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  • 1

    Solution Methods

    Steady convection-diffusion equation

    Lecture 05

  • Navier-Stokes equation

  • • This is the actual form of the conservation equations solved by finite

    volume based CFD programs to calculate the flow pattern and

    associated scalar fields.

    ACV

    Integral form

    • The key step of the finite volume method is to integrate the differential

    equation shown in the previous slide, and then to apply Gauss’

    divergence theorem, which for a vector a states:

    ∫divadV = ∫n⋅adA

    • This then leads to the following general conservation equation in integral

    form:

    Suggested reading: Gauss divergence theorem

  • The steady convection-diffusion equation

    Formal integration over a control volume gives

    • This equation represents the flux balance in a control volume. • The left hand side gives the net convective flux and the right hand

    side contains the net diffusive flux and the generation or destruction of the property  within the control volume.

  • • The principal problem in the discretisation of the convective terms is the calculation of the value of transported property  at control volume faces and its convective flux across these boundaries.

    • The central differencing method was used for obtaining discretised equations for the diffusion and source terms.

    • It would seem obvious to try out this practice, which worked so well for diffusion problems, on the convective terms.

    • However, the diffusion process affects the distribution of a transported quantity along its gradients in all directions, whereas convection spreads influence only in the flow direction.

    • This crucial difference manifests itself in a stringent upper limit to the grid size, that is dependent on the relative strength of convection and diffusion, for stable convection-diffusion calculations with central differencing.

    Discretization scheme

  • Steady one-dimensional convection and diffusion

    • In the absence of sources, the steady convection and diffusion of a property  in a given one-dimensional flow field u is governed by

    • The flow must also satisfy continuity so

  • A control volume around node P

    • Integration of transport equation over the control volume of figure above gives

    • And integration of continuity equation yields

    • To obtain discretised equations for the convection-diffusion problem we must approximate the terms in equations above.

    • It is convenient to define two variables F and D to represent the convective mass flux per unit area and diffusion conductance at cell faces:

    Discretization

  • • The cell face values of the variables F and D can be written as

    • We develop our techniques assuming that Aw = Ae = A and employ the central differencing approach to represent the contribution of the diffusion terms on the right hand side.

    • The integrated convection-diffusion equation can now be written as

    • and the integrated continuity equation

    • We also assume that the velocity field is 'somehow known', which takes care of the values of Fe and Fw.

    • In order to solve above equation we need to calculate the transported property  at the e and w faces. Schemes for this purpose are discussed.

    Discretization…

  • The central differencing scheme

    • The central differencing approximation has been used to represent the diffusion terms which appear on the right hand side of convection-diffusion equation.

    • It is logical to try linear interpolation to compute the cell face values for the convective terms on the left hand side of this equation. For a uniform grid we can write the cell face values of property  as

    On substitution,

    And re-arranging, gives

  • • Identifying the coefficients of W and E as aW and aE the central differencing expressions for the discretised convection-diffusion equation are

    Where,

    The central differencing scheme…

    • It can be easily recognised that above equation for steady convection-diffusion problems takes the same general form as for pure diffusion problems.

    • The difference is that the coefficients of the former contain additional terms to account for convection.

    • To solve a one-dimensional convection-diffusion problem we write discretised equations above for all grid nodes.

    • This yields a set of algebraic equations that is solved to obtain the distribution of the transported property .

  • Equation for pure diffusionEquation for convection-diffusion

    Comparison

  • A property  is transported by means of convection and diffusion through the one- dimensional domain sketched in figure below. The governing equation below; boundary conditions are 0 = 1 at x = 0 and L = 1 at x = L . Using five equally spaced cells (for first two cases) and the central differencing scheme for convection and diffusion calculate the distribution of  as a function of x for cases: (i) Case 1: u = 0.1 m/s, using 5 cells (ii) Case 2: u = 2.5 m/s, using 5 cells (iii) Case 3: u = 2.5 using 30 cells

    and compare the results with the analytical solution given below. The following data apply: length L = 1.0 m, p = 1.0 kg/m3,  = 0.1 kg/m/s.

    Assignment 7 Page 106, Versteeg

  • Case 1: u = 0.1 m/s, 5 cells Case 2: u = 2.5 m/s, 5 cells Case 3: u = 2.5, 30 cells

    F/D = 0.2 F/D = 5 F/D = 1.25

    • F/D = uL/ = Peclet number (Pe); • Pe = Re Pr; • Re is Reynolds no. and Pr is Prandl number

    Note: For high Pectlet number, result oscillates (case 2)

    Result comparison

  • Properties of discretisation schemes

    • The central differencing fails in in certain cases involving combined convection and diffusion schemes.

    • In theory numerical results may be obtained that are indistinguishable from the 'exact' solution of the transport equation when the number of computational cells is infinitely large irrespective of the differencing method used.

    • However, in practical calculations we can only use a finite - sometimes quite small - number of cells and our numerical results will only be physically realistic when the discretisation scheme has certain fundamental properties.

    • The most important ones are: • Conservativeness • Boundedness • Transportiveness • Accuracy

  • Conservativeness

    • To ensure conservation of  for the whole solution domain the flux of  leaving a control volume across a certain face must be equal to the flux of  entering the adjacent control volume through the same face.

    • To achieve this the flux through a common face must be represented in a consistent manner - by one and the same expression - in adjacent control volumes.

    Example of consistent specification of diffusive fluxes

  • • The fluxes across the domain boundaries are denoted by qA and qB.

    • Four control volumes are considered and apply central differencing to calculate the diffusive flux across the cell faces.

    • The expression for the flux leaving the element around node 2 across its west face is w2 (2 - 1) and the flux entering across its east face is e2 (3 - 2).

    • An overall flux balance may be obtained by summing the net flux through each control volume taking into account the boundary fluxes for the control volumes around nodes 1 and 4.

    Conservativeness…

  • Conservativeness… • Boundary fluxes for the control volumes around nodes 1 and 4.

    • Since e1, = w2, e2, = w3 and e3, = w4 the fluxes across control volume faces are expressed in a consistent manner and cancel out in pairs when summed over the entire domain.

    • Only the two boundary fluxes qA and qB remain in the overall balance so above equation expresses overall conservation of property .

    • Flux consistency ensures conservation of  over the entire domain for the central difference formulation of the diffusion flux.

  • Boundedness

    • The discretised equations at each nodal point represent a set of algebraic equations that needs to be solved.

    • Iterative numerical techniques are used to solve large equation sets. • These methods start the solution process from a guessed distribution of the variable 

    and perform successive updates until a converged solution is obtained. • Scarborough (1958) has shown that a sufficient condition for a convergent iterative

    method can be expressed in terms of the values of the coefficients of the discretised equations:

    • Here a‘P is the net coefficient of the central node P (i.e. aP - Sp) and the summation in the numerator is taken over all the neighbouring nodes (nb).

    Scarborough criteria

  • Boundedness…

    • If the differencing scheme produces coefficients that satisfy the above criterion the resulting matrix of coefficients is diagonally dominant.

    • To achieve diagonal dominance we need large values of net coefficient (aP - Sp) so the linearisation practice of source terms should ensure that SP is always negative. If this is the case -Sp is always positive and adds to aP.

    • Diagonal dominance is a desirable feature for satisfying the 'boundedness‘ criterion.

    • This

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