The Finite Volume Method Finite Volume... · Chapter 4 M. Darwish ([email protected]) American...

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Chapter 4 M. Darwish ([email protected] ) American University of Beirut MECH 663 The Finite Volume Method

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  • Chapter 4M. Darwish ([email protected])

    American University of BeirutMECH 663

    The Finite Volume Method

  • The General Scalar Equation

    ∂ ρφ( )∂t

    transient term

    + ∇. ρuφ( )convective term

    = ∇ ⋅ Γ∇φ( )diffusion term

    + Qφ φ( )source term

    Time derivative

    Source term

    Control VolumeCV

    Advection

    Diffusion

    The scalar equation is a balance equation written in differential form

  • Balance Form

    ∂ ρφ( )∂t

    dVV∫ + ρuφ( ) ⋅ dS

    ∂V∫ = Γ∇φ( ) ⋅ dS

    ∂V∫ + Q φ( )dV

    V∫

    Time derivative

    Source term

    Control VolumeCV

    Advection

    Diffusion

    ∂V

    V

    ∂ ρφ( )∂t

    dVVP

    ∫ + J ⋅ dSf∫

    ⎝ ⎜ ⎜

    ⎠ ⎟ ⎟

    f = nb VP( )∑ = Qφ φ( )dVP

    VP

    To recover the balance form we integrate over some CV

    ∂ ρφ( )∂t

    dVV∫ + ρuφ( ) ⋅ dS

    f∫

    ⎝ ⎜ ⎜

    ⎠ ⎟ ⎟

    f = nb V( )∑ − Γ∇φ( ) ⋅ dS

    f∫

    ⎝ ⎜ ⎜

    ⎠ ⎟ ⎟

    f = nb V( )∑ = Q φ( )dV

    V∫

    J = JC + JD

    = ρuφ( ) − Γ∇φ( )PSf1f1

    Sf2

    f2

    f3 f4

    Sf4

    Sf3

    discrete CV

  • Flux Integration

    ρuφ( ) ⋅ dSf∫

    ⎝ ⎜ ⎜

    ⎠ ⎟ ⎟

    f = nb VP( )∑ − Γ∇φ( ) ⋅ dS

    f∫

    ⎝ ⎜ ⎜

    ⎠ ⎟ ⎟

    f = nb VP( )∑

    P

    Nf

    P

    Nf

    P

    Nf

    1 point gauss integration

    2 point gauss integration

    3 point gauss integration

    J ⋅ dSf∫ = J ⋅S( )ip

    ip( f )∑

    = Jip ⋅ wipS fip( f )∑

    ρuφ( ) f ⋅S f

    ρuφ( )1 f( ) ⋅ w1 f( )S f + ρuφ( )2 f( ) ⋅ w2 f( )S f

  • Volume Integration

    P

    1 point gauss integration

    4 point gauss integration

    9 point gauss integration

    P P

    Q φ( )dVPVP

    ∂ ρφ( )∂t

    dVVP

    = QipVip( )ip(P )∑

    QPVP

    Q1 P( )V1 P( ) + Q2 P( )V2 P( ) + Q3 P( )V3 P( ) + Q4 P( )V4 P( )

  • Semi-Discretized Equation

    wip ρuφ( )ip ⋅S f( )ip~ ip( f )∑

    f = nb VP( )∑ − wip Γ∇φ( )ip ⋅S f( )

    ip~ ip( f )∑

    f ~ faces VP( )∑ = wipQipΩP

    ip~ ip VP( )∑

    PN1

    N4

    N3

    N2

    J1

    J2

    J3

    J4

  • ρuφ −Γ∇φ( )b ⋅Sb

    BC: Flux Specified

    bnt

    P

    Sb

    e

    dPb

    qb,specified

    −Γ∇φ( )b ⋅nbSpecified Flux

    = qb,specified

    ub = 0

    = −Γ∇φ( )b ⋅nbSb

    = qb,specified Sb

    Wall

  • BC: Value Specified

    b

    n

    tP

    Sb

    Φb,specified

    Inlet

    φb = φb,specified

    ρu( )b known

    Jb ⋅Sb = ρuφ( )b ⋅Sb

    No Diffusion

    = ρbub ⋅Sb( )φb,specified

    = Fbφb,specified

  • Spatial Variation

    φ(x) = φP + x − xP( ) ⋅ ∇φ( )P + 12 x − xP( )2 : ∇∇φ( )P

    + 13! x − xP( )3 :: ∇∇∇φ( )P + …

    + 1n! x − xP( )n:::n(∇∇∇

    n φ)P + …

    φ(x) = φP + x − xP( ) ⋅ ∇φ( )P + O Δx 2( )ΦP

    Φ(x)

    ∇ΦPxP

    x

    ∇∇ΦP

  • Mean Value Theorem

    φ P =1ΩP

    φ( )dΩΩP

    = 1ΩP

    φP + x − xP( ) ⋅ ∇φ( )P + O Δx2( )[ ]dΩ

    ΩP

    = φP1ΩP

    dΩΩP

    ∫ + 1ΩPx − xP( )dΩ

    ΩP

    ∫ ⋅ ∇φ( )P +1ΩP

    O Δx 2( )dΩΩP

    = φP + O Δx2( )

    ΦP

  • Problem 3.4Consider one-dimensional diffusion in the calculation domain shown in the figure below with ∆x =1 for all control volumes. Assume that the source term Q=50x, and that Γ is constant with a value Γ=1. Let the boundary values of f be Φ0=100 and Φ4=500a Write the discrete equations for each of the cells 1, 2 and 3.Solve the discrete equation set using Gauss-Seidel iteration and report the resulting cell-centroid valuesCompute the diffusion flux at each of the cell faces f0, f12, f23 and f4 using the same discretization approximations made in obtaining the discrete equationsShow that the conservation principle is satisfied on each discrete cell and on the whole domainFind the exact solution to this problem. Find the percentage error in the computed solution at each cell centroid

    21 3

    (δx) (δx)

    (∆x)

    (δx)/2 (δx)/2

    (∆x)(∆x)

    Φ0=1000 4

    Φ4=500

  • Problem 3.5Consider the computational domain in the figure below. Using the conservation of mass principle compute the mass flux at faces f12, f23 and f4.

    Write the steady state transport equation for a scalar, Φ, being advected through the domain with no diffusion and no source terma. Write the discrete equations for cells 1, 2 and 3 (Use the upwind profile),b. If we assume that (dΦ/dx)4 = 0 compute the value of Φ4. c. If (∆x=1) and (∆y=0.5) compute the values of Φ1, Φ2, Φ3 and Φ4.

    21 3

    (δx) (δx)

    (∆x)

    (δx)/2 (δx)/2

    (∆x)

    (∆x)

    0 4(∆y)

    (3∆y)

  • Problem 3.6Consider the computational domain shown below. Write the diffusion equation for this domain for Γ=1 and Q=0Write discrete equations for cells 1,2 and 3Can you find a unique solution for Φ1, Φ2 and Φ3. Explain ?

    21 3

    (δx) (δx)

    (∆x)

    (δx)/2 (δx)/2

    (∆x)(∆x)

    0 4