discretization of convection-diffusion type equations by finite
TRANSCRIPT
7 T H INDO GERMAN WINTER ACADEMY- 2008
Discretization of convection-diffusion type equations by
Finite Volume Method
Ritika Tawani
Department of Chemical Engineering Indian Institute of Technology, Bombay
Guides: Prof. Suman Chakraborty, IIT-Kharagpur
Prof. Vivek V. Buwa, IIT-Delhi
Contents
⯠The Convection Diffusion Equation ⯠Finite Volume Method
⯠Four basic rules ⯠Central Differencing Scheme ⯠Upwind Differencing Scheme ⯠Exact Solution ⯠Exponential Scheme ⯠Hybrid Scheme ⯠Power Law Scheme ⯠Higher Order Differencing Schemes ⯠QUICK Scheme ⯠Discretization Equations for 2-D, 3-D ⯠Handling the Source term ⯠Handling the Unsteady term
⯠False Diffusion
The Convection Diffusion Equation
⯠The general differential equation, for the conservation of a physical property, !
⯠The 4 terms are: Unsteady term, Convection term, Diffusion term and Source term
⯠In general, ! = !(x, y, z, t) = !(x, y, z, t) (x, y, z, t) âŻ! is the diffusion coefficient corresponding to the particular property !, S is the corresponding source term , S is the corresponding source term
⯠As ! takes different values we get conservation equations for different quantities
eg: !=1: Mass conservation =1: Mass conservation !=u: x-momentum conservation =u: x-momentum conservation !=h: Energy conservation =h: Energy conservation
Finite Volume Method
⯠Key concept: Integration of differential equation over Control Volume
⯠For simplification, we first do finite volume formulation for 1-D steady state equation(with no source term)
⯠The flow field should also satisfy continuity equation
Finite Volume Method
⯠Control Volume(CV) to be used:
⯠Integration of transport equation for the shown CV gives
⯠Derivatives for diffusion term are calculated assuming piecewise linear profile of !
,
Finite Volume Method
⯠Assuming , the integral of transport equation becomes,
where,
⯠Also, from continuity equation, we have
⯠There are various methods to calculate the Convection term and will be discussed after the four basic rules
Four Basic Rules
⯠For solutions to be: 1. Physically realistic 2. Satisfy overall balance (conservative) There are some basic rules that need to be satisfied by the discretization equations
Standard form of discretization equations(1-D): Rule 1: Flux consistency at CV faces
When a face is common to two adjacent control volumes, flux across it must be represented by the same expression in discretization equations for both the control volumes
Rule 2: Positive coefficients All coefficients must always be of same sign because an increase in must lead to increase in
Four Basic Rules
Rule 3: Negative slope linearization of source term If source term is dependent on !, it is linearized as:
This will then appear in along with other terms. To ensure remains positive, must be negative or zero
Rule 4: Sum of neighbour coefficients If governing differential equation contains only derivatives of !, both ! and !+c will satisfy the equation. In this case,
Central Differencing Scheme
⯠The Convective term is evaluated using piecewise linear profile of !
⯠Transport equation becomes,
Central Differencing Scheme
⯠Discretization equation can be written as
where
Assessment ⯠Conservativeness : Uses consistent expressions to evaluate convective
and diffusive fluxes at CV faces. Unconditionally Conservative
⯠Boundedness : will become negative if Scheme is conditionally bounded ( )
Central Differencing Scheme
⯠Transportiveness : The CDS uses influence at node P from all directions. Does not recognize direction of flow or strength of convection relative to diffusion
Does not possess Transportiveness at high Peclet Numbers
⯠Accuracy : Second Order in terms of Taylor series Stable and accurate only if Now,
For stability and accuracy, either velocity should be very low or grid spacing should be small
Upwind Differencing Scheme
⯠The diffusion term is still discretized using piecewise linear profile of ! ⯠For convection term, ! at interface is equal to ! at the grid point on
the upwind side
is defined similarly ⯠Define , then, upwind scheme gives
⯠Discretization equation:
Upwind Differencing Scheme
Assessment ⯠Conservativeness : It is conservative
⯠Boundedness : When flow satisfies continuity equation, all coefficients are positive. Also, which is desirable for stable iterative solutions of linear equations
⯠Transportiveness : Direction of flow inbuilt in the formulation, thus, accounts for transportiveness
⯠Accuracy : When flow is not aligned with the grid lines, it produces false diffusion, which will be discussed later
Exact Solution
⯠The governing transport equation:
⯠If ! = constant, the equation can be solved exactly
⯠Boundary conditions: ,
⯠Solution:
where,
Exponential Scheme
⯠Define
⯠Our transport equation becomes,
⯠Integrating over CV,
⯠The exact solution derived above can be used as profile assumption with
⯠Substitution gives
where
Exponential Scheme
⯠After substitution of similar expression for , equation in our standard form can be written as:
⯠Merit: Guaranteed to produce exact solution for any Peclet number for 1-D steady convection-diffusion
⯠Demerits: 1. exponentials expensive to compute 2. not exact for 2-D, 3-D
Hybrid Scheme
⯠In exponential scheme,
Hybrid Scheme
⯠From Figure, we can see that 1.
2.
3.
The 3 straight lines representing these limiting cases are shown in figure ⯠The hybrid scheme is made up of these 3 straight lines
,
Hybrid Scheme
⯠Standard Discretization equation
⯠Significance of HDS: 1. Combines advantages of both CDS and UDS 2. Identical to CDS for -2 ⤠⤠2 3. Outside this range, it reduces to UDS with diffusion set equal to zero
⯠Disadvantage: First order accuracy in terms of Taylor Series
Power Law Scheme
⯠Similar to HDS but more accurate ⯠Diffusion is set equal to zero for >10 or < -10 ⯠Otherwise diffusion is calculated from a polynomial expression
⯠Discretization equation
Higher Order Differencing schemes
⯠CDS has second order accuracy but does not posses transportiveness property.
⯠Upwind, hybrid schemes are very stable and obey transportiveness but are first order in terms of Taylor series truncation error which makes them prone to diffusion errors.
⯠Such errors minimized by employing higher order discretisation. ⯠Higher order schemes involve more neighbour points and reduce
discretization errors by bringing wider influence. ⯠Formulations that do not take into account the flow direction are
unstable and, therefore, more accurate higher order schemes, which preserve upwinding for stability and sensitivity to flow direction, are needed.
Quadratic upwind differencing scheme (QUICK)
⯠Quadratic upstream interpolation for convective kinetics(QUICK) ⯠3 point upstream-weighted quadratic interpolation used for cell face
values
⯠For
,
QUICK Scheme
⯠Diffusion terms are evaluated using gradient of the appropriate parabola (For uniform grid, gives same results as CDS for diffusion)
⯠Discretized convection diffusion transport equation:
⯠Standard form of discretized equation
⯠Similarly, coefficients can be obtained for
QUICK Scheme
Assessment ⯠Conservativeness : Ensured ⯠Boundedness : For , is always negative, can
become negative for , thus the scheme is conditionally stable.
⯠Transportiveness : Built in because the quadratic function is based on 2 upstream and 1 downstream node
⯠Accuracy : Third order in terms of Taylor series truncation error on a uniform mesh
⯠Another feature : Discretization equations not only involve immediate neighbour nodes but also nodes further away, thus TDMA methods are not applicable
QUICK Scheme
⯠QUICK scheme above can be unstable due to negative coefficients ⯠Reformulated in different ways- Formulations involve placing -ve
coefficients in source term to retain +ve main coefficients ⯠The Hayse et el(1990) QUICK scheme is summarized as:
⯠Discretization equation:
QUICK Scheme
Summarizing: ⯠Has greater formal accuracy than central differencing or hybrid
schemes and it retains upwind weighted characteristics ⯠But, can sometimes give minor undershoots and overshoots(example
given later) Other higher order schemes: ⯠Use increases accuracy ⯠Implementation of Boundary Conditions can be problematic ⯠Computation costs also need to be considered ⯠To avoid undershoots and overshoots(get oscillation free solution),
class of TVD(Total variation diminishing) schemes have been formulated.
Discretization Equations for 2-D, 3-D
Discretization Equation for 2-D
Discretization Equation for 3-D
The coefficients for 2-D, 3-D for hybrid differencing scheme are shown on next page
Coefficients for 2-D, 3-D(HDS)
Summary
Handling the Source term
⯠For 1-D, Discretization equation simply becomes,
⯠If the source term is a constant , then all other coefficients remain same and,
⯠If source term is dependent on !, linearization is done as:
In this case, b and become,
All other coefficients remain same In a similar way, Source term can be incorporated in 2-D, 3-D
Handling the Unsteady term
⯠For handling the unsteady term we will look at 1-D unsteady equation without convection(without source term), later we can extend the concept to convection-diffusion equations of all kinds
⯠Integration over the 1-D CV gives
Handling the Unsteady term
⯠Density remains constant(from continuity equation)
⯠Now we need an assumption for with t, We assume
⯠We use similar formulas for and
Handling the Unsteady term
⯠Final Discretization Equation:
where,
Handling the Unsteady term
⯠If f=0: Scheme is explicit ⯠If f=0.5: Crank Nicholson Scheme ⯠If f=1: Implicit Scheme ⯠Variation of Temperature with time for the three schemes is :
Handling the Unsteady term
Analysis: ⯠Explicit Scheme:
⯠The coefficient of becomes negative if exceeds
⯠For uniform conductivity and equal grid spacing, scheme is stable if
⯠Crank Nicholson Scheme: Coefficient of is
Handling the Unsteady term
⯠Even in Crank- Nicholson Scheme, if the time step is not sufficiently small, the coefficient of will become negative
⯠Crank Nicholson Scheme is also conditionally stable
⯠Implicit Scheme: Only in this case, the coefficient of is always positive. Thus, fully implicit scheme satisfies requirements of simlicity and physically realistic behavior.
⯠However, at small time steps, Crank Nicholson scheme is more accurate than fully implicit scheme
⯠Reason: Temperature time curve is nearly linear for small time intervals which is exactly what we assumed in Crank Nicholson scheme
False Diffusion - Common View
⯠CDS has 2nd order accuracy while UDS has 1st order accuracy : From Taylor series expansion
⯠UDS causes severe false diffusion : UDS is equivalent to replacing ! in the CDS by !+!u!x/2
â CDS is better than UDS (misleading, true only for small Pe) Problem with this view: ⯠Truncated taylor series ceases to be a good representation(except for
small !x or small Pe), since !~x variation is exponential ⯠We assumed CDS as standard, then compared diffusion coefficient of
UDS with that of CDS ⯠The so called false diffusion coefficient !u!x/2 is indeed desirable at
large Peclet numbers
False Diffusion - Proper View
⯠Important only for large Pe(for small Pe, real diffusion is large enough) ⯠Multidimensional phenomena ⯠Consider example: 2 parallel streams with equal velocity, nonequal
Temperature contacted ⯠If !â 0, mixing layer forms where T changes from higher to lower value ⯠If !=0, T discontinuity persists in streamwise direction =0, T discontinuity persists in streamwise direction
To observe false diffusion: set !=0, If numerical solution produces smeared T profile(characteristic of !â 0), it entails false diffusion
False Diffusion - Proper View
⯠CDS: For !=0, it gives non unique or unrealistic solutions ⯠UDS:
1. Uniform flow in x-direction: !=0 and y-direction velocity = 0
Thus, given upstream value on each horizontal line gets established at all points on that line
No false diffusion
False Diffusion - Proper View
2. Uniform flow at 45°togrid lines(say, âx=ây)
Results obtained are shown in adjacent figure Thus, false diffusion is observed
For no false diffusion: !=100 above the diagonal !=0 below the diagonal
False Diffusion - Proper View
⯠The above problem solved for different grid sizes gives the following results
False Diffusion - Proper View
Conclusions ⯠Occurs when flow is oblique to grid lines and nonzero gradient exists in
direction normal to flow ⯠False diffusion reduction: Use smaller âx and ây, allign grid lines more
in direction of flow ⯠Enough to make false diffusion << real diffusion ⯠CDS is no remedy for false diffusion. At high Pe, it produces unrealistic
results ⯠Basic Cause: Treating flow across each CV as locally 1-D ⯠For less false diffusion: Scheme should take account of
multidimensional nature of flow. Also, involve more neighbours in discretisation equation.
QUICK Scheme (contâŚ)
⯠The above problem if solved on a 50*50 grid using Upwind and QUICK schemes gives the following results.
⯠Notice the undershoots and overshoots by the QUICK scheme
References
⯠Ferziger J. H. and Peric M. Computational Methods for Fluid Dynamics
⯠Patankar S.V. Numerical Heat Transfer and Fluid Flow ⯠Versteeg H. K. and Malalasekera W. An introduction to
computational fluid Dynamics: The finite volume method
Thank You