Quantitative techniques in fire safety engineering 5 U01620 710. Computational fluid dynamics CFD fo

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Quantitative techniques in fire safety
engineering 5 (U01620) 7-10. Computational
fluid dynamics (CFD) for fire

• Stephen Welch
• S.Welch_at_ed.ac.uk
• School of Engineering and Electronics
• University of Edinburgh

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CFD for fire lectures (L7-10)

• Scope
• L7 Introduction to CFD
• L8 Introduction to Turbulence Modelling
• L9 Introduction to Combustion Modelling
• L10 Introduction to Radiation Modelling
• General overview and familiarity with some of the
  terms
• Covering CFD in general but with an eye on fire
  applications

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Quantitative techniques in fire safety
engineering 5 (U01620) 7. Introduction to CFD

• Stephen Welch
• S.Welch_at_ed.ac.uk
• School of Engineering and Electronics
• University of Edinburgh

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Introduction to CFD Scope (1)

• Governing equations
• Navier Stokes equations
• Discretisation of transport equations
• Finite Volume Methods (FVM)
• Other methods (FDM, FEM)
• Grids, co-ordinate systems
• Interpolation
• Numerical issues
• Errors and stability
• False Diffusion

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Introduction to CFD Scope (2)

• Pressure correction methods
• Numerical solvers
• Relaxation
• Convergence
• Boundary conditions
• Static pressure boundaries
• Guidance on use of methods
• Example application

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Objective

• Introduction to terminology
• Know where to look
• Know what questions to ask!
• Grid sensitivity
• Interpolations schemes
• Boundary treatments
• Unstructured
• Newtonian fluid

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What is CFD?

• Most general sense
• All aspects of computer-based simulation of fluid
  flow phenomena
• More specifically
• Computer simulations solving fluid flow
  conservation equations
• Using established numerical methods for
  second-order partial differential equations

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Whats the trouble?

• Partial differential equations
• Non-linear in all real situations
• Due to convective acceleration
• Acceleration associated with change of velocity
  over position
• Part of cause of turbulence
• Hard to solve
• Turbulence
• Time-dependent chaotic behaviour
• Effect of time dependent and convective
  acceleration
• Due to inertia of fluid as a whole
• Navier-Stokes equations model turbulence properly

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What do we need it for?
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What do we need it for?
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What do we need it for?
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Navier Stokes system of equations

• Continuity
• Momentum
• Stress tensor
• Energy
• NB - repeated subscript implies summation

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Navier Stokes system of equations

• History
• Claude-Louis Navier (1785-1836)
• French engineer and physicist
• George Gabriel Stokes (1819-1903)
• Irish mathematician and physicist
• 1822
• Established by Navier

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Continuity

• Simple mass balance

? - density u, v, w - velocities
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Momentum equations

• Momentum
• Viscous stress tensor
• P pressure
• D dilation term
• ? viscosity

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Energy equation

• Energy
• h enthalpy

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Governing equations - Generalised

• Generalised
• ? general unknown
• ? diffusion coefficient
• s Schmidt/Prandtl number

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Generalised transport equation

• Unknown variable, ?
• ? is a diffusion coefficient
• S is a general source term
• For continuity, ? 1, ? 0, S0
• For momentum, ? u, v, w, ?f(µ), S
• For energy, ? e (or h), ?k/cp, , Sradiation

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Notations

• Cartesian
• Cylindrical

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Notations

• Vector notation
• Tensor notation

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Newtonian fluids

• Stress-strain curve linear and goes through O
• e.g. water
• flows, irrespective of forces on it
• Viscosity f(p,T) only!
• cf. Non-Newtonian
• Sand
• Non-drip paints
• Oobleck (starch-water mix solid if you squeeze
  it)
• Glurch (starch-glue-colouring polymerizes)

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Special forms

• For low Mach numbers, pressure-density coupling
  can be neglected
• Incompressible
• Density varies with temperature only
• Pressure effects tiny by comparison, of order
  0.01
• Boussinesq approximation
• Euler equations
• Remove viscosity terms
• Not relevant to fire flows

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Finite volume method (FVM)

• Integrate the generalised transport equation
• Volume, ?
• Surface, S
• Foundation of the method
• Everything that goes in must come out!
• Conservation properties guaranteed
• Complex geometries are easily accommodated

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Other methods Finite Difference

• Finite difference method (FDM)
• Easy to formulate
• Mesh must be structured, i.e. regular
• Curved meshes transformed to orthogonal
• Gradient boundary conditions can only be
  approximated, not enforced
• FVM is an integral version of the FDM
• Chung (2002) shows how FVM can be formulated from
  FDM

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Other methods Finite Element

• Finite element method (FEM)
• Formulation requires mathematical rigor
• Complex geometries can be easily accommodated
• Unstructured meshes allowed
• No need to transform curved meshes
• Gradient boundary conditions enforced exactly
• FVM is a special case of FEM weighted residuals
• weighting function 1 within cell volume, 0
  elsewhere
• Chung (2002) shows how FVM can be formulated from
  FEM

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Discretisation

• To solve the integral form of generalised
  transport equations first represent field by
  discrete values
• Nodes in a computational grid
• Assemble algebraic equations which can be solved
  for the nodal values
• Other values are undefined and if needed have to
  be obtained by interpolation

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Discretisation - Example

• For example, continuity (2D steady-state)
• In finite difference form

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Discretisation - Cell location

• Two principal cell definition methods
• Grid based (staggered)
• Nodal value stored at intersection of grid lines
• Control volume surfaces defined midway between
  grid lines
• Cell centred (co-located)
• Construct control volumes and assign central
  nodes
• Intuitively consistent with integral approach
• Assume hereafter

j
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Discretisation Grid structure

• Structured
• Unstructured

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Discretisation Grid structure

• Two types of grid structure
• Structured
• Physical geometry mapped to a regular
  computational space
• Typically accessed via conventional array indices
  representing physical coordinates
• Less general, but assume hereafter for simplicity
• Unstructured
• Individual cells may be arbitrarily subdivided
• Control volumes related to each other using an
  array of neighbour pointers
• Well suited to complex geometries
• Accuracy of spatial derivatives can be diminished
• Program data structure complex, not easy to
  vectorise

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Discretisation Coordinate system

• Two structured grid coordinate systems
• Curvilinear
• Most general
• Can deal with more complex geometries
• Equations far more complex and can be
  non-conservative
• Computational cost of additional terms
• Hard to make the mesh!
• Cartesian
• Mutually orthogonal unit vectors
• Easier to understand

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Discretisation Transport Equation

• Cartesian coordinates
• A is cell face area
• V is cell volume
• Assumes variables are functions of normal
  direction only, for surface integrals
• Assumes variables are constant in cells, for
  volume integrals
• Remaining unknowns found by interpolation

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Interpolation

• Need a method of approximating the convective and
  diffusive fluxes at cell faces
• An interpolation technique relates value to nodes
• At first sight would seem best to use linear
  interpolation, i.e. central differencing
• Stable for 2nd derivatives (diffusion terms)
• May be unstable for 1st derivatives (convection
  terms)
• Particular problem at high Reynolds numbers

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Interpolation Example

• Considering 1D convection/diffusion equation
• Where
• G mass flow
• D diffusive flow
• E, P, W are nodal values

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Interpolation Central Differencing

• Using linear interpolation for convective terms
• Assuming uniform mesh spacing

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Interpolation Central Differencing

• Algebraic equations can be solved if coefficient
  matrix is diagonally dominant
• All coefficients need to be same sign (i.e.
  positive)
• AP must be greater than neighbour coefficients
• Diffusion coefficients unconditionally positive
• Mass flux terms may be negative if
• Peclet number is a grid Reynolds number
• Only satisfied if ?x small

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Interpolation Upwind Differencing

• Allow flow to be influenced only by conditions
  upstream (pig pen theorem)
• All coefficients unconditionally positive
• Diagonal dominance
• Unconditionally stable

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Interpolation Hybrid Differencing

• Upwind is stable but can be inaccurate for
  non-convection dominated flows
• Use Central up to Peclet number limit and Upwind
  thereafter
• Good choice for low speed flows

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Errors and stability

• Two sources of numerical error
• Computer round-off
• Truncation errors
• If not limited, solution may be
• Unstable
• Spurious oscillations

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Perturbation analysis

• Convection-diffusion equation
• AP lumps all terms related to nodal variable,
• S is for neighbours
• For stability errors must decay
• AP lt 0
• Implies negative feedback
• For diffusion term
• For convection term
• Hence no damping, and errors can arise

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False diffusion

• First-order upwind schemes have a second-order
  truncation error
• Substituting back into convection term truncation
  error contributes additional diffusion
  coefficient of
• Need to be sure this term is ltlt real diffusion
• Very dependent on bulk flow direction
• Errors worst when grid lines at 45o to bulk
  convective flow
• Take care in setting up problem

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Interpolation higher order

• Higher-order schemes reduce truncation error
• Expanding the second-order truncation error
• Problems
• Not conservative in FVM
• Can be unstable (no damping effect from
  truncation term!)
• TVD flux-limited second-order scheme (Van Leers)
• Third-order scheme, QUICK
• Conservative, but more cumbersome and may be
  unstable

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Interpolation higher order

• Higher-order schemes inherently
• More unstable
• Overshoot where there are steep gradients
• Fitting a curve with a polynomial of finite order
• Impossible to get an exact match at corners
• Can mitigate by adjusting diffusion
• Introducing artificial diffusion
• Problem is in knowing how much to add!
• Combining higher and lower order schemes
• Using a blending factor
• Shown to work but at additional computational cost

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Interpolation summary

• Define normalised variables
• Where
• C is downstream node
• D is upstream node
• U is next upstream node
• Hence Giving

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Discretisation Scalar Transport

• Expand into form suitable for numerical solution
• Each cell face term substituted from appropriate
  differencing scheme
• Mass flux linear
• Diffusion central
• Convection - upwind

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Discretisation Scalar Transport

• Algebraic equation for steady-state flow
• Coefficients, A, contain contributions from
• Convection terms
• Diffusion terms
• Transported source term, S, may be non-linear
• Chemical rates
• Radiative loss (?T4)
• Hence requires special attention linearisation
• Time splitting (or lagging)
• Newton-Raphson

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Discretisation Momentum

• Analogous procedure to scalar transport equation
• Equations are all non-linear and closely coupled
• Discretisation of pressure derivative term gives
  rise to unique problem
• Pressure decoupling
• pi,j cancels so there is no link to nodal value
• Chequer boarding
• Two solutions
• Staggered grid
• Momentum interpolation

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Staggered Grid

• Exploits fact that there is no need to use same
  control volumes for all variables
• Store velocities on faces
• More complex code
• especially for boundary conditions!

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Colocated Grid

• Exploits efficiencies in storing all variables at
  same place (cell centre)
• Can use same code routines for all variables
• Consistent with curvilinear grids
• Relies on an interpolation procedure
• Velocity terms modified by a third-order
  derivative of pressure (Rhie Chow, 1982)
• Artificial damping thereby introduced

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Pressure equation

• Pressure is derived from an equation of state
• Variations are small, but still important in
  driving flow
• Fundamental problem in solving discretized
  equations
• Velocity and pressure in momentum equation
  interact
• equations are very tightly coupled!
• Concept of pressure correction
• Most popular for incompressible flows

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Pressure correction

• Iterative procedure for solving momentum
  transport equations
• Guess a pressure field
• Solve momentum equations to obtain approximate
  velocity field
• Correct the approximate velocity and pressure
  fields such that continuity is satisfied
• SIMPLE (Patankar Spalding)
• Correction to velocities obtained by subtracting
  approximate and exact pressure equations, i.e.
  determining pressure correction terms

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Pressure correction

• Improved methods
• SIMPLEC (SIMPLE Consistent)
• Faster convergence
• Many other variants (subject of a lot of
  research)
• Time-dependent algorithms
• SIMPLER (SIMPLE Revised)
• PISO (Pressure Implicit Split Operator)

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Numerical solvers

• Direct solution of full matrix of solution
  variables is computationally intractable, so
• Segregate solution variables and solve each in
  turn
• Subdivide computational space
• Coefficient matrix has a diagonal structure
• Due to each node referencing only a small number
  of neighbours
• Allows some memory savings

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Relaxation and convergence

• Equations need to be under-relaxed
• Solution would diverge
• Not too much, or solution may freeze!
• Convergence assessed by examining residuals
• Normalised numerical errors
• Generally require progressive reduction in
  errors, stability of values and achieving certain
  minimum error (e.g. 0.1 mass imbalance)

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Boundary conditions

• Require values on all boundaries for solution
• Fixed values at inlets
• Dirichlet boundary conditions
• Gradients elsewhere
• Neumann boundary conditions
• To avoid over constraining a problem
• Mirror symmetry boundaries
• Used to reduce size of simulation domain
• Half
• Quarter

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Guidance on use of codes

• Cox Kumar (2003)
• Resist temptation to do 2D simulations
• Choose mesh appropriate to main features of flow
• Keep cell aspect ratio lt50
• Demonstrate insensitivity to mesh resolution
• Try to use higher order schemes to improve
  accuracy
• Demonstrate insensitivity to numerical timestep
• Solved variable residuals lt1
• Residuals progressively decreasing (may
  oscillate)
• Solved variable values at monitoring points
  stable
• Global heat and mass balance should be gt99

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Advanced fire applications

• Distributed burning
• Fires are often ventilation controlled and
  combustible volatiles may finally oxidise well
  outside an enclosure
• Heat source models are poor representations
• Smoke concentrations
• Strong coupling of smoke production, radiative
  heating and combustion
• Temperature-dependent material properties
• Extending up to fire temperatures!
• Moisture/intumescent effects

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Pros and cons of CFD

• Pros
• Fine detail
• High accuracy
• Versatile
• Less assumptions
• Complex geometries
• Transient phenomena
• Fully coupled

• Cons
• Problem setup mesh
• Long simulation times
• Hardware requirement
• Expertise requirement
• Post-processing burden

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References

• Chung, T.J. Computational Fluid Dynamics,
  Cambridge University Press, 2002
• Cox, G. Combustion Fundamentals of Fire,
  Academic Press, 1995 (chapters 1, 6)
• Moss, J.B. Turbulent diffusion flames, chapter
  4 in above, 1995
• Cox, G. Kumar, S. Modelling enclosure fires
  using CFD, in SFPE Handbook 3rd edition, 2003
• http//www.cfd-online.com/Wiki/

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Codes

• CFD codes list
• http//www.fges.demon.co.uk/cfd/CFD_codes.html
• Fire codes
• http//www.firemodelsurvey.com/
• SOFIE RANS code
• http//www.hull.ac.uk/php/331346/sofie.htm
• NIST Fire Dynamics Simulator (FDS) LES code
• http//fire.nist.gov/fds/