Spectral solvers with Variable Robin Boundary Conditions.

HMM

Hmm in this case gives us a lot of control. As the number of micro domains tends to infinity, and their sizes tends to zero, we will converge very close to the true solution (depending on how accurate our boundary extrapolation is). Where do we then choose the cutoff?

Wall models for CFD

  • Wall $f\colon [0,1] \to \mathbb{R}$ is viewed as a function of a random variable $Y = f(X)$, where $X\sim U[0,1]$. Moments of $f$ will hence be of the form \(\mathbb{E}[\phi(Y)] = \int_0^1 \phi(f(x))\mathrm{d}x\) These are interpreted as shape parameters. For example, the moments $\mathbb{E}[|Y|^m]$ are often used. A nice interpretation is that the integer moments are adding information about the measure $\mu_Y = \mu_x \circ f^{-1}$, since \(\hat p_Y(\omega) = \sum_{n=0}^{\infty} \frac{\mathbb{E}[Y^n]\omega^n}{n!}.\) However, it does not provide information about the regularity of $f$, such as it’s slope characteristics (for example, changing the period does not affect the moments). For such information, one can instead look at the moments of $f’$. Wiki article on surface roughness. We can likely choose a scale below which we interpret as roughness. How do we choose such a scale? Density parameters say little to nothing about the frequency.
  • Reynolds number Re is $\frac{U\cdot L}{\nu}$, where $U$ is characteristic velocity, $L$ is characteristic scale, and $\nu$ is the kinematic viscocity. We see that $Re$ increases if $U$ increases, but also if $L$ increases. For example, flow around a large cylinder has to have lower rate to be laminar than flow around a tiny grain of dust. Wetted surface is the part that is in contact with the flow.
  • Multi Scale Coupling for Rough Reactive Boundary. Calculate an effective wall height where they then put no-slip conditions. This is an alternative way. Note that the computational domain will change.
  • Wall bounded turbulence. Prandtl was first to suggest that the viscous term must persist at the boundary, otherwise no-slip cannot be satisfied. Eulerian flow can satisfy no-penetration, but will necessitate slippage at the boundary.
  • Navier slip condition states that the tangential velocity is proportional to the normal derivative of the velocity.The constant of proportionality is called the “slip length”, we call it $\alpha$. Holds for laminar incompressible flows (Re < 1). This typically holds when the roughness does not protrude into the viscous or turbulent sublayer. Instead, we need to assume that stationary stokes flow holds.
  • Nice blog explaining how wall models are used. For finite volume methods, estimate the flow quantities in a cell at the boundary by extrapolating using known information about the shear stress at the boundary, and finite volume averages.
  • Review on turbulent flow over rough surfaces Wall modeling using equivalent grain roughness. Obtained from the distribution of the wall function using different shape related quantities like variance, skewness, kurtosis. GPR has been used to obtain estimates of grain roughness with high accuracy (<10%) for CFD.
    • Viscous (sub)layer = laminar flow
    • Buffer layer (neither applies)
    • Log layer = log law
    • Typically cells at the log layer scale?
  • Navier Stokes rough surface book (chapter 12)

Multi Scale problems

Below is a list of useful literature on the subject, along with comments.

Literature

  • Tutorials in multi scale problems by Runborg et al. Some tutorials on Multi Scale Simulations (FMM, Wavelet analysis.) The one on Wavelet analysis is interesting (Runborg.)

  • Introductory book
    Homogenization recap in Appendix B: Typically a problem of the form \(\nabla \cdot (a(x/\epsilon) \nabla u(x)) + a_0(x/\epsilon) u(x) = f(x), \implies L_\epsilon u = f\) where $a$ and $a_0$ are 1-periodic functions. Goal is to study the limit as $\epsilon \to 0$. Homogenisation theory is not what we are interested in. Instead, we want to minimize error of a low resolution approximation, at a nonzero $\epsilon$. Of course we still want convergence to homogenized solution as $\epsilon \to 0$. The 1-d homogenization consists of taking the harmonic average.

    The standard assumption seems to be periodic multi scale:
    \(u(x) = \tilde{u}(x) + \phi(x/\epsilon),\) where $\tilde{u}$ and $\phi$ are periodic, and $\tilde{u}$ is band limited to some large scale $»1/\epsilon$.