M093 · Fluid mechanics

Continuity & Bernoulli Flow Bench

Shape a transparent Venturi tube, move a pressure probe and compare energy and hydraulic grades. Acquire raw pressure and speed readings to infer flow and a Darcy friction factor.

Inference status $\text{Ready}$

Accepted transport flow $\text{Ready}$

Candidate probe speed $\text{Ready}$

Candidate probe gauge pressure $\text{Ready}$

Candidate probe absolute pressure $\text{Ready}$

Minimum candidate absolute pressure $\text{Ready}$

Minimum margin above vapor pressure $\text{Ready}$

Probe centreline elevation $\text{Ready}$

Tube centreline length $\text{Ready}$

Candidate probe energy head $\text{Ready}$

Candidate probe hydraulic head $\text{Ready}$

Candidate total loss head $\text{Ready}$

Candidate energy budget residual $\text{Ready}$

Acquired stations $\text{Ready}$

Flow inferred from raw readings $\text{Ready}$

Fit coordinate residual $\text{Ready}$

Single-phase status $\text{Ready}$

Darcy factor inferred from data $\text{Ready}$

GEOMETRY / PRESSURE / ACQUIRE

Can pressure recover after the throat?

Prescribed flow

Q

6.00\,L/s

Throat diameter ratio

D_t/D

0.50\,

Outlet elevation change

\Delta z

0.00\,m

Inlet gauge pressure

p_1

30.00\,kPa

Tube inner diameter

D

70.00\,mm

Reference Darcy friction factor

f_D

0.00\,

Known diffuser loss coefficient

K

0.00\,

Pressure noise bound

\epsilon_p

0.00\,kPa

Probe path fraction

\xi_p

0.50\,

Liquid density

\rho

1000.00\,kg/m^3

Gravity

g

9.81\,m/s^2

Ambient absolute pressure

p_a

101.33\,kPa

Specified vapor pressure

p_v

2.34\,kPa

Steady incompressible flow with mean speed and unit kinetic correction. The upstream source imposes flow and gauge pressure; outlet pressure is a result. Darcy friction is prescribed, not a Reynolds correlation. The known diffuser loss uses throat speed. Separation and transverse pressure variation are omitted.

Electronic probes report centreline-equivalent pressure; fixtures and sensors are schematic. Pressure and speed share one pipe geometry. Absolute pressure below the specified vapor pressure rejects single-phase transport and acquisition; dashed curves remain candidate references.

Acquire raw pressure and speed readings. Their area-weighted speed estimates flow, then an affine energy fit estimates Darcy friction using the known diffuser loss. Moving the probe retains data; apparatus and noise edits clear it. CSV excludes reference flow and friction. Repeated stations replace old readings; noise is reproducible and bounded.

Steady incompressible one-dimensional flow with a prescribed Darcy factor and known diffuser loss. Single-phase candidates below the vapor threshold are excluded from transport and acquisition.

Start with the essentials

Focus question: Can the pressure recover while energy is still lost?
One-sentence intuition: Constriction trades pressure head for speed; dissipative losses reduce the total head.

Core mathematical model

Continuity

Q=A(s)v(s),\qquad A(s)=\pi r(s)^2

Every cross-section carries the same prescribed volume flow.

Energy and hydraulic grades

\begin{aligned}H(s)&=z(s)+\frac{p(s)}{\rho g}+\frac{v(s)^2}{2g}\\H_{\mathrm{hyd}}(s)&=H(s)-\frac{v(s)^2}{2g}\end{aligned}

Pressure is gauge pressure; the source sets inlet head.

Loss with local speed

h_L(s)=f_D\!\int_0^s\!\frac{v(u)^2}{2gD(u)}\,du+K\chi(s)\frac{v_t^2}{2g}

The known diffuser loss accumulates after the throat. Darcy loss uses each local diameter and speed.

Infer from the sensor record

\begin{aligned}\widehat Q&=\frac1n\sum_i A_i v_i\\\widetilde H_i&=b-f_D X_i\end{aligned}

Correct acquired total head for known diffuser loss, then regress against wall-loss coordinate computed from inferred flow.

Single-phase screen

\min_s\!\left[p_a+p(s)\right]>p_v

Crossing the threshold rejects the candidate. It does not solve a cavitating flow.

Common difficulties

Gauge and absolute pressure

Typical misconceptionA negative gauge pressure always means the liquid cannot flow.

Better mental modelAdd ambient pressure before comparing with the specified vapor pressure. The entire path is checked.

A fit is an inference

Typical misconceptionThe fitted coefficient is a reference slider in disguise.

Better mental modelClear the record and the estimate disappears. Noise can yield an unphysical negative coefficient; unconstrained fits are not clipped or safety certificates.

Run the experiment

01
Locate the throat
Move the gold probe handle through the narrow section.
What to observe: Speed rises and pressure drops; energy head remains flat without loss.
02
Separate recovery from loss
Select With losses and compare inlet, throat and outlet.
What to observe: Pressure recovers downstream, but the energy grade stays below its inlet value.
03
Acquire and infer
Acquire the profile, add bounded noise and export the raw record.
What to observe: Pressure and velocity data estimate flow and friction with no reference flow in the fit.
04
Test the domain
Select Pressure boundary or reduce ambient pressure.
What to observe: A smooth candidate can violate the vapor threshold; acquisition is disabled.