FVM LITERATURE REVIEW AND INTRODUCTION TO FLUX LIMITERS

In computational fluid dynamics there are 3 numerical method based solvers to solve the governing navier stokes equation they are

1. Finite difference method : The grid is linear and aligned so that all the grid points lie in a striaght line and the properties are considered to be concentrated at a single point.
2. Finite volume method : The method is applied over a volume such that the governing equation is valid over the entire volume and also the properties are assumed to be concentrated at the geometric center of the volume concerned.
3. Finite element method : The properties are valid over an elemental area and are assumed to be concentrated at the area center.

Drawbacks of Finite difference method :

1. We cannot use FDM directly to a body whose grid points are not aligned properly along x,y,z directions , these may occur due to complex folds , cuts in the structure.
2. As it is not aligned we cannot calculate the properties at the next adjacent point exactly, but we can take some correction factor and approximate it, this method is called as 'variable Transformation method' which basically seeks to create grid using different variable space which look very similar to cartesian grid.
3. Variable transformation method gives an approx solution of property but it is time consuming and complex
4. For more complex profiles FDM is less preferred

so Finite volume method is used

Finite volume method:

It is method for representing and evaluating partial differential equations in the form of algebric equations. In the FVM 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 entering a given volume , these methods are conservative . Another advantage of FVM is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamic packages. Finite volume refers to the small volume surrounding each node point on a mesh.

In FVM instead of writing the equation in a single point , the equations are written in an entire volume. It means for a considered volume no assumptions are made over the shape of the volume . Inside the particular volume the flow is conserved, flow quantities are conserved , energy is conserved , mass is conserved , momentum is conserved and so on . This is called finite volume representation which begins with integral form of governing equation.

Steps :

1. Take the integral form of governing equation
2. Apply guass divergence theorem
3. A discretized form is obtained which can be solved using computer("Flux Terms")

Consider a 1 Dimensional steady state heat conduction equation with heat generation

Where

S is the source term

Governing equation is valiad at every point in domain , which means it is also valid in entire control volume.

If small control volume dv is considered and it is multiplied with the governing equation and integrated for entire volume then that should be also equal to zero i,e the Integral form of governing equation.

Multiplying the equation with dv and integrating it

S̄ is the average value os source term(S) which when multiplied by dv is going to be equal to

Assuming Alpha as a constant and not a function of temperature

The source term which is to be added in a cell , If it is a function of temperature it has to be linearized

There are different schemes used to solve the Equation they are

1. Gradient scheme :

From the above figure the Temperature gradient is calculated at the East face , Basically temperature is used from the neighbouring point trying to evaluate or numerically represent Gradient of temperature is called Gradient scheme

2. Interpolation scheme :

In FVM 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 os steady 1D convection and diffusion

Where Φ is our solution variable(scalar) and ρ is the density and Γ is the diffusion coefficient

The above figure shows the 3 neighbouring control volumes, which are going to use for deriving the discretization equation.

Integration of governing equation over the cv aroung the grid point p yields :

Evaluation of the left hand side of the equation requires the values of Φ, e and w. These values are estimated by using certain interpolation schemes.

Some interpolation schemes used in CFD :

1. Upwind Interpolation scheme
2. Linear Interpolation scheme
3. Quadratic upwind interpolation scheme for convective kinematics (Quick schemes)
4. Hybrid scheme
5. Total variation Diminishing scheme (TVD scheme)

Upwind interpolation scheme :

The upwind interpolation 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

If the fluid is moving from west to east then

If the fluid is moving from east to west then

Inaddition to the gradient schemes and interpolation schemes they are flux limiters

Flux limiters :

While calculating the variable gradients they tend to toggle and they are not bounded to a particular value and keeps jumping , this occurs due to discontinuity due to shock occuring , having the flux limiter will help tobound the gradient value they by giving a stable solution

Numerical approximation using higher order spatial discretization schemes result in better accuracy but exhibits spurious oscillations near discontinuties such as shocks, on the other hand lower order spacial discretization schemes remain stable near discontinuties but dampen out the distributions because of their dissipative nature , hence in order to capture discontinuties ,higher resolution schemes are desired which are highly accurate and stable through out the domain.

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

Fe represents flux at face e calculated using high resolution scheme and Le represents flux at e calculated using lower order scheme and He represents flux at e using higher order scheme then Ψ(r) represents flux limiter function

Flux limiter depends on the smoothness parameter r

The flux limier function is constrained to be greater than or equal to zero. It assumes a value of 0 near discontinuity and a value of 1 away from discontinuity. Hence it switches between higher order scheme and lower order scheme for numerical representation of flux, there by keeping the scheme table accurate through out

Flux limiters are also called as slope limiters because they both have the same mathematical form and both have the effect of limiting the solution gradient near shocks or discontinuties.

The term flux limiters are used when limiter acts on the system fluxes and slope limiters are used when limiter acts on system states (pressure, velocity ..etc)

Consider 1D semi discrete scheme

are the edge fluxes for the ith cell

f high = low precision and high resolution flux

flow = high precision and low resolution flux

ϕ(r)= Flux limiter function

r represents the ratio of succesive gradients on the solution mesh

ϕ(r)>=0

This is the limiter function case

If limiter function =0 then there will be a low resolutiion and sharp gradient

If limiter function >0 then there will be a high resolution and +ve gradient