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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How to calculate the force of solid applied by fluid? Using finite difference method, DNS, staggered grid, SIMPLE algorithm, immersive boundary

Problem I am using finite difference method to solve classic problem of flow around cylinder, for validation of my group's immersive boundary method. The common way to validate numerical method is ...
CheapMeow's user avatar
2 votes
1 answer
96 views

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

Consider this Stokes equations, $$ \left\{ \begin{array}{r} - \mu \Delta \vec{u} + \nabla P = \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{array} \right. $$ Weak form I is: $$ \...
Hao's user avatar
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How to approximate the flux when using finite volumes?

How to approximate flux $F(u)\cdot n$ where $n$ denotes the unit normal outward when using finite volumes? $$\int_{\sigma} F(u) \cdot \boldsymbol{n}_{K, \sigma} \mathrm{d} \gamma(x)$$
Chems Eddine's user avatar
3 votes
1 answer
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Non-standard boundary condition for incompressible Navier Stokes

I am having difficulties applying the boundary condition $$\frac{\partial \vec{V}}{\partial t} + u\frac{\partial \vec{V}}{\partial x} = \frac{1}{\operatorname{Re}}\frac{\partial ^2 \vec{V}}{\partial y^...
2Napasa's user avatar
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Using FEM for Navier-Stokes equation

I'm typically using FEM for solid mechanics problems and when I look into performing fluid-structure interaction, I saw they use FV or SPH method for the fluid domain. I'm not an expert in fluid ...
kstn's user avatar
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How can i model upward natural convection in various angles (0 - 90)?

I am looking for a way to numerically solve the Naiver-Stokes equations for steady incompressible flow using FDM over a surface with various angles?
user16829029's user avatar
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How to address the element face adjacent to boundaries when the finite difference method and marker-and-cell scheme are used to solve the Stokes flow?

The Stokes equations are $$-\Delta \mathbf u + \nabla p = f \text{, in }\Omega,$$ and $$ -\nabla \cdot \mathbf u = g, \text{ in } \Omega$$ where $\mathbf u = \left( u, v \right)$ is the flow ...
Tingchang Yin's user avatar
3 votes
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181 views

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

My question is in the context of the finite element method. Incompressible Navier-Stokes equations can be solved using the coupled method or projection/fractional step methods. Each method has its ...
Chenna K's user avatar
  • 934
2 votes
1 answer
124 views

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

Sorry for the screenshot but I don't want to try to format this on latex: We have this annotation of the Navier-Stokes equations: I am particularly puzzled by the viscosity/stress term. For an ...
Makogan's user avatar
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How to calculate the reynolds number for airborne wind energy system?

I am asking for some help on how to calculate the reynolds number for an AWES. I know the formula of Re=rhovl/viscosity. I have selected the air foil as Clark y with chord length 3.72m (which will be ...
Yash Shah's user avatar
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navier-stokes equations in semi-discrete form

Can someone point me to where the Navier-Stokes equations in 2D, both compressible and incompressible are written in semi-discrete form? I'm doing reduced order modelling and I need to write them down ...
NNN's user avatar
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Partition of unity in FEM with bubble functions

In FEM with bubble functions, the field ($\boldsymbol{u}$) is approximated as a linear combination of the standard one ($\tilde{\boldsymbol{u}}$) plus the bubble field ($\boldsymbol{u}^b$). That is, \...
Chenna K's user avatar
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How to implement pressure stabilization in matlab while solving steady stokes equation

the equation is $$ \left\{\begin{array}{l} -\nabla \cdot \mathbb{T}(\mathbf{u}, p)=\mathbf{f} \text { in } \Omega, \\ \nabla \cdot \mathbf{u}=0 \text { in } \Omega, \\ \mathbf{u}=\mathbf{g} \text { on ...
吴yuer's user avatar
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How to specify the value of pressure in a point in solving steady stokes equation numerically?

I try to solve the steady-Stokes equation numerically on , that is \begin{aligned} -\nabla \cdot \mathbb{T}(\mathbf{u}, p) & =\mathbf{f} \quad \text { on } \Omega=[0,1]*[-0.25,0], \\ \nabla \cdot \...
吴yuer's user avatar
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Decoupling Stokes problem into two problems: velocity and pressure, using FEM

I have seen finite difference methods for fluid equations (Stokes and Navier--Stokes) that solve a pressure problem first and then a fluid problem. That is, although they solve two different problems, ...
yemino's user avatar
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The mathematical meaning of a zero gradient pressure boundary condition in the Navier-Stokes equations

I would like to solve the Navier-Stokes equations for the unsteady problem of the flow around a circular cylinder. I would like to understand how to write mathematically the boundary condition for the ...
Saddam N Y Hijazi's user avatar
2 votes
0 answers
75 views

How to assign initial velocity field and handle pressure-velocity coupling in FVM?

I am trying to solve the 2D incompressible Navier-Stokes equations for laminar flow over a backward facing step using the finite volume method. This is the plot that I generated of a generic mesh ...
4th_ord_Padme_scheme's user avatar
1 vote
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Slow convergence of Stokes solver used with the Immersed Boundary method

I am using Immersed Boundary Method to simulate elastic particles in 3D Stokes flow. Specifically, one has $\nabla ^2 \mathbf{u}-\nabla p + \mathbf{f}(t) = 0$, $\nabla \cdot \mathbf{u} \; $, where $\...
P. Trinli's user avatar
3 votes
1 answer
180 views

Discontinuous pressure elements for incompressible Navier-Stokes

I am looking for some LBB-stable velocity-pressure combinations for incompressible Navier-Stokes where the pressure space is element-wise discontinuous, preferably with a linear variation elementwise. ...
Chenna K's user avatar
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Good non oscilliatory derivatives for an exsisting grid

I'm calculating the entropy production of a shockwave by utilizing the equations: \begin{equation} \sigma = J'_q\frac{\partial}{\partial x}\left(\frac{1}{T}\right) +\frac{1}{T}\frac{4\eta}{3}\left(\...
Twm1995's user avatar
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3 votes
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Discretizing the viscous component in 1 - D Navier stokes compressive flow

I've been working on modelling the NS equations in order to simulate shock waves. The equations are set up on the form: \begin{equation} \frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x}...
Twm1995's user avatar
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5 votes
1 answer
263 views

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

Consider the Navier stokes equation after the discretization with conforming finite elements with source term $f=0$. We have the algebraic structure of a saddle point problem: $$M \dot{u} = f- Au -B^...
FEGirl's user avatar
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1 answer
139 views

Locking phenomena for $P1 - P0$ elements

Consider the Stokes problem and the usual divergence operator $B:V \rightarrow Q'$, $\langle Bv, q\rangle = b(v,q)=(\operatorname{div} v,q)$ and its discrete versione $B_h : V_h \rightarrow Q_h'$. In ...
FEGirl's user avatar
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5 votes
0 answers
159 views

About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation

I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$: $$A u + B^t p = f$$ $$Bu = g$$ where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
FEGirl's user avatar
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195 views

Understanding inf-sup conditions for classical saddle point problems

I'm studying the inf-sup conditions for saddle point problems. I'm referring to the usual one $$\begin{cases}Au + B^t p = f \\Bu=g \end{cases}$$ In the book I'm using (Ern - Guermond: Theory and ...
FEGirl's user avatar
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4 votes
1 answer
294 views

Discrete divergence free functions

I'm studying the weak formulation of NS equations. During the analysis, the book I'm using (Quarteroni-Valli, page 301-302), defined $$Z_h=\{v_h \in V_h: (\operatorname{div}(v_h),q_h)=0 \quad \forall ...
FEGirl's user avatar
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3 votes
1 answer
242 views

Time discretization Navier Stokes equation

This question is a follow-up of this one. The weak form of Navier Stokes equation is (assuming $v,q$ test functions for the velocity and the pressure, respectively) $$(\frac{du}{dt},v)_{\Omega} + (\...
Vefhug's user avatar
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2 votes
0 answers
147 views

Confusion about preconditioner for incompressible Navier-Stokes equation with implicit-explicit method

Consider the time-dependent Navier-Stokes equation $$u_t + (u \cdot \nabla) u - \Delta u + \nabla p = f$$ $$\operatorname{div}(u)=0$$ Looking at deal.ii tutorials, I've notice that there are ...
Vefhug's user avatar
  • 309
1 vote
2 answers
364 views

What is the rationale of second-order finite volume discretization?

When it comes to a second-order accurate finite volume discretization of Navier-Stokes equations, which one of the two following rationales is adopted? 1- Second-order accuracy is a direct consequence ...
Naghi's user avatar
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0 answers
61 views

Why does including the pressure in this FVM for Stokes 2nd Problem lead to wrong solutions?

I'm trying to learn how to use finite volume methods and I want to solve a more general case of Stokes' second problem i.e. an infinite half-plane oscillating harmonically with no-slip boundary ...
Wihtedeka's user avatar
  • 136
6 votes
1 answer
248 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. $$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$ ...
2Napasa's user avatar
  • 362
3 votes
2 answers
151 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 & ...
Chenna K's user avatar
  • 934
1 vote
0 answers
236 views

Lumped mass matrices for higher-order finite elements for CFD

Given that some of the mass lumping techniques, for example, row-sum lumping does not produce practically viable lumped mass matrices for all the element shapes, what are the techniques used for mass ...
Chenna K's user avatar
  • 934
2 votes
1 answer
101 views

Fix for FD WENO method for multi-component compressible flows

I'm solving two-dimensional four-component compressible Navier-Stokes equations with finite-difference WENO approach. The equations are pretty standard: $$ \frac{\partial U}{\partial t} + \frac{\...
omican's user avatar
  • 347
1 vote
2 answers
461 views

How to apply central difference to viscous fluxes in 2D Navier-Stokes equations?

I'm trying to solve 2D unsteady compressible Navier-Stokes equations with finite-difference or finite-volume method. Here is the system, it's pretty standard: $$ \frac{\partial U}{\partial t} + \frac{...
omican's user avatar
  • 347
2 votes
2 answers
161 views

Different form of the Navier--Stokes equations

Normally I write the incompressible Newtonian isothermal flow Navier--Stokes equations as follows: $$\displaystyle \frac{\partial v}{\partial t} -\nu\Delta v +\color{red}{(\nabla v)v} +...
yemino's user avatar
  • 515
3 votes
0 answers
99 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) $$ \Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac {N^2 \mu}{h}} $$ ($...
Zxcvasdf's user avatar
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2 votes
0 answers
78 views

Haw to apply central difference to viscous flux in energy equation?

In many modern papers Navier-Stokes equations are solved with finite-difference or finite-volume methods using WENO reconstruction for non-viscous fluxes and central differences for viscous ones. It ...
omican's user avatar
  • 347
2 votes
2 answers
166 views

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

I have quite a lot of experience solving unsteady Euler equations, including multi-component ones, with in house-coded finite-difference and finite-volume methods, including MacCormack and MUSCLE ...
omican's user avatar
  • 347
2 votes
0 answers
65 views

Solving Stokes Equations in 3D - Do I need to treat pressure-velocity coupling iteratively?

I need to develop a code to solve Stokes Equations in 3D in cubic geometries (structured grid, uniform mesh spacing). My code needs to take a pressure gradient in one direction as a BC (pinlet=p1, ...
Rafael March's user avatar
3 votes
0 answers
140 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}^...
Pepe's user avatar
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1 vote
0 answers
100 views

Stokes problem with imposed acceleration on boundaries (projection scheme)

I am trying to solve FSI problems with finite elements and using a projection scheme (I am taking as reference the review of Guermond: Guermond, J. L.; Minev, P.; Shen, Jie, An overview of ...
gc11's user avatar
  • 11
0 votes
1 answer
184 views

FDM on nonlinear PDEs

I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$. In order to perform time discretization with FDM (finite ...
Rubi C.g.'s user avatar
5 votes
1 answer
308 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 ...
user35908's user avatar
0 votes
0 answers
80 views

Numerical flow visualization in 2D for a moving boundary,

I have a rigid body that moves according to a set of governing ODEs, and I'd like to numerically visualize the vortices that are shed by this object. How could I proceed? I've been reading up on the ...
user35764's user avatar
0 votes
0 answers
75 views

The relation between PDE order and discretization order

In Jasak's Ph.D. thesis (2000), a notion is given about discretization of a transport equation: For good accuracy, it is necessary for the order of the discretization to be equal to or higher than the ...
Naghi's user avatar
  • 235
0 votes
1 answer
146 views

How to visualize the vorticity / flow for a rigid body moving through a fluid?

How can I write down two-dimensional Navier-Stokes equations for a simple rigid object immersed in a flow and freely falling due to gravity? I'm trying to view the vorticity that's induced by the ...
user35678's user avatar
1 vote
0 answers
106 views

Pressure boundary conditions in Stokes Equation in 2D

I am solving the steady-state incompressible Stokes equations in 2D: \begin{equation} \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0, \end{equation} \begin{equation} \mu\left[\...
Rafael March's user avatar
2 votes
1 answer
248 views

What is the correct way to calculate deviatoric stress tensor in lattice Boltzmann method?

Due to my previous question, where I asked about flux calculation in lattice Boltzmann (LB) method here, I have more or less same question for deviatoric stress tensor calculation due to pseudo-...
Mithridates the Great's user avatar
4 votes
3 answers
3k views

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

Note, there is already a question here, however the answers don't answer the original question, let alone specific considerations when dealing with nearly in-compressible fluids (water). Another ...
Krupip's user avatar
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