Week 9 - FVM Literature Review
FINITE VOLUME METHOD(FVM)
-The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume 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 at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes.The method is use many computational fluid dynamics packages. Finite volume refers to the small volume surrounding each node point on a mesh.
-Finite volume method is like FEM, is based on an unstructured mesh. Therefore, it is suitable for irregular and complex geometries. FVM has another advantage over FEM for fluid mechanic problems. So far, the numerical methods that we presented have been based on PDEs. By contrast, FVM is based on the integral form of the conservation laws, rather than their differential form. This leads to more accuracy/stability, especially for sharp gradients inside a domain, which is also called shock-capturing property. To explain this more clearly, as we mentioned before, the dynamics of flow can be described by conservation of mass, momentum, and energy. These conservation laws can be written as a system of PDE.For instance, the differential form of the continuity equation can be written as
`(delrho)/(delt) + (del(rhou))/(delx) +(del(rhov))/(dely)+(del(rhow))/(delz) =0`
where u, v, and w are the components of flow velocities in the x, y, and z directions, and ρ is the density. The integral form of continuity for a control volume or a finite volume can be written as,
`del/(delt)intrhodV +intrhou.ds = 0`
where V is the control volume, S is source term of control surface u = (u, v, w).
Interpolation scheme to solve the equation-
-Interpolation schemes are used to transform cell-centre quantities to face centres. The operation is used in many of the finite volume calculations, for the calculation of gradient, divergence, and Laplacian terms.
Some Interpolation schemes use in openFoam are given below,
Upwind interpolation scheme-
- The upwind interpolation scheme takes into account of fluid flow and based on it , the value of solution variable at the face is approximates by its value at the upstream computational node,for instance if the fluid is moving from west to east then `phi approx phiP` and if the fluid is moving east to west then `phiundersete``approx phiE` .
Linear interpolation scheme-
-linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
Quadratic upstreme interpolation scheme-
It is ahigher order differencing scheme that considers a three-point upstream weighted quadratic interpolation for the cell face values. In computational fluid dynamics there are many solution methods for solving the steady convection diffusion equation.
- In order to interpolation scheme there are Flux limiter to limit the solution gradient near the shocks instead of allowing it to grow which would have failed to reach a solution.
FLUX LIMITERS-
-The main idea behind the construction of flux limiter schemes is to limit the spatial derivatives to realistic values – for scientific and engineering problems this usually means physically realisable and meaningful values. They are used in high resolution schemes for solving problems described by PDEs and only come into operation when sharp wave fronts are present. For smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher order approximations without introducing spurious oscillations.
- flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have the effect of limiting the solution gradient near shocks or discontinuities. In general, the term flux limiter is used when the limiter acts on system fluxes, and slope limiter is used when the limiter acts on system states.