---
title: Fluid Flow
module: Gravitation and Matter
moduleNumber: 6
lessonNumber: 5
order: 605
summary: >
  Two accounting rules carry most of steady flow: mass cannot pile up, so the same
  volume crosses every section each second, and mechanical energy is conserved along
  a streamline when the fluid is ideal. From those we get continuity, Bernoulli's
  relation between pressure, speed, and height, and the results that follow —
  Torricelli's efflux speed, the Venturi meter, the Pitot tube. We then let go of
  the ideal assumptions one at a time: viscosity adds wall shear and head loss,
  Reynolds number decides laminar versus turbulent, and Mach number marks where a
  gas stops behaving as incompressible.
topics: [Gravitation and Matter]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 13 — Fluids; §13-4"
---

## Continuity, streamlines, and Bernoulli flow

The velocity field gives fluid velocity at each position and time. A streamline is
tangent to velocity at every point. In steady flow, the streamline pattern is fixed,
and a flow tube bounded by streamlines has no mass flow through its sides. Mass
entering one cross section must leave another. Thus

$$
\rho_1A_1v_1=\rho_2A_2v_2.
$$

For incompressible flow, density is constant and the volume flow rate is

$$
Q=Av=\text{constant}.
$$

A narrow section therefore has a higher speed. Continuity alone does not establish
a pressure difference; energy and height must also be considered.

$$
% caption: The same volume crosses every section each second, so the fluid speeds up where the tube narrows: $A_1v_1=A_2v_2$ forces $v_2>v_1$ when $A_2<A_1$.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
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\draw[thick] (0,0.9)--(2.2,0.9)--(3.4,0.45)--(6,0.45);
\draw[thick] (0,-0.9)--(2.2,-0.9)--(3.4,-0.45)--(6,-0.45);
\draw[->,black,thick] (0.5,0)--(1.5,0) node[right] {$v_1$};
\draw[->,acc,thick] (4.1,0)--(5.6,0) node[right] {$v_2$};
\node at (1.1,1.25) {$A_1$};
\node at (4.8,0.82) {$A_2$};
\end{tikzpicture}
$$

**Bernoulli equation.**

For steady, incompressible, nonviscous flow along one streamline, energy per unit
volume is constant:

$$
p+\frac12\rho v^2+\rho gy=\text{constant}.
$$

Pressure, kinetic, and gravitational terms each have pressure units. The equation
comes from pressure work on a fluid element and its change in kinetic and
gravitational energy. It does not say that fast fluid always has low pressure:
height differences, pumps, turbulence, and viscosity can alter the comparison.

In horizontal ideal flow, a smaller section has higher speed and lower static
pressure. A Venturi meter uses this pressure difference to determine flow rate.

$$
% caption: In horizontal ideal flow the fast throat has lower static pressure; the standpipe over the wide section stands higher ($p_1>p_2$), which a Venturi meter reads to infer the flow rate.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[thick] (0,0.9)--(2.3,0.9)--(3.2,0.4)--(3.9,0.4)--(4.8,0.9)--(6.4,0.9);
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\node[acc] at (3.55,2.4) {$p_2$};
\draw[->,black,thick] (0.4,0)--(1.2,0);
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$$

**Efflux and viscosity.**

Between a tank free surface and a small hole at depth $h$, equal atmospheric
pressures cancel and surface speed is negligible for a large tank. Bernoulli gives

$$
v=\sqrt{2gh}.
$$

This Torricelli result is an ideal exit speed. Real jets contract and lose energy
to viscosity, so measured flow rate is smaller than $Av$ without a discharge
coefficient. Viscosity is resistance to relative layer motion. Laminar Newtonian
flow in a pipe obeys

$$
Q=\frac{\pi R^4\Delta p}{8\eta L}.
$$

The fourth-power radius dependence makes narrow passages strongly resistive.

$$
% caption: A jet leaves a hole at depth $h$ below the free surface with the ideal Torricelli speed $v=\sqrt{2gh}$, then follows a projectile path.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0,2.0)--(0,-2.4)--(2.4,-2.4)--(2.4,-0.3);
\draw[thick] (0,1.2)--(2.4,1.2);
\draw[<->,black] (0.55,1.2)--(0.55,-0.3) node[midway,left] {$h$};
\draw[->,acc,thick] (2.4,-0.3)--(3.2,-0.3) node[above] {$v$};
\draw[acc,thick] plot[domain=0:1.7,samples=60] ({\x+3.2},{-0.3-0.32*\x*\x});
\end{tikzpicture}
$$

Reynolds number $\mathrm{Re}=\rho vD/\eta$ compares inertial and viscous
effects. Low values favor laminar flow; high values favor turbulence, with the
transition also depending on geometry and disturbances.

## Derivations and elementary flow rates

During a short interval $\Delta t$, fluid advances distances $v_1\Delta t$ and
$v_2\Delta t$ through two cross sections. The corresponding masses are

$$
\Delta m_1=\rho_1A_1v_1\Delta t,
\qquad \Delta m_2=\rho_2A_2v_2\Delta t.
$$

Steady flow does not accumulate mass in the tube, so these expressions are equal.
Division by $\Delta t$ yields the continuity equation. A compressible gas requires
the density factors; setting $A_1v_1=A_2v_2$ is justified only when density change
is negligible. In high-speed gas flow, density variation can be central even in a
pipe of constant area.

Mass flow rate is $\dot m=\rho Q$. It is often the natural quantity for pumps,
engines, and chemical systems, while $Q$ determines reservoir filling and pipe
velocity. A stated cross section must be perpendicular to the average flow. Using
the outer diameter of a pipe instead of its internal diameter gives the wrong area
and can produce a large velocity error because area scales as diameter squared.

**Derivation of Bernoulli's equation.**

Take a small fluid volume $\Delta V$ moving from point 1 to point 2. Pressure on
its rear face does work $p_1\Delta V$; pressure at its front face does negative
work $-p_2\Delta V$. The net pressure work changes kinetic and gravitational
energy:

$$
(p_1-p_2)\Delta V
=\frac12\rho\Delta V(v_2^2-v_1^2)+\rho\Delta Vg(y_2-y_1).
$$

Division by $\Delta V$ and rearrangement gives Bernoulli's equation. The derivation
assumes the same parcel enters and leaves without viscous energy conversion or
shaft work. The word _along_ one streamline is essential in rotational flow: two
different streamlines can have different Bernoulli constants.

Static pressure is the thermodynamic pressure of the moving fluid. Dynamic pressure
$\rho v^2/2$ is an energy-density term, not an additional pressure exerted in all
directions. A Pitot tube brings fluid to rest at a forward-facing opening; the
difference between stagnation and static pressure is dynamic pressure under ideal
conditions.

$$
% caption: The forward-facing opening brings fluid to rest and reads stagnation pressure; a side port reads static pressure, and their difference is the dynamic pressure $\rho v^2/2$.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
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$$

**Flow-rate examples and losses.**

> **Worked example.** Water ($\rho=1000\ \mathrm{kg\,m^{-3}}$) flows through a
> horizontal pipe whose area contracts from $A_1=8.0\ \mathrm{cm^2}$ to
> $A_2=2.0\ \mathrm{cm^2}$. The upstream speed is $v_1=0.50\ \mathrm{m\,s^{-1}}$.
> Find the downstream speed and the pressure change.
>
> Continuity fixes the speed from the area ratio:
>
> $$
> v_2=v_1\frac{A_1}{A_2}=(0.50)\frac{8.0}{2.0}=2.0\ \mathrm{m\,s^{-1}}.
> $$
>
> With no height change, Bernoulli gives the pressure change directly:
>
> $$
> p_2-p_1=\tfrac12\rho\left(v_1^2-v_2^2\right)
> =\tfrac12(1000)\left(0.50^2-2.0^2\right)=-1.88\times10^3\ \mathrm{Pa}.
> $$
>
> Pressure falls in the fast section. This is not lost energy: pressure energy has
> become kinetic energy. A downstream expansion recovers only part of it, since
> turbulence and viscosity convert some mechanical energy into heat.

Real engineering flow equations include a head-loss term. Per unit mass, one form
is

$$
\frac p\rho+\frac{v^2}{2}+gy+w_{\rm pump}-w_{\rm turbine}=h_L,
$$

when comparing two sections with appropriate signs. The loss $h_L$ represents
dissipation, usually positive. This extended energy balance shows why a pump can
raise pressure despite acceleration and why a turbine can extract mechanical work.

**Laminar profiles and model limits.**

Poiseuille flow has a parabolic velocity profile: fluid at the wall is stationary
in the no-slip model, while centreline speed is largest. The average speed is half
the centreline speed. The relation $Q\propto R^4$ assumes a long straight tube;
entrance effects, fittings, pulsation, roughness, and non-Newtonian viscosity alter
the profile and resistance.

Blood, paint, and polymer solutions can be non-Newtonian, with viscosity changing
with shear rate. Treating them as Newtonian may be acceptable only over a stated
shear range. Reynolds number, surface roughness, and geometry determine the onset
of turbulence. A slowly moving viscous oil can have low Reynolds
number, while fast air around a large vehicle can have very high Reynolds number.

**Dimensional checks.**

Every Bernoulli term has units of pascals. The Poiseuille equation has units
$\mathrm{m^3\,s^{-1}}$: radius to the fourth power multiplied by pressure and
divided by viscosity times length gives volume per time. These checks expose common
errors such as using diameter in place of radius or using gauge pressure where a
pressure difference was required. The ideal model is most reliable when losses are
small compared with the retained pressure, kinetic, and height terms.

## Control volumes, pumps, and losses

Continuity is most reliable when written for a named control volume. A fixed
surface encloses a portion of space; the mass inside can change only through the
fluxes crossing that surface. For steady one-dimensional flow, the mass flux at
each section is $\rho A\bar v$, where $\bar v$ is the area-averaged speed. The
average matters because real pipe flow usually has a velocity profile rather than
one speed everywhere across the section.

Laminar pipe flow illustrates the distinction. The no-slip condition sets fluid
speed to zero at a stationary wall, while the centre moves fastest. The volume
flow rate is the integral of velocity over the cross-sectional area. Replacing the
profile by a uniform speed is acceptable only when that speed is explicitly the
cross-sectional average. Using the centreline speed in $Q=Av$ overestimates the
flow rate for a parabolic profile by a factor of two.

Control-volume balances also expose storage. A tank being filled is not steady as
a whole: inflow exceeds outflow and the liquid volume changes. A pipe can be
treated as steady over a time interval only when its density field and contained
mass do not change appreciably. Compressible systems need additional care because
mass conservation can hold while volume flow changes substantially between
sections.

The sign convention should be fixed before substitution. One common convention
treats flux outward through the control surface as positive. The rate of mass
change inside then equals inflow minus outflow. This bookkeeping handles reservoirs,
nozzles, pumps, and branching networks, where a verbal statement such
as "the same flow goes through every part" is not generally true.

**Bernoulli balance with pumps and losses.**

The ideal Bernoulli equation is a statement of mechanical-energy conservation
along a streamline. Real flow systems often require pump work, turbine work, and
losses from viscosity, fittings, sudden expansions, and turbulence. A convenient
head form divides energy by weight, so each term has units of length. Between two
sections, pressure head, velocity head, elevation head, pump head, turbine head,
and loss head are balanced with a declared sign convention.

Head loss is not a pressure at one point. It is the irreversible conversion of
organized mechanical energy into internal energy between selected sections. A
longer rough pipe, a sharp bend, or a constriction can increase the required pump
head even when inlet and outlet elevations are equal. The pressure drop measured
across a component includes both recoverable changes associated with velocity and
irrecoverable losses; the distinction depends on the locations chosen for the
comparison.

Pumps and turbines delimit the simple fast-flow/low-pressure rule. A pump can
raise both pressure and elevation while adding shaft energy. A
turbine can reduce pressure while extracting shaft work. Bernoulli remains valid
when these contributions are written explicitly, but it cannot determine
their values without a pump curve, turbine characteristic, or loss model.

Energy balances should be checked against limiting cases. With no pump, no turbine,
no height change, and negligible speed change, a positive loss corresponds to a
pressure decrease downstream. With an ideal large reservoir at both ends, pipe
speed near each free surface is negligible, so the available elevation difference
is consumed by loss and any extracted work.

## Flow regimes and measurement

Reynolds number compares an inertial scale with a viscous scale,
$\mathrm{Re}=\rho vD/\eta$. It does not label a flow turbulent or laminar by
itself; the relevant transition range depends on geometry, surface roughness,
inlet disturbances, and whether the flow is internal or external. In a long smooth
circular pipe, laminar flow is common at low Reynolds number, while transitional
and turbulent behaviour become increasingly likely as the number rises.

Laminar and turbulent models predict different pressure-flow relations. In fully developed
laminar pipe flow, pressure drop is proportional to volume flow rate. In many
turbulent pipe flows, the pressure drop grows more nearly with the square of flow
rate, with a resistance coefficient depending on roughness and Reynolds number.
Using the fourth-power Poiseuille law in a turbulent system can therefore give a
large error even when the pipe diameter and viscosity are known accurately.

Determine the flow regime from observations. A thin dye streak remains ordered in
laminar flow and becomes irregular after transition. Differential-pressure
measurements at several flow rates provide another test; a log-log plot determines
whether the observed scaling is closer to linear or quadratic. The
test must use a stable fluid temperature, because viscosity can be strongly
temperature dependent.

Turbulence also limits Bernoulli measurements. A single local speed may fluctuate
rapidly, and a pressure port can sample a time-varying signal. Mean quantities may
still satisfy an energy balance when losses are included, but the simple
inviscid streamline derivation no longer describes every instantaneous motion.

$$
% caption: At low Reynolds number a dye streak stays ordered along the pipe; above transition the streak breaks up into irregular mixing that marks turbulent flow.
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**Flow measurement and calibration.**

No single instrument measures every flow quantity. A Venturi meter infers
average speed from a pressure difference and a known area ratio. A Pitot-static
probe infers local speed from stagnation and static pressure. A collecting tank
and timer measure volume flow rate directly. Each method has a calibration range,
installation requirement, and sensitivity to velocity profile, alignment, bubbles,
or temperature.

Differential-pressure meters need a discharge coefficient because real flow
separates, loses energy, and may not have a uniform velocity profile. The
coefficient is determined by calibration against a reference flow. A calibration
curve should be built from multiple stable flow settings, with the reference
method, fluid temperature, and pressure-tap arrangement stated. Reversing the
flow or stepping it upward and downward can reveal zero shift and hysteresis.

Uncertainty in a computed flow rate often comes from several sources: pressure
difference, density, throat area, upstream area, and the discharge coefficient.
The coefficient can dominate even when electronic pressure resolution is fine.
Retain raw pressure readings, temperature, timing data, and the calculation method
so geometry or calibration corrections can be recomputed.

The measured range should be kept distinct from extrapolation. A meter calibrated
in steady water flow may not provide the same coefficient in pulsating, gas-liquid,
or highly viscous flow. An energy balance still organizes the calculation, but
the instrument response must be established under conditions close to those of the
intended measurement.

## Viscosity and hydraulic resistance

Viscosity is the link between a velocity gradient and a tangential stress. In a
Newtonian fluid, the shear stress magnitude is proportional to the rate at which
neighbouring layers slide past one another. Between parallel plates separated by a
small gap, a stationary lower plate and a moving upper plate produce an
approximately linear velocity profile. The fluid adjacent to each plate shares its
speed in the no-slip model, while intermediate layers interpolate between them.

In a circular pipe driven by a pressure difference, the wall is stationary and
the profile is parabolic rather than linear. The pressure force on a cylindrical
fluid core is balanced by viscous shear on its lateral surface. Integrating these
balances gives the Hagen--Poiseuille law for a long, straight, circular pipe with
steady laminar Newtonian flow:

$$
Q=\frac{\pi R^4}{8\eta L}\Delta p.
$$

The assumptions are part of the result. The radius must be the internal hydraulic
radius; deposits or a slightly narrowed tube can strongly alter flow because of
the fourth power. The pressure difference is the difference between two sections
of developed flow, not a pressure tap placed directly in a disturbed entrance or
fitting. The viscosity is the value at the actual fluid temperature and shear
range. A non-Newtonian fluid such as a polymer solution or blood can have a
shear-dependent effective viscosity and therefore a different relation between
pressure difference and flow rate.

Poiseuille flow converts mechanical energy into internal energy continuously along
the pipe. Unlike an ideal Venturi acceleration, the pressure decline is not
recovered by returning to the original diameter. This distinction is visible in
the velocity profile: the gradient and associated shear persist at the wall. A
smooth straight tube and a stable low Reynolds number are required before the
parabolic profile can be used as a quantitative model.

$$
% caption: Fully developed laminar pipe flow has a parabolic velocity profile: zero speed at the no-slip walls, maximum on the axis. The pressure drop and this profile set the Poiseuille volume rate.
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**Pipe networks and hydraulic resistance.**

With a fixed fluid and laminar regime, a pipe can be treated as a hydraulic
resistance $R_h=8\eta L/(\pi R^4)$, so that $\Delta p=R_hQ$. Hydraulic resistance
puts series and parallel pipe networks in a common algebraic form. In series, the same
volume flow rate passes through every element and pressure drops add. In parallel, the
pressure difference across each branch is the same and the branch flow rates add.
The relations resemble electrical circuits, but the analogy should not hide the
fluid assumptions that produced the linear resistance law.

Networks containing fittings, valves, porous sections, or turbulent branches are
not generally linear. Their pressure-flow curves can depend on direction, valve
position, Reynolds number, or fluid temperature. A network model should therefore
assign a pressure-drop relation to every component, then impose continuity at each
junction and an energy balance around each path. A single resistance value is
valid only over the range where the observed curve is approximately linear.

Parallel paths redistribute flow when one branch changes. Closing a valve in one
branch increases its resistance and shifts more flow to other branches; the total
flow may also change if the pump operating point changes. In a laboratory network,
measure pressure at junctions and flow in each branch rather than inferring every
branch flow from one total reading. This identifies leaks, unintended bypasses,
or a branch whose resistance model has failed.

**Pump curves and system operating points.**

A pump does not impose an arbitrary independent pressure rise and flow rate. At a
fixed rotation speed, its available head usually decreases as flow rate increases.
The connected piping system requires a head that often increases with flow because
frictional losses grow. The operating point is the intersection of the pump curve
and the system curve. Changing a valve, pipe diameter, elevation difference, or
pump speed shifts one of these curves and therefore shifts both the achieved flow
and pressure rise.

This graphical view prevents a common error in energy-balance problems. A pump's
catalogue shutoff head applies near zero flow; it is not the head supplied at every
flow rate. Likewise, a system curve measured with one fluid may change when
viscosity or density changes. A pressure reading at one location is insufficient
to identify the operating point unless the reference elevations, velocities, and
losses are accounted for.

Series pumps add head at approximately the same flow, while parallel pumps add
available flow at approximately the same head. These rules apply only when
the pumps have compatible curves and the connecting network does not force one
unit into an unstable operating range. Cavitation, inlet starvation, and transient
surge require further modelling beyond a steady one-dimensional curve.

**Energy-loss measurement and uncertainty.**

Energy loss is often determined from a differential-pressure measurement across a
pipe section or component. The pressure taps should be far enough from a fitting
that the intended comparison sections are clear, yet close enough that unrelated
pipe loss is not added. Their elevations must be recorded. For horizontal sections
of equal diameter, a pressure difference is directly associated with loss under
steady conditions; when diameter or elevation changes, kinetic and gravitational
terms also enter the energy balance.

Loss coefficients are obtained by measuring pressure difference over several flow
rates and comparing the results with an appropriate scaling law. In laminar flow,
a line through the origin in pressure difference versus flow is expected for a
stable Newtonian fluid. For many turbulent components, fit pressure difference as a
function of $Q^2$. Accept that scaling only when residuals about the fit are
consistent with measurement uncertainty and show no systematic trend with flow rate;
otherwise test an alternative loss model or measurement bias.

Uncertainty sources include pressure-transducer calibration, zero drift, density,
temperature-dependent viscosity, flow reference, diameter, and pressure-tap
location. Repeated electronic readings can reduce random noise but do not remove a
common calibration error. A sound result states the loss coefficient's reference
diameter, Reynolds-number range, fluid condition, and whether the reported value
is based on gauge or absolute pressure differences.

### Validating a network or pump model

Network calculations should be tested at more than one operating condition. A
model that reproduces one pressure and one total flow can still assign incorrect
branch resistances or an incorrect pump characteristic. Measure junction pressures
and, where possible, individual branch flow rates while changing a valve setting
or pump speed. The resulting changes are often more diagnostic than one nominal
operating point because the model predicts how flow redistributes when resistance
is altered.

The measured pump curve should be referenced to a stated speed, impeller diameter,
fluid, inlet condition, and pressure datum. A difference between the measured and
catalogue curve may arise from instrument calibration, air entrainment, internal
wear, inlet losses, or a mismatch between the stated and actual rotation speed.
Simply adding an arbitrary loss term can hide the distinction. A residual plot
against flow, temperature, or valve position helps locate the condition under
which the model begins to fail.

Transient observations require additional caution. A rapid valve closure can
create pressure waves, and a pulsating pump can make instantaneous pressure and
flow differ from their time averages. A steady Bernoulli or resistance model may
still describe average operation after sufficient averaging, but the averaging
interval and sensor response must be stated. Where peak pressure controls safety,
the transient rather than the average model is the relevant one.

The final result should distinguish a calibrated empirical relation from a
first-principles prediction. Each has a defined role. The former is valid over the
tested range; the latter identifies the variables and scaling expected when
geometry or fluid conditions change. Keeping that distinction explicit prevents a
loss coefficient measured in one apparatus from being applied uncritically to a
different regime.

## Boundary layers and scaled flow

The no-slip condition creates a boundary layer next to a solid surface. At a
stationary wall, fluid speed is zero relative to the wall. Farther from the wall,
the speed approaches the outer-flow value. The region across which this adjustment
occurs is the boundary layer. Its thickness grows downstream from a leading edge
because viscosity transfers momentum between adjacent layers. The outer flow may
be approximated as nearly inviscid even while viscous effects within the thin
boundary layer control drag, pressure loss, and separation.

An adverse pressure gradient makes the boundary layer vulnerable to separation. If
pressure rises in the direction of flow, low-momentum fluid near the wall can slow,
reverse locally, and detach from the surface. The separated region contains
recirculating, unsteady flow and usually causes a large pressure drag. A smooth
streamlined body can have less drag than a blunt body not because viscosity has
disappeared, but because the body delays or reduces separation and its wake is
smaller.

Boundary-layer state matters. A laminar boundary layer has orderly velocity
variation and lower wall shear under some conditions, but it can separate readily.
A turbulent boundary layer mixes high-momentum outer fluid toward the wall. This
increases skin friction yet can remain attached through a stronger adverse pressure
gradient. Surface roughness, leading-edge disturbances, and flow history influence
which state occurs. The practical drag of a body is therefore not determined by
one Reynolds number alone.

The distinction between skin friction and pressure drag sets a modelling check. A
long smooth pipe is dominated by wall shear and pressure loss. A bluff
body at high Reynolds number is often dominated by pressure difference between
front and separated wake. Adding a fairing may lower pressure drag while increasing
wetted area, so the net effect depends on the balance between mechanisms. A model
that uses only inviscid Bernoulli pressure around a separated body cannot predict
the wake loss correctly.

Boundary layers also constrain flow measurement. A Pitot probe placed too near a
wall samples a lower local speed than the area average. A pressure tap can be
affected by nearby separation or by a rough edge. Instrument locations and upstream
straight length belong in the measurement specification, especially when a
calibration is transferred from one installation to another.

$$
% caption: The no-slip boundary layer thickens downstream as near-wall speed rises from zero to the outer flow; an adverse pressure gradient can reverse the near-wall fluid and separate the flow.
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\draw[black,very thick] (0,0)--(6,0);
\draw[->,thick] (0.5,1.5)--(1.6,1.5);
\draw[->,thick] (2.3,1.5)--(3.4,1.5);
\draw[->,thick] (4.1,1.5)--(5.2,1.5);
\draw[black,thick] (0.2,0.05) .. controls (1.8,0.35) and (3.6,0.9) .. (5.4,1.35);
\node[black] at (2.5,1.05) {boundary layer};
\draw[->,acc,thick] (5.15,0.3) .. controls (5.45,0.15) and (5.7,0.2) .. (5.85,0.5);
\node[acc,right] at (5.5,0.85) {separation};
\node[black,below] at (3.0,0) {wall};
\end{tikzpicture}
$$

**Dimensional similarity and scaled flow.**

Dimensional similarity determines whether a model experiment can represent a
larger flow system. Geometric similarity preserves shape ratios, but it does not
guarantee matching force balances. The governing dimensionless groups must also
be comparable. Reynolds number controls the ratio of inertia to viscosity;
Froude number compares inertia with gravity-wave effects; Mach number compares
speed with the speed of sound. Surface-tension effects introduce another ratio
when capillary forces are relevant.

The appropriate group depends on the question. A model spillway concerned with
free-surface waves must usually preserve a Froude-type ratio. A small model of a
submarine concerned with viscous drag must address Reynolds similarity. One fluid
and gravity setting may not match both ratios simultaneously. The experimenter then
identifies the dominant mechanism, changes fluid properties or model scale where
possible, and reports the unmatched ratio as a limitation.

Buckingham-pi analysis organizes this procedure. List the variables, express their
dimensions, and form independent dimensionless combinations. For a drag force on
a body in a fluid, a common result is a drag coefficient that depends on Reynolds
number and shape parameters. Plotting the coefficient against the governing
dimensionless parameters permits comparison across sizes, speeds, and densities. Scatter that fails to
collapse can indicate roughness, transition, compressibility, or an omitted
geometric variable.

Similarity is also a design constraint for flow meters. A discharge coefficient
calibrated at one Reynolds range may drift when the range changes. A Venturi meter
with the same shape but a much smaller diameter can operate in a different viscous
regime. Calibration data should therefore state fluid, diameter, roughness, and
Reynolds range instead of presenting one coefficient as a universal constant.

Scale arguments can be checked before fabrication. If a geometric scale factor is
$\lambda$, area scales as $\lambda^2$ and volume as $\lambda^3$. Holding one
dimensionless group fixed may require a speed that differs from the speed required
by another group. The incompatibility identifies effects that cannot all be
reproduced in one simple model.

## Compressibility, visualization, and validation

The incompressible approximation assumes that density changes are too small to
affect the desired result. It is often excellent for liquids and for low-speed gas
flow, but it is not a definition of a fluid. In a gas, pressure changes and speed
changes can alter density enough that $A_1v_1=A_2v_2$ fails even though mass
continuity remains exact. The correct steady relation retains density:
$\rho_1A_1v_1=\rho_2A_2v_2$.

Mach number $\mathrm{Ma}=v/c$ compares flow speed with the local speed of sound.
As Mach number rises, density changes, temperature changes, and compressibility
effects become increasingly important. Set the compressibility criterion from the
required accuracy, local Mach number, and flow geometry. A low-speed pressure
calibration may be adequate for nearly incompressible air but can fail in a
high-speed nozzle, where density and temperature variations alter the sensor
interpretation and continuity model.

In a converging nozzle carrying compressible gas, increasing the upstream-to-
downstream pressure difference raises speed until a limiting sonic condition can
occur at the narrowest section. Further reduction of downstream pressure does not
increase the mass flow through that throat without changing upstream conditions or
nozzle geometry. This choked-flow condition has no analogue in ordinary
incompressible pipe flow and shows why a single Bernoulli expression cannot cover
all gas-nozzle regimes.

Compressible-flow energy balances include thermodynamic state changes. Pressure
work can change internal energy and temperature as well as kinetic energy. A
simple isothermal or adiabatic relation may be appropriate only when the heat
transfer and time scale justify it. The correct model names its equation of state,
heat-transfer assumption, and reference conditions before pressure and speed data
are converted into mass flow.

Experimental signs of compressibility include density-sensitive mass-flow results,
temperature change across a restriction, and pressure ratios that no longer match
an incompressible calibration. These observations should prompt a change of model,
not an ad hoc correction factor added to a liquid-flow formula.

$$
% caption: In a converging nozzle the gas accelerates from subsonic ($M<1$) toward the throat, where it can reach the sonic limit $M=1$; the mass flow is then fixed by upstream state and throat area, not by lowering the downstream pressure further.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[thick] (0,1.3)--(4.5,0.45);
\draw[thick] (0,-1.3)--(4.5,-0.45);
\draw[dashed,black] (4.5,-0.8)--(4.5,0.85);
\draw[->,black,thick] (0.7,0)--(2.0,0) node[above] {$M<1$};
\draw[->,acc,thick] (3.2,0)--(4.3,0) node[above] {$M=1$};
\node[black,below] at (4.5,-0.8) {throat};
\end{tikzpicture}
$$

**Flow visualization and experimental evidence.**

Visualization resolves structure unavailable from a single pressure or flow-rate
reading. Dye injection can trace streamlines in steady laminar liquid flow; smoke,
tufts, bubbles, or particle tracking can show separation and wake motion in other
settings. The visual method must be chosen so that the tracer follows the fluid
without substantially altering density, viscosity, surface tension, or local
momentum. A heavy dye jet can sink through a tank and create a misleading path;
large particles can lag rapid velocity changes.

Images require a spatial and temporal reference. A video-based velocity estimate
needs a scale in the image plane, a known frame interval, a camera orientation,
and a statement of how a feature was tracked. Perspective distortion, out-of-plane
motion, shutter blur, and illumination changes can be comparable with the flow
variation being measured. A qualitative image can establish separation or mixing
without supporting a precise numerical velocity claim.

Visualization should be paired with a quantitative model check.
A dye streak that broadens downstream may support diffusion or turbulent mixing;
pressure taps can test the associated loss. A separation line observed on a model
body can be compared with drag data and surface-pressure measurements. Agreement
between different methods is stronger evidence than a visually appealing flow
pattern alone.

The record should retain the unprocessed images, tracer conditions, camera setup,
and the criterion used to identify a boundary or wake. Image enhancement can make
features easier to see but can also create apparent gradients or erase small-scale
structures. The displayed image should be traceable to the recorded observation.

| observation method | structure resolved directly | calibration required for a numerical result | common interpretation limit |
| --- | --- | --- | --- |
| dye, smoke, or tufts | streamline direction, separation, wake topology | tracer response and image scale | tracer inertia or buoyancy can depart from the fluid |
| particle tracking | local displacement and velocity | camera geometry, frame interval, depth estimate | out-of-plane motion biases a two-dimensional speed |
| pressure taps | static-pressure distribution | tap location, zero, density, reference section | separation can invalidate a uniform-section interpretation |
| force or loss measurement | integrated drag or pressure loss | load calibration, flow reference, temperature | one integral value does not locate the separation point |

Visual evidence and instrument data constrain different parts of the model. Record
both at the same operating condition rather than assigning a
quantitative speed to an uncalibrated image.

**Uncertainty and validation of flow models.**

Flow models combine measured quantities with assumptions that can be tested. A
Venturi calculation combines differential pressure, density, area ratio, and a
discharge coefficient. A Poiseuille calculation combines radius, length,
viscosity, and pressure difference. The sensitivity can be highly uneven: the
fourth-power radius dependence means a small relative radius error creates a much
larger relative error in predicted laminar flow. A reported uncertainty should
therefore be built from the governing relation rather than copied from the most
precise sensor.

Correlations matter. Pressure readings from one transducer share zero and gain
uncertainty; several flow points from one pipe share diameter and roughness
uncertainty. Repeating samples can reduce random noise but does not remove those
common components. Temperature can be a shared hidden variable because it changes
both density and viscosity. Recording temperature at every flow setting makes a
later correction or sensitivity analysis possible.

Validation should compare predictions with data not used to tune the model.
Residuals plotted against Reynolds number, valve opening, temperature, or flow
direction can reveal a regime boundary that a single summary error misses. A
linear pressure-flow fit may work at low rate and systematically underpredict loss
after transition. A calibrated discharge coefficient may be stable in one
installation yet change after a rough fitting is added upstream.

The final conclusion should state the geometry, fluid, temperature range,
Reynolds or Mach range, reference sections, and measurement method. Within that
scope, a model may be accurate enough for design or measurement. Outside it, the
proper response is a new calibration, a different loss relation, or a
compressible-flow treatment—not a claim that Bernoulli or continuity has failed.

### Selecting the appropriate flow model

Flow analysis is strongest when the hierarchy of approximations is stated before
the calculation. Mass conservation is usually retained first. The next question is
whether density can be treated as constant. A liquid in a moderate-pressure pipe
often permits that simplification; a high-speed gas nozzle may not. The next
questions concern steadiness, viscosity, rotational effects, free surfaces, and
heat transfer. Each omitted mechanism should be small compared with a retained term
over the stated operating range.

An ideal Bernoulli calculation is appropriate for relating pressure, speed, and
height along a region of steady, nearly inviscid flow when losses are demonstrably
small. A resistance or head-loss model is appropriate when wall shear, fittings,
or separation consume a material fraction of the available energy. A Poiseuille
model is appropriate only for developed laminar Newtonian flow in a long circular
pipe. Compressible-flow relations are required when density changes alter mass
flux or when pressure work changes temperature and sound speed. These are not
competing formulas for the same situation; they describe different physical
levels of approximation.

| regime | governing representation | evidence required before use |
| --- | --- | --- |
| constant-density stream | $A_1\bar v_1=A_2\bar v_2$ | density variation is below the stated accuracy |
| low-loss streamline | $p+\frac12\rho v^2+\rho gz=\text{constant}$ | steady flow, common streamline, negligible dissipative loss |
| developed laminar pipe | $Q=\pi R^4\Delta p/(8\mu L)$ | Newtonian fluid, long circular pipe, laminar profile |
| fitting or separated network | measured $\Delta p(Q)$ or a stated head-loss relation | wall shear, fittings, or separation affect the operating range |
| compressible gas flow | $\dot m=\rho A\bar v$ with a thermodynamic state model | density and temperature changes affect mass flux |

A calculation selects one row for each region of the apparatus. Adjacent regions can
require different descriptions, joined by measured pressure, mass-flow, and energy
conditions at their reference sections.

The reference sections used in an energy balance are part of the model. A section
inside a separated elbow does not have a well-defined uniform velocity profile, so
an average speed based only on pipe area may hide the kinetic-energy correction
and local loss. Moving the sections to straight, fully developed regions can make
the same balance interpretable. If this cannot be done, a calibrated component
loss relation is usually more defensible than an ideal streamline calculation.

Dimensionless estimates give a rapid screening tool. Reynolds number identifies
whether a laminar profile is plausible. A ratio of roughness height to diameter
helps assess wall effects in turbulent internal flow. Mach number identifies when
gas compressibility may matter. A ratio of pressure-loss head to available
elevation head indicates whether neglecting losses is credible. The purpose of
these estimates is not to classify a flow perfectly but to prevent a model from
being chosen without a scale comparison.

| dimensionless or scaled quantity | comparison made | modelling decision |
| --- | --- | --- |
| $\mathrm{Re}$ | inertia against viscosity | assess developed laminar assumptions and transition risk |
| relative roughness | wall scale against pipe diameter | select or calibrate an internal-flow loss relation |
| $\mathrm{Ma}$ | speed against sound speed | retain density and thermodynamic changes when required |
| $h_L/\Delta z$ | dissipative head loss against elevation head | include loss terms in an energy balance |
| $\mathrm{Fr}$ | inertia against gravity-wave effects | preserve free-surface similarity in a scaled model |

The screening values belong to the stated fluid, temperature, geometry, and reference
sections. A coefficient transferred outside those conditions is an empirical
extrapolation, not a consequence of similarity alone.

**Reproducible flow measurements.**

A reproducible flow result preserves both the measurement chain and the model
chain. Raw observations include pressure signals, time intervals, collection
volumes, temperatures, valve settings, pump speed, and geometry measurements.
The model chain specifies calibration constants, density and viscosity values,
reference elevations, cross-sectional areas, loss coefficients, and any filters
or averaging applied to time-varying signals. The final reported flow rate or
pressure loss should be traceable back through these two chains without relying on
unstated spreadsheet steps.

Independent checks are valuable. A timed collection measurement can test a
differential-pressure meter. A pressure drop measured over two lengths of the same
pipe can test whether loss grows with length as expected. A dye observation can
test whether a supposedly laminar calibration has developed unstable mixing.
Agreement does not prove every assumption, but disagreement often identifies a
specific failure: an air bubble in a pressure line, an incorrect diameter, a
temperature-dependent viscosity change, or a meter installed too close to a bend.

The uncertainty statement should match the intended use. A pump-control setting
may need only a repeatable relative flow indication, whereas an experiment testing
Poiseuille scaling needs calibrated absolute pressure, diameter, temperature, and
flow. Reporting more digits than the calibration supports does not improve the
model. Reporting the tested range, residual pattern, and dominant uncertainty does.
The model prediction, experimental test, and stated operating limits then remain
linked in the final result; operation outside those limits requires a new
calibration.