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.
149
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105
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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 ...
- 229
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votes
0
answers
28
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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?
0
votes
1
answer
43
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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 ...
- 323
3
votes
1
answer
163
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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 ...
- 875
2
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1
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119
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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 ...
- 263
1
vote
0
answers
52
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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 ...
- 11
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0
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39
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What is the boundary condition of $\varepsilon $ for $k-\varepsilon $ turbulence model?
I'm working on the numerical solution of systems of equations, you can use this link to access it, the system of equations is
$\begin{aligned} \partial_t \theta+u \nabla \theta-\nabla \cdot\left(\...
- 51
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0
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91
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navier-stokes equations in semi-discrete form
Can someone point me to where the Navier-Stokes equations in 2D, both compressible and incompressible are written in semi-discrete form? I'm doing reduced order modelling and I need to write them down ...
- 668
1
vote
0
answers
160
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Partition of unity in FEM with bubble functions
In FEM with bubble functions, the field ($\boldsymbol{u}$) is approximated as a linear combination of the standard one ($\tilde{\boldsymbol{u}}$) plus the bubble field ($\boldsymbol{u}^b$). That is,
\...
- 875
1
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1
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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 ...
- 51
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0
answers
49
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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 \...
- 51
0
votes
1
answer
91
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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, ...
- 515
0
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0
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535
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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 ...
2
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0
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73
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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 ...
1
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0
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73
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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 $\...
- 21
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1
answer
174
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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. ...
- 875
1
vote
0
answers
61
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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(\...
- 55
3
votes
1
answer
113
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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}...
- 55
5
votes
1
answer
248
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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^...
- 281
1
vote
1
answer
131
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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 ...
- 281
5
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0
answers
153
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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 $A: V \rightarrow V', B:V \rightarrow Q'$ ...
- 281
2
votes
0
answers
179
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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 ...
- 281
4
votes
1
answer
266
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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 $$Z_h=\{v_h \in V_h: (\operatorname{div}(v_h),q_h)=0 \quad \forall ...
- 281
3
votes
1
answer
242
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Time discretization Navier Stokes equation
This question is a follow-up of this one.
The weak form of Navier Stokes equation is (assuming $v,q$ test functions for the velocity and the pressure, respectively)
$$(\frac{du}{dt},v)_{\Omega} + (\...
- 309
2
votes
0
answers
138
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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 ...
- 309
1
vote
2
answers
312
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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 ...
- 235
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0
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60
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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 ...
- 136
6
votes
1
answer
244
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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.
$$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$
...
- 384
3
votes
2
answers
145
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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 & ...
- 875
1
vote
0
answers
219
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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 ...
- 875
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:
$$
\frac{\partial U}{\partial t} +
\frac{\...
- 337
1
vote
2
answers
442
views
How to apply central difference to viscous fluxes in 2D Navier-Stokes equations?
I'm trying to solve 2D unsteady compressible Navier-Stokes equations with finite-difference or finite-volume method. Here is the system, it's pretty standard:
$$
\frac{\partial U}{\partial t} +
\frac{...
- 337
2
votes
2
answers
158
views
Different form of the Navier--Stokes equations
Normally I write the incompressible Newtonian isothermal flow Navier--Stokes equations as follows:
$$\displaystyle
\frac{\partial v}{\partial t}
-\nu\Delta v
+\color{red}{(\nabla v)v}
+...
- 515
3
votes
0
answers
99
views
Explicit DG time step restriction for compressible Navier-Stokes equations
Hesthaven's book 1 mentions the following time step restriction for Navier-Stokes equations (see (7.32) in 2008 edition)
$$
\Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac
{N^2 \mu}{h}}
$$
($...
- 141
2
votes
0
answers
76
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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 ...
- 337
2
votes
2
answers
164
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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 ...
- 337
2
votes
0
answers
63
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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, ...
- 31
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0
answers
132
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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}^...
- 459
1
vote
0
answers
100
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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 ...
- 11
0
votes
1
answer
180
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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 ...
5
votes
1
answer
300
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 ...
- 53
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votes
0
answers
79
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 ...
0
votes
0
answers
73
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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 ...
- 235
0
votes
1
answer
146
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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 ...
- 11
1
vote
0
answers
104
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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[\...
- 31
2
votes
1
answer
243
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-...
- 2,332
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 ...
- 161
1
vote
0
answers
26
views
Equi-order in pressure correction schme of Navier-Stokes equation
I am wondering if there is an stabilized equi-order scheme in pressure correction scheme in solving Navier-Stokes equation? Usually P2-P1 element combination is used to solve NS equation, and a PSPG ...
- 11
2
votes
0
answers
125
views
Governing equations vs Transport equations
This is a basic question. But I did not find any explanations for this. How are governing equations, like mass, momentum, energy conservations equations, different from 'Transport equation'?. Is a ...
- 225
0
votes
0
answers
95
views
Stokes Equation fails to converge for an ellipse
This might be because of the mesh, but the following code blows up for all values of b not 1. Does anybody have any experience working with the ellipse mesh in Fenics?
...
