Discretization of convection-diffusion type equations by Finite
Transcript of Discretization of convection-diffusion type equations by Finite
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) ! 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
⇒ 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 !≠ 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 !≠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, ∆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
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 ∆x and ∆y, 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 (cont…)
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