Direct Numerical Simulation

Direct Numerical Simulation (DNS) solves the time dependent Navier-Stokes equations, resolving from the largest length scale of a computational domain size to the smallest length scale of turbulence eddy (Kolmogorov length scale).

Considering a vast range of length scales within the computational domain, you can rightly claim that huge computer resource would be required for DNS. Examining the computer resource requirements for DNS will support this claim.

For a computational square box domain, the number of grid points (

n

) in one direction depends on the grid spacing

Δ L

. (1)

n = \frac{L}{Δ L}

In order to resolve the Kolmogorov length scale (

n

), the grid spacing

Δ L

and the Kolmogorov length scale should have the same order of magnitude. (2)

n ~ \frac{L}{η}

Since the Kolmogorov length scale is a function of the fluid kinematic viscosity (

ν

) and the turbulent dissipation rate (

(ε) η = {(\frac{ν^{3}}{ε})}^{1 / 4}

), the number of grid points in one direction can be estimated from (3)

n ~ L \frac{ε^{1 / 4}}{ν^{3 / 4}}

Utilizing

ε ~ \frac{U^{3}}{L}

the equation for the grid size can be rewritten as (4)

n ~ {(\frac{U L}{ν})}^{3 / 4} ~ {Re}^{3 / 4}

where $U$ is the characteristic velocity.

For three-dimensional flows, the total grid size for DNS can be computed as (5)

N = n^{3} ~ {Re}^{9 / 4}

For a flat plate, turbulent flow occurs when the Re > 500,000 (Schlichting and Gersten, 2000). The above relation shows the estimation of six trillion nodes, which easily exceeds the capacity of even the most advanced high performance computers.

In addition to the demanding grid size requirements of DNS, the computing time step size (

Δ t

) should be small enough to resolve the Kolmogorov time scales. Since the Kolmogorov time scale depends on the Kolmogorov length scale and the characteristic velocity (

U

), the time step size can be approximated as (6)

Δ t ~ \frac{η}{U}

Given this, the number of time steps (

n_{t}

) can be estimated from the total duration of the simulation (

T

) and the time step size (

Δ t

)(7)

Δ N_{t} = \frac{T}{Δ t} ~ \frac{T}{η / U}

Utilizing

η = {Re}^{- 3 / 4} L

the equation of the number of the time steps can be estimated as (8)

N_{t} ~ \frac{T}{L / U} {Re}^{3 / 4} \propto {Re}^{3 / 4}

Finally, the number of floating point operations needed to perform DNS can be computed from the multiplication of total grid size (

N

) and the number of the time steps (9)

N_{f} = N N_{t} \propto {Re}^{3}

Therefore, the computational cost for DNS is very expensive, confirming that DNS is not feasible for high Reynolds Number turbulent flows.

In addition, high-order (third-order or higher) numerical schemes are commonly used in order to reduce the numerical dissipation and to keep the problem size tractable. These include spectral methods or spectral element methods. Although these methods are very efficient in resolving the small scales of turbulence, they require that the computational domain be relatively simple. The direct result is that these high-order schemes have little flexibility in dealing with complex industrial geometries because of the structured (blocked) mesh approach. Finally, DNS requires special treatments for realistic initial and boundary conditions

Considering these observations it can be concluded that DNS attempting to resolve all turbulent length and time scales is restricted to the low Reynolds number range and is impractical for industrial flows due to huge computing resource requirements. Most DNS applications are served as benchmark databases for tuning turbulence models and have been used for fundamental turbulent flow studies, including homogeneous turbulent flows with mean strain, free shear layers, fully developed channel flows, jets and so on.