dongkesun(孙东科 [email protected] and icme/06th steady convective... · introduction outline 1...

稳态对流扩散问题 · 基本格式Steady Convective Diffusion Problems

Dongke Sun (孙东科)[email protected]

东南大学机械工程学院School of Mechanical Engineering

Southeast University

January 31, 2019

Introduction

OUTLINE

1 Introduction

2 Steady 1D convection and diffusion

3 The central differencing scheme

4 Properties of discretisation schemesConservativenessBoundednessTransportiveness

5 Assessment of central differencing scheme

6 The upwind differencing scheme

Dongke Sun (Southeast University) January 31, 2019 2 / 92

Introduction

Introduction

The steady convection-diffusion equation can be derived from the transportequation for a general property 𝜑 by deleting the transient term

div(𝜌𝑢𝜑) = div(Γ grad𝜑) + 𝑆𝜑 (1)

Formal integration over a control volume givesˆ𝐴𝑛 · (𝜌𝜑𝑢)d𝐴 =

ˆ𝐴𝑛 · (Γ grad𝜑)d𝐴 +

ˆ𝐶𝑉

𝑆𝜑d𝑉 (2)

This equation represents the flux balance in a control volume.


Steady 1D convection and diffusion

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1 Introduction









In the absence of sources, steady convection and diffusion of a property 𝜑in a given one-dimensional flow field 𝑢 is governed by

d

d𝑥(𝜌𝑢𝜑) =

d

d𝑥

(Γ

d𝜑

d𝑥

)(3)

The flow must also satisfy continuity, so

d(𝜌𝑢)

d𝑥= 0 (4)

We consider the 1D control volume shown in the figure




Our attention is focused on a general node 𝑃 .

Integration of transport equation (3) over the control volume

(𝜌𝑢𝐴𝜑)𝑒 − (𝜌𝑢𝐴𝜑)𝑤 =

(Γ𝐴

d𝜑

d𝑥

)𝑒

−(

Γ𝐴d𝜑

d𝑥

)𝑤

(5)

And integration of continuity equation (4) yields

(𝜌𝑢𝐴)𝑒 − (𝜌𝑢𝐴)𝑤 = 0 (6)

We must approximate the terms in equation (5).




Our attention is focused on a general node 𝑃 .

Define 𝐹 and 𝐷 to represent the convective mass flux per unit area anddiffusion conductance at cell faces:

𝐹 = 𝜌𝑢 and 𝐷 =Γ

𝛿𝑥(7)

The cell face values of the variables 𝐹 and 𝐷 can be written as

𝐹𝑤 = (𝜌𝑢)𝑤 𝐹𝑒 = (𝜌𝑢)𝑒, 𝐷𝑤 =Γ𝑤

𝛿𝑥𝑊𝑃𝐷𝑒 =

Γ𝑒

𝛿𝑥𝑃𝐸(8)




We develop our techniques assuming that

𝐴𝑤 = 𝐴𝑒 = 𝐴.

The integrated convection-diffusion equation (5) can now be written as

𝐹𝑒𝜑𝑒 − 𝐹𝑤𝜑𝑤 = 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 ) (9)

and the integrated continuity equation (6) as

𝐹𝑒 − 𝐹𝑤 = 0 (10)

We also assume that the velocity field is "somehow known", which takescare of the values of 𝐹𝑒 and 𝐹𝑤.In order to solve equation (9) we need to calculate the transported property𝜑 at the 𝑒 and 𝑤 faces.


The central differencing scheme

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1 Introduction









For a uniform grid we can write the cell face values of property 𝜑 as

𝜑𝑐 = (𝜑𝑃 + 𝜑𝐸) /2, 𝜑𝑤 = (𝜑𝑊 + 𝜑𝑃 ) /2 (11)

Substitution of the above expressions into the convection terms of (9) yields

𝐹𝑒

2(𝜑𝑃 + 𝜑𝐸) − 𝐹𝑤

2(𝜑𝑊 + 𝜑𝑃 ) = 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 ) (12)

This can be rearranged to give[(𝐷𝑤 − 𝐹𝑤

2

)+

(𝐷𝑒 +

𝐹𝑒

2

)]𝜑𝑃 =

(𝐷𝑤 +

𝐹𝑤

2

)𝜑𝑊 +

(𝐷𝑒 −

𝐹𝑒

2

)𝜑𝐸[(

𝐷𝑤 +𝐹𝑤

2

)+

(𝐷𝑒 −

𝐹𝑒

2

)+ (𝐹𝑒 − 𝐹𝑤)

]𝜑𝑃

=

(𝐷𝑤 +

𝐹𝑤

2

)𝜑𝑊 +

(𝐷𝑒 −

𝐹𝑒

2

)𝜑𝐸




Identifying the coefficients of 𝜑𝑊 and 𝜑𝐸 as 𝑎𝑊 and 𝑎𝐸 , the centraldifferencing expressions for the discretised convection-diffusion equation are

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 (14)

where

Equation (14) for steady convection–diffusion problems takes thesame general form as equation (11) for pure diffusion problems.The difference is that the coefficients of the former contain additionalterms to account for convection.

It yields a set of algebraic equations that is solved to obtain the distributionof the transported property 𝜑.




Example 1A property 𝜑 is transported by means of convection and diffusion throughthe one-dimensional domain sketched in the figure. (cited by slides 38, 52)

The governing equation isd

d𝑥(𝜌𝑢𝜑) =

d

d𝑥

(Γ

d𝜑

d𝑥

)The boundary conditions are

𝜑0 = 1 at 𝑥 = 0,

and𝜑𝐿 = 0 at 𝑥 = 𝐿.




Example 1Using five equally spaced cells and the central differencing scheme forconvection and diffusion, calculate the distribution of 𝜑 as a function of 𝑥for

Case 1: 𝑢 = 0.1 m/s,Case 2: 𝑢 = 2.5 m/s, and compare the results with the analyticalsolution

𝜑− 𝜑0

𝜑𝐿 − 𝜑0=

exp(𝜌𝑢𝑥/Γ) − 1

exp(𝜌𝑢𝐿/Γ) − 1, (15)

Case 3: recalculate the solution for 𝑢 = 2.5m/s with 20 grid nodesand compare the results with the analytical solution.

The following data apply:

𝐿 = 1.0m, 𝜌 = 1.0 kg/m3, Γ = 0.1 kg/ms.




SolutionThe domain has been divided into five control volumes giving 𝛿𝑥 = 0.2 m:𝐹 = 𝜌𝑢,𝐷 = Γ/𝛿𝑥, 𝐹𝑒 = 𝐹𝑤 = 𝐹 and 𝐷𝑒 = 𝐷𝑤 = 𝐷 everywhere.

The value of 𝜑 is given at the west face of this cell (𝜑𝑤 = 𝜑𝐴 = 1).Thisyields the following equation for control volume 1:

𝐹𝑒

2(𝜑𝑃 + 𝜑𝐸) − 𝐹𝐴𝜑𝐴 = 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝐴(𝜑𝑃 − 𝜑𝐴) (16)

For node 5, the 𝜑-value at the east face is known (𝜑𝑒 = 𝜑𝐵 = 0). So

𝐹𝐵𝜑𝐵 − 𝐹𝑤

2(𝜑𝑃 + 𝜑𝑊 ) = 𝐷𝐵(𝜑𝐵 − 𝜑𝑃 ) = 𝐷𝑤(𝜑𝑃 − 𝜑𝑊 ) (17)





Rearrangement of these two equations, noting that𝐷𝐴 = 𝐷𝐵 = 2Γ/𝛿𝑥 = 2𝐷 and 𝐹𝐴 = 𝐹𝐵 = 𝐹 , gives discretised equationsat boundary nodes of the following form:

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝑆𝑢 (18)

with central coefficient𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + (𝐹𝑒 − 𝐹𝑤) − 𝑆𝑃





𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝑆𝑢




Case 1𝑢 = 0.1m/s: 𝐹 = 𝜌𝑢 = 0.1, 𝐷 = Γ/𝛿𝑥 = 0.1/0.2 = 0.5 gives thecoefficients as summarised in the table.

The matrix form of the equation set using 𝜑𝐴 = 1 and 𝜑𝐵 = 0 is⎡⎢⎢⎢⎢⎣1.55 −0.45 0 0 0−0.55 1.0 −0.45 0 0

0 −0.55 1.0 −0.45 00 0 −0.55 1.0 −0.450 0 0 −0.55 1.45

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

𝜑1

𝜑2

𝜑3

𝜑4

𝜑5

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣1.10000

⎤⎥⎥⎥⎥⎦ (19)




The solution to the above system is⎡⎢⎢⎢⎢⎣𝜑1

𝜑2

𝜑3

𝜑4

𝜑5

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣0.94210.80060.62760.41630.1579

⎤⎥⎥⎥⎥⎦

The exact solution of the problem:

𝜑(𝑥) =2.7183 − exp(𝑥)

1.7183

The numerical and analytical solutions are compared in the table.




Comparison with the analytical solution:

Given the coarseness of the grid the central differencing (CD) scheme givesreasonable agreement with the analytical solution.Dongke Sun (Southeast University) January 31, 2019 17 / 92



Case 2𝑢 = 2.5m/s: 𝐹 = 𝜌𝑢 = 2.5, 𝐷 = Γ/𝛿𝑥 = 0.1/0.2 = 0.5 gives

By the same method used in Case 1, the analytical solution for the datathat apply here is

𝜑(𝑥) = 1 +1 − exp(25𝑥)

7.20 × 1010.

The numerical and analytical solutions are compared in the table.




Case 2𝑢 = 2.5m/s: 𝐹 = 𝜌𝑢 = 2.5, 𝐷 = Γ/𝛿𝑥 = 0.1/0.2 = 0.5 gives




The numerical and analytical solutions are shown in the figure.

The solution appears to oscillate about the exact solution. Theseoscillations are often called "wiggles" in the literature.Dongke Sun (Southeast University) January 31, 2019 19 / 92



Case 3𝑢 = 2.5m/s: a grid of 20 nodes gives 𝛿𝑥 = 0.05, 𝐹 = 𝜌𝑢 = 2.5, 𝐷 = Γ/𝛿𝑥= 0.1/0.05 = 2.0. The coefficients are summarised in

Comparison of the data for this case with the one computed on thefive-point grid of Case 2

Case 2𝑢 = 2.5m/s: 𝐹 = 𝜌𝑢 = 2.5, 𝐷 = Γ/𝛿𝑥 = 0.1/0.2 = 0.5.

shows that grid refinement has reduced the 𝐹/𝐷 ratio from 5 to 1.25.




The resulting solution is compared with the analytical solution.

The central differencing scheme seems to yield accurate results when the𝐹/𝐷 ratio is low.Dongke Sun (Southeast University) January 31, 2019 21 / 92

Properties of discretisation schemes

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1 Introduction









The failure of central differencing in certain cases involving combinedconvection and diffusion forces us to take a more in-depth look at theproperties of discretisation schemes.

In theory numerical results may be obtained that are indistinguishable fromthe nexacto solution of the transport equation when the number ofcomputational cells is infinitely large, irrespective of the differencingmethod used.

However, in practical calculations we can only use a finite-sometimes quitesmall-number of cells, and our numerical results will only be physicallyrealistic when the discretisation scheme has certain fundamental properties.The most important ones are:

ConservativenessBoundednessTransportiveness


Properties of discretisation schemes Conservativeness

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1 Introduction








Properties of discretisation schemesConservativeness

Consider the 1-D steady state diffusion problem without source terms

An overall flux balance may be obtained by summing the net flux for thecontrol volumes around nodes 1 and 4:[

Γ𝑒1

(𝜑2 − 𝜑1)

𝛿𝑥− 𝑞𝐴

]+

[Γ𝑒2

(𝜑3 − 𝜑2)

𝛿𝑥− Γ𝑤2

(𝜑2 − 𝜑1)

𝛿𝑥

]+

[Γ𝑒3

(𝜑4 − 𝜑3)

𝛿𝑥− Γ𝑤3

(𝜑3 − 𝜑2)

𝛿𝑥

]+

[𝑞𝐵 − Γ𝑤4

𝜑4 − 𝜑3

𝛿𝑥

]= 𝑞𝐵 − 𝑞𝐴




Consider the 1-D steady state diffusion problem without source terms

Since Γ𝑒1 = Γ𝑤2 , Γ𝑒2 = Γ𝑤3 and Γ𝑒3 = Γ𝑤4 , the fluxes across controlvolume faces are expressed in a consistent manner and cancel out inpairs when summed over the entire domain.Only the two boundary fluxes 𝑞𝐴 and 𝑞𝐵 remain in the overall balance,so the above equation expresses overall conservation of property 𝜑.

Inconsistent flux interpolation formulae give rise to unsuitable schemes thatdo not satisfy overall conservation.




Consider the situation where a quadratic interpolation formula

As shown, the resulting quadratic profiles can be quite different.The flux values calculated at the east face of CV 2 and the west faceof CV 3 may be unequal if the gradients of the two curves are different.If this is the case the two fluxes do not cancel out when summed andoverall conservation is not satisfied.




Consider the situation where a quadratic interpolation formula

As shown, the resulting quadratic profiles can be quite different.The example should not suggest to the reader that quadraticinterpolation is entirely bad.Further on we will meet a quadratic discretisation practice - theso-called QUICK scheme - that is consistent.


Properties of discretisation schemes Boundedness

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1 Introduction








Properties of discretisation schemesBoundedness

Scarborough (1958) has shown that a sufficient condition for aconvergent iterative method can be expressed in terms of the values ofthe coefficients of the discretised equations:∑

|𝛼𝑛𝑏|𝛼′𝑃

{≤ 1 at all nodes

< 1 at one node at least(22)

Here 𝛼′𝑃 is the net coefficient of the central node 𝑃 (i.e. 𝑎𝑃 - 𝑆𝑃 ), and

the summation in the numerator is taken over all the neighbouring nodes(𝑛𝑏).

If the differencing scheme produces coefficients that satisfy the abovecriterion the resulting matrix of coefficients is diagonally dominant.To achieve diagonal dominance we need large values of net coefficient(𝑎𝑃 - 𝑆𝑃 ) so the linearisation practice of source terms shouldensure that 𝑆𝑃 is always negative.



Properties of discretisation schemesBoundedness

Scarborough (1958) has shown that a sufficient condition for aconvergent iterative method can be expressed in terms of the values ofthe coefficients of the discretised equations:∑

|𝛼𝑛𝑏|𝛼′𝑃

{≤ 1 at all nodes

< 1 at one node at least(22)

Here 𝛼′𝑃 is the net coefficient of the central node 𝑃 (i.e. 𝑎𝑃 - 𝑆𝑃 ), and

the summation in the numerator is taken over all the neighbouring nodes(𝑛𝑏).

Diagonal dominance states that in the absence of sources theinternal nodal values of 𝜑 should be bounded by its boundary values.Another essential requirement for boundedness is that all coefficientsof the discretised equations should have the same sign (usually allpositive, "+").


Properties of discretisation schemes Transportiveness

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1 Introduction







Properties of discretisation schemes Transportiveness

Properties of discretisation schemesTransportiveness

Considering the effect at a point 𝑃 due to two constant sources of 𝜑 atnearby points 𝑊 and 𝐸 on either side.

We define the non-dimensional cell Peclet number

𝑃𝑒 =𝐹

𝐷=

𝜌𝑢

Γ/𝛿𝑥(23)

where 𝛿𝑥 = characteristic length (cell width). Two extreme cases:no convection and pure diffusion (𝑃𝑒 → 0);no diffusion and pure convection (𝑃𝑒 → ∞).


Assessment of central differencing scheme

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1 Introduction







Assessment of central differencing scheme

Assessment of the upwind differencing scheme

Conservativeness: The central differencing scheme uses consistentexpressions to evaluate convective and diffusive fluxes at the controlvolume faces. The scheme is conservative.Boundedness: The internal coefficients of the central differencingscheme satisfy the Scarborough criterion. Given that 𝐹𝑤 > 0 and𝐹𝑒 > 0 (i.e. the flow is unidirectional), for 𝑎𝐸 to be positive, 𝐷𝑒 and𝐹𝑒 must satisfy the following condition:

𝐹𝑒/𝐷𝑒 = 𝑃𝑒𝑒 < 2 (24)

Transportiveness: The scheme does not recognise the direction ofthe flow or the strength of convection relative to diffusion. It does notpossess the transportiveness property at high 𝑃𝑒.Accuracy: The Taylor series truncation error of the centraldifferencing scheme is second-order. The scheme will be stable andaccurate only if 𝑃𝑒 = 𝐹/𝐷 < 2.


The upwind differencing scheme

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1 Introduction









One of the major inadequacies of the central differencing scheme is itsinability to identify flow direction.

The value of property 𝜑 at a west cell face is always influenced byboth 𝜑𝑃 and 𝜑𝑊 in central differencing.In a strongly convective flow from west to east, the above treatment isunsuitable because the west cell face should receive much strongerinfluencing from node 𝑊 than from node 𝑃 .

The upwind differencing scheme takes into account the flow direction whendetermining the value at a cell face.




When the flow is in the positive direction, 𝑢𝑤 > 0, 𝑢𝑒 > 0 (𝐹𝑤 > 0,𝐹𝑒 > 0), the upwind scheme sets

𝜑𝑤 = 𝜑𝑊 and 𝜑𝑒 = 𝜑𝑃 (25)

𝐹𝑒𝜑𝑃 − 𝐹𝑤𝜑𝑊 = 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 ) (26)

(𝐷𝑤 + 𝐷𝑒 + 𝐹𝑒)𝜑𝑃 = (𝐷𝑤 + 𝐹𝑤)𝜑𝑊 + 𝐷𝑒𝜑𝐸

to give

[(𝐷𝑤 + 𝐹𝑤) + 𝐷𝑒 + (𝐹𝑒 − 𝐹𝑤)]𝜑𝑃 = (𝐷𝑤 + 𝐹𝑤)𝜑𝑊 + 𝐷𝑒𝜑𝐸 (27)




When the flow is in the negative direction, 𝑢𝑤 < 0, 𝑢𝑒 < 0 (𝐹𝑤 < 0,𝐹𝑒 < 0), the upwind scheme sets

𝜑𝑤 = 𝜑𝑃 and 𝜑𝑒 = 𝜑𝐸 (28)

𝐹𝑒𝜑𝐸 − 𝐹𝑤𝜑𝑃 = 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 ) (29)

(𝐷𝑤 + 𝐷𝑒 − 𝐹𝑤)𝜑𝑃 = 𝐷𝑤𝜑𝑊 + (𝐷𝑒 − 𝐹𝑒)𝜑𝐸

to give

[𝐷𝑤 + (𝐷𝑒 − 𝐹𝑒) + (𝐹𝑒 − 𝐹𝑤)]𝜑𝑃 = 𝐷𝑤𝜑𝑊 + (𝐷𝑒 − 𝐹𝑒)𝜑𝐸 (30)




Identifying the coefficients of 𝜑𝑊 and 𝜑𝐸 as 𝑎𝑊 and 𝑎𝐸 , equations (27)and (30) can be written in the usual general form


with central coefficient𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + (𝐹𝑒 − 𝐹𝑤)

and neighbour coefficients

A form of notation for the neighbour coefficients of the upwind differ-encing method that covers both flow directions is given below:




Example 2

Solve the problem considered in Example 1 (see slide 12) using the upwinddifferencing scheme for (i) 𝑢 = 0.1m/s, (ii) 𝑢 = 2.5m/s with the coarsefive-point grid.

SolutionThe grid shown in Example 1 is again used here for the discretisation.

The discretisation equation at internal nodes 2, 3 and 4 and the relevantneighbour coefficients are given by 𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 and itsaccompanying tables.




Note that, 𝐹 = 𝐹𝑒 = 𝐹𝑤 = 𝜌𝑢 and 𝐷 = 𝐷𝑒 = 𝐷𝑤 = Γ/𝛿𝑥 everywhere.

At the boundary node 1, the use of upwind differencing for the convectiveterms gives

𝐹𝑒𝜑𝑃 − 𝐹𝐴𝜑𝐴 = 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝐴(𝜑𝑃 − 𝜑𝐴) (32)

And at node 5

𝐹𝐵𝜑𝑃 − 𝐹𝑤𝜑𝑊 = 𝐷𝐵(𝜑𝐵 − 𝜑𝑃 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 ) (33)

At the boundary nodes we have

𝐷𝐴 = 𝐷𝐵 = 2Γ/𝛿𝑥 = 2𝐷, and 𝐹𝐴 = 𝐹𝐵 = 𝐹.




Note that, 𝐹 = 𝐹𝑒 = 𝐹𝑤 = 𝜌𝑢 and 𝐷 = 𝐷𝑒 = 𝐷𝑤 = Γ/𝛿𝑥 everywhere.

The BCs enter the discretised equations as source contributions:

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝑆𝑢, with 𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + (𝐹𝑒 − 𝐹𝑤) − 𝑆𝑃 .




Case 1𝑢 = 0.1m/s:

𝐹 = 𝜌𝑢 = 0.1, 𝐷 = /𝛿𝑥 = 0.1/0.2 = 0.5

So 𝑃𝑒 = 𝐹/𝐷 = 0.2

The results are summarised in the table




Comparison of the upwind difference numerical results and the analyticalsolution for case 1.

The UD scheme produces good results at this cell Peclet number.




Case 2𝑢 = 2.5m/s:

𝐹 = 𝜌𝑢 = 2.5, 𝐷 = /𝛿𝑥 = 0.1/0.2 = 0.5

Now 𝑃𝑒 = 𝐹/𝐷 = 5.0

The results are summarised in the table




Comparison of the upwind difference numerical results and the analyticalsolution for case 2.

The UD scheme produces a more realistic solution than the CD scheme.



The upwind differencing schemeAssessment of the upwind differencing scheme

Conservativeness: The upwind differencing scheme utilises consistentexpressions to calculate fluxes through cell faces: therefore it can beeasily shown that the formulation is conservative.Boundedness: The coefficients of the discretised equation are alwayspositive and satisfy the requirements for boundedness. When the flowsatisfies continuity the term (𝐹𝑒 − 𝐹𝑤) in 𝑎𝑃 is zero and gives𝑎𝑃 = 𝑎𝑊 + 𝑎𝐸 , which is desirable for stable iterative solutions. All thecoefficients arepositive and the coefficient matrix is diagonallydominant, hence no "wiggles" occur in the solution.Transportiveness: The scheme accounts for the direction of the flowso transportiveness is built into the formulation.Accuracy: The scheme is based on the backward differencing formulaso the accuracy is only first-order on the basis of the Taylor seriestruncation error.




False diffusionBecause of its simplicity the upwind differencing scheme has beenwidely applied in early CFD calculations.A major drawback of the scheme is that it produces erroneous resultswhen the flow is not aligned with the grid lines.The upwind differencing scheme causes the distributions of thetransported properties to become smeared in such problems.The resulting error has a diffusion-like appearance and is referred to asfalse diffusion.




To identify the problem of False diffusion due to the upwind scheme, apure convection process is considered without physical diffusion.

A domain where the velocity fieldis uniform and parallel to thediagonal (solid line) across the grid(𝑢 = 𝑣 = 2m/s).

The boundary conditions for thescalar are 𝜑 = 0 along the south andeast boundaries, and 𝜑 = 100 on thewest and north boundaries.

At the first and the last nodeswhere the diagonal intersects theboundary a value of 50 is assigned toproperty 𝜑.




Grid refinement can overcome the false diffusion problem but expensive.

In high Reynolds number flows, false diffusion can be large enough togive physically incorrect results (Leschziner, 1980; Huang et al., 1985).

ConclusionThe upwind differencing scheme is not entirely suitable for accurate flowcalculations and considerable research has been directed towards findingimproved discretisation schemes.


稳态对流扩散问题 · 高级格式Steady Convective Diffusion Problems

Dongke Sun (孙东科)[email protected]

东南大学机械工程学院School of Mechanical Engineering

Southeast University

January 31, 2019

The hybrid differencing scheme

OUTLINE

7 The hybrid differencing schemeAssessment of the hybrid differencing schemeFor multi-dimensional convection-diffusion

8 The power-law scheme

9 Higher-order differencing schemesQuadratic upwind differencing scheme: the QUICK schemeAssessment of the QUICK schemeStability problems of the QUICK scheme and remediesGeneral comments on the QUICK differencing scheme

10 TVD schemes

11 Summary




The hybrid differencing scheme of Spalding (1972) is based on acombination of central and upwind differencing schemes.The Peclet number is evaluated at the face of the control volume. Forexample, for a west face,

𝑃𝑒𝑤 =𝐹𝑤

𝐷𝑤=

(𝜌𝑢)𝑢Γ𝑤/𝛿𝑥𝑊𝑃

(35)

The hybrid differencing formula for the net flux per unit area through thewest face is as follows:

𝑞𝑤 = 𝐹𝑤

[1

2

(1 +

2

𝑃𝑒𝑤

)𝜑𝑊 +

1

2

(1 − 2

𝑃𝑒𝑤

)𝜑𝑃

]for − 2 < 𝑃𝑒𝑤 < 2

𝑞𝑤 = 𝐹𝑤𝜑𝑊 for 𝑃𝑒𝑤 ≥ 2

𝑞𝑤 = 𝐹𝑤𝜑𝑃 for 𝑃𝑒𝑤 ≤ −2




It can be easily seen that for low Peclet numbers this is equivalent to usingcentral differencing for the convection and diffusion terms, but when

|𝑃𝑒| > 2,

it is equivalent to upwinding for convection and setting the diffusion tozero. The general form of the discretised equation is


The central coefficient is given by

𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + (𝐹𝑒 − 𝐹𝑤)

The neighbour coefficients for the hybrid differencing scheme (1-D)



The hybrid differencing schemeExample

Solve the problem considered in Case 2 of Example 1 (see slide 12) usingthe hybrid scheme for 𝑢 = 2.5m/s. Compare a 5-node solution with a25-node solution.

SolutionIf we use the 5-node grid and the data of Case 2 of Example 1 and𝑢 = 2.5m/s we have:

𝐹 = 𝐹𝑒 = 𝐹𝑤 = 𝜌𝑢 = 2.5, and 𝐷 = 𝐷𝑒 = 𝐷𝑤 = Γ/𝛿𝑥 = 0.5

and hence a Peclet number

𝑃𝑒𝑤 = 𝑃𝑒𝑒 = 𝜌𝑢𝛿𝑥/Γ = 5.

Since the cell Peclet number 𝑃𝑒 is greater than 2, the hybrid scheme usesthe upwind expression for the convective terms and sets the diffusion tozero.Dongke Sun (Southeast University) January 31, 2019 52 / 92


The hybrid differencing schemeSolution

The discretised equation at internal nodes 2, 3 and 4 is defined by

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸

and its coefficients.We also need to introduce boundary conditions at nodes 1 and 5, whichneed special treatment. At the boundary node 1 we write

𝐹𝑒𝜑𝑃 − 𝐹𝐴𝜑𝐴 = 0 −𝐷𝐴(𝜑𝑃 − 𝜑𝐴) (38)

and at node 5

𝐹𝐵𝜑𝑃 − 𝐹𝑤𝜑𝑊 = 𝐷𝐵(𝜑𝐵 − 𝜑𝑃 ) − 0 (39)

It can be seen that the diffusive flux at the boundary is entered on the righthand side and the convective fluxes are given by means of the upwindmethod.Dongke Sun (Southeast University) January 31, 2019 53 / 92



Noting that 𝐹𝐴 = 𝐹𝐵 = 𝐹 and 𝐷𝐵 = 2Γ/𝛿𝑥 = 2𝐷, we have

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 +𝛼𝐸𝜑𝐸+𝑆𝑢 with 𝛼𝑃 = 𝛼𝑊 +𝛼𝐸+(𝐹𝑒−𝐹𝑤)−𝑆𝑃 (40)

Substitution of numerical values gives the coefficients




The matrix form of the equation set is⎡⎢⎢⎢⎢⎣3.5 0 0 0 0−2.5 2.5 0 0 0

0 −2.5 2.5 0 00 0 −2.5 2.5 00 0 0 −2.5 2.5

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

𝜑1

𝜑2

𝜑3

𝜑4

𝜑5

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣3.50000

⎤⎥⎥⎥⎥⎦ (41)

The solution to the above system is⎡⎢⎢⎢⎢⎣𝜑1

𝜑2

𝜑3

𝜑4

𝜑5

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣1.01.01.01.0

0.7143

⎤⎥⎥⎥⎥⎦ (42)




Comparison with the analytical solution

The solution obtained with the fine grid is remarkably good.




The numerical results are compared with the analytical solution:

Since the cell Peclet number is high, they are the same as those forpure upwind differencing.When the grid is refined to an extent that the cell 𝑃𝑒 < 2, the schemereverts to central differencing and produces an accurate solution.

ConclusionFor 𝑃𝑒 = 1, the hybrid scheme reverts to central differencing scheme.


The hybrid differencing scheme Assessment of the hybrid differencing scheme

OUTLINE




10 TVD schemes

11 Summary


The hybrid differencing scheme Assessment of the hybrid differencing scheme

The hybrid differencing schemeAssessment of the hybrid differencing scheme

The hybrid difference scheme exploits the favourable properties of theupwind and central differencing schemes. It switches to upwind differencingwhen central differencing produces inaccurate results at high 𝑃𝑒 numbers.

The scheme is fully conservative and since the coefficients are alwayspositive it is unconditionally bounded.It satisfies the transportiveness requirement by using an upwindformulation for large values of Peclet number.The scheme produces physically realistic solutions and is highly stablewhen compared with higher-order schemes such as QUICK to bediscussed later in the chapter.The disadvantage is that the accuracy in terms of Taylor seriestruncation error is only first-order.

Hybrid differencing has been widely used in various CFD procedures andhas proved to be very useful for predicting practical flows.


The hybrid differencing scheme For multi-dimensional convection-diffusion

OUTLINE




10 TVD schemes

11 Summary



The hybrid differencing schemeFor multi-dimensional convection-diffusion

The hybrid differencing scheme can easily be extended to 2-D and 3-D.

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝛼𝑆𝜑𝑆 + 𝛼𝐵𝜑𝐵 + 𝛼𝑇𝜑𝑇 (43)

with central coefficient

𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + 𝛼𝑆 + 𝛼𝑁 + 𝛼𝐵 + 𝛼𝑇 + ∆𝐹.

In the above expressions the values of 𝐹 and 𝐷 are calculated with



The hybrid differencing schemeFor multi-dimensional convection-diffusion


The power-law scheme

OUTLINE




10 TVD schemes

11 Summary




The power-law differencing scheme of Patankar (1980) is a more accurateapproximation to the 1-D exact solution and produces better results thanthe hybrid scheme.

In this scheme diffusion is set to zero when cell 𝑃𝑒 exceeds 10.If 0 < 𝑃𝑒 < 10 the flux is evaluated by using a polynomial expression.

For example, the net flux per unit area at the west control volume face is

𝑞𝑤 = 𝐹𝑤[𝜑𝑊 − 𝛽𝑤(𝜑𝑃 − 𝜑𝑊 )] for 0 < 𝑃𝑒 < 10

𝑞𝑤 = 𝐹𝑤𝜑𝑊 for 𝑃𝑒 > 10

where 𝛽𝑤 = (1 − 0.1𝑃𝑒𝑤)5/𝑃𝑒𝑤

The central coefficient 𝑎𝑃 = 𝑎𝑊 + 𝑎𝐸 + (𝐹𝑒 − 𝐹𝑤), and


Higher-order differencing schemes

OUTLINE




10 TVD schemes

11 Summary


Higher-order differencing schemes Quadratic upwind differencing scheme: the QUICK scheme

OUTLINE




10 TVD schemes

11 Summary



Higher-order differencing schemesQuadratic upwind differencing scheme: the QUICK scheme

The quadratic upstream interpolation for convective kinetics (QUICK)scheme of Leonard (1979) uses a three-point upstream-weighted quadraticinterpolation for cell face values.

The face value of 𝜑 is obtained from a quadratic function passing throughtwo bracketing nodes (on each side of the face) and a node on theupstream side (See the figure).




For a uniform grid the value of 𝜑 at the cell face between two bracketingnodes 𝑖 and 𝑖− 1 and upstream node 𝑖− 2 is given by the following formula:

𝜑𝑓𝑎𝑐𝑒 =6

8𝜑𝑖−1 +

3

8𝜑𝑖 −

1

8𝜑𝑖−2 (45)

When 𝑢𝑤 > 0, the bracketing nodes for the west face 𝑤 are 𝑊 and 𝑃 , theupstream node is 𝑊𝑊 (See Figure) and

𝜑𝑤 =6

8𝜑𝑊 +

3

8𝜑𝑃 − 1

8𝜑𝑊𝑊 (46)

When 𝑢𝑒 > 0, the bracketing nodes for the east face 𝑒 are 𝑃 and 𝐸, theupstream node is 𝑊 , so

𝜑𝑒 =6

8𝜑𝑃 +

3

8𝜑𝐸 − 1

8𝜑𝑊 (47)




The discretised form of the one-dimensional convection–diffusiontransport equation may be written as[

𝐹𝑤

(6

8𝜑𝑃 +

3

8𝜑𝐸 − 1

8𝜑𝑊

)− 𝐹𝑤

(6

8𝜑𝑊 +

3

8𝜑𝑃 − 1

8𝜑𝑊𝑊

)]= 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 )

which can be rearranged to give[𝐷𝑤 − 3

8𝐹𝑤 + 𝐷𝑒 +

6

8𝐹𝑒

]𝜑𝑃

=

[𝐷𝑤 +

6

8𝐹𝑤 +

1

8𝐹𝑒

]𝜑𝑊 +

[𝐷𝑒 −

3

8𝐹𝑒

]𝜑𝐸 − 1

8𝐹𝑤𝜑𝑊𝑊




This is now written in the standard form for discretised equations:

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝛼𝑊𝑊𝜑𝑊𝑊 (49)

where

For 𝐹𝑤 < 0 and 𝐹𝑒 < 0:The flux across the west and east boundaries is given by the expressions

𝜑𝑤 =6

8𝜑𝑃 +

3

8𝑊− 1

8𝜑𝐸

𝜑𝑒 =6

8𝜑𝐸 +

3

8𝑃− 1

8𝜑𝐸𝐸

(50)




This is now written in the standard form for discretised equations:

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝛼𝑊𝑊𝜑𝑊𝑊 (49)

where

For 𝐹𝑤 < 0 and 𝐹𝑒 < 0:After rearrangement as above, we obtain the following coefficients




The QUICK scheme for 1-D convection–diffusion problems is

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝛼𝑊𝑊𝜑𝑊𝑊 + 𝛼𝐸𝐸𝜑𝐸𝐸 (51)

with central coefficient

𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + 𝛼𝑊𝑊 + 𝛼𝐸𝐸 + (𝐹𝑒 − 𝐹𝑤)

and neighbour coefficients

where𝛼𝑤 = 1 for 𝐹𝑤 > 0 and 𝛼𝑒 = 1 for 𝐹𝑒 > 0,𝛼𝑤 = 0 for 𝐹𝑤 < 0 and 𝛼𝑒 = 0 for 𝐹𝑒 < 0.



The QUICK schemeExample

Using the QUICK scheme solve the problem considered in Example 5.1 for𝑢 = 0.2 m/s on a five-point grid.

Mirror node treatment at the boundary



The QUICK schemeExample

It can be easily shown that the linearly extrapolated value at the mirrornode is given by

𝜑0 = 2𝜑𝐴 − 𝜑𝑃 (52)

The extrapolation to the "mirror" node has given us the required 𝑊 nodeforthe formula (47) that calculates 𝜑𝑒 at the east face of control volume 1:

𝜑𝑒 =6

8𝜑𝑃 +

3

8𝜑𝐸 − 1

8(2𝜑𝐴 − 𝜑𝑃 )

=7

8𝜑𝑃 +

3

8𝜑𝐸 − 2

8𝜑𝐴

(53)

At the boundary nodes the gradients must be evaluated using an expressionconsistent with formula (53).




It can be shown that the diffusive flux through the west boundary is givenby

Γ𝜕𝜑

𝜕𝑥

𝐴

=𝐷*

𝐴

3(9𝜑𝑃 − 8𝜑𝐴 − 𝜑𝐸) (54)

The discretised equation at node 1 is

𝐹𝑒

[7

8𝜑𝑃 +

3

8𝜑𝐸 − 2

8𝜑𝐴

]− 𝐹𝐴𝜑𝐴

= 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷*

𝐴

3(9𝜑𝑃 − 8𝜑𝐴 − 𝜑𝐸)

(55)

At control volume 5, the 𝜑-value at the east face is known ( 𝜑𝑒 = 𝜑𝐵 ) andthe diffusive flux of 𝜑 through the east boundary is given by

Γ𝜕𝜑

𝜕𝑥

𝐵

=𝐷*

𝐵

3(8𝜑𝐵 − 9𝜑𝑃 + 𝜑𝑊 ) (56)




At node 5 the discretised equation becomes

𝐹𝐵𝜑𝐵 − 𝐹𝑤

[6

8𝜑𝑊 +

3

8𝜑𝑃 − 1

8𝜑𝑊𝑊

]=

𝐷*𝐵

3(8𝜑𝐵 − 9𝜑𝑃 + 𝜑𝑊 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 )

(57)

At node 2 we have

𝐹𝑒

[6

8𝜑𝑃 +

3

8𝜑𝐸 − 1

8𝜑𝐴

]− 𝐹𝑤

[7

8𝜑𝑊 +

3

8𝜑𝑃 − 2

8𝜑𝐴

]= 𝐷𝑒(𝜑𝐸 − 𝜑𝑃 ) −𝐷𝑤(𝜑𝑃 − 𝜑𝑊 ) (58)

The discretised equations for nodes 1, 2 and 5 are now written as

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝑊𝜑𝑊𝑊 + 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝑆𝑢 (59)

with 𝛼𝑃 = 𝛼𝑊𝑊 + 𝛼𝑊 + 𝛼𝐸 − 𝑆𝑃




Substitution of numerical values gives the coefficients:




The matrix form of the equation set is⎡⎢⎢⎢⎢⎣2.175 −0.592 0 0 0−0.7 1.075 −0.425 0 00.025 −0.675 1.075 −0.425 0

0 0.025 −0.675 1.075 −0.4250 0 0.025 −0.817 1.925

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

𝜑1

𝜑2

𝜑3

𝜑4

𝜑5

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣1.583−0.05

000

⎤⎥⎥⎥⎥⎦The solution to the above system is⎡⎢⎢⎢⎢⎣

𝜑1

𝜑2

𝜑3

𝜑4

𝜑5

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣0.9648−0.87070.73090.52260.2123

⎤⎥⎥⎥⎥⎦Dongke Sun (Southeast University) January 31, 2019 77 / 92




The QUICK solution is almost indistinguishable from the exact solution.





The table confirms that the errors are very small even with this coarsemesh.

The sum of absolute errors in the above table indicates that

ConclusionThe QUICK scheme gives a more accurate solution than the centraldifferencing scheme.


Higher-order differencing schemes Assessment of the QUICK scheme

OUTLINE




10 TVD schemes

11 Summary


Higher-order differencing schemes Assessment of the QUICK scheme

Higher-order differencing schemesAssessment of the QUICK scheme

The scheme uses consistent quadratic profiles – the cell face values offluxes are always calculated by quadratic interpolation between twobracketing nodes and an upstream node – and is therefore conservative.

Since the scheme is based on a quadratic function its accuracy in termsof Taylor series truncation error is third-order on a uniform mesh.The transportiveness property is built into the scheme as the quadraticfunction is based on two upstream and one downstream nodal values.If the flow field satisfies continuity the coefficient a 𝑃 equals the sumof all neighbour coefficients, which is desirable for boundedness.The QUICK scheme is therefore conditionally stable.

Another notable feature is the fact that the discretised equations involvenot only immediate-neighbour nodes but also nodes further away.Tri-diagonal matrix solution methods (see Textbook) are not directlyapplicable.Dongke Sun (Southeast University) January 31, 2019 81 / 92

Higher-order differencing schemes Stability problems of the QUICK scheme and remedies

OUTLINE




10 TVD schemes

11 Summary



Higher-order differencing schemesStability problems of the QUICK scheme and remedies

The Hayase et al. (1992) QUICK scheme can be summarised as follows:

𝜑𝑤 = 𝜑𝑊 +1

8[3𝜑𝑃 − 2𝜑𝑊 − 𝜑𝑊𝑊 ] for 𝐹𝑤 > 0

𝜑𝑒 = 𝜑𝑃 +1

8[3𝜑𝐸 − 2𝜑𝑃 − 𝜑𝑊 ] for 𝐹𝑒 > 0

𝜑𝑤 = 𝜑𝑃 +1

8[3𝜑𝑊 − 2𝜑𝑃 − 𝜑𝐸 ] for 𝐹𝑤 < 0

𝜑𝑒 = 𝜑𝐸 +1

8[3𝜑𝑃 − 2𝜑𝐸 − 𝜑𝐸𝐸 ] for 𝐹𝑒 < 0

(61)

The discretisation equation takes the form

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝑆 (62)

The central coefficient is

𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + (𝐹𝑒 − 𝐹𝑤)



Higher-order differencing schemesStability problems of the QUICK scheme and remedies

and

where𝛼𝑤 = 1 for 𝐹𝑤 > 0 and 𝛼𝑒 = 1 for 𝐹𝑒 > 0,𝛼𝑤 = 0 for 𝐹𝑤 < 0 and 𝛼𝑒 = 0 for 𝐹𝑒 < 0.


Higher-order differencing schemes General comments on the QUICK differencing scheme

OUTLINE




10 TVD schemes

11 Summary


Higher-order differencing schemes General comments on the QUICK differencing scheme

Higher-order differencing schemesGeneral comments on the QUICK differencing scheme

The QUICK differencing scheme has greater formal accuracy than thecentral differencing or hybrid schemes, and it retains theupwind-weighted characteristics.

Comparison of QUICKand upwind solutionsfor the 2D testconsidered in the caseof False Diffusion.

The resultant false diffusion is small, and solutions achieved withcoarse grids are often considerably more accurate than those of theupwind or hybrid schemes.


TVD schemes

OUTLINE




10 TVD schemes

11 Summary


Summary

OUTLINE




10 TVD schemes

11 Summary


Summary

Summary

The discretised equations for a general internal node for the central,upwind and hybrid differencing and the power-law schemes of a 1-Dconvection–diffusion problem take the following form:


with 𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + (𝐹𝑒 − 𝐹𝑤)

The neighbour coefficients for these schemes are

The boundary conditions enter the discretised equations via sourceterms. Their treatment is specific to each discretisation scheme.


Summary

Summary

Discretisation schemes that possess conservativeness, boundedness andtransportiveness give physically realistic results and stable iterativesolutions:

The central differencing method is not suitable for general-purposeconvection–diffusion problems because it lacks transportiveness andgives unrealistic solutions at large values of the cell Peclet number.Upwind, hybrid and power-law differencing all possessconservativeness, boundedness and transportiveness and are highlystable, but suffer from false diffusion in multi-dimensional flows if thevelocity vector is not parallel to one of the co-ordinate directions.

The discretised equations of the standard QUICK method of Leonard(1979) have the following form for a general internal node point:

𝛼𝑃𝜑𝑃 = 𝛼𝑊𝜑𝑊 + 𝛼𝐸𝜑𝐸 + 𝛼𝑊𝑊𝜑𝑊𝑊 + 𝛼𝐸𝐸𝜑𝐸𝐸 (87)

where𝛼𝑃 = 𝛼𝑊 + 𝛼𝐸 + 𝛼𝑊𝑊 + 𝛼𝐸𝐸(𝐹𝑒 − 𝐹𝑤)


Summary

Summary

The neighbour coefficients of the standard QUICK scheme are

with 𝛼𝑤 = 1 for 𝐹𝑤 > 0 and 𝛼𝑒 = 1 for 𝐹𝑒 > 0,𝛼𝑤 = 0 for 𝐹𝑤 < 0 and 𝛼𝑒 = 0 for 𝐹𝑒 < 0.

Higher-order schemes, such as QUICK, can minimise false diffusionerrors but are less computationally stable.Dongke Sun (Southeast University) January 31, 2019 91 / 92

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