G OVERNING H YDRODYNAMIC E QUATIONS AND
3.3 E QUATIONS OF C ONTINUITY AND M OTION OF F LOW
3.5.3 Turbulence Model Classification
3.5.3.3 Two-equation Models
The length scale characterising the size of the large, energy-containing eddies is subject to transport processes in a similar manner to the energy k. Other processes influencing the length scale are dissipation, which destroys the small eddies and thus effectively increases the eddy size, and vortex stretching connected with the energy cascade, which reduces the eddy size. The balance of all these processes can be expressed in a model transport equation for L, which can then be used to calculate the distribution of L. The difficulties in finding widely valid formulae for prescribing or calculating L, have stimulated the use of such a length scale equation.
3.5.3.3.1 ε-equation
At high Reynolds numbers where local isotropy prevails, the rate of dissipation, ε, is equal to the molecular kinematic viscosity times the fluctuating vorticity
ui xj
2 . An exact transport equation can be derived from the Navier-Stokes equations for the fluctuating vorticity, and thus for the rate of dissipation (Tennekes and Lumley, 1972). This equation contains complex correlations whose behaviour is little known and for which fairly drastic model assumptions must be introduced in order to make the equation tractable. When the turbulence is considered to be locally isotropic the equation contains terms representing the rate of change, advection, diffusion, generation of vorticity due to vortex stretching connected with the energy cascade, and viscous destruction of vorticity. The diffusion, generation and destruction terms require model assumptions. Usually, the diffusion is modelled with the gradient assumption. The generation and destruction terms cannot be modelled separately; it is their difference that has to be modelled.The outcome of the modelling is the ε-equation presented in the following section.
Together with the k-equation and the Kolmogorov-Prandtl expression it forms the so-called k-ε turbulence model.
3.5.3.3.2 The k-ε Turbulence Model
The k-ε model has been included into the numerical model presented herein.
The equations which comprise the complete model are as follows (Rodi, 1984):
T c k2 (3.42)
T
generationdestruction
2 buoyant situations also c . The empirical constants recommended by Launder and 3 Spalding are given in Table (3.1) as follows:
Table (3.1) - Values of the constants in the k-ε model (adapted from Rodi, 1984)
c c 1 c 2 k
0.09 1.44 1.92 1.0 1.3
These values are based on extensive examination of free turbulent flows, but they can also be used for wall flows. A sensitivity study has shown that the calculations are most sensitive to the values of c and 1 c . Complete universality of 2 the constants given in Table (3.1) should not be expected. Experience has shown that even in certain fairly simple flows some of the constants require different values. The range of applicability of the k-ε model can be extended when some of the constants are replaced by functions of suitable flow parameters.
The standard k-ε model is based on the assumption that the eddy viscosity is the same for all Reynolds stresses (isotropic eddy viscosity). To allow for the non-isotropic nature of the eddy viscosity in such cases, the k-ε model is refined by introducing an algebraic stress model to replace the eddy-viscosity relation (Eq. 3.32) and the Kolmogorov-Prandtl expression (3.37). This model relates the individual stresses uiuj to mean-velocity gradients, k and ε by way of algebraic expressions by simplifying the transport equations for uiuj.
Mass transfer is calculated in the same way as in zero- and one-equation models via the turbulent Prandtl/Schmidt number. Buoyancy effects can be accounted for in the k-ε model at two levels. The first level is to simply include the buoyancy terms in the k- and ε-equations as shown in equations (3.44) and (3.45) and
to leave constants unaffected by buoyancy. However, there is an additional buoyancy constant, c , in the ε-equation, and its value is somewhat controversial, and this is 3 due to the definition of the flux Richardson number of Rf to which c is a 3 multiplier. If a positive Rf becomes large enough, it leads to complete suppression of all turbulence. Observations have shown that turbulence cannot be maintained if
2 .
0
Rf approximately (Tennekes and Lumley, 1972). Usually Rf is defined as minus the ratio of buoyancy production of k to stress production, -G/P. With this definition, various researchers found that c should be close to zero and unity for 3 vertical and horizontal buoyant shear layers respectively (Rodi, 1984). c3 1 implies that there is no buoyancy term in the ε- equation while c3 0 implies that the buoyancy-production term is multiplied with the same constant as the stress production. In order to resolve this difficulty Rodi suggested to replace in the Rf definition, the buoyancy production G of the total turbulent energy k by the buoyancy production Gv2 of only the lateral energy component v2 and to write
) 2 /(
1 G 2 P G
Rf v . In horizontal shear layers then Rf G/(PG) and in vertical layers Rf 0. With this definition, a single value can be used for c in 3 both vertical and horizontal layers, c3 0.8.
The second level of accounting for buoyancy in the k-ε model makes use of the algebraic stress model approach. Modelled transport equations for the stress ui uj and the mass flux ui C are simplified to yield algebraic relations. The transport equations contain buoyancy terms, which appear also in the algebraic relations, leading effectively to non-isotropic eddy viscosities and diffusivities as functions of some local Richardson number. This modelling automatically yields a buoyancy influence on c and the turbulent Prandtl/Schmidt number T.
Rodi (1987) employed the following ε- equation:
c k G c k P x c
x u x
t i
T i i i
2 2 3
1
(3.46)
On discussion for determining c , according to test calculations, he stated that in 3 situations where G is a positive term, as in unstably stratified flows, c should take 3
a value of 1, while in stably stratified shear layers, where G is a negative term, c 3 should be chosen near zero. Successful calculations for the latter flows have been obtained with c in the range 0-0.2. Further, the turbulent Prandtl/Schmidt number 3
T as well as c have also been observed to depend on buoyancy effects and are not really a constant under stratification conditions. An extended version of the k-ε model has been introduced by Rodi (1987), in which the constants c and T are replaced by functions of suitable stratification parameters. These functions have been derived by simplifying a complex stress/flux-equation model. Yu and Li (1998) adapted the modifications suggested by Viollet (1990) on the standard k-ε model and used c = 3 c when 1 G0, and c = 0, when 3 G0. In a study conducted by Verdier-Bonnet et al. (1999), c was given the value 0, as was suggested by Rodi 3 (1987) for stable flows. Launder and Spalding (1972) stated that it is likely that a two-equation model provides the best starting point, and perhaps the best finishing point as well. In spite of all significant advances in computer technology, their statement still might be valid at least for turbulence modelling of large water bodies.
3.5.4 Boundary Conditions
In this part only physical boundaries are discussed. Turbulent water flow may be bounded by a solid wall, a free surface or by non-turbulent flow. The location of wall and free surface boundaries is well defined, that of a free boundary is not because the interface between turbulent and non-turbulent fluid is highly indented and unsteady. For practical purposes, a free boundary is defined as the location where the velocity (or sometimes a scalar quantity) is nearly equal to its free-stream value (Rodi, 1984). The implementation of boundary conditions in the numerical model has been discussed in Chapter Four.