Hi, can somebody suggest some references where the definition of streamline for 3-D flow can be found? Are there some mathematic definitions for it? Thanks

(1). There is no difference between a streamline for 2-D flow and a streamline for 3-D flow. (2). So, you need to go back to the Introduction to Fluid Mechanics text book and try to find the definition of a streamline.(2). There is easier way to define a streamline. First you inject fluid dye at certain point in the flow field as the starting point of this streamline. The dye will give you the fluid particle path called a streakline. (3). Now, if this streakline does not change with time, that is, steady-state, then it becomes a streamline. (4). At any point on the streakline, the tangent of the line is the same as the direction of the velocity vector at that instant. (5). If the velocity vectors in the velocity field all have the same direction, then the streamline will be straight lines. So, streamlines in 2-d as well as in 3-D are very important subject which requires extensive study. (6). As the velocity vectors start to change direction, there is another very important variable which plays key role in the flow field, that is the vorticity. (7). So, basically, numerical analysis in fluid mechanics is almost equal to the study of the streamlines and the vorticity distribution in the flow field. I say "almost" because there is another variable "Mach number" which can provide new flow features such as the shock waves and expansion waves in the flow field. (8). So, in a way, the definition of a streamline is very simple. But the understanding of the streamline and the related vorticity field is not easy to do. (9). Based on my recent experience, it seems to me that the vacuum of 90's in CFD development has started to show its effect in the recent stock market down fall. In other words, the vacuum created a generation gap between the traditional designer and the advanced CFD research. It is something like trying to integrate the dot.com business. (10). The designer is being fed by the garbage generated from various cfd codes, and the cfd researchers are busy cranking out more cfd codes for designers. And both seem to forget about the definition of the streamline and the vorticity. (11). But as I say, it is just the begining, you will see more failure in business and in product design. Remember that over-supply of printed money will lead to inflation, and over-supply of the product will lead to depression. In my visit to the flea market, the brand new watch is now selling at $5.00 each, while an old automatic Hamilton watch is still asking for $120.00 So, my suggestion is: use your brain (a very advanced computer) actively, instead of your brute force. If you still can't find the definition of a streamline or the vorticity, then please come back here, we will talk about it in more detail.

Thank you, John. In fact, I have already got the streamlines by Tecplot from my computed results. I just wanna present my streamline figures in a mathematical way. And hopefully I can capture some special streamlines during experiments. My idea is I can capture what happens along the streamlines by Laser Doppler Velocimeter (LDV), although I can visulize the streamlines by laser sheet or dying. Then, my question is to get the appropriate definition for streamlines and apply it to the experiments. Thanks for your help. Best regards.

(1). In 2-D, a streamline can be represented by the stream function. (2). But in 3-D, you can only do the streamline tracing for the steady-state flow. In that case, the streamline is tangent to the local velocity vector and you have to give it a starting point. (3).In 3-D, the streamlines distribution can be very complex. It is commonly used to study the flow separation and the secondary flow behavior.

pages 162-164 of Principles of Idean-Fluid Aerodynamics by K. Karamcheti... Krieger Publishing Company, FL, 1980.... it has the generic definition of a streamline without restricting to the particular consideration of a two-dimensional flow, and expressed it in terms of a mathematical relation.

Definition:

dx/dt = u, dy/dt = v, dz/dt = w

Solve x, y, and z and you get you streamline.

(1). In the 1966 version of the book published by John Wiley and Sons, p-162, the equation in vector form is ds x V =0, V is the velocity vector, ds is the tangent vector of the streamline. (2). On p-163, the Cartesian form is (dx/u = dy/v = dz/w). where dx, dy, dz are components of the tangent vector of the streamline, and u, v, w are components of the velocity vector at the same location. (3). It is really very simple. So, keep it handy.