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Direct numerical simulation
 
 
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A direct numerical simulation (DNS) is a simulation
Simulation

Simulation is the imitation of some real thing, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviors of a selected physical or abstract system....
 in computational fluid dynamics
Computational fluid dynamics

Computational fluid dynamics is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows....
 in which the Navier-Stokes equations
Navier-Stokes equations

The Navier?Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances, that is substances which can flow....
 are numerically solved without any turbulence
Turbulence

In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time....
 model. This means that the whole range of spatial and temporal
Time

Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
 scales of the turbulence must be resolved. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (Kolmogorov microscales
Kolmogorov microscales

Andrey Kolmogorov microscales are the smallest scale s in Turbulence. They are defined bywhere is the average rate of energy dissipation per unit mass, and is the kinematic viscosity of the fluid....
), up to the integral scale L, associated with the motions containing most of the kinetic energy.






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A direct numerical simulation (DNS) is a simulation
Simulation

Simulation is the imitation of some real thing, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviors of a selected physical or abstract system....
 in computational fluid dynamics
Computational fluid dynamics

Computational fluid dynamics is one of the branches of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows....
 in which the Navier-Stokes equations
Navier-Stokes equations

The Navier?Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances, that is substances which can flow....
 are numerically solved without any turbulence
Turbulence

In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time....
 model. This means that the whole range of spatial and temporal
Time

Time is a component of the measurement used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects....
 scales of the turbulence must be resolved. All the spatial scales of the turbulence must be resolved in the computational mesh, from the smallest dissipative scales (Kolmogorov microscales
Kolmogorov microscales

Andrey Kolmogorov microscales are the smallest scale s in Turbulence. They are defined bywhere is the average rate of energy dissipation per unit mass, and is the kinematic viscosity of the fluid....
), up to the integral scale L, associated with the motions containing most of the kinetic energy. The Kolmogorov scale,, is given by where ? is the kinematic viscosity
Viscosity

Viscosity is a measure of the Drag of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness"....
 and e is the rate of kinetic energy
Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
 dissipation. On the other hand, the integral scale depends usually on the spatial scale of the boundary conditions.

To satisfy these resolution requirements, the number N of points along a given mesh direction with increments h, must be , so that the integral scale is contained within the computational domain, and also , so that the Kolmogorov scale can be resolved.

Since , where u is the root mean square
Root mean square

In mathematics, the root mean square , also known as the quadratic mean, is a statistics measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids....
 (RMS) of the velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
, the previous relations imply that a three-dimensional DNS requires a number of mesh points satisfying where Re is the turbulent Reynolds number
Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
: .

Hence, the memory storage requirement in a DNS grows very fast with the Reynolds number. In addition, given the very large memory necessary, the integration of the solution in time must be done by an explicit method. This means that in order to be accurate, the integration must be done with a time step, ?t, small enough such that a fluid particle moves only a fraction of the mesh spacing h in each step. That is, (C is here the Courant number). The total time interval simulated is generally proportional to the turbulence time scale given by .

Combining these relations, and the fact that h must be of the order of , the number of time-integration steps must be proportional to . By other hand, from the definitions for Re, ? and L given above, it follows that , and consequently, the number of time steps grows also as a power law of the Reynolds number.

One can estimate that the number of floating-point operations required to complete the simulation is proportional to the number of mesh points and the number of time steps, and in conclusion, the number of operations grows as Re3.

Therefore, the computational cost of DNS is very high, even at low Reynolds numbers. For the Reynolds numbers encountered in most industrial applications, the computational resources required by a DNS would exceed the capacity of the most powerful computers currently available
Supercomputer

A supercomputer is a computer that is at the frontline of current processing capacity, particularly speed of calculation. Supercomputers introduced in the 1960s were designed primarily by Seymour Cray at Control Data Corporation , and led the market into the 1970s until Cray left to form his own company, Cray Research....
. However, direct numerical simulation is a useful tool in fundamental research in turbulence. Using DNS it is possible to perform "numerical experiments", and extract from them information difficult or impossible to obtain in the laboratory, allowing a better understanding of the physics of turbulence. Also, direct numerical simulations are useful in the development of turbulence models for practical applications, such as sub-grid scale models for Large eddy simulation
Large eddy simulation

Large eddy simulation is a numerical technique used to solve the partial differential equations governing turbulence fluid dynamics.It was formulated in the late 1960s and became popular in the later years.It was first used by Joseph Smagorinsky to simulate atmospheric air currents, so its primary use at that time was for meteorological calc...
 (LES) and models for methods that solve the Reynolds-averaged Navier-Stokes equations
Reynolds-averaged Navier-Stokes equations

The Reynolds-averaged Navier?Stokes equations are time-averagedequations of motion for fluid flow. They are primarily used while dealing with turbulent flows....
 (RANS). This is done by means of "a priori" tests, in which the input data for the model is taken from a DNS simulation, or by "a posteriori" tests, in which the results produced by the model are compared with those obtained by DNS. The biggest DNS in the world, up to this date, used 40963 mesh points. It was carried out in the Japanese Earth Simulator
Earth Simulator

The Earth Simulator was the fastest supercomputer in the world from 2002 to 2004. The system was developed for Japan Aerospace Exploration Agency, Japan Atomic Energy Research Institute, and Japan Marine Science and Technology Center in 1997 for running global climate models to evaluate the effects of global warming and problems in solid ear...
 supercomputer in 2002.

See also

  • Large eddy simulation
    Large eddy simulation

    Large eddy simulation is a numerical technique used to solve the partial differential equations governing turbulence fluid dynamics.It was formulated in the late 1960s and became popular in the later years.It was first used by Joseph Smagorinsky to simulate atmospheric air currents, so its primary use at that time was for meteorological calc...
  • Reynolds-averaged Navier-Stokes equations
    Reynolds-averaged Navier-Stokes equations

    The Reynolds-averaged Navier?Stokes equations are time-averagedequations of motion for fluid flow. They are primarily used while dealing with turbulent flows....


External links

  • at CFD-Wiki