17

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 ...


16

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 ...


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 ...


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, ...


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.


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

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

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

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 ...


4

One approach to convert this into an ODE is with index reduction methods. These allow you to convert high-index DAEs into low-index DAEs or ODEs. See section VII.2 of "Solving Ordinary Differential Equations II" by Hairer and Wanner. Consider a generic, Hessenberg index-2 DAE $$ \begin{align} y' &= f(y, z) \\ 0 &= g(y) \end{align} $$ ...


3

Assuming you choose stable pairings of elements, the main consideration is which variable you are more interested in. For example, if you choose the $H(div),L^2$ formulation, then you have the choice of piecewise constants for the pressure and a BDM element for the velocity. This yields only first order accuracy for the pressure in the $L^2$ norm -- in other ...


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 ...


3

For the incompressible Navier-Stokes equations, in practice, where ever there is a Dirichlet boundary condition on the velocity, Neumann boundary condition is applied for the pressure. It does not matter if it no-slip or "velocity-inlet" or "velocity-outlet". You should not specify the pressure where the velocity is specified. Your figure shows velocity ...


3

It depends on how you set your Dirichlet conditions, but they are often equivalent to setting $\phi_j=0$ on those nodes, so you shouldn't have to worry about that equation as long as you set the Dirichlet condition correctly. Now, that being said, I've never been a huge fan of pinning one pressure node in traditional mixed formulations for Navier-Stokes ...


3

If you have a vector of coefficients $U$ for the velocity approximation, then your velocity field is given by $\vec u_h = \sum_j U_j \vec\varphi_j$. The divergence is then clearly $$ \text{div} \;\vec u_h = \sum_j U_j\;\text{div}\,\vec\varphi_j $$ and similarly for the curl. The thing to realize, though, is that even if you are using continuous finite ...


3

Note: I originally posted an answer that I was not 100% pleased with, so I have revised it heavily. 1/20/2017 The projection method is not in an exact approximation to the full system in general. There are a few reasons for this, but the most immediate has to do with the boundary conditions. Let's look at the continuous setting first. Consider the unsteady ...


3

The first thing to keep in mind is that there is no one turbulence model that works well in all situations. You need to choose the right model for the right situation. See this link for reference: https://engineering.stackexchange.com/questions/336/which-turbulence-models-are-suitable-for-cfd-analysis-on-a-streamlined-vehicle-b/344#344 There are many ...


3

There are two aspects to your question I think. 1) Do your equations match the physical problem you're trying to model? 2) Do your finite difference equations converge to the continuous ones as dx and dt approach zero? First, I'd like to address question 1, since this affects question 2. Question 1: Assuming you're looking for a general (developing) flow ...


3

For a vector field $\vec{F}$ that satisfies a couple of conditions (smooth, decaying rapidly enough) Helmholtz's theorem states that you can write $\vec{F}$ as $$\vec{F} = \nabla \times \vec{A} - \nabla \phi.$$ The 'curl field' $\nabla \times \vec{A}$ is equal to the divergence-free part of $\vec{F}$, since we have: $$\nabla \cdot (\nabla \times \vec{A}) = ...


3

Water (and other fluids that are considered incompressible) are of course in reality compressible -- just not very much, at the pressures and speeds involved in the flow we are modeling. Within the ranges of pressures we consider, a good approximation is that $$ \rho = \rho_0 (1+\alpha p) $$ where $\rho_0$ is the reference density at reference pressure, $p$...


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