Various pipe flow types exist, each with distinct physics

• Single phase pipe flows

• Completely filled with liquid (or gas)
• Pressure difference ($p_2 - p_1$) drives flow

• Open channel pipe flow

• Partially filled with liquid
• Gravity drives flow

• Multiphase pipe flow

• Either partially or full of liquid (or gas)
• Gravity drives flow
• Complex physics (bubbles, particles, droplets etc)

Numerical simulations provide details on the process, which can be difficult to capture experimentally.

• Turbulent pipe flow
• Solid particles with $d_p = 10$ μm
• Particle agglomeration (sticking) occurs due to van der Waals and electrostatic forces

Video: Agglomeration of particles in a turbulent pipe flow

Characterised by different flow regimes:

• Turbulent flow: chaotic and irregular (mixing occurs)
• Transitional flow: between laminar and turbulent
• Laminar flow: smooth and orderly (no mixing)

Region close to inlets

• (1): Inviscid (enters with near uniform velocity aka plug flow)
• (2): Reaches fully-developed (boundary layer reached centre)
• (3): Remains fully developed and nothing changes
• (4): Skewed and starts to develop again
• (5): Reaches fully-developed again
• (6): Remains fully-developed and nothing changes

Entrance length $l_e$ depends on Reynolds number:

• Laminar: \( l_e/D \approx 0.06 \text{Re}_D \)
• Turbulent: \( l_e/D \approx 4.4 \text{Re}_D^{1/6} \)

Pressure varies in fully-developed laminar flow linearly along the pipe

• $\partial p/\partial x= -\Delta p/p < 0$

Pressure drop in pipe flows is balanced by:

• Pressure drop due to friction (viscous effects)
• Pressure drop due to acceleration/deceleration of flow (inertia effects)
• Hydrostatic pressure variation due to elevation changes

Recall from analytical solutions earlier:

\(Q = \dfrac{\pi R^4}{8\mu} \left(-\dfrac{\partial p}{\partial z}\right)\) \(= \dfrac{\pi D^4}{128\mu} \left(-\dfrac{\partial p}{\partial z}\right)\) \(= \dfrac{\pi D^4 \Delta p}{128\mu l}\)

• Valid for laminar, steady, horisontal, fully-developed flow

For non-horisontal pipes:

\( Q =\dfrac{\pi D^4(\Delta p - \rho g l \text{sin}\theta)}{128\mu l}\)

Happens gradually between $Re \approx 2100$ and $Re \approx 4000$

• Small bursts of turbulence start at $\text{Re} \approx 2100$
• Bursts grow in size and frequency with increasing $\text{Re}$
• Increased mixing and momentum transfer occurs
• Higher heat transfer rates and pressure drop

Velocity field $\textbf{\textit{V}}$ changes from being 1-D to 3-D

• Laminar: $\textbf{\textit{V}}=v_x\cdot \hat{\textbf{i}}$
• Turbulent: $\textbf{\textit{V}}=v_x\cdot \hat{\textbf{i}} + v_y\cdot \hat{\textbf{j}} + v_z\cdot \hat{\textbf{k}}$

Consider a turbulent flow with instantaneous velocity $u(t)$

• Time-averaged velocity component $\overline{u}$

$$\overline{u}(x,y,z) = \dfrac{1}{T} \int_{t_0}^{T+t_0} u(x,y,z,t) \, \text{d}t$$

• $T$ is the averaging time period (should be long enough to capture the relevant flow structures)
• Instantaneous velocity can then be decomposed as:

$$u(x,y,z,t) = \overline{u}(x,y,z) + u'(x,y,z,t)$$

• Fluctuating velocity component $u'$

$$u'(x,y,z,t) = u(x,y,z,t) - \overline{u}(x,y,z)$$

Relevant to quantify strength of fluctuations

• Clearly, time-averaging fluctuations $\overline{u'}$ does not make sense

$$\overline{u'}(x,y,z) = \dfrac{1}{T} \int_{t_0}^{T+t_0} u'(x,y,z,t) \, \text{d}t = 0$$

• What if we take square fluctuations and then time-average?

$$\overline{(u')^2}(x,y,z) = \dfrac{1}{T} \int_{t_0}^{T+t_0} u'(x,y,z,t)^2 \, \text{d}t > 0$$

• To get "back" to same order of magnitude, we now take the square root of fluctuations

$$u_\text{rms}(x,y,z) = \sqrt{\overline{(u')^2}(x,y,z)}$$

• This is called the root-mean-square (RMS) of the fluctuations
• Often normalised by the time-averaged velocity to get a relative measure, aka turbulence intensity

$$I(x,y,z) = \dfrac{u_\text{rms}(x,y,z)}{\overline{u}(x,y,z)}$$