Very Important FVM
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Transcript of Very Important FVM
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7T H INDO GERMAN WINTER ACADEMY- 2008
Discretization of convection-diffusiontype equations by
Finite Volume Method
Ritika Tawani
Department of Chemical EngineeringIndian Institute of Technology, Bombay
Guides:Prof. Suman Chakraborty, IIT-Kharagpur
Prof. Vivek V. Buwa, IIT-Delhi
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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
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The Convection Diffusion Equation
The general differential equation, for the conservation of a physicalproperty,!
The 4 terms are: Unsteady term, Convection term, Diffusion term andSource term
In general,! =!(x, y, z, t)=!(x, y, z, t)(x, y, z, t)
! 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 differentquantities
eg:!=1: Mass conservation=1: Mass conservation
!=u: x-momentum conservation=u: x-momentum conservation
!=h: Energy conservation=h: Energy conservation
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Finite Volume Method
Key concept: Integration of differential equation over Control Volume
For simplification, we first do finite volume formulation for 1-D steadystate equation(with no source term)
The flow field should also satisfy continuity equation
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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 linearprofile of!
,
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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
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Four Basic Rules
For solutions to be: 1. Physically realistic2. Satisfy overall balance (conservative)
There are some basic rules that need to be satisfied by the discretizationequations
Standard form of discretization equations(1-D):
Rule 1: Flux consistency at CV facesWhen a face is common to two adjacent control volumes, flux across it
must be represented by the same expression in discretization equationsfor both the control volumes
Rule 2: Positive coefficients
All coefficients must always be of same sign because an increase inmust lead to increase in
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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 ensureremains 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,
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Central Differencing Scheme
The Convective term is evaluated using piecewise linear profile of!
Transport equation becomes,
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Central Differencing Scheme
Discretization equation can be written aswhere
Assessment
Conservativeness : Uses consistent expressions to evaluate convectiveand diffusive fluxes at CV faces.
Unconditionally Conservative
Boundedness : will become negative ifScheme is conditionally bounded ( )
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Central Differencing Scheme
Transportiveness : The CDS uses influence at node P from alldirections. Does not recognize direction of flow or strength ofconvection relative to diffusion
Does not possess Transportiveness at high Peclet Numbers
Accuracy : Second Order in terms of Taylor seriesStable and accurate only if
Now,
For stability and accuracy, either velocity should be very low or gridspacing should be small
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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:
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Upwind Differencing Scheme
Assessment Conservativeness : It is conservative
Boundedness : When flow satisfies continuity equation, all coefficientsare positive. Also, which is desirable for stable iterativesolutions 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 falsediffusion, which will be discussed later
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Exact Solution
The governing transport equation:
If! = constant, the equation can be solved exactly
Boundary conditions: , Solution:
where,
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Exponential Scheme
Define Our transport equation becomes,
Integrating over CV,
The exact solution derived above can be used as profile assumptionwith
Substitution giveswhere
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Exponential Scheme
After substitution of similar expression for , equation in our standardform can be written as:
Merit: Guaranteed to produce exact solution for any Peclet number for1-D steady convection-diffusion
Demerits: 1. exponentials expensive to compute
2. not exact for 2-D, 3-D
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Hybrid Scheme
In exponential scheme,
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Hybrid Scheme
From Figure, we can see that1.
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,
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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
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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
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Higher Order Differencing schemes
CDS has second order accuracy but does not posses transportivenessproperty.
Upwind, hybrid schemes are very stable and obey transportiveness butare first order in terms of Taylor series truncation error which makesthem 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 areunstable and, therefore, more accurate higher order schemes, whichpreserve upwinding for stability and sensitivity to flow direction, are
needed.
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Quadratic upwind differencing scheme(QUICK)
Quadratic upstream interpolation for convective kinetics(QUICK) 3 point upstream-weighted quadratic interpolation used for cell face
values
For,
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QUICK Scheme
Diffusion terms are evaluated using gradient of the appropriateparabola (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
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QUICK Scheme
Assessment Conservativeness : Ensured Boundedness : For , is always negative, can
become negative for , thus the scheme isconditionally stable.
Transportiveness : Built in because the quadratic function is based on 2upstream and 1 downstream node
Accuracy : Third order in terms of Taylor series truncation error on auniform mesh
Another feature : Discretization equations not only involve immediateneighbour nodes but also nodes further away, thus TDMA methods are
not applicable
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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:
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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(examplegiven 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.
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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 onnext page
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Coefficients for 2-D, 3-D(HDS)
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Summary
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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
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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
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Handling the Unsteady term
Final Discretization Equation:
where,
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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 :
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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
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Handling the Unsteady term
Even in Crank- Nicholson Scheme, if the time step is not sufficientlysmall, the coefficient of will become negative Crank Nicholson Scheme is also conditionally stable
Implicit Scheme: Only in this case, the coefficient of is alwayspositive. Thus, fully implicit scheme satisfies requirements of simlicityand physically realistic behavior.
However, at small time steps, Crank Nicholson scheme is more accuratethan fully implicit scheme
Reason: Temperature time curve is nearly linear for small timeintervals which is exactly what we assumed in Crank Nicholson scheme
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False Diffusion - Common View
CDS has 2
nd
order accuracy while UDS has 1st
order accuracy : FromTaylor series expansion
UDS causes severe false diffusion : UDS is equivalent to replacing! inthe CDS by!+!u!x/2
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 forsmall!x or small Pe), since!~x variation is exponential
We assumed CDS as standard, then compared diffusion coefficient ofUDS with that of CDS
The so called false diffusion coefficient!u!x/2 is indeed desirable atlarge Peclet numbers
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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! 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 directionTo observe false diffusion: set!=0, If numerical solution producessmeared T profile(characteristic of!0), it entails false diffusion
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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
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False Diffusion - Proper View
2. Uniform flow at 45togrid lines(say, x=y)
Results obtained are shown in
adjacent figure
Thus, false diffusion is
observed
For no false diffusion:
!=100 above the diagonal
!=0 below the diagonal
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False Diffusion - Proper View
The above problem solved for different grid sizes gives the followingresults
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False Diffusion - Proper View
Conclusions
Occurs when flow is oblique to grid lines and nonzero gradient exists indirection normal to flow
False diffusion reduction: Use smaller x and y, allign grid lines morein direction of flow
Enough to make false diffusion
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QUICK Scheme (cont)
The above problem if solved on a 50*50 grid using Upwind and QUICKschemes gives the following results.
Notice the undershootsand overshoots by the
QUICK scheme
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References
Ferziger J. H. and Peric M. Computational Methods for FluidDynamics
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
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Thank You