# Approximation Schemes for convective term - structured grids - Summary of Discretizations Schemes and examples

### From CFD-Wiki

(→Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem)) |
(→Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem)) |
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:<math> | :<math> | ||

v = -2x \left( 1 - y^{2} \right) | v = -2x \left( 1 - y^{2} \right) | ||

+ | </math> | ||

+ | </td><td width="5%">(1)</td></tr></table> | ||

+ | |||

+ | The scalar <math>\boldsymbol{\phi}</math> is solved over the domain, with the value of <math>\boldsymbol{\phi}</math> being prescribed at the inlet and on the left, right and top boundaries, whilst on the outlet the derivative of <math>\boldsymbol{\phi}</math> normal to the boudary is set to zero. The inlet profile is given as | ||

+ | |||

+ | <table width="100%"><tr><td> | ||

+ | :<math> | ||

+ | \phi = 1 + \tanh \left[ \alpha \left( 2x + 1 \right) \right] \ \ \ : \ \ \ y=0 \ \ \ -1 \leq x \leq 0 | ||

</math> | </math> | ||

</td><td width="5%">(1)</td></tr></table> | </td><td width="5%">(1)</td></tr></table> |

## Revision as of 22:52, 29 September 2005

*When we shall fill this page, I offer to make common identifications, because in different issues was used different notation.*

*Also we beg everybody to help me with original works. Later I shall write, what is necessary. If anyone have literature connected with convective schemes, please drop me a line.*

*We shall be very glad and grateful to hear any critical suggestion (please drop a few lines at Wiki Forum)*

*It is just a skeleton, but we hope that it will be developed into the good thing*

## Contents |

## Discretizations Schemes Estimation of order

## Discretizations Schemes Estimation of error

## Selection advice

## Comparison of Discretizations Schemes

## Numerical examples

### Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem)

**R.M.Smith and A.G.Hutton (1982)**, "The numerical treatment of advection: A performance comparison of current methods", Numerical Heat Transfer, Vol. 5, p439.

This was the test problem devised for evaluating a range of numerical models of convection at the third meeting of the International Association for Hydraulic Research Working Group on Refined Modelling of flow

Sometimes it was used scalar profile with a discontinuity at

We shall use here more smooth inlet profile

| (1) |

provides a simple problem with a strong discontinuity in a scalar profile and flow that is not parallel to the boundaries of the domain being tested. As such it should reveal the poor convergence of the first order schemes, which exhibit false diffusion on flow that is not parallel to the grid, whilst the sharp gradient should generate oscillations in the solutions generated using the second and third order schemes.

The steady transport equation is solved in the region \ , \ , with the streamfunction being specified as

| (1) |

which is shown in figure below. This streamfunction gives a velocity field of

| (1) |

| (1) |

The scalar is solved over the domain, with the value of being prescribed at the inlet and on the left, right and top boundaries, whilst on the outlet the derivative of normal to the boudary is set to zero. The inlet profile is given as

| (1) |

### Square Lid-driven cavity flow

## Example code for solving Smith-Hutton problem

*Dear friends*

*It's just a scrap. Later I'll correct it, although it's a complete working code*

*Michail*

Sample code for solving Smith-Hutton test - Fortran 90

It's a results, obtained using this code (**UDS** and **HLPA** schemes)

Below it's cleary seen the numerical diffusion impact, comparing the contour fields obtaining using the UDS and HLPA. A bit later we shall place here a solution gained with QUICK scheme, and it will be seen the osscilations.