steady state diffusion equation - dr. om prakash ... worked examples: one-dimensional steady state...

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  • 1

    Solution Methods

    Lecture 04

    Steady State Diffusion Equation

  • Solution methods

    • Focus on finite volume method.

    • Background of finite volume method.

    • Discretization example.

    • General solution method.

    • Convergence.

    • Accuracy and numerical diffusion.

    • Pressure velocity coupling.

    • Segregated versus coupled solver methods.

    • Multigrid solver.

    • Summary.

  • Overview of numerical methods

    • Many CFD techniques exist.

    • The most common in commercially available CFD programs are:

    • The finite volume method has the broadest applicability (~80%). • Finite element (~15%).

    • Here we will focus on the finite volume method.

    • There are certainly many other approaches (5%), including:

    • Finite difference method (FDM). • Finite element method (FEM). • Spectral method. • Boundary element method (BEM). • Vorticity based methods. • Lattice gas/lattice Boltzmann. • And more!

  • 4

    Finite difference method (FDM)

    • Historically, the oldest of the three.

    • Techniques published as early as 1910 by L. F. Richardson.

    • Seminal paper by Courant, Fredrichson and Lewy (1928) derived

    stability criteria for explicit time stepping.

    • First ever numerical solution: flow over a circular cylinder by

    Thom (1933).

    • Scientific American article by Harlow and Fromm (1965) clearly and publicly expresses the idea of “computer experiments” for the

    first time and CFD is born!!

    • Advantage: easy to implement.

    • Disadvantages: restricted to simple grids and does not conserve

    momentum, energy, and mass on coarse grids.

  • • The governing equations (in differential form) are

    discretized (converted to algebraic form).

    • First and second derivatives are approximated by

    truncated Taylor series expansions.

    • The resulting set of linear algebraic equations is

    solved either iteratively or simultaneously.

    Finite difference: basic methodology

    • The domain is discretized into a series of grid points.

    • A “structured” (ijk) mesh is required.

  • Subtract:

    Taylor Series: central finite difference method

    • Central difference formula • 2nd order accurate

  • • Earliest use was by Courant (1943) for solving a torsion problem.

    • Clough (1960) gave the method its name.

    • Method was refined greatly in the 60’s and 70’s, mostly for

    analyzing structural mechanics problem.

    • FEM analysis of fluid flow was developed in the mid- to late 70’s.

    • Advantages: highest accuracy on coarse grids. Excellent for

    diffusion dominated problems (viscous flow) and viscous, free

    surface problems.

    • Disadvantages: slow for large problems

    and not well suited for turbulent flow.

    Finite element method (FEM)

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    • First well-documented use was by Evans and Harlow (1957) at Los

    Alamos and Gentry, Martin and Daley (1966).

    • Advantage: Was attractive because while variables may not be continuously differentiable across shocks and other discontinuities; mass, momentum

    and energy are always conserved.

    • FVM enjoys an advantage in memory use and speed for very large

    problems, higher speed flows, turbulent flows, and source term

    dominated flows (like combustion).

    • Late 70’s, early 80’s saw development of body-fitted grids. By early 90’s,

    unstructured grid methods had appeared.

    • Advantages: basic FV control volume balance does not limit cell shape;

    mass, momentum, energy conserved even on coarse grids; efficient,

    iterative solvers well developed.

    • Disadvantages: false diffusion when simple numerics are used.

    Finite volume method (FVM)

  • • Integrate the differential equation over the control volume and

    apply the divergence theorem.

    • To evaluate derivative terms, values at the control volume faces

    are needed: have to make an assumption about how the value


    • Result is a set of linear algebraic equations: one for each control


    • Solve iteratively or simultaneously.

    Finite volume: basic methodology

    • Divide the domain into control volumes.

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    Cells and nodes

    • Using finite volume method, the solution domain is subdivided

    into a finite number of small control volumes (cells) by a grid.

    • The grid defines the boundaries of the control volumes while the

    computational node lies at the center of the control volume.

    • The advantage of FVM is that the integral conservation is

    satisfied exactly over the control volume.

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    Typical control volume

    • The net flux through the control volume boundary is the sum of

    integrals over the four control volume faces (six in 3D). The

    control volumes do not overlap.

    • The value of the integrand is not available at the control volume

    faces and is determined by interpolation.

  • The Finite Volume Method for Diffusion Problems

  • • Consider the 1D diffusion (conduction) equation with source term S

    Finite Volume method

    Another form,

    • where  is the diffusion coefficient and S is the source term.

    • Boundary values of  at points A and B are prescribed.

    • An example of this type of process, one-dimensional heat conduction in a


  • Step 1: Grid generation

    • The first step in the finite volume method is to divide the domain into discrete control


    • Place a number of nodal points in the space between A and B. The boundaries (or faces)

    of control volumes are positioned mid-way between adjacent nodes. Thus each node is

    surrounded by a control volume or cell.

    • It is common practice to set up control volumes near the edge of the domain in such a

    way that the physical boundaries coincide with the control volume boundaries.

    • A general nodal point is identified by P and its neighbours in a one-dimensional

    geometry, the nodes to the west and east, are identified by W and E respectively.

    • The west side face of the control volume is referred to by 'w' and the east side control

    • volume face by ‘e’.

    • The distances between the nodes W and P, and between nodes P and E, are identified by

    xWP and xPE respectively. Similarly the distances between face w and point P and

    between P and face e are denoted by xwP and xPe • The control volume width is x = xwe

  • Step 2: Discretisation

    • Integrate over the control volume, from west to east face

    • The key step of the finite volume method is the integration of the governing equation (or equations) over a control volume to yield a discretised equation at its nodal point P.

    = 0

    • Here A is the cross-sectional area of the control volume face, V is the volume and S is the average value of source S over the control volume. It is a very attractive feature of the finite volume method that the discretised equation has a clear physical interpretation.

    • Above equation states that the diffusive flux of  leaving the east face minus the diffusive flux of  entering the west face is equal to the generation of , i.e. it constitutes a balance equation for  over the control volume.

  • Step 2: Discretisation…

    • In order to derive useful forms of the discretised equations, the interface diffusion coefficient  and the gradient d/dx at east (‘e’) and west ('w') are required.

    • The values of the property  and the diffusion coefficient are defined and evaluated at nodal points.

    • To calculate gradients (and hence fluxes) at the control volume faces an approximate distribution of properties between nodal points is used. Linear approximations seem to be the obvious and simplest way of calculating interface values and the gradients.

    • This practice is called central differencing. • In a uniform grid linearly interpolated values for e and w are given by

    • And the diffusive flux terms are evaluated as

  • • In practical situations, the source term S may be a function of the dependent variable. In such cases the finite volume method approximates the source term by means of a linear form:

    Step 2: Discretisation…

    Therefore, equation become


    • Identifying the coefficients of W and E as AW and AE and the coefficient of P as AP , the above equation can be written as


    Discretised form of diffusion equation

  • Step 3: Solution of equations

    • Discretised equations of the form above must be set up at each of the nodal points in order to solve a problem.

    • For control volumes that are adjacent to the domain boundaries the general discretised equation above is modified to incorporate boundary conditions.

    • The resulting system of linear algebraic equations is then solved to obtain the distribution of the property  at nodal points.

    • Any suitable matrix solution technique


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