Week 6 Challenge-FVM Fundamentals 101

Aim : FVM Fundamentals 101

1. Derive the expression for the ∂f∂x∂�∂� for a forward difference of second order

The Taylor expansion of `f _(i + 1)` is given by:

`f _(i + 1) = f _(i) + f ' _(i) Δ x + f ' ' _(i) Δ x^ 2/ (2 !) + f ' ' ' _(i) Δ x^ 3 /(3 !)` + ... (1)

Similarly, the Taylor expansion of `f _(i - 1)` is given by:

`f _(i - 1) = f _(i) - f ' _(i) Δ x + f ' ' _(i) Δ x^ 2/ (2 !) - f ' ' ' _(i) Δ x^ 3 /(3 !)` + ... (2)

Now subtracting Eqn(1) with Eqn(2) and neglecting the higher-order terms, we get:

`f_( i + 1) − f _(i − 1) = 2 f '_ i Δ x + O ( Δ x^ 3 )` (3)

Now, finally dividing Eqn(3) with `2 x` on both sides, we get,

`(f _(i + 1) − f _i )/(2 Δ x) = f ' i + O ( Δ x^ 2 )`

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2. Apply the finite volume formula to the equation `(∂ ρ ϕ) / (∂ t) + ∇ . ( ρ u ϕ ) = 0`

For Finite Volume Formulation , here we would be using Gauss Divergence theorem, which helps us convert volume integrals into surface integrals and vice versa

The given differential equation is of the form '(∂ ρ ϕ) / (∂ t)+ ∇. (ρ u ϕ) = 0' (1)

Applying Intergration on Eqn(1) on both sides, we get:

`∫ ( ∂ ρ ϕ ) /( ∂ t) + ∫ ∇ . ( ρ U ϕ ) = 0` (2)

Now, multiplying by dV on both sides, we get:

`∫ ( ∂ ρ ϕ ) /( ∂ t) dV + ∫ ∇ . ( ρ U ϕ ) dV = 0` (3)

Now, we can apply Gauss Divergence Theorem and re-write the Eqn(3) as,

`∂/ (∂ t) ∰ ρ ϕ d V + ∯ ρ Uϕ^ → d S^→ = 0` (4)

The above Eqn(4) is the required equation

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3. Write a note on the van Leer TVD scheme ((Hint: Use Sweby Diagram)

Total Variation Diminishing :

• Total Variation Diminishing (TVD) schemes are numerical methods used in computational fluid dynamics to solve hyperbolic partial differential equations
• TVD schemes are crucial for advection problems, offering stability without introducing numerical artifacts such as excessive diffusion or compression
• In OpenFOAM, convection schemes, including TVD, are based on Normalized Variable (NV) and TVD methods, emphasizing their significance in various simulations

Van-Leer TVD Scheme :

• It is the original limited advection scheme described by the limiter function

• `\beta = (r+\|r|)/(1+|r|)`
• Since `\beta -> 2` as `r -> \infty`, the van Leer scheme uses relatively strong downwind interpolation.