Dear Sirs,

Could someone please advise me how to constructe a random divergence-free velocity noise for the initial turbulent velocity field ?

Thx in advance.

Tony

Hi, Tony.

I am not sure what is usually done, but I would first try a two-streamfunction approach.

Let

u=dPsi/dy

v=-dPsi/dx+dPhi/dz

w=-dPhi/dy

in Cartesian coordinates. Then (u,v,w) is divergence-free (check youself, do not trust me!), and Psi(x,y,z) and Phi(x,y,z) are two streamfunctions, which can be any random functions, produced with a random-number generator in any form. Sure, you still have to define what probability density distribution you wish to have.

Hope this helps.

Sergei

Hi there,

In addition to generating random fluctuations of the velocity field using the stream function, you might want to have a given initial spectral energy distribution.

Turbulent flow have a spectral energy distribution with different slopes. For example 3D homogeneous turbulence have a Kolmogorov spectrum, etc... So it is wise to start with a given slope of the energy spectrum and check how it evolves.

The energy is E = 1/2 v**2

and its Fourier transform (in one dimension) is given by the notation E(k)

For example you might want to start with

E(k) proportional to k/(1+k/k0)**4 , where k0 is a parameter to be adjusted, etc.. All this depends also if the flow is 2D or 3D.

There are more general forms for different initial conditions. In some cases the results might depend on the initial condition.

See e.g.

Cho and Polvani, 1996, Physics of Fluids, vol8, no.6, 1531.

McWilliams, J. Fluid Mech., 146, 21 (1984) McWilliams, J. Fluid Mech., 219, 361 (1990).

I hope this helps as a start.

PG.

Hi Sergei...

you mentioned random streamfunction... right!! are they smooth enough to be differentiable? ( I mean if the stream function is random in space it's derivatives are going to be very large, am i right?)

if one is really not concerned about the energy spectrum as mentioned by patrick and only cares about divergence free condition then a simple control volume based mass conservation can tell you what the third component of velocity should be if you have determined first two components by totally arbitrary random fields (i think you can work that out!!)

But the point mentioned by patrick is really important, the real fields do have some form of energy spectrum... One can prescribe the absolute energy spectrum (1-D or 3-D) reflecting the amount of energy in variuos wave numbers. Then multiply each amplitude by random phase. Then extend the signal using its conjugate in negative wave numbers (note that applying FFT or inverse FFT to such a signal should yield a real signal). again construct two components of velocity field by this procedure and the third component by simple CV based mass balance. (I think have assumed your familiarity with spectrum manipulations and signal construction... anyway there is huge literature on that topic)

regards Mayank Tyagi

Hi, Mayank,

>I mean if the stream function is random in space it's

>derivatives are going to be very large, am i right?

Well, random in space is not so simple a notion, really. If you have a velocity distibution which is random in space and if you integrate it to obtain the streamfunctions, will the streamfunctions be random in space (the answer is yes)?

To be on a safe side, one has to use an ensemble (spelling?) approach: you have a pool of functions, all smooth, and choose them at random to obtain an object which can be truly considered as a random function.... It is a big topic. And, sure, energy spectrum is important, as well as other characteristics, say, correlation length etc. A complete description of a random function is it's probability density functional. But, really, you can use any sufficiently complicated initial conditions, and then in a short time DNS code will select the proper things itself, I believe.

Why not to go to ERCOFTAG database site and take the distribution provided by other people's DNS (Kim at el.) Then it will be OK with all those characteristics, I think.

Sergei

The initial energy spectrum has usually a given slope in the longest wavelengths (largest scale) and a sharper slope in the shortest wavelengths (small scale). As a result of the hyperviscosity, usually the shortest scales are quickly damped and the slope there is even futher increased. This makes the function (velocities) actually continuous, because if a function has a sharp jump or transition region, then its Fourier transform (or any other decomposition of the function in orthogonal polynomial) does not converge. The coefficients of the FFT a(k) decreases only as approximatively 1/k instead of a higher power. Due to the viscosity, in the smallest scales, a(k) decreases much faster, like 1/k**n, where n=3,4, and more. So the derivatives should also have roughly the same behaviour, and discountinuous derivatives should not be a concern.

Patrick

Hi ..

Refering to Sergei's explaination of random stream function, i still feel uncomfortable. One can perform INTEGRATION operation on random velocity fields to obtain a "random" stream function (I have no problem with that, in fact integration is less restrictive, all it needs is some "bound" on the integrand, which one will have on such estimated random velocity fields). But going other way, to obtain velocity field from stream function through DIFFERENTIATION is a severe operation.

Again, notion of ensemble of stream functions seems to be close to finding distributional derivative of such function. I would request Sergei to elaborate a little on how to implement such an alogrithm. I think it may not be a feasible approach.

Remark on energy spectrum by Patrick were good. It boils down to the fact that governing equations take care of "discontinous derivatives" through viscosity. This evolution of random initial conditions subjected to specific boundary conditions can be easily verified. Try Box-Muller algorithm for gaussian deviates to generate normal distribution for first two velocity components and solve for third using mass conservation.

However, the question about generating realistic initial condition is bypassed in our discussion (Sergei has mentioned a practical way... use DNS solution available closest to your simulation conditions ... unfortunately that's not always possible !!). Hopefully, somebody would like to suggest another way to incorporate physical information in random fields as much as possible without actually performing a DNS first.

regards

Mayank

>Again, notion of ensemble of stream functions seems to be close to finding distributional derivative of such function.

Well, I do not know what a distributional derivative is, sorry. What I implied that a single function, say u(x,y,z) cannot, or is difficult to, be considered as random in a rigorous mathematical way. You just need an ensemble to be able to talk about randomness. You, obviously, do not distinguish between non-differentiable and random functions. I do. Now, turbulent flow fields can be considered as random in time and space using special approaches like discrete sampling and ideas from the theory of dynamical systems etc. (this is outside the scope here), but they are certainly differentiatable! Yes, the space behaviour of a velocity field can be irregular on large scales, but at the viscosity length scale they are smooth. And successful DNS is supposed to resolve this scale. So, there are no problem with differentiatio, really.

>I would request Sergei to elaborate a little on how to implement such an alogrithm.

Take psi as a truncated Fourier series with coefficients denerated with a random number generator. This will be a random function and nicely differentiatable in space, even explicitly.

Have to run. Good luck again. Sergei

(1). When we take hotwire measurement of a turbulent flow field, the signal is continuous and irregular ( or random). (2). With the LDA measurement of the same turbulent flow field, the signal is digitally random. (3). In both cases, there is no continuity of the flow information available. In other word, we can't say that the hotwire, or the LDA data are divergence free velocity, even though these are definitely random ( either continuous or digital ). (4). So, that is our understanding of the real world. My feeling is that there is no need to impose the continuity equation or the momentum equation in the initial velocity noise, unless one can verify that the hotwire, or LDA signals are divergence-free. (5). It is almost impossible to impose the continuity and momentum equations on the random velocity noise. It is hard to verify it.

Dear John,

I posted this question since I read the journal paper of L. Shen et. al on the DNS of free-surface turbulent flow (JFM 386:167-212).

In their simulation, the initial turbulence field is implemented by the superposition of divergence-free random velocity noise upon the mean flow. I was wondering how they did that.

Thx.

Tony

(1). Thank you for the information about the paper. (2). Assuming that the random velocity noise is divergence-free, there must be a unique way to verify it. This depends on how the subsequent code handle the continuity equation. (3). I mean that the divergence-free must be at the finite-difference, or finite-volume level to be consistent with the subsequent formulation. (4). So, the question you have really is related to whether the divergence-free constraint was implemented at the finite-difference level, or at the finite-volume level, or at the finite-element level, or at the analytical level. (5). At this point, I don't think it is important whether the random initial velocity field is under specific constraints or not. When you use it in the solver ( continuity, momentum, and energy equations), it will have to re-adjust itself. At that point, the physics of the flow will start taking shape. (6). Maybe I am wrong, then there must be a universal method to specify the divergence-free random initial velocity field. This is because divergence-free is universal, and random is also universal. (7). So, if you feed a divergence-free random initial velocity field into the INVISCID solver, the solution will be unique to this initial flow field. But if you feed the same initial velocity field into the Navier-Stokes solver, the situation will be different. And we hope that the final solution will approach the real flow situation. If it still remember the original flow field, then each simulation will be unique and different. And the simulation of the real turbulent flow will fail. (8). I think, in my mind, the turbulent flow is the one which is not sensitive to the initial random field. In other word, the turbulent flow should be insensitive to a random input at any time. ( it should be sensitive to input at specific frequency) (9). I think, the answer to the question is inside the paper you cited. (if the divergence-free was implemented at the analytical level and the solver was not analytical, then you know that the divergence-free itself is not consistent with his formulation.)