Commit e9210df4 authored by Olivier's avatar Olivier

Chapter 9: completed re-structuring

* Flow parameters (Re, Ma etc) come first, then force coefficients
* New brief section on building models
* Moved flow-parameters-as-force-ratios section to appendix
* Overall re-write & strenthening
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Thus, we can see that if we follow a particle along its path, in a steady, incompressible, frictionless flow with no heat or work transfer, its change in kinetic energy is due only to the result of gravity and pressure, in accordance with the Navier-Stokes equation.
\section{Flow parameters as force ratios}
\coveredin{Massey \cite{massey1983}}
Instead of the mathematical approach covered in \S\ref{ch_non_dim_ns} p.\pageref{ch_non_dim_ns}, the concept of \vocab{flow parameter} can be approached by \emph{comparing forces} in fluid flows.
Fundamentally, understanding the movement of fluids requires applying Newton’s second law of motion: the sum of forces which act upon a fluid particle is equal to its mass times its acceleration. We have done this in an aggregated manner with integral analysis (in \chapterthreeshort, eq.~\ref{eq_rtt_linearmom} p.\pageref{eq_rtt_linearmom}), and then in a precise and all-encompassing way with differential analysis (in \chaptersixshort, eq.~\ref{eq_navierstokes} p.\pageref{eq_navierstokes}). With the latter method, we obtain complex mathematics suitable for numerical implementation, but it remains difficult to obtain rapidly a quantitative measure for what is happening in any given flow.
In order to obtain this, an engineer or scientist can use force ratios. This involves comparing the magnitude of a type of force (pressure, viscous, gravity) either with another type of force, or with the mass-times-acceleration which a fluid particle is subjected to as it travels. We are not interested in the absolute value of the resulting ratios, but rather, in having a measure of the parameters that influence them, and being able to compare them across experiments.
\subsection[Acceleration vs. viscous forces: the Reynolds number]{Acceleration vs. viscous forces:\\ the Reynolds number}
The net sum of forces acting on a particle is equal to its mass times its acceleration. If a representative length for the particle is $L$, the particle mass grows proportionally to the product of its density $\rho$ and its volume $L^3$. Meanwhile, its acceleration relates how much its velocity $V$ will change over a time interval~$\Delta t$: it may be expressed as a ratio $\Delta V/ \Delta t$. In turn, the time interval~$\Delta t$ may be expressed as the representative length $L$ divided by the velocity $V$, so that the acceleration may be represented as proportional to the ratio $V \Delta V/L$. Thus we obtain:
|\text{net force}| = |\text{mass} \times \text{acceleration}| &\sim& \rho L^3 \frac{V \Delta V}{L}\\
|\vec{F}_\net| &\sim& \rho L^2 V \Delta V
We now observe the viscous force acting on a particle: it is proportional to the the shear effort and a representative acting surface $L^2$. The shear can be modeled as proportional to the viscosity $\mu$ and the rate of strain, which will grow proportionally to $\Delta V/L$. We thus obtain a crude measure for the magnitude of the shear force:
|\text{viscous force}| &\sim& \mu \frac{\Delta V}{L} L^2\\
|\vec{F}_\text{viscous}| &\sim& \mu \Delta V L
The magnitude of the viscous force can now be compared to the net force:
\frac{|\text{net force}|}{|\text{viscous force}|} &\sim& \frac{\rho L^2 V \Delta V}{\mu \Delta V L} = \frac{\rho V L}{\mu} = \re \label{eq_re_forces}
and we recognize the ratio as the Reynolds number (\ref{eq_def_re} p.\pageref{eq_def_re}). We thus see that the Reynolds number can be interpreted as the inverse of the influence of viscosity. The larger $\re$ is, and the smaller the influence of the viscous forces will be on the trajectory of fluid particles.
\subsection{Acceleration vs. gravity force: the Froude number}
The weight of a fluid particle is equal to its mass, which grows with $\rho L^3$, multiplied by gravity $g$:
|\text{weight force}| = |\vec{F}_\text{W}| &\sim& \rho L^3 g
The magnitude of this force can now be compared to the net force:
\frac{|\text{net force}|}{|\text{weight force}|} &\sim& \frac{\rho L^2 V^2}{\rho L^3 g} = \frac{V^2}{L g} = \fr^2
and here we recognize the square of the Froude number (\ref{eq_def_fr} p.\pageref{eq_def_fr}). We thus see that the Froude number can be interpreted as the inverse of the influence of weight on the flow. The larger $\fr$ is, and the smaller the influence of gravity will be on the trajectory of fluid particles.
\subsection{Acceleration vs. elastic forces: the Mach number}
In some flows called \vocab{compressible flows} the fluid can perform work on itself, and and the fluid particles then store and retrieve energy in the form of changes in their own volume. In such cases, fluid particles are subject to an \vocab{elastic force} in addition to the other forces. We can model the pressure resulting from this force as proportional to the bulk modulus of elasticity $K$ of the fluid (formally defined as $K \equiv \rho \ \partial{p}/\partial{\rho}$); the elastic force can therefore be modeled as proportional to $K L^2$:
|\text{elasticity force}| = |\vec{F}_\text{elastic}| &\sim& K L^2
The magnitude of this force can now be compared to the net force:
\frac{|\text{net force}|}{|\text{elasticity force}|} &\sim& \frac{\rho L^2 V^2}{K L^2} = \frac{\rho V^2}{K}
This ratio is known as the Cauchy number; it is not immediately useful because the value of $K$ in a given fluid varies considerably not only according to temperature, but also according to the type of compression undergone by the fluid: for example, it grows strongly during brutal compressions.
During isentropic compressions and expansions (isentropic meaning that the process is fully reversible, i.e. without losses to friction, and adiabatic, i.e. without heat transfer), %
%we will show in chapter~9 (with eq.~\ref{eq_speed_sound_elasticity_two} p.\pageref{eq_speed_sound_elasticity_two})
it can be shown that the bulk modulus of elasticity is proportional to the square of the speed of sound~$c$:
K|_\text{reversible} &=& c^2 \rho \label{eq_speed_sound_elasticity_one}
The Cauchy number calibrated for isentropic evolutions is then
\frac{|\text{net force}|}{|\text{elasticity force}|_\text{reversible}} &\sim& \frac{\rho V^2}{K} = \frac{V^2}{c^2} = \ma^2
and here we recognize the square of the Mach number (\ref{eq_def_ma} p.\pageref{eq_def_ma}). We thus see that the Mach number can be interpreted as the influence of elasticity on the flow. The larger $\ma$ is, and the smaller the influence of elastic forces will be on the trajectory of fluid particles.
\subsection{Other force ratios}
The same method can be applied to reach the definitions for the Strouhal and Euler numbers given in \S\ref{ch_scaling_flows} p.\pageref{ch_scaling_flows}. Other numbers can also be used which relate forces that we have ignored in our study of fluid mechanics. For example, the relative importance of surface tension forces or of electromagnetic forces are quantified using similarly-constructed flow parameters.\\
In some applications featuring rotative motion, such as flows in centrifugal pumps or planetary-scale atmospheric weather, it may be convenient to apply Newton’s second law in a rotating reference frame. This results in the appearance of new reference-frame forces, such as the Coriolis or centrifugal forces; their influence can then be studied using additional flow parameters.
In none of those cases can flow parameters give enough information to predict solutions. They do, however, provide quantitative data to indicate which forces are relevant in which places: this not only helps us understand the mechanisms at work, but also distinguish the negligible from the influential, a key characteristic of efficient scientific and engineering work.
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