Direct numerical simulation

Under Direct Numerical Simulation, DNS short, refers to the computational solution of the full unsteady Navier -Stokes equations. It differs from other methods of calculation of fluid mechanics by the fact that small-scale turbulent fluctuations are numerically resolved in space and time and not represented by turbulence models.

Due to the time-varying, spatially small fluctuations in a turbulent flow unsteady-state approach for the detailed description of the physical processes is required in addition to the high spatial resolution. The DNS is thus the most accurate method to calculate currents; However, it also provides the highest demands on the numerical method as well as the available computing power. Therefore, the DNA is mainly used in basic research use. A major application area is the laminar-turbulent envelope under which refers to the transition from a smooth steady flow in the quasi- chaotic turbulent state. Furthermore, the DNA is used in the fields of separation bubbles, fully turbulent flows and aeroacoustics and also serves the development of turbulence models.

The underlying equations

Flow fields behave according to the Navier -Stokes equations. For low speeds at a Mach number less than 0.3, it is usually sufficient to the Navier -Stokes equations for incompressible fluids apply. They consist essentially of three transport equations, the momentum equations in the three spatial directions. From the mass conservation is obtained for incompressible flows the divergence freedom of speed. With increasing speed, compressibility effects play an increasingly important role, so that the Navier -Stokes equations are applicable for compressible fluids. These are also necessary if, for example, aero-acoustic phenomena to be investigated, as there is a finite speed of sound only when taking into account the compressibility. As the Navier-Stokes equations for compressible flows are continuous transport equations, there are no terms in the elliptical unlike the equations for incompressible fluids. Depending on the application, additional equations can be included. Thus, chemical reactions such as dissociation of molecules or ionization of atoms play about hypersonic at high Mach number a role.

Physical processes

Since in a DNS the flow field to be calculated without turbulence model, the persons targeted by the process fluid mechanical processes that can be described using the laminar- turbulent transition most clearly. The term envelope, also called transition, is understood to be the transition from an originally laminar steady-state flow to a turbulent state, which is dominated by small-scale almost chaotic fluctuations. This is due to an instability of the flow to small disturbances that exist depending on the flow condition at a certain Reynolds number. Already using the linear stability theory can be the frequencies of the fanned disorders and their amplification rates very well predict for a given stationary flow field. For an incompressible plate boundary layer without pressure gradient, for example, fanned interference from a Reynolds number of Re ≈ 91,000 x ⋅ rates, where x is the coordinate in the flow direction with its origin is at the plate leading edge. Instead of Stromabkoordinate the local displacement thickness of the boundary layer δ1 is often used. This gives a critical Reynolds number of Reδ1 = 520, exist from the first time fanned disorders. This fanned errors, called Tollmien -Schlichting waves grow in downstream direction exponentially. Reaching the instability waves an appreciable amplitude, they are no longer considered independently but interact with each other. In the Navier -Stokes equations, this follows from the non-linear terms. If one divides a variable U in their stationary component u0 and the non-stationary part u, it follows from a non-linear term

Assuming for the unsteady share a wave of the form

α with the wave number, the result is the product of u with itself

That is, it can be generated at twice the wave number of waves. The nonlinear generation causes an increase also of actually linearly damped disturbances. Increasingly shorter wavelength interference reach nonlinear amplitudes and draw energy from the coarser scales from. Because the second spatial derivative of the wave is proportional to the square of the wave number, effect for shorter-wavelength disturbances but increasing the viscosity of the generation counter.

Depending on the introduced disturbances, different vortex structures in the nonlinear region. So at the interface between K- type (English for wrong ) distinction (after Philip Klebanoff ), H type (after Thorwald Herbert) and Oblique. In the late stages of transition, the large eddy structures are breaking up and the result is a chaotic picture of small vortices. Two-dimensional DNS are useful only in exceptional cases. In general, a 2D turbulence does not exist. Therefore it is necessary for a DNS in general, to simulate the full three-dimensional Navier -Stokes equations, resulting in a correspondingly high computational complexity.

Comparison with other methods

From a DNS is used if all relevant scales are resolved in space and time. This means that wave numbers must be mapped cleanly to the extent that counteracts the more influence the viscosity of the non-linear generation up only. Other methods of computational fluid dynamics consider the transient processes of turbulence either restricted or not at all. In the large-eddy simulation (LES, in German also large eddy simulation ) we restrict ourselves to the rough turbulent scales by considering spatially filtered flow variables. The effect of no longer resolved scales is represented by so-called Subgridscale models. Since these models essentially bring artificial viscosity, under -resolved DNS can also be viewed as LES.

In return to this, in RANS calculations (English for Reynolds - averaged Navier -Stokes equations) resolved turbulent fluctuations but no fully modeled by turbulence models. With the time-averaged Navier -Stokes equations and other equations than with DNS and LES to be solved, so a RANS result does not converge to the exact solution of the direct numerical simulation even at arbitrarily fine resolution.

Discretization in the DNS

Task of a numerical method for the direct numerical simulation, therefore, is sufficiently fine to resolve unsteady wave processes in space and time. In addition, make sure that the appropriate boundary conditions, to require an unnecessarily large computational domain and to avoid reflections at the boundaries. Typically disorders are additionally initiate. Although in principle disorders are already inherent in the method include (eg discretization approximations in the initial and boundary conditions or rounding errors), but the result then depends on the computational grid used for rounding errors and even from the computer used. In addition, you want to investigate the influence of different disorders in a DNS.

Due to the high computational cost usually is limited to the DNS on periodic boundary conditions in the third dimension. For the boundary layer on a flat plate, this means that it is assumed of an infinite extension in the transverse or spanwise direction. With rotationally symmetrical bodies, the spanwise direction corresponds to the circumferential direction, then. The picture shows an example of an integration area for the plate boundary layer.

Since the success of a DNS much depends on the ability to adequately convey exactly waves, such requirements must be met by the numerics. In the space discretization good dispersion and dissipation properties are to achieve, which is why almost exclusively high-order methods are used. For the spanwise direction Spektralansatz is suitable due to the periodicity, as this is ideal for the wave transport. In non- periodic directions in space, a spectral approach is possible, however, discretization methods that are not based on periodic boundary conditions, more practical. Examples include finite difference or finite volume. In DNA codes finite differences are often used because they are the fastest method for calculating space derivatives of high order on structured grids.

Due to the continuous generation of the higher harmonic waves, it is necessary for the stability of a calculation to remove short-wavelength disturbances. The problem of the short-wave components is that only waves can be resolved to a certain wave number on a given grid. However, when waves generated by wave numbers above this limit, then these mapped to certain shares to lower wave numbers (known as aliasing), which generally leads to a crash of the process. Also for the wave transport properties of each spatial discretization, there exists a limit aliasing. For wavenumbers above this limit aliasing results in a negative group velocity, which means that these shares unsteady run counter to the physically correct direction of transport. At what wavenumber this occurs depends on the type and order of the proceedings. With increasing order of the aliasing limit moves to higher wave numbers. In principle, removal of the short-wave components is also possible by a correspondingly high spatial resolution. In this case, the viscosity for an attenuation of these shares provides. However, this is not practical because this requires too high spatial resolution. Common methods for Dealiasing are spatial filtering with a low pass filter or a Upwinding alternate weighting of discretization. At a spectral discretization of the units as a function of the wave number are given. Dealiasing done simply by zeroing high wave numbers (usually from 2/ 3 of the maximum wave number. )

From the Navier -Stokes equations is obtained by means of the spatial discretization, the time derivative of the individual variables. In principle, the time integration can be performed with explicit or implicit methods. For an explicit time integration, the time step limit for a stable simulation is observed. Depending on whether convective or viscous terms dominate, the time step proportionally scaled, or the square of the spatial resolution. Often, the Runge- Kutta method of 4th order for the time integration applied because it has excellent properties with respect to accuracy and stability. Although implicit methods have the advantage of ( at least in principle ) to have no time limit step, however, they require a much higher computational complexity per time step. If we choose the time step so rough, so that an implicit worth compared to an explicit method, the implicit integration attenuates almost all instability waves away. For this reason, implicit integration method for DNA codes can not be applied generally. An exception to represent compressible bills is at low Mach number, in which the time step of an explicit procedure must be very fine.

Due to the high computational complexity, it is usually essential to count on multiple processors. Since DNA codes typically operate on structured grids and can be vectorized so good, can on vector computers (eg Earth Simulator or NEC SX8 ) high computing power are achieved, partially achieved over 50 % of the theoretical computing power. Unstructured process could not prevail so far in the DNA code, because the greater flexibility is in terms of the geometry at the expense of computational speed.

Initial and boundary conditions

Spatial and temporal simulations

As in the linear stability theory, a distinction between temporal and the spatial problem. In a temporal simulation disorders who rely on the Ausströmrand the domain of integration, re-introduced at the inflow boundary. This means that disturbances grow in time until saturation. Background of this approach is the fact that from a mathematical point of view must be specified because of the parabolic nature of the viscous terms and the Ausströmrand boundary conditions. Through a periodic approach in the flow direction, the problem is bypassed to specify the usually unknown to suitable boundary conditions. A problem in this case is naturally the fact that the flow at the Ausströmrand does not have the same state as at the inflow edge. For example, a boundary layer grows in the downstream or in a channel, a pressure gradient ensures that the flow does not come to a standstill. Therefore, the disturbances are to scale accordingly before they can be imposed on the inflow boundary. Despite the problems and limitations of the temporal model, it is still used today because of a relatively small area of ​​integration a turbulent flow can be calculated.

In contrast, a spatial simulation inputs and Ausströmrand be considered separately. This means that faults grow spatially to ( with the exception of an absolute instability occurs) which significantly better reflects the reality. In a boundary layer about Tollmien -Schlichting waves are transported downstream as they grow, which corresponds to an increase in the amplitude in downstream direction. After sufficient simulation time, the disturbances reach a periodic state in time in time- periodic disturbance excitation. Is particularly important in a spatial simulation modeling a suitable Ausströmrandes because impingement of non-linear disturbances would cause the Ausströmrand reflections, which in turn results in unphysical results. Initially, the region of integration has been extended over time in downstream direction, so that disturbances were never able to achieve backward running Ausströmrand. This method is, however, with increasing simulation time more complex. The breakthrough of spatial simulation was achieved only with the development of suitable damping zones before Ausströmrand. Nowadays the most part, the spatial model is used.

Initial conditions

As an initial condition typically a solution of the boundary layer equations is used, since this represent a good approximation of a laminar frictional flow. In a plate boundary layer self-similar profiles ( Blasius or Falkner -Skan ) can be interpolated on the computational grid. If it is more complex geometries, the boundary layer equations are to be integrated in the downstream direction. If this is not possible, eg due to reverse flow, the flow field can be initialized with simple assumptions, as in clean conditions initial perturbations from the region of integration should run out.

Boundary conditions

For a wall, the boundary conditions can be relatively easy to formulate. Due to the friction are the three components of velocity at the wall is zero ( no slip condition ). In the compressible case, another boundary condition for the temperature is to be given, which may be isothermal or adiabatic optional. Isotherm indicates that the temperature is required determine adiabatically, that the heat flow and thus the normal wall temperature gradient is zero. Further actuators may be placed on the wall, such as a noise bars for fault excitation to investigate the behavior of certain spurious modes.

At the inflow boundary, all flow variables are imposed when incompressible calculations and supersonic. With compressible subsonic bills, for example, to ensure, through a characteristic boundary condition that current accruing sound waves to exit the domain of integration. The inflow boundary is also defined disorders contribute. For small disturbances can be used for this purpose, for example, on the amplitude and phase characteristics of the linear stability theory.

Problems are basically boundary conditions which only arise due to the finite computational grid. While RANS calculations typically only the stationary free-flow conditions may be imposed due to the transient solution, the choice of appropriate boundary conditions for the success of a DNS is critical. A special importance is attached to the Ausströmrand because reflections can significantly distort the result because of the nonlinear fluctuations. To this end, various damping zones have been developed in the past, in which, for example, the solution is drawn on a stationary base flow. Especially for aeroacoustic calculations arise particularly strict requirements on the Ausströmrand because the sound is considered smaller than the fluid-mechanical fluctuations by orders of magnitude. One possibility here is the combination of grid extension and spatial low-pass filter, which unsteady shares are successively removed in the damping zone from the solution before they cause reflections at the very edge and can contaminate the delicate acoustic field.

Free stream margins are generally less critical than Ausströmränder because the amplitudes occurring are significantly lower than Ausströmrand. However, inappropriate boundary conditions can lead to incorrect amplification rates of the interfering waves or the right solution requires a too large integration area. Because of the small variations in the free- stream edges are based typically on linearized formulations, such as decay or characteristic conditions. If periodicity in the transverse direction adopted, the question of appropriate boundary conditions at this point does not arise.

Scaling of the problem Reynolds number

The resolved scales are determined by the physics. The computational grid defines which scale can be resolved at a particular numerical method. The resulting required step size must therefore be of the order of ( proportional to and not equal to) be the smallest scales, the Kolmogorov length scale

Are determined, the kinematic viscosity and ε is the dissipation of kinetic energy. At a spatial increment h is the number of points n in one direction in space

The dissipation rate ε of the kinetic energy can be with the RMS value of the velocity fluctuation u ' to assess

Thus, for the number of points required in one spatial direction

The Reynolds number

Subject to all three spatial directions of the scaling with the Reynolds number, the total number of points required grows with RE9 / 4 Assuming periodic boundary conditions in spannweitiger direction, so is a scaling of the points in this direction with Re3 / 4 is not absolutely necessary, thereby reducing the overall resolution scaled only with Re3 / 2. This can be further reduced, as for example, the boundary layer thickness does not increase linearly with the downstream direction, which is why the number of points in wall- normal direction is weaker than with Re3 / 4 can scale. The time to be simulated is proportional to τ of the turbulent length scale, by

Is given. Assuming that the time step limit is dominated by convective terms (At ~ h ~ η / u '), the number of time steps yields to

That is, the number of time steps computing scaled to the power ¾ of the Reynolds number. Thus, the total computational effort is proportional to Re3 for a fully three-dimensional domain of integration without spanwise periodicity.

With a scaling factor of the overall problem of RE9 / 4 or Re3 an application of DNA on real-world problems at first glance appear as excluded. As an evaluation of large data sets (note that this is non-stationary problems ) also does not appear feasible. In the DNS, however, is not about to simulate about a complete aircraft, but one is interested in the physical basis, through the understanding of a complete system with high Reynolds numbers, such as the profile of a wing can be improved. Develop about methods for laminar, so it is sufficient to simulate the relevant region of the boundary layer in order to reduce the resistance of the entire aircraft.

Interpretation of results

Since one unsteady from the DNS data of large areas receives (around 100 million grid points for spatial resolution are quite common ), incurred huge amounts of data from several gigabytes. This, of course, are not directly derivable statements about the physical processes, it is necessary to evaluate the data (post processing). Since the spatial model of the time course after sufficient time steps is calculated periodically or at least quasi - periodic, it is advisable to examine the data using Fourier analysis. It is often carried out a double- spectral analysis of periodic boundary conditions in the spanwise direction. The resulting modes is designated ( h, k ), where h is a multiple of the fundamental frequency, and k is a multiple of the basic spanwise wavenumber

Is what results from the spanwise extent of the area of ​​integration λz. Thus, the modes (0,1), (0,2), etc. denote stationary disorders are modeled by the third spatial direction. Accordingly, a two-dimensional interference with the fundamental frequency with (1.0 ) and its higher harmonics (2.0 ), (3.0 ), so called. The growth of the amplitude in the direction of flow as well as the amplitude or phase profile normal thereto may be compared with results of the linear stability theory.

In the turbulent region amplitude gradients are less reliable in general, since a large number of modes has reached saturation. Therefore vortices are visualized, eg by means of lambda2 criterion to get a better insight into the fluid mechanical mechanisms. The picture shows an example of three-dimensional vortex structures in a shear layer by means of isosurfaces of lambda2 criterion. By introducing stationary spannweitiger disorders longitudinal vortices are generated, leading to the collapse of the Kelvin - Helmholtz vortices. To represent the sound radiation at the aeroacoustic calculations can be represented, the printing itself, but also the expansion, the divergence of the velocity field. Have a nice impression mediate density, since one can thus generate Schlieren images. An evaluation is eg EAS3, which is used in various universities to evaluate DNS data.

History of DNS

The beginning of computational fluid mechanics, the calculation of a circular cylinder with Re = 10 from the 1933 dar. Thom scored the solution by hand calculation using a difference method, which was already amazingly accurate. From a first DNS in the real sense one can speak for the simulation of Orszag & Patterson in 1972, the isentropic turbulence at Re = 35 calculated on a 323 lattice with spectral methods. The first spatial DNS comes from Fasel from the year 1976. During this growth of small disturbances has been investigated in a boundary layer and compared with the linear stability theory. The turbulent flow in a plane channel with Re = 3300 and periodic boundary conditions in the flow direction was by Kim et al. calculated on a grid with already 4 million points in 1987. In 1988 published Spalart DNS results of a turbulent boundary layer with Reθ = 1410 ( θ stands for the momentum thickness of the boundary layer ), which is also the temporal model is based. A breakthrough in the application of the spatial model succeeded in the early 1990s with the development of suitable damping zones before Ausströmrand, eg by Kloker et al. Using appropriate boundary conditions could Colonius et al. 1997 perform one of the first acoustic DNS, the radiated sound of the simulated free shear layer was calculated directly using the Navier -Stokes equations and not by an acoustic analogy. The largest to date DNS, based on the spatial resolution is a statement of Kaneda and Ishihara from the year 2002, which was performed on the Earth Simulator in Japan. They used 40963 ≈ 68.7 billion grid points for the simulation of isentropic turbulence in a periodic domain of integration. The great progress that has been achieved in the area of ​​DNA, of course, is based partly on the ever-increasing computer capacities, but also on the developed numerical methods that allow effective use of these resources only.

Areas of application

An example of the application is the separation of a boundary layer flow due to the adverse pressure gradient. By energizing the boundary layer with certain disorders is an attempt to reduce separation bubbles, or to avoid it altogether. Examples of applications are as turbine blades or the wings of the aircraft. You would be able to avoid the stall at high angles of attack, higher lift coefficients could be achieved and you could do without flaps.

Another issue is the laminar flow, which attempts to keep the boundary layer above the natural range beyond laminar and herauszuzögern the envelope from laminar to turbulent. This happens eg with extraction or the introduction of longitudinal vortices in the boundary layer. Since a laminar boundary layer has a lower resistance than turbulent, thus the fuel consumption of aircraft could be reduced by up to 15 %.

In aeroacoustics, the mechanisms of sound generation caused by fluid mechanical processes studied. The aim is to reduce the sound radiated by appropriate actuators. Aeroacoustics is a relatively new field in the area of ​​DNA, since it is a multi -scale problem is relatively difficult to solve. The fluidic fluctuations of, for example, in a free shear layer to have large amplitudes with a small spatial extent. The radiated sound, however, is relatively long wavelengths with a very low amplitude. It follows that special requirements (boundary conditions, accuracy) to be made on a numerical method in order not to distort the results, for example by reflections. An important aspect is the jet noise, as it one of the main sources of noise of an aircraft - is - especially during startup. A breakthrough in this field would increase the quality of life of many people living near airports. Due to restrictions such as night flight ban or noise-related Start-/Landegebühren but also airlines and airport operators are interested in a reduction of aircraft noise.

In the area of ​​over-and hypersonic it is necessary to analyze not only time-averaged quantities, but also non-stationary processes in the boundary layer, as high thermal stresses can destroy the structure. The DNS is used here for basic investigations of the laminar- turbulent transition or to the development of cooling concepts. To account for effects in hypersonic, more complex equations are now used to take into account the chemical reactions such as dissociation or ionization, and thermal non-equilibrium.

The DNS is also an important tool for modeling. The high-resolution unsteady flow data are used for the development of turbulence models in order to improve the less computationally intensive methods such as large-eddy simulation or RANS calculations. Thus addition, the results of new methods of fluid dynamics are validated.