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Objective : 1. Describe the difference between the FDM and FVM. 2. Need of the interpolation schemes. 3. Flux limiters in Finite volume method. …
Epuri Yoga Narasimha
updated on 16 Apr 2021
Objective : 1. Describe the difference between the FDM and FVM.
2. Need of the interpolation schemes.
3. Flux limiters in Finite volume method.
4. How to find the gradient at the cell centroid.
Finite Difference Method :
The finite-difference method is the most direct approach to discretizing partial differential equations. consider a point in space where we take the continuum representation of the equations and replace it with a set of discrete equations, called finite-difference equations. The finite-difference method is typically defined on a regular grid and this fact can be used for very efficient solution methods. The method is therefore not usually used for irregular CAD geometries, but more often for rectangular or block-shaped models.
There is a connection with the finite-element method: Certain formulations of the finite-element method defined on a regular grid are identical to a finite-difference method on the same grid. Regular grids can often be used in meteorological, seismological, and astrophysical simulations, for example.
Delaunay triangulations are used to build meshes for space-discretized solvers for the finite-element and finite-volume methods
Finite Volume Method :
The finite-volume method is similar to the finite-element method in that the CAD model is first divided into very small but finite-sized elements of geometrically simple shapes. Apart from this, the finite-volume method is very different from the finite-element method, beginning with the concept of elements, which are instead referred to as cells.
The finite-volume method is based on the fact that many physical laws are conservation laws—what goes into one cell on one side needs to leave the same cell on another side. Following this idea, end up with a formulation that consists of flux conservation equations defined in an averaged sense over the cells. This method has been very successful in solving fluid flow problems
Finite volume methods can be compared and contrasted with the Finite Difference Method which approximate derivatives using nodal values, or Finite Element method , which create local approximations of a solution using local data, and construct a global approximation by stitching them together.
In contrast a finite volume method evaluates exact expressions for the average value of the solution over some volume, and uses this data to construct approximations of the solution within cells.
For the approximations within the cell we use the interpolation schemes.
The discretization procedure used in the finite volume method is distinctive and involves two basic steps.
In the first step, the partial differential equations are integrated and transformed into balance equations over an element (or finite volumes) into which the domain has been subdivided, then the Gauss Divergence theorem is applied to transform the volume integrals of the convection and diffusion terms into surface integrals. The result is a set of semi-discretized equations.
In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations.
The advantage of FVM is that it allows easily to formulate unstructured mesh, also FVM formulation itself gives physical intuition. In many CFD packages FVM method is used.
Two types in FVM :
1. Cell centered. ( Data structure is at the centroid of the cell ).
2. Vertex centered. ( Data structure is at the vertex ).
Types of Interpolation schemes :
1. Linear/central differencing.
2. Upwind.
3. Linear Upwind.
--> Advance Interpolation schemes:
4. Quick scheme.
--> Hybrid Interpolation Schemes :
5. TVD scheme.
6. Gamma Differencing.
q1. Why do we need Interpolation schemes?
-> CFD codes computes the values at the cell centroid.
-> Need to know the values at the cell faces (or) within the faces.
-> Same analysis for all faces.
-> Each face connects an owner cell (P) and the Neighbour cell(N).
-> Don't care about number of faces here , this analysis is valid for all polyhedral cells.
-> Also valid for the hanging nodes.
-> We only need an owner and neighbour cell centrois and a connecting face.
-> Interpolation schemes are only for the internal cells.
-> For the boundary cells -> wall functions.
Analysis set up:
1. Linear/Central Differencing :
ϕf=ψ⋅ϕN+(1-ψ)⋅ϕP
ψ=|xf-xP||xN-xP|
0≤ψ≤1
In this interpolation scheme , all the flied variables follows the linear relation accross the cell.
Second order accurate.
Scheme is Unbounded scheme. (Wiggles in the simullation).
Used under special circumtances only.
Central Differencing is consitionally stable (Pecelet Number < 2).
Using central differencing interpolation scheme , No Limitations ,
For using the central differencing for the convection term , limitations arising. -> (Origin of problem).
So we need to assume a another profile for the convection terms.
2.Upwind Differencing :
1. This scheme depends on the direction of the mass flux.
Ff=ρA⋅Af⋅(Uf⋅ˆn)
Ff>0 => Indicates mass flow out of the cell.
Ff<0 => Indicates mass flow into the cell.
First order accurate.
Upwind scheme is not accurate but stable , use it for the initial condition , we get the initial condition in the case of bad mesh , unstable simulation , difficult in the convergence ,solution diverging quickly.
Upwind scheme -------> Initial condition --------------> Most accurate scheme ----------------> Accurate solution.
3. Linear Upwind Differencing :
1. Most accurate than the Upwind scheme.
2. Nominally second order accurate.
3. Variation b/w cell centroid and the face is linear.
4. Use (∇ϕ)cellcentroid to improve the accuracy of the extrapolation`
5. Linear upwind scheme is only intended for the convection term.
∇ϕ at cell centroid is calculated by the gradient techniques
1. Green gauss cell based and node based gradient.
2. Least squares Gradient scheme.
** Gradient Limiters :
-> We have often have to limit the gradient (∇ϕ)_p to prevent the local maxima/minima.
-> control of the steepness.
-------->
Hybrid Interpolation scheme:
These schemes are the combination of the central differecing(for accurate) and he Upwind differencing(for stable).
ϕf=ξ⋅ϕUD+(1-ξ)⋅ϕCD
ξ is a blending function to switch between schemes.
If the mesh is unstable -> Towards the UD for stable.
If the mesh is stable -> Towards the CD for better accuracy.
Overall we get an accurate and the stable solution using the advance interpolation schemes.
1. Total variation Diminishing.
2. Gamma Differencing.
QUICK Interpolation scheme:
-> Higher Order schems , use three grids for the evalutation of cell-face value.
-> Considered the quadratic profile b/w the three cell centers.
-> QUICK -> Quadratic Upstream Interpolation for Convective Kinematics.
-> Third Order Accurate.
->
UU = WW.
DD = EE.
Even a quick scheme shows the wiggles around the discontunities , won't capture the correct physical behaviour of the problem.
To get rid of overshoot and the undershoot problems in the solution , we use TVD scheme , an hybrid scheme.
Flux Limiters in FVM :
-> For a convective fluid flow , Low order accurate schmes are stable , but unstable during the discontinuity or shocks , where as the higher order schemes unbalanced or unstable in nature because they show High oscillations near the discontinuities or shocks.
-> These are used in the High Resolution schemes . Helps in limiting the solution gradient near shocls and disocntinuties.
-> Objective is to control teh spatial derivatives to the realistic values.
-> Help in limiting the gradient near shocks or discontinuities.
-> The flux limiters is used when the limiter acts on the system and the slope limiters is used when the limiter acts on the system(Like pressure and velocity).
-> Flux limiter are selected based on the problem and the scheme considered for the solution.
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