# Part XIII: Boundary Layers, Lift & Drag

Now that we’ve said everything we can about flows far from immersed objects and flows near immersed objects, it makes sense to try and understand the flow everywhere in between. Even though we won’t be able to make too many quantitative statements about it—the math is way too complicated in that region—we will be able to make a couple of general statements about it.

The term boundary layer is often used as a catch-all term for any region between the surface of the object and the background flow at infinity, but here we use it specifically to refer to the regions in the fluid that are neither described by near or far field flow assumptions. Most of the time, this region is illustrated or described as a velocity field that monotonically increases in magnitude from 0 at the surface of the object to the background speed. Such a description is both shockingly common and always incorrect.

Although this can be proved rigorously, it suffices to look at the form of far field flow we derived two Parts ago; the velocity in the far field increases as we get closer to the object. As a result, the flow speed can’t just increase slowly until it gets to the background flow speed as we move away from the object. It needs to increase in the near field, and then keep increasing until it gets to a speed that’s larger than the background speed somewhere within the boundary layer, and then decay into the background speed in a way consistent with our expressions for far field flow. There isn’t a consistent name for this, but I like to call it the speed bump.

Another qualitative feature of boundary layer flow that plays an important role in fluid dynamics is that it can reverse directions. In such a scenario, the near field flow around the object is pointing in the opposite direction as the far field flow associated with the object/flow, and so the flow needs to “turn around” somewhere in the boundary layer—this was the case for the sphere with a vortex pair behind it that we saw in the last Part, for example. Because the velocity changes continuously, as the flow “turns around”, there has to be some point at which the flow velocity becomes zero when moving away from the object. Around these points, the fluid effectively behaves as if it were near the surface of an imaginary object, and so any fluid bounded by the object and these imaginary surfaces effectively becomes locked in within it. This phenomenon is called flow separation, as the fluid flow appears to “separate” off of the object, and the flow bounded within the zero-velocity surfaces is said to be recirculating flow. Unsurprisingly, the region of fluid bounded by these surfaces is called the recirculation zone.

This is by-and-large all we can generically say about steady flow in the boundary layer. That being said, we’re going to find that—just as in pipe flow—flow in the boundary layer always becomes turbulent and unsteady once certain conditions are met, and we can find those conditions using the same technique we did for pipe flow; dimensional analysis.

To make things a little easier, let’s take a look at our tried-and-true scenario of a fixed rigid object immersed in a fluid with density $\rho$ and viscosity $\mu$, with a background speed $U$ and background pressure $p_\infty$. But this time, we’ll specifically make the immersed object a sphere with radius $R$. These parameters define everything about the flow around the sphere.

As we saw for pipe flow, the Navier-Stokes equation (and by extension the physics of this problem) only involves pressure gradients, not the absolute pressure. And because the total pressure in the system is the sum of an absolute background pressure that is uniform everywhere and a perturbation pressure that isn’t, none of the pressure gradients are affected by the value of the background pressure! Consequently, the system is independent of the background pressure as long as the pressure isn’t so low that the liquid boils.

That leaves us with 4 dimensional parameters that define the flow based on 3 units of measurement—length, mass, and time. If we tried to construct a turbulence index $\zeta$ as a function of these four parameters, we’d find that (thanks to the Buckingham Pi theorem) the turbulence index can only be a function of a single dimensionless number—the Reynolds number $Re$.

$\displaystyle \zeta = f\left(Re\right) = f\left(\frac{\rho U R}{\mu}\right)$

Just like we saw in pipes, whether or not the flow past an immersed sphere is steady or turbulent is entirely decided by the Reynolds number. But the transition to turbulence in flow past spheres (and immersed objects in general, for that matter) has a very rich structure that doesn’t just go from “nice and clean” to “chaotic and unpredictable”. This structure is largely present in all flow transitions for flows past immersed objects.

For steady flow past a sphere, the flow takes one of two forms—flow with no recirculation zone at the lowest Reynolds numbers, or flow with a small recirculation zone in the back consisting of two counter-rotating vortices at the small-but-not-smallest Reynolds numbers.

When the Reynolds number increases enough, we find that the flow fails to be steady, but in a peculiar way—the vortices in the recirculation zone begin to wiggle perpendicular to the flow, and “detach” from the object, forming a streak of vortices called a von Karman vortex street. These emitted vortices travel downstream many distances longer than the radius of the sphere, and are the beginning of what we might recognize in common parlance as a wake. Although this flow is turbulent, it is quite simple and structured, and can be studied theoretically (although we will not do so here).

As the Reynolds number increases further, the structure of the vortices being emitted from the back of the object decompose into a turbulent chaotic mess, and the flow appears to look well-behaved everywhere except for in a long messy “tail” immediately behind the object—this is a proper wake. Finally, increasing the Reynolds number even more results in the flow becoming turbulent everywhere in the boundary layer, not just in the wake. In this final stage, the flow has become fully turbulent and behaves very much as it does in turbulent pipe flow, where it possesses very little structure or order anywhere. These last two stages of turbulence are commonly referred to as subcritical and supercritical, respectively.

Given that dimensional analysis gave us a way to at least classify observed flows past an immersed sphere, perhaps it will be useful to answer another extremely important question about flow past immersed objects; the forces that fluids exert on them.

Usually, the forces on an immersed object are split up into components that are parallel and perpendicular to the background flow direction—the parallel component is usually called the drag force, and the perpendicular component is called lift force. Drag and lift are both just different components of the same force, caused by both pressure-induced tractions and flow-induced molecular interaction tractions on the surface of the object. The general equation for the force on an immersed object is then:

$\displaystyle \vec{F} = \int_S \vec{t}_s\ dA= \int_S -p \hat{n} + \boldsymbol \tau \cdot \hat{n}\ dA$

$\displaystyle \vec{F} = \int_S -p_\epsilon \hat{n} + \mu \left(\nabla \vec{v}_\epsilon + \nabla \vec{v}_\epsilon^T\right) \cdot \hat{n}\ dA$

Notice the similarity to the expression we found in Part II for buoyant forces on a submerged object; this formula is a generalized version of it accounting for fluid flow.

However, we can’t solve for the flow in the boundary layer, and so we don’t really have a chance to derive anything for the net force acting on the object and its lift/drag components using this expression. But we can use dimensional analysis, coupled with some situational assumptions, to get surprisingly general expressions.

For example, we can try to find how many independent things with units of force we can construct out of the set of 4 dimensional parameters we have available for flow past a sphere. It turns out we can only make two, $\mu U R$ and $\rho U^2 R^2$. This means that either the lift or the drag force on a sphere can be represented in the following form:

$\displaystyle F_d\; \text{or}\; F_l = \sum\limits_{n \in \mathbb{R}} \alpha_n \left(\rho U^2 R^2 \right)^n \left(\mu U R \right)^{1-n}$

This is correct, but essentially useless; we need some other assumptions to get a workable, specific expression. One assumption we could make is that the surface tractions coming from the viscous effects are much smaller than those coming from the pressure perturbations; mathematically, that $\mu |\left(\nabla \vec{v} + \nabla \vec{v}^T\right)| \cdot \hat{n} << |p| \hat{n}$ on the surface of the sphere. This is true if the viscosity is very small, or if the average velocity gradient on the surface of the sphere is on average zero because the flow in the boundary layer is chaotically flipping direction. In either scenario, the Reynolds number would be very large.

In this scenario, the drag/lift force is independent of the viscosity, and so anything dependent of it must have no bearing on the drag/lift force—namely, the $\mu U R$ term. As a result, for high Reynolds numbers, we expect the drag and lift to have the following form:

$\displaystyle F_d\; \text{or}\; F_l = \alpha_1\rho U^2 R^2$

where $\alpha_1$ is a numerical constant independent of any of the other physical parameters in the system, a consequence of the dimensional crisis caused by removing the viscosity. This kind of drag, which is quadratic in the velocity, is usually called Newtonian drag.

Alternatively, we can consider what happens when the viscosity is very large, or when the Reynolds number is very low. In such a scenario, the terms dependent on the viscosity in the above sum should dominate, and anything independent of the viscosity shouldn’t influence the drag force; particularly, the $\rho U^2 R^2$ term. Removing the only parameter that solely appears in this expression, the density, we find the following expression for the drag/lift force at low Reynolds numbers:

$\displaystyle F_d\; \text{or}\; F_l = \alpha_0 \mu U R$

where $\alpha_0$ is also just a number. This kind of drag is called Stokes drag, and it is linear in the velocity of the object/background flow. If one attempts to solve for Stokes flow over a sphere assuming the near field flow equations apply everywhere (which is incorrect), one would find that $\alpha_0 = 6 \pi$ for the drag force relationship.

Usually, fluid mechanicians plot the drag (or lift) force in terms of a nondimensional number called a drag (or lift) coefficient $C_d\; \text{or}\; C_l$, which is the drag force divided by $\rho U^2 R^2$ or some multiple of it:

$C_d\; \text{or}\; C_l = \frac{F_d\; \text{or}\; F_l}{\rho U^2 R^2}$

This nondimensionalization of the drag/lift force is convenient because this nondimensional number can only be a function of nondimensional parameters constructed from the 4 physical variables that fully describe the physical set-up of flow past a sphere. And there’s only one nondimensional parameter we can make with those four variables; the Reynolds number.

If you were to plot the drag coefficient versus the Reynolds number $Re$ for a sphere based on experimental results, this is what you’d find:

This is all consistent with what we expected from dimensional analysis! Now we might ask, how do any of these results change when the object isn’t a sphere, but something totally different?

A simple way to illustrate this might be by considering what would happen if I put a spherical “nose” on the sphere with a radius $R_2$, directly in the front of the sphere. This parameter $R_2$ defines everything about the nose, and $R$ along with $R_2$ fully define the geometry of this new class of immersed object.

If I did the whole dimensional analysis rigmarole I did for the sphere for this new object, I’d find the same things except for the presence of another, new dimensionless parameter: $\frac{R_2}{R}$. More importantly, if I tried to find expressions for the drag/lift force in the limits of low and high Reynolds numbers, I’d get stuck with a bunch of horrid sums again:

$\displaystyle F_d\; \text{or}\; F_l\; \text{for low Re}\; = \sum\limits_{n \in \mathbb{R}} \alpha_n \left(\mu U R \right)^n \left(\mu U R_2 \right)^{1-n}$

$\displaystyle F_d\; \text{or}\; F_l\; \text{for high Re}\; = \sum\limits_{n \in \mathbb{R}} \alpha_n \left(\rho U^2 R^2 \right)^n \left(\rho U^2 R_2^2 \right)^{1-n}$

However, in either case, the $\alpha$‘s can only be a function of the only dimensionless objects we can construct; the radii ratio $\frac{R_2}{R}$. Because of that, we can actually factor out a couple of things out of the sum! Simplfying a bit, we’ll find that everything that isn’t a function of either radius winds up on the outside:

$\displaystyle F_d\; \text{or}\; F_l\; \text{for low Re}\; = \mu U \sum\limits_{n \in \mathbb{R}} \alpha_n R^n R_2^{1-n}$

$\displaystyle F_d\; \text{or}\; F_l\; \text{for high Re}\; = \rho U^2 \sum\limits_{n \in \mathbb{R}} \alpha_n R^{2^n} R_2^{2^{1-n}}$

Everything that’s within the sums is only a function of the geometry of the object, and so can be grouped into a geometric factor $\Gamma$ which we choose to have units of length. With this, we find that:

$\displaystyle F_d\; \text{or}\; F_l\; \text{for low Re}\; = \mu U \Gamma_{\text{low}}$

$\displaystyle F_d\; \text{or}\; F_l\; \text{for high Re}\; = \rho U^2 \Gamma_{\text{high}}^2$

The great thing about this is that, if we were to add another geometric feature to the immersed object (say, “ears”), we’d wind up finding the exact same thing, albeit with a presumably more complicated geometric factor involving more geometric parameters. As a result, one can construct whatever kind of immersed object one wants using features, each defined by a single length, and the above results will still hold! For example, this indicates that at the very low and very high Reynolds number limits, the drag/lift force dependence on the viscosity, density, and speed are independent of the shape of the object. Anywhere in between, the geometry of the object does affect those relationships by affecting the values of the $\alpha_n$‘s.