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

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== Discretizations Schemes - Estimation of critical Peclet number  ==

== 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) ===

{{reference-paper | author=R.M.Smith and A.G.Hutton | year=1982 | title=The numerical treatment of advection: A performance comparison of current methods | rest=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 <math> \boldsymbol{x=-0.5}</math>

We shall use here smooth inlet profile

This is 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 <math>-1\leq x \leq 1</math> \ , \ <math>0 \leq y \leq 1</math>, with the streamfunction being specified as 


<table width="100%"><tr><td>
:<math> 
\psi = - \left( 1 - x^{2} \right) \left( 1 - y^{2} \right)
</math>
</td><td width="5%">(1)</td></tr></table>

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

<table width="100%"><tr><td>
:<math> 
u = 2y \left( 1 -  x^{2} \right)
</math>
</td><td width="5%">(1)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
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>
</td><td width="5%">(1)</td></tr></table>

where <math>\boldsymbol{\alpha}</math> is a parameter which defines the sharpness of the inlet profile. The outer boundaries are prescribed as 

<table width="100%"><tr><td>
:<math> 
\phi = 1 - \tanh \alpha	 \ \ \ :  
\begin{cases}
x = -1  &   0 \leq y \leq 1  \\ 
y =  1  & - 1 \leq x \leq 1  \\  
x =  1  &   0 \leq y \leq 1  
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

Thus <math>\boldsymbol{\phi}</math> is <math>\boldsymbol{0}</math> on <math>\boldsymbol{x\pm 1 }</math> and <math>\boldsymbol{y=1}</math>, and is <math>\boldsymbol{2}</math> at the origin. At the outlet a zero normal derivative is prescribed

<table width="100%"><tr><td>
:<math> 
\frac{d \phi}{d y}= 0 \ \ \ \ : \ \ \ \ y=0 \ \ \ \  \ 0\leq x \leq 1 
</math>
</td><td width="5%">(2)</td></tr></table>

The two parameters which define the scalar field are the Peclet number, which specifies the diffusivity of the problem, and <math>\boldsymbol{\alpha}</math> which is a parameter that defines the sharpness of the inlet profile.

=== 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.
 


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