Questions tagged [navier-stokes]

Questions about solution methods of the Navier-Stokes equations, related physical constants and non-dimensional number. Also special methods to solve the equations including the assumptions and their implementation in order to simplify them. Also, questions regarding modelling of the non-linear term, coefficients of these model can be subjective of this title.

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16
votes
1answer
244 views

Usefulness of elements with mesh-dependent stability

After doing some mathematics related to the stability of elements in 3D Stokes problem I was slightly shocked to realize that $P_2-P_1$ is not stable for an arbitrary tetrahedral mesh. More precisely, ...
14
votes
1answer
17k views

How to formulate lumped mass matrix in FEM

When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...
13
votes
1answer
4k views

Pressure as a Lagrange Multiplier

In the incompressible Navier-Stokes equations, \begin{align*} \rho\left(\mathbf{u}_t + (\mathbf{u} \cdot \nabla)\mathbf{u}\right) &= - \nabla p + \mu\Delta\mathbf{u} + \mathbf{f}\\ \nabla\cdot\...
11
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2answers
347 views

What is the underlying structure of scientific code performance?

Consider two computers with different hardware and software configurations. When running the exact same serial Navier-Stokes code on each platform it takes x and y time to execute one iteration for ...
10
votes
3answers
1k views

manufactured solutions for incompressible Navier-Stokes — how to find divergence-free velocity fields?

In the method of manufactured solutions (MMS) one postulates an exact solution, substitutes it in the equations and calculates the corresponding source term. The solution is then used for code ...
9
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2answers
619 views

Time discretization of the variational formulation of the Navier-Stokes equation

I've asked this question on mathoverflow too. Let $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):...
7
votes
3answers
225 views

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

In Theorem 6.2 of their 2003 paper on "Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(6), 2171–2194", D. Schötzau, Ch. Schwab, and A. Toselli prove a bound of the $\inf$-$\sup$ ...
7
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5answers
2k views

Open source codes for 2D instationary Navier Stokes equations

What open source tool can be recommended for solving 2D instationary Navier Stokes equations (in simple geometries, but with high Reynolds numbers)? Most packages I found, I'm not very lucky with. ...
7
votes
3answers
522 views

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

I am currently working on my bachelor’s diploma. The research concerns mixed finite element method for the 2D Stokes system $$ - \Delta \boldsymbol u + \nabla p = \boldsymbol f, \quad \boldsymbol x \...
7
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2answers
1k views

What are some of the differences between using a Lagrangian and Eulerian framework to quantify passive scalar dynamics?

On one hand, one may seed the domain with particles and track their trajectories in the Lagrangian sense by implementing a Lagrangian particle tracking model. On the other hand, one may use the ...
7
votes
2answers
4k views

FEniCS: separate boundary conditions in normal and tangential direction of mesh boundary

Given a vector-valued PDE, I'd like to enforce the boundary conditions $$ \vec{n}\cdot u = g\\ \vec{n}\cdot \nabla (\vec{t}\cdot u) = 0 $$ on the solution $\vec{u}$. If the boundary happens to align ...
7
votes
2answers
956 views

Feasibility and software for acoustic simulation

I'm looking at doing a finite-element simulation of air flow essentially for the purposes of approximating the response to an external audio impulse of a smallish (~10-30 cm scale), stationary 3D-...
6
votes
1answer
7k views

Type of Navier-Stokes equation

What type equation Navier-Stokes is: Elliptic, parabolic, or hyperbolic? Should it give always the same answer no matter what is the initial condition? How these statements could be proved?
6
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2answers
2k views

Finite elements for Stokes with traction boundary conditions

Suppose we are given the Stokes equations with Neumann conditions on part of the boundary: $-\nabla\cdot\boldsymbol{\sigma} = \mathbf{f}, \quad \text{and} \quad \nabla\cdot \mathbf{u} = 0 \quad \...
6
votes
1answer
122 views

L1 functional setting for Navier-Stokes with finite elements

Typical finite element problems assume $L^2$ which is a Hilbert space, but I've heard that $L^1$ for Navier-Stokes results in less overshoots/undershoots, but $L^1$ is not Hilbert. The dual space for $...
6
votes
2answers
438 views

Implementing the pressure correction method using finite elements

Ok so I am nearing the completion of my finite element Navier-Stokes solver that uses the $\theta$-method for time stepping and the pressure correction method for the pressure. I am following the ...
6
votes
3answers
778 views

Fluid-structure interaction solver for cardiovascular applications

I would like to start running FSI simulations for cardiovascular applications. More precisely, I'm interested in the behaviour of aortic valve under physiological flow conditions in which the ...
6
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0answers
98 views

Benchmarks or generic configurations for optimal flow control

I am about to test my algorithms for solving optimal control problems of type: Find an input $u$, such that for a time interval $(0,T]$ the cost functional $$J(v,u) = \mathcal M(v(T)) + \int_0^T\...
5
votes
1answer
269 views

What kind of a researcher am I?

So far, I've worked a bit in modeling, simulations and simple lab experiments, and I've really enjoyed all three research methods to approach a single research question. I can write tricky (in terms ...
5
votes
2answers
883 views

Incompressible Navier-Stokes equations: Is projection method exact?

Is the projection method of integrating Navier-Stokes equations exact? Take the incompressible flow equations: $$ \frac{\partial\mathbf{u}}{\partial t} = -\mathbf{u}\cdot \nabla \mathbf{u} -\nabla ...
5
votes
2answers
476 views

Choice of spaces for mixed formulation for Poisson Equation Or Darcy equation

Consider the mixed formulation of the Poisson/Darcy system for a region $\Omega$: $\alpha \mathbf{v} + \nabla p = f \\ \mathrm{div}[\mathbf{v}] = 0 $ with the boundary conditions $\mathbf{v}\cdot ...
5
votes
2answers
154 views

Stokes Equation in “two-fold saddle point” form?

Are there papers that deal with the (nondimensionalized) Stokes equation for incompressible fluid flow in a "doubly mixed" form like the following? \begin{align*} 0&=\underline{\epsilon} + \frac{...
5
votes
2answers
3k views

Recovering pressure from velocity or streamfunction fields

I am interested in 2D channel flow of an incompressible Stokes fluid (Re << 1), with periodic boundary conditions in the x-direction and no-slip at the walls in the y-direction. I have existing ...
5
votes
0answers
129 views

How can Navier--Stokes equations have asymmetric solutions such as Karman vortex streets

The Navier--Stokes equations are axially symmetric, so with symmetric boundary conditions, how can features such as Karman vortex streets develop? I understand that in reality symmetry does never ...
5
votes
0answers
188 views

Time-stepping for coupled nonlinear PDEs

What are good references for time-stepping of the coupled incompressible Navier-Stokes-heat equation (Boussinesq flow), $$ \begin{cases} \rho\left(\dot{\mathbf{u}} + \mathbf{u}\cdot\nabla \mathbf{u}\...
4
votes
4answers
3k views

How to deal with nonlinear term in Navier Stokes equations (finite element code)

I am trying to solve the Navier Stokes equations using the finite element method. I plan on using the pressure correction method to deal with the pressure and an implicit time stepping scheme for ...
4
votes
2answers
2k views

Software for 3D Navier-Stokes equation

What is the best software for solving and simulating the 3D Navier-Stokes equation for incompressible laminar non-Newtonian fluid flow?
4
votes
1answer
106 views

Is stabilization of energy equation needed when momentum equation needs it?

When SUPG/PSPG stabilization is added to momentum equation of flow problem, is needed stabilization for energy equation also? I would guess that when stabilization for velocity works fine so one gets ...
4
votes
2answers
2k views

Projection Method: Boundary condition on intermediate velocity field

I'm trying to solve variable density and viscosity Navier-Stokes equation using lagged pressure projection method. I'm solving for cavity problem as a test case now (once I get projection right, I ...
4
votes
1answer
336 views

How to go from turbulent RANS to laminar Navier-Stokes and Euler

SU2 is an open-source CFD suite that is built around a RANS-solver. The main PDE that is solved, is the following: $$ \frac{\partial}{\partial t} \mathbf{U} + \nabla \cdot \mathbf{F^c} - \nabla \cdot ...
4
votes
2answers
545 views

Lid-driven Cavity benchmark in 3D. Classical paper to compare

I'm looking for a benchmark for the lid-driven cavity problem in 3D to compare the results of my code. In 2D I used: U. K. N. G. Ghia, K. N. Ghia and C. T. Shin (1982) High-Re solutions for ...
4
votes
2answers
565 views

Manufactured solution for pressure based 3d incompressible Navier-Stokes solver with wall boundaries

I already successfully verified my solver (SIMPLE-type FVM-method) with the following manufactured solution (3d Taylor-Green vortex) on the solution domain $[-1,1]^3$ with Dirichlet boundary ...
4
votes
1answer
2k views

Comparison of Lattice Boltzmann Method vs Traditional Navier-Stokes based Methods

I have a choice of two options, analysing and implementing Lattice Boltzmann methods or traditional Navier Stokes based methods. I'm a CFD newbie and I have a rough idea (though not rigorous enough to ...
4
votes
1answer
128 views

Projection method FVM poisson part, adding source term

The idea of the method is to decompose the Navier-Stokes equation into the solenoidal and irrotational parts. $$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$ ...
4
votes
0answers
215 views

Reflecting boundary condition posed as a Riemann problem

I am trying to implement a solver for the Euler/Navier Stokes equations. I have a problem implementing boundary conditions for the wall. I am using an unstructured solver. A lot of literature says ...
3
votes
2answers
96 views

Efficient schemes for solving the extended Saddle point problem

I am interested in knowing some efficient techniques for solving the following extended Saddle point problem. \begin{align} \begin{bmatrix} A & B^T & C^T \\ B & 0 & 0 \\ C & ...
3
votes
1answer
267 views

precision loss in non-trigonometric, periodic functions using FFTW and NaNs after marching forward in time (Fortran)

I have developed a pseudospectral solver of the Navier-Stokes equations using FFTW. I tested my formulation of right hand sides (RHS) of the NS equations against standard trigonometric functions (...
3
votes
2answers
2k views

Fractional-step method

I am about to start my journey into the world of CFD and wanted to start with the Fractional-step method for solving the incompressible Navier-Stokes equations. Could you perhaps suggest some articles ...
3
votes
3answers
906 views

Solving Stokes flow with walls using Oseen tensor

Introduction I've developed a code to solve for generalised, incompressible 2D Stokes flow $\eta \nabla^2 \mathbf{v} - \nabla p + \mathbf{S} = 0$ $\nabla . \mathbf{v} = 0$ where $\mathbf{S}$ can ...
3
votes
3answers
139 views

Test case suggestion for incompressible flow with ALE method on deforming grids

I'm working on a finite-volume discretization method with implicit time-integration for the incompressible Navier-Stokes equations in Arbitrary Lagrangian-Eularian (ALE) form. So far I've tested the ...
3
votes
1answer
381 views

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

A common approach says an incompressible flow has velocity divergence of 0; use this to solve for pressure in the Navier Stokes momentum equation. Or, using the Helmholtz Decomposition "project" the ...
3
votes
1answer
208 views

Is this finite difference approach correct?

I am solving incompressible 2D Navier-Stokes equations with zero y-component velocity. The geometry is a simple 2D pipe of a length $L$ and diameter $W$ and there is only two boundary conditions: Non-...
3
votes
1answer
154 views

Decaying turbulence and simulation

I am a beginner in CFD having written few codes for laminar flow cases using SIMPLE and some other explicit solvers. Now, I want to use my solvers and some other models to solve for the turbulent flow ...
3
votes
1answer
727 views

Pressure projection method boundary conditions

When using the pressure projection method to solve the incompressible Navier-Stokes equations do we apply Neumann boundary conditions for pressure only where there are associated no-slip velocity ...
3
votes
2answers
343 views

Looking for reference on Streamline Upwind Petrov Galerkin finite elements for incompressible unsteady Navier-Stokes

I am looking for a relatively simple book/paper that explains the basic Streamline Upwind Petrov Galerkin (SUPG) method for solving the incompressible unsteady Navier-Stokes equations. Most of the ...
3
votes
1answer
183 views

Relaxation Parameters for Steady Navier-Stokes

I am working on a project involving steady solutions for the Navier-Stokes Equations. In the past I've only worked with the unsteady Navier-Stokes, so some of this is new to me. In particular, at ...
3
votes
1answer
716 views

2D Poisson Solver for Taylor Green Vortex Problem

I am trying to write a 2D Navier Stokes solver using an RK3 for time advancement in python. For debugging, I have converted the RK3 to an Euler step for simplicity. Checking my divergence for my ...
3
votes
2answers
591 views

Solving Navier Stokes Eq using Gauss–Seidel and Finite Difference [closed]

I am trying to solve the following set of equations using a finite-difference approach and an iterative solver (Gauss-Seidel). Continuity Equation: $\frac{\partial V_x}{\partial x} + \frac{\partial ...
3
votes
0answers
61 views

Explicit DG time step restriction for compressible Navier-Stokes equations

Hesthaven's book 1 mentions the following time step restriction for Navier-Stokes equations (see (7.32) in 2008 edition) $$ \Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac {N^2 \mu}{h}} $$ ($...
3
votes
0answers
65 views

How to construct a Fortin Operator for Crouzeix-Raviart Element?

I want to prove the LBB condition for the Stokes Equations discretised by the Crouzeix-Raviart element. The continuous Stokes Equation in the weak formulation is Find $u \in H_0^1(\Omega, \mathbb{R}^...