# What's the shear rate in a turbulent flow?

The explanation of shear rate in laminar flow is straightforward: We imagine small layers of fluid that glide on each other. Now, in turbulent flow, this does not work as there are no layers. I'm not even sure that shear rate is a meaningful concept in turbulent flow.

If I want to know the apparent viscosity of a shear thinning (or other non-Newtonian) liquid, I need to know the shear rate. How do I know the shear rate in turbulent flow?

For Newtonian fluids (such as water and air), the viscous stress tensor, $T_{ij}$, is proportional to the rate of deformation tensor, $D_{ij}$:

$$D_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)$$

$$T_{ij} = \lambda\Delta\delta_{ij} + 2\mu D_{ij}$$

where $\Delta \equiv D_{11} + D_{22} + D_{33}$. The Navier-Stokes equation for Newtonian fluids can then be written as:

$$\rho\left(\frac{\partial v_i}{\partial t} + v_j\frac{\partial v_i}{\partial x_j}\right) = -\frac{\partial p}{\partial x_i} + \rho B_i + \frac{\partial T_{ij}}{\partial x_j}$$

The Navier-Stokes equation above governs both laminar and turbulent flow using the same stress tensor. This shows that the definition of shear rate is the same in both laminar and turbulent flows, however, their values will be very different.

For non-Newtonian fluids, the same is true. Instead of the stress tensor defined above, replace it with a non-Newtonian stress tensor. Still the same governing equation applies to laminar and turbulent flows so the definition of shear rate is the same for both regimes.

As you mention, turbulent flow does not have nice, orderly layers. As a result, there can be acute stress localizations.

• I should mention that for non-Newtonian fluids you should replace the total stress tensor, not just the viscous part of it. – OSE Nov 1 '13 at 13:58
• IMHO you have just highlighted the beauty of the Navier-Stokes Equation. Everything is in there, all spatial scales, all temporal scales, from laminar flows to fully turbulent flows. Flow in a pipe or breaking waves, the NS equation covers it all. If only we knew how to solve it in full... – Isopycnal Oscillation Nov 1 '13 at 22:48
• so what is the shear rate in turbular flow in a pipe? – mart Nov 3 '13 at 9:36
• @mart The shear rate is the rate of deformation tensor, $D_{ij}$, mentioned above. The off-diagonal terms are the shearing terms whereas the diagonal terms contribute to dilatation. I focused mainly on the Navier-Stokes equation because you really need it to talk about turbulence. Turbulence is mainly driven by the nonlinear convective terms on the left hand side of the equation. – OSE Nov 4 '13 at 14:29
• Damn. I had hoped to go through life as an engineer without understanding Navier Stokes. – mart Nov 4 '13 at 15:00