Advanced workshop on Computational Fluid Dynamics

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Advanced workshop on Computational Fluid Dynamics

Discretization

Schemes

Dr.D.Prakash

Senior Assistant Professor, School of Mechanical Engineering,

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• Domain is

discretized into a finite set of control volumes

or cells. The discretized

domain is called the “grid” or the “mesh.”

• General conservation (transport) equations for mass,

momentum, energy, etc., are discretized into algebraic equations.

• All equations are solved to render flow field.

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Advanced workshop on Computational Fluid Dynamics

Status of FDM, FEA in CFD

FDM

Very simple to get discretized equation

Suitable for simple geometry

Not convenient to solve problem with complex geometry

Mathematical sense

FDM rely on expansion of a function in Taylor’s series.

Requirement of discretization

Conservativeness

Boundedness

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FEA

Relatively more complicated than FDM

Have strong mathematical sense- Principle of error minimization

Principle

Minimization of function is the minimization of potential energy

Eg- Structural problem: Structure at equilibrium tends to minimization

of potential energy.

But the governing principles of a fluid flow , heat transfer and mass

transfer is the conservation of mass momentum , energy.

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1-D Steady state heat conduction problem

Governing differential equation: 𝑑

𝑑𝑥 𝑘 𝑑𝑡

𝑑𝑥 + 𝑆 = 0

Step 1: Divide into no of subdomains

Centroid

Isolate any control volume from the domain

In addition to 3 centroid, two grind point at the boundaries should be considered.

P E

W

Boundary grid point

1 2 3 4 5 6

w e

Faces of the control volume – e and w

Main grind points- Upper case letter

x

xw xe Step 2: Integrate the GDE for each

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Step 3: Convert to algebraic equation using profile variation function

Assume the variation of T wrt x

(1) Simple assumption as piece wise linear (Constant)

W w P e _E

(2) Non constant linear

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W w P e _E

(3) Piece wise linear between control grids

𝑘_𝑒 𝑇𝐸_𝑥− 𝑇𝑃

(4) Higher order interpolation function

W

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• The net flux through the control volume boundary is the sum of

integrals over the four control volume faces (six in 3D). The

control volumes do not overlap.

• The value of the integrand is not available at the control volume

faces and is determined by interpolation.

Typical control volume

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Discretization example

• To illustrate how the conservation equations used in CFD can be

discretized

• The continuity equation is given by:

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Introduce control volume integral- Gauss divergence theorem

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Discretization example - continued

The face velocity are located midway between each of the control

volume centroid.

The face velocities are determined from the values located at the

centroids

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11 Discretization example - continued

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Discretization (Interpolation

Methods)

• Field variables (stored at cell centers) must be interpolated to the faces of

the control volumes in the FVM:

• FLUENT

_{offers a number of interpolation schemes:}

–

First-Order Upwind Scheme

• easiest to converge, only first order accurate.

–

Power Law Scheme

• more accurate than first-order for flows when Re

_cell

< 5 (typ. low Re

flows).

–

Second-Order Upwind Scheme

• uses larger ‘stencil’ for 2nd order accuracy, essential with tri/

tet

mesh or when flow is not aligned with grid; slower convergence

–

Quadratic Upwind Interpolation (QUICK)

• applies to quad/hex and hyrbid meshes (not applied to

tri’s

), useful

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First order upwind scheme

P e E

(x)

_P _e

_E

Flow direction

• This is the simplest numerical scheme.

• Assume that the value of  at the upstream of the face is the same as the cell centered value in the cell

• The main advantages are that it is easy to implement and that it results in very stable calculations, but it also very diffusive. Gradients in the flow field tend to be smeared out

• This is often the best scheme to start calculations

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Central differencing scheme

• This scheme determines the value of  at the face by linear

interpolation between the cell centered values.

• This is more accurate than the first order upwind scheme, but it leads to oscillations in the solution or divergence if the local Peclet number is larger than 2.

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15 • This is based on the analytical

solution of the one-dimensional convection-diffusion equation.

• The face value is determined from an exponential profile through the cell values. The exponential profile is approximated by the following power law equation:

• Pe is again the Peclet number.

• For Pe>10, diffusion is ignored and first order upwind is used.

Power law scheme

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Second-order upwind scheme

• We determine the value of  from the cell values in the two cells

upstream of the face.

• This is more accurate than the first order upwind scheme, but in

regions with strong gradients it can result in face values that are outside of the range of cell values. It is then necessary to apply limiters to the predicted face values.

• There are many different ways to implement this, but second-order upwind with limiters is one of the more popular numerical schemes because of its combination of accuracy and stability.

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QUICK scheme

• QUICK stands for Quadratic Upwind Interpolation for Convective

Kinetics.

• A quadratic curve is fitted through two upstream nodes and one

downstream node.

• This is a very accurate scheme, but in regions with strong gradients, overshoots and undershoots can occur. This can lead to stability

problems in the calculation. P

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Accuracy of numerical schemes

• Each of the previously discussed numerical schemes assumes some shape of the function . These functions can be approximated by Taylor series

polynomials:

• The first order upwind scheme only uses the constant and ignores the first derivative and consecutive terms. This scheme is therefore considered first order accurate.

• For high Peclet numbers the power law scheme reduces to the first order upwind scheme, so it is also considered first order accurate.

• The central differencing scheme and second order upwind scheme do include the first order derivative, but ignore the second order derivative. These

schemes are therefore considered second order accurate. QUICK does take the second order derivative into account, but ignores the third order derivative. This is then considered third order accurate.

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Solution accuracy

• Higher order schemes will be more accurate. They will also be less

stable and will increase computational time.

• It is recommended to always start calculations with first order

upwind and after 100 iterations or so to switch over to second

order upwind.

• This provides a good combination of stability and accuracy.

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Finite volume solution methods

• The finite volume solution method can either use a “segregated”

or a “coupled” solution procedure.

• With segregated methods an equation for a certain variable is

solved for all cells, then the equation for the next variable is

solved for all cells, etc.

• With coupled methods, for a given cell equations for all variables

are solved, and that process is then repeated for all cells.

• The segregated solution method is the default method in most

commercial finite volume codes. It is best suited for

incompressible flows or compressible flows at low Mach number.

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consistency

stability

convergence

Discretized equation is a good replica of PDE

Check the Discretized equation is having a property of amplifying

errors or errors get suppressed

Confirm that the computed solution is how much close to

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