I have a velocity field v in an inertial frame of reference (it can be non-inertial too but not important). I want to analyze the same field observed by a rotating & translating observer. The classical space transformation is given as:
y = Q(t)*x + c(t) (1)
where x is the position of a material point in the original frame of reference, y is the position of a material point seen by the rotating & translating observer in the original frame of reference, Q(t) is a proper orthogonal tensor (rotation tensor), c(t) is the translation vector.
Let us see how the velocity field will transform under such a transformation, i.e. how the velocity field will be seen by the rotating & translating observer.
\dot{y} = \dot{Q(t)} * x + Q(t) * \dot{x} + \dot{c(t)} (2)
It seems easy, but I code this up, and it does not work.
I agree with Yuri it the important thing to undergrad in physics
Dear Mirlan Karimov and Readers,
The answer is provided by Stoyanov In the note: “Transformation of kinematical quantities from rotating into static coordinate system”
https://arxiv.org/pdf/0806.2529.pdf
I hope that this text is sufficient in order to provide a simple explanation. Please let me know if You need any further information and good luck in exploring the fundamental concepts of mechanics.
If a CFD-related viewpoint interests you, the transformation(s) are discussed in Numerical Computation of Internal and External Flows, Vol. 1 - Fundamentals of Numerical Discretization. Chapter 1, Section 1.4 deals with rotating frames. Unfortunately, it's a bit difficult.
Hi all,
@Douglas, I would not say it is difficult but it is definitely not what I am looking for. Thank you anyway!
@Janusz, I read through but could not find anything new there. These derivations are given in all continuum mechanics books.
@Yuri, thank you for the detailed reply, but I do not think you address the right point. My background is dynamical systems, so I have a core understanding of governing equations behind these systems and what is actually going on. Please just give me a piece of code that would take the velocity field v(x,t) to v(y,t) if you can.
To my knowledge it should be as follows: (MATLAB)
syms c t om
v_in_x_frame = [ - sin(ct) , cos(ct) - om/2 ; cos(ct) + om/2 , sin(ct)];
Q = [cos(ct/2) ,sin(ct/2); -sin(ct/2) , cos(ct/2)];
Qtr = transpose(Q); assume(Q, 'real');
v_in_y_frame = Q*v_in_x_frame + diff(Q,t,1)*x;
x = Qtr*x; %to change the coordinates
v_in_y_frame = Q*v_in_x_frame + diff(Q,t,1)*(Qtr*x);
Dear Mirlan Karimov and Readers,
@Mirlan
I suggest to change your derivation based on the equations discussed in the attached file. I have included also a set of figures illustrating the action of a transformation on a simple flow field. I hope that the attached information answers your question.
look at this book may help you '' Computational Fluid Dynamics'' by John F. Wendt
After reading the book '' Computational Fluid Dynamics'' by John F. Wendt I was not able to find the answer to the question posed in this thread. The problem of transformation of flow field between different frames is discussed in detail in the excellent text by Ottino (1989) (The kinematics of mixing: stretching, chaos and transport, Cambridge University Press). For derivation of the explicit transformation equation please kindly consult material in section 3.7 (page 49).
Dear Mirlan Karimov ,
Use Basic advances of turbulence modeling book as a reference. This book is good for transform a flow field to rotating &transforming frame of reference.
Here, book can explain you using Galilean transformation equation which is valid for the mathematical form of time independent conservation laws of newtonion physics in the presence of relative motion between two frames of reference. Must go ahead with Galilean and frame rotation inverience of the mean rate of strain(deformation) tensor. Equation is proposed for velocity field, rotation variation under the coordinate transformation. It has explained the boussinesq hypothesis on Reynolds stress tensor.
Ashish
Dear Prof. Janusz, your explanation in pdf clarified the problem. I could not be more thankful! You knew where the problem exactly was instead of offering me to read basics :)
Dear Mirlan, Thanks for the message, I'm glad to hear that the comments have been helpful. Ottino's book is particularly relevant in research aimed at improving our understanding of the complicated behavior of fluids. The methods of dynamical systems created by H. Poincare and applied further to geophysical fluids by E. Lorenz are particularly well explained in this text.
For interesting review of the related concepts in a very broad context (excellent in my opinion) please see A history of chaos theory by C. Oestreicher
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3202497/
Dear Prof. Janusz, after I looked inside the book by Ottino I immediately realized how valuable it is. I ordered it from the library and reading at the moment. It is well structured and rich in content. However, it is not a book for a beginner to either of Fluid Dynamics and Theory of Dynamical Systems. One needs to have at least introductory (in a mathematical way) understanding of important concepts in both fields to be able to enjoy Ottino's book. Tbh, while reading it I feel like it is a collection of the concepts that I learned each by digging extensive literature...It helps me to connect dots.
I will also check the Oestreicher. I really appreciate your advise!
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