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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 transport equations for any additional

models such as heat transfer and/or turbulence

models

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

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

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)

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

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?

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

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.

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

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

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

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

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

Oscillatory behaviour at wave fronts

- Fletcher, 1, p314

Numerical or False Diffusion

- It creates most problems in flow simulations

where real diffusion is small. Why? Fletcher 1 p

285

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)

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

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

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

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

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.

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

Laminar Falling Film

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)

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

Fluent Prediction using a tri-paved grid

Fluent prediction with mapped hex grid

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

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

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

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?

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?

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.

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

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

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

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

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

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