direct numerical simulations

direct numerical simulation: direct solution of the Navier-Stokes equations without turbulence modeling (in the turbulent flow regime). Typically, (almost) all spatial and temporal scales are resolved

reasons for performing DNS

• analysis of otherwise inaccessible data
• physical insights into unknown flows
• guidance for turbulence modeling
• validation of turbulence models
• test and validation of experimental techniques

challenges of DNS

• high computational cost
• limited $Re$ and physical modeling
• stability issues
• challenging data management and analysis
• highly specialized (unique) tool chain

numerical techniques

DNS flow solvers are highly specialized and often unique.

considerations

• accuracy (convergence order)
• stability
• complexity
  • implementation
  • setup (geometry, BC, ...)
• scalability (parallelization)
• ...

finite difference method: approximation of partial derivatives based on Taylor series

• higher-order possible
• no inherent conservation/boundedness
• very easy to implement
• limited to simple geometries
• moderate scaling

$\rightarrow$ sometimes used for DNS

finite volume method: divergence theorem and Taylor series approximations

• limited to first/second order
• inherently conservative, often bounded
• relatively easy to implement
• complex geometries (meshing moderate)
• moderate scaling

$\rightarrow$ rarely used for DNS

spectral method: global basis basis functions substituted in PDE; optimization of coefficients

• higher-order accuracy
• no inherent conservation/boundedness
• moderate complexity
• very limited geometry/Bcs
• good scalability

$\rightarrow$ initial go-to method for DNS

spectral element method: similar to spectral method but with local basis functions

• higher-order accuracy
• no inherent conservation/boundedness
• high complexity
• moderately complex geometries
• good scalability

$\rightarrow$ reasonable choice for DNS

discontinuous Galerkin method: local, non-continuous basis functions

• higher-order accuracy
• conservative, shock handling
• high complexity (inter-element coupling)
• moderately complex geometries
• good scalability

$\rightarrow$ reasonable choice for DNS

resolution requirements

strategy

1. determine size of smallest scales $\eta$, $u_\eta$, $t_\eta$
2. estimate largest scales $l_0$, $u_0$, $t_0$
3. infer resolution requirements

Kolmogorov hypotheses (assuming large $Re$)

1. local isotropy: small scale motions $l\ll l_0$ are statistically isotropic
2. dynamic equilibrium: statistics of small scale motion determined by viscosity $\nu$ and dissipation of turbulent energy $\varepsilon$
3. inertial subrange: motion of scales $\eta \ll l \ll l_0$ determined only by $\varepsilon$

Kolmogorov scales

1. $\eta = (\nu^3/\varepsilon)^{1/4}$
2. $u_\eta = (\varepsilon \nu)^{1/4}$
3. $t_\eta = (\nu/\varepsilon)^{1/2}$

only based on dimensionality analysis

estimate for dissipation

$$ \varepsilon \approx \frac{u_0^3}{l_0} $$

ratio of larges to smallest scales

1. $\eta/l_0 \approx Re^{-3/4}$
2. $u_\eta / u_0 \approx Re^{-1/4}$
3. $t_\eta / t_0 \approx Re^{-1/2}$

very rough estimate

estimate for computational cost

1. resolve smallest scales (Kolmogorov)
2. at least one duration of largest time scale

doubling $Re$ yields roughly $8\times$ comp. cost

Decaying homogeneous isotropic turbulence in a box; $\overline{u}_i = 0$, $\overline{u_x^\prime u_x^\prime} = \overline{u_y^\prime u_y^\prime} = \overline{u_z^\prime u_z^\prime}$, fluctuations uncorrelated.

How to verify Kolmogorov's hypotheses experimentally?

$\rightarrow$ Taylor hypothesis $\partial_t(\dots) = -\overline{u}\partial_x(\dots)$

summary

• DNS is essential for guiding and validating turbulence modeling
• DNS solvers are often highly customized and have a high convergence order
• based on Kolmogorov's hypotheses, we can determine the ratio between the largest and the smallest scales of space, velocity, and time
• Kolmogorov's scales provide a guidance for the spatial and temporal resolution of a DNS
• Taylor's hypothesis connects fluctuations in space to fluctuations in time (frozen turbulence)