Week 9 - FVM Literature Review
INTRODUCTION: When we find the values of any variable at different control volumes by finite volume method, we actually consider small finite volumes across the whole volume where each cell shares their faces with other blocks. These faces are the internal common faces ,( not the outer most boundary faces of the whole geometry) .We can calculate the value of any variable eg. Temperature, pressure at the centroid of each control volume, ie every information is calculated for this centroid but the face which is shared by 2 blocks , value of variable at the face is dependent on both those cells that share that face. So how do we calculate the value of the any variable we want at the centre of these common internal faces?
If we consider any variable whose value we want to calculate at the face, eg Temperature or pressure, lets take that variable as phi. We know the value of the variable at the centroid of the cells, so we interpolate the value of the variable at the faces which connects the two blocks .
We have different interpolation schemes for finding value of variable phi at the common face:
INTERPOLATION SCHEMES
There are 4 types of interpolation schemes,
-upwind differencing
-Linear differencing
-Linear upwind differencing
-QUICK
-Hybrid differencing
Consider a cell or a small volume in 2D, each cell is sharing its boundary with 4 neighbouring cells, we name them as per the 4 directions north, west, south, east.
Our focus is only the block P, it shares its faces with 4 cells and its corner with the north-east block and similarly with other blocks at corners, but that corner is just a one common point, we are here considering the blocks with which it share the faces .
The boundary shared between volume P and volume E =e face
The boundary shared between volume P and volume W =w face
The boundary shared between volume P and volume N =n face
The boundary shared between volume P and volume S = s face
P,E,N,S,W are the centroid points of their respective volumes and w, s, e, n are the centre of the respective internal common faces.
-Linear differencing/ Central differencing
According to this scheme, there is a linear relation between value of phi or any variable at the centroid of the cell P and the value at centroid of cell N
ϕf=value of variable at face f( f can be any face ie e, w ,n ,s, here it is e face )
ϕP= value of variable at centroid of block P
ϕE = value of variable at centroid of block E( we can do that for other faces as well)
ψ= Parameter whose value lies between 0 and 1,
Xf- XP is the cell distance between the face centre and the centroid of the P block
XE-XP is the cell distance between the centroid of the neighbouring block ie block E here and the centroid of block P.
ψ= 0 at the starting , ie when the value of Xf=XP
and ψ=1 at the end when value of Xf=XE
When ψ=0, ϕf=ϕP , i.e value of variable at the face = value of variable at centroid of P block
When ψ=1,ϕf=ϕE i.e value of variable at the face= value of variable at centroid of E block (ie the neighbour block )
This is linear variation , much like a straight line.
This is second order accurate. For transient solutions it becomes unstable.
Now if we think it as straight line in a plane, then
f is a face that is shared by the 2 block, value of a variable at centroid of P is ɸP and value at centroid of neighbouring cell E is ɸE , so marking a line between these 2 co-ordinates, we can say that value at face f centre i.e ϕf should be somewhere between ϕP and ϕE because face f centre lies between P and E.
Central Differencing is mostly used in the diffusion terms of the equation.
UPWIND DIFFERENCING
This depends on the mass flux flowing direction. Let mass flux be F.
So, mass flux through face Ff is =density*area*velocity
Where ρf = density through f
Af= area through f
Uf= velocity through the face
n= unit vector in direction of velocity component perpendicular to A
Ff= mass flux going through face f i.e from cell P to neighbouring cell ie cell E here
If Ff>0 , that means flux is going out of the cell P
If Ff<0 , that means flux is going into the cell P
If we have to find the value of any variable ϕ at the face f, then
ϕf=ϕP when Ff>0 ie flux going out of cell P
ϕf=ϕE when Ff<0 i.e flux going into cell p from E here
This scheme is mostly important in convection terms of the navier stokes equation and not in the diffusion term.
So the value of variable at the common face is just a projection of the value of variable at the centroid of the block from which flux is coming out.
This is first order accurate.
LINEAR UPWIND DIFFERENCING
Linear scheme is accurate but not very stable, Upwind scheme is stable but not very accurate.
But the linear upwind differencing scheme is both stable and accurate.
If ϕf is the variable whose value we want to calculate at the common face connecting 2 blocks,
ϕP is the value of variable at the centroid of block P
ϕE is the value of variable at the centroid of block E (ie any neighbouring block here)
∇ϕP=gradient of phi at P ie slope of the value phi at P
∇ϕE= gradient of phi at E i.e slope of value phi at E
r= vector from block block P centroid to face centre
If Ff>0 i.e mass going out from cell P , then
., that means with ɸP term which is constant , a gradient term is also added .Here r is the direction vector from block P to face centre.
If Ff<0, ie mass going into the cell P, then
, that means with ɸE term which is constant, a gradient term is also added . Here r is the direction vector from block E to adjacent face centre.
The value of the gradient can be only upto a certain limit as the value of ɸf lies between ɸ P and ɸE, so the slope also has a limit.For this we use FLUX LIMITER which limits the value of gradient . So,
Flux limiter=Ø
and the value of Ø lies between 0 and 1.
At Ø=0, the scheme becomes same as upwind differencing
HYBRID INTERPOLATION
This is a mixture of linear interpolation and upwind differencing scheme,
ɸf= value of variable phi at the face
ϕ">ϕud = value of variable phi at the face by upwind differencing
ϕ">ϕcd = value of variable phi at the face by central differencing or linear interpolation
ψ= blending function to switch between schemes
QUICK (QUADRATIC UPWIND INTERPOLATION IN CONVECTIVE KINEMATICS)
Here we use one downstream and 2 upstream node points i.e three computational nodes.
The equation for derivation that we assume here is quadratic that is why it is called quadratic interpolation. This is a third order accurate solution. This is mostly used for convection terms.
The general equation is:
fe=value of value of variable at common face centre,
fu= value of variable at upstream node point
fD=value of variable at downstream node point
fUU= value of variable at the point upstream of the upstream point
If Velocity >0 and flow is from left to right,
Then,
Here EE= east of the east block
We want to find value of variable at face common to cell P and cell E ie face e
So, here
fD ie value at the centroid of the downstream cell, becomes= here ( ie value at centroid of E cell)
fU i.e value at the centroid of the upstream cell becomes= here (ie value at centroid of cell P)
fUU = here (i.e value at centroid of cell W which is at the upstream of the upstream side)
Taking the 1D linear heat conduction equation,
If the equation is steady and the source term is constant or if not then linearized, then
Integrating this equation with volume,
Taking small volume as dxdydz, then dz and dy do not matter as it is 1D equation(they are taken constant), and dx varies from w to e( ie. West face to east face in the P cell),
We have applied source term linearization in above equation.
Now, descritization can be done here, dT/dx at e is
dT/dx at w face is
but conductivity at face e and face w have to be found by interpolation here, so we use linear interpolation here,
Sc and Sp , are constant here, so integration over dx gives, Sc.∆x calculated over w to e.
Sp.Tp.∆x where Sp.∆x over w to e is a constant.
Then the whole equation becomes,
∂XE,∂XW is constant, so this means the whole equation can be written in terms of,
This means the equation is now of the form, AX=B, ie algebraic form and this matrix can be solved.
CONCLUSION: Interpolation schemes play very much important role in finding solution in FVM. They are useful in finding the value of variable at the common face of cells.
- The linear interpolation scheme is used mostly for the diffusion terms, it is accurate but less stable. It becomes oscillating and unstable when the mesh is coarser.
- The upwind differencing scheme is more stable, but less accurate, it is first order accurate. It is mostly used in convection terms.
- Linear upwind differencing , uses flux limiter term to correct the range of the solution.
- The QUICK interpolation scheme is mostly used for convection terms and it is 3rd order accurate.