fzd theory seminar series cfd simulations for single and

Institute of Safety Research � FWSS � Dr. Thomas Höhne � www.fzd.de � 19.03.2010 1

Nuclear Safety Research

Thomas HThomas Hööhnehne

FZDFZDDresdenDresden--Rossendorf, GermanyRossendorf, Germany

CFD simulations for CFD simulations for single and multisingle and multi--phase phase flowsflows

FZD Theory Seminar SeriesFZD Theory Seminar Series


Nuclear Safety ResearchWhat is CFD?What is CFD?

� CFD (Computational Fluid Dynamics) is the simulation of fluids

engineering systems using modeling (mathematical physical

problem formulation) and numerical methods (discretization

methods, solvers, numerical parameters, and grid generations,

etc.)

� CFD made possible by the advent of digital computer and

advancing with improvements of computer resources (500

Floating Point Operations per Second (flops), 1947

�1 Petaflops, 2009)

Jugene am FZ Jülich


Nuclear Safety ResearchWhere is CFD used?Where is CFD used?

Where is CFD used?

– Aerospace

– Automotive

– Biomedical

– Chemical

Processing

– HVAC

– Hydraulics

– Marine

– Oil & Gas

– Power

Generation

– Sports

Aerospace

Power Generation

Automotive

Sports


Nuclear Safety ResearchCFD ModelingCFD Modeling

�CFD Modeling is the mathematical physics problem formulation in the form of Partial Differential

Equations (PDEs) with appropriate boundary conditions and initial conditions.

�Modeling includes:

1. Geometry and domain

2. Coordinates

3. Governing equations

4. Flow conditions

5. Initial and boundary conditions

6. Selection of models for different applications


Nuclear Safety ResearchCFD Methods - Definition

• Basis � Continuum

mechanics

• Conservation laws for– Mass

– Momentum

– Energy

– Species concentration

– …

• Equation characteristics– Geometry independent

– Galilean invariant

– Boundary conditions


Nuclear Safety ResearchCFD & System Codes

• System codes

• Geometry & flow approximation: 1-D

• Less computing time

• Less computer memory

• Larger assemblies �whole systems

• Increased empirical input

• Application dependent

• CFD codes

• Geometry & flow field resolution

• High computing times

• Large computer memory

• Smaller assemblies �

components

• Reduced empirical input

• Application ‘independent’



Geometry ModellingGeometry Modelling


Nuclear Safety ResearchGeometry Geometry ModellingModelling

� Simple geometries can be easily created by few geometric parameters (e.g. circular pipe)

� Complex geometries must be created by importing the database of the geometry(e.g. airfoil) into commercial software

� Typical approaches

�Geometry approximation

�CAD/CAE integration:

use of industry standards

such as Parasolid, ACIS,

STEP, or IGES, etc.


Nuclear Safety ResearchGrid Generation

• Geometry & grid generation:– Time-consuming

– Labour-intensive � expensive

• CFD result quality &

computing times � strongfunction of grid quality

• Grid generation process:– High intellectual & cognitive

demands �

– Difficult to ‚automate‘

• ‚Critical Path‘ for closer

integration of CFD in designand optimization procedures


Nuclear Safety ResearchGrid Requirements

• Grid widths sufficiently small

� Discretisation errorssufficiently small (accuracy)

• Grid widths sufficiently large

� Computer memory & computing time limitations

• Grid point arrangement �Minimisation of discretisation

errors

• Discretisation errors: Difference between

numerical and exact solution � infinitely fine grid

h h exe f f= −

-0,300

-0,200

-0,100

0,000

0,100

0,200

0,300

0 5 10 15 20 25 30 35 40

x/H, -

Skin

fri

cti

on

co

eff

icie

nt,

-

Grid 1

Grid 2

Grid 3

Grid 4


Nuclear Safety ResearchElement Types

• Common 3-D element types:

PyramidPrismTetrahedron (tet)Hexahedron (hex)

• General polyhedra, …

• Difference between control volumes & elements


Nuclear Safety ResearchElements & Control Volumes

• “Cell-Centred”

• Element = Control volume

• “Vertex-Centred”

• Element-vertex-method

• Element � Control volume



Mathematical ModelsMathematical Models


Nuclear Safety ResearchModeling (governing equations)Modeling (governing equations)

Navier-Stokes equations (3D in Cartesian coordinates)

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∂

∂+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂+

∂

∂2

2

2

2

2

2ˆ

z

u

y

u

x

u

x

p

z

uw

y

uv

x

uu

t

uµρρρρ

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∂

∂+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂+

∂

∂2

2

2

2

2

2ˆ

z

v

y

v

x

v

y

p

z

vw

y

vv

x

vu

t

vµρρρρ

( ) ( ) ( )0=

∂

∂+

∂

∂+

∂

∂+

∂

∂

z

w

y

v

x

u

t

ρρρρ

RTp ρ=

Convection Pressure gradient Viscous termsLocal

acceleration

Continuity equation

Equation of state

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∂+

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∂

∂2

2

2

2

2

2ˆ

z

w

y

w

x

w

z

p

z

ww

y

wv

x

wu

t

wµρρρρ


Nuclear Safety ResearchConservation Equation

( ) ( )j j

jF U F D S

t xρ ρ ρ

∂ ∂+ + =

∂ ∂

Concentration

Energy

Momentum

Mass

Variable F

1

jU

E

C

( )j ij ij iq P Uτ δ+ +

ij ijPτ δ+

0

jD

jJ

S

0

ig

i ig U

R


Nuclear Safety ResearchCFD Solver

Numerical

Algorithms

Numerical

AlgorithmsComputer

Architecture

Computer

Architecture

Mathematical

Models

Mathematical

Models

CFD SolverCFD Solver


Nuclear Safety ResearchMathematical Models

• 5 Conservation laws– 1 × Mass

– 3 × Momentum

– 1 × Energy

• 5 Unknowns– U, V, W, P, E

• Closed system– Laminar flows &

– Turbulent flows



TurbulenceTurbulence


Nuclear Safety ResearchTurbulenceTurbulence

�Turbulent flows:� Three-dimensional

� Unsteady-state

� Irregular

� Large spectrum of length and time scales

�„Huge“ computational effort �

�Turbulence models� Statistical models

� …

� Scale-resolving models


Nuclear Safety ResearchTubulence Spectrum


Nuclear Safety ResearchTurbulenceTurbulence ParametrisationParametrisation

• Energy-containing large eddies– Velocity scale, Vc

– Length scale, Lc( )2 2 2

1 2 3

3 2

1

2

c

c

V k

k u u u

kL

ε

=

′ ′ ′= + +

=


Nuclear Safety ResearchModelling Strategies

SASSolution controlled

DESGrid controlled

(U)RANSAllNone

LESSmallLarge

DNSNoneAll

ModelResolve


Nuclear Safety ResearchURANS Equations

( )0

j

j

U

t x

ρρ ∂∂+ =

∂ ∂

( ) ( ) ( )i j i ij i

j

j

i j

U U U P

x

u

x

u

t x

ρ ρ τ ρ∂ ∂ ∂ +∂+ = −

∂

′−

∂ ∂

′

∂

jiij

j i

UU

x xτ µ

� �∂∂= − + ∂ ∂� �


Nuclear Safety ResearchURANS: Eddy Viscosity Models

• Eddy viscosity:

t c cV Lµ ρ=

D�kk-�

2-Eqn

A & D�m�t1-Eqn

D� & �kSST

D�kk-�

A�mMixing length

0-Eqn

Algebraic or Diff.

LcVcNameClass


Nuclear Safety ResearchTurbulence Turbulence ModelingModeling withwith RANS modelsRANS models

�� ωωωω �� ωωωω ��

�� ωωωω��

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ωρβσ

µµρ

ρkPkUk

t

kk

k

t ′−+��

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�∇�

��

�+⋅∇=⋅∇+

∂

∂)(

)(

2)()(

ωβρω

αωσ

µµωρ

ρω

ω

kPk

Ut

kt −+�

�

��

�∇�

��

�+⋅∇=⋅∇+

∂

∂


Nuclear Safety ResearchNear Wall Turbulence Modeling

Standard Log-Law Wall

Functions

• The k-ε model uses log-law wall

functions to provide the

hydrodynamic wall shear stress.

• The near wall tangential velocity is

related to the wall shear stress by

means of a logarithmic function.

- where κ and C are constants

depending on wall roughness

Log-law Wall Function

Cyy += ++ )ln(1

κ


Nuclear Safety ResearchScaleScale--resolvingresolving model model -- LESLES

Large eddies of the turbulence are computed and only the smallest eddies

are modeled.

� Filtering operation of the physical quantities which preserves only their

large scale components. Usually, the computational grid serves as a low

pass filter and only the subgrid scale turbulent phenomena are modeled

� Smagorinsky subgrid scale model; it is an eddy viscosity model, that is

based on the assumption, that the effect of the small scales eddies can be

accounted for by adding a contribution to the momentum diffusivity

( ) ijijSt SSC ⋅⋅⋅∆= 22

ν�

��

�+=

i

j

j

iij

x

U

x

US

∂∂

∂∂

21

The characteristic length scale ∆ refers to the filter width and corresponds to the mesh spacing. CS is the Smagorinsky constant (CS = 0.18), reduction

of the eddy viscosity near walls with Van Driest damping functions

rate-of-strain tensor


Nuclear Safety ResearchTubulence Spectrum

Modeled URANS

DNS computed

LES computed LES modeled


Nuclear Safety ResearchExamples of modeling (Turbulence)Examples of modeling (Turbulence)

LES, Re=105, Iso-surface of Q criterion (0.005)

for turbulent flow in a T-junction

URANS, Re=105, Iso-surface of Q criterion

(0.005) for turbulent flow in a T-junction

Visualization of flow structure by isosurfaces of Q-criteria - the value of Q is a measure for visualized scales



Multiphase FlowMultiphase Flow


Nuclear Safety ResearchMulti-Phase Flow Simulation

• Euler-Lagrange:– Continuous ‚carrier‘ phase �

Euler

– Tracking of single particles or

particle groups

– Interaction with carrier phase

– Limited to disperse flows

• Euler-Euler:– Interpenetrating continua

– Phase indicator function

– Phase weighted averaging

– Additional unknowns �

consequence of averaging

– Empirical closure

– More ‚general‘ approach

Multi-Phase Flow CFD


Nuclear Safety ResearchPhase Indicator Function

t

Mk

0

1

‘Phase sensor’


Nuclear Safety ResearchMass Conservation Equation


Nuclear Safety ResearchMomentum Conservation Equation



Numerical MethodsNumerical Methods


Nuclear Safety ResearchCFD Solver: Elements

Discretization –

Solution domain

Discretization –

Equations

Equation coupling

Matrix solvers

Parallelisation

Model equations


Nuclear Safety ResearchSegregated Solution

• Generic coupled system(e.g. velocities U and V):

• Segregated solution:

uu uv u

vu vv v

A A BU

A A BV

� � � �� ⋅ = � ��

k

uu uv uCalculate A , A V , B k 1Solve for U +

k 1

vv vu vCalculate A , A U , B+k 1Solve for V +


Nuclear Safety ResearchCoupled Solution 1

uu uv u

vu vv v

A A BU

A A BV

� � � �� ⋅ = � ��

uu uv vu vv u vCalculate A , A , A , A , B , B k 1 k 1Solve for U ,V+ +

• Generic coupled system

(e.g. velocities U and V):

• Coupled solution:


Nuclear Safety ResearchCoupled Solution 2

• Coupled solution of linear equation systems:

– The solution variables of all coupled equations are always on the

same time/iteration

– Avoid unphysical over- and undershoots

– Improved robustness:

• Coupling between velocity and pressure

• Coupling of velocity components for rotating systems

• Coupling of the species in a combustion calculation

• Coupling of different phases in a multiphase flow calculation

– More effort per iteration but less iteration required in order to reach

convergence

– 1 ‘big’ matrix instead of several ‘small’ matrices

• Increased memory requirements

– Allows larger time steps


Nuclear Safety ResearchILU – The Smoother

11

11

11

11

A L U

• Incomplete Lower/Upper-Decomposition

– forward and backward substitution

– ILU factorisation only modifies the main diagonal

� efficient storage: original matrix + one nodal array

– tends toward a 1-D Trid-Diagonal Matrix Algorithm (TDMA) for

equations with very large coefficients

� well suited as smoother for multigrid solver



Segregated Coupled

Coupled Volume Fractions


Nuclear Safety ResearchMatrix Solvers

• Objective:– Efficient solution of sparse

system

• Scalability

• Relative effort for 2-D-Poisson equation

8 sN log(N)Multigrid

5 minN1.25SSOR-PCG

0.5 hN1.5SOR, CG

2.5 hN2Gauss-Seidel

5 hN2Jacobi

24 hN2Gauss elimination(direct)

CPU-tOperation

Count Method



• Simple relaxation methods are

good at reducing error

components that have short

wavelengths with respect to the

grid spacing.

• In Multigrid, a hierarchy of grids is

constructed, each coarser than

the previous grid.

• Applying the simple relaxation

method to each grid results in a

reduction of all components of

error in the final solution.

The left side of the figure shows the grids; the right side shows the error components that are most effectively treated on that grid.

Algebraic Multi-Grid Method


Nuclear Safety ResearchAlgebraic Multi-Grid Method

• Dynamic multigrid hierarchy

• Scalable parallelization

Fine grid

Coarsest grid

Coarse grid...

...

V-cycle W-cycle Others ...


Nuclear Safety ResearchHigh performance computingHigh performance computing

• CFD computations are usually very expensive which requires

parallel high performance supercomputers with the use of multi-

block technique.

multi-block technique

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Nuclear Safety ResearchParallel Computing

• Single Program Multiple Data (SPMD):– Identical code @ all processors

– Communication between processors: PVM, MPI, …

• “Domain-Decomposition”

Inner node Cut element Core elementOverlap node

Original gridPartitioned grid



1.1.ExampleExample

CFD CFD Calculation of horizontal Calculation of horizontal

gas/liquid flow experimentsgas/liquid flow experiments

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Nuclear Safety ResearchIntroduction and motivationIntroduction and motivation

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Nuclear Safety ResearchHot leg and pressure chamberHot leg and pressure chamberH

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Nuclear Safety ResearchThe hot leg model: Types of experimentsThe hot leg model: Types of experiments

• Co-current flow experiments:

air inlet

water inlet

air

outlet

SG inlet chamber & separator

RPV

simulator

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Nuclear Safety ResearchExample of coExample of co--current flow experimentcurrent flow experimentH

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Nuclear Safety ResearchCFD simulation of twoCFD simulation of two--phase cophase co--current flowcurrent flow

• Boundary conditions:

– pressure: 3.14 bar

– temperature: 22.3 °C

– air flow rate: 0.036 kg/s

– water flow rate: 0.902 kg/s

• CFD modeling:

– Euler-Euler two fluid model

– interphase transfer model: mixture model using the AIAD model

– fluid dependent k-� turbulence

model with turbulence damping functions and automatic wall functions

experiment simulation



2. Example2. Example

Coolant MixingCoolant Mixing

(Boron Dilution Transients)(Boron Dilution Transients)

Co

ola

nt

Mix

ing

Co

ola

nt

Mix

ing



Basic PhenomenonBasic Phenomenon

Boron 10 = strong thermal

neutron absorber

� Used as boric acid solved in

the coolant of PWRs to

compensate excess reactivity

� inadvertent or unavoidable

decrease of boron

concentration (boron dilution)

might result in a reactivity

transient

� Power peak depends on

coolant mixing in cold leg,

downcomer lower plenum

� Density differences can

strongly influence the mixing

Steam Generator

Pressurizer

RPV

Main Coolant Pump

deborated slug

ECCS

Scheme of the primary circuitScheme of the primary circuit

NPP PhilipsburgNPP Philipsburg

Motivation Motivation –– Safety of Nuclear Power PlantsSafety of Nuclear Power PlantsC

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ReactorReactorPressurePressureVesselVessel

SteamSteamGeneratorGenerator

Main CoolantMain Coolant

PumpPump

DeboratedDeborated SlugSlug


Nuclear Safety ResearchROCOM: ROCOM: ROROssendorfssendorf COCOolantolant MMixing Test Facilityixing Test FacilityC

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Wire Mesh Sensor (Pos. 1)

Pos. 1Pos. 1

Pos. 2Pos. 2

Pos. 4Pos. 4

Pos. 3Pos. 3

Dimensionless Mixing Scalar:Dimensionless Mixing Scalar:

ΘΘ x,y,zx,y,z (t) = (c (t) = (c x,y,zx,y,z -- ccrefref)/(c)/(cslugslug -- ccrefref))

cref – Reference Concentration Reactor (0)

cslug – Concentration Slug (1)

c x,y,z – Local Concentration

Conductivity Measurements with Wire Mesh SensorsConductivity Measurements with Wire Mesh SensorsC

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� Mesh: 3.6 million nodes and 6.5

million hybrid elements

� Combination of Hexahedral and

Tetrahedral cells, mesh

refinement at the perforated

drum, in the lower support plate

and at the wall regions of the

cold legs

BDT: Grid (ICEMBDT: Grid (ICEM--CFD)CFD)C

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Nuclear Safety ResearchBDT: Transport of the Slugs (i)BDT: Transport of the Slugs (i)C

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MovieMovie


Nuclear Safety ResearchBDT: Transport of the Slugs (ii)BDT: Transport of the Slugs (ii)C

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MovieMovie



0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 10 20 30 40 50 60 70 80 90 100

time / s

theta

/ -

Experiment

CFX

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 45 90 135 180 225 270 315 360

azimuthal position / °

the

ta / -

Experiment

CFX

t=40.0 s

CL 1 CL 4

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ConcludingConcluding

RemarksRemarks

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��Computational fluid dynamics (CFD)Computational fluid dynamics (CFD) is one of the branches of is one of the branches of fluid mechanicsfluid mechanics that uses that uses numerical methodsnumerical methods and and algorithmsalgorithms to to solve and analyze problems that involve fluid flows. solve and analyze problems that involve fluid flows.

��Computers are used to perform the millions of calculations Computers are used to perform the millions of calculations required to simulate the interaction of liquids and gases with required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. surfaces defined by boundary conditions.

�� Even with highEven with high--speed speed supercomputerssupercomputers only approximate only approximate solutions can be achieved in many cases. solutions can be achieved in many cases.

��Ongoing research, however, may yield software that improves Ongoing research, however, may yield software that improves the accuracy and speed of complex simulation scenarios such as the accuracy and speed of complex simulation scenarios such as transonic or transonic or turbulentturbulent flows. flows.

�� Validation and verification of such software is necessary using Validation and verification of such software is necessary using high resolution experiments.high resolution experiments.

Concluding RemarksConcluding RemarksC

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Thank You!Thank You!

fzd theory seminar series cfd simulations for single and

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