OUTLINE 6-1 Introduction 6-2 An Illustrative Example 6...

Chapter 6: Finite-Volume (Control-Volume) Method-Introduction

Applied Computational Fluid Dynamics Y.C. Shih Sep 2012

Chapter 6 Finite-Volume (control-Volume) Method -Introduction

OUTLINE 6-1 Introduction 6-2 An Illustrative Example 6-3 The Four Basic Rules

6-1



6-1 Introduction (1)

In developing what has become known as the finite-volume method, the conservation principles are applied to a fixed region in space known as a control volume, are somewhat interchangeably in the literature.

6-2




In the finite-volume approach, a point of view is taken that is distinctly different from finite-difference method(or Taylar-series method ). In the Taylor-series method, we accepted the PDE as the correct and appropriate from of the conservation principle(physical law) governing our problem and merely turned to mathematical tools to develop algebraic approximations to derivatives. We never again considered the physical law represented by the PDE. In the finite-volume method, the conservation statement is applied in a form applicable to a region in space (control volume).

6-3




This integral form of the conservation statement is usually well known from the first principles, or it can in most cases, be developed from the PDE form of the conservation from.

6-4




The feature of the FV method is shared in common with the finite-element methods. The FV procedure can, in fact, be considered as a variant of the finite-element method, although it is, from another point of view, just a particular type of finite-difference method.

6-5




As an example, consider unsteady 2-D heat conduction in a rectangular-shaped solid. The problem domain is divided up into control volume with associated points. We can establish the control volumes first and place grid points in the centers of the volumes (cell-centered method) or establish the grid first and then fix the boundaries of the control volumes (cell-vertex method) by, for example placing the boundaries halfway between grid points.

6-6




The General Differential Equation The differential equation obeying the generalized

conservation principle can be written by the general differential equation as

( ) ( ) ( ) )1(−−−−+∇Γ⋅∇=∇+∂

∂ svt

φφρρφ

:dependent variable, such as velocity components (u,v,w), h or T, k, ε concentration, etc.

φ

: diffusion coefficients s : source term

Γ6-7




The four terms of eq.(1) are the unsteady term, the convection term, the diffusion term and the source term.

*Note: The “conservation form” of the PDE is also referred to as “conservation law form” or “divergence form”, i.e., all spatial derivatives appear purely as divergences.

6-8




Conservation form of the governing equations of fluid flow

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) C

T

M

SCDcvtcSpecies

STkhvthEnergy

SvpvvtvMometum

vt

Mass

+∇∇=∇+∂

∂

+∇∇=∇+∂

∂

+∇∇+−∇=∇+∂

∂

=∇+∂∂

ρρ

ρρ

µρρ

ρρ

:

:

:

0:

6-9




One-way and two-way coordinates : 1. Definitions: a two-way coordinate is such

that the conditions at a given location in that coordinate are influenced by changes in conditions on either side of that location. A one-way coordinate is such that the conditions at a given location in that coordinate are influenced by changes in the conditions on only one side of that location.

6-10




2. Examples: one-dimensional steady heat conduction in a rod provides one example of a two-way coordinate. The temperature of any given point in the rod can be influenced by changing the temperature of either end. Normally, space coordinates are two-way coordinates. Time, on the other hand, is always a one-way coordinate. During the unsteady cooling of a solid, the temperature at a given instant can be influenced by changing only these conditions that prevailed before that instant.

6-11




3. Space as a one-way coordinate: If there is a strong unidirectional flow in the

coordinate direction, then significant influences travel only from upstream. The conditions at a given point are then affected largely by the upstream conditions, and very little by the downstream ones. It is true that convection is a one-way process, but diffusion (which is always present) has two-way influences. However., then the flow rate is large, condition overpowers diffusion and thus make the space coordinate nearly one-way.

6-12




4. Parabolic, elliptic, hyperbolic: a) The term parabolic indicates a one way behavior,

while elliptic signifies the two-way concept. b) It would be more meaningful if situations were

described as being parabolic or elliptic in a given coordinate. Thus, the unsteady heat condition problem, which is normally called parabolic, is actually parabolic in time and elliptic in all coordinate. A two-dimension boundary layer is parabolic in the stream wise coordinate and elliptic in the cross-stream coordinate

6-13




c) A hyperbolic problem has a kind of one-way behavior, which is, however, not along coordinate directions but along special-lines called characteristics.

d) A situation is parabolic if there exists at least one one-way coordinate: otherwise, it is elliptic.

e) A flow with one one-way space coordinate is sometimes called a boundary-layer-type flow, while a flow with all two-way coordinate is referred to as a recirculating flow.

6-14




5. Computational implications: The motivation for the foregoing discussion about one-

way and two-way coordinates is that, it a one-way coordinate can be identified in a given situation, substantial economy of computer storage and computer time is possible.

6-15



6-2 An Illustrative Example (1)

The FV method used the integral form of the conservation equation(eq.1) as the starting point:

( ) ( ) )2(−−−−−+∇Γ•=+∇Γ•∇ ∫∫∫∫CVACVCV

dVsdAndVsVd ϕϕ φφ

• Let us consider steady one-dimensional heat conduction governed by

)3(0 −−−−−=+

s

dxdTk

dxd

6-16




1. Preparation: To derive the discrerization equation, we shall employ the grid-point cluster shown in Fig.1. We focus attention on the grid point P, which has the grid points E and W as its neighbors.(E denotes the east side, while W stands for the west side). The dashed lines show the faces of the control volume. The letters e and w denote these faces.

6-17




For one-dimensional problem under consideration, we shall assume a unit thickness in the y and z directions. Thus, the volume of the cv shown is ∆x × 1 ×1. If we integrate eq(3) over the cv, we get

W P

E

(δx)w (δx)e

∆ x

)4(0 −−−−−=+

∂∂

−

∂∂

∫e

wwe

sdxxTk

xTk

Fig. 1

w e

6-18




2. Profile assumption: To make further further progress, we need a profile assumption or an interpolation formula. Here, linear interpolation functions are used between the grid points, as shown in Fig 2.

T x

Fig. 2

W

E

w e δxw δxe

Δx

P

6-19




3. The discrerization equation: If we evaluate the derivatives dT/dx in eq.(4) from the piecewise-linear profile, the resulting equation will be

( )( )

( )( ) )5(0 −−−−−=∆+

−−

−xs

xTTk

xTTk

w

wpw

e

pEe

δδ

6-20




( ) ( ) xsbaaaxka

xkaWhere

bTaTaTa

s

WEPw

wW

e

eE

WWEEpp

∆=+===

−−−−++=

,,,

)6(:form following theinto

(5) eqtion discretiza cast the touseful isIt cv. over the s of valueaverage theis where

δδ

6-21



6-2 An Illustrative Example (7) 4. Comments:

a) In general, it is convenient to extend eq.(6) into multidimensional form as

where nb denotes a neighbor, and the summation is to be taken over all the neighbors.

b) In deriving eq (6), we have used the simplest profile assumption that enabled us to evaluate dT/dx. Of course, many other interpolation functions would have been possible.

c) Further, it is important to understand that we need not use the same profile for all quantities.

d) Even for given variable, the same profile assumption need not be used for all terms in the equation.

)7(−−−−−+= ∑ bTaTa nbnbPP

6-22




5. Treatment of source term: The discretization equations will be solved by the

techniques for linear algebraic equations. The procedure for “linearizing” a given S~T relationship is necessary. Here, it is sufficient to express the average value S as

PPC TSSS +=

6-23




where Sc stands for the constant part of S, while Sp is the coefficient of Tp. With the linearized source expression, the discretization equation will become

( )

( )

xSbxsaaa

xka

xkawhere

bTaTaTa

C

pWEP

w

wW

e

eE

WWEEPP

∆=

∆−+=

=

=

++=

δ

δ

6-24



6-3 The Four Basic Rules (1)

Rule 1: Consistency at a control-volume face -When a face is common to two adjacent control volumes, the flux across it must be represented by the same expression in the discretization equations for the two control volumes

Rule 2: Positive coefficients -All coefficients (ap and neighbor coefficients anb) must always be positive.

6-25



6-3 The Four Basic Rules (2)

Rule 3:Negative-slope linearization of the source term

-When the source term is linearized as S=SC+SPTP , the coefficient SP must always be less than or equal to zero.

Rule 4:Sum of the neighbor coefficients -We require ∑= nbP aa

6-26



Introduction to Moving Zones











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OUTLINE 6-1 Introduction 6-2 An Illustrative Example 6...

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