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

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

Also we beg everybody to help us with original works. Please see section about what we need. If anyone have literature connected with convective schemes, please drop us a line. Of course You are welkome to participate in Wiki

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 critical Peclet number

Discretizations Schemes - Estimation of order - Richardson extrapolation

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 \boldsymbol{x=-0.5}

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 -1\leq x \leq 1 \ , \ 0 \leq y \leq 1, with the streamfunction being specified as


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

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

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

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

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

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

\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}
(2)

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

\frac{d \phi}{d y}= 0 \ \ \ \ : \ \ \ \ y=0 \ \ \ \ \ 0\leq x \leq 1
(2)

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

NM convectionschemes struct grids Smith hutton Countor HLPA. probe 01.jpg

NM convectionschemes struct grids Smith hutton Countor UDS 02 probe 03.jpg

NM convectionschemes struct grids Smith hutton Countor HLPA 02 probe 03.jpg

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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