flowfluid dynamicsturbulence
(Left is a laminar separation, Right is a turbulent separation)
I have been having a bit of trouble understanding this concept. How come the laminar separation involves a much larger wake than the turbulent one? I would think that it would be the other way around due to the fact that a laminar flow is more organized and less chaotic that turbulent. Any explanation would be greatly appreciated.
Thanks.
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.
This is a very good question! Drag due to viscous effects can be broken down into 2 components:
$$D = D_f + D_p$$
where $D$ is the total drag due to viscous effects, $D_f$ is the drag due to skin friction, and $D_p$ is the drag due to separation (pressure drag).
The equation above demonstrates one of the classic compromises of aerodynamics. As you mention, laminar boundary layers reduce the skin friction drag but are more prone to flow separation. Turbulent boundary layers have higher skin friction but resist flow separation.
$$D \quad\quad\quad=\quad\quad\quad D_f \quad \quad\quad+ \quad\quad \quad D_p\quad\quad\quad\quad\quad$$ $$\quad\quad\text{less for laminar}\quad\quad\text{more for laminar}$$ $$\quad\quad\text{more for turbulent}\quad\quad\text{less for turbulent}$$
Generally speaking the more "blunt" the body is (such as a golf ball) the more likely adding dimples to trip the boundary layer will reduce drag. Airplane wings are less prone to separation since they aren't as "blunt" and as a result skin-friction drag is more important.
For more information see Section 4.21 of Introduction to Flight by John D. Anderson
EDIT:
Laminar and turbulent boundary layers are fundamentally different in many ways but the important aspect for flow separation is how "full" the profile is. The figure below is a schematic comparing the mean velocity profile of a turbulent boundary layer to that of a laminar one. $V$ is the velocity tangent to the surface and $\eta$ is the distance away from the surface. As you can see, for turbulent boundary layers, the fluid close to the wall is moving faster than for the laminar profile.
What causes the flow to separate is an adverse pressure gradient, or $dp/dx < 0$ where $x$ is the coordinate along the surface. Generally fluid moves from high to low pressure. In the case of a boundary layer that is on the verge of separating, the flow is locally going from low to high pressure. The figure below illustrates the effect this has on the boundary layer. When the flow near the wall begins to reverse, the flow is beginning to separate. Because the fluid in a turbulent boundary layer near the surface is moving faster, a turbulent boundary layer is better able to resist an adverse pressure gradient than a laminar boundary layer.
Most objects that are designed with aerodynamics in mind are slender. This is done specifically to reduce the adverse pressure gradient ($dp/dx$) over the surface of the object and reduce the possibility of flow separation.
Figures are from Fundamentals of Aerodynamics by John D. Anderson.
Best Answer
Deliberately introducing turbulence can often reduce overall drag, counterintuitive as it seems. On the wing of this aircraft, you can see vortex generators fitted along it's length.
From Wikipedia Turbulent Flow and Drag
Vortex generators, small metal plates, are often fitted to aircraft wings to control where on the wing the laminar flow will separate.
No matter what you do, the laminar flow will eventually separate, so it's better to control where it happens, especially if it can help in increasing the efficiency of control surfaces.
Turbulent flow resists change better than laminar flow because, in a way, it has a life of it's own, and small airflow changes may get damped out, but with laminar flow, it is much easier to disturb it and cause it to separate from the wing.
Golf balls and aircraft wings explains this idea in much more detail, (and is much better than my answer all round.)