Week 9 - FVM Literature Review
Finite volume method:Finite volume method is a domain discretization technique used alternatively to finite difference method and finite element method where small volume surrounding a grid point in control volumes is discretized and solved for the governing equations.
It solves governing equtions in the integral form and satisfies the mass conservation property up to a great extent when compared with finite difference and finite element method.This method is more efficient and widely used when grids are not aligned with the cartesian axes.
Interpolation schemes:
The approximation of surface and volume integrals may require values of variables at different locations other than computational nodes of control volume, values at these locations are found by using interpolation schemes.
Different types of interpolation schemes are,
1) Upwind interpolation scheme:
This scheme is equivalent to forward or backward difference approximation method depending on the flow direction.It is first order accurate and is numerically diffusive with a coefficient of numerical difusion.
Linear Interpolation :
we can approximate the value of the variable at control volume face centre by linear interpolation of the values at two nearest computational nodes.Thus ,at location 'e' on a cartesian grid,the variable value is approximated by
`f_e=f_Elambda_e+f_p(1-lambda_e), lambda_e=(x_e-x_p)/(x_E-x_p);`
The linear interpolation is equivalent to the use of central difference formula of the first order derivative and hence this scheme is also termed as central differnce scheme.This scheme is secnd order accurate and may produce oscillatory solutions.
Quadratic upwind interpolation(QUICK):
A Quadratic interpolation scheme can be derived using polynomial fitting.The QUICK interpolation on a uniform cartesian grid is given by
`phi_e=6/8phi_U+3/8phi_D-1/8phi_(UU)`
where D,u and UU denote the downstream,the first upstream and the second upstream node respectively(E,P and W or P,E and EE depending on the flow direction),this is a third order accuarate scheme prone to oscillations in solution.
Explained with 1D heat conduction equation at steady state, assumed alpha as a function of temperatute
`alpha_e.A.(dt/dx)_e-alpha_w.A.(dt/dx)_w+q.dx.A`
`implies (alpha_E-alpha_P).(T_E-T_P)/dx.A-(alpha_P-alpha_W).(T_P-T_W)/dx.A+q.A.dx=0`
Flux:
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. In vector calculus flux is a scalar term defined as a surface integral of perpendicular component through a surface .In transport phenomena it is a vector having both magnitude and direction
Flux limiters:
Flux limiters are used in high resolution schemes(which are generally second or higher order accurate) – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics described by partial differential equations (PDE's). They are used in high resolution schemes, to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution scheme, make the solutions total variation diminishing (TVD).
Note that 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 (like pressure, velocity etc.).