19

This is most easily seen by considering the stationary Stokes equations $$ -\mu \Delta u + \nabla p = f \\ \nabla \cdot u = 0 $$ which is equivalent to the problem $$ \min_u \frac\mu 2 \|\nabla u\|^2 - (f,u) \\ \text{so that} \; \nabla\cdot u = 0. $$ If you write down the Lagrangian and then the optimality conditions of this optimization problems, ...


15

I do not think that there is a definite answer to this, because it might change from one topic to other (and also depends on the type of elements you are using). There are some recent papers talking about that, as well [2]. So, it is not a closed discussion. Furthermore, you can have different inertial components (at least in mechanics), when you have ...


14

The question assumes that there is a strict delineation between equations, but there isn't. On paper, of course, the Navier-Stokes equations have a parabolic character because there is a non-zero diffusion term. But, in reality, we say that equations are "hyperbolic" when we mean that they are advection dominated, and "parabolic" when they are diffusion ...


13

This list is nowhere near complete, but hopefully the size of it will give a hint as to the scale of possible factors. I am assuming you are compiling the code from source on your platform of choice. Software Standard Library Performance Lin. Alg. Library Performance (if the software links to outside libraries) Compiler Choice Compiler Optimization ...


8

You can use a Nitsche-type method for this. See the following reference: J. Freund, R. Stenberg. On weakly imposed boundary conditions for second order problems. Proceedings of the Ninth Int. Conf. Finite Elements in Fluids, Venice 1995. M. Morandi Cecchi et al., Eds. pp. 327-336. I have implemented this a while ago in some simple FEniCS-Code to deal with ...


8

Up until a couple of decades ago, science was based on two large pillars. Those were theory and actual physical experiments. It is an exciting time to see a third pillar arise with numerical simulations. In between pure theory and expensive real-world experiments, we can now run simulations! When it comes to these simulations, you may observe two types of ...


7

Use a vector streamfunction or take the cross product of two gradients. I.e.: $$ \boldsymbol{u}=\nabla\times\boldsymbol{A} $$ where $\boldsymbol{A}$ is a vector field of your choosing, or $$ \boldsymbol{u}=\nabla f\times\nabla g $$ where $f$ and $g$ are two scalar fields of your choosing. It's hard both have the velocity be divergence-free and prescribe ...


6

I think this greatly depends on what kind of physics you are trying to model even though for some problems both approaches are viable. Lagrangian vs. Eulerian Framework For certain problems Lagrangian frameworks are better suited than their Eulerian counterparts. For instance, if one is interested in studying the current patterns or sedimentation problems ...


6

Take a look at some of the papers by Jean-Luc Guermond over the past 10 or so years that deal with solving PDEs my minimizing the residual in the L1 norm. His website is here.


6

You're learning that the two forms (with $-\Delta u$ and with $-2\nabla \cdot \varepsilon(u)$) are not equivalent. They lead to different boundary terms in the bilinear form after integration by parts. In the first case, you get a term involving $n\cdot \nabla u$, in the latter $2n \cdot \varepsilon(u)$, and the "natural" boundary conditions you can enforce ...


6

There are three issues that are likely to cause such problems in pseudospectral methods: Gibbs oscillations Aliasing Time step too large In any case you likely develop oscillations in the solution until some point ends up with a negative density, resulting in a NaN when computing the pressure or sound speed or some other term. The solution to 3 is obvious, ...


5

For interior penalty type HDG methods we have recently shown that $1/\sqrt{k}$ is possible (Proposition 6.10) and this bound is valid for several element types. In Remark 6.11 there is also a discussion about known results from the literature for other methods. To my knowledge there exist no uniform results for hp-DG methods yet. I would however also be ...


5

This is not a general answer, but for the Navier-Stokes equations, there are manufactured solutions that describe real flow. For example, the Kovasznay flow field is a popular choice: http://link.springer.com/article/10.1007/BF00948290 The original reference is: Kovasznay L.I.G., "Laminar flow behind a two-dimensional grid". Proc. Cambridge Philos. Soc., ...


5

Look at the cell Péclet number $$\mathrm{Pe}_h = h v / K$$ where $h$ is mesh size, $v$ is the magnitude of velocity, and $K$ is diffusivity. It is analogous to cell Reynolds number for the momentum equation and is small when "thermal diffusivity is large compared to advection". It is common common in macro-scale fluid dynamics that thermal diffusivity $K$...


5

Check out An overview of projection methods for incompressible flows; Guermond, Minev, Shen; Comput. Methods Appl. Mech. Engrg., 195 (2006); http://www.math.ust.hk/~mawang/teaching/math532/guermond-shen-2006.pdf It gives a fairly good overview over a bunch of solver classes, amongst others the fractional-step methods such as pressure-correction (e....


5

The problem was solved recently: Lederer/Schöberl: Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations


5

The topic of linear solvers for the Stokes equation is pretty well discussed in the literature, and the common consensus is that (i) using GMRES as the outer solver, (ii) the Silvester-Wathen approach to forming a preconditioner for the outer solver, (iii) using multigrid as the inner solver or preconditioner for the elliptic block is the way to go. This is ...


5

When we say that the pressure is only defined "up to a constant", what we mean is this: If $u,p$ is a solution to the Navier-Stokes equations, then $u,p+c$ for any constant $c$ is also a solution. In other words, the solution is not unique, but the non-uniqueness has a very particular structure: two solutions $u,p$ and $u,p'$ can only differ in a way where $...


4

Before I answer your question, I just want to clarify one key point: Solving PDE's, like Navier Stokes equations, requires a two step process: Geometry Discretization (meshing) Solution of the linear equations Usually this is done with two separate programs. Meshing can be a very costly operation by itself and should be done carefully according to ...


4

If you are using an ideal gas equation of state, only the ratio of specific heats should appear in the nondimensional governing compressible Navier-Stokes equations. Knowing that you want to simulate (say) air, which provides gamma, is sufficient. If this is your setting and my answer unclear, please let me know and I can point you to a write up.


4

You can absolutely use the Newton method to linearize the system of equations that results after you have discretized in time. It's a pretty common approach. It might be overkill in lots of cases, but it's not inappropriate. Freezing like you suggest and not iterating at all within a timestep can lead to some pretty awful solutions unless your timestep is ...


4

The projection scheme is indeed analytic, provided that the initial conditions are incompressible. You can derive it by taking the divergence of the evolution equation for $u$. Finding the projection P can be tricky; it's easier with some methods like spectral / pseudospectral methods. But one has to be careful that numerical error doesn't move the ...


4

@JannisTeunissen gave an answer for infinite domains. However, for finite domains, the Helmholtz decomposition is not unique, and the scalar potential $\phi$ can take many forms. First, it is true that the curl part $\nabla\times\vec A$ is of course divergence free. But if $\vec F$ is divergence free, all you know is that $\phi$ has to satisfy $-\Delta \phi=...


4

Physically, vorticity can only be created (as opposed to transported, stretched or intensified after being created) either by the appearance of a boundary layer on a solid surface, or through the pressure gradient not being parallel to the density gradient. For the shallow water equation without viscosity neither mechanism is available, so congratulations, ...


4

So first, represent your system as the following in your case: \begin{align} \frac{\partial q_k}{\partial t} - q_k \nabla \cdot F_k(\boldsymbol{q}) &= 0 & \forall k \in \lbrace 1, 2 \rbrace \end{align} where $\boldsymbol{q} = [q_1, q_2]^T = [f, g]^T$, $F_1(\boldsymbol{q}) = q_2 \hat{e}_1$, and $F_2(\boldsymbol{q}) = q_1 \hat{e}_1$. Then you just ...


4

For flow solvers, the general rule is that the time step needs to satisfy some kind of "CFL condition", named after Courant, Friedrichs, and Lewy. This means that $$ \Delta t \le C \min_{K} \frac{h_K}{\|\mathbf u\|_{L^\infty(K)}} $$ In other words, the time step must be proportional to the (minimum over all cells $K$) of the ratio of the mesh size $h_K$ ...


4

First, I'm not sure why you emphasize on water? I mean I understand that you are looking for a CFD scheme that works for your special case, but you need to know that water fluid is not a special fluid at all. Water is a incompressible fluid and you can simulate its movement by using incompressible Navier-Stokes equation as long as your Mach number is not ...


4

For both schemes: I am going to lump $B$ and $C$ into a single matrix $B$ for convenience of communicating the idea. Likewise, $y$ and $z$ are now $y$, and $F_y$ and $F_z$ are $F_y$. $Ax+B^Ty=F_x$ or $x = A^{-1}(F_x-B^Ty)$ $Bx=F_y$ or $BA^{-1}(F_x-B^Ty)=F_y$ $BA^{-1}B^Ty=BA^{-1}F_x-F_y$ Note that $BA^{-1}B^T$ is symmetric and invertible, and can be ...


3

This "model" is the incompressible (constant density) Navier Stokes problem, the second equation being the mass balance: $$ \frac{\partial\rho }{\partial t}+\nabla\cdot(\rho v)=0 $$ I have worked in the past with Comsol, and I believe that the Navier Stokes weak forms are readily implemented in the CFD module as states the Comsol modeling manual in this LINK....


3

OpenFOAM has turbulence models built in, which is helpful. I think it's easier to use OpenFOAM as a black-box solver than to modify its internals (which I found was very difficult given the lack of developer documentation and OpenFOAM coding style). If you're solving the Navier-Stokes equations, there's probably already a solver available in OpenFOAM, in ...


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