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Solution Techniques

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Title: Solution Techniques


1
Solution Techniques

2
What CFD packages do
  • Aim is to solve, numerically, the equations of
    motion (continuity and motion) for a fluid for a
    given flow geometry
  • Plus transport equations for any additional
    models such as heat transfer and/or turbulence
    models

3
Transport equations
  • These are the modelling basis for all engineering
    processes involving momentum, mass or heat
    transfer
  • They have a general form.
  • However the notation used in engineering
    literature varies. Most references use some
    shorthand form because the full equation set
    (particularly the momentum equation) is tedious
    to write

4
For transport of a scalar quantity ?
  • Full notation (pain-full) For a 3d problem there
    will be 3 PDEs too!
  • Cartesian tensor notation (Fluent and general CFD
    literature) summation over co-ordinates implicit.
    Multiple PDEs implicit too

5
For transport of a scalar quantity ?
  • Vector notation (BSL and also other CFD
    literature)
  • integral form (over a control volume and used in
    Fluent documentation)

6
Transport equations
  • Express the conservation relationship for
    intensive properties (T, c, u) in terms of
    accumulation, advection, diffusion, generation
    and destruction
  • Since the important adjective is conservation,
    this is also one important criteria of the
    solution
  • they are also known as convection diffusion
    equations

7
Three questions
  • What does the time derivative imply?
  • What process does a first order spatial
    derivative represent?
  • What process does a 2nd order spatial derivative
    represent?

8
but
  • Transport equations are generally second order
    non linear differential equations for which a
    general analytical solution is not known
  • so we usually have to solve them numerically
  • however analytical solutions exist for particular
    simplified problems ( BSL)
  • Steady 1d flow has an analytical solution

9
An aside...
  • Some parts of a CFD flow problem may approximate
    conditions where an analytical solution of the
    equations of motion does exist or can be
    approximated.
  • Such an exact solution can be used for
    validation. I will use this later in this lecture
  • ? is a scalar. The momentum equation uses u
    which is a vector.

10
Numerical Solution of Transport Equations
  • Numerical solution involves discretizatng the
    transport equations in time and space (along a
    grid)
  • Discretization can be viewed as a mathematical
    operator which transforms the transport equation
    (a PDE) into a set of coupled ODEs which express
    conversation over the set of finite volumes as
    defined by the grid
  • This involves replacing spatial derivatives by
    algebraic approximations based on grid values in
    the same and adjacent grid volumes

11
Finite differences
  • This approach uses numerical approximations to
    the spatial derivatives. Example is a finite
    difference approximation to a 1st order
    derivative at a mesh point

12
Finite Volumes
  • Integrates the transport equation over a finite
    volume defined by a grid element and then assumes
    a mean value of ?v exists in the volume
  • This integration means that any first order
    spatial derivatives are replaced by fluxes across
    the boundary. (why?) This flux is calculated
    based on a value of ?f at the boundary
  • But we only know an average value of ?v for each
    volume, so ?f must be calculated by an
    approximation

13
Finite Volumes
  • Second order derivatives are integrated to first
    order derivatives.
  • These have to be approximated by finite
    differences
  • The time derivative still appears in the
    discretized equations so the operator Dis results
    in a set of coupled ordinary differential
    equations in time

14
Discretization is an approximation
  • And an approximation will introduce errors
  • These errors will introduce oscillatory behaviour
    to wave fronts and false diffusion
  • There is a large body of literature on techniques
    which aim to minimise these problems
  • A good starting point is Fletcher

15
Oscillatory behaviour at wave fronts
  • Fletcher, 1, p314

16
Numerical or False Diffusion
  • It creates most problems in flow simulations
    where real diffusion is small. Why? Fletcher 1 p
    285

17
Fluents Finite Volume approach
  • Fluent uses the finite volume approach to
    discretization
  • The transport equations (PDEs) are then
    integrated over each finite volume as defined by
    the grid and an average value of ?v is assumed
    over the volume. (Equivalent to a well mixed
    assumption)

18
Discretization
  • However since the boundary of each finite volume
    consists of a finite set of flat faces of known
    area, the discretized equations become
  • To solve these equationswe need the values of ?f
    at each face

19
Calculation of ?f by Upwinding
  • Upwinding calculates the value of ?f based on ?v
    in cells upstream of the face f
  • 1st order upwinding proposes that ?f is equal to
    the value of ?v in the cell that adjoins f
    upstream
  • 2nd order upwinding calculates ?f based on a
    Taylor series expansion of the upstream cell
    centered-solution about the cell centroid

20
Calculation of ?f by Upwinding
  • And
  • 1st order upwinding in 1d is equivalent to the 2
    point finite difference equation using values
    upstream of the grid point j

21
Power law and Quick
  • Power Law scheme uses an analytical solution to
    the steady 1d convection diffusion equation - See
    Ch 17 Fluent documentation
  • Quick Scheme uses a weighted average of second
    order upwinding and central differencing
  • The higher order schemes are more accurate but
    this can be at the expense of numerical stability

22
Which is best?
  • 1st order upwinding works best in problems which
    are dominated by real diffusion and will be
    accurate where the real diffusion exceeds the
    numerical diffusion. Also useful as a starting
    point in solution
  • Most CFD problems are dominated by advection
    (flow) and hence higher order schemes are
    probably going to be the most accurate. So you
    will probably use at least 2nd order upwinding.
  • Many problems will converge well with the higher
    order schemes anyway
  • However this is also very dependent on grid
    quality. The higher order schemes will give the
    most improvement in accuracy on poorer quality
    grids.

23
For Example Laminar falling film
  • Laminar flow of a falling film of water on flat
    plate
  • Consider a case where the plate is nearly
    horizontal, the film depth is very small (say
    10mm) and the plate is very long (say 4m) and the
    plate is very wide.
  • The plate is wide enough that we treat the flow
    as a 2d problem in length z and water depth x
  • Since the plate is very long, a point in z will
    be reached where the flow will be fully developed
    and we can reasonably assume this will be near
    the exit

24
Laminar Falling Film

25
Laminar falling film
  • In the fully developed region any spatial
    derivatives along the plate (in z) are zero
  • With this simplification we can derive an
    analytical solution for the velocity profile from
    the momentum equation, provided that we know the
    film thickness and the angle of inclination (See
    BSL pp38 -39)

26
Laminar Falling Film
  • This velocity profile has been verified
    experimentally
  • Note that this is a laminar boundary layer
    problem!
  • Here are Fluents predictions of the z velocity
    profile vs height x from the plate near the film
    surface for a 10mm thick film along a 4m long
    plate that is nearly horizontal at a point which
    is near the exit from the flow

27
Fluent Prediction using a tri-paved grid
28
Fluent prediction with mapped hex grid
29
Laminar Falling Film - Conclusions
  • For a mapped hex mesh all discretisation
    techniques give much the same answer for the
    velocity profile
  • For a triangular paved mesh the higher order
    schemes show a definite improvement

30
If you want to read more
  • Ch 17 of the Fluent documentation
  • See some of the references at the end of the
    Fluent documentation
  • Fletcher C. A.J. Compuational Techniques for
    Fluid Dynamics Volumes 1 and II, Springer 1991

31
Solvers
  • Discretisation results in a set of coupled ODEs
    in time
  • The unsteady solvers in Fluent integrate these
    ODEs in time using a Kutta Merson algorithm
  • The steady solvers assume that the time
    derivative to zero. The result is a set of
    non-linear simultaneous equations. These are
    solved iteratively

32
Solvers - Unsteady or Steady?
  • Remember that flow through the tap is a steady
    flow problem but filling the sink is not.
  • But the sink may be filled to overflowing and the
    flow onto the floor appears to be constant. Is
    this a steady flow?
  • What if there are waves on the surface of the
    water which propagate out from the point of
    impact of the water stream from the tap?
  • Will a steady solver be able to get a steady
    solution for this sort of behaviour? Why or why
    not?

33
Solvers - Steady or Unsteady?
  • Your problem may actually have a periodic
    characteristic that you may not be aware of. You
    may be trying to get a steady state solution and
    this may be why you arent getting convergence
  • Consider flow in a slow moving river or stream.
    This is actually turbulent flow and is
    characterised by eddies which well up to the
    surface.
  • Even though the water flow down the stream is
    constant , is this a steady flow?

34
Solvers - Steady or Unsteady?
  • Turbulence is intrinsically a dynamic phenomena.
    It might be more chaotic than periodic but the
    flow domain contain effects where the velocity
    obviously varies in time
  • A steady solution therefore only exists in a time
    averaged sense for turbulent flows. We will
    discuss this in a later lecture
  • Large Eddy Simulation (and DNS) must use the
    unsteady solver because they resolve the path of
    these eddies
  • k-e, RSM can use the steady solver but this
    doesnt mean you should be using it.

35
Solvers - Steady
  • The simulation might have a steady solution but
    you might still have to integrate in time from an
    initial condition to get using the unsteady
    solver to get to the steady state
  • This situation will probably arise in multiphase
    flow problems. Particularly where the simulation
    is tracking a phase boundary

36
Fluents Unsteady Solver
  • The unsteady solver integrates the ODEs using a
    Kutta Merson integrator
  • If you have a small number of small grid points,
    this may introduce stiffness into the equation
    set
  • See Johnson and Reiss, Numerical Analysis,
    Addison Wesley, 1982 for a discussion of the
    effects of stiffness

37
Fluents Steady Solvers
  • With the steady solvers the time derivative is
    assumed to be zero
  • The ODEs then reduce to a set of coupled non
    linear simultaneous equations
  • These are linearised. This can be by an implicit
    of explicit form

38
Fluents Steady Solvers
  • Segregated Solver
  • The default option
  • solves the momentum, continuity and then other
    transport equations sequentially
  • because of this the solver must iterate until the
    solution is converged
  • Coupled Solver
  • Solves the momentum, continuity, energy and
    species transport as one equation set
  • however the equations are still non linear and
    the solution is still iterative

39
What does this mean?
  • The various steady solvers and the linearization
    techniques affect convergence and stability in
    particular problems
  • The various solvers will however work across a
    variety of problems. These will be discussed in
    detail later

40
Under-relaxation
  • This is a technique which controls the change in
    a variable during iteration
  • It is the main technique by which the solver is
    stabilised
  • Reducing the under-relaxation factor ? will
    stabilise the solver at the expense of slowing
    the rate of convergence
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