Finite Volume Method is a sub domain method with piecewise definition of the
field ... In the implementation of the finite volume method, the heat fluxes are.
FINITE VOLUME METHOD: BASIC PRINCIPLES AND EXAMPLES SHRUTI JAIN B.Tech III Year, Electronics and Communication IIT Roorkee
Tutors: Professor G. Biswas Professor S. Chakraborty
ACKNOWLEDGMENTS I would like to thank my mentor, Professor G. Biswas without whose help, this presentation would not have been possible. My presentation is a result of his lecture notes and his consistent guidance. Participating in the winter academy has been a learning experience and I would like to thank all the members of the academy for giving me this opportunity. It has been a pleasure working on this presentation.
FINITE VOLUME METHOD: AN OUTLINE Introduction to Finite Volume Method 2 approaches
-Weighted Residual Approach -Physical Approach Boundary Conditions in Finite Volume Method Equations with First Order Derivatives Equations with Second Order Derivatives
FINITE VOLUME METHOD Finite Volume Method is a sub domain method with piecewise definition of the field variable in the neighborhood of chosen control volumes. The total solution domain is divided into many small control volumes which are usually rectangular in shape. Nodal points are used within these control volumes for interpolating the field variable and usually, a single node at the centre of the control volume is used for each control volume. This method was developed by Patankar and Spalding and they proposed the use of physical approach for deriving the nodal equations.
FINITE VOLUME METHOD Consider a 2-D, steady heat conduction in rectangular geometry (Figure 1).The 2-D heat conduction equation is (1)
Where T(x,y) is the temperature field, k is the thermal conductivity and Q is the heat generation per unit volume. The two alternative ways of settings up the nodal equations are the weighted residual approach and the physical approach
WEIGHTED RESIDUAL APPROACH Weighted residual approach for 2-D heat conduction equation:
(2)
Where the weight Wi =1 within the ith control volume Wi =0 outside the ith control volume Thus, we get, for each i= 1,…n (3)
Integrating the above equation by parts, we get:
WEIGHTED RESIDUAL APPROACH
Typical control volume
Figure 1: Grid Arrangement for the Finite Volume Method
Where the Gauss divergence theorem has been used to convert the volume integral to a surface integral. (4)
WEIGHTED RESIDUAL APPROACH The meaning of Equ. (4) is that the net heat generation rate in the control volume is equal to the net sum of the rate of heat energy going out of the control volume )
where ci is the boundary of the control volume
PHYSICAL APPROACH
Figure 2: Balance of Heat Flux in a Control Volume
PHYSICAL APPROACH Considering the balance of heat flux in Figure 2 for the control volume (for unit depth in z direction):
Net rate of heat leaving the control volume through the boundary = Rate of heat generation within the control volume (CV) at steady state: (5)
which is the same equation as obtained before.
PHYSICAL APPROACH In the implementation of the finite volume method, the heat fluxes are represented in terms of nodal temperatures (e.g. TE etc. at the CV centers). Thus , assuming temperature to have linear variation between points E and P, the heat flux qe can be evaluated as follows:
(6)
While deriving equation (6) it has been assumed that the cell size is, constant in x-direction. Similarly, qw is given by (7)
PHYSICAL APPROACH Using similar expressions for qn and qs also, the nodal equation for point P becomes:
(8)
This equation can be rewritten in the familiar form used in finite difference as:
Where,
(9)
During numerical implementation, the subscripts E, W, etc. will be changed to numerical indices of i, j and solved in the some way.
BOUNDARY CONDITIONS: When heat flux at a boundary is specified
Figure 3: Prescribed Heat Flux at the Boundary
The control volumes adjacent to the x = 0 boundary as shown Figure 3, the term qw will be substituted by q0 in equation (5).Thus,
(10)
BOUNDARY CONDITIONS: When boundary temperature is specified When boundary temperature is specified, the control volume shapes near the boundary can be changed to facilitate the implementation of the boundary conditions. For example: consider the BC, T=TL at boundary x=L. For nodes on boundary an imaginary extension of CV’s outside the actual domain can be considered. We consider the boundary to be the center of the boundary cells and their width to be ∆x/2, thus reducing the width of the adjacent cells to 3∆x/4. Figure 4: Boundary Condition, at x=L, T= TL
BOUNDARY CONDITIONS The nodal equation for the adjacent cell P will be written considering a shortened control volume
(11)
Where Note that TL has been substituted instead of TW is the above equation, so the boundary condition is being directly applied.
EQUATION WITH FIRST DERIVATIVES General first-order equation: (12)
For the above equation is the inviscid momentum equation in the x-directions.
In a similar manner, for x direction viscous momentum equation, (13)
EQUATION WITH FIRST DERIVATIVES
Figure-5: Finite Volume Mesh System
EQUATION WITH FIRST DERIVATIVES We consider the area integral of general first order equation over Ωp (14)
Ωp is the finite volume (quadrilateral) ABCD shown in figure 5 Green’s theorem: (15)
EQUATION WITH FIRST DERIVATIVES Using Green’s theorem the equation (14) becomes: (16)
where H = (F,G) and n is the outward unit normal of segment ds. On a counter clockwise contour n dyi dx j , where ds = dx dy 2
2
ds
Hence H.n ds = F dy – G dx. The finite volume method is a discretization of the governing equation in integral form, in contrast to the finite difference method, which is unusually applied to the governing equation in differential form. (17)
EQUATION WITH FIRST DERIVATIVES Where A is the area of the quadrilateral ABCD in Figure-5, and the average value of E over the quadrilateral is represented by Ei,j and the remaining terms are approximations for the line integral over segments AB, BC, CD and DA respectively
Figure-5: Finite Volume Mesh System
EQUATION WITH FIRST DERIVATIVES With similar expressions for time, then:
, etc. If A is not a function of
(18)
If the grid-mesh is uniform and coincides with lines of constant x and y, the above equation (18) becomes: (19)
which coincides with a central difference representation for the spatial terms in equation (12)
EQUATIONS WITH SECOND DERIVATIVES Let us consider Laplace’s equation: (20)
The finite volume method follows the application of the subdomain method to equation (20).
where Green’s theorem has again been used and now: (21)
EQUATIONS WITH SECOND DERIVATIVES The above equation can be evaluated approximately over the segments AB, BC,CD and DA by:
(22)
EQUATIONS WITH SECOND DERIVATIVES
Figure 4: Finite Volume for a Curvilinear Grid
Here
is evaluated as the mean value over the area . in Figure 4.
EQUATIONS WITH SECOND DERIVATIVES Thus we can write: (23) (24)
Using Green’s theorem:
A similar expression can be written for
dx
x A'B ' B xB 'C ' i , j xC 'D ' A xD ' A'
i , j 1
A' B 'C ' D '
EQUATIONS WITH SECOND DERIVATIVES If the mesh is not too distorted,
y A'B ' yC 'D ' y AB y B 'C ' y D ' A' y j 1, j x A'B ' xC 'D ' x AB xB 'C ' xD ' A' x j 1, j S A, B S A'B 'C 'D ' | A' B' B' C ' | Hence, (25)
EQUATIONS WITH SECOND DERIVATIVES Therefore the equation (23) & (24) becomes: (26)
(27)
Now, the first line of equation (22) can be evaluated as: (28)
EQUATIONS WITH SECOND DERIVATIVES In a similar manner, one can evaluate the following: (29)
(30)
EQUATIONS WITH SECOND DERIVATIVES If the mesh is not too distorted,
The equivalent expressions for similar terms in equation (22), finally yields:
and other
(31)
EQUATIONS WITH SECOND DERIVATIVES Where the geometrical parameters are:
In equation (31) are evaluated as the average of the four surrounding nodal values. Thus
EQUATIONS WITH SECOND DERIVATIVES Substitution into equation (31) generates the following nine-points discretization of equation (20):
(32)
We need to calculate the geometric quantities like MAB, NAB etc. only once for a given grid and can use the obtained values for all subsequent calculations.
EQUATIONS WITH SECOND DERIVATIVES Equation (32) is solved conveniently using a Successive OverRelaxation (SOR) technique.
(33)
and the improved better value is: (34)
where is the relaxation parameter. The discretised equation (32) reduces to the centre finite difference scheme on a uniform rectangular grid (35)
SUMMARY Finite Volume Method is a sub domain method with piecewise definition
of the field variable in the neighborhood of chosen control volumes. The total solution domain is divided into many small control volumes which are usually rectangular in shape. Boundary Conditions may be applied in finite volume equations by substituting for heat fluxes at boundaries or by changing the control volume spacing (while substituting for boundary temperatures). The finite volume method is a discretization of the governing equation in integral form, in contrast to the finite difference method, which is unusually applied to the governing equation in differential form. Finite volume method can be applied in first and second order equations and the discretized equation finally reduces to the central finite difference scheme on a uniform rectangular grid. One attractive feature of the finite volume method is that it can handle Neumann boundary condition as readily as the Dirichlet boundary condition.
REFERENCES Lecture notes of Professor G. Biswas Fletcher, 'Computational Techniques for Fluid Dynamics‘ Peyret and Taylor, ‘Computational Methods for Fluid Flow'
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