Week 9 - FVM Literature Review
MFV consists on three steps like MDF ,the difference between them appears in the step of grid generation and equation obtained as a result of this grid form, and the general differences resumed on:
-In the MDF we we solve the equation which describe the process f some phenomena on singular points ,but we cannot use it directly, when the grids are not aligned along x,y orz direction
-instead of writing equation in a point as in the MDF, we write it for a volume, then we apply a few mathematical techniques to help us solve these equations for any type of grids .
-Also ,the finite volumere presentation preserves the concept of conservation.
-inside this control volume mass is conserved
,in other hand in the MVF we solve through integring the general equation by applying the gauss divergence theorem the equation which describes the process in a control volume, we can use the 1D linear heat conduction as an example for illustraion:
equation for local point:
The equation is validated at a single point then it's validated on the entire volume, and we just summing it up, and we basicaly say in the entire volume the property is conserved.
in the common use we approximate the source term mean as follow:
.
heatfluxout-heatfluxin+Source=0:
Then we proceed for the discretisation using the taylor series :
The discretisation process has yielded one equation for each of the nodal points 1 to 5:
where phi in this case is temerature.
So we can apply this equation on all the nodes ,then we get the algebraic system. And , in the end of the day we reach the solution.
the need for interpolation schemes :
The approximation of the area and volume integrals may require values of the variable at locations other than the CV compute nodes. The values at these locations are obtained using interpolation formulas
the need offlux limiters in FVM:
The flow limiter functions are used to tune higher order and lower order patterns so that the resulting pattern gives high order accuracy in the smooth region of the flow and maintains first order accuracy in the vicinity of shocks and discontinuities.