Discretization of Convection Diffusion type equation

Transcription

1 Discretization of Convection Diffusion type equation 10 th Indo German Winter Academy 2011 By, Rajesh Sridhar, Indian Institute of Technology Madras Guides: Prof. Vivek V. Buwa Prof. Suman Chakraborty IIT Delhi IIT Kharagpur

2 Agenda: Introduction Finite Volume Discretization Introduction Spatial Discretization Spatial Discretization for 2-D and 3-D problems Source Term Discretization Unsteady Term Discretization False Diffusion

3 Introduction: General transport equation for the conservation of property is, Unsteady term Convection term Diffusion term Source term Γ is the diffusion coefficient corresponding to and S is the source term. The above equation transforms into certain famous equations for different s. = 1 = v = e Mass Conservation Equation Momentum Conservation Equation Energy Conservation Equation

4 Finite Volume Method: Key concept in FVM: Integration of the differential equation over the Control Volume. Solution Procedure: Grid Generation. Discretization of differential equation into algebraic equations. Simultaneous solution of the algebraic equations.

5 Grid Generation: Divide the domain into discrete control volumes. Place a number of nodes between the system boundaries. Boundaries of control volumes are placed midway between the adjacent nodes.

6 Discretization: The general form of discretized equation is, a P ϕ P = Σ a nb ϕ nb + b where, Σ indicates summation over the neighboring nodes, a nb is the neighboring node coefficient, Φ nb is the value of property at the neighboring node, b is a constant.

7 Discretization Schemes: For the solutions to be physically realistic, a discretization scheme should satisfy the following requirements: 1. Conservativeness 2. Boundedness 3. Transportiveness Conservativeness: Flux consistency at the CV faces. i.e. Flux of ø leaving a control volume = Flux of ø entering the adjacent control volume through the same face

8 Boundedness: Discretization Schemes: The discretized equations are solved by iterative methods. Sufficient condition for a convergent iterative solution is, Σ a nb < 1 a P where, a P is the net coefficient of the central node. i.e. a P S P Negative Slope Linearization of Source term: If the source term depends on ϕ, it can be linearized as, S = S C + S P ϕ P

9 Discretization Schemes: Boundedness: Σ a nb a P < 1 This constraint can be satisfied if a P is made as large as possible. This can be achieved if the scheme can ensure, i. S P is always negative. All coefficients should be of the same sign, preferably positive. i.e. an increase in ϕ at one node should result in an increase in ϕ at neighbouring nodes. If S P becomes zero, then both ϕ and ϕ + c will satisfy the equation.

10 Transportiveness: Discretization Schemes: A non-dimensional Peclet number is defined to measure the relative strengths of convection and diffusion. where, F Pe = = D ρu Γ/δx F = convective mass flux per unit area ρ = fluid density Γ = diffusion coefficient D = diffusion conductance u = fluid flow velocity δx = characteristic length Primary Objective: The relationship between the magnitude of Peclet Number and the directionality of influencing.i.e. Transportiveness should be borne out of the discretization scheme.

11 Discretization Schemes: Consider the general contours of constant ϕ for a constant ϕ source at P for different values of Pe depicted below. For Pe = 0.i.e pure diffusion, the contours are concentric circles for constant values of ϕ because diffusion tends to spread out ϕ evenly in all directions. As Pe increases, the curve becomes more elliptic and the value of ϕ at E node is more influenced by the upstream node. At Pe =.i.e pure convection, the contours are completely outstretched and value of ϕ at E is affected only by upstream conditions.

12 Spatial Discretization: Consider the steady convection and diffusion of a property ϕ in a onedimensional flow field u. The flow must also satisfy continuity equation. Integration of the transport equation over the control volume yields, Integration of the continuity equation yields,

13 Spatial Discretization: Assuming Ae = Aw = A and employing central differencing approach to represent the diffusion term, we get The integrated continuity equation, where, Schemes for calculating the value of ϕ at the cell faces will be discussed now.

14 Spatial Discretization Schemes: Central Differencing Scheme. First Order Upwind Scheme Exact Solution Exponential Scheme Hybrid Scheme Power Law Scheme QUICK Scheme

15 Central Differencing Scheme: Here, we use linear interpolation for computing the cell face values. ϕ W ϕ w ϕ P ϕ e ϕ E W w P e E ϕ e = (ϕ E + ϕ P ) ϕ w = (ϕ P + ϕ W ) Transport equation becomes, 2 2

16 Central Differencing Scheme: The final discretized equation can be written as, where, Assessment: Conservativeness: Uses consistent expressions to evaluate convective and diffusive fluxes at CV faces. Unconditionally Conservative.

17 Boundedness: Central Differencing Scheme: a E = D e - F e 2 < 0 Pe > 2 a W or a E will become negative if Pe > 2, depending on the flow direction. Scheme is conditionally bounded ( Pe < 2). 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.

18 Accuracy: Central Differencing Scheme: Second Order in terms of Taylor series. Stable and accurate only if Pe < 2. For a low value of Pe, either velocity should be less or the grid spacing should be small.

19 First Order Upwind Scheme: Here, ϕ W ϕ w ϕ P ϕ e ϕ E ϕ e = ϕ P if, F e > 0 ϕ e = ϕ E if, F e < 0 W w P e E ϕ w is defined similarly. Now, we can define [[ ]] operator as, [[ A, B ]] = MAX (A,B). The final discretization equation is, where,

20 Assessment: First Order Upwind Scheme: Conservativeness: Utilises consistent expressions to calculate through cell faces. Hence, conservative. Boundedness: When flow satisfies continuity equation, all coefficients are positive. Also, a P = a E + a W 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 form last part of our discussion.

21 Exact Solution: The transport equation, If Γ is constant, the above equation can be solved exactly for the boundary conditions, ϕ = ϕ 0 x = 0 ϕ = ϕ L x = L Solution: where, P = ρul Γ

22 Exponential Scheme: Here, we define a variable J such that J = ρuϕ Γ dϕ dx Now, our transport equation can be expressed as, dj dx = 0 Integrating over the CV, we get J e = J w The exact solution derived in the previous slide can be used as a profile assumption between P and E with ϕ 0 being replaced by ϕ P,ϕ L by ϕ E,L by δx e resulting in where,

23 Exponential Scheme: Similar substitution for J w results in the standard discretization equation, with the values of coefficients, Merits: Guaranteed to produce exact solution for any Peclet number for 1-D steady convection-diffusion. Demerits: Exponentials are expensive to calculate. Not exact for 2-D and 3-D cases.

24 Hybrid Differencing Scheme: Combination of Upwind and Central Differencing Scheme. Central Differencing Scheme (2 nd order accurate) for small Pe (Pe < 2). Upwind Scheme (transportive) for large Pe (Pe > 2). Simplification of the exponential scheme for easier computation. a E D e Exponential scheme Hybrid Scheme -2 2 Pe

25 Hybrid Differencing Scheme: From exponential scheme, So, we can define a hybrid piece-wise linear function as, a E D e = 1 - Pe 2 0 -Pe -2 < Pe < 2 Pe > 2 Pe < 2

26 Hybrid Differencing Scheme: The final discretization equation along with the value of the coefficients is given below: Assessment: Combines advantages of both Upwind and CDS schemes. Conservative and unconditionally bounded. Satisfies transportiveness by using Upwind scheme for large Pe. Only disadvantage is accuracy in terms of Taylor series expansion which is first order.

27 Power Law Scheme: Similar to Hybrid Differencing Scheme, but more accurate and produces better results. Diffusion is set equal to zero for Pe > 10.

28 Higher Order Differencing Schemes: Need for Higher Order Schemes: Accuracy of Upwind and Hybrid schemes is only first order in terms of Taylor Series truncation error, leading to numerical diffusion errors. Central differencing scheme is second order accurate but doesnot possess transportiveness property, hence unstable. Errors reduced by bringing in a wider influence involving more neighbour points. 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.

29 QUICK Scheme: QUICK stands for Quadratic Upwind Interpolation for Convective Kinetics. 3 point upstream-weighted quadratic interpolation used for cell face values ϕ WW ϕ W ϕ w ϕ P ϕ e ϕ E ϕ EE u w u e WW W P E EE w e When F e > 0, When F w > 0, ϕ e = 6 ϕ P + 3 ϕ E - 1 ϕ W ϕ w = 6 ϕ W + 3 ϕ P - 1 ϕ WW 8 8 8

30 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: F e ( 6 ϕ P + 3 ϕ E - 1 ϕ W ) F w ( 6 ϕ w + 3 ϕ P - 1 ϕ WW ) = D e (ϕ E - ϕ P ) - D w (ϕ P - ϕ W ) Standard form of discretized equation: Similar expressions can be obtained for F e < 0 and F w < 0.

31 QUICK Scheme: General expressions valid for both positive and negative flow directions is given below: with central coefficient, and neighbouring coefficients,

32 Assessment of QUICK Scheme: Conservativeness: Uses consistent quadratic profiles. Hence, conservative. Boundedness: Conditionally stable. For u w > 0 and u e > 0, a WW is negative and a E can become negative for Pe > 8/3. 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. Note: Discretization equations not only involve immediate neighbour nodes but also nodes further away, thus TDMA methods are not directly applicable.

33 QUICK Summary: Has greater formal accuracy than central differencing or hybrid schemes and it retains upwind weighted characteristics. But, can sometimes give minor undershoots and overshoots. 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.

34 Spatial Discretization for 2-D and 3-D: Discretization in 2D: Discretization in 3D:

35 Spatial Discretization for 2-D and 3-D: Coefficients for HDS in 2D and 3D cases:

36 Spatial Discretization Summary:

37 Source Term Discretization: In a one-dimensional case, the discretization equation simply becomes, If the source term is a constant S = S C then all other coefficients remain same and, If source term is dependent on ϕ, linearization is done as: In this case, b and a P become, a P ϕ P = a E ϕ E + a W ϕ W + b b = S C x S = S C + S P ϕ P All other coefficients remain same. In a similar way, Source term can be incorporated in 2-D and 3-D.

38 Unsteady Term Discretization: Consider a 1-D unsteady diffusion equation. Integration over the control volume gives, w e t+δt t+δt ϕ = Γ E - ϕ P e ϕ - Γ P - ϕ W w δx e δx w t t

39 Unsteady Term Discretization: Now we need an assumption for ϕ with t, we assume where 0< f <1 ϕ P f =1 Implicit f = 0.5 Crank-Nicholson f = 0 Explicit t t+δt

40 Unsteady Term Discretization: The final discretization equation becomes, a P ϕ P 1 = a E {fϕ P1 +(1-f)ϕ E0 } + a W {fϕ W 1 +(1-f)ϕ W0 } + {a P0 -(1-f)a E (1-f) a W } ϕ P 0 where,

41 Asssesment: Unsteady Term Discretization: Explicit Scheme: a P ϕ P 1 = a E ϕ E 0 + a W ϕ W0 + {a P0 - a E - a W } ϕ P 0 The coefficient of ϕ P 0 becomes negative if a P0 exceeds a E + a W. For uniform conductivity and equal grid spacing, scheme is stable if Crank Nicholson Scheme: Coefficient of ϕ P 0 is

42 Unsteady Term Discretization: Even in Crank-Nicholson Scheme, if the time step is not sufficiently small, the coefficient of ϕ P0 will become negative. Crank Nicholson Scheme is also conditionally stable. Implicit Scheme: Only in this case, the coefficient of ϕ P0 is always positive. Thus, fully implicit scheme satisfies requirements of simplicity and physically realistic behaviour. However, at small time steps, Crank Nicholson scheme is more accurate than fully implicit scheme.

43 False Diffusion Usually it is stated that the Central difference scheme is superior to the Upwind scheme because it is second-order accurate whereas the Upwind scheme is only first-order accurate. But when we compare the Central difference and Upwind schemes: This added diffusion is considered bad at small Peclet numbers, however it actually corrects the solution at large Peclet number.

44 False Diffusion False diffusion is a numerically introduced diffusion and arises in convection dominated flows, i.e. high Pe number flows. False diffusion is a multidimensional phenomenon and occurs when the flow is perpendicular to the grid lines. X Consider the following example: If there was no false diffusion, then the temperature will be exactly 100 ºC everywhere above the diagonal and exactly 0 ºC everywhere below the diagonal. Hot Fluid T = 100ºC Cold Fluid T = 0ºC X

45 False Diffusion

46 False Diffusion Upwind Scheme:

47 It diverges! False Diffusion QUICK Scheme:

48 False Diffusion - Summary: Occurs when flow is oblique to grid lines and nonzero gradient exists in direction normal to flow. False diffusion reduction: Use smaller Δx and Δy, align grid lines more in the 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. Involve more neighbours in discretisation equation.

49 References Patankar S.V. Numerical Heat Transfer and Fluid Flow Versteeg H. K. and Malalasekera W. fluid Dynamics: The finite volume method An introduction to computational Lecture Notes by Dr. Shamit Bakshi, IIT Madras.

50 Thank You!