Week 9 - FVM Literature Review

Assignment name-literature review of finite volume method

1.What is finite volume method-

FVM is computational discretization method and it is used in CFD and it is based on control volume formulation of analytical fluid dynamics. In FVM we divide the domain into number of control volume where interested or calculated variables are in the centre of the control volume and then integrated the differential form of governing equations over each control volume in a domain because if we consider control volume valid in one variable so it will be valid entire domain. FVM satisfy conversation results or quantities like mass, momentum, energy.

2.Difference between finite volume method (FVM) & finite Difference method (FDM)-

• FVM is similar to FDM but the FVM integrate differential equation over the control volume and discretising domain. FVM accuracy is based on order of polynomial used.
• In FDM the grid is linear and aligned so that all grid point in straight line and the properties are consider to concentrated at single point
• The properties are valid over an elemental area and properties are assumed at area centre point
• FVM computes solution at each control volume whereas FDM computes solution at each point.
• FVM converse the result better than FDM

3.stpes to solve FVM

Find out integral for of governing equation

Apply gauss divergence theorem

Used suitable method for calculating flux term

4.interpolation scheme-

• It is used to transform cell centre quantities to face centre
• Examples-the calculation of gradient, divergence and Laplacian term

5.Types of Interpolation Scheme-

Upwind interpolation Scheme (UDS)-

consider the following one-dimensional linear advection equation

which describes a wave propagating along the x-axis witha α velocity.Consider a typical grid point i in the domain. In a one-dimensional domain, there are only two directions associated with point i – left (towards negative infinity) and right (towards positive infinity). If α is positive, the traveling wave solution of the equation above propagates towards the right, the left side of i is called upwind side and the right side is the downwind side. Similarly, α is negative the traveling wave solution propagates towards the left, the left side is called downwind side and right side is the upwind side. If the finite difference scheme for the spatial derivativecontains more points in the upwind side, the scheme is called an upwind scheme

linear interpolation scheme (CDS)-

linear interpolation is a simple technique used to estimate unknown values that lie between known values. The concept of linear interpolation relies on the assumption that rate of change between the known values is constant. And the calculate this value form simple slope formula.

Quadratic upwind interpolation scheme-

For the one-dimensional domain shown in the figure the Φ value at a control volume face is approximated using three-point quadratic function passing through the two bracketing or surrounding nodes and one other node on upstream side. In the figure, in order to calculate the value of the property at the face, we should have three nodes i.e. two bracketing or surrounding nodes and one upstream node.

1. Φ_wwhen u_w > 0 and u_e > 0 a quadratic fit through WW, W and P is used,
2. Φ_ewhen u_w > 0 and u_e > 0 a quadratic fit through W, P and E is used,
3. Φ_wwhen u_w < 0 and u_e < 0 values of W, P and E are used,
4. Φ_ewhen u_w < 0 and u_e < 0 values of P, E and EE are used.

Let the two bracketing nodes be i and i − 1 and upstream node i – 2 then for a uniform grid the value of φ at the cell face between the three nodes is given by:

Hybrid interpolation scheme-It is combination of CDS (linear interpolation) and UDS(upwind interpolation)

6.Gradient scheme-

In a gradient scheme finding the temperature at east face e we get information from neighbouring point E

7.Flux Limiter in FVM

They are used high resolution Schemes, such as MUSCL scheme, to avoid the spurious oscillation(wiggles) that would otherwise occur with high order spatial discretization scheme due to shock, discontinuities or sharp changes in the solution domain. Flux limiter also referred as slope limiter

8.example-linear heat conduction equation

Note-taking two images because of clear image

Form above equations we can say that, finding the temperature at east face e we get information from neighbouring point E called as gradient scheme and if we can find values at unknown location we need or require interpolation scheme