Week 9 - FVM Literature Review
Finite Volume Method.
To give a introductory part on which method suits the best for simulation is quite difficult and depending upon the assumption made by the engineer. As we know that the conservational laws of fluid motion are given either in differential form or in integral form.
When any numerical scheme is applied to the differential equation, the domain in which the solution will be divided results in the discrete domain upon which finite difference equation are solved. On the contrary, when the integral form of equation are solved the domain is divided into small volumes. Hence this conservation laws can be applied to this elementary volumes. This integral method includes the finite element and finite volume methods.
If we recall the finite difference equation in which partial differential equation are solved on the rectangular domain which is equally spaced points. As we know most of the physical domain are irregular in shape, the conservation of physical space to computation space is performed where the computation space is rectangular for most of the case. To cope with the problem highly irregular bodies co-ordinate transformation are applied, however even with that it would not give proper accuracy and the convergence for the simulation. This is happen because metrices and the jacobian transformations and the corresponding gradients which are used in the equation are may include much more discontinuities if the computational grid is not smooth enough.
This weakness accosiated with Finite difference approach is not reflected in the Finite volume method because the independet variable are integrated directly on the volume of domain or physical domain hence grid smoothness is no longer required for finite volume method. Also for finite volume method it is not required to always go for structured grid system as in the case of finite difference approach hence can be used for unstructured grid. But if we have structured grid for complicated shape also it is always recommended to go over finite difference method due to its efficiency over finite volume and element method.
Calculation of Result for domain:
In Finite volume method two type of schemes are used one is "cell centered" and other one is "nodal point" scheme. To know how they applied we will just consider an equation:
`(delQ)/(delt) + (delE)/(delx) + (delF)/(dely) = 0`
We will consider an quadrilateral over which above equation can be solved.
From the equation we can see that Q is the quantity changing with respect time hence need to be evaluated. Therefore the equation is going to integrated over an element as shown in figure.
`int_(abcd)(delQ)/(delt) dxdy = - int_(abcd) ((delE)/(delx) + (delF)/(dely))dxdy`
For right hand side to change the area integral to line integral we will apply Green's theorem,
`int_(abcd)((delQ)/(delt))dxdy = - oint_(abcd) (Edy - Fdx)`
The dependent variabel Q is solved at the point 5 which at the center of cell, hence the scheme is known as cell centered scheme. The scheme can be explicit or implicit depending upon how the value of E and F are evaluated.
`((Q_5^(n+1) - Q_5^(n))/(Deltat))A_(abcd) = -[E_iDeltay_(ab) + E_jDeltay_(bc)+E_mDeltay_(cd)+E_nDeltay_(da)]+[F_iDeltax_(ab) + F_jDeltax_(bc)+F_mDeltax_(cd)+F_nDeltax_(da)]`
Where `A_(abcd)` is the area of quadrilateral and points i,j,m and n are midpoint of edge ab,bc,cd and da respectively.
The value of E and F can be calculated by averaging value from two control points located on other side of corresponding edges can be given as:
`E_i = 1/2*(E_5^star + E_2^star) ,`
`E_j = 1/2*(E_5^star + E_3^star)`
`E_m = 1/2*(E_5^star + E_4^star)`
`E_n = 1/2*(E_5^star + E_1^star)`
`F_i = 1/2*(F_5^star + F_2^star)`
`F_j = 1/2*(F_5^star + F_3^star)`
`F_m = 1/2*(F_5^star + F_4^star)`
`F_n = 1/2*(F_5^star + F_1^star)`
Depending upon the value of `E_5^star & F_5^star` the scheme can be implicit and explicit.
For Nodal scheme we will consider below figure:
In nodal point scheme the the dependent variable are calculated at vertices of cells. The governing equation for nodal scheme is written as:
`[((Q_a + Q_b + Q_c + Q_d)^(n+1) - (Q_a + Q_b + Q_c + Q_d)^n)/(4Deltat)]A_(abcd) = -[E_iDeltay_(ab) + E_jDeltay_(bc)+E_mDeltay_(cd)+E_nDeltay_(da)]+[F_iDeltax_(ab) + F_jDeltax_(bc)+F_mDeltax_(cd)+F_nDeltax_(da)]`
The value for E and F are calculated from the cell distinct points which are given as:
`E_i = 1/2*(E_a^star + E_b^star)`
`E_j = 1/2*(E_b^star + E_c^star)`
`E_m = 1/2*(E_c^star + E_d^star)`
`E_n = 1/2*(E_d^star + E_a^star)`
`F_i = 1/2*(F_a^star + F_b^star)`
`F_j = 1/2*(F_b^star + F_c^star)`
`F_m = 1/2*(F_c^star + F_d^star)`
`F_n = 1/2*(F_d^star + F_a^star)`
Interpolation Schemes for Finite Volume Methods:
The interpolation schemes used for the finite volume methods are:
1) Upwind Interpolation schemes.
2) Hybrid Interpolation schemes.
3) QUICK Interpolation schemes.
4) TVD Interpolation schemes.
Let discuss this schemes in more details.
1) Upwind Interpolation scheme.
In the central difference scheme there is major problem because of its inability to identify the flow direction. In the consideration of central difference scheme the face value is always influenced by the both central points of volume. But in the case of pure convection problem in which the transport properties are moving from west to east then the face value should have major proportion of W node in the scheme. The convected value at the face of domain is equal to the value at the upstream function.
When the flow is in positive direction, `u_w>0, u_e > 0` Then the upwind scheme is given for face value as:
`phi_w = phi_W and phi_e = phi_P`
and the discretized eqaution for one dimensional steady state flow equation without source term is given as:
`F_ephi_p - F_wphi_W = D_e(phi_E-phi_P)-D_w(phi_P - phi_W)`
where `F = rhou` and `D = (Gamma/(delx))`
When the flow in is negative direction, `u_w<0 , u_e < 0` the scheme takes the value as
`phi_w = phi_P and phi_e = phi_E`
and the discretized eqaution for one dimensional steady state flow equation without source term is given as:
`F_ephi_E - F_wphi_P = D_e(phi_E-phi_P)-D_w(phi_P - phi_W)`
The above scheme is based on the backward difference formula hence the accuracy is first-order approximate. The major drawback of the upwind interpolation scheme is that it produce error results when the flow is not aligned with the grid lines.
2) Hybrid Interpolation scheme
The hybrid interpolation scheme utilizes the both central difference scheme and upwind interpolation scheme. The central difference scheme which is second-order accurate is applied to small peclet number (Pe < 2) and the upwind difference scheme which is first order accurate is applied to the large peclet number (Pe >= 2).
The net flux value from each face is calculated from piecewise formulae based on peclet number which make hybrid interpolation scheme very much stable. The peclet number is calculate as for west face:
`Pe_w = (F_w)/(D_w) = (rhou)_w/(Gamma_w/(delx_WP)`
The hybrid scheme is use both advantages of central difference and upwind scheme. The hybrid interpolation scheme switches to upwind differencing when central difference scheme giving the incorrect values. The only disadvantages is that accuracy in terms of Taylor series is only first order.
3) QUICK Interpolation Scheme
The quadratic upstream interpolation for convective kinetics (QUICK) use three-point upstream weighted quadratic interpolation for cell face values. The face value phi is calculated from the quadratic function passes through two bracketing nodes which are on each side of face and other node from upstream side.
For example, when `u_w > 0 and u_e> 0` the quadratic fit is running through WW in above figure, W & P are used to calculate the `phi_w` and the quadratic fit running through W, P and E are used to calculate the `phi_e`
Again when `u_w <0 and u_e<0` the value of W, P and E will be used to calculate the `phi_w` and the P, E and EE are used to calculate the `phi_e`
If we have the uniform grid then face value phi is calculated as:
`phi_(face) = 6/8 phi_(i-1) + 3/8phi_i + 1/8 phi _(i-2)`
Therefore, when `u_w >0` the face value is given as
`phi_w = 6/8phi_W + 3/8phi_P - 1/8 phi_(WW)`
When `u_e > 0` then face value is given as:
`phi_e = 6/8phi_P + 3/8phi_E - 1/8 phi_W`
As the above scheme is based on the quadratic function it's accuracy is in terms of Taylor's series is third order if any uniform mesh is present.
4) TVD Interpolation scheme
The TVD schemes are specially developed to remove the oscillation from the solution. The higher order scheme will create some problem over the boundary condition hence it is required to develope the second-order scheme to avoid the problem.
In TVD (Total Variation Diminishing) to counter the oscillations by artificial diffusion fragment or by adding a weighting toward the upwind scheme.
Flux Limiters
As we have seen from the above methods that lower order scheme are unconditionally stable but gives more error when any discontinuity or shock is present. Similarly, the higher order scheme are more accurate but have oscillation or wiggles when come across higher peclet number.
The flux limiter function are used to provide the smooth operation over the both higher order scheme and lower order schemes. The flux limiters function can be applied to TVD schemes are given as:
`F(u_(i + 1/2)) = f^l (i+1/2)-phi(ri)(f^l(i+1/2) - f^h(i+1/2))`
`F(u_(i + 1/2)) = f^l (i-1/2)-phi(r_i - 1)(f^l(i-1/2) - f^h(i-1/2))`
where,
`f^l` = Low precision flux
`f^h` = High precision flux
`phi(r)` = flux limiter function.
`r = (u_i-u_(i-1))/(u_(i+1)-u_i)`