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8 votes
Accepted

What kind of a researcher am I?

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 ...
MPIchael's user avatar
  • 2,935
6 votes
Accepted

Boundary condition for Pressure in Navier-Stokes equation

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 ...
Wolfgang Bangerth's user avatar
5 votes

Has a uniform estimate in k of the inf-sup constant for hp-DG methods for the Stokes problem been established?

The problem was solved recently: Lederer/Schöberl: Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations
Guido Kanschat's user avatar
5 votes
Accepted

Mixed Finite Element Method for the Stokes System—Some Implementation Details

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 ...
Wolfgang Bangerth's user avatar
5 votes
Accepted

Can we simulate compressible flows by simple direct explicit calculation, without solving systems of linear equations (such as Poisson eq)?

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 ...
Philip Roe's user avatar
  • 1,154
5 votes
Accepted

Projection method FVM poisson part, adding source term

The smooth solution turned out to have BC's applied in the following way: Walls and inlet: $\frac{\partial p}{\partial n}=0$ Outlet: $p=0$ Actually thought that we need only one value of P to pin, not ...
2Napasa's user avatar
  • 362
5 votes
Accepted

How do the navier stoke equations model materials who "forget" their original form?

The key difference between fluids and solids is that in a fluid, the stress is $$ \nu \nabla \mathbf v, $$ (or some variation involving the symmetric gradient instead of the gradient) which involves ...
Wolfgang Bangerth's user avatar
5 votes

Why can the weak forms of the Stokes and continuity equations be combined into a single equation?

To get from weak form I to II, just add the two equations of weak form I. To see the other direction, note that weak form II has to hold for all test functions $\phi$ and $\psi$. That is, you can test ...
cos_theta's user avatar
  • 451
4 votes

Incompressible Navier-Stokes equations: Is projection method exact?

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. ...
A.Vigs's user avatar
  • 71
4 votes
Accepted

What is the difference between the curl component, and the divergence-free component, of a vector field?

@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 ...
Wolfgang Bangerth's user avatar
4 votes
Accepted

Galerkin method for a system of nonlinear PDEs

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 \...
spektr's user avatar
  • 4,248
4 votes

Efficient schemes for solving the extended Saddle point problem

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 $...
Charlie S's user avatar
  • 661
4 votes
Accepted

Write incompressible Navier Stokes as ODE in $(\mathbf{u},p)$

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 ...
Steven Roberts's user avatar
4 votes
Accepted

Decoupling Stokes problem into two problems: velocity and pressure, using FEM

Such splitting methods as you described, known as projection or fractional-step methods, are indeed available for FEM. The Characteristic Based Split (CBS) method is quite popular among such methods, ...
Chenna K's user avatar
  • 944
4 votes

Projection (or fractional-step) methods Vs coupled method for incompressible Navier-Stokes

Let us consider the system of equations describing viscous incompressible flow in a unit cube $0\le x\le 1, 0\le y \le 1, 0 \le z \le 1$, and in time interval $0\le t \le 1$, we have \begin{equation}\...
Alex Trounev's user avatar
3 votes

What is the difference between the curl component, and the divergence-free component, of a vector field?

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} - \...
Jannis Teunissen's user avatar
3 votes
Accepted

Is this finite difference approach correct?

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 ...
Charles's user avatar
  • 619
3 votes

Decaying turbulence and simulation

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: ...
Paul's user avatar
  • 12k
3 votes
Accepted

Can we simulate incompressible flows using the (slight) density changes to give pressure?

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 ...
Wolfgang Bangerth's user avatar
3 votes
Accepted

CFL condition in Stokes equation

Essentially, the time dependent Stokes equation looks like the heat equation: $$ \frac{\partial u}{\partial t} - \nu\Delta u = f-\nabla p, $$ plus the incompressibility condition $\nabla \cdot u=0$ ...
Wolfgang Bangerth's user avatar
3 votes

Automatic timestep adjustment in a CFD solver

Yes! Normally what's done is called Method of Lines. Essentially, you discretize in space to get all of your operators, but instead of discretizing the time component, you leave that derivative along. ...
Chris Rackauckas's user avatar
3 votes

Automatic timestep adjustment in a CFD solver

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{...
Wolfgang Bangerth's user avatar
3 votes
Accepted

Lattice Boltzmann methods vs Navier stokes/ other eulerian methods for *water* simulation

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 ...
Mithridates the Great's user avatar
3 votes

Lattice Boltzmann methods vs Navier stokes/ other eulerian methods for *water* simulation

I disagree with the answer given by Alone Programmer. His reasoning is not objective and seems to be based on a very primitive lattice-Boltzmann model with BGK collision operator. LBM has ...
2b-t's user avatar
  • 148
3 votes
Accepted

Simplest way to "upgrade" from Euler equations to Navier-Stokes equations in FV or FD framework

Generally the step from compressible Euler equations to the Navier-Stokes equations is not that hard, at least the coding part. If you want to implement it with an explicit scheme you have to ...
ConvexHull's user avatar
  • 1,335
3 votes

Different form of the Navier--Stokes equations

The operation in the second equation is the divergence of the dyadic product $vv$.The dyadic product of two vectors is a square matrix. In this case, a $3\times3$ matrix where $(vv)_{ij}=v_iv_j$. The ...
boltz's user avatar
  • 83
3 votes
Accepted

Discretizing the viscous component in 1 - D Navier stokes compressive flow

Consider a FV method as an approximation of the integral conservation law. Starting from the one-dimensional, scalar conservation equation \begin{equation} u_t + f(u,\nabla u)_x =0, \end{equation} ...
ConvexHull's user avatar
  • 1,335
3 votes

Discontinuous pressure elements for incompressible Navier-Stokes

There are many variations of the idea of Taylor-Hood elements that remain stable. The Taylor-Hood element on quadrilateral and hexahedral elements are generally understood (though historically ...
Wolfgang Bangerth's user avatar
3 votes
Accepted

Non-standard boundary condition for incompressible Navier Stokes

I managed to resolve the issue, Need to compute the values only for the inner part of the domain, then apply BC for the boundary values. The only issue is that there is a backward scheme at the ...
2Napasa's user avatar
  • 362
2 votes

Recovering pressure from velocity or streamfunction fields

So in case if you want to get pressure field there are two variants. 1) Simple one As you have already solve the problem in terms of stream function ($\psi$), you get velocity field from the ...
madmech's user avatar
  • 23

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