Week 9 - FVM Literature Review
Objective:
To describer the need for interpolation scheme and flux limiter in FVM.
Basics:
In computation fluid dynamics there are 3 numerical method based solvers to solve the governing navier stokes equations these are:
1. Finite Difference Method
2. Finite Element Method
3. Finite Volume Method
In Finite Difference Method:
The grid is linear and aligned so that all grid points lie in a straight line. The properties are considered to be concentrated at a single Point.
In Finite Volume Method:
The method is applied overa volumae such that the governing equation is valid over the entrie volume, as also the properties are assumed to be concentrate at the geometric centre of the volume concerned.
In Finite element method:
The properties are valid over an elemental area. The properties assumed to be concentrated at area center.
Drawbacks of Finite Difference Method:
We cannot use FDM directly to a body whose grid points are not aligned properly along the x,y directions. These may occur due to complex folds, cuts in the structure.
As it is not aligned we cannot calculate the properties at the next adjacent point exactly, but we can make some correction factor and approximate it, this method is called as Variable Transformation Method which basically seeks to create grid using different variabel space which look very similar to cartesian grid.
Variable transformation method gives an approx solution of property but it is time consuming and complex
For Complex profiles this method is less preferred.
That's why Finite Volumet Method (FVM) is used.
Finite Volume Method
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 is it easily formulated to allow for unstructural meshes. The method is used in many computational fluid dynamics packages. 'Finite Volume' refers to the small volume surrounding each node point on a mesh.
In FVM instead of writing the equation in a single point, equations are writted in an entire volume. It means for a considered volume no assumptions are made over the shape of the volume. Inside the particular volume the flow is conserved, flow quantites are conserved , energy is conserved, mass is conservedm momentum is conserved and so on. This is called as Finite Volume Representation which begins with Integral form of Governing Equation
Steps:
1. Once we have this Intgeral form of equation
2. Apply Gauss Divergency Theorem
3. We get descretized form which can be solved using Computer ('Flux Terms")
Consider a steady heat conduction equation with heat generation:
`del/(delx).(alpha.(delT)/(delx))+S=0`
S is source term
alpha is thermal diffusivity term
If small control volume(dv) is considered and it is multiplies with the governing equation and integrated it for entire volume then that should be also equal to Zero. i.e the integral form of giverning equation.
Taking the conservation equation validated at single point and if it is validated at single point then it is validated at entire volume and in this entire volume properties are considered.
`int(del/(delx).(alpha.(delT)/(delx))+S)dv=0`
`int(del/(delx).((alpha.delT)/(delx))dv+Sdv=0`
dv=A.dx
S is an average value of Source term(S) which when multiplied by dv is going to be equal to Integral Sdv
If we assume alphs as a constant and not a function of temperature
`alphaA[(Te-Tp)]/(Deltax)-alphaA[(Tp-Tw)]/(Deltax) + Sdv=0`
Heat Flux out - Heat Flux in + Sourceterm = 0
The amount of heat that is removed from a particular box is conserved.
Different Schemes used to solve the equation:
1. Gradient Scheme
2. Interpolation Scheme
Gradient Scheme
From the above figure when we calculate temperature gradient in the east face, we basically use temperature from neighboring point trying to evaluate or numerically represent gradient of temperature is called as Gradient Scheme.
2. Interpolation Scheme
In the finite volume formulation, evaluation of surface and volume integrals sometimes requires the values of unknown variables at locations other than the computational nodes of the control volume. These values are obtained by employing certain interpolation schemes.
As an example let us consider the simple case of steady one-dimensional convection and diffusion.
`(d(puphi))/dx = d/dx(T(dphi)/dx)`
where phi is our solution variable, rho is density, u is the velocty, I is the diffusion coefficient.
Somr Interpolation schemes used in CFD are as follows
1. Upwind Interpolation Scheme
2. Linear Interpolation Scheme
3. Quadratic Upwind Interpolation for Convective Kinematics.
3. Hybrid Scheme
5. Total Variation Diminshing
The Upwind Interpolation scheme takes into account the direction of fluid flow and based on it, the value of solution variable at the face is approximately by its value at the upstreat computational node. For instance, if the fluid is moving from west to east.
`phie approx phip`
and if it is from east to west
`phie approx phiE`
Flux Limiters
While calculating the variable gradients they tend to toggel and are not bounded to a particular value and keeps jumping, this occurs due to discontinuity due to shock occuring, having a fluw limiter will help to bound the gradient value there by giving a stable solution.
Numerical approximation using higher order spatial discretization scheme results in better accuracy but exhibits spurious oscillations near discontinuities such as shocks. On the other hand, lower order spatial discretization schemes remain stable near discontinuites but dampen out the distributions because of their dissipative nature. Hence in order to caputre discontinuites high resolution schemes are desired which are highle accurate and stable throught the domain.
Flux Limiters operate functionally as a swicth and tune the numerical fluxes obtained from lower order scheme and higher order scheme such that near the discontinuity lower order(stable) scheme is employed and away from it higher order (accurate) scheme is employed.
Suppose represent flux at face e calculated using high resolution scheme, Le represents flux at e calculated using lower order scheme and He represents flux at e calculated using higher order scheme. The phi(r) represents the flux limiter function such that:
`Fe=Le + psi(r)(He-Le)`
The flux limiter function is constrained to be greater than or equal to zero. It assues a value of 0 near discontinuites and a value of 1 away from them. Hence it switches between higher order scheme and lower order scheme for numerical representation of flux, thereby keeping the scheme stable and accurate throughout.
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 discontinuites. In genral the term flux limiteer is used when the limter acts on system fluxes and slope limiter is used when the limiter acts on system states
Conisder 1D-Semi Discrete Scheme
r represents the ratio of successive gradients on the solution mesh
phi(r)>0
this is the limiter function case
If limiter function = 0, then there will be a low resolution and sharp gradient.
If limter function > 0 then there will be a high resolution and positive gradient.