Part 2 CFD basics Pt 2(1).pdf
- 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
- 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