Finite Volume Method FVM Literature Review

Aim : To describe the need for interpolation schemes and flux limiters in Finite Volume Method (FVM).

Theory :

In computational fluid dynamics (CFD) there are 3 commonly used numerical method based solvers to solve the governing equations i.e Navier Stokes Equations. These 3 methods are:

• Finite Difference Method :

Finite-difference methods (FDM) are discritizations used for solving differential equations by approximating them with difference equations that finite differences approximate the derivatives.

FDMs convert a linear ordinary differential equations (ODE) or non-linear partial differential equations (PDE) into a system of equations that can be solved by matrix algebra techniques. For FDMs the grid is linear and should be aligned in such a way that all grid points lie in a straight line. The properties and governing equations are considered to be concentrated at a single point.

• Finite Volume Method :

This method is applied over a volume so that the governing equation is valid over the entire volume. The properties are assumed to be concentrated at the centre area. The volume integrals in a partial differential equation containing a divergence term are converted into surface integrals, using the divergence theorem. The obtained terms are considered as fluxes at the surfaces of each finite volume because when we look through these discretised equations the flux entering a given volume is identical to that leaving the adjacent volume, which is known as conservative. FVM is widely used as it can be applied for complex geometry and moving parts.

• Finite Element Method :

The FEM is a particular numerical method for solving PDEs in two or three space variables. For solving a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. The finite element method finally results in a system of algebraic equations. This method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem.

Drawbacks of Finite Difference Method:

• We cannot use FDM directly to a body whose grid points are not aligned correctly along the x, y or z directions. These situations may occur due to complex designs, folds and cuts in an geometry.

• Since it is not aligned we cannot calculate the properties at the next adjacent point exactly (since there is an inclination between 2 mesh points), but we can use some correction factor in order to approximate the difference with that adjacent point. This Method is known as Variable Transformation Method. This approach seeks to create a grid using other space variables such as eta and epsilon. These space grids looks similar to a cartesian grid. A suitable mapping function is defined and this function can be easily transformed through cartesian to new space vaiables (eta,epsilon) and vice versa.

• The Variable Transformation Method gives an approximate solution of the property and also time consuming and complex.

• For complex profiles and moving parts the above method is less preferred.

Due to this Finite Volume Method (FVM) is used.

Finite Volume Method :

The finite volume method (FVM) is a method for representing and evaluating partial differential equations PDEs in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. \"Finite volume\" refers to the small volume surrounding each node point on a mesh.

In FVM instead of writing equation for a single point, we write equations for an entire volume. Also for a considered volume no assumptions are made over the shape of the volume. Inside this particular volume the flow is conserved, flow quantities are conserved, energy, mass, momentum is conserved and so on. This is called as Finite Volume Representation. The starting point of FVM is with \'Integral form of governing equations\'.

Steps for FVM discretisation :

• We will write the required governing equation in its integral form.
• Apply Gauss Divergence Theorem.
• We obtain discretised form which can be solved using a computer (\"Flux Terms\").

Consider a steady state heat conduction equation with a source term (heat generation source) :

where

S is the Source term.

`alpha` is thermal diffusivity term.

`(delT)/(delx)` is Temperature gradient along x direction.

The above governing equation is valid at every point in the domain, meaning it is valid for the entire control volume.

If we consider a small control volume (dv) and it is multiplied with the governing equation and integrated for entire volume then that should also be equal to zero i.e. the integral form of governing equation.

Taking the conservation equation validated at single point and if it is validated at single point then it is validated atentire volume and in this entire volume the properties are conserved.

Multiplying the equation by \'dv\' and integrating it, we obtain

`bar S` is the average value of source term (S) which when multiplied by dv will be equal to `intSdv`.

If we assume `alpha`as a constant and not as a function of temperature,

The source term which is to be added in a cell, if it is a function of temperature we have to linearize it. If we consider `alpha`as a function of temperature we have to linearize it as well.

(`barS` linearization `rarr` `S_u + S_pT_p`)

The amount of heat that is removed from a particular box is conserved.

Different Schemes are used to solve the equation:

• Gradient Scheme :

From the above figure when we calculate temperature gradient on the east face, we basically use temperature from neighbouring point trying to evaluate or numerically represent gradient of temperature which is called as gradient scheme.

• Interpolation Scheme :

In the finite volume formulation, evaluation of surface and volume integrals sometimes requires the values of unknown variables at locations other than the computational nodes of the control volume. These values are obtained by employing certain interpolation schemes.

As an example let us consider the simple case of steady state one dimensional convection and diffusion.

where

`phi` is our solution variable (scalar)

`rho` is the density

`u` is the velocity

`Gamma` is the diffusion coefficient.

The above figure represents the three neighbouring control volumes, which we are going to use for deriving the discretization equation.

Integration of the governing differential equation over the control volume around grid point P yeilds:

Evaluation of LHS of the above equation requires value of `phi` over faces e and w. These values are estimated by employing certain interpolation schemes.

Some interpolation schemes used in CFD are as follows:

1. Upwind interpolation scheme.
2. Linear interpolation scheme.
3. Quadratic Upwind interpolation for convective Kinematics (QUICK) scheme.
4. Hybrid Scheme.
5. Total Variation Diminishing (TVD) scheme.

• Upwind Interpolation Scheme:

This scheme takes into account the direction of fluid flow and based on it the value of solution variable at the face is approximated by its value at the upstream computational node. For example, if the fluid is moving from west to east, then:

`phi_e ~~ phi P`

if the fluid is moving from east to west , then:

`phi_e ~~ phi E`

• Linear Interpolation Scheme:

In order to approximate the value of variable at control volume face center by two nearest computational nodes location e. This is equivalent to a central difference scheme of first order derivative. It is second order accurate and produce oscillatory solutions. The relation for linear interpolation is given below:

Apart from Gradient scheme and Interpolation schemes there are something called Flux Limiters.

Flux Limiters

While calculating the variable gradients they tend to toggle and are not bounded to a particular value and keep jumping. This occurs due to discontinuity, having a flux limiter will help to bound the gradient value giving a stable solution.

Numerical approximation using higher order spatial discretization schemes results in better accuracy but exhibits spurious oscillations near discontinuities such as shocks. On other hand, lower order spatial discretization schemes remain stable near discontinuities but dampen out the distributions due to their dissipative nature. Hence to capture discontinuities, high resolution schemes are required which are highly accurate and stable throughout the domain.

Flux Limiters operate functionally as a switch and tune the numerical fluxes obtained from a lower order scheme and higher order scheme such that near the discontinuity lower order (stable) scheme is employed and away from it higher order (accurate) scheme is employed.

Suppose a flux at face e is calculated using high resolution scheme, `L_e`represents flux at e calculated using low order scheme and `H_e`represents flux at e calculated using higher order scheme. Then `psi(r)` represents the flux limiter function such that:

`F_e = L_e + psi(r)(H_e-L_e)`

Flux limiter function depends upon the smoothness parameter r which is defined as:

The flux limiter function is constrained to be greater than or equal to zero. It assumes a value of 0 near discontinuities anda value of 1 away from them. Hence it switches between higher order scheme and lower order scheme for numerical representation of flux, thereby keeping the scheme stable and accurate throughout.

The flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have the effect of limiting the solution gradient near shocks or discontinuities. In general, the term flux limiter is used when the limiter acts on system fluxes and slope limiter is used when the limiter acts on system states (like pressure,velocity etc.).

Consider 1D-semi discrete scheme,

`f^(high)` - low precision, high resolution flux

`f^(low)` - high precision, low resolution flux

`phi(r)` - flux limiter function

\'r\' represents the ratio of successive gradients on the solution mesh.

`phi(r) >= 0`

This is the limiter funcyion case

if limiter function = 0, then there will be a low resolution and sharp gradient.

if limiter function > 0, then there will be a high resolution and positive gradient.