I have seen people nondimensionalize navier-stokes, dividing u by U and t by U/l, etc etc. How do we know that these are the right things to divide by? what happens if I divide by something else?
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:
When nondimensionalizing the Navier-Stokes equations, the choice of characteristic scales for velocity, length, and time is crucial because it determines the form of the nondimensional parameters that emerge, such as the Reynolds number. The characteristic scales are chosen based on the physical context of the problem to capture the dominant effects and simplify the analysis.
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Velocity Scale (U): This is typically chosen as a representative velocity of the flow, such as the free-stream velocity in external flows or the average velocity in internal flows. Dividing the velocity field \boldsymbol{\vec{u}} by U nondimensionalizes the velocity, allowing us to express the flow field in terms of a dimensionless velocity \boldsymbol{\vec{u}}^* = \boldsymbol{\vec{u}} / U.
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Length Scale (\ell): This is often chosen as a characteristic length of the geometry, such as the diameter of a pipe or the chord length of an airfoil. Dividing spatial coordinates by \ell nondimensionalizes the spatial domain, leading to dimensionless coordinates like x^* = x / \ell.
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Time Scale (\ell / U): This is derived from the velocity and length scales, representing the time it takes for a fluid particle to traverse the characteristic length at the characteristic velocity. Dividing time by \ell / U nondimensionalizes time, resulting in a dimensionless time t^* = t / (\ell / U).
If you choose different scales, the nondimensional parameters will change, potentially obscuring the dominant physics or complicating the analysis. For example, using an inappropriate velocity scale might lead to a Reynolds number that does not accurately reflect the balance between inertial and viscous forces. The goal is to choose scales that highlight the key nondimensional parameters relevant to the flow regime and problem at hand.
The nondimensionalization process is not arbitrary; it is guided by the need to simplify the governing equations and highlight the important dimensionless groups that govern the flow behavior.
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 (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
- Appendix C.4: (HIGHLY RELEVANT)
- Appendix C.3: How to Form Nondimensional Parameters Part 7 Part 1 (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.
The first question is – why would we nondimensionalize an equation. There are two key reasons:
- reduce the equation to fewer parameters, so for example if Navier–Stokes is a function of U, \rho, \mu, \ell, t, it can be reduced to two nondimensional parameters.
- give some physical insight into the relative importance of different terms in the equation, especially if some terms can be neglected in some contexts.
PK Bot’s answer isn’t wrong but I am not convinced that saying “nondimensionalize the equation with a characteristic velocity U” actually explains anything. That simply leads to a followup question – what does it mean for a velocity to be “characteristic” such that I use it when nondimensionalizing?
So, let’s consider the two goals above.
- reduce parameters. If the problem has parameters U, \rho, \mu, \ell, t, the nondimensional parameters \mathrm{Re}=\rho U \ell / \mu and \mathrm{St} = \ell/t U are enough to specify this problem, as would be any of an infinite number of combinations of these parameters, raised to any power and multiplied by any numerical constants. The two I list are the ones most often used.
- physical insight. If the goal is to provide physical insight, especially in the context of the Navier–Stokes equations, then we would like to rewrite the equation such that nondimensional parameters describe which terms in the equations are important. For example, take the Navier–Stokes equations with no body force term:
\rho \frac{\partial \vec{u}}{\partial t} +\rho \vec{u} \cdot \nabla \vec{u} = -\nabla p + \mu \nabla^2 \vec{u}
if we define u^*=u/U, p^*=p/\rho U^2, t^*=tU/\ell, \nabla^* = \ell\nabla, which is what is most often done by mechanical engineers, we can rewrite this equation as
\frac{\partial \vec{u^*}}{\partial t^*} +\vec{u^*} \cdot \nabla^* \vec{u^*} = -\nabla^* p^* + \frac{1}{\mathrm{Re}}\nabla^{*^2} \vec{u^*}
The purpose of this form is to show us that \mathrm{Re} tells us which terms are important.This form of the equation is useful if and only if it does that correctly, which is true if \frac{\partial \vec{u^*}}{\partial t^*}, \vec{u^*} \cdot \nabla^* \vec{u^*}, \nabla^* p^*, and \nabla^{*^2} \vec{u^*} are of the same order of magnitude, roughly \mathcal{O}(1). If that is the case, then this form of the equation shows us that when \mathrm{Re} is high, the viscous term can be neglected.
The notion that the nondimensional terms (absent premultipliers) should be of same order is how we tell what is “characteristic” and what is useful for analyzing a system in general or useful for nondimensionalizing the equation.
For example, we find that if \ell properly describe the shortest length over which velocities change in the system, the nondimensional terms tend to be of similar order. A tube that is L long with a diameter d has speeds that vary from its center to its outer wall, and d is often a good length to use. If the flow is turbulent and has a thin boundary layer, the boundary layer thickness \delta might be even better. L is a bad length to use. So one way to nondimensionalize the equation but get a misleading result is to use a characteristic length or velocity that leads to nondimensional terms that are of different orders.
Similarly, we could normalize p by \rho U^2, or we could nondimensionalize it by \mu U/\ell. Either is dimensionally correct, but the two results we get are different by approximately a factor of \mathrm{Re}. And, no matter which one we pick, the \nabla^*p^* ends up being of the wrong order at some value of \mathrm{Re}. So actually we find that if we want to use the nondimensional NS equations to tell us which terms are important, the pressure term is tricky. So I usually use a trick and nondimensionalize p by \rho U^2+\mu U/\ell. That way the value of \nabla^*p^* is always of the correct order.
if we define u^*=u/U, p^*=p/(\rho U^2+ \mu U/\ell), t^*=tU/\ell, \nabla^* = \ell\nabla, which is the way I nondimensionalize the equation, we can rewrite this equation as
\mathrm{Re}\,\frac{\partial \vec{u^*}}{\partial t^*} +\mathrm{Re}\,\vec{u^*} \cdot \nabla^* \vec{u^*} = -(1+\mathrm{Re})\nabla^* p^* + \nabla^{*^2} \vec{u^*}
now, if all of the nondimensional terms ( \frac{\partial \vec{u^*}}{\partial t^*}, \vec{u^*} \cdot \nabla^* \vec{u^*}, \nabla^* p^*, and \nabla^{*^2} \vec{u^*}) are of the same order, then the magnitude of \mathrm{Re} tells you which terms in the equation are important. If \mathrm{Re}\gg 1, then the nondimensional Navier–Stokes equation simplifies to
\frac{\partial \vec{u^*}}{\partial t^*} +\vec{u^*} \cdot \nabla^* \vec{u^*} = -\nabla^* p^*
which are the inviscid (Euler) equations. If \mathrm{Re}\ll 1, then the nondimensional Navier–Stokes equation simplifies to
0 = -\nabla^* p^* + \nabla^{*^2} \vec{u^*}
which are the Stokes equations.
what is St? you never used it
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 Strouhal number, St, is a nondimensional parameter that arises when time is an independent parameter in a flow problem. In the context where the flow is described by parameters such as U, \rho, \mu, \ell, and t, unit elimination shows that two independent nondimensional groups can be formed; one is the Reynolds number, \mathrm{Re}=\rho U\ell/\mu, and the other is the Strouhal number, defined here as
\mathrm{St}=\frac{\ell}{Ut}\,.
In physical terms, \mathrm{St} compares the convective time scale, \ell/U, with the characteristic time scale t describing the unsteadiness of the flow. A value of \mathrm{St} of order unity indicates that the time scale of the flow’s unsteadiness is comparable to the time it takes a fluid particle to traverse the characteristic length. Although in the previous discussion the focus was on the Reynolds number for highlighting viscous versus inertial effects, the Strouhal number is equally important when the temporal aspects—such as oscillatory or transient phenomena—need to be captured in the nondimensional formulation of the equations.
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:
- Appendix C.4: (HIGHLY RELEVANT)
- Chapter 6 Fundamentals: Reynolds Number (HIGHLY RELEVANT)
- Appendix C Fundamentals: Reynolds Number (HIGHLY RELEVANT)
- Chapter 10 Fundamentals: Reynolds Number (HIGHLY RELEVANT)
- Chapter 10.3: Boundary Layers: Overview Part 4 (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
the strouhal number is really important when there is a characteristic time with which the boundary conditions change. Implicitly, most NS eqn nondimensionalizations assume that the BCs don’t change at all or change on a time scale that isn’t any faster than \ell/U, so they just assume that \mathrm{St}=1. However, if the BCs change faster than that, the temporal term can become more important. So for example if you vibrate a piezo really fast in a puddle of honey, the nondimensionalization I use above with \mathrm{St}=1 will incorrectly tell you that the unsteady term can be neglected. But properly capturing the temporal effects by nondimensionalizing the time by 1/f of the piezo will show you \mathrm{St}\neq1 and that the unsteady term in that case is still important.