Prandtl mixing length and sudden expansion loss

Prandtl's mixing length theory

In turbulent flow, fluid lumps move across the layers and carry momentum with them. Prandtl assumed that a lump keeps its identity over a distance $l$ (the mixing length) and then mixes with the new layer, like the mean free path in kinetic theory.

• Velocity difference between layers a distance $l$ apart: $u' \approx l\,\dfrac{du}{dy}$
• The transverse fluctuation $v'$ is of the same order as $u'$, so $v' \approx l\,\dfrac{du}{dy}$
• Turbulent (Reynolds) shear stress: $\tau_t = -\rho\,\overline{u'v'} = \rho\, l^2 \left(\dfrac{du}{dy}\right)^2$

Near a wall the eddy size grows with distance from the wall, so Prandtl took $l = \kappa y$, where $\kappa \approx 0.4$ is von Karman's constant. Total shear stress is $\tau = \mu\dfrac{du}{dy} + \rho l^2\left(\dfrac{du}{dy}\right)^2$; away from the wall the viscous part is negligible.

Logarithmic velocity distribution in a pipe

Near the wall the shear stress is nearly constant and equal to the wall shear $\tau_0$. Let $u_* = \sqrt{\tau_0/\rho}$ be the shear velocity and $y$ the distance from the wall.

$$\tau_0 = \rho\,\kappa^2 y^2\left(\frac{du}{dy}\right)^2 \;\Rightarrow\; \frac{du}{dy} = \frac{u_*}{\kappa\, y}$$

Integrating:

$$u = \frac{u_*}{\kappa}\ln y + C$$

so velocity varies with the logarithm of the distance from the wall. The constant $C$ is fixed by the wall condition:

• Smooth pipe: $\dfrac{u}{u_*} = 5.75\log_{10}\dfrac{u_* y}{\nu} + 5.5$
• Rough pipe: $\dfrac{u}{u_*} = 5.75\log_{10}\dfrac{y}{k} + 8.5$

Here $\dfrac{1}{\kappa}\ln y = \dfrac{2.303}{0.4}\log_{10} y = 5.75\log_{10} y$. Measured velocities in pipes follow this law over almost the whole cross-section, and for $y = R$ at the axis, $\dfrac{u_{max}-u}{u_*} = 5.75\log_{10}\dfrac{R}{y}$.

Head loss due to sudden expansion

   1          2            Section 1: A1, V1, p1 (small pipe)
  ____       _______       Section 2: A2, V2, p2 (large pipe)
 |    |_____|      
 |  V1   ~~~~ eddies  V2  Control volume between 1 and 2
 |____       _______      (eddy zone pressure = p1 on the annulus)
      |_____|

Take a control volume from section 1 (just before the expansion) to section 2 (where the jet has spread over the full area). Eddies in the corner zone keep the pressure on the annular area $(A_2 - A_1)$ equal to $p_1$. Friction on the short length is neglected.

Momentum equation (flow direction):

$$p_1 A_2 - p_2 A_2 = \rho Q (V_2 - V_1), \qquad Q = A_2 V_2$$ $$\frac{p_1 - p_2}{\gamma} = \frac{V_2(V_2 - V_1)}{g}$$

Energy equation between 1 and 2 (same level), with head loss $h_e$:

$$\frac{p_1}{\gamma} + \frac{V_1^2}{2g} = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + h_e \;\Rightarrow\; h_e = \frac{p_1-p_2}{\gamma} + \frac{V_1^2 - V_2^2}{2g}$$

Substituting the momentum result:

$$h_e = \frac{V_2^2 - V_1V_2}{g} + \frac{V_1^2 - V_2^2}{2g} = \frac{2V_2^2 - 2V_1V_2 + V_1^2 - V_2^2}{2g}$$ $$\boxed{h_e = \frac{(V_1 - V_2)^2}{2g}}$$

Using continuity $V_2 = V_1A_1/A_2$, this is also $h_e = \left(1 - \dfrac{A_1}{A_2}\right)^2\dfrac{V_1^2}{2g}$.