the trhust F of a propeller is a fn of its diameter D and angular velocity omega, forward speed u, and density rho and viscosity mu of the fluid. rewrite the function in nondimensional form.
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
-
PURPOSE OF THE EXERCISE:
Demonstrate how to collapse a function with physical variables into a relationship in terms of nondimensional groups via systematic nondimensionalization. -
ENGINEERING CONTEXT:
Propeller performance assessment frequently relies on scaling laws that relate thrust to operating and geometric parameters; mastering this nondimensional form is essential for design and performance comparison among different propeller configurations. -
PHYSICAL CONTEXT:
The dominant physics involves momentum transfer from the rotating blade (characterized by angular speed and diameter) interacting with the incoming flow (characterized by forward speed) in a viscous, incompressible fluid. -
ASSUMPTIONS:
Assume steady, incompressible flow with constant fluid properties; such assumptions simplify the dimensional analysis without compromising the scaling insight. -
SOLUTION TECHNIQUE:
We begin with the dimensional dependency
F = f(D,\;\omega,\;u,\;\rho,\;\mu) \,,
where the thrust F has dimensions \mathrm{kg\,m/s^2}, diameter D has dimensions \mathrm{m}, angular velocity \omega has dimensions \mathrm{1/s}, forward speed u has dimensions \mathrm{m/s}, density \rho has dimensions \mathrm{kg/m^3}, and viscosity \mu has dimensions \mathrm{kg/(m\,s)}.
The dimensional matrix is
\left[
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 1 & 1 \\
1 & 1 & 0 & 1 & -3 & -1 \\
-2 & 0 & -1 & -1 & 0 & -1
\end{array}
\right]
and its nullspace is
\left[
\begin{array}{ccc}
0 & -1 & -1 \\
-1 & 4 & 2 \\
-1 & 2 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right]
or, equivalently,
\left[
\begin{array}{ccc}
0 & 1 & 0 \\
-1 & -4 & 2 \\
-1 & -2 & 1 \\
1 & 0 & 0 \\
0 & -1 & -1 \\
0 & 0 & 1
\end{array}
\right]
Which leads to the following nondimensional groups:
• A thrust coefficient:
C_T = \frac{F}{\rho\, \omega^2 D^4} \,.
• An advance ratio:
J = \frac{u}{\omega D} \,.
• And a Reynolds number based on tip speed:
Re = \frac{\rho\, \omega D^2}{\mu}\,,
Thus, the nondimensional form of the thrust function becomes
\frac{F}{\rho\, \omega^2 D^4} = f\Biggl(\frac{u}{\omega D},\; \frac{\rho\, \omega D^2}{\mu}\Biggr)
\,.
- REFLECTION:
The analysis yields a compact nondimensional representation of the propeller thrust. This form isolates the effects of forward speed and viscous effects from the fundamental scaling due to density, angular speed, and size, allowing engineers to compare designs independent of size and operating conditions.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 2 How Stuff Works: How a propeller works (HIGHLY RELEVANT)
- Appendix C.3: How to Form Nondimensional Parameters Part 3 Part 1 (HIGHLY RELEVANT)
- Chapter 2.2: Nondimensional Turbomachine Parameters Part 1 (HIGHLY RELEVANT)
- Chapter 7.1: Helicopter/Quadcopter Rotors Part 5 (HIGHLY RELEVANT)
- Appendix C Fundamentals: Example Nondimensional Groups (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
Professor Kirby Bot: you should explain why the two nullspaces are equivalent, and also why you switched from the first nullspace (which came from reduced row echelon form) to the second nullspace which you said was equivalent
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
The key point is that the nondimensional groups arise from the nullspace of the dimensional matrix, so any basis for that nullspace is valid as long as it spans the same subspace. In our example a nullspace computed in reduced row echelon form yielded one set of basis vectors while an alternative—but linearly equivalent—set was later presented. The two sets are equivalent because each basis vector of one set can be expressed as a linear combination of the basis vectors in the other; in other words, the two bases span the same nullspace.
We switched from the first representation to the second primarily for clarity and to steer the groups toward forms commonly used in turbomachinery analysis. The second set aligns directly with the conventional definitions of the thrust coefficient, advance ratio, and Reynolds number based on tip speed. Although the RREF-provided nullspace is mathematically correct, it may not immediately suggest the familiar engineering nondimensional groups. The alternate representation is obtained by an invertible change of basis, which preserves the subspace property while making the physical interpretation more transparent.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Appendix C.1: Introduction (RELEVANT)
- Appendix C.3: How to Form Nondimensional Parameters Part 3 Part 1 (RELEVANT)
- Appendix C Fundamentals: Example Nondimensional Groups (RELEVANT)
- Appendix D Fundamentals: Why … (RELEVANT)
- Appendix F.3: Volumetric Viscous Force Part 1 (RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
OK that was a bit algebra-y. You can multiply any columns in the nullspace and you still have a proper basis for the nullspace. That is like raising a nonedimensional parameter to a power. You can also add a column to another column. That is like multiplying nondimensional parameters. Here we effectively took the parameters that fell out of the nullspace (which are perfectly mathematically valid) and massaged them so that they are the exact parameters that by convention are typically used.