Before we start talking about the aspects of viscous flow, let’s first define what it is. When air flows over surfaces that have friction (aka, they have a non-zero friction coefficient, $ \mu $), something interesting happens. Instead of behaving like inviscid flows, where air flows over perfectly smooth surfaces, with viscous flows, the air will interact with the surface and in that moment, the effects of viscosity, thermal conduction, and mass diffusion become important.
Fig. 1 - Effect of viscosity on an airfoil in a moving fluid.
We will not talk much about mass diffusion, but we will certainly talk about viscosity and thermal conduction. For the effects of viscosity, you can think of two containers: one filled of honey and one filled of water. Now imagine you drop a weight in each container, the weight in the honey will accelerate more slowly than the one dropped in the water. That is because the viscosity effect of the honey is greater than the water.
Now imagine this, but for air. The viscosity of air is effected by temperature change, so as the temperature decreases, the air’s viscosity would decrease as well. There is actually a relation between viscosity and temperature: this is called Sutherland’s Law.
As the air flows over a surface, let’s say the top of an airfoil, in a viscous flow, the surface will experience a “tugging” force in the same direction of the flow, tangent to the surface. This is shown in Figure 1 after Insert (a), and this “tugging” force is called shear stress, which is denoted by $ \tau $ and has units of pressure.
At the surface of the airfoil where the air interacts, the surface feels a retarding force that decreases the local flow velocity. This in turn will create a velocity of zero, $ v = 0 $, at the surface of the airfoil; this is called the no-slip condition. Now for some notation clarity, let n be the distance away from and normal to the surface such that velocity can be written in terms of n: $ v = v(n) $ where $ v = 0 $ at $ n = 0 $ and velocity increases as n increases.
So far, we have learned that friction generates shear stress, but it also creates another interesting occurrence. Let’s say there is an infinitesimally small flow element that starts at a point $ s_1 $ and as the flow continues along the surface, the pressure increases such that for points $ s_3 $, $ s_2 $, and $ s_1 $, $ P_3 > P_2 > P_1 $. This section of increasing pressure is called an adverse pressure gradient.
Fig. 2 - Separated flow induced by an adverse pressure gradient.
In Figure 2, this depicts an adverse pressure gradient. Notice that the velocity at point 2 is less than point 1, and by the time the flow element reaches point 3, it can have reversed flow. This phenomena corresponds with flow that separates from the surface. At the point of separation, $ \frac{\partial v}{\partial n} = 0 $ at the surface.
So in addition to creating shear stress, friction can cause the flow to separate from a surface. When this separation occurs, there is actually a change in pressure difference. This can be seen in Figure 3.
Fig. 3 - Example of pressure distributions for attached and separated flow on a body.
To understand the above figure, I like to think about flow over an airfoil like this: first picture the flow is attached to the top and bottom of the airfoil. Due to an airfoil’s shape, the intent is to speed up the flow on the top of the airfoil and slow the flow beneath. This in turn creates the dotted pressure line labeled Attached Flow.
Now imagine the flow on the top of the airfoil separates at some point. When the flow separates, vortices form. Due to this formation, the airflow will slow compared to the attached flow scenario. This will lower the pressure difference between the top off the airfoil and the bottom, giving the solid line labeled Separated Flow.
With the presence of viscosity comes the production of two types of drag:
Pressure drag, sometimes called form drag, is the drag created due to a pressure difference. This can be pictured using Figure 4 below.
Fig. 4 - Example of pressure drag on a plate.
As the pressure difference decreases and the pressures get closer to one another on either side of the wall, the drag will decrease as well.
Skin friction drag is the created drag due to the shear stress $ \tau $ over the body. The sum of $ D_P + D_f $ is called the profile drag for 2-D bodies, and is called parasite drag for 3-D bodies.
One analogy that is often brought up in physics for describing friction is when you rub your hands together, they start to get warmer. This is because some of the energy that is put into rubbing your hands together is dissipated by friction.
This same concept can be applied to air flowing over a body. The flow has kinetic energy, and like discussed before, the flow will slow due to friction. Now with the lost kinetic energy of the flow, this is added to the flow’s internal energy, hence increasing the temperature of the flow. This occurrence is called viscous dissipation.
Now as the temperature increases in the flow, there will be heat transfer between the air and the surface of the body. For faster flow, the dissipated kinetic energy of the flow will increase, hence increasing the temperature of the flow further. This can get to extreme temperatures for hypersonic aircraft!
Flow can move in two configurations: laminar and turbulent flow, where each have their own dependencies with shear stress, flow separation, and aerodynamic heating.
Fig. 5 - Depiction of laminar and turbulent flow.
You can think of laminar flow as smooth and continuous; this can be seen in Figure 5 (a). As for turbulent flow, found in Figure 5 (b), it is chaotic and jagged-looking. For turbulent flow, it will have a higher velocity when closer to the surface because of the agitated motion, the higher-energy fluid elements are pushed closer to the surface.
Fig. 6 - Diagram of velocity profiles for laminar and turbulent flow.
One thing to note is that directly above the surface of the airfoil, the change in velocity is larger for turbulent flow rather than laminar flows.
[ 1 ]
$$ \begin{bmatrix} \begin{Bmatrix} \frac{\partial v}{\partial n} \end{Bmatrix}_{n=0} \end{bmatrix}_{turbulent} > \begin{bmatrix} \begin{Bmatrix} \frac{\partial v}{\partial n} \end{Bmatrix}_{n=0} \end{bmatrix}_{laminar} $$
Due to the difference in velocity gradients, turbulent flow will have more severe frictional effects. Due to this, shear stress and heating will be larger for turbulent flow compared to laminar flow. Although turbulent flow may sound less appealing than laminar flow, turbulent flow is less likely to separate from the surface. As a result, the pressure drag, $ D_P $ would be smaller for a turbulent flow.
Whenever we see a difference in qualities or effects, our minds should think “but in which scenario is each better?”. That’s a great question!
For larger blunt bodies, such as a circle, turbulent flow is preferred because the turbulent flow would be more likely to stay attached to the regions behind the circle. Whereas for more slender bodies such as an airfoil, laminar flow is more desirable.
Fig. 7 - Drag on slender and blunt bodies.
And since we are dealing with two types of flow which can occur along the same surface of a body, there must be a point where they transition. This is depicted below in Figure 8.
Fig. 8 - Transition from laminar to turbulent flow.
Notice how as the flow continues further along the flat plate, the influence of friction causes the flow to slow and grow higher above the surface. Now I also want to point out the region that is labeled Transition region; this is the region where the flow transitions from laminar to turbulent. However, oftentimes models will treat the transition region as a single point, $ x_{cr} $. This point’s position along the surface depends on many different factors:
Given the transition point, $ x_{cr} $, we can calculate the critical Reynolds number, $ Re_{cr} $, which is the Reynolds number at the point of transition. It is defined as
[ 2 ]
$$ Re_{cr} = \frac{\rho_{\infty} v_{\infty} x_{cr}}{\mu_{\infty}} $$
Now we have a solid foundation of what viscous flow is and some of the effects of having a viscous flow rather than the prior inviscid flows.