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Question about different kinds of Boundaries and Boundary Conditions 

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April 1, 2022, 00:45 
Question about different kinds of Boundaries and Boundary Conditions

#1 
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Mandeep Shetty
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I am having a hard time understanding these different terms: Physical Boundary (or Boundary Condition), Numerical/Artificial/Computational Boundry (or Boundary condition), Open Boundary (or Boundary condition), Closed Boundary or (Boundary condition). I am having the problem understanding because of how these boundaries/boundary conditions are related to the idea of Spatial Dimensions, Temporal dimensions, and (I read) spatial direction which acts like timelike direction. Here is an example from a document:
insert1.png Now, I understand what a Physical/Real Boundary is and also what a Physical Boundary Condition means. For ex: "Wall" is a Physical Boundary/Boundary Condition. The Wall boundary of a computational domain is usually 'closed' to flow but 'open' to say heat flux. But when I considered this picture for propagation problem: insert2.png the time direction is given as the open boundary, and it has conditions (initial condition) only at t=0 and then the time direction stretches to infinity without any 'condition' at the end, which is the definition of an open boundary. Now if I consider x=0 as the inlet and x=L as the outlet, the flow fluxes do pass through the inlet and outlet. So I am confused with are inlet and outlet an 'open boundary' too?, and if they are open boundaries how can we specify some condition (boundary conditions) at, say, x=L (outlet) as we don't specify any 'end condition' for the time direction which is a given as a 'open boundary in the figure. Last edited by granzer; April 1, 2022 at 05:02. 

April 1, 2022, 03:56 

#2 
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Filippo Maria Denaro
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The attachment are not working


April 1, 2022, 05:02 

#3 
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Mandeep Shetty
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April 1, 2022, 05:08 

#4 
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Filippo Maria Denaro
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As example, think about the Prandtl equations. They are parabolic in the momentum, that is the xdirection can be assumed to be a timelike direction: the flow evolves along x as a temporal evolution.
The domain for parabolic equation is open, you cannot prescribe mathematically the BCs over all boundaries. The figure you posted could be rotated right and showing the domain of integration for a BL. In general, numerical BCs are sometime required owing to the specific algorithm. 

April 1, 2022, 05:19 

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Lucky
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You have to clarify what you mean exactly by inlet and outlet if we are talking about NavierStokes, no these are not always "open boundary conditions." In general, there are constraints at inlets and outlets: fixed pressure, fixed velocity, fixed mass flow rate, temperature, many other transported variables are explicitly declared at inlets and outlets, any linear combination of fixed values and their gradients. In some cases with supersonic flow at an outlet, it can behave like an open boundary condition but still not quite.
"Open boundary conditions" don't have any conditions. The solution evolves in that direction/dimension forever according to the other specifications in the problem. But other governing equations are parabolic as mentioned. A lot of loworder models or reduced order models have ode's in space that depend on only the inlet boundary condition and not the outlet boundary conditions (most thermodynamic system calculations are in this category) and you solve these by simple marching. 

April 1, 2022, 05:53 

#6  
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Let's consider first the question of the time variable and how, for certain problems, one or more spatial variables behave in the same manner. I think you euristically understand why the time variable is such that it doesn't require/allow boundary conditions at the end of the interval. Without going into the math, the fact that certain spatial coordinates in certain equations are similar to the time variable is something that becomes clear from the equation itself. As mentioned by Filippo, in the boundary layer equations, advancement along the streamwise direction (because of simplifications in the original NS equations) is similar to advancement in time. In certan conditions, similar NS approximations are possible also in pipes and similar geometries. Euristically, for space coordinates this happens when they only appear with first derivative and there is convection from one boundary to opposite one. Mathematically things are more complex, but that's not the point here. For the question of the open boundary, the wording, in my opinion, is officially reserved for geometrical boundaries (i.e., in space) that are numerically needed to truncate a computational domain in order to make a finite computation. They are in contrast to physical boundaries that, as you recognize, are walls. Now, what happens at open boundaries? It depends from the equations, the local state and how you geometrically defined the boundary with respect to certain directions. A so called parabolized equation like the boundary layer ones is such that if the boundary is perpendicular to the advancement spatial direction, then everything is extrapolated at the boundary, no physical information is expected to travel in the domain from it. Just like time. If an equation has only a second derivative in a certain spatial dimension, you are sure that, in that direction, your equation needs boundary conditions at both ends of the domain. For a system of equations like the NS ones, where you have both diffusion and convection, things are much more complex. The usual approach is to consider only the inviscid part for the analysis, as diffusion away from physical boundaries should be negligible (but that is questionable in general). Then convection for a system of equations implies a system of waves, some going forward, some backward, or all forward, or all backward. So, your boundary conditions at such boundaries just reflect that. For example, a pressure outlet, which is a classical open boundary at the exit for subsonic flows, implies that one wave is entering while all the other are exiting. So, it is still an open boundary but some information is traveling in the domain, that's why you still need a boundary condition there for the pressure. 

April 1, 2022, 08:08 

#7 
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Sayan Bhattacharjee
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Adding to what FMDenaro said, the boundary conditions for parabolic equations are based on their "characteristics".
Elliptic and hyperbolic have different characteristics i.e the solution in the domain depends on different sections in the domain and even time. You have to get a feeling for how they are, and it might not be possible to completely understand everything about them. For that, I will leave this quote from Von Neumann here for you : "Young man, in mathematics you don't understand things. You just get used to them.” 

April 11, 2022, 03:43 

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Mandeep Shetty
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Thank you all.I get why this question and the answer to it is better coming from a more general, mathematical side (by studying analytical solutions to "types" of PDEs) than just looking at it just from a CFD perspective, but fluid flow does provide some really good realworld examples for different types of equations and boundary conditions! For anyone looking here are a few references that I am currently still studing:
J.D Hoffman's books on Numerical Methods for a general explanation on BCs and how analytical BCs can change/adopted when solving Numerically. A Good explanation of boundary conditions and how they are related to the 'types' of PDEs is Morse Feshbach's book on modeling and Solving Physics problems. Also, Arnold Sommerfeld's book is really good too. Any good CFD book like Perić's or Hirsch or Anderson's has good reference information specific for application in CFD. Quote:
Quote:
So, there is two 'types' of open condition. One in time where it is 'truly open' condition, ie we can give till what time to run the simulation..but we don't give any 'condition' at the said end time. It is 'truly open' because we don't have any condition at all at the end time and it can be extended up to infinity. But in space, its difficult to get a 'truly open' condition because the space domain cannot be extended up to infinity and we artificially cut off the space domain and create boundaries. So, if we want an 'open' condition on these artificially created boundaries we have to come up with a condition that would 'act' like an open condition but it wouldn't be 'truly open' since we are giving conditions on the 'artificial' boundary. Ex: For a 1D supersonic flow, at the outlet should be 'open' ie conditions at the outlet shouldn't affect the interior flow. To have a 'Truly Open' condition means we don't give any condition at all. But instead, we do give a condition to 'extrapolate' the interior solution at the outlet, which 'acts' like an open condition. Please do correct me if wrong. Quote:
I do have a specific question here though. If we consider a simple 1d flow, say in a pipe, we have truncated the space domain at 2 points. If the flow is subsonic the equation governing the flow would be hyperbolic and consider one of the characteristics at the outlet would be coming into the domain, meaning some condition at the outlet does affect the interior. This would mean we would have to give condition to one of the flow variable at the Outlet and can directly extrapolate the values of other variables to the Outlet. I know that extrapolating the solution from the interior domain acts like an 'open' condition, but we are also giving a condition to one of the variables. So, is the Outlet here considered "Open" or "Closed" OR is it called 'closed' in terms of the variable to which we have given the condition and 'open' in terms of variables we are extrapolating? [I realize that the 'condition' that we do give at the outlet could be affecting the interior flow, and so due to the coupling of equations, it could also be affecting the variable that we are extrapolating at the outlet] Quote:


April 11, 2022, 04:10 

#9 
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I would separate the problem related to how this or that reference uses a specific wording to treat this or that boundary condition (and, again, my preferred choice is to have the "open boundary" name for anything non wall, without specific reference to the characteristics) from the actual math problem.
The NS system of equations has a mixed nature depending on the conditions. Not purely hyperbolic nor purely parabolic, less than ever purely elliptic. In your pipe case, for example, it is not hyperbolic, exactly because a characteristic is entering the domain for subsonic flows. It is hyperbolic instead in the supersonic case. As I said above, my preferred wording, however, is still "open boundary" in both cases, exactly because there are characteristics passing trough it. Their direction, in my opinion, is really irrelevant here. Said otherwise, if by your wording you want to communicate information on the characteristic direction, I think that "open boundary" is not sufficient and you need to be more specific. In those cases we typically rely on expressions like "subsonic/supersonic inlet/outlet" or similar, which leave no doubt about their meaning. 

April 11, 2022, 08:51 

#10  
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Mandeep Shetty
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Quote:


April 11, 2022, 09:05 
How to understand this boundary condition table?

#11 
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Mandeep Shetty
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insert3.png
I am trying to understand this table. but haven't been making any headway. Can you please help me how to understand this?. i) What does 'one' or 'two' closed boundary mean? I read that 'closed boundary means it should completely cover the domain. ii)According to the table a parabolic equation with Dirichlet open boundary condition is 'solvable'. But Dirichlet BC means we are specifying the value at the boundary. How can the boundary be 'open' if we specify the value on it? 

April 11, 2022, 09:51 

#12  
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Roughly speaking, you have: convection > hyperbolic diffusion > parabolic take away the time(like) derivative from the parabolic and you have elliptic. I have no idea what the terms in your table refer to 

April 11, 2022, 10:11 

#13  
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Filippo Maria Denaro
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April 11, 2022, 20:47 

#14  
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Mandeep Shetty
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I think the table is about the classic secondorder linear DE with constant coefficients. So they can have 3 distinct 'types'. But the table can be used with any particular type arising in any equation to get an idea about the BC needed for the problem to be wellposed. 

April 11, 2022, 20:54 

#15 
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Mandeep Shetty
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Was going through the method of characteristics for a general firstorder linear coupled system. Would study the Euler system and the Eigenvalue problem next.


April 12, 2022, 04:39 

#16  
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April 12, 2022, 16:39 

#17 
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Filippo Maria Denaro
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April 12, 2022, 17:27 

#18 
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Lucky
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I really don't like this open vs closed boundary nomenclature and some of the other text being quoted because it seems to be very author specific. It's definitely not widely adopted and can't be understood in a general setting without the context already being established, which the rest of us here are missing. Most people never refer to a boundary as open or closed anyway. We just say where is the boundary and what is the boundary condition, and there is no confusion.
Take this one versus two boundaries for example... A general problem has n boundaries (not always 1 or 2, it can be 3 or 4 or 5). What use is this table there? Obviously this table is applicable to only the onedimensional domain and is using nomenclature that can only be interpreted within this landscape. But since we're already here... in the context of the text being presented here (and only in this context and not anywhere else, because normal people don't talk this way), an open boundary is a boundary condition at infinity. Certain types of boundary value problems admit solutions at infinity (these have exponential eigenfunctions) and certain problems don't (these have sines and cosines as eigenfunctions or hyperbolic sines and cosines). And never say that a problem has no analytical solution... 

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