Can someone please provide me with the equation for generating numerical shadowgraphs for experimental comparison?

Thanks Shuo

A thought:

What is a shadowgraph using/capturing in the flowfield to give it's effect? Is it density, for instance?

If you try to understand the physics the technique, then you could try & capture the appropriate cfd data.

As an example, I used density in some comparative work I did in high-speed flow comparisons of shadow-graph plots. The result was pretty effective.

I hope this helps.

mw...

<www.adthermtech.com> <www.adthermtech.com/wordpress3>

If I remember correctly, a shadowgram can be approximated by a plot of the Laplacian of the density.

Dino is right, shadowgraph gives you: nabla**2(rho)

So you should make black&white/greyscale contourplots of this variable.

Does anyone have the underlying theory behind this concept of using the density Laplacian?

mw...

<www.adthermtech.com>

Shadowgraph normally captures the second derivative of density, while schlieren takes the first derivative as it uses a knife-edge. For this reason, shadowgraph is being used where there are high denisty gradients in the flow and schlieren is used to capture minor density variations in the flow field and for precise information. You can refer to any basic book which deals with flow visualiztaion techniquues in high-speed flow.As far as I remember, there is a book particularly on schlieren methods which expalins with the theory. I cannot remember the name..I will also serach for that.

hope this helps R

Yes, Its "Schlieren and Shadowgraph Methods" by G.S. Settles

Many thanks R, for your information. Very interesting.

mw...

Thank you all for your helpful responses Shuo

In the field of Experimental Fluid Dynamics, Laser light is used to produce Schlieren images, Laser light travels in straight lines in homogeneous matter, it deviates when it encounters a region of different density, that is why in numerical work, the density gradient is used to trace zones of S.Waves, but honestly speaking I have no physical explanation for the laplacian of density, Could any one of the previous posts shed light on this point (articles, papers are welcom)

Shadowgraph is the simplest form of the optical visualization systems, where an intense collimated light passes through the flow field normal to the flow direction and falls directly on the screen.The idea is to measure the illumintaion of the light on the screen. For a uniform denisty flow field, the illumintaion is directly proportional to the brightness of the light source and the source area, and inversely proportional to the square of the focal length of the collimating lense. Whenever there is a density difference in the field, refraction occurs in such a way that the new illumintaion is proportional to the second spatial derivative of the density of the flow field. Thus the change in illumination is proportional to the integrated second derivative of the density. While in schlieren, the total angular deflection of the ray is measured. Owing to a density change in the flow field, the collimated high intensity light (need not be laser) will refract and will be dflected from the staright path.The light rays deflected from the path will be cut by a knife-edge. Here, the deflection of the light path is measured, and which from Snell's law is inversely proportional to the first derivative of the refractive index of the medium, which in turn is proprtional to the density of the flow field. In short, shdowgraph gives the direct indication of change of the illumintaion of the light on the screen (proportional to second derivative of density) while schlieren measures the deflection of light (proportional to the first derivative of density) due to the density change with the use of a knife-edge..

Unfortunately, I don't have a link/paper on this. Many basic books explains the mathematical theory in more detail

Rajesh wrote:

Thus the change in illumination is proportional to the integrated second derivative of the density.

So, it is the integrated second derivative (?integrated Laplacian?), we are after, not the Laplacian. Is this correct?

mw...

<www.adthermtech.com/wordpress3>

As far as I know, the change in illumination is the integrated second derivative of density. Laplacian doesn't come into picture. There is a paper by Prof.Takayama on the theoretical aspects of numerical schlieren. I found it somewhere long back..cannot recall now..will try to google it..

Simultaneous Shadow, Schlieren and Interferometric Visualization of Compressible Flows Harald Kleinea, Hans Grönigb and Kazuyoshi Takayama

Optics and Lasers in Engineering Volume 44, Issues 3-4, March-April 2006, Pages 170-189

This is not the paper that I have read..but hope to serve the purpose..

If you don't have access, I can mail you this paper..

Rajesh

I'd really appreciate a copy of this paper - thanks Rajesh.

e-mail: <des@adthermtech.com>

Many thanks, mw...

Sent....

Thanks so much, Rajesh. A very informative paper. Some lovely pictures.

mw...

<www.adthermtech.com/wordpress3>

I appreciate your answer, I will dig for information

http://www.math.uio.no/~jks/talk2.pdf a nice freely available article

http://www.potto.org/text.pdf