Part XI: Far Field & Irrotational Flow

Having said essentially everything we could about pipe flow, we now seek to characterize the last lingering question of basic fluid dynamics; flow past immersed objects.

In most descriptions of flow past immersed objects, and the only scenario we’ll cover here, we have some solid, rigid, stationary object surrounded by what can be approximated as an infinite amount of fluid. These types of flows tend to be called unbounded, or are said to be in an infinite domain, as they are defined over an infinite amount of space. That doesn’t necessarily mean we seek to characterize flow over all of space—just an infinite amount of it. All the way out at infinity, we’ll assume that the flow is a uniform flow, with a constant velocity in a single direction and a constant pressure; the presence of an embedded object then “bends” the flow close to it in a way consistent with the Navier-Stokes equation and with conservation of mass.

This physical set-up is convenient because it is equivalent to another ubiquitous scenario in fluid mechanics, that of a rigid object moving at a constant velocity through an infinitely large medium of otherwise stationary fluid. This scenario closely approximates the dynamics of a plane or bird in the sky, or of a submarine or fish in the ocean. The correspondence between this scenario and the first one I first mentioned comes from uniformly adding or subtracting the background flow; in this latter case, the flow will be a perturbation of a completely stationary flow by a rigid object moving with a velocity equal to the negative of the background velocity in the former case.

Two equivalent unbounded flows. On the left-hand side, a stationary object is placed in an infinite fluid with a uniform background flow and constant pressure at infinity. On the right-hand side, a rigid object is moving with a fixed speed and constant direction in a stationary background fluid with constant pressure at infinity. The velocity of the object in the right-hand scenario is equal and opposite to the velocity of the flow at infinity in the left-hand scenario. A spherical coordinate system at the origin has been illustrated for convenience.

With this in mind, one can consider the description of the velocity field/pressure field/flow to be a combination of the background flow and of some other velocity/pressure field which decays to nothing as you get further and further away from the object. Formally, this is called a flow perturbation, in the sense that the object “perturbs” the background flow only in a region close to the object. The velocity and pressure field would then be of the form:

\vec{v} = \vec{v}_b + \vec{v}_\epsilon

p = p_\infty + p_\epsilon

where the perturbation velocity field \vec{v}_\epsilon and perturbation pressure field p_\epsilon need to become negligibly small relative to the background velocity \vec{v}_b and pressure p_\infty respectively as we move further and further away from the immersed object.

With either set-up in mind, we could try to perform a mathematical analysis like we did for pipe flow and solve directly for the pressure and velocity everywhere. However, in this case, we don’t have nearly any of the simplifying assumptions we had with pipe flow; even though we can assume the flow is steady in the stationary immersed object case, the flow can potentially be in all three coordinate directions, and it can potentially change in all three directions. In addition, we know that turbulence is baked into the Navier-Stokes equations somewhere, meaning that even the steady assumption is bound to fail at some point. What can we do?

Well, here’s an idea; we could try analyzing the behavior of the flow only in a region “almost at infinity”. We can’t do it all the way at infinity, since all we’d get is the background flow by definition—but we can try to get a sense of what the Navier-Stokes equations and conservation of mass tell us about the perturbation flow as we move out further and further away from the object. Formally, we are attempting to understand the asymptotic behavior of the perturbation flow near infinity, in a region commonly referred to as the far field.

A still object is submerged in an unbounded background flow of uniform speed and constant pressure—a spherical shell around it has been constructed “almost at infinity”. We want to identify the characteristics of the flow in the question mark regions outside the shell, representing the region “almost at infinity” called the far field, using asymptotic methods. A spherical coordinate system has been drawn for convenience.

Solving the full Navier-Stokes equation plus conservation of mass in this scenario isn’t an option—not even the best mathematicians can do it—so what we’ll do is come up with an additional assumption about the perturbation velocity field far from the object that, when plugged in, makes the Navier-Stokes equation easier to solve. Such an assumption is commonly referred to as a kinematic constraint by dynamicists.

We want this kinematic constraint to have a couple of key features:

  1. It’s consistent with the perturbation velocity decaying to zero as we move further and further away from the object.
  2. It reduces the Navier-Stokes equation to something tractable.
  3. It doesn’t violate conservation of mass.

The idea is that now we’ll have three equations we can work with; and if we make the right kinematic constraint, we’ll be able to solve for the velocity or pressure straightforwardly and use the Navier-Stokes equation to calculate the fluid quantity we didn’t calculate before. Because we only care about the flow outside of the boundary of the spherical shell we drew above, the flows we obtain only need to be correct in the far field; the flows we obtain from this process are far field flows.

Let’s begin. By inserting the expression \vec{v} = \vec{v}_b + \vec{v}_\epsilon and p = p_\infty + p_\epsilon into the Navier-Stokes and incompressible conservation of mass equations for the scenario where we have a uniform background flow of constant speed and pressure, and under the assumption of no external forces and steady flow, we get:

\displaystyle \rho \nabla \vec{v}_\epsilon \cdot \left(\vec{v_b} + \vec{v_\epsilon}\right) = - \nabla p_\epsilon\ +\    \mu \nabla^2 \vec{v_\epsilon}

\displaystyle \nabla \cdot \vec{v}_\epsilon = 0

The criteria for the kinematic constraint on the velocity we listed above, namely the first one regarding the flow decay at infinity, induce a very specific behavior in the perturbation flow at the far field. Informally, the idea is that the flow far from the object should be unaffected by viscous effects—only the presence of the object is triggering viscous effects in the flow, and far enough away from it, those viscous effects should be trivial. This suggests that the ratio of convective to viscous effects (i.e. the Reynolds tensor) will be large in the far field, and so \nabla^2 \vec{v}_\epsilon \sim 0*. The \sim symbol indicates “asymptotic” equivalence, i.e. that the objects are identical as one approaches the asymptotic limit, which is at r \rightarrow \infty.

*I really wish I could prove this rigorously, but every strategy to do so that I could come up with is either pointlessly contrived or way too elaborate to show here. Oh well.

This creates a problem for our strategy of calculating far field flows; we know that our biggest obstacle to a straightforward solution is that nonlinear term, \displaystyle \rho \nabla \vec{v}_\epsilon \cdot \left(\vec{v_b} + \vec{v_\epsilon}\right), so it would make sense to try to get rid of it first. However, we know that in the far field, \nabla^2 \vec{v}_\epsilon \sim 0, so we can’t just get rid of it outright; it would make the Navier-Stokes equation just an equation for the pressure and not the velocity in the far field, which we don’t want. Somehow, we have to come up with a kinematic constraint that both simplifies the nonlinear convective term and completely gets rid of the viscous term. How are we going to do that?

The secret is in that alternative form of the Navier-Stokes equations I kept showing before—the Lamb form. If I plug in my expression for the velocity, I wind up with the following representation of the Navier-Stokes equation:

\displaystyle \rho\left(\frac{\nabla \left( \vec{v}_\epsilon \cdot \vec{v}_\epsilon \right)}{2} -  \vec{v}_\epsilon \times\left(\nabla\times\vec{v}_\epsilon \right) \right) =  -\nabla  p_\epsilon\ -\   \mu \nabla \times \left(\nabla \times \vec{v}_\epsilon \right)

This form makes a possible kinematic constraint obvious, one which simplifies the nonlinear terms and kills off the viscous term; the constraint that \nabla \times \vec{v}_\epsilon = 0. In general, the quantity \nabla \times \vec{v} is referred to as the vorticity \vec{\omega}, as it is proportional to the local rotation rate of the fluid. Small objects embedded in flows that satisfy this kinematic constraint do not rotate, and the flow itself is automatically unaffected by friction and rotation-related convective effects, as our Reynolds tensor calculation required for the far field.

If we take this as our constraint, we find the following system of equations:

\displaystyle \rho\left( \frac{\nabla \left( \vec{v} \cdot  \vec{v}\right)}{2}\right) =  -\nabla  p

\displaystyle \nabla \cdot \vec{v}_\epsilon = 0

\nabla \times \vec{v}_\epsilon = 0

Flows derived from this system of equations are alternatively referred to as either irrotational flows or potential flows. The latter naming convention comes from the fact that, thanks to the kinematic constraint, the velocity can be described as the gradient of a scalar quantity known as the velocity potential \Phi that satisfies the following equation:

\displaystyle \nabla^2 \Phi = 0,\quad \left(u = \nabla \Phi\right)

This equation is known as Laplace’s equation, and it is perhaps the easiest partial differential equation to solve (although solving it is still relatively complicated). In spherical coordinates, the general solution can be written as the sum of a bunch of different functions—to save you the trouble, here is the general solution to the Laplace equation in sum form:

\displaystyle \Phi = \sum \limits_{\ell=0}^{\infty}  \sum\limits_{m=-\ell}^{\ell} \left(A_{\ell m}r^\ell + \frac{B_{\ell m}}{r^{\ell+1}}\right) Y^m_{\ell}(\theta,\varphi)

where the A_{\ell m} and B_{\ell m} terms represent constant coefficients, and the Y^m_{\ell}(\theta,\varphi) terms are functions of only the angles called real spherical harmonics. As either of the indices in the sum (\ell or m) increases, these spherical harmonics become “wavier”. This type of sum expression is called a multipole expansion, as the waviness causes crests, or poles, in the resulting functions.

Taking the gradient of the velocity potential to find the velocity field, we find the following general expression for the components of all irrotational/potential flows:

\displaystyle v_{r_{\text{irrot.}}} =  \sum \limits_{\ell=0}^{\infty}  \sum\limits_{m=-\ell}^{\ell} \left(A_{\ell m} \ell r^{\ell-1} - \frac{B_{\ell m}\left(\ell + 1\right)}{r^{\ell+2}}\right) Y^m_{\ell}(\theta,\varphi)

\displaystyle v_{\theta_{\text{irrot.}}} =  \sum \limits_{\ell=0}^{\infty}  \sum\limits_{m=-\ell}^{\ell} \left(A_{\ell m}r^{\ell-1} + \frac{B_{\ell m}}{r^{\ell+2}}\right) \frac{\partial Y^m_{\ell}(\theta,\varphi)}{\partial \theta}

\displaystyle v_{\varphi_{\text{irrot.}}} =  \sum  \limits_{\ell=0}^{\infty}  \sum\limits_{m=-\ell}^{\ell} \left(A_{\ell  m}r^{\ell-1} + \frac{B_{\ell m}}{r^{\ell+2}}\right)\frac{1}{\sin{\theta}}\frac{\partial  Y^m_{\ell}(\theta,\varphi)}{\partial \varphi}

This all just seems like useless mathematical gobbledygook. But that’s because we haven’t restricted our form of the potential flow by a key principle—that the velocity decay in the far field. On inspection, this immediately indicates that every A_{\ell\geq1  m} term has to be zero, as the flow would fail to decay if they weren’t thanks to the r^{\ell-1} term. The flow coming from the A_{00} term is also identically zero, as Y^0_{0} is just a constant that is independent of either angle and the radial part of the flow is nullified by multiplication of the \ell = 0 factor. Finally, for a rigid object, B_{0m} = 0, as the flow generated by this term fails to satisfy conservation of mass in this scenario. This leaves us with:

\displaystyle v_{r_{\epsilon}} =  \sum \limits_{\ell=1}^{\infty}  \sum\limits_{m=-\ell}^{\ell} -\frac{B_{\ell m}\left(\ell + 1\right)}{r^{\ell+2}}Y^m_{\ell}(\theta,\varphi)

\displaystyle v_{\theta_{\epsilon}} =  \sum \limits_{\ell=1}^{\infty}  \sum\limits_{m=-\ell}^{\ell}\frac{B_{\ell m}}{r^{\ell+2}} \frac{\partial Y^m_{\ell}(\theta,\varphi)}{\partial \theta}

\displaystyle v_{\varphi_{\epsilon}} =  \sum  \limits_{\ell=1}^{\infty}  \sum\limits_{m=-\ell}^{\ell} \frac{B_{\ell m}}{r^{\ell+2} \sin{\theta}}\frac{\partial  Y^m_{\ell}(\theta,\varphi)}{\partial \varphi}

This still looks like a lot of terms. However, remember we only care about the behavior in the far field; and in the far field, no matter what nonzero values the B_{\ell m} coefficients have, the terms that decay the slowest will always eventually become much larger than any of the other terms. As a result, asymptotically, every far field flow always looks like the flow generated by the surviving terms with the lowest radial order—which for a steadily moving rigid object, are the B_{1m} terms. Even better, the spherical harmonics corresponding to Y^m_{1} are really just the same function rotated 90 degrees in either angle, so we can always rotate our coordinate system to get rid of the Y^{-1}_{1} and Y^{1}_{1} terms. This means we get the following universal form of far field perturbation flows for rigid immersed objects moving at a constant speed:

\displaystyle v_{r_{\epsilon}} \sim -\frac{2B_{1 0}}{r^{3}}\cos{\theta}

\displaystyle v_{\theta_{\epsilon}} \sim -\frac{B_{1 0}}{r^{3}}\sin{\theta}

\displaystyle v_{\varphi_{\epsilon}} \sim 0

Notice the velocity doesn’t depend on \varphi; this is because of the coordinate rotation trick I mentioned above.

I like to call this flow a dipole flow, in reference to the fact that the mathematical form of the flow field is identical to that of an electric dipole. It is important to mention that because of the analysis above, every flow perturbation caused by a steadily-moving rigid object, or by a rigid object embedded in a background flow, looks like a dipole flow in the far field. Here’s what that looks like in the latter case:

A far field flow corresponding to an object embedded in a uniform background flow. The flow is axisymmetric around the background flow axis, and is a sum of a uniform background flow and a dipole flow. The opaque grey sphere represents the near field, where the true flow behavior would be inconsistent with this flow. The background flow is deflected by the object about the background velocity axis, with the magnitude of the deflection dropping rapidly as one moves further from the object.

You might notice that the velocity vectors almost appear to be flowing over a sphere, as if the embedded object were a sphere itself. This is because every rigid object deflects flow in the far field as if it were a sphere—a mathematical consequence of the dipole flow dominating in the far field. As a result, every arbitrarily-shaped object has an effective radius r_{\text{eff}} such that a sphere with that radius deflects flow in the far field identically to the original object.

Switching over to the rigid object moving with constant velocity in a quiescent fluid case, here’s what that would look like:

A far field flow generated by a rigid object moving with a constant speed through a quiescent fluid. The flow is an axisymmetric dipole flow with an origin moving at a uniform speed. The fluid is pushed in the direction of the object’s motion in front of the object, pulled backwards in the opposite direction when near the object, and pulled forward again towards the object when behind it.

In the region immediately behind the object, the fluid is being pulled towards it; this phenomenon is called slipstreaming, and is exploited by nature and by humans to increase aerodynamic performance when traveling in groups. For example, this is why geese fly behind each other in V formations, or why cyclists and race car drivers like to drive immediately behind another car when they need a speed boost. The flow patterns of all objects moving steadily while immersed in quiescent fluids behave the same way far from the objects, so it is hopefully now unsurprising that the phenomenon is ubiquitous. They can only ever be different by the scaling factor B_{10}, which is called a dipole moment in electromagnetism contexts, and a quick fly-by dimensional analysis indicates that it must be proportional to the background/object speed U times the effective radius of the object cubed r_{\text{eff}}^3.

Now that we have our expression for the perturbation velocity field, we can use the (simplified) Navier-Stokes equation to get the expression for the pressure field in the static object case:

\displaystyle \rho\left( \frac{\nabla \left( \vec{v} \cdot  \vec{v}\right)}{2}\right) =  -\nabla  p

It might look like we’ll have to solve a partial differential equation, but notice that we’re taking the gradient of a scalar in all of the terms in this equation. As a result, we can collect all the terms inside of a single gradient operation and then “integrate it out”:

\displaystyle \nabla\left(p + \rho\frac{\vec{v} \cdot  \vec{v}}{2}\right) = 0

\displaystyle p + \rho\frac{\vec{v} \cdot  \vec{v}}{2} = C

where C is a constant.

This last equation is called Bernoulli’s equation, and provides a direct algebraic relationship between the pressure and the velocity of a fluid in the case where the viscous effects are negligible, as is the case (we claim) in the far field.

Because the equation holds at every point in the far field, including at infinity, we find that Bernoulli’s equation is satisfied even when the background flow is the only non-trivial flow:

\displaystyle p + \rho\frac{\vec{v} \cdot  \vec{v}}{2} =  p_\infty + \frac{\rho U^2}{2} = C

Noting that the total pressure is the sum of the background and perturbation pressures, and that the total velocity is the sum of the background and perturbation velocities, we can finally get an expression for the perturbation pressure:

\displaystyle p_\epsilon = \frac{B_{10} \rho  \left(2 r^3 U (3 \cos (2 \theta )+1)-B_{10} (3 \cos (2 \theta )+5)\right)}{4 r^6}

Although it may not be immediately obvious from the expression above, this perturbation pressure is always negative, which appears to be nonsense. However, notice that the total pressure, which is the thing we need to keep positive, isn’t necessarily always negative; what this result for the perturbation pressure indicates is that the effect of the object’s presence on the background pressure is to decrease the pressure, more and more as one gets closer to the object. Eventually, if the other assumptions we made for the flow don’t break down as we get closer to the object, the total pressure drops so much that it must become negative—indicating that our solution for the flow can’t be valid. This is yet another example of an existence failure for solutions of the Navier-Stokes equations, which we had seen before in pipe flow.

Practically, however, the liquid would boil due to the pressure reduction long before reaching that point. This occurs when the total pressure in the flow is equal to the vapor pressure p_{\text{vap}}. Using the equation for the perturbation pressure above, one can solve for the distance at which the liquid would vaporize as a function of the other system parameters:

r_{\text{vap}} = \sqrt[3]{\frac{\sqrt{B_{10}^2 \rho  \left(4 (3 \cos (2 \theta )+5) p_{\text{vap}}+\rho  (3 U \cos (2 \theta )+U)^2\right)}-B_{10} \rho  U (3 \cos (2 \theta )+1)}{4p_{\text{vap}}}}

For objects/background flows moving sufficiently fast, the pressure drop due to the presence of the object generates a vapor shell (also called a vapor cone) encasing the object, whose shape and location is described by the equation above. The equation for the vapor shell above gives us a lower estimate on what the “boundary” of the far field is, as the assumption we made when solving for far field flow clearly break down once we get inside the vapor shell due to the density change of the fluid. That being said, vapor shells rarely manifest except in the case of exceedingly rapid objects, in which case one needs to incorporate thermodynamics into the analysis above. In most cases, viscosity begins to become relevant long before hitting the vapor shell region.

The shape of the vapor shell predicted by the far field dipole pressure perturbation against a uniform background flow for one set of system parameters, while ignoring pressure-temperature coupling. Changing the parameters of the system causes the shape to change, but it is always roughly spheroidal. The vapor shell serves as an estimate of the boundary of the far field, as our flow assumptions fail within it.

This asymptotic analysis, using the irrotational kinematic constraint on the flow field, has netted us some very nice, universal results for flow far from an object embedded in a fluid. However, as the existence failure for the pressure near the object demonstrates, it’s not enough to get us some of the most important physical results we need to design objects in fluids for engineering purposes; namely, what the force on the object from the fluid flow is, and the nature of turbulence in flows over immersed objects. For this, we’ll need every trick in the book so far.

Things To Think About

1. To derive our kinematic constraint, we took the assumption that the viscous effects drop to zero in the far field. As a result, the Reynolds tensor should increase as one moves out into the far field. Is this correct, given the form of dipole flow?

2. Is the sum of two dipole flows also a potential flow? How would this be useful for calculating the flow caused by the motion of a group of distant objects, like airplanes?

3. What do you think would be the dominant behavior in the far field if we allowed the immersed object to expand/contract? You can make your life easier if you assume the expansion is really slow, so that the flow is still approximately steady.

4. Why can’t we use potential flow to describe behavior in the near field, i.e. very close to the object?

5. Instead of picking the irrotational kinematic constraint, we could have also simply stated \nabla^2 \vec{v} = 0s=3. Why didn’t I do that?

5. Instead of all this math, we could have tried doing a control volume analysis over some volume enclosing the object to get all the relevant information we needed. Why wouldn’t that be a good idea?

6. For the moving object case, consider a weather vane stuck on some fixed point in the far field while the object moves past it. What do you intuitively think happens to the weather vane as the objects moves past it, and how does the animation above help validate your intuition?

7. How do some of the other irrotational flows look like? Use the equations above to get a sense of what happens when you increase \ell.

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