CFD « Mesh Mess

Guidelines for CFD papers

Posted in Engineering, Flows, Numerical by Yi Zhang on 02/02/2011

An interesting editorial from Journal of Fluids Engineering by Malcolm J. Andrews, essentially about is worth of showing to people in numerical modeling of flows, contains some guidelines for submitting numerical results to JFE, such as:

Providing complex figures may be nice for a presentation to an audience or sponsor, but are often not scientific or quantitative unless great care is taken in their presentation/discussion or they illustrate a novel aspect of the work, and rarely provide much detailed insight into the problem under consideration. The usual x-y plots are often more illuminating, but also require the author to spend more time thinking about what is important and why which helps make the article more archival.

Simply reporting one parameter when, in fact, there are multiple parameters that self-interact suggests that the author does not understand the diagnostic, or its proper use, and also the basic elements of the flow itself e.g., simply reporting pressure, and not associated velocity fields, might indicate a lack of basic understanding. The article should include a detailed description of the results, their consequences, and their importance i.e., simply stating values or shapes does not warrant archival.

Nondimensional parameters serve not only to collapse data but they demonstrate an understanding of the basic parameters that control the processes of interest and form the basis of generality that can underlie resulting archival value formulas. Not expressing results in nondimensional form substantially weakens the archival value, suggests that the author does not understand the fundamental flow physics, and also suggests that the results have no generality or archival value.

It is crucial to provide the applicable parameter ranges for the commercial software or diagnostic and ensure that they are met in the current application e.g., this might mean answering the question about an appropriate use of a turbulence model, the Reynolds number range of the experiment, or the Stokes relaxation time of particle in the flow relative to the time scale of interest in the flow.

Tagged with: CFD, research

leave a comment

Log law at a smooth plate

Posted in FEM, Flows by Yi Zhang on 07/22/2010

Log-law is THE most important and elegant analytical result in turbulence, considering many others are dauntingly complicated, though justified. What log law says is the fact that for a turbulent flow over flat plate, there is this section of flow away from the wall, within the turbulent boundary layer, in which the relation between the distance and mean flow velocity is simply of log function. This conclusion could be drawn by an asymptotic analysis matching outer flow and inner flow. Any CFD trying tempted to address turbulence models should recover the “law of the wall“, among which log law is a part of, before moving forward to more advanced problems.

The delicacy of the modeling in this case is the boundary condition (shocking!) at the wall, because log law is only valid when y+ is approximately above 30 ( the other cap depends on Reynolds number ), and the boundary conditions there for unknown variables in turbulence models are generally not known a priori, where is all kinds of wall function come into play.

Channel Flow

Plot of CFD result against log law

Tagged with: CFD, FEM

leave a comment

Across an interface

Posted in Flows by Yi Zhang on 03/09/2010

I was reading this interesting paper by Idelsohn et al when I noticed that for the Navier-Stokes equation the velocity boundary condition in normal and tangential direction have different physical meanings. We know that the NSE requires complete (against normal) velocity profile to determine the problem. Through a point in incompressible flow domain, we draw an arbitrary straight (to eliminate tension’s effect) line as interface separating two sides of the domain. In order to determine the problem on side A, velocity at interface should be applied as boundary condition. Now, normal component of velocity at that point should be continuous across the interface (since it’s arbitrary instead of physical), and it corresponds to the incompressibility condition: what goes in should equal to what goes out. It is easy to see when we imagine a slice of spacial volume (against material volume) along the interface, and its thickness is much less than its span. The match of normal velocity indicates the conservation of mass within that volume.

On the other hand, the match of tangential component of the velocity means something else. Imagine that they does not match across the interface, what would happen? Well, the gradient of tangential velocity would be infinite (remember that interface has no thickness, so the dx in the derivative goes to zero), and the shear stress in that direction would be infinite (Newtonian fluid), and apparently it’s physically impossible. This is like a boundary layer with zero thickness. For a boundary layer, viscous effect dominates and causes relatively large viscous force. In our imagined case this means the infinite viscous force and unbalance the momentum conservation.

So here are two conservation laws hidden in one boundary condition. Mathematically it suffices to see this when we degenerate from NSE to Euler equation. By losing one order of the PDE, the equation can not meet both normal and tangential boundary condition, and this corresponds to the slip condition when only normal velocity is prescribed (usually zero) out side the boundary layer.

Tagged with: CFD, coupling, FSI

2 comments

Being irrotational and viscous

Posted in Flows by Yi Zhang on 01/03/2010

I have been working on something involving both potential flows and turbulent flows, and this is how I found fascinating work by Daniel D. Joseph. In one of his papers, Potential flow of viscous fluids: Historical notes, one can get a peek of the missing link in our textbooks. I have also learned a lot in his web page about the same topic. His other two books I can put my hands on, Stability of Fluid Motions, Elementary Stability and Bifurcation Theory, are the standard textbooks for those topics as well. Shame on me, even though I read those two books, I am not aware of this potential viscous flow, making me wonder how fast I am drifting away from theoretical fluid dynamics with only the computing stuff in my mind.

Tagged with: CFD, free surface flow, FSI

leave a comment

Saddle point problem in Navier-Stokes equation

Posted in Flows by Yi Zhang on 12/21/2009

It is well known that the the optimization problem

$\min J(x)= \frac{1}{2}(Ax, x)-(f, x)$

is equal to solving a linear operator equation

$Ax=f$

provided that $A$ is symmetric and positive definite. Here $x\in X$ which is a Hilbert space, and $f\in X'$ while $A$ is a bilinear operator.

This is a very general problem considered widely in many areas. However, in the real world, some further condition are to be applied on the space where $x$ lies on. If this is described as constraint equation

$Bx=g$

then we are dealing with constraint optimization problem ,to which Lagrange multiplier is usually the first one can come up. We then try to solve the stationary point of Lagrangian

$L(x,\lambda)=\frac{1}{2}(Ax, x)-(f, x)+\lambda (Bx-g)$

Now supposing we set $X=R^n$ , using stationary point condition on above equation, we arrive at

$\left[\begin{array}{cc} A & B^T\\ B & O \end{array}\right]\left[\begin{array}{c} x\\ y\end{array}\right]=\left[\begin{array}{c} f\\ g\end{array}\right]$

where $\lambda$ is replaced by $y$ indicating the parallel unknown as $x$ in the problem. The stationary point nature of this solution gives the name of saddle point problem, whose general case is

$L(x,v)\le L(x,y) \le L(u,y), \forall u,v \Leftrightarrow \min_u\max_v L(u,v)=L(x,y)=\max_v\min_u L(u,v)$

So, putting a constraint to a minimization problem, one try to solve a saddle problem instead of a simple linear system with SPD matrix.

For a linear system, one can always decompose it in to the form of

$\left[\begin{array}{cc} A & B_1^T\\ B_2 & -C \end{array}\right]\left[\begin{array}{c} x\\ y\end{array}\right]=\left[\begin{array}{c} f\\ g\end{array}\right]$

and this is the most general form of saddle problem.

For incompressible Navier-Stokes equation, we have (without mentioning the boundary conditions)

$\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u}-\nu\Delta \mathbf{u}+\nabla p/\rho=\mathbf{f}$

$\nabla\cdot\mathbf{u}=0$

Following $\theta$ -scheme by Glowinski, one has the time discretized plan as:

$(\mathbf{u^{n+\theta}}-\mathbf{u}^n)/(\theta\Delta t)-\alpha\nu\Delta\mathbf{u}^{n+\theta}+\nabla p^{n+\theta}=\mathbf{f}^{n+\theta}+\beta\nu\Delta \mathbf{u}^n-(\mathbf{u}^n\cdot\Delta)\mathbf{u}^n$

$\nabla\cdot\mathbf{u}^{n+\theta}=0$

then

$(\mathbf{u^{n+1-\theta}}-\mathbf{u}^{n+\theta})/((1-2\theta)\Delta t)-\beta\nu\Delta\mathbf{u}^{n+1-\theta}+(\mathbf{u}^{n+1-\theta}\cdot\Delta)\mathbf{u}^{n+1-\theta}=\mathbf{f}^{n+\theta}+\alpha\nu\Delta \mathbf{u}^{n+\theta}-\nabla p^{n+\theta}$

then

$(\mathbf{u}^{n+\theta}-\mathbf{u}^{n+1-\theta})/(\theta\Delta t)-\alpha\nu\Delta\mathbf{u}^{n+1}+\nabla p^{n+1}=\mathbf{f}^{n+1}+\beta\nu\Delta \mathbf{u}^{n+1-\theta}-(\mathbf{u}^{n+1-\theta}\cdot\Delta)\mathbf{u}^{n+1-\theta}$

$\nabla\cdot\mathbf{u}^{n+1}=0$

The first and last step in above scheme are to solve Stokes problem like

$-\Delta\mathbf{u}+\nabla p=\mathbf{f}$

$\nabla\cdot \mathbf{u}=0$

Multiply $\mathbf{u}$ on both sides of the first equation, by using Green’s formula, we can see it is just the saddle point problem, with $A$ as $-\Delta$ and $B$ as $\nabla$ :

$\left[\begin{array}{cc} -\Delta & \nabla\\ \nabla & O \end{array}\right]\left[\begin{array}{c} \mathbf{u}\\ p\end{array}\right]=\left[\begin{array}{c} \mathbf{f}\\ 0\end{array}\right]$