numerics
Uniform meshes are considered for this test case with resolutions of 10 × 10, 20 × 20,
40 × 40, 80 × 80 and 160 × 160. Numerical parameters and main features of the
numerical scheme are gathered in the following table.
Initial time 0
Time step ∆t = 0.1
The computation is stopped when the L 2 norm of the
difference between two consecutive time steps of both
Final time
the velocity and the temperature, normalized by the
L2 norm of the solution, falls below < 10 −5 ∆t
semi-implicit fractional step scheme, obtained from
Solution algorithm
equations (3b, 3d-3e) by setting ρ = 1
Time discretization first order (backward Euler) scheme
upwinding or hybrid approximation for the convective
Spatial discretization terms of the energy balance, no discrete upwinding for
Navier-Stokes equations
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 9Verification and Validation of the ISIS CFD Code
C - Results
For the finite volume approximation of the temperature, we define the discrete L 2D
error norm by: sZ
L2D (φ) = (Sh (x) − φh (x))2 (9)
Ω
where Sh (x) refers to the projection of the analytical solution in the finite volume
discrete space (i.e. piecewise constant functions over each control volume) and φ h
to the computational solution.
The behavior of the L2 error norm for velocity and pressure and of the discrete L 2
error norm for temperature as a function of grid spacing is reported in Fig. 2, for
the upwind (Fig. 2a) and hybrid (Fig. 2b) approximation of the convective terms
in the energy balance.
With logarithmic scales, the upwind scheme exhibits a slope of unity (first order
convergence) whereas the slope is two (second order convergence) for the hybrid
approximation scheme. Results are thus in agreement with theoretical expectations,
since formal convergence order is recovered.
0.1 0.1
0.01
Error norm
Error norm
0.01
0.001
0.001
1e-04
velocity L2 error norm velocity L2 error norm
pressure L2 error norm pressure L2 error norm
temperature L2D error norm temperature L2D error norm
1e-04 1e-05
0.01 0.1 0.01 0.1
Grid spacing Grid spacing
(a) (b)
Figure 2: Error norm for velocity, pressure and temperature. (a): upwind approximation scheme,
(b): second order approximation scheme for the transport equation of the reduced temperature.
4.3 A comparison to a benchmark solution
Benchmark solutions are solution to physical problems which may be considered as
accurate and reliable, because obtained and published by a wide panel of researchers
using different and often high-performance numerical methods [17]. However, these
verification tests, in most cases and up to now, are in two dimensions and essentially
address laminar and incompressible flows.
A - Problem description
This test case is concerned with a differentially heated square cavity at high Rayleigh
number with large temperature differences. Because of this last aspect, the Boussi-
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 10Verification and Validation of the ISIS CFD Code
nesq approximation is no longer valid and the use of a low Mach number formulation
seems to be justified considering the low-level of the fluid velocity. Performances of
the ISIS code in term of accuracy and grid convergence are compared to a reference
solution recently given by Le Quéré et al. [14]. The computational domain is a
two-dimensional square enclosure [0,L] × [0,L], as sketched in Fig. 3. The left and
right vertical walls are respectively heated to T h and cooled down to Tc , and a zero
heat flux is imposed at the two horizontal walls. A no-slip condition is prescribed
for the velocity field v.
v=0
Th g Tc
v=0 v=0
v=0
L
Figure 3: Configuration of thermally driven cavity.
The governing equations for this test case read:
∂ρ
+ ∇ · ρv = 0 (10a)
∂t
∂ρv
+ ∇ · (ρv ⊗ v) = −∇p + ∇ · τ + (ρ − ρ0 )g (10b)
∂t
∂ρh dPth µ
+ ∇ · ρvh = +∇· ∇h (10c)
∂t dt Pr
Pth
ρ= (10d)
RT
where t stands for the time, v for the fluid velocity vector, p for the dynamic pressure,
and ρ for the fluid density. The tensor τ is the viscous part of the stress tensor for
a Newtonian fluid, given by the following expression:
t 2
τ = µ ∇v + ∇ v − (∇ · v) I
3
The dynamic viscosity µ follows the Sutherland’s law:
3
µ(T ) T 2 T∗ + S
=
µ∗ T∗ T +S
where T ∗ = 273 K, S = 110.5 K and µ∗ = 1.68 × 10-5 kg/(m.s). In the energy
balance equation (10c), the enthalpy h is defined as h = c p (T − T0 ), with T0 a
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 11
Verification and Validation of the ISIS CFD Code
reference temperature and cp the specific heat. The Prandtl number, P r, is assumed
to remain constant with respect to the temperature and is defined as:
µ(T )
P r = cp
λ(T )
where λ(T ) is the thermal conductivity. In the low Mach approximation used here,
the thermodynamic pressure, Pth , is constant in space and the dynamic part of
pressure is neglected in the equation of state (10d). The mass conservation principle
yields the following additional equation, which is used to compute the evolution of
the thermodynamic pressure:
Z
1
Pth (t) = m0 dx
Ω RT (x, t)
where m0 is the initial mass.
This problem involves two main parameters, the Rayleigh number Ra and the tem-
perature difference parameter ε defined by:
gρ20 (Th − Tc )L3
Ra = P r = 106 (11)
T0 µ20
and
Th − T c
ε= = 0.6 (12)
Th + T c
where ρ0 = ρ(T0 , P0 ) and µ0 = µ(T0 ) describe the density and shear viscosity at
a given reference temperature T0 and thermodynamic pressure P0 . Heat transfers
to the hot and cold wall are represented by the local and average Nusselt numbers,
respectively defined by:
L ∂T 1 y=L
Z
N u(y) = k |w , N u = N u(y)dy (13)
k0 (Th − Tc ) ∂x L y=0
where k is the thermal conductivity, defined by k(T ) = c p µ(T )/P r, k0 = k(T0 ).
At steady-state and with the prescribed boundary conditions, integration of the
energy equation over the volume of the square cavity implies that the average Nusselt
numbers on the two vertical walls are equal:
N uh = N u c (14)
where subscripts h and c refers to the hot and cold wall.
The parameters of the simulation are:
- Gas constant: R = 287 J/(kg.K)
- Ratio of specific heat: γ = 1.4
- Specific heat: cp = γR/(γ − 1) J/(kg.K)
- Prandtl number: P r = 0.71
- Gravity: gy = −9.81 m/s2
- Reference temperature: T0 = 600 K
- Reference pressure: P0 = 101325 Pa
Spatially uniform initial conditions are imposed: ∀(x, y) ∈ [0, L] 2 , T (x, y) = T0 and
v(x, y) = 0. The prescribed temperature on vertical walls are T h = T0 (1 + ε) and
Tc = T0 (1 − ε).
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 12Verification and Validation of the ISIS CFD Code
B - Numerics
Various uniform grid size are used, obtained by successive refinements. Numerical
parameters and main features of the numerical scheme are gathered in the following
table.
Initial and final time 0 s, 50 s
Time step ∆t = 0.05 s
semi-implicit fractional step scheme, obtained from
Solution algorithm equations (3a-3e), where the variable φ is to be
understood as the temperature
Time discretization first order (backward Euler) scheme
hybrid approximation for the convective terms of the
Spatial discretization energy balance, no discrete upwinding for Navier-
Stokes equations
C - Results
One of the outputs of the benchmark was a reference value for the Nusselt number
and the thermodynamic pressure at steady state, commonly agreed up to 5 digits:
N uh = 8.6866, P/P0 = 0.9244
These results, obtained for different grid size, are reported in table 1 with the relative
error between the computational solution and the reference solution. Fig. 4 shows
an approximately second order convergence for the Nusselt number.
#dofs N uh EN u h Pth /P0 EPth /P0
1 600 8.4087 3.19e-2 0.9571 3.54e-2
6 400 8.6099 8.83e-3 0.9335 9.89e-3
25 600 8.6641 2.58e-3 0.9270 2.6e-3
57 600 8.6646 1.24e-3 0.9257 1.40e-3
102400 8.6801 7.39e-4 0.9251 8.64e-4
160 000 8.6783 4.89e-4 0.9249 5.99e-4
313 600 8.6822 2.59e-4 0.9247 3.60e-4
Table 1: Mean Nusselt number and thermodynamic pressure obtained with different grid sizes.
5 Validation tests
Validation is defined as the process of determining the degree to which a model is
an accurate representation of the real world from the perspective of the intended
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 13Verification and Validation of the ISIS CFD Code
1
Relative error on the mean Nusselt number
0.1
0.01
0.001
1.e-04
1.e-04 0.001 0.01
Grid spacing
Figure 4: Relative error on the mean Nusselt number function of the grid spacing.
uses of the model in reference documents [11, 17]. Main concepts interacting in the
validation pocess are largely clarified in [17] and this discussion is far beyond the
scope of the present paper. Instead, we focus here on the implementation of the so-
called ”validation phases method” for the fire simulation ISIS code. This method,
also called the building-block approach [17, 16], consists in breaking up the complex
engineering system of interest in several sub-systems of less complexity. Generally
speaking, this simplification process generates three levels of decreasing complexity:
into the first class, corresponding to the highest level of complexity, fall the so-
called ”sub-system cases”; the second class gathers ”benchmark cases” and the last
one ”unit problems”. Simplification may consist of uncoupling physical phenomena,
adressing steady state cases instead of transient ones, approximating boundary con-
ditions by simpler ones, idealizing the geometry . . . For each sub-system, benchmark
case and unit problem, code results will have to be compared to experimental data
or published reliable solutions.
The validation process may include a calibration work, a definition of which can be
found in [17, section 4.2]:
”Calibration: the process of adjusting numerical or physical modelling param-
eters in the computational model for the purpose of improving agreement with
experimental data.”
For the particular topic of fire simulation, the main concerned physical models are
the turbulence and reaction rate modelling. As far as possible, the calibration phase
has to be gone through first, as any change in a basic modelling may make another
iteration of at least a part of the validation process necessary.
The validation process of the ISIS code has just begun at IRSN, and will be pursued
over next years, together with the realization of supporting experimental programs.
An example of what could be a building-block approach developed to this purpose
is sketched on Fig. 5; of course, the choice of the different blocks of each three tiers
is not unique. We restrict here the presentation to two validation works. The first
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 14Verification and Validation of the ISIS CFD Code
one falls in the class of unit problems; in the second one, we address a full-scale
experimental fire in a compartment mechanically ventilated.
Fire simulation in a
compartment mechanically
ventilated
buoyancy flows in
steady state transient buoyancy
mechanically ventilated
room fires reactive flows
compartments
natural convection of room fires with buoyant turbulent
turbulent buoyancy flows a volumetric heat source non-premixed flames
incompressible turbulent jet
turbulent flows diffusion flames
Figure 5: Validation phases
5.1 A unit problem
Turbulent flows over backward facing steps are widely used benchmark problems to
evaluate the performance of turbulence models in the prediction of separated flows.
This test focuses on the specific case of the incompressible flow over a backward-
facing step at a Reynolds number of 44 500, based on the step height and the flow
velocity uref just before the step. Under these conditions, the flow everywhere may
be taken as fully turbulent.
A comparison of various commercial CFD code on this benchmark problem is de-
scribed in [10], together with main experimental results. Code benchmarking sup-
plements the code-to-experience comparison with respect to two aspects. First,
the comparison between CFD codes allows to identify slight model variants (small
change of parameters, different implementation of boundary conditions,...) which
may be difficult to uncover in models presentations in the literature and codes doc-
umentation, while strongly modify the results. In addition, provided that model
identity can be checked, it yields an additional verification test (i.e. checking the
correctness of the numerical solution of the model set of PDE).
A - Problem description
The computational domain is presented in Fig. 6.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 15Verification and Validation of the ISIS CFD Code
wall
inlet h1
outlet
l h
wall
L
Figure 6: Backward facing step configuration. h = 3.8096 cm, L = 76.2 cm, l = 19.054 cm, and
h1 = 7.6204 cm.
The flow is governed by the following set of balance equations:
∇·v =0 (15a)
∂ρv
+ ∇ · (ρv ⊗ v) = −∇p + ∇ · τ (15b)
∂t
∂ρk µe
+ ∇ · ρvk = ∇ · ∇k + Gk − ρε (15c)
∂t σk
∂ρε µe ε
+ ∇ · ρvε = ∇ · ∇ε + (cε1,1 Gk − cε2 ρε) (15d)
∂t σε k
where µe = µ + µt is the effective viscosity. The eddy viscosity model is used to
express the turbulent viscosity as: µ t = ρCµ k 2 / and the Reynolds stress tensor, τ ,
in the momentum equation is modelled with the Boussinesq hypothesis:
2
τ = µe ∇v + ∇t v − ρkI
3
The generation of turbulent kinetic energy results of the interaction of the turbulent
shear stress and mean velocity gradients:
Gk = τ ⊗ ∇v
The model constants are set to the following standard values:
σk = 1., σε = 1.3, cµ = 0.09, cε1,1 = 1.44, cε2 = 1.92
Fluid density and viscosity are respectively:
ρ = 1.0 kg/m3 , µ = 1.101 × 10−5 kg/m/s.
Boundary conditions are set as follows:
- inlet : u = 13. m/s, v = 0, k = 0.7605 m2 /s2 , ε = 31.78 m2 /s3 .
- outlet: homogeneous Neumann condition for u, v, k, and ε.
- all walls: turbulent wall functions.
B - Numerics
We use the same grid as the reference data [10], which is rather coarse (2734 meshes).
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 16Verification and Validation of the ISIS CFD Code
The steady state is approached by a transient computation, starting from the fol-
lowing initial conditions:
v(x) = 0, k(x) = 0.01, and ε(x) = 0.01
Numerical parameters and main features of the numerical scheme are gathered in
the following table.
Initial and final time 0., 1.
Time step ∆t = 0.001
semi-implicit fractional step scheme, obtained from
equations (3b, 3d-3e), where the variable φ stands for
Solution algorithm the turbulent kinetic energy k and its dissipation rate
ε; implicit coupling between these two equations is
obtained by a fixed point algorithm
Time discretization first order (backward Euler) scheme
upwinding approximation for the convective terms of
Spatial discretization
the k − ε and Navier-Stokes equations
C - Results
Velocity profiles
In Fig. 7 and Fig. 8, a good agreement can be observed between results obtained
with the ISIS code and reference data (Ref represents average results obtained with
SMARTFIRE, PHOENICS and CFX).
0.12
ISIS
Ref
0.1
Y Displacement
0.08
0.06
0.04
0.02
0
-4 -2 0 2 4 6 8 10 12 14
U Velocity
Figure 7: Velocity profiles 0.285 m downstream of inlet.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 17Verification and Validation of the ISIS CFD Code
0.12
ISIS
Ref
0.1
Y Displacement
0.08
0.06
0.04
0.02
0
0 2 4 6 8 10 12
U Velocity
Figure 8: Velocity profiles at the outlet.
Reattachment length
The reattachment length is determined by locating the region where the stream-
wise velocity component is changing from positive to negative (i.e., u =0.0) along
the lower duct wall. Results of the benchmark [10] obtained with SMARTFIRE,
PHOENICS and CFX are reported below, Table 2. S represents the reattachment
length from the step. The reduced experimental value is S/h = 7.2.
Reat. length SMARTFIRE PHOENICS CFX ISIS
S 0.2217 0.2587 0.1967 0.2124
S/h 5.82 6.79 5.16 5.57
Table 2: Prediction of reattachment length for CFD codes
The code predicts the reattachment length with a relative error of 22.5 %. This is
coherent with the fact that the standard k − ε model with wall functions is known
to underpredict the reattachment length in the backward facing step by an amount
in the order of 20-25 %.
5.2 A complex system
We address in this section the simulation of a real scale experimental fire [15]. This
test has been performed at IRSN, as a part of an experimental program performed
to provide data for the validation of computational tools simulating fires in mechan-
ically ventilated compartments, with first application to nuclear power plant.
This test turns out to be particularly difficult, for essentially two reasons. The first
one is that the large scale geometry of the studied problem as well as the duration of
the transient of interest make the computational requirements considerable; our first
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 18Verification and Validation of the ISIS CFD Code
concern will then be to assess the code stability and convergence for such systems.
Second, the flow results from an intricate coupling between non-linear phenomena,
as turbulence, combustion and buoyancy effects; separate validation of each single
model will then be clearly out of reach, and we rely for this purpose on the previously
described building-block approach. In the same direction, note in addition that the
knowledge of initial and boundary conditions, together with the characterization of
the flow are necessarily less comprehensive than in experiments carried out at the
laboratory scale, which even reinforces the interest of validating each ”elementary”
model using simpler experiments.
A - Problem description
The experiment consists in a confined ethanol pool fire in a compartment mechan-
ically ventilated with a metal cupboard close to the fire. The schematic diagram
of the compartment fire is shown in Fig. 9. The dimensions are for the x, y and
z directions, respectively, Lx = 9 m, Ly = 6 m and Lz = 7.5 m. The different
walls, the floor and the ceiling are 0.25 m thick concrete walls. The compartment is
connected to a ventilation network including a forced ventilation supply inlet and a
forced ventilation exhaust vent (Fig. 9) with respectively dimensions of 0.3 m 2 and
0.4 m2 . The ventilation rate is 5 h-1 and the depression -200 Pa. The pool fire is a
square of surface 1 m2 and height 0.13 m, located at the center of the compartment.
The fire heat release rate, defined as the product of the fuel mass loss rate and the
heat of combustion of ethanol reaches 563 kW during the stationary combustion
phase, Fig. 10.
exhaust vent
7.5 m
cupboard
6m
supply inlet
9m
Figure 9: Experimental fire case geometry.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 19Verification and Validation of the ISIS CFD Code
0.03
Fuel mass loss rate (kg/(s.m2)
0.025
0.02
0.015
0.01
0.005
0
0 500 1000 1500 2000 2500
Times (s)
Figure 10: Fuel mass loss rate in the experimental fire case.
The system describing the turbulent reactive flow in the low Mach number regime is
presented below. The approximation turbulence resort to the mass-weighted averag-
ing, also called the Favre averaging. A modified k −ε model based on the Boussinesq
hypothesis and the eddy viscosity model is used for turbulence closure. To model
the turbulent combustion process, we use a fast chemistry assumption and the con-
served scalar approach; we keep as unknowns variables the mixture fraction variable
z and the fuel mass fraction Yf . Removing for short in the notations the Favre or
Reynolds turbulence averaging operators, governing equations read:
- mass balance:
∂ρ
+ ∇ · ρv = 0
∂t
- momentum balance:
∂ρv
+ ∇ · (ρv ⊗ v) = −∇p + ∇ · τ + (ρ − ρ0 )g
∂t
- turbulent kinetic energy balance:
∂ρk µe
+ ∇ · ρvk = ∇ · ∇k + Gk + Gb − ρε
∂t σk
- viscous dissipation balance:
∂ρε µe ε
+ ∇ · ρvε = ∇ · ∇ε + (cε1,1 Gk + cε1,2 Gb − cε2 ρε)
∂t σε k
- enthalpy balance:
∂ρh µe dPth
+ ∇ · ρvh = ∇ · ∇h +
∂t σh dt
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 20Verification and Validation of the ISIS CFD Code
- mixture fraction balance:
∂ρz µe
+ ∇ · ρvz = ∇ · ∇z
∂t σz
- fuel mass fraction balance:
∂ρYF µe
+ ∇ · ρvYF = ∇ · ∇YF + ω̇F
∂t σ YF
The Reynolds stress tensor, τ , appearing in the momentum equation, is expressed
as:
t 2 2
τ = µe ∇u + ∇ u − (∇ · u) I − ρkI
3 3
Turbulent production terms are defined as:
µt
Gk = τ ⊗ ∇v, Gb = ∇ρ · g
ρσg
where the term Gb stands for the generation and destruction of turbulence due
to buoyancy forces. In multicomponent mixtures, the density of the mixture is
evaluated by:
N
Pth W 1 X Yk
ρ= , with =
Ru T W Wk
k=1
where W is the mean molar weight of the mixture and Y k and Wk respectively stand
for the mass fraction and the atomic weight of species k (i.e. fuel, . . . ). The fuel
burning rate is calculated according to:
YO
ω̇F = −CEBU ρ min YF ,
k s
where CEBU is a model constant commonly taken of the order of four but which can
be modelled by a viscous mixing model; here, we use the first option: C EBU = 4.
To deal with radiative losses, we use the so-called Markstein model, so the specific
enthalpy is linked to the temperature by the following relation:
h = cP (T − T0 ) + ∆Hc (1 − χr )YF
where T0 is a reference temperature, ∆Hc is the heat of combustion and χr is the
fraction of energy of combustion lost by radiative transfer; χ r is set to 0.25 in this
simulation. The model constants have the following standard values:
cµ = 0.09, cε1,1 = 1.44, cε2 = 1.92, cε1,2 = 1.44,
σk = 1., σε = 1.3, σh = σz = σYF = 0.71
The thermodynamic pressure of the room is computed by solving a simplified (0
D) momentum balance equation for the system composed of the confined compart-
ment and the ventilation network. In this modelling, a Bernoulli general equation
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 21Verification and Validation of the ISIS CFD Code
describes each branch i of the network, which is, in this particular case, connected
to the compartment:
Li ∂Qi
= Pth − Pnode,i − f
Si ∂t
where Qi is the flow rate in the branch i, Pnode,i is the pressure at extremity of
the branch which is not located at the compartment wall and f is an aerodynamic
resistance. The geometrical dimensions L i and Si are respectively the length and
the surface of the branch i. This system must be supplemented by the overall mass
balance equation of the compartment:
X
∂ Pth W
Z
+ Qi = 0
Ω ∂t RT
i
Geometrical and material properties are gathered in the following tables.
- gas mixture:
dynamic viscosity µ = 1.68 × 10−5 kg/(m.s)
thermal conductivity λ = 0.018 W/(m.K)
specific heat cP = 1100 J/(kg.K)
Prandtl number P r = 0.71
heat of combustion ∆Hc = 2.56 × 107 J/kg
- concrete walls:
density ρw = 2430 kg/m3
thermal conductivity λw = 1.5 W/(m.K)
specific heat cP,w = 736 J/(kg.K)
walls thickness ew = 0.25 m
- metal cupboard:
density ρc = 7801 kg/m3
thermal conductivity λc = 43 W/(m.K)
specific heat cP,c = 473 J/(kg.K)
walls thickness ec = 0.25 m
Initial conditions are given by:
v = p = h = z = YF = 0,
k = 10−12 m2 /s2 , ε = 10−9 m2 /s3 , T = 290.K, ρ = ρair
To define the boundary conditions, three different surfaces are considered:
- fire:
k 3/2
v = (0, 0, wF ), h = ∆Hc (1−χr ), z = YF = 1, k = 0.1wF2 , ε = Cµ
lε
where wF is function of the fuel mass loss rate, w F = ṁF /ρF and lε ∼ 0.07L
with L a characteristic length scale, equivalent to fire radius.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 22Verification and Validation of the ISIS CFD Code
- walls: conduction in walls is accounted for in the energy balance equation.
A log-law wall function is used for the momentum and turbulence balance
equations.
- supply inlet and exhaust vent: a prescribed velocity is computed to match the
flow rate in each branch; fuel is supposed to remain in the compartment.
B - Numerics
Varous non-uniform grid are tested with 8 500, 68 000 and 240 000 meshes named
thereafter mesh1, mesh2 and mesh3. Numerical parameters and main features of
the numerical scheme are gathered in the following table.
Initial time 0., 1.
Final time 2800 s for mesh1, 1000 s for mesh2 and 250 s for mesh3
Time step ∆t = 0.1 s for mesh1 and mesh2, and 0.05 s for mesh3
semi-implicit fractional step scheme implemented for
transport equations similar to (3e) for k, ε, h, z and
Solution algorithm Yf variables; implicit coupling between the k − ε two
equations and the two transport equations for z and
YF is obtained by a fixed point algorithm
Time discretization first order (backward Euler) scheme
upwinding approximation for the convective terms of
Spatial discretization
the k − ε and Navier-Stokes equations
C - Results
Thermodynamic pressure and mass flow rate
The evolution of the thermodynamic pressure in the compartment fire as a function
of time (see Fig. 11) is in a good agreement with the experiment data. During the
combustion stationary phase, the calculated pressure oscillates around -2 hPa and
the amplitude of the oscillations is less strong than in the experiment. A strong
overpressure at ignition and a weak depression at extinction are observed in both
cases.
The evolution of mass flow rate at the supply inlet according to time is shown in Fig.
12. It stabilizes around 0.7 kg/s after a short period corresponding to the ignition
phase. The mass flow rate at exit, not shown here, is almost identical. These results
lie in the uncertainty range of the experimental data.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 23Verification and Validation of the ISIS CFD Code
0
ISIS
Exp.
-1
Pth -Pref (hPa)
-2
-3
0 500 1000 1500 2000
Time (s)
Figure 11: Thermodynamic pressure vs. time.
0.8
Mass flow rate (kg/s)
0.6
0.4
ISIS
Exp.
0.2
0 500 1000 1500 2000 2500
Time (s)
Figure 12: Mass flow rate at admission vs. time.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 24Verification and Validation of the ISIS CFD Code
Temperature in the plume region
The evolution of temperature with time in the plume region, 3 meters above the fire,
obtained with mesh1 is presented in Fig. 13. for the whole transient and for the
first 250 s in Fig. 14. After the time t = 250 s, both the computed and experimental
temperature oscillate around similar values until the end of fire. The predicted peak
of temperature in the early transient, not observed in experiments, could be due
to the absence of models in the code to simulate combustion in the transition from
the laminar to the turbulent flow regime, as occurs at ignition. Fig. 14 points out
strong temporal variations of temperature corresponding to a rotation of the flame
around its axis. This flame rotatory behavior is also observed in experiments with
an almost equal frequency but with an amplitude of temperatures variations twice
stronger.
1000
ISIS
Exp.
750
T-T0
500
250
0
0 500 1000 1500 2000 2500
Time (s)
Figure 13: Temperature vs. time to 3 meters above the fire.
As far as numeric are concerned, the code shows quite satisfactory stability prop-
erties, allowing, in particular, the limitation of the time step to be based only on
accuracy considerations. As expected, spatial convergence cannot be considered as
achieved, even for the finer mesh; qualitative results however remain similar. Studies
are ongoing to overcome this latter limitation, by extensive use of parallelism and
development of more accurate spatial discretizations.
ISO/TC 92/SC4 Workshop on Assessment of Calculation Methods in FSE 25Verification and Validation of the ISIS CFD Code
300
T-T0
200
ISIS
Exp.
100
500 1000 1500
Time (s)
Figure 14: Temperature vs. time to 3 meters above the fire.
6 Conclusion
We have presented in this paper some verification and validation tests of the ISIS
CFD code, following the strategy recommended by the reference ISO document [11],
DoD [6, 7], Roache [21] and the AIAA guide [17].
Three verification cases are presented, with the purpose of illustrating the main tech-
niques developed for the purpose of code verification, i.e. comparison to analytical,
manufactured and benchmark solutions.
The validation of the ISIS code is now underway, following the strategy of the so-
called building-block approach. Two preliminary validation tests are presented. The
first one falls in the class of unit tests problems, as the second one address a real-
scale realistic experiment. The complexity of this last exercise clearly highlights the
interest of the chosen progressive approach.
Acknowledgements
The practical implementation of the ISIS CFD code is based on the software object-oriented
component library PELICANS, developed at IRSN. The authors would like to thank B.
Piar and D. Vola for their support, in particular concerning the use of this computational
platform, and for useful discussions. We also thank E. Garnier and M. Jobelin for their
contributions to this work, in particular for the LIC1.14 validation case.
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