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• Measurement • Analysis of results

• Formulation of the governing equation 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.