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Fluid Mechanics II

    Lecture 11
 Muhammad Usman
Experimental Techniques
• In certain situation, an experimental
  investigation involving full-scale equipment
  can be used to predict how the equipment
  would perform under given conditions.
  However in most engineering applications,
  such full scale tests or either difficult or
  very expensive to perform or not possible
  at all.
Analytical Techniques
• Analytical models work out the consequences of a
  mathematical model which represents the behavior of a
  system. The mathematical model representing the
  physical process mainly consist of a set of differential
  equations. If classical mathematics were used to solve
  these equations, we call the approach as analytical
  approach.
• In most engineering applications, various assumptions
  and simlifications need to be made to enable the
  analytical solution of the differential equations
  representing the physical solution. This at one hand
  limits the applicability of these methods to simple type
  problems, or limits the validity of the solutions if too
  many assumptions and simplifications are made.
Computational Fluid Dynamics
• It is used to calculate the approximate
  solutions to wide variety of fluid mechanics
  problems.
• Replacing partial differential equation with
  discreted algebric equation. These
  equations are then used to calculate the
  solution at discrete points in space or in
  time.
• The analytical solution for navier stokes
  equation are available for only limited
  number of simplified flow geometries.
• The CFD Simulation solves for the
  relevent flow variables only at discrete
  points. Interpolation are used to obtain the
  values for non grid location.
Numerical Experiements Vs CFD
• Modeling              • Formulation of the
• Measurement             governing equation
• Analysis of results     and development of
                          the numerical
                          algorithm.
                        • Running an algorithm
                          in the computer
                        • Analysis of results
Discretization Techniques for
         Numerical Solution
• Finite Difference Method.
• Finite Element ( Volume ) Method.
• Boundary Element Method.
Finite Element Method
• Flow field is broken into a set of small fluid
  elements.
• The conservation equations ( Conservation of
  mass, momentum and energy ) are written for
  each of the element.
• For flows with complex boundaries, the number
  of algebric equations must be solved also
  inceases.
• Commonly problems include the formation of 1
  million gird cells.
Boundary Element Method
• Boundary of the flow field is broken into
  discrete segments.
• It requires less time and space then finite
  element method.
Finite Difference Method
• The method of using Taylor’s Series expansion
  to obtain discrete algebric equations is called
  finite difference method.
• Along with this approximation comes some
  amount of error, this type of error is called
  tuncation error, because in taylor’s series
  expansion higher order terms are ignored.
• The tuncation error tends to zero as the grid is
  refined by making Δx and Δy smaller.
• The larger the number of grid points used the
  larger the number of equations that must be
  solved.
Example
Example
• The equations can then be solved through
  computational techniques and the
  solutions between these six nodes can be
  obtained through interpolation.
Grids
• The arrangment of the discrete points in the flow
  domain is called grid.
• The grid must represent the geometry of the
  correctly since an error in this representation can
  have significant error.
• The grid must also have suffient grid resolution.
• It is usually necessary to increase the number of
  grid points where large gradient are to be
  expected as in the boundary layer of the solid
  surfaces.
Type of Grids
• Structured
• Structured grid has some type of regular
  coherent structure to the mesh layout that
  can be defined mathematically.
Types Of Grids
• The grid spacing in the normal direction
  increases as one moves away from the
  surface.Such kind of variable grid spacing
  is used where there is need to increase
  the grid resolution
 and is termed
as grid stretching.
Types of the Grids
• Unstructured
• The grid cell arrangment is irregular and
  has no systamatic pattern.
• It consists of trianles for 2D patterns
• And tetrahedron for 3D patterns.
• Each grid cell and connection information
  to the neighboring cell is defined
  separatly.
• This can be applied to complicated
  geometeries.
Types of Grids
• Finite difference method is restricted to
  structured grids, whereas finite element
  method is can be used either for
  structured or unstructured grids.
Types Of Grids
• Hybrid,
Combination of rectangles and triangles
Moving
Used for flows having time dependent
  geometry
Adaptive
This type of grid adapt itself during the
  simulation.
Area of Applications
Automotives
Biomedical Applications
• CFD can be used to model the flow of
  blood in heart and valves.
• The use of CFD reduces the need of the
  tests on the human being.
Softwares Used In CFD
•   Abaqus CAE,
•   Matlab,
•   Flowlab,
•   Fluent.

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Lecture 11

  • 1. Fluid Mechanics II Lecture 11 Muhammad Usman
  • 2. Experimental Techniques • In certain situation, an experimental investigation involving full-scale equipment can be used to predict how the equipment would perform under given conditions. However in most engineering applications, such full scale tests or either difficult or very expensive to perform or not possible at all.
  • 3. Analytical Techniques • Analytical models work out the consequences of a mathematical model which represents the behavior of a system. The mathematical model representing the physical process mainly consist of a set of differential equations. If classical mathematics were used to solve these equations, we call the approach as analytical approach. • In most engineering applications, various assumptions and simlifications need to be made to enable the analytical solution of the differential equations representing the physical solution. This at one hand limits the applicability of these methods to simple type problems, or limits the validity of the solutions if too many assumptions and simplifications are made.
  • 4. Computational Fluid Dynamics • It is used to calculate the approximate solutions to wide variety of fluid mechanics problems. • Replacing partial differential equation with discreted algebric equation. These equations are then used to calculate the solution at discrete points in space or in time.
  • 5. • The analytical solution for navier stokes equation are available for only limited number of simplified flow geometries. • The CFD Simulation solves for the relevent flow variables only at discrete points. Interpolation are used to obtain the values for non grid location.
  • 6. Numerical Experiements Vs CFD • Modeling • Formulation of the • Measurement governing equation • Analysis of results and development of the numerical algorithm. • Running an algorithm in the computer • Analysis of results
  • 7. Discretization Techniques for Numerical Solution • Finite Difference Method. • Finite Element ( Volume ) Method. • Boundary Element Method.
  • 8. Finite Element Method • Flow field is broken into a set of small fluid elements. • The conservation equations ( Conservation of mass, momentum and energy ) are written for each of the element. • For flows with complex boundaries, the number of algebric equations must be solved also inceases. • Commonly problems include the formation of 1 million gird cells.
  • 9. Boundary Element Method • Boundary of the flow field is broken into discrete segments. • It requires less time and space then finite element method.
  • 10. Finite Difference Method • The method of using Taylor’s Series expansion to obtain discrete algebric equations is called finite difference method. • Along with this approximation comes some amount of error, this type of error is called tuncation error, because in taylor’s series expansion higher order terms are ignored. • The tuncation error tends to zero as the grid is refined by making Δx and Δy smaller. • The larger the number of grid points used the larger the number of equations that must be solved.
  • 12. Example • The equations can then be solved through computational techniques and the solutions between these six nodes can be obtained through interpolation.
  • 13. Grids • The arrangment of the discrete points in the flow domain is called grid. • The grid must represent the geometry of the correctly since an error in this representation can have significant error. • The grid must also have suffient grid resolution. • It is usually necessary to increase the number of grid points where large gradient are to be expected as in the boundary layer of the solid surfaces.
  • 14. Type of Grids • Structured • Structured grid has some type of regular coherent structure to the mesh layout that can be defined mathematically.
  • 15. Types Of Grids • The grid spacing in the normal direction increases as one moves away from the surface.Such kind of variable grid spacing is used where there is need to increase the grid resolution and is termed as grid stretching.
  • 16. Types of the Grids • Unstructured • The grid cell arrangment is irregular and has no systamatic pattern. • It consists of trianles for 2D patterns • And tetrahedron for 3D patterns. • Each grid cell and connection information to the neighboring cell is defined separatly. • This can be applied to complicated geometeries.
  • 17. Types of Grids • Finite difference method is restricted to structured grids, whereas finite element method is can be used either for structured or unstructured grids.
  • 18. Types Of Grids • Hybrid, Combination of rectangles and triangles Moving Used for flows having time dependent geometry Adaptive This type of grid adapt itself during the simulation.
  • 21. Biomedical Applications • CFD can be used to model the flow of blood in heart and valves. • The use of CFD reduces the need of the tests on the human being.
  • 22. Softwares Used In CFD • Abaqus CAE, • Matlab, • Flowlab, • Fluent.