It is importatnt

1. Dr Patrick Geoghegan
Book: H. Versteeg and W. Malalasekera An Introduction to
Computational Fluid Dynamics: The Finite Volume Method Chapter 2
FEA/CFD for
Biomedical
Engineering
Week 8: CFD –
Continuity

2. CFD Basics Part 2

3. Flow conditions and fluid properties
1. Flow conditions: inviscid, viscous, laminar, or turbulent, etc.
2. Fluid properties: density, viscosity, and thermal conductivity, etc.
Selection of models: Different models usually fixed by
codes, though some options for user to
choose
Initial and Boundary Conditions: Not fixed by codes, user needs
specify them for different
applications.
The Physics

4. 𝜌𝜌
𝜕𝜕𝐮𝐮
𝜕𝜕𝑡𝑡
+ 𝐮𝐮 � 𝛻𝛻𝐮𝐮 = −𝛻𝛻𝑝𝑝 + 𝛻𝛻 � 𝜇𝜇 𝛻𝛻𝐮𝐮 + 𝛻𝛻𝐮𝐮 𝑻𝑻 −
2
3
𝜇𝜇 𝛻𝛻 � 𝐮𝐮 𝐈𝐈 + 𝐒𝐒𝐌𝐌
The Physics – Model Equations
Built upon Navier Stokes equations
The inertial forces, pressure forces, viscous forces, and the external
forces (e.g. gravity) applied to the fluid.
These equations are always solved together with the continuity
equation:
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝛻𝛻 � 𝜌𝜌𝐮𝐮 = 0
The Navier-Stokes equations represent the conservation of momentum, while the continuity
equation represents the conservation of mass.
Most commercial CFD codes solve the continuity, Navier-Stokes, and energy equations, which
form coupled, non-linear, partial differential equations (PDEs)

5. Broadly two methods of approaching the solving of the PDEs
Finite Difference Method:
Replace the derivatives with ratios of differences at points
within the grid
Finite volume method:
Apply “conservation laws” to the small volumes created by
the grid (most common in CFD software)
The Maths -Discretisation

6. Finite difference methods describe the unknowns of the flow problems by means
of point samples at the node points of a grid
The governing equations (NSE) are converted to algebraic form, allowing the first
and second derivatives to be approximated using Truncated Taylor series
expansions.
The resulting linear algebraic equations can then be solved iteratively of
simultaneously
The Maths –Finite Difference Method

7. The domain is divided into a number of control volumes (aka cells,
elements) -the unknown of interest is located at the centroid of the
control volume.
The differential form of the governing equations are integrated over
each control volume.
The Maths –Finite Volume Method

8. Finite Volume approximations are substituted for the terms in
the integrated equations (discretization) - converts the
integral equations into a system of algebraic equations.
One equation for each control volume results in a set of
algebraic equations which can be solved by an iterative
method, or simultaneously
The Maths –Finite Volume Method

9. The computational domain is what is Discretised into a “grid”
or “mesh”, which is formed of a finite set of control volumes
or cells.
The Model –Computational Domain

10. Cell Types:
The Model -Discretization

11. Grid Types: Structured
All cells have the same number of nodes
All grid lines must pass through all of domain (forms a grid index)
Restricted to simple geometries
The Model -Discretization

12. Structured Grids

13. Grid Types: Un-structured
Cells can be arranged in arbitrary fashion
No grid index and therefore no constraints on cell layout
Can therefore be used in complex geometry
The Model -Discretization
Human Lung

14. Unstructured Grids
Structured-Unstructured
Nozzle Grid

15. 3D Meshes can be formed of
Quad or Hex meshes, which
can be useful for simple
geometry.
For complex geometries, tri or
tetra - meshes may be more
suitable
The Model -Discretization

16. As with FEA, the Boundary Conditions setup how the model interacts with the environment.
Examples:
• Wall interaction (stress induced in fluid)
• Specified suction or blowing at interfaces
• Inflow/Outflow pressure
• Interface Condition, e.g., Air-water free surface
• Symmetry and Periodicity
The Boundary Conditions

17. In addition to the boundary conditions, the model can also have some
Initial Conditions defined such as whether it is a Steady/unsteady flow,
ambient temperature.
Initial conditions should not affect final results and only affect
convergence path, i.e. number of iterations (steady) or time steps
(unsteady) need to reach converged solutions.
They can help speed up the convergence
The Boundary Conditions

18. The discretized conservation equations are solved iteratively. A number
of iterations are usually required to reach a converged solution.
Convergence is reached when:
• Changes in solution variables from one iteration to the next are
negligible.
• The solution no longer changes with additional iterations.
• Mass, momentum, energy and scalar balances are obtained.
Analysis

19. Convergence can be monitored by the residuals in the
solutions, which are a measure of the imbalance (or error) in
the conservation equations
The accuracy of a converged solution is dependent upon:
• Appropriateness and accuracy of the physical models.
• Grid resolution and independence.
• Problem setup.
Analysis

20. Once the analysis is complete, the results require
examination
This can allow the problem to be explored, asking questions
such as:
What is the overall flow pattern?
What is the pressure at the outlet?
Post-Processing

21. CFD packages will provide several “user friendly” ways to look at the results of a
simulation:
• Vector Plots
• Contour Plots
• Particle Tracking
Post Processing

22. Following analysis of the results, decisions on
whether to re-run the analysis can be taken.
Reasons to do this may be:
Unexpected flow conditions –are the boundary
conditions correct?
Post-Processing