Baumgarten

64 3 Basic Equations

Using Eqs. 3.61, 3.64, 3.67, 3.68, and 3.69 in Eq. 3.59 and applying the product rule results in

Because of the continuity equation, the first term in Eq. 3.70 is zero. The use of

This equation is further modified by using the momentum equation. Multiplication of Eq. 3.49 with the velocity vector yields the second term on the left hand side of Eq. 3.73,

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3.1.4 Turbulent Flows

3.1.4.1 RANS Equations

The Navier-Stokes equations can be solved directly without averaging or using turbulence models if the grid spacing is fine enough to resolve the smallest eddies in a flow field. This approach is called direct numerical simulation (DNS). The size of the smallest eddies is proportional to the Kolmogorov length scale, which decreases with increasing Reynolds numbers. For a cylinder volume of 1.0 liters at least 1012 grid points are needed due to the highly turbulent flow, where the smallest eddies are in the range of 0.01 mm. The requirements for the application of DNS to in-cylinder processes, concerning computer speed and memory, exceed the power of today’s processors by far. Up to now, DNS simulations have not been suitable for solving engineering problems and are limited to basic research applications with low Reynolds numbers and small geometric domains.

The large eddy simulation (LES) distinguishes between large and small eddies. Only the large eddies, which contain most of the energy and thus are much more important concerning the transport of the conserved quantities than the small ones, are resolved by the grid. The effect of the small eddies is described by sub-models. The separation of small and large eddies is obtained by filtering the flow field using an appropriate threshold value of the eddy size. Because the small eddies are described by so-called subgrid-scale Reynolds stress models, the Navier–Stokes equations are expressed in terms of averaged quantities, similar to the RANS equations described in the following. The numerical algorithms are usually specially adapted to the individual problem and the geometry. Although the large eddy simulation consumes less time and computational power than DNS, it is still not suitable for engineering applications. A more detailed description of DNS and LES is given in [6] for example.

Today, the computation of technical flow fields is performed using the socalled Reynolds averaged Navier-Stokes equations (RANS equations) in combination with an appropriate turbulence model. Following the basic approach of Reynolds (1895), the instantaneous values of the turbulent flow quantities are split into a mean and a fluctuating component (Reynolds decomposition):

The overbar denotes the time-average, and the superscript (´) denotes the superimposed fluctuation, see Fig. 3.9. The time-average of a flow quantity, e.g. a velocity component, is

The time interval t must be large compared to the relevant period of the fluctuations, but small enough to map the time dependence of an unsteady mean flow.

Fig. 3.9. Unsteady turbulent flow

The RANS equations are obtained by substituting the Reynolds decomposition terms into the instantaneous conservation equations, and then by averaging the entire equations over time. Now the turbulent flow is described by the conservation equations in terms of time average quantities. Due to the time-averaging, the information about the turbulent fluctuating quantities is lost. This effect is shown in additional terms in the momentum and energy equations, the Reynolds stresses, and the turbulent heat flux, which have to be described by a turbulence model in order to close the system of equations again.

In general, thermodynamic properties like Π, cp, and Ο can also fluctuate due to the fluctuation of pressure and temperature, but these fluctuations are usually small, and they are neglected for the treatment of turbulence in this book. Special attention must be given to the density fluctuations. Neglecting the density fluctuations does not mean that the density is constant. It simply expresses the fact that the effect of turbulent density variations is not included; the mean density variations however may be large. Should fluctuations in density be considered, the appropriate equations can be derived either by simple time-averaging (Reynolds averaging, Eq. 3.81), or by a mass-weighted time-averaging procedure (Favreaveraging). Both kinds of averaging are in use today and will be described in the following.

Substituting the Reynolds decomposition terms into the instantaneous continuity equation (here only shown for two-dimensional flow) and averaging over time (denoted by a long overbar) yields

where f and g are two turbulent flow quantities, the continuity equation for a compressible flow becomes

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Compared to the original continuity equation, there are additional terms expressing the fact that the mean flow quantities alone no longer express the mass conservation principle. For incompressible (constant density) flow, the fluctuation terms disappear, and the continuity equation reduces to

The momentum and energy equations are treated accordingly. Replacing the instantaneous values by the Reynolds decompositions, averaging over time, separating components, and dropping all terms that contain only one fluctuating quantity (these terms are zero), the x1-momentum equation for a compressible flow in the absence of external forces becomes (here only shown for two-dimensional flow)

The fluctuating parts of the viscous stress tensor disappear entirely upon timeaveraging. The x2- and x3- momentum equations as well as the energy equation may be treated in the same manner. Compared to the original equations, there are again a lot of extra terms due to additional turbulent momentum and heat transfer, complicating the system of conservation equations.

In the case of incompressible flow however, the resulting momentum and energy conservation equations reduce to

where j = 1, 2, 3 indicates that there are three momentum equations, each for one coordinate direction. These equations are very similar to the original set of conservation equations. However, two additional terms have been added as a result of the averaging process. In the momentum equation, this is the so-called turbulent stress tensor,

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The general form of this equation is identical to that of the original continuity equation. The Navier-Stokes equations and the energy equation are treated accordingly (e.g. [16]). Again, the general form of the resulting equations is identical to the original form, except for an additional turbulent stress term in the momentum equations and an additional turbulent heat flux in the energy equation:

The components of the stress tensor in Eq. 3.98 are obtained by using the Favreaveraged decomposition terms in Eq. 3.51, but in contrast to simple timeaveraging the fluctuation terms do not disappear: