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Solution Techniques

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Solution Techniques 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 ... – PowerPoint PPT presentation

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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
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

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
• 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
• 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?
solution for this sort of behaviour? Why or why
not?

33
• 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
• 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
• 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
• 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
• 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
• 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