[lWiNdY]

Following the book of 'Katz & Plotkin, Low Speed Aerodynamics', I would like to construct a 3D panel method for a wind turbine in yaw. However, at one point they suggest both source and doublet terms on the panels.

How does one determine the source strength distribution? Isn't this a non-unique solution?

[lWiNdY]

I mean, in 9.2b of 'K&P, LSA' it is stated that the source distribution must be such that . But then, how is the doublet distribution determined?

If the net flow through the body is already zero (especially at t=0, when there is no induced velocity yet), the resulting doublet terms are also zero right? Otherwise, we end up with non zero flow through the boundary? This will repeat for all time steps..

[blackjack]

In 1976, Harold Levine at Stanford taught a mathematics course, "Methods of Mathematical Physics". In the first week, he showed that the Laplace eqn combined with Green's (3rd) Identity leads to an equation relating the potential, phi, to its normal derivative, d(phi)/dn.

[A] {phi} = [B] {d(phi)/dn}

In 1979, I attended a training session for SOUSSA (Steady, Oscillatory, Unsteady, Subsonic and Supersonic Aerodynamics) written by Luigi Morino and his students based on the same equation. I recommend getting: Morino & Kuo, Potential Flow: a General theory, in the AIAA Journal (about 1972).

In Morino's method, each panel has two unknowns (phi and d(phi)/dn) with two equations: (1) Neumann boundary condition, (2) the equation relating phi and d(phi)/dn. The solution is unique for an external flow because the potential is unique for an external Neumann problem.

If you make the mistake of including a fictitious flow inside the body, the doublets and sources are not unique. If you realize the flow inside an airplane has no effect on the external pressures, the potential (aka doublets) are determined entirely by the Neumann boundary condition (aka sources).

For a wind turbine in yaw, the flow is unsteady, and you should write the Neumann B.C. as

d(phi)/dn = n (dot) (Qinf - Qsurface)

If the airfoil is sitting still in calm air, d(phi)/dn=0 and, yes, phi=0.
If the airfoil is moving with the wind, phi=0.
If the airfoil is sitting still, but the wind is blowing d(phi)/dn is not zero, and phi is not zero. Whether it's a model in a wind tunnel or a model in flight, Morino's method produces the same answer.

You can analyze a wind turbine in yaw with Morino's method. I do.

[lWiNdY]

Originally Posted by blackjack

In 1976, Harold Levine at Stanford taught a mathematics course, "Methods of Mathematical Physics". In the first week, he showed that the Laplace eqn combined with Green's (3rd) Identity leads to an equation relating the potential, phi, to its normal derivative, d(phi)/dn.

[A] {phi} = [B] {d(phi)/dn}

In 1979, I attended a training session for SOUSSA (Steady, Oscillatory, Unsteady, Subsonic and Supersonic Aerodynamics) written by Luigi Morino and his students based on the same equation. I recommend getting: Morino & Kuo, Potential Flow: a General theory, in the AIAA Journal (about 1972).

In Morino's method, each panel has two unknowns (phi and d(phi)/dn) with two equations: (1) Neumann boundary condition, (2) the equation relating phi and d(phi)/dn. The solution is unique for an external flow because the potential is unique for an external Neumann problem.

If you make the mistake of including a fictitious flow inside the body, the doublets and sources are not unique. If you realize the flow inside an airplane has no effect on the external pressures, the potential (aka doublets) are determined entirely by the Neumann boundary condition (aka sources).

For a wind turbine in yaw, the flow is unsteady, and you should write the Neumann B.C. as

d(phi)/dn = n (dot) (Qinf - Qsurface)

If the airfoil is sitting still in calm air, d(phi)/dn=0 and, yes, phi=0.
If the airfoil is moving with the wind, phi=0.
If the airfoil is sitting still, but the wind is blowing d(phi)/dn is not zero, and phi is not zero. Whether it's a model in a wind tunnel or a model in flight, Morino's method produces the same answer.

You can analyze a wind turbine in yaw with Morino's method. I do.

Okay, to sum up: The source distribution is chosen such that the zero flow through panel condition is met. In other words, the relative velocity (wind + rotation + induction) is taken care of by the source terms. Then, from potential flow theory we know that the sources + doublets must add up to zero. Hence, the doublets equal the negative source terms. The latter are again the relative velocity, or equal to.

What is then the use of the source terms?

[blackjack]

I completely disagree with your summation.

I find it clearer to not talk about sources and doublets. The second equation I wrote is the Neumann Boundary condition. It's the input to the problem, i.e., you know it before solving the problem. Green's Identity (Eq. 1) tells you how to calculate the surface potential from the Neumann Boundary condition. Green's Identity is a compatibility equation between the normal derivative of phi on the surface and the potential on the surface. Step 3 is to determine the surface gradient of phi (velocity) and hence the pressure on the surface.

The relative velocity is not "taken care of by the source terms". Every panel with non-zero d(phi)/dn induces a normal velocity on every other panel. The sum of the source influence alone does not give you tangent flow, but a non-zero interference normal velocity. The doublets don't equal the negative source terms. The doublets reduce the interference velocity.

Please read Morino's paper(s). I find it is not a needlessly complicated explanation.

[adrin]

James has provided an excellent set of explanations and answers. I, too, dislike the "source and doublet" approach and worked with phi and its normal gradient since day one of using the boundary element method for solving the laplace equation (I prefer the BEM terminology to panel method, partly because it's mathematically more precise to me)!

To reiterate, doublets do *not* cancel out sources (per panel)! For that matter, neither do phis cancel their normal gradients. What this view is missing is that [A].doublets = [B].sources (or phi and dphi/dn). As James clarifies, it's the sum contributions of all doublets at a point (and that involves surface integrations that include the Green's function, and not just the variable itself) that are equal to the sum contributions of all sources at that point!

Also, a comment was made in the original question about the "net sources" being zero for this (incompressible flow) problem. The keyword here is *net*! Yes, the *net* mass flow (sources) into the object is zero. However, that doesn't automatically imply that the local flow into/out of the body is also zero! There is, indeed, a source/flow distribution around the object determined by Q_inf.n (for a fixed object). It is a very simple matter to verify that Q_inf.n is generally non-zero pointwise, giving rise to a distribution, which, nevertheless, has a net value of zero when contributions (by just Q_inf.n, excluding the Green's function) by all panels are included.

BTW, as a sanity check, the sum of Q_inf.n had better be equal to zero, or you're doing something wrong! Also, due to precision or other careless programming issues you may end up with a slightly non-zero value for this sum. This may either lead to an incorrect distribution of phi (if the sum is not too far from machine precision zero) or even numerical blow up (because you're violating the necessary compatibility condition)

Best of luck

adrin

[sbaffini]

Apologies if I make confusion over the comments of people knowing better than me but, maybe, what you miss is that a single source is just assigned to balance the local Vinf*n term. That is, all the other sources will still induce a non zero velocity which must then be compensated by the unknown doublets. Only in the case of Vinf=0 you obviously have both distributions equal to 0

[adrin]

Test ... I wrote a fairly long response to "what you miss is that a single source is just assigned to balance the local Vinf*n term" but it got lost upon submission Trying this test first!

arin

[adrin]

OK, trying again ...

strictly speaking "what you miss is that a single source is just assigned to balance the local Vinf*n term" is incorrect!

The generalized BIE equation for the velocity anywhere in the field (inside and outside the body) is:

u(x) = Curl[ Int[ ( gamma(x') - u(x') ^ n(x') ) G(x, x') dS(x') ] ] + Grad[ Int[ u(x') . n(x') G(x, x') dS(x') ] ], where gamma(x') is the vortex sheet strength that must be generated at the boundary to enforce (both) slip and flow-through boundary conditions, u(x') ^ n(x') and u(x') . n(x'), respectively.

gamma(x') is the only unknown and in 3-D it's usually obtained by projecting u(x) (on LHS) tangential to the boundary and applying the BC accordingly, the discussion for which is beyond the scope of intro here.

This is a generalized equation that actually can be shown to include all previously proposed and used formulations involving sources and doublets. Depending on the algorithm, however, the sources can actually be considered as unknowns (or even both sources and doublets be considered as unknowns and solve for 3 simultaneous system of equations that give the most accurate results for the lowest possible panel resolution!) In this case, even the source where Vinf*n is evaluated will generally NOT be equal to Vinf*n!

If one opts _not_ to solve for the sources, then the self-influence would be zero at the collocation point IF the panel is piecewise flat! However, this is generally no longer true if the panel is curved!

Hope this helps

adrin

[sbaffini]

Ok, again apologies for the confusion, but Katz and Plotkin describe exactly this approach in the first 4 paragraphs of chapter 9. They even mention issues when the body gets too thin. So, is that part wrong?

Of course I understand that the specific selection of sources and doublets is arbitrary and non unique. But once the specific choice above is made, which I thinks is what op did, isn't this anymore correct?

[adrin]

I'm sorry, but you're *very* vague in your assertions and questions, which leads only to further confusion. For example, "Katz and Plotkin describe exactly this approach in the first 4 paragraphs of chapter 9". Exactly which approach are you referring to? What I wrote earlier or something else? And, there are no clear "4 paragraphs" in chapter 9 of the 2nd edition I have! Are you implying 4 sub-sections? And, even then, you need to be specific as to which exact method and assumption you're referring to -> as you acknowledge yourself there are different solution strategies, so, each method has its own unique nuances and corresponding assumptions and performance benefits/pitfalls! Your original question seems to generalize to imply that all these methods have the same common characteristics, which is simply not true!

Further, "... which I thinks is what op did, isn't this anymore correct?". What or who is op? I must be missing something here! Also, what does _this_ in "isn't this" refer to? I honestly do not understand the question, and will not even try to guess because it is ripe with further confusions down the line!

With all due respect, in math and science one must be painfully specific in expressing one's thoughts so as to avoid _any_ room for misinterpretation by others!

Best

adrin

[blackjack]

Adrin, you placed importance on having the net flow through the body be zero.

Lighthill mentioned coupling potential flow with the boundary layer by modeling the displacement effect with transpiration through the body surface. This yields a non-zero net flow. Please comment on this approach.

I am familiar with Drela's codes, xfoil and mses, which use the boundary layer displacement.

[adrin]

"Lighthill mentioned coupling potential flow with the boundary layer by modeling the displacement effect with transpiration through the body surface. This yields a non-zero net flow. Please comment on this approach."

OK, so, now, the topic has completely changed which is fine! Initially, we were discussing the solution of the Laplace equation (potential flow), which has a global compatibility (continuity) requirement that the integral over the surface of the body of the velocity in/out of the object be zero! In fact, if for whatever reason this compatibility condition is violated your matrix will either not converge to the correct answer or, based on my experience, it will simply blow up (depending on the severity of the violation of the compatibility condition).

I'm afraid I don't have Lighthill handy in front of me and I don't precisely remember the underlying problem assumptions ... But, I can imagine that, as a first cut, he'd be dealing with incompressible flow, which implies - boundary layer modeling included or not - that continuity must be satisfied! So, whereas there may be (is) local transpiration, the compatibility constraint (integral of fluxes over the proper control volume) would still have to add up to zero (let's remember the important keyword "net", not necessarily the local)

Also, it is *crucial* to understand and remember physics when implementing numerics! The boundary layer transpiration model is applied _in_ the fluid domain, not _at_ the solid wall. Physically, the boundary layer develops _in_ the fluid and _above_ the wall. Yes, for high Re number the boundary layer is so thin such that we can _model_ the boundary layer (and the continuity associated with it) as fluid transpiration across (and _at_) the wall. But, remember this is a model. In that model, there already is an *implicit* application of no-through wall boundary condition _at_ the wall! Take a 2D control volume abutting the solid wall and inside the fluid domain, apply no-flow at the bottom, _at_ the wall, inlet and outlet flow at the sides, and outflow (or inflow) at the top (vertical velocity due to continuity), and then take the thickness of that control volume to its limit of zero, which should hopefully clarify this misconception that there is flow through the body! There is only fluid flow above the body, which has a transpiration looking vertical velocity at the top surface due to continuity (actually implicitly enforcing no-flow across the wall).

Note that the above discussion is valid only for incompressible flow, and *impermeable* walls! Obviously, both compressible flow and/or permeable walls would have non-zero flux conditions to satisfy. But, for _incompressible permeable_ flow, ask yourself how it is possible to continuously "leak" (i.e., transpire) water (because we're dealing with incompressible flow) across the wall without drying up an internal source! The only way I can think of right now that you can indefinitely "leak" water out of the body somewhere is if you "infuse" the same amount of water _into_ the body. That is, once again, continuity dictates that what leaks out *must* be equal to what enters the body! Armed with this important and simple concept you should be able to understand Lighthill's or others' boundary layer models (IF the models violate continuity throw them away!)

Hope this long-winded answer helps, partially

adrin

[sbaffini]

Originally Posted by adrin

I'm sorry, but you're *very* vague in your assertions and questions, which leads only to further confusion. For example, "Katz and Plotkin describe exactly this approach in the first 4 paragraphs of chapter 9". Exactly which approach are you referring to? What I wrote earlier or something else? And, there are no clear "4 paragraphs" in chapter 9 of the 2nd edition I have! Are you implying 4 sub-sections? And, even then, you need to be specific as to which exact method and assumption you're referring to -> as you acknowledge yourself there are different solution strategies, so, each method has its own unique nuances and corresponding assumptions and performance benefits/pitfalls! Your original question seems to generalize to imply that all these methods have the same common characteristics, which is simply not true!

Further, "... which I thinks is what op did, isn't this anymore correct?". What or who is op? I must be missing something here! Also, what does _this_ in "isn't this" refer to? I honestly do not understand the question, and will not even try to guess because it is ripe with further confusions down the line!

With all due respect, in math and science one must be painfully specific in expressing one's thoughts so as to avoid _any_ room for misinterpretation by others!

Best

adrin

Apologies again if my intervention seemed out of place or contentious, the questions in my last answer were genuine. I want to understand what you two are saying because, now, I don't understand.

In forums, and the like, OP typically stands for "Original Poster", which is something I don't typically use (abbreviations in general), but "lWiNdY" was difficult to remember or copy/paste from the phone app.

Now, let me clarify my line of tought. The starting point, for me, is this one:

Originally Posted by lWiNdY

I mean, in 9.2b of 'K&P, LSA' it is stated that the source distribution must be such that . But then, how is the doublet distribution determined?

If the net flow through the body is already zero (especially at t=0, when there is no induced velocity yet), the resulting doublet terms are also zero right? Otherwise, we end up with non zero flow through the boundary? This will repeat for all time steps..

I understand that in English-speaking countries "paragraph" and "section" have different meanings, but the world at large uses "paragraph" for what the English speaking countries use "section" for.

Chapter 9 of the first 2 editions of K&P is "Numerical (Panel) Methods" and the first 4 sections/paragraphs are (for both editions):

9.1 - Basic Formulation
9.2 - The Boundary Conditions
9.3 - Physical Considerations
9.4 - Reduction of the Problem to a Set of Linear Algebraic Equations

and while I didn't literally check word by word the two editions, they are sufficiently similar for what we need here. It is in these very pages that lWiNdY mentions he found a difficult passage, something he seems to not understand.

Now, you adrin and blackjack, really know stuff better than me, but as I have coded this, and reading the question by lWiNdY, I thought you guys really went astray in your answers.

I then tried to give my contribution:

Originally Posted by sbaffini

maybe, what you miss is that a single source is just assigned to balance the local Vinf*n term. That is, all the other sources will still induce a non zero velocity which must then be compensated by the unknown doublets. Only in the case of Vinf=0 you obviously have both distributions equal to 0

but you guys suggested it is, strictly speaking, incorrect. Then, I read again the 4 sections above from K&P and wrote a second post whose meaning was:

"Katz and Plotkin describe exactly this approach (of assigning the local source on the panel equal to the local Vinf*n term) in the first 4 paragraphs of chapter 9. They even mention issues when the body gets too thin. So, is that part of the book wrong?

Of course I understand that the specific selection of sources and doublets is arbitrary and non unique. But once the specific choice above is made, which I thinks is what op did (that is, sources are assigned the local value of Vinf*n, as mentioned explicitly by lWiNdY in his question), isn't this answer anymore correct? That is, isn't it true in this specific case we are discussign about?"

That is, I want to understand which part of the book is wrong or which part I am reading wrong, which seems a legit question to me.

Again, let me stress again that I know the solution is not unique in terms of singularity selection. For example, instead of the local Vinf*n term, one could first solve the non lifting problem with only sources as unknown, and then use their so calculated value in the lifting problem with only doublets as unknown. The problem mentioned by lWiNdY would still not exists, because the lifting problem still has to satisfy kutta at trealing edges and to shed a wake (or just has a wake). Yet, in this example the sources would exactly give zero net flux if considered alone.

[blackjack]

Adrin,

Do I understand you correctly?

I could model an inflating balloon by specifying a uniform, positive normal velocity on a sphere. The net normal velocity would be far from zero, which you claim means it cannot be solved. A balloon is impermeable, and at Mach numbers below 0.001 air can be regarded as incompressible.

It would be trivial for me to model such a case.

[adrin]

James,

>> I could model an inflating balloon by specifying a uniform, positive normal velocity on a sphere.

Is this a closed sphere? (because *real* balloons that I'm aware of aren't closed!). You can _model_ anything you want and you will get "an answer" so long as your math at least satisfies certain physics correctly! The statement I quote above is one clear example that you're suggesting you're going to model a certain physics with an equation that should (will) not work without modification! Ask yourself the simple question I already asked earlier: If you have air permeating out of the (presumably) closed balloon, and all velocity vectors (correctly) are pointing outward (pure outflow), where is that air coming from? Surely, you're not creating that outflow "out of thin air" )) To put it differently, there *must* - at least initially - be a source of air inside the balloon. So, there is a mass source, initially, inside the balloon, and now you let it out. And, if you paid attention to what I was careful to say, continuity would have to be satisfied within the proper control volume containing this system! I'm sure we both agree that continuity is fundamental, and you can't wiggle around it - it doesn't matter what example you can verbally come up with; how are you going to convert that verbal example into a mathematically valid and consistent formulation that will at least satisfy continuity?

So, are you going to solve the inflating balloon problem without *any* initial mass (i.e., source) of air inside the (closed) balloon? And are you actually going to use Laplacian(phi) = 0, with simply Q.n BC? And, will this model of yours be able to inflate the balloon indefinitely (will assume the material can stretch forever) because you can apply the outflow velocity BC indefinitely after the initial mass source has been depleted? (and that's even assuming you accounted for some initial mass source!)

If you can do such a model successfully, I'd truly appreciate learning something new that shatters everything I've been taught about continuity! I would very much appreciate seeing what exact equations and assumptions you'd be using (and/or have used) for this problem. My experience over decades has taught me that there is, quite often, a long road connecting the verbal expression of a problem and the finalized version of the same in terms of a mathematical model!!!! In that road, there are lots of hidden assumptions and "buts and ifs" that pop up, which modify the original statement considerably!

PS - As I finished reading my answer, I already realized you've thrown at me a _new_ problem that is _different_ from the premise of the original as well as my response (which I was quite careful to be specific). In all past discussions, we have talked about fixed (as in non-expanding) objects that could potentially transpire. And, yes, in this case, my personal experience is that if the surface integral of u.n is non-zero for the Laplacian(phi) = 0, then the matrix solution *will* blow up! Now, James, you've posed a completely different problem - one that involves a moving (non-fixed) body surface, which is *not* what I talked about earlier; this is a *new* problem altogether! Since I have not solved this problem myself I will not give a definitive answer; however, Laplacian(phi) = 0 is a kinematic (continuity) equation, devoid of time-evolving dynamics (such as the balloon expanding). So, as such, I will go on a limb and still claim that *no* you can't solve Laplacian(phi) = 0 with Q.n > 0 all around the sphere IF (and this puts me back to insisting that we're extra specific with our words in math and science) you apply just simple, typical BCs of the type that the original question was posing! I know one can apply quite complicated non-linear and time-dependent BCs that "meld" dynamics into the kinematics of L(phi) = 0. But, that would already violate the whole premise of this discussion (neither would it challenge anything I said regarding net flux out of a selected control volume being zero in incompressible flow)!

Best

adrin

[adrin]

Hi Paolo,

Thank you so much about the extensive clarifications. I truly appreciate it. Please allow me some time to go over the material in the references you pointed out (paragraph/section locations), and I will hopefully get back with my thoughts on this soon. My short answer to one question, at least, is: no, Katz and Plotkin are not wrong! I would personally not challenge them

Best

adrin

[blackjack]

Paolo,

I take your concerns seriously.

As I stated in my first post, I use Luigi Morino's method based on Green's identity.
In this method, the input is the Neumann input, d(phi)/dn. The output is the Neumann output, phi. If an internal flow is analyzed the input must conform to the Neumann compatibility condition that the net mass flow into the volume is zero.

The input to Morino's method is not the Dirichlet input, phi. and the output is not the Dirichlet output, d(phi)/dn. The method does not solve a Dirichlet problem. So the method has all the characterisitics of a Neumann solver and none of a Dirichlet solver, but in the first edition of LSA K&P refer to this method as "based on the Dirichlet boundary condition".

Morino's method has (per body panel) two unknowns, phi & d(phi)/dn, and two equations. LSA describes a method with six unknows: real (phi & d(phi)/dn), imaginary (phi & d(phi)/dn), source & doublet strenth. with five equations. The sixth condition is the statement that setting the imaginary potential to zero implies that the imaginary d(phi)/dn is zero. This statement is correct, but Program Number 8 in the LSA appendix is a simple fortran program that is incapable of implying anything. It obeys the rules of linear algebra that require an equation to determine d(phi)/dn. After assuming an imaginary flow exists, K&P then set both imaginary phi & d(phi)/dn to zero. This has nothing to do with the work of Dirichlet.

You mention choosing the singularities. Do you see any rules in LSA for choosing the singularities? I've encountered more than one reader of LSA who is confused as to how the choice is made. Who chooses? The program developer or the user? Program No. 8 does not allow the user (the person who prepares the input) to choose the singularities. From the user's perspective, why all the discussion about imaginary quantities they have no ability to change? Where in Program No. 8 is the Dirichlet boundary condition?
Where in the derivation of Morino's method does one choose the singularities?

K&P have said several times that the control point is on the inside surface of the panel.
The result is exactly the same if the control point is on the outside surface. Needless and irrelevant complexity.

Morino's method is physical. If you test a wind tunnel model made of solid wood or metal, it isn't necessary to imagine that inside the model is freestream flow.

I have applied Morino's method to a thousand geometries, because it applies to arbitrary geometries. I have applied it to steady, unsteady and oscillatory subsonic flow and added mass calculations. I have applied it to external and internal flows. Like Tranair, Morino's method can be extended to transonic flow by recognizing that at high Mach number the Laplacian of the potential is not zero.

LSA 1st ed. did not provide useful descriptions of oscillatory flow, added mass, or internal flow.

What analyses have you performed based on the methods discussed in LSA?

[adrin]

BTW, as to why I insist on a simple mathematical fact. The problem at hand (with simple BCs) is Laplacian[phi] = 0. Now, let's take a volume integral:

Int [ Laplacian[phi] . dV ] = 0

Int [ n . Grad[phi dS ] = Int [ n . Q dS] = 0

Q.E.D.

On the other hand, for the more general problem of a Poisson equation, we have

Laplacian[phi] = epsilon (as in source of air for the balloon)

Int [ Laplacian[phi].dV ] = Int [ epsilon.dV ]

Int [ n . Grad[phi] dS ] = Int [ n . Q dS ] = Int [ epsilon.dV] = epsilon.Volume = mass/density (assuming eps is constant)

The above mandatory compatibility requirement clearly shows the net flux is _non-zero_, but only if there is a source! (that's the initial air in the closed balloon, consistent with what we know intuitively about physics)

adrin

[FMDenaro]

Originally Posted by adrin

BTW, as to why I insist on a simple mathematical fact. The problem at hand (with simple BCs) is Laplacian[phi] = 0. Now, let's take a volume integral:

Int [ Laplacian[phi] . dV ] = 0

Int [ n . Grad[phi dS ] = Int [ n . Q dS] = 0

Q.E.D.

On the other hand, for the more general problem of a Poisson equation, we have

Laplacian[phi] = epsilon (as in source of air for the balloon)

Int [ Laplacian[phi].dV ] = Int [ epsilon.dV ]

Int [ n . Grad[phi] dS ] = Int [ n . Q dS ] = Int [ epsilon.dV] = epsilon.Volume = mass/density (assuming eps is constant)

The above mandatory compatibility requirement clearly shows the net flux is _non-zero_, but only if there is a source! (that's the initial air in the closed balloon, consistent with what we know intuitively about physics)

adrin

I have just read the discussion, what you are addressing is exactly the same argument present in the pressure equation for incompressible flows.

Note that the BCs are expressed in terms of non-homogeneus Neuman conditions fulfilling the compatibility condition. On the other hand, you can see easily that homogeneous Neumann BCs and a modified source term produce exactly the same mathematical problem.


The ballon inflaction can be modelled with a single source in a 1d radial problem. However, you have to consider the surface of the ballon moving at a fixed velocity not related to the real physics of a compression of the flow. That is, in this model the ballon has only an immaginary surface. At least if I understand you.