2.9. Direct Numerical Simulation (DNS)

So, another method apart from Reynolds average Navier Stokes equation for solving the turbulence, for solving the computational fluid dynamics Navier Stokes equation is called direct numerical simulation. So, in direct numerical simulation DNS Navier Stokes equation or simulated numerically without any turbulence model.

Direct Numerical Simulation (DNS) is a computational technique used to solve the Navier-Stokes equations, which describe how fluids move. Unlike other methods that rely on turbulence models, DNS calculates the behavior of the fluid directly. This means that all scales of motion, from large to small, are resolved in the simulation.

Think of DNS as a high-definition film of fluid flow. Just as a high-definition camera captures every detail of a scene, DNS captures every detail of fluid movement without any shortcuts or approximations.

So, in this modeling Reynolds number is expressed as UL by nu, where R e represents the ratio of the inertial forces to viscous forces or Reynolds number, use the characteristic velocity L is the characteristic length or Reynolds number can also be written as square by nu divided by L by U.

The Reynolds number (Re) is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is calculated by the formula Re = (UL)/ν, where U is the characteristic velocity of the fluid, L is a characteristic length (such as diameter of a pipe), and ν is the kinematic viscosity. A high Reynolds number indicates that inertial forces dominate over viscous forces, which is typical in turbulent flow.

Imagine driving a car through a light mist versus a heavy rainstorm. In the mist (low Reynolds number), viscosity (the thick, sticky feel of the water) is noticeable. In the storm (high Reynolds number), the dominance of the car's speed (inertial forces) greatly outweighs the effects of the sticky rain.

The magnitude of inertial terms are much higher than viscous term in high Reynolds number flows that is true, because it is the ratio of viscous diffusion because the ratio of the inertial forces to viscous forces.

In high Reynolds number flows, the inertial forces, which are related to the motion of the fluid, become much stronger than the viscous forces, which resist motion. This indicates that in turbulent flows, fluid motion is primarily driven by its inertia rather than the friction caused by viscosity.

Envision a fast-moving river. The powerful current (inertia) swiftly carries leaves and small debris downstream, while the friction from the riverbed (viscous forces) has a significantly lesser impact on their movement compared to the force of the water's motion.

To sustain the fluctuations the supply of kinetic energy must be balanced by these dissipation of turbulent energy.

In a turbulent flow, the continuous fluctuations in velocity require a constant supply of energy. This energy comes from the system, and to maintain flow stability, it must be dissipated, mostly as heat through viscous effects. In other words, the energy needed to keep the turbulent flow active must be equal to the energy lost due to friction and dissipation.

Imagine blowing air into a balloon. The faster you blow, the more energy you put into the balloon. If there are tiny holes in the balloon (like dissipation), the air bounces around inside, but eventually, without enough energy supplied, the balloon will deflate. Maintaining equilibrium between energy in and energy loss is crucial.

So, from the above consideration what we get is all definitely outside the scope the derivation of this that L by eta is of the order of Reynolds number to the power 3 by 4 this eta is a Kolmogorov length scale.

The Kolmogorov length scale (η) is a small length scale in turbulent flows where the kinetic energy is dissipated due to viscosity. The relationship L/η being proportional to Re^{3/4} indicates that as the turbulent intensity (Re) increases, the distance over which energy dissipates decreases significantly.

Consider a rough ocean wave. The larger waves (characteristic length L) break down into smaller ripples, occurring at much smaller scales (Kolmogorov scale, η) where the energy is finally dissipated as heat. The relationship shows that in very turbulent waters, these smaller dissipative scales become quite crucial.

The implication of this is that for 3 dimensional simulation of turbulent flow we will require at least L by eta to the power 3 that means, Reynolds number 2 the power 9 by 4 grid points.

In 3D simulations utilizing DNS, the total number of required grid points increases significantly, particularly as the Reynolds number increases. The grid points needed scale with Re^{9/4}, indicating that high Reynolds numbers create a computational demand for very fine grids to capture smaller turbulent structures effectively.

Imagine trying to capture a highly detailed sculpture. If you're using a low-resolution camera, you might miss fine details. But if you use a high-resolution camera that captures every bit, it increases the number of pixels needed to form the image. Similarly, in DNS, to accurately simulate turbulent flow, an enormous number of computing resources are necessary.

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